Electromagnetic Theory and Plasmonics for Engineers

This book presents the theory of electromagnetic (EM) waves for upper undergraduate, graduate and PhD-level students in engineering. It focuses on physics and microwave theory based on Maxwell’s equations and the boundary conditions important for studying the operation of waveguides and resonators in a wide frequency range, namely, from approx. 10**9 to 10**16 hertz. The author also highlights various current topics in EM field theory, such as plasmonic (comprising a noble metal) waveguides and analyses of attenuations by filled waveguide dielectrics or semiconductors and also by conducting waveguide walls. Featuring a wide variety of illustrations, the book presents the calculated and schematic distributions of EM fields and currents in waveguides and resonators. Further, test questions are presented at the end of each chapter.

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Liudmila Nickelson

Electromagnetic Theory and Plasmonics for Engineers

Electromagnetic Theory and Plasmonics for Engineers

Liudmila Nickelson

Electromagnetic Theory and Plasmonics for Engineers

123

Liudmila Nickelson Faculty of Electronics, Department of Electronic systems Vilnius Gediminas Technical University Vilnius, Lithuania

ISBN 978-981-13-2350-8 ISBN 978-981-13-2352-2 https://doi.org/10.1007/978-981-13-2352-2

(eBook)

Library of Congress Control Number: 2018952889 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is created on the basis of lectures on electromagnetic (EM) field theory for Erasmus undergraduate (BA degree) students at Vilnius Gediminas Technical University (VGTU), Lithuania, and on the basis of lectures for Ph.D. students at Linköping University, Sweden. Some calculations in Chaps. 7 and 8 were completed for BA degree practice at VGTU by Dr. Artūras Bubnelis. A number of calculations in Chap. 9 were carried out for a BA degree at VGTU by Engineer Ekaterina Nadopta. This book focuses on physics and microwave theory based on Maxwell’s equations and boundary conditions which are important for studying the operation of waveguides and resonators in the wide frequency range, namely, from about 109 to 1016 hertz (Hz). This book contains such main topics as the interaction of EM waves with different substances, plane EM wave propagation, reflection and transmission thereof, boundary value problems, solutions of Maxwell’s equations under certain boundary conditions, dispersion equations, behaviors of waveguide modes as transverse electric and magnetic ones, surface plasmon polariton mode as well as oscillations in resonators, etc. Taking into account the fact that the operation of all electrical, electronic, and plasmonic devices is based on the knowledge of the EM field theory, the study of the subject gives a powerful tool for understanding the operation of any EM device. During the last decades, the development of the EM field theory was basically carried out in two directions, namely developing microwave devices as well as developing devices on the basis of optic laws due to the formal point of view since light is actually an EM wave, and the Maxwell’s equations with the corresponding boundary conditions can thoroughly describe optical phenomena. Throughout history, the development of microwave waveguide and antenna technologies evolved from relatively low frequencies (radio and microwave frequencies) to higher ones. Here the frequency measurement is traditionally made in Hz (as well as in multiple Hz units such as GHz, THz, and PHz).

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Historically, optical devices operate at relatively high frequencies (infrared (0.3 THz–0.43 PHz), visible (0.43–0.79) PHz, and ultraviolet (0.79–30) PHz), and there is a tendency to develop devices that can also operate at lower frequencies. The optics was the basis for successful development of such advanced scientific research area as plasmonics and photonics. Therefore, in plasmonics, which we will be considering in this book, the measurement of frequency usually occurs in Hz (hertz), eV (electron volt), cm‒1 (centimeter to the power mines one), and K (kelvin)—as in optics. In this way, we see a different approach in microwave and optical studies even in frequency units. So the terminology and approaches to EM problems used in microwave theory and plasmonics (optics) may differ. Here we note a few differences important for the book: (1) Since this book contains plasmonics, we wish to note that everywhere the dispersion characteristics are depicted in a way which is accepted in the microwave theory (electrodynamics), i.e., propagation constants are located on the ordinate axis and frequencies are placed on the abscissa axis. (2) Dependences on time and longitudinal propagation constant are also usually taken into account in technical literature by different multiplier ei ðxthzÞ in electrodynamics and ei ðhzxtÞ in optics. Due to differentiation, this difference leads to non-matching signs in some expressions of intermediate values. (3) The approach to the definition of circular right- and left-polarized waves in electrodynamics and optics is also different. In this book, we have chosen the approach recommended by the Institute of Electrical and Electronic Engineers (IEEE) for the engineering community. Being aware of the large workload of students and specialists as well as significant volume of material for studies, the author gives detailed explanations of derivation of many expressions. The book can be useful for Erasmus students with English language as a foreign language; therefore, the author sometimes additionally gives the synonyms of a word (in round brackets) which are possible to use for the specific item under analysis. Chapter 6 can be of particular use for specialists in Telecommunications Engineering since this chapter presents a detailed analysis of various questions about polarization of reflected waves from dielectric and metal surfaces. The contents of this chapter cover issues encountered by Prof. L. Nickelson while implementing the project for the Communications Regulatory Authority of the Republic of Lithuania. This book sometimes presents several equivalent terminologies related to the same value or to the same phenomena used nowadays. This may assist in gaining a rapid understanding of technical literature as articles use a wide range of terminologies.

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The main motivation for writing this book was the author’s wish to pass the knowledge accumulated throughout a span of approximately 50 years on to those who are now starting their studies in the EM field theory. Field research activities of author: Electromagnetic field theory and application thereof to various physical and engineering problems. She with co-authors developed methods based on the Theory of Singular Integral Equations with the help of which two-dimensional and three-dimensional waveguides and structures with very complicated topology can be calculated. Prof. L. Nickelson has also researched different-layered waveguides containing strong lossy materials such as metamaterials, onion-like carbon, silicon carbide (SiC), gyrotropy plasma. Together with co-authors, she has published the total of five books, including this one, and plus two chapters of INTECH books, eight patents, and more than 100 articles. It is also important to note the principle of formula numbering in this textbook. The first digit of numbering corresponds to the number of certain chapter while the second digit indicates the section number of that chapter and the third digit corresponds to the number of the formula in this section. For example, if we take formula (3.9.7), the digit “3” represents the third chapter, the digit “9” means the ninth section, and the digit “7” signifies the seventh formula in Sect. 9. Vilnius, Lithuania

Liudmila Nickelson

Acknowledgements

I am particularly thankful to Associate Professor Juozas Bucinskas (Vilnius University, Faculty of Physics, Department of Theoretical Physics) who carefully reviewed every chapter of this textbook as they were being prepared during several years of work on this book. I am also grateful to him for the additional review of the whole textbook. Associate Professor J. Bučinskas made a number of valuable comments and remarks which were taken into account and improved the overall quality of the textbook. I appreciate very valuable comments made while reviewing the entire textbook by Professor Vytautas Urbanavičius (Vilnius Gediminas Technical University, Faculty of Electronics, Department of Electronic Systems). I am very thankful for very valuable comments regarding Chap. 9 made by Associate Professor Zigmas Balevičius (Center for Physical Sciences and Technology, Bio-Nanotechnology Lab, Senior Researcher, Vilnius, Lithuania and Vilnius Gediminas Technical University, Faculty of Electronics, Department of Computer Science and Communications Technologies). In addition, I would like to extend my sincere thanks to Engineer Ekaterina Nadopta for her long-term and substantial help. She put all figures in this book in an electronic format. I am very thankful to Dr. Artūras Bubnelis who let me use his BA degree practice results in Chaps. 7 and 8 of this textbook. Moreover, I am grateful to Ms. Aistė Šimkonienė for English language corrections made throughout the textbook. My additional thanks go to Dr. Loyola D’Silva (Publishing Editor of Springer) and Mr. Ravi Vengadachalam (Project Coordinator of Springer). They have provided useful information during the period of four years and have contributed significantly to the final stages of textbook preparation. My sincere gratitude for the highly qualified help to Mr. Arunkumar Raviselvam (Project Manager) and his team at Scientific Publishing Services. Finally, I am very grateful to my academic supervisor of the first thesis—Prof. Victor Shugurov—who read and commented on Chaps. 1 and 9 of the textbook.

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Contents

1 Vector Analysis in Annex to Electromagnetics . . . . . . . . . . . . . . 1.1 Introduction to Vector Analysis . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Addition of Two Vectors . . . . . . . . . . . . . . . . . . . . 1.2.3 Base Vectors and Vector Components . . . . . . . . . . . 1.2.4 Vectors in Three Dimensions . . . . . . . . . . . . . . . . . 1.2.5 Rectangular Components in 2D . . . . . . . . . . . . . . . . 1.2.6 Position and Displacement Vectors . . . . . . . . . . . . . 1.2.7 Vector Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Dot Product of Two Vectors . . . . . . . . . . . . . . . . . . 1.2.9 Cross Product of Two Vectors . . . . . . . . . . . . . . . . 1.2.10 Triple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Differential and Integral Calculus . . . . . . . . . . . . . . . . . . . . . 1.3.1 Differential Length Vector . . . . . . . . . . . . . . . . . . . 1.3.2 Differential Surface Vector . . . . . . . . . . . . . . . . . . . 1.3.3 Total and Partial Derivatives . . . . . . . . . . . . . . . . . . 1.3.4 The Operator r . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Gradient of a Scalar Field . . . . . . . . . . . . . . . . . . . . 1.3.6 Divergence of a Vector Field and Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.7 Curl of a Vector Field and Stokes’ Theorem . . . . . . 1.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Geometric Representation of the Complex Numbers . 1.4.2 Trigonometric Form of the Complex Numbers . . . . . 1.4.3 Cross and Dot Products of Two Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Examples and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fields and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dielectrics and Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Electric Polarization and Dipole Moment . . . . . . . . . 3.2.2 Polarization of Nonpolar Dielectrics . . . . . . . . . . . . 3.2.3 Polarization of Polar Dielectrics . . . . . . . . . . . . . . . 3.2.4 Electrostatic Attraction and Repulsion Forces . . . . . . 3.2.5 Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Atomic Structure of Linear Dielectrics . . . . . . . . . . . . . . . . . 3.3.1 Nonpolar and Polar Covalent Bonds . . . . . . . . . . . . 3.3.2 Ionic Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Types of Dielectric Polarization . . . . . . . . . . . . . . . . . . . . . . 3.5 Polarization Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Electronic Polarization . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Atomic Polarization . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Ionic Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Dipolar Polarization . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Predominant Type of Polarization Depending on the Range of Frequencies . . . . . . . . . . . . . . . . . . . . . . .

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2 Static and Stationary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Classification of Fields . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Distinction Between Stationary and Static Fields . 2.2.3 Fields and Examples . . . . . . . . . . . . . . . . . . . . . 2.2.4 Main Expressions for Stationary and Static Fields 2.3 Basis Laws and Concepts of Electrostatics . . . . . . . . . . . . 2.3.1 About Electrostatic Fields . . . . . . . . . . . . . . . . . . 2.4 Electric Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Force Between Electric Charges and Coulomb’s Law . . . . 2.6 Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Gradient of Electrostatic Potential . . . . . . . . . . . . . . . . . . 2.9 Electric Flux, Electric Flux Density and Gauss’s Law . . . . 2.9.1 Divergence of Electrostatic Field . . . . . . . . . . . . . 2.9.2 Curl of Electrostatic Field . . . . . . . . . . . . . . . . . . 2.10 Divergence of Electric Flux Density . . . . . . . . . . . . . . . . 2.11 Work and Potential Energy in Electrostatics . . . . . . . . . . . 2.12 Electrostatic Potential Energy . . . . . . . . . . . . . . . . . . . . . 2.13 Laplace’s and Poisson’s Equations . . . . . . . . . . . . . . . . . . 2.14 Stationary-in-Time Fields Arising of Steady Current . . . . . 2.15 Examples and Problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.6 3.7 3.8 3.9

3.10 3.11 3.12 3.13

3.14

3.15 3.16

3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26

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Electric Flux Density, Permittivity, and Susceptibility . . . . . . . Relationships Between Volume Charge Densities . . . . . . . . . . Electric Fields Created by Free and Bound Charges . . . . . . . . Electric Field into a Dielectric Insert . . . . . . . . . . . . . . . . . . . 3.9.1 Examples of Extraneous Charges . . . . . . . . . . . . . . . 3.9.2 Surface Charge Density . . . . . . . . . . . . . . . . . . . . . . 3.9.3 Electric Field in Dielectric Insert Placed Between Two Plates of a Capacitor . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 Consideration of Non-uniform Polarization . . . . . . . . Relative Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement and Polarization Currents . . . . . . . . . . . . . . . . . Notes About Electric Currents . . . . . . . . . . . . . . . . . . . . . . . . 3.13.1 Conduction Currents . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Convection Currents . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.3 Advection Currents . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.4 Eddy Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.5 Electrolytic Currents . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.6 Magnetization Currents . . . . . . . . . . . . . . . . . . . . . . . Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14.1 Extraneous Currents and Forces . . . . . . . . . . . . . . . . 3.14.2 Electromotive Force . . . . . . . . . . . . . . . . . . . . . . . . . 3.14.3 Magnetomotive Force . . . . . . . . . . . . . . . . . . . . . . . . Electrical Conductivity and Resistivity . . . . . . . . . . . . . . . . . . Magnetic Field Sources and Concepts . . . . . . . . . . . . . . . . . . 3.16.1 Right-Hand Grip Rule . . . . . . . . . . . . . . . . . . . . . . . 3.16.2 Left-Hand Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulates of Magnetostatics in Free Space . . . . . . . . . . . . . . . Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Field Intensity and Magnetic Susceptibility . . . . . . . Boundary Conditions for Steady Electric Current Density . . . . Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . . . . . . . Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biot–Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Flux Density Around a Straight Wire . . . . . . . . . . . Magnetic Flux Density Due to Circular Current Loops . . . . . . Main Types of Magnetic Materials . . . . . . . . . . . . . . . . . . . . 3.26.1 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26.4 Ferrimagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26.5 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . .

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3.27 Constitutive Relations for Media . . . . . 3.27.1 Isotropic Media . . . . . . . . . . . 3.27.2 Anisotropic Media . . . . . . . . . 3.27.3 Permittivity Tensor of Crystals 3.27.4 Biisotropic Media . . . . . . . . . . 3.27.5 Bianisotropic Media . . . . . . . . 3.27.6 Notes About Media . . . . . . . . 3.28 Review Questions . . . . . . . . . . . . . . . .

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4 Maxwell’s Equations and Boundary Conditions . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Faraday’s Law of Electromagnetic Induction . . . . . . . . . 4.2.1 Fundamental Postulate for Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Lenz’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Maxwell’s Equations in Differential Form . . . . . 4.3.2 The Sense of Maxwell’s Equations . . . . . . . . . . 4.3.3 Co-dependence of Maxwell’s Equations . . . . . . 4.3.4 Maxwell’s Equations in Large-Scale Form . . . . . 4.3.5 Faraday’s Law in Integral Form . . . . . . . . . . . . 4.3.6 Ampere’s Law in Integral Form . . . . . . . . . . . . 4.3.7 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Magnetic Source Law . . . . . . . . . . . . . . . . . . . . 4.3.9 Table of Maxwell’s Equations . . . . . . . . . . . . . . 4.4 Maxwell’s Equations for the Time–Periodic Case . . . . . . 4.5 Source-Free Fields in Nonconducting Media . . . . . . . . . 4.6 Classification of Media Based on the Conductivity . . . . . 4.7 Concept of Boundary Conditions . . . . . . . . . . . . . . . . . . 4.8 Derivation of Electromagnetic Boundary Conditions . . . . 4.9 Boundary Conditions for Et and Dt Components . . . . . . 4.10 Boundary Conditions for Ht and Bt Components . . . . . . 4.11 Boundary Conditions for Dn and En Components . . . . . . 4.11.1 Explanation About Surface Charges on Interface of Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2 Boundary Conditions for Polarization Vectors . . 4.12 Boundary Conditions for Bn and Hn Components . . . . . . 4.13 Deformation of E and D Vectors at a Dielectric Sphere Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.1 Solution for the Radial Part of Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.2 Solution for the Angular Part of Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.13.3 4.13.4 4.13.5 4.13.6

4.14 4.15 4.16 4.17

4.18

Constructing of the General Solution . . . . . . . . . Satisfying of Boundary Condition . . . . . . . . . . . Potentials Inside and Outside the Sphere . . . . . . Electric Field Inside and Outside the Sphere and Bounded Surface Charges . . . . . . . . . . . . . . . . . 4.13.7 Force Lines of Electric Field and Electric Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Conditions for E and D in Pictures . . . . . . . . Boundary Conditions for H and B in Pictures . . . . . . . . Summarizing Boundary Conditions Between Lossless Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Conditions for Dielectric-Conductor . . . . . . . . 4.17.1 Boundary Condition at the Interface of Two Conductor Media . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Plane Electromagnetic Wave Propagation . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Phase Shift—Leading and Lagging Phases of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spherical and Cylindrical Electromagnetic Waves . . . . . . . . . . 5.4 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Relations Between Electric and Magnetic Fields of a Plane Wave in a Vacuum . . . . . . . . . . . . . . . . . . . . . 5.5 Complex Amplitudes and Attenuation Constant . . . . . . . . . . . 5.6 Presentation of Maxwell’s Equations for Plane Waves . . . . . . 5.7 Intrinsic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Time-Periodic Case for a Plane Wave . . . . . . . . . . . . . . . . . . 5.9 Plane Waves in Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Plane Waves in Good Conductors . . . . . . . . . . . . . . . . . . . . . 5.10.1 Phase Difference Between E and H Fields in Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Plane Waves in Low-Loss Dielectrics . . . . . . . . . . . . . . . . . . 5.12 Flow of Electromagnetic Power and the Poynting Vector . . . . 5.12.1 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Time-Averaged Poynting Vector . . . . . . . . . . . . . . . . . . . . . . 5.14 Phase and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . 5.15.1 Constructive and Destructive Interference . . . . . . . . . 5.16 Plane Wave and Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 5.16.1 Common Notes About Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Approach to Polarization from Electrodynamics and Optics . . .

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288 289 291 292 296 298 303

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308 309 310 314 315 319 326 327 328

. . 332 . . 333

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5.18 Linear Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Circularly Polarized Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Elliptically Polarized Wave . . . . . . . . . . . . . . . . . . . . . . . . . 5.20.1 Polarization Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 General Description of Polarizations . . . . . . . . . . . . . . . . . . 5.21.1 Shape of Curve Drawn by the End of E-Vector Depending on the Phase Shift . . . . . . . . . . . . . . . . . 5.22 Problem Related to Linear Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23 Problem Related to Circular Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.24 Problem Related to Elliptical Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25 Standing Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . 5.26 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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338 340 340 341 346

. . . 351 . . . 352 . . . 354 . . . 356 . . . 358 . . . 360

6 Reflection and Transmission of Plane Electromagnetic Waves . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Main Definitions About Incident Plane Waves . . . . . . . . . . . . 6.2.1 Perpendicular and Parallel Polarizations . . . . . . . . . . . 6.2.2 Perpendicular Polarization . . . . . . . . . . . . . . . . . . . . . 6.2.3 Parallel Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Fresnel’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Total Internal Reflection and Critical Angle . . . . . . . . . . . . . . 6.5 Brewster’s Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Determination of a Phase of Reflected Wave . . . . . . . . . . . . . 6.7 Oblique Incidence of Perpendicular Polarized Wave on Dielectric Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Oblique Incidence of Parallel Polarized Wave on Dielectric Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Normal Incidence of Parallel Polarized Wave on Dielectric Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Normal Incidence of Parallel Polarized Wave on Conductor Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1 Calculations of Electric and Magnetic Fields of Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Oblique Incidence of Perpendicular Polarized Waves on Conducting Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Determination of the Phase of Reflected Wave from a Perfect Conductor Plane . . . . . . . . . . . . . . . . . . . . . 6.11.2 Distributions of Total Magnetic Field . . . . . . . . . . . . 6.11.3 Pattern of Electromagnetic Field and Current Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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363 363 365 367 371 371 372 376 379 380

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6.12 Oblique Incidence of Parallel Polarized Waves on Conducting Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12.1 Phase Velocity of Electromagnetic Waves Directed by the Conducting Plane . . . . . . . . . . . . . . . . . . . . . . 6.12.2 Distributions of Total Electric Field . . . . . . . . . . . . . . 6.12.3 Pattern of Electromagnetic Field, Current and Charge Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Types of Waves Propagating Between Two Conductor Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Rectangular Hollow Metallic Waveguides and Resonators . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Classification of Propagating Modes . . . . . . . . . . . . . . . . . . 7.3 Electromagnetic Wave Propagation in a Source-Free Area . . 7.4 General Characteristics of Waveguide Modes . . . . . . . . . . . . 7.4.1 Helmholtz’s Equations . . . . . . . . . . . . . . . . . . . . . . 7.4.2 TEM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 General Behaviors of TE and TM Modes . . . . . . . . . . . . . . . 7.5.1 Transverse Electric Modes . . . . . . . . . . . . . . . . . . . 7.5.2 Transverse Magnetic Modes . . . . . . . . . . . . . . . . . . 7.5.3 Expressions Corresponding to TE and TM Modes . . 7.5.4 A Cutoff Frequency and Wavelength of Waveguide Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Phase and Group Velocities of Propagating Modes . 7.5.6 Wave Impedance of TE and TM Modes . . . . . . . . . 7.6 Solutions for TEmn Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Instantaneous Field Expressions for TEmn Modes . . 7.6.2 Explanation of TEmn Mode Indexes . . . . . . . . . . . . . 7.6.3 Dispersion Characteristics and Cutoff Frequencies of TEmn Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Main Mode TE10 of a Rectangular Waveguide . . . . 7.6.5 Poynting Vector and Energy Transportation of TE10 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.6 Calculations of Electric and Magnetic Field Vectors . 7.6.7 Distribution of Surface Conduction Currents of TE10 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.8 3D Distribution of Electric Field Intensity of TE10 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.9 Displacement and Surface Conduction Currents . . . . 7.6.10 Schematic Sectional View for TE10 . . . . . . . . . . . . .

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425 425 426 429 430 433 437 440 441 443 445

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445 449 450 452 459 460

. . . 462 . . . 464 . . . 466 . . . 467 . . . 471 . . . 472 . . . 473 . . . 475

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Contents

Solutions for TMmn Modes . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Instantaneous Field Expressions for TMmn Modes . 7.7.2 Explanation of TMmn Mode Indexes . . . . . . . . . . . 7.7.3 Dispersion Characteristics and Cutoff Frequency of TMmn Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.4 Joint Dispersion Characteristics of TEmn and TMmn Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Calculations of Electric and Magnetic Fields of TM11 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.6 Distribution of Surface Conduction Currents of TM11 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.7 3D Distribution of Electric Field Intensity of TM11 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.8 Schematic Sectional View for TM11 Mode . . . . . . Attenuation in Rectangular Waveguides . . . . . . . . . . . . . . . 7.8.1 Attenuation by a Filled Waveguide Dielectric . . . . 7.8.2 Attenuation by Conducting Waveguide Walls . . . . Excitation of Waveguides by Probes, Loops, and Slots . . . . 7.9.1 Excitation by a Probe . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Excitation by a Loop . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Excitation by a Slot . . . . . . . . . . . . . . . . . . . . . . . Rectangular Cavity Resonators . . . . . . . . . . . . . . . . . . . . . 7.10.1 Common Principles About Resonators . . . . . . . . . . 7.10.2 Transverse Electric TEmnp Modes . . . . . . . . . . . . . 7.10.3 Resonance Frequency of TEmnp Mode . . . . . . . . . 7.10.4 Schematic Sectional View of TE101 Mode . . . . . . . Transverse Magnetic TMmnp Resonator Modes . . . . . . . . . . 7.11.1 Resonance Frequency of TMmnp Mode . . . . . . . . . 7.11.2 Transverse Magnetic TM111 Mode . . . . . . . . . . . . 7.11.3 Schematic Sectional View for TM111 Mode . . . . . . Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitation of Rectangular Resonators by Probes, Loops, and Slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions About Resonators . . . . . . . . . . . . . . . . . . . . . Examples and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 477 . . . . 481 . . . . 482 . . . . 483 . . . . 485 . . . . 485 . . . . 487 . . . . . . . . . . . . . . . . . . .

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489 490 491 492 494 497 498 500 501 503 504 510 517 522 523 529 532 534 535

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538 539 540 542

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Solution of Helmholtz’s Equation in Cylindrical Coordinates 8.3 Bessel Functions and Their Properties . . . . . . . . . . . . . . . . . 8.4 Solutions for TEmn Modes in Circular Waveguides . . . . . . . . 8.5 Instantaneous Field Expressions for TEmn Modes . . . . . . . . .

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545 545 547 552 556 559

7.8

7.9

7.10

7.11

7.12 7.13 7.14 7.15 7.16

Contents

8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 8.32 8.33

xix

Explanation of TEmn Mode Indexes . . . . . . . . . . . . . . . . . . Expressions Corresponding to Transverse Electric Modes . . Dispersion Characteristics of TEmn Modes . . . . . . . . . . . . . Main Mode TE11 of Cylindrical Waveguides . . . . . . . . . . . Electromagnetic Field Distributions of TE11 Mode . . . . . . . Schematic Sectional Views for TE11 Mode . . . . . . . . . . . . Solutions of Helmholtz’s Equation for TEmn Modes . . . . . . Instantaneous Field Expressions for TMmn Modes . . . . . . . Explanation of TMmn Mode Indexes . . . . . . . . . . . . . . . . . Expressions Corresponding to Transverse Magnetic Modes . Dispersion Characteristics and Cutoff Frequency of TMmn Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Dispersion Characteristics of TEmn and TMmn Modes . Formulae for TM01 Mode . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of TM01 Field Distributions . . . . . . . . . . . . . . . Schematic Sectional Views for TM01 Mode . . . . . . . . . . . . Circular Cylindrical Cavity Resonators . . . . . . . . . . . . . . . . Expressions for Electromagnetic Field Components of TEmnp Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instantaneous Field Expressions for Resonator TEmnp Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Frequency of TEmnp Modes . . . . . . . . . . . . . . . Electromagnetic Field Calculations of TE111 Mode . . . . . . . Schematic Sectional Views of TE111 Mode . . . . . . . . . . . . Transverse Magnetic TMmnp Resonator Modes . . . . . . . . . . Instantaneous Field Expressions for Resonator TMmnp Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Frequency of TMmnp Modes . . . . . . . . . . . . . . . Resonator Mode TM010 . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculations of TM011 Field Distributions . . . . . . . . . . . . . Schematic Sectional View of TM011 . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Chronological Development of Plasmonics . . . . . . . . 9.3 Definitions Used in Plasmonics . . . . . . . . . . . . . . . . 9.4 General Concepts of Plasmonics . . . . . . . . . . . . . . . 9.4.1 Dispersion Equations for Two Types of Electromagnetic Waves . . . . . . . . . . . . . 9.4.2 Plasmons with the Longitudinal Component of Electric Field . . . . . . . . . . . . . . . . . . . . . 9.4.3 High- and Low-Frequency Plasmons . . . . . .

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560 562 563 563 565 567 569 571 572 573

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575 576 576 578 579 581

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586 587 587 592 594

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597 597 598 600 604 607

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611 611 612 616 619

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Contents

9.5 9.6

9.7 9.8 9.9 9.10 9.11 9.12

9.13 9.14 9.15 9.16

9.17

9.4.4 Plasmons in Complex Media . . . . . . . . . . . . . . . . . . Optical Properties of Metals . . . . . . . . . . . . . . . . . . . . . . . . . Models of Dielectric Permittivity Determination . . . . . . . . . . . 9.6.1 Drude and Drude–Sommerfeld Models . . . . . . . . . . . 9.6.2 Dependence of Current Density on the Loss Term . . . 9.6.3 Solution of Homogeneous Differential Equation . . . . . 9.6.4 Case When the Electric Field Is a Stationary Field . . . 9.6.5 Case When the External Electromagnetic Field Is a Time-Harmonic Monochromatic . . . . . . . . . . . . . . . . 9.6.6 Notes About the Conductivity . . . . . . . . . . . . . . . . . . 9.6.7 Realization of Drude–Sommerfeld Model . . . . . . . . . Drude–Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . Determination of Permittivity by the Drude–Lorentz Model . . Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume Plasmons in “Jelly” Model . . . . . . . . . . . . . . . . . . . . Surface Plasmon Polariton at a Flat Metal Interface . . . . . . . . 9.11.1 General Characteristics of SPPs . . . . . . . . . . . . . . . . Dispersion Equations of SPPs at a Flat Interface . . . . . . . . . . . 9.12.1 TM and TE Mode Components . . . . . . . . . . . . . . . . . 9.12.2 Helmholtz’s Equations for TM and TE Modes . . . . . . 9.12.3 Solution and Dispersion Equation for TM Modes . . . . 9.12.4 Solution for TE Modes and Dispersion Equation . . . . Dispersion Equations of Conducting Cylindrical Waveguides . Principles of SPPs Excitations . . . . . . . . . . . . . . . . . . . . . . . . Application of Plasmonic Phenomena . . . . . . . . . . . . . . . . . . Examples and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.16.1 Drude–Sommerfeld Model . . . . . . . . . . . . . . . . . . . . 9.16.2 Dispersion Characteristics of SPP Mode at Au and Ag Flat Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.16.3 Dispersion Characteristics of Au Cylindrical Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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626 627 629 630 633 634 635

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636 641 642 643 644 645 648 649 654 654 660 663 664 665 669 671 682 685 688 689

. . 690 . . 693 . . 694

List of Appendixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 Brief CV of Prof. Liudmila Nickelson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Table of Main Designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

Abbreviations

1D 2D 3D a.r. a.u. AC Chap. Chaps. dB DC DIN e.g. EHF EHz EM etc. eV Fig. Figs. Hz i.e. IEEE IR ISO ITU K KHz LDF LF LSPs

One dimension Two dimensions Three dimensions Atomic radius Atomic units Alternating current Chapter Chapters Decibel Direct current German institute for standardization For example Extremely high frequency (30–300 GHz) Exahertz (1018 Hz) Electromagnetic Et cetera, continuing in the same way Electron volt Figure Figures Hertz, unit of frequency “that is” (id est) Institute of Electrical and Electronics Engineers Infrared range f * 0.3–430 THz Internal System of Quantity International Telecommunications Union Kelvin Kilohertz (103 Hz) London dispersion forces Low frequency (30–300 kHz) Localized surface plasmons

xxi

xxii

m MHz mm nm Np PEC PHz pm RF s SERS SHF SI units SPPs SPR TE TEM THF THz TIRE TM UV VGTU VIS viz. lm

Abbreviations

Meter is unit of length Megahertz (106 Hz) Millimeter (10‒3 m) Nanometer (10‒9 m) Neper Perfect electric conductor Petahertz (1015 Hz) Picometers (10‒12 m) Radio frequency (20 kHz–300 GHz) Second, unit of time Surface-enhanced Raman scattering spectroscopy Super high frequency (3–30 GHz) International System of Units Surface plasmon polaritons Surface plasmon resonance Transverse electric Transverse electromagnetic Tremendously high frequency (300–3000 GHz) Terahertz (1012 Hz) Total internal reflection ellipsometry Transverse magnetic Ultraviolet (*0.79–30 PHz) Vilnius Gediminas Technical University Visible range (*0.43‒0.79 PHz) “namely,” “that is to say” Micrometer (10‒6 m)

Abstract

The textbook Electromagnetic Theory and Plasmonics for Engineers (L. Nickelson) contains nine chapters, 51 tables, more than 350 figures and 2000 formulae, 749 pages. This textbook focuses on physics and microwave theory based on Maxwell’s equations and boundary conditions which are important for the purposes of studying the operation of waveguides and resonators in wide frequency range, namely, from approx. 109 to 1016 hertz. This textbook includes physical explanations of electromagnetic (EM) phenomena and explores the derivation of most expressions of physical quantities in great detail. The chapter about plasmonics is written in a style characteristic for microwave electrodynamics. The textbook has a fairly large number of figures and tables to ease the understanding of the topics discussed therein. In addition, there are numerous review questions after each chapter. The main topic of this textbook is the propagation of EM waves on waveguides or surfaces (Chaps. 7–9). All previous chapters prepare the reader for a more in-depth understanding of these last three chapters.





Keywords Electromagnetics Plasmonics Electromagnetic waves Maxwell’s equations Boundary conditions Constitutive relations Dispersion equations Dielectrics Conductors Complex media Lossy media Permittivity Permeability Linearly polarized EM waves Circularly polarized waves Elliptically polarized waves Standing wave Incident wave Reflected wave Refracted wave Perpendicular and parallel polarized waves Phase and group velocities Hollow metallic waveguides Propagation constant Complex amplitudes Phase constant Attenuation constant Impedance Cavity resonator TE mode TM mode Surface plasmon polaritons SPP waves Cylinder and flat metal interface Mode excitations Conduction current Displacement current Skin depth Cutoff frequency

 







 





 

















 













xxiii

Chapter 1

Vector Analysis in Annex to Electromagnetics

Abstract This chapter provides basic knowledge of vector analysis as EM-fields are actually vector values. This chapter considers such operators as divergence and curl because ЕМ theory is based on Maxwell’s equations containing vector operators. In case of propagation of EM waves in strongly absorbent media, the electric and magnetic fields of waves are described by using complex values. Therefore, the first chapter also deals with complex numbers. At the end of the chapter, the author gives 18 tasks with respective solutions.

1.1

Introduction to Vector Analysis

The classical electromagnetic (EM) field theory (or electrodynamics) is a division of natural science which describes the EM force interaction between electric charges or/and currents as well as the processes of excitation and propagation of EM waves in different media, EM devices, and different appliances. Electromagnetism is associated with charged bodies in motion and varying electric and magnetic fields. Because of the fact that a moving charge produces a magnetic field, electromagnetism is concerned with effect such as magnetism, EM radiation, and EM induction, including such practical applications as electric generators and electric motors. EM theory deals with the study of electric and magnetic fields. It covers wide region of EM phenomena from the propagation of waves in the outer or near-Earth spaces to the processes in the EM devices. In this book, we present the classical EM theory which was first systematically explained by the famous physicist James Clerk Maxwell. James Clerk Maxwell (1831–1879) was a Scottish physicist and mathematician. His most significant achievement was the development of the classical EM theory, synthesizing all previous related observations, experiments, equations of electricity and magnetism. His set of equations which were named Maxwell’s equations demonstrated that electricity, magnetism, and even light are all manifestations of the same phenomenon related to the EM field. Maxwell’s equations are a set of differential equations. © Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_1

1

2

1 Vector Analysis in Annex to Electromagnetics

The course of technical electrodynamics includes the study of the theory of electromagnetic processes and electrodynamics of devices technique. It covers wide region of electromagnetic phenomena from the propagation of waves in the near-Earth space or the interstellar one to the processes in the EM devices, including   lasers and different plasmonic phenomena at very high frequencies  1016 Hz : A more recent development is quantum electrodynamics, which was formulated to explain the interaction of EM radiation with matter and to which the laws of the quantum theory apply. Quantum mechanics (also known as quantum physics or quantum theory) is a branch of physics dealing with physical phenomena at microscopic scales (typically, this means distances of nanometers or less, i.e. atomic and subatomic length scales). The physicists P. A. M. Dirac, W. Heisenberg, and W. Pauli were the pioneers in the formulation of quantum electrodynamics. Because of the fact that the velocities of charged particles can become comparable with the speed of light, corrections involving the theory of relativity must be made. This branch of the theory is called relativistic electrodynamics. Electric and magnetic fields are vector quantities, and their behavior is governed by Maxwell’s equations. The mathematical formulation of Maxwell’s equations requires that we first learn the basic rules pertinent to mathematical manipulation involving vector quantities. We will study Maxwell’s equations in detail later on. Here, we only write the set of the equations in some formal form to show that we need some knowledge about the vector algebra. Maxwell’s equations are set of four fundamental equations governing the behavior of electric and magnetic fields. This is the set of two curl and two divergence equations of Maxwell’s ones: r  E ¼  @B @t ; r  H ¼ J þ @D @t ; r  D ¼ q; r  B ¼ 0:

ðFaraday's lawÞ ðAmpere's law) ðGauss's law) ðno name)

ð1:1:1Þ

Here are Maxwell’s equations written in the traditional way, where E is a vector of electric field strength (intensity), D is a vector of electric flux density (electric induction, electric displacement), H is a vector of magnetic field strength (intensity), B is a vector of magnetic flux density (magnetic induction), J is a vector of current density, q is a density of electric charge, r is a vector differential operator that acts upon any scalar and vector fields, and the nabla symbol r denotes the mathematical “del” operator in vector calculus. In terms of vector spaces, the mapping from one vector space to another may be also presented as an operator. r  E is the curl of the electric field, r  H is the curl of the magnetic field, r  D is the divergence of the electric flux density, and r  B is the divergence of the magnetic flux density. So we see here that we need to know the basic principles of the algebraic operations.

1.1 Introduction to Vector Analysis

3

In the next paragraph, we will briefly recall the main points of vector algebra. The vector algebra is very important in the study of EM-field theory.

1.2

Basic Vector Operations

When we go 2 km from the point A to the south and after those 3 km from the point B to the east, we will have gone the total of 5 km and our final point will be C (Fig. 1.1). In this example, we are dealing with vectors because the straight-line segments (displacement) going from points A to B, from B to C, and from point A to C have a direction and a magnitude. Both the magnitude and the direction must be specified for a vector quantity (or vector fields, vectors), in contrast to a scalar quantity (scalar fields, scalars) which can be defined with just a magnitude (or numerical value). In general, a field is a vector or scalar quantity that can be specified everywhere in space as a function of position or other variable. Examples of vector quantities (fields) are force, velocity, acceleration, momentum, electric field strength, magnetic field strength, displacement (a displacement vector is the shortest distance from the initial (origin) to the final (terminal) position), Poynting vector, current density, polarization (dipole moment per unit volume), magnetization (magnetic dipole moment per unit volume). Examples of scalar quantities (fields) are length, work, power, pressure, energy, charge (of electron, hole, etc.), mass (of particle or other body, etc.), temperature, concentration of particles, permittivity or permeability of free space, speed of light (not to be confused with the velocity of light), volume, area, resistance, conductivity, impedance, capacitance, etc. The main vector operations are graphically described in Fig. 1.2. The process of splitting a vector into various parts or components is called the resolution of the vector. These parts of a vector may act in different directions and are called components of the vector. The resolution (decomposition) of vector into components can be done in 2D space Fig. 1.3a and in 3D space Fig. 1.3b. There are three components of a vector, viz. component along x-axis called x-component, component along y-axis called y-component, and component along z-axis called z-component.

Fig. 1.1 Illustration of vectors

A

B

C

4

1 Vector Analysis in Annex to Electromagnetics Resolution (decomposition) of vector into component

Vector Operation Vector (cross) product of vectors

which is used in calculation of

which is used in calculation of

Scalar (dot, inner) product of vectors

Scalar multiplication (vector multiplication by a constant)

Addition and subtraction of vectors

Work, magnetic potential energy

Torque, magnetic force

x X component

z

x

Θ

X component

nt

φ

ne

O

ori

po

r

Θ O

gin

y

m

r

g ori

or

ect

v al

co

l

ina

y

(b)

Z

or

ct ve

Y component

(a)

Y component

Fig. 1.2 Picture of vector operations

Fig. 1.3 Resolution (decomposition) of vector into components may be done in a 2D space and b 3D space

The process of finding the unit vector ^r of vector r is called normalization. As mentioned in scalar multiplication, multiplying a vector by constant C will result in the magnitude being multiplied by C. Any number of vector quantities of the same type (i.e. same units) can be combined by basic vector operations. Vector operations are extension of the laws of elementary algebra to vectors. In vector operations, four vector operations can be defined, i.e. addition (subtraction) and three types of multiplication. Since vector may have in general an arbitrary orientation in three dimensions, we need to define a set of three reference directions at every point in space in terms of which we can describe vector drawn at the point. Two combinations are possible for the orientation of the coordinate system (Figs. 1.4 and 1.5). The positive rotation of the coordinate system about the y- axis is clockwise for the left-handed coordinate system when the positive x-, y-, and z- axes point right, up, and forward, respectively (Fig. 1.5a). The positive rotation of the coordinate system about the y-axis is counterclockwise for the right-handed coordinate system when the positive x- and y- axes

1.2 Basic Vector Operations

(a)

5

y

(b)

y z

x z

x

Fig. 1.4 Coordinate system a left-handed, b right-handed

(a)

(b)

+y

+y

+z

–z

O

–z

Θ +x

+z

–y

O –y

Θ

+x

Fig. 1.5 Positive rotation a left-handed, b right-handed

point right and up, and the negative z- axis points forward (Fig. 1.5b). The right-handed coordinate system will be used in Chap. 7. In the Cartesian coordinate system, the origin is the point where the axes of the system intersect. Please pay attention that scalars are one-component quantities that are invariant under rotations of the coordinate system.

1.2.1

Unit Vectors

A unit vector is a vector of unit length. A unit (or basis, or coordinate) vector is sometimes denoted by replacing the arrow on a vector with a sign “^” (with a “hat” as “^”) or just adding a “^” on a boldfaced character (i.e. ^e). The unit vector in a normed vector space is the vector of length 1.

6

1 Vector Analysis in Annex to Electromagnetics

Therefore, the magnitude of unit vector is ^e ¼ j^ej ¼ 1:

ð1:2:1Þ

Any vector r can be converted into a unit vector by dividing the vector by its length. If our vector is r, then our unit vector ^e is comfortable to be denoted by ^r (the same letter as the vector), ^e ¼ ^r, the unit vector ^r and its module j^rj are: j^rj ¼ 1; ^r ¼

r ; j rj

ð1:2:2Þ

where module jrj is also known as the norm (or the length, or the magnitude) of vector r: The vector can be written through the unit vector ^r and the vector length r (not a boldfaced character): r ¼ r^r:

ð1:2:3Þ

Any vector can be fully represented by providing its magnitude and a unit vector along its direction (Fig. 1.6). The starting point of a vector is known as the initial point, and the endpoint is known as the terminal point. We reserve vectors ^x, ^y, and ^z as unit vectors in the x-, y-, and z-directions in the 3D Cartesian (right-handed, Fig. 1.5b) coordinate system (also known as Cartesian system, or Cartesian coordinates) (Fig. 1.7). In the literature, you can often meet the notations of unit vectors as “i; j, and k”; instead of ^ x, there is “i”; instead of ^ y, there ^ is “j”; and instead of z, there is “k.”

1.2.2

Addition of Two Vectors

As we mentioned, a vector is a mathematical object that has a magnitude and a direction. A line of given length and pointing along a given direction, such as an

Fig. 1.6 Representation of any vector, r ¼ r^r and ^r is a unit vector

r r

l ^r

1.2 Basic Vector Operations

7

y

Fig. 1.7 Unit vectors in the Cartesian system coordinate

y^

^z

O

x^

x

z

arrow, is the typical representation of a vector. Typical notation to designate a vector is a boldfaced character, or a character with arrow on it (i.e. A, ~ A). The magnitude of a vector is its length and is normally denoted by jAj or A (when A is not a boldfaced character). We are going to use here boldface characters (A, B, F, E, H, p, v, etc.) for vectors and ordinary type for scalars (A, B, F, E, q, c, e, etc.). The sum of two vectors A and B (Fig. 1.8a) is another vector C (Fig. 1.8b, c). The sum of two vectors can be obtained graphically in two ways. The first way is by the parallelogram rule. The resultant vector C is the diagonal vector of the parallelogram formed by vectors A and B drawn from the same point, as we can see in Fig. 1.8b. The second way is by the head-to-tail rule (also known as the tip-to-tail rule). The addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as shown in Fig. 1.8c. We receive the sum vector (A + B) when we place the tail of B at the head of A. The following rules apply in vector algebra: Commutative Law The commutative law says we can change number over and still get the same answer when we add: A þ B ¼ B þ A;     A þ B ¼ ^xAx þ ^yAy þ ^zAz þ ^xBx þ ^ yBy þ ^zBz ;

(a)

(b)

ð1:2:4Þ ð1:2:5Þ

(c)

tip

B

B tail

C

C B

A tail

tip

A

A

Fig. 1.8 Illustration of addition of two vectors a vectors A and B, b vector addition using parallelogram rule, c head-to-tail rule

8

1 Vector Analysis in Annex to Electromagnetics

  A þ B ¼ ^xðAx þ Bx Þ þ ^y Ay þ By þ ^zðAz þ Bz Þ:

ð1:2:6Þ

Associative Law The associative laws say that it does not matter how we group the terms (i.e. which we calculate first) when we add: ðA þ BÞ þ C ¼ A þ ðB þ CÞ;

ð1:2:7Þ

where A, B, and C are vectors. Distributive Law kðA þ BÞ ¼ kA þ kB:

ð1:2:8Þ

The multiplication of a vector by a scalar can be defined as: kA ¼ kA^aA

ð1:2:9Þ

where ^aA is the unit vector in the vector A direction. Magnitude A has the unit and dimension of vector A. If a number k is a positive scalar, the magnitude of A will be changed by k times without changing the direction. Subtraction Subtraction of two vectors is just a special case of addition. The vector ðA  BÞ is defined as: A  B ¼ A þ ðBÞ;

ð1:2:10Þ

where the term −B is the negative of vector B, i.e. vector −B has the same magnitude as B but its vector direction is opposite to vector B (Fig. 1.9). In Fig. 1.9b, the sum and subtraction of two vectors are shown. A vector has an initial (origin) point at “O” and a terminal (final) point at the end of the vector.

(a)

y

(b) A B

A+

A–B

B

–B

A–B

x

O

B A

Fig. 1.9 Illustration of subtraction of two vectors using a the parallelogram, b triangle methods

1.2 Basic Vector Operations

1.2.3

9

Base Vectors and Vector Components

Base vectors are a set of vectors selected as a base for representation of all the other vectors. Each vector is constructed from the addition of vectors along the base directions. For example, the vector r ¼ r^r in Fig. 1.10 can be written as the sum of the three vectors r1 ; r2 and r3 , each along the direction of one of the base vectors ^r1 ; ^r2 and ^r3 , so we can write: r ¼ r1 þ r2 þ r3 :

ð1:2:11Þ

Each one of the vectors r1 ; r2 and r3 is parallel to one of the base vectors and can be written as a scalar multiplier of that base. Let r1 ; r2 and r3 denote these scalar multipliers so that one has (Fig. 1.11). r1 ¼ ^r1 r 1 ; r2 ¼ ^r2 r 2 ; r3 ¼ ^r3 r 3 :

ð1:2:12Þ

The original vector r can now be written as r ¼ ^rr ¼ ^rjrj and can be expressed as: r ¼ ^r1 r 1 þ ^r2 r 2 þ ^r3 r 3 :

ð1:2:13Þ

The scalar multipliers r 1 ; r 2 , and r 3 are known as the components of r in the base described by the base vectors ^r1 ; ^r2 , and ^r3 . A vector can be resolved (decomposed) into its components along any two directions in a plane containing it. A 2D vector can be decomposed in many ways. In the Cartesian coordinate system, the vector is decomposed into a portion along the ^x- and the ^y-directions. This rule also applies to 3D space. ^r

Fig. 1.10 Construction of any vector

r

r

3

^r

3

r

r

1

2

^r

^r

1

2

^r

Fig. 1.11 Construction of vectors

1

^r

1

1

3

r^ r

r^r

3

r=^rr

^r

3

^r r

2 2

^r

2

10

1.2.4

1 Vector Analysis in Annex to Electromagnetics

Vectors in Three Dimensions

In three dimensions (3D), a vector can be resolved along any three non-coplanar lines. Coplanar lines are lines that lie on the same plane. Figure 1.12 shows how a vector can be resolved (decomposed) along the three directions by first finding a vector in the plane of two directions (e.g. in plane xOy) and then resolving the vector along the other two directions in the other plane (e.g. xOz). When vectors are represented in terms of base vectors and components, the addition of these vectors is the addition of the components at the same base vectors. Therefore, if two vectors A and B are represented by their components Ax , Ay , Az and Bx , By , Bz : A ¼ Ax ^x þ Ay ^y þ Az^z;

ð1:2:14Þ

B ¼ Bx ^x þ By ^y þ Bz^z;

ð1:2:15Þ

  A þ B ¼ ðAx þ Bx Þ^x þ Ay þ By ^y þ ðAz þ Bz Þ^z:

ð1:2:16Þ

then we can write:

1.2.5

Rectangular Components in 2D

The base vectors of a rectangular Oxy coordinate system are given by the unit vectors ^x and ^y along the x- and y-directions, respectively (Fig. 1.13). Using the base vectors, one can represent any vector F as: F ¼ ^xF x þ ^yF y : Fig. 1.12 Resolution of a vector in three directions

ð1:2:17Þ

y Ay A

^ ^y A O ^z Az z

^x

Ax

x

1.2 Basic Vector Operations

11

Fig. 1.13 Vectors in the Oxy coordinate system

y

F y^

O

Fy

Θ x^

x

Fx

We will define the resultant vector as the vector sum of two or more vectors. Due to the orthogonally of the bases, the magnitude of the resultant vector is: F ¼ jFj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2x þ F 2y :

ð1:2:18Þ

Here, F is a module jFj of the resultant vector F. The components of the vector can be written from Fig. 1.13: F x ¼ F cosðHÞ;

ð1:2:19Þ

F y ¼ F sinðHÞ;

ð1:2:20Þ

tanðhÞ ¼ F y =F x ;

ð1:2:21Þ

  H ¼ atan F y =F x ;

ð1:2:22Þ

where the angle H determines the direction of the vector sum Eq. (1.2.17).

1.2.6

Position and Displacement Vectors

A position or a position vector, also known as a location vector or a radius vector, is a vector that represents the position of a point P in space (Fig. 1.14). Let us denote the position vector by r. The position vector (known also as radius vector) is: r ¼ x^x þ y^y þ z^z:

ð1:2:23Þ

A vector is usually written in the literature thought its projections as follows: r ¼ ðx; y; zÞ ¼ hx; y; zi;

ð1:2:24Þ

12

1 Vector Analysis in Annex to Electromagnetics

Fig. 1.14 Vectors in the Oxyz coordinate system

y (0,y,0)

ry

β ^r

O

P(x,y,z)

r α

rx

x

rz z

(x,0,z)

^; y ^; ^z (sometimes in the litx; y; z are projections of vector r along directions x erature the projections r x ; r y ; r z are denoted with the same letter as the vector   r ¼ rx ; ry ; rz . Examples, r ¼ ð1; 2; 2Þ ¼ h1; 2; 2i; x ¼ 1; y ¼ 2; z ¼ 2. The unit vectors in Cartesian coordinates ^x ¼ ð1; 0; 0Þ, ^y ¼ ð0; 1; 0Þ, ^z ¼ ð0; 0; 1Þ. The length or magnitude of any r vector (also known as the module or the norm of a vector) is denoted as jrj, or r (not bold), and sometimes in the literature, it is also denoted as krk: j rj ¼ r ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y2 þ z 2 :

ð1:2:25Þ

r r ¼ : j rj r

ð1:2:26Þ

The unit vector of this r vector is: ^r ¼

The radius vectors (position vectors) r1 and r2 can be written as follows: r1 ¼ ^xx1 þ ^yy1 þ ^zz1 ;

ð1:2:27Þ

r2 ¼ ^xx2 þ ^yy2 þ ^zz2 :

ð1:2:28Þ

The displacement r12 ¼ r2 r1 is a straight-line segment going from the point P1 to P2: r12 ¼ r2  r1 ¼ ^xðx2  x1 Þ þ ^yðy2  y1 Þ þ ^zðz2  z1 Þ:

ð1:2:29Þ

1.2 Basic Vector Operations

13

The magnitude (length, module) of vector r12 is: jr12 j ¼ r 12 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2  x1 Þ2 þ ðy2  y1 Þ2 þ ðz2  z1 Þ2 :

ð1:2:30Þ

The displacement has direction ^r12 : ^r12 ¼

r12 x^ðx2  x1 Þ þ y^ðy2  y1 Þ þ ^zðz2  z1 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r j 12 j ð x2  x1 Þ 2 þ ð y2  y1 Þ 2 þ ð z 2  z 1 Þ 2

ð1:2:31Þ

The displacement vector r12 is very similar to position vectors, but instead of being a directed line segment from the origin “O” to a point the displacement vector is the line segment between any two coordinate points, e.g. points P1 and P2. The unit vector ^r12 of vector r12 is directed along the line drawn from P1 to P2; see Fig. 1.15 and Eq. (1.2.31). One should keep in mind that distance and displacement are two quantities that have distinctly different definitions and the sense: (a) Distance is a scalar quantity that refers to a number of how far apart objects are situated. The distance is measured in kilometers, miles, etc. Scalar quantities such as distances are ignorant of direction. (b) Displacement is a vector quantity that refers to the shortest distance from the initial to the final position of the object’s overall change in its position. Vector quantity such as a displacement has a length and a direction.

y y2

Fig. 1.15 Vector r12 is between two points P1 and P2 in the Oxyz coordinate system

y1 1 1,z 1,y P1(x r12

r1

z1 z2 z

y^ O ^z

)

P2(x2,y2,z2)

r2 x^

x1

x2

P1(x1,0,z1) P2(x2,0,z2)

x

14

1 Vector Analysis in Annex to Electromagnetics

1.2.7

Vector Transform

We use the coordinate to describe position in space but there is the specific geometrical transformation law for converting vector components from one coordinates to another. Let us say the coordinate system x0 Oy0 is rotated by angle H relative to xOy (Fig. 1.16). We have a vector A with components Ax and Ay in coordinate xOy:   A ¼ Ax ^x þ Ay ^y :

ð1:2:32Þ

We have in new coordinate system x0 Oy0 :   A0 ¼ A0x x^0 þ A0y y^0 :

ð1:2:33Þ

The relations between vectors A and A0 can be written in the form: ^ A0 ¼ RA;

ð1:2:34Þ

 cos H sin H ^ is the tensor. The sign of “hat” as in Sect. 1.2.1 ,R  sin H cos H is usually placed at the top of the tensor symbol (it is not a vector. In the literature, ^¼ where R



$

tensors are also sometimes denoted as R ). The tensor presents a matrix that depicts linear relations between vectors and other tensors. In this case, the relation Eq. (1.2.34) can be written as follows: 

A0x A0y



 ¼

cos H sin H

sin H cos H



Ax Ay

 ð1:2:35Þ

We can write from Eq. (1.2.35) a more general expression by using other notations:

Fig. 1.16 Geometrical transformation law for converting vector components from one coordinate xOy to another x0 Oy0 geometrical transformation law for converting vector components from one coordinate to other

y

y'

A y^

y^ ' O

x^ ' x^

x' Θ x

1.2 Basic Vector Operations

15



A01 A02



 ¼

R12 R22

R11 R21



A1 A2

 ð1:2:36Þ

 P R11 R12 , A0i ¼ 2j¼1 Rij Aj . R21 R22 Vector transforms in 3D space We can write the transformation law for the rotation about an arbitrary axis in three dimensions in the following form: ^¼ where R



0

1 0 A01 R11 @ A02 A ¼ @ R21 A03 R31

R12 R22 R32

10 1 R13 A1 R23 A@ A2 A R33 A3

ð1:2:37Þ

and Eq. (1.2.37) in a more compact form: A0i ¼

3 X

Rij Aj ;

ð1:2:38Þ

j¼1

where index 1 is for x-component, and indexes 2 and 3 are for y- and z-components.

1.2.8

Dot Product of Two Vectors

The dot product (also known as scalar or inner product) of two vectors A and B is a scalar which equals the product of magnitudes of vectors A and B as well as cosine of the angle between them. A  B ¼ jAjjBj cos HAB ;

ð1:2:39Þ

where HAB is the angle between the vectors A and B which is less than 180° (Fig. 1.17). Magnitudes jAj and jBj are also called the modules (sometimes called the norms and denoted as kAk; kBk in technical literature):

Fig. 1.17 Scalar projection AB ¼ jAj cos HAB of a vector A in the direction of vector B

A ΘAB |A|cosΘAB

B

16

1 Vector Analysis in Annex to Electromagnetics

jAj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2x þ A2y þ A2z ; jBj ¼ B2x þ B2y þ B2z ;

ð1:2:40Þ

The dot product has the commutative property: A  B ¼ B  A:

ð1:2:41Þ

Typically, one of the following symbols is used: A  B or ðA; BÞ. Scalar product is an algebraic operation that takes two vectors and returns a single number. This operation can be defined either algebraically or geometrically. The scalar projection (or scalar component) of a vector A in the direction of a vector B is given by: AB ¼ jAj cos HAB ;

ð1:2:42Þ

where HAB is the angle between A and B. It follows that A  B ¼ 0 if A is perpendicular (orthogonal, normal) to B. The dot product therefore has the geometric interpretation as the length of the projection of ^ when the two vectors are placed so that their tails coincide A onto the unit vector B (Fig. 1.17). If vectors are expressed in terms of unit vectors ^ x; ^ y; ^z along the x-, y-, and z-directions, the scalar product can be also expressed in the following form: A  B ¼ Ax Bx þ Ay By þ Az Bz ¼ B  A;

ð1:2:43Þ

where A ¼ ^xAx þ ^yAy þ ^zAz and B ¼ x^Bx þ y^By þ ^zBz . So when A  B ¼ 0 and Ax Bx þ Ay By þ Az Bz ¼ 0: There are some rules about the dot product of two vectors: (1) The dot product is less than or equal to the product of their magnitudes. (2) The dot product can be either a positive quantity or a negative one, depending on whether the angle between them is smaller (Fig. 1.18a) or larger (Fig. 1.18b) than p=2 radians. Figure 1.19 shows a unit circle with a radius of one.

(a)

(b) A

B ΘAB

ΘAB |A|cosΘAB

B

|B|cosΘAB

A |A|

Fig. 1.18 Dot product of two vectors A and B, a when HAB  p=2 and HAB  p=2

1.2 Basic Vector Operations

17

Fig. 1.19 Cosine function sign in four quarters of unit circle

+y

+

Θ –x

O

+

x,y +x

–y (3) The dot product is equal to the product of magnitude of one vector and the projection of the other vector upon the first one. (4) The dot product is equal to zero; when the vectors are perpendicular to each other, then the angle between them is 90°, and the expression is: ^x  ^y ¼ ^x  ^z ¼ ^y  ^z ¼ 0:

ð1:2:44Þ

(5) The dot product is equal to one when the vectors are parallel to each other: ^x  ^x ¼ ^y  ^y ¼ ^z  ^z ¼ 1

ð1:2:45Þ

The angle between the vectors A and B is equal: HAB

   AB 1 Ax Bx þ Ay By þ Az Bz ¼ cos ¼ cos AB jAj jBj   Ax Bx þ Ay By þ Az Bz ¼ Arc cos : AB 1



ð1:2:46Þ

The distributive property also holds for the dot product: A  ðB þ CÞ ¼ A  B þ A  C:

ð1:2:47Þ

This means that the projection of B þ C onto A is equal to the sum of the projection of B and C onto A. Question to students: The dot product A  B  C is meaningless, WHY?

18

1.2.9

1 Vector Analysis in Annex to Electromagnetics

Cross Product of Two Vectors

The second multiplication of two vectors is called the cross (also known as the vector or the area) product and is defined as: ^: A  B ¼ jAjjBj sin HAB n

ð1:2:48Þ

The cross product is denoted as A  B or ½A; B and illustrated in Fig. 1.20. The multiplication of vectors A and B results is the vector C ¼ A  B with the magnitude equal to the area of the parallelogram ds ¼ jAj jBj sin HAB and with the ^ (Fig. 1.20a). direction as the indicated unit vector n We can write: ^ ds: AB¼n

ð1:2:49Þ

The cross product C whose direction is determined by the right-handed screw rule is shown in Fig. 1.20b. The right-handed screw rule states (Fig. 1.20b) that if ^ will the screw rotates in the direction from vector A to vector B, then unit vector C point to the direction of the movement of the screw. The vector C is: ^x C ¼ A  B ¼ Ax Bx

^y Ay By

^z Az ; Bz

ð1:2:50Þ

    C ¼ Ay Bz  Az By ^x þ ðAz Bx  Ax Bz Þ^y þ Ax By  Ay Bx ^z:

ð1:2:51Þ

In general form for orthogonal system formula (1.2.51), we can be written as follows:

(a)

(b)

C=AxB

^ n

B ^ n O

AxB

AREA

Θ A

O

B ΘAB

A

^, and its module Fig. 1.20 Cross product of two vectors A and B a direction of vector A  B is n is jAjjBj sin HAB , b by the right-handed screw rule

1.2 Basic Vector Operations

19

C¼AB¼

3 X

eijk ^ei Aj Bk ;

ð1:2:52Þ

i¼1 j¼1 k¼1 where basis ^ei is one of unit vectors ^x, ^y, ^z,where three indices i; j; k correspond to the coordinates. We can write a term: eijk

8 0

(b)

Δ

D0 B

B

Fig. 3.23 Demonstration of magnetic field existence by using compass needles which are tangentially directed to the magnetic field lines: a current is absent, b current is present

3.18

Ampere’s Law

161

Fig. 3.24 Amperian loop path around the wire with current I

permeability l0 times electric current I enclosed in the loop “c” (Eq. (3.18.1) and Fig. 3.24). n  X

 Bk j d‘j ¼ l0 I;

ð3:18:1Þ

j¼1

where the projection Bk is in the same plane as the vector d‘ which is the differential length vector with regard to the closed-loop path (Sect. 1.3.1). The integral around any closed Amperian loop path “c” of the tangent component of the magnetic flux density to the direction of the path Bk equals l0 times the electric current intercepted by the area within the Amperian loop. We have broken down the path into segments for calculations. We can decompose the magnetic flux density into components (Fig. 3.24) at the corresponding point of the curve as follows: B ¼ Bk þ B? :

ð3:18:2Þ

The integral form of the original circuital law Eq. (3.18.1) is a line integral of magnetic flux density around some arbitrary closed curve (also known as closed-loop path, Amperian loop, closed contour) “c”: I B  d‘ ¼ l0 I: ð3:18:3Þ c

Equation (3.18.3) in the simplified scalar form is: I Bk d‘ ¼ l0 I; c

we received the expression like in Eq. (3.18.1).

ð3:18:4Þ

162

3 Fields and Materials

In the cylindrical coordinates for a length element d‘ of any closed-loop path “c” at the current flowing in the z-axis direction, we can write this: I 

I

 I l0 I lI lI du ¼ 0 2p ¼ l0 I enc ; rdu ¼ 0 2pr 2p 2p

B  d‘ ¼ c

c

ð3:18:5Þ

c

^ rdu is an infinitesimal element (differential) of the curve “c”, u ^ is the where d‘ ¼ u unit vector, I enc ¼ I total is the total current enclosed by the curve “c” or the current that penetrates surface “S” enclosed by the curve “c” (Fig. 3.24), the index “enc” means “enclosed.” So, we receive the same result in Cartesian and cylindrical coordinate systems. I

Z H  d‘ ¼

c

Jfree  ds ¼ I free ;

ð3:18:6Þ

S

where H is the magnetic field in A/m, Jfree is the free current density only (in A/m2) through the surface S, ds is a differential vector area element of surface with infinitesimally small magnitude and direction normal to surface (see Table 3.4). The integral form of Ampere’s circuital law is a line integral of the magnetic field H around some arbitrary closed curve “c”. The curve “c” restricts a surface “S” through which the electric current passes, and it encloses the current. The statement of the Ampere’s circuital law is a relation between the total amount of magnetic field circulation around some path (line integral) due to the current which passes through this enclosed path (surface integral). The main designations in Table 3.4 are: I enc ¼ I total is the total current, which is H enclosed in an Amperian loop path “c”, I free is the free current only, c is the closed R line integral around the closed curve “c”, S is the surface integral over area S enclosed by the curve “c”, d‘ is a vector with magnitude which equals to the length of an infinitesimal element of the curve “c” (Sect. 1.3.1) and with a direction given by the tangent to the curve “c”, ds is a vector area with the magnitude which is equal to the area of the infinitesimal surface element and with the direction Table 3.4 Ampere’s circuital law written in SI units Form of law Differential form of the circuital law Integral form of the circuital law

Expressed by the magnetic flux density B

Expressed by the magnetic field strange H

r  B ¼ l0 Jtotal (3.18.7) H c

B  d‘ ¼ l0

R S

Jtotal  ds ¼ l0 Ienc (3.18.9)

r  H ¼ Jfree (3.18.8) H c

H  d‘ ¼

R S

Jfree  ds ¼ Ifree (3.18.10)

3.18

Ampere’s Law

163

normal to the surface S. (Sect. 1.3.2). The direction of the normal must correspond with the orientation of curve “c” according to the right-hand rule. Designations in Table 3.4 are: J is the total current density (in A/m2), Jfree is the electric current density of free charges only (free current, drift current, conduction current (Sects. 3.12 and 3.13).

3.19

Magnetic Field Intensity and Magnetic Susceptibility

Magnetostatics is used at the examination of magnetic fields in electric circuits when the electric currents are steady (not varying in time). Electric currents are produced by moving charges. The magnetization current is produced if magnetic dipoles are non-uniformly distributed. We can write on the base of Eq. (2.14.3) the principal in non-time-varying case (in steady state), i.e. when @=@t ¼ 0, in the following form: @q þ r  J ¼ r  J ¼ 0; @t

ð3:19:1Þ

q is the charge density ðC=m3 Þ; and J is the electric current density in ðA=m2 Þ; @q=@t ¼ 0. The charge continuity equation Eq. (3.19.1) expresses the law in terms of charge density and electric current density J. The first term on the left of Eq. (3.19.1) is the change of the charge density q at a point. The next term r  J is the divergence of the current density J at the same point. A static magnetic field can also be produced by a non-uniform magnetization M that produces a magnetization current:   Jm ¼ r  M A=m2 :

ð3:19:2Þ

The magnetization current is a divergence-free vector field: r  Jm ¼ r  ðr  MÞ 0;

ð3:19:3Þ

because the divergence-free vector field (also known as a solenoidal or a transverse vector field) is a vector field ðr  MÞ with divergence which always equals zero at all points in the field (Sect. 1.3.6). The field B exerts a force F perpendicular to the velocity v of electric charges q. The Lorentz force law describes the force that acts on a moving charged particle; see expression (3.17.1).

164

3 Fields and Materials

There are two basic equations in magnetostatics (Table 2.4): r  B ¼ 0;

ð3:19:4Þ

r  H ¼ J;

ð3:19:5Þ

which we are going to use. Since we consider magnetostatics, the current J is the steady electric current density. We investigate now the behavior of this current in a magnetic material. In a magnetic material, after its placing into the external magnetic field, two phenomena arise: an alignment of the atom (molecule) magnetic dipole moments and an induced magnetic moment. In this case, the resultant magnetic flux density will be different in comparison with the magnetic flux density in a vacuum when the material is absent. We can write from Eqs. (3.19.5) and (3.13.17): r

B ¼ J þ Jm ¼ J þ r  M; l0

ð3:19:6Þ

and from (3.19.6), we get: 

   B J¼r  M A=m2 : l0

ð3:19:7Þ

We denote the magnetic field strength H as:  H¼

 B  M ðA=mÞ: l0

ð3:19:8Þ

Combining Eqs. (3.19.7) and (3.19.8), we obtain:   J ¼ r  H A=m2 ;

ð3:19:9Þ

where J ¼ Jfree is the volume density of free charge current (also known as drift current, free current, conduction current; see Sect. 3.13). We would like to note that in general for very complicated medium a total current can be the sum of many different currents such as conduction, magnetization, polarization, by changing of an electric field @E=@t; diffusion (see Sect. 3.13). The magnetization M is directly proportional to the applied magnetic field intensity H: M ¼ vm H;

ð3:19:10Þ

where vm is the magnetic susceptibility. The magnetic susceptibility vm (from Latin: susceptibilis means “receptiveness”) is the proportionality constant that indicates the level of magnetization of a

3.19

Magnetic Field Intensity and Magnetic Susceptibility

165

magnetic material in response to an applied magnetic field. The magnetic susceptibility vm is closely related to the permeability which expresses total magnetization of material. The value vm is a dimensionless value. Most materials can be classified as diamagnetic, paramagnetic, or ferromagnetic ones. Diamagnetic materials have a weak, negative susceptibility to a magnetic field. Paramagnetic materials have a small, positive susceptibility, and ferromagnetic materials have a large, positive susceptibility to an external magnetic field. When a material is placed within an external magnetic field, the magnetic forces of the material’s electrons will be affected. However, materials can react quite differently to the presence of an external magnetic field. This reaction is dependent on a number of factors such as the atomic and molecular structure of the material, and the net magnetic field associated with atoms. Electrons, which are in a pair, rotate in opposite directions. So, when electrons are paired together, their opposite spins cause their magnetic fields cancel each other. Therefore, no net magnetic field exists. Materials with some unpaired electrons will have a net magnetic field and will react more to an external field. Using (3.19.8) and (3.19.10):   B ¼ l0 ð1 þ vm ÞH ¼ l0 lr H ¼ lH Wb=m2 ;

ð3:19:11Þ

where lr ¼ ð1 þ vm Þ ¼ l=l0 ;

ð3:19:12Þ

here lr is the relative permeability of material (dimensionless), l ¼ lr l0 is the absolute permeability, henry per meter ðH=mÞ in the SI system, l0 is the vacuum permeability (permeability of free space, magnetic constant) see in Eq. (2.5.2). The permeability l is the measure of the ability of a material to support the formation of a magnetic field within the material. In other words, it is the degree of magnetization that a material obtains in response to an applied external magnetic field. Value l is the ratio of the magnetic flux density in a material to the external magnetic field strength: B ¼ lH. The relative permeability lr is the ratio l=l0 . For most substances, lr has a constant value. Diamagnetic material has l\1, paramagnetic one has l [ 1;, and ferromagnetic material possesses l  1. The values of relative permeability of materials are given in Table 3.5. Table 3.5 Relative permeability of materials

Materials

Relative permeability lr

Material

Gold Copper Aluminum Titanium Nickel Iron

0.99996 0.99999 1.000021 1.00018 250 4000

Diamagnetic Diamagnetic Paramagnetic Paramagnetic Ferromagnetic Ferromagnetic

166

3 Fields and Materials

From (3.19.11): 1 H ¼ B: l

ð3:19:13Þ

Devices used to measure the magnetic field are called magnetometers, e.g. SQUID magnetometers, which can measure about 5  1018 ðTÞ at present. SQUID is a superconducting quantum interference device, which consists of two superconductors separated by thin insulating layers to form two parallel Josephson junctions. Human brain magnetic field is about 1012 ðTÞ. The Earth’s magnetic field is 105 ðTÞ on the equator. The strength of some refrigerator magnet is 103 ðTÞ. A maximum pulsed magnetic field that was obtained in our days in scientific laboratories has the order of 103 ðTÞ. The magnetic field of some astronomical objects as neutron stars (magnetars) can be in range 108 1011 ðTÞ.

3.20

Boundary Conditions for Steady Electric Current Density

The electric current obliquely flows and crosses the interface between two conductors with different electric conductivities r1 and r2 (Fig. 3.25). Current density vector J1 changes its direction and magnitude after crossing the boundary separating the surface (interface) between two conductive materials. At media boundaries (interface), the field and current vectors can be discontinuous and their behaviors across the boundaries are governed by boundary conditions. We will consider the boundary conditions for the steady current density. When an isolated physical system does not change over time and a charge q cannot be created or destroyed, then rsour ¼ 0 and from the equation of continuity (2.14.3), we can express: rJ¼

@q : @t

ð3:20:1Þ

Using Eq. (3.19.1) for the steady current @q=@t ¼ 0 and r  J ¼ 0. The curl equation is obtained by the combination of the equation r  E ¼ 0 from Table 2.4 and Ohm’s law J ¼ rE from Eq. (3.14.12). We get the curl equation: r  ðJ=rÞ ¼ 0. Eqs. (3.20.2)–(3.20.5) for the steady current in differential and integral forms are placed in Table 3.6. Here “S” is the cross-sectional area of the circuit through which J flows, a curve “c” is the path of the line integral, the curve “c” of a circuit is bordering the surface “S”, r is the electric conductivity of a material, d‘ is an infinitesimal element of the curve “c”, and ds is the vector area of an infinitesimal element of surface “S” (i.e. a vector with magnitude equal to the area of the infinitesimal surface element and to the direction normal to surface “S”).

3.20

Boundary Conditions for Steady Electric Current Density

Jn1 Jt1

Ϭ1

Ϭ2

167

J1 θ1 P1 ^ n1 P2 Jn2

θ2

Interface

J2 Jt2

Fig. 3.25 Boundary conditions at the interface of two conducting media with conductivity r1 and r2 Table 3.6 Equations for steady current density Differential form Integral form

r  J ¼ 0 (3.20.2) H S J  ds ¼ 0 (3.20.4)

r  ðJ=rÞ ¼ 0 (3.20.3) H c ðJ=rÞ  d‘ ¼ 0 (3.20.5)

The boundary condition for normal and tangential components of a steady current density can be expressed in the following way. The normal component of a divergenceless (solenoidal) vector field, see Eq. (3.20.3) in Table 3.6, is continuous: J n1 ¼ J n2 ðA=mÞ2 :

ð3:20:6Þ

The tangential component of a curl-free vector field, see Eq. (3.20.3), is continuous across an interface (Fig. 3.25): J t1 r1 ¼ : J t2 r2

ð3:20:7Þ

Equation (3.20.7) states that the ratio of tangential components of steady current densities J1 and J2 in points P1 and P2 (Fig. 3.25) at two sides of interface is equal to the ratio of the conductivities. We can write from Eqs. (3.20.6) and (3.20.7): J 1 cos h1 ¼ J 2 cos h2 ;

ð3:20:8Þ

r2 J 1 sin h1 ¼ r1 J 2 sin h2 ;

ð3:20:9Þ

where J 1 ¼ jJ1 j and J 2 ¼ jJ2 j are magnitudes (modules) of current density vectors J1 and J2 : J 1 ¼ jJ1 j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 2n1 þ J 2t1 and J 2 ¼ jJ2 j ¼ J 2n2 þ J 2t2 :

ð3:20:10Þ

168

3 Fields and Materials

From Eqs. (3.20.8) and (3.20.9): tan h2 r2 ¼ : tan h1 r1

ð3:20:11Þ

If a steady current flows across the interface between two different lossy dielectrics with constitutive parameters: permittivity e1 and e2 and, respectively, conductivity r1 and r2 , tangential components of the electric field E ¼ En þ Et are satisfied in terms of equality: E t1 ¼ Et2 :

ð3:20:12Þ

The last equality of Et , taking into account Eq. (3.14.12), is equivalent to Eq. (3.20.7). We can write, using Eqs. (3.14.12) and (3.20.6), the following expressions: J n1 ¼ J n2 ) r1 E n1 ¼ r2 En2 :

ð3:20:13Þ

When the surface charge density qs exists at the interface, we have: Dn1  Dn2 ¼ qs ) e1 E n1  e2 En2 ¼ qs :

ð3:20:14Þ

The relations are equal: r2 =r1 ¼ e2 =e1 if qs ¼ 0:

ð3:20:15Þ

We can write surface charge density qs by using (3.20.13) and (3.20.14) at r1 ¼ 6 0 and r2 6¼ 0:   r1 qs ¼ e1  e2 En1 : r2

ð3:20:16Þ

If r2  r1 , then r1 =r2 ! 0 and from Eq. (3.20.16) we get: qs ¼ e1 E n1 ¼ Dn1 :

ð3:20:17Þ

The surface charge density is equal to the normal component of electric flux density. The boundary conditions at the surface of a perfect electric conductor (PEC) are: Et ¼ 0; H t ¼ J s ; Dn ¼ qs ; Bn ¼ 0;

ð3:20:18Þ

where J s is the surface current density, qs is the surface charge density, E t is the tangential component of the electric field vector, H t is the tangential component of the magnetic field, Dn is the normal component of the electric flux density, and Bn is the normal component of the magnetic flux density.

3.21

3.21

Scalar and Vector Potentials

169

Scalar and Vector Potentials

The magnetic scalar potential w and magnetic vector potential A are used to calculate magnetic fields. Magnetic potential A is defined along with the electric potential u by the equations: B ¼ r  A; E ¼ ru

@A ; @t

ð3:21:1Þ ð3:21:2Þ

where B is the magnetic field and E is the electric field. As we have mentioned before, there is no time variation @=@t ¼ 0 in magnetostatics (Table 2.4). We take the divergence from Eq. (3.21.1) and take into account Eq. (1.3.33): r  B ¼ r  ðr  AÞ ¼ 0:

ð3:21:3Þ

Then we take the curl from Eq. (3.21.2):   @A r  E ¼ r  ru ; @t

ð3:21:4Þ

where the divergence of the curl of any vector field is equal to zero (see Sect. 1.3.7). Laplace’s equation for the electrostatic potential u in a charge-free space (Eq. 2.13.5) is r2 u ¼ 0, then we can express: rE¼

@ @B ðr  A Þ ¼  : @t @t

ð3:21:5Þ

The magnetic vector potential depicts a magnetic field that has curl wherever there is a current density JðrÞ. The magnetic vector potential allows calculating the magnetic field intensity directly from a given source current. We can write the curl of magnetic field in the space free of current ðJ ¼ 0Þ from Eq. (3.19.9): r  H ¼ 0;

ð3:21:6Þ

where the curl-less fields are r  H ¼ 0 everywhere, H is a gradient of some scalar potential (Sect. 1.3.7) and thus magnetic field H is derivable there from the gradient of a magnetic scalar potential: H ¼ rw:

ð3:21:7Þ

The magnetic scalar potential w can nicely depict the magnetic field for permanent magnets.

170

3 Fields and Materials

We can write from (3.21.3) and (3.19.11) for a linear media: r  ðlHÞ ¼ 0:

ð3:21:8Þ

If we take the gradient from Eq. (3.21.8), we receive: r2 w ¼ Dw ¼ 0:

ð3:21:9Þ

The magnetic scalar potential w obeys Laplace’s equation.

3.22

Magnetization

Different kinds of magnetic properties in different magnetic materials can be explained as the response of the materials to external magnetic fields. Magnetic materials can be classified into five main groups: diamagnetic, paramagnetic, ferromagnetic, ferrimagnetic, and antiferromagnetic. Some materials possess much more magnetic properties than others. The distinction of magnetic behaviors occurs because in some materials there is no collective interaction of atomic magnetic moments, and in other materials, there is a very strong interaction between atomic moments. We will now consider the phenomenon of magnetism in detail. We have indicated previously (Sect. 3.1) that the elementary atomic model of materials is based on the opinion that materials are composed of atoms, and each of them consists of a positively charged nucleus surrounded by a number of electrons moving counterclockwise in circular orbits around the nucleus. We can take as an example a hydrogen atom in which a nucleus with a single positively charged proton is orbited by one electron (see Fig. 3.1). Any moving electron (or any other electric charge) creates an electric current. The magnetic moments occur due to microscopic electrical currents resulting from the orbital rotation of electrons around an atomic nucleus or the spin (rotation) of the electrons or the nuclei around its own axis. The magnetic dipole moment of a spinning nucleus is usually negligible in comparison with the one of orbiting or spinning electron. The orbiting electron (or electrons) causes circulating current and build microscopic magnetic dipoles. Characteristics of magnetic materials by their magnetic moments are given in Table 3.7. Here the magnetic flux densities Bint and Bappl have indexes “internal” and “applied,” mspin is the magnetic moment of spinning electrons and morb is the magnetic moment of orbiting electrons. : In Table 3.7, the sign “¼” means equal by definition because the magnetic susceptibility vm value which indicates the level of magnetization is very small for antiferromagnetic materials (see Sect. 3.26).

3.22

Magnetization

171

Table 3.7 Magnetic moments of materials Material

Magnetic moments, m

Magnetic flux densities

Diamagnetic Paramagnetic Ferromagnetic

mspin þ morb ¼ 0 mspin þ morb ¼ small mspin  jmorb j mspin [ jmorb j mspin  jmorb j

Bint \Bappl Bint [ Bappl Bint  Bappl

Ferrimagnetic Antiferromagnetic

Bint [ Bappl : Bint ¼ Bappl

A single electron has a magnetic moment (also known as an electron magnetic dipole moment) due to its rotation around the nucleus and the approximate value is 9:3  1024 ðA m2 Þ. For comparison, the Earth has an approximate value of the magnetic moment of about 8  1022 ðA m2 Þ. A magnetic dipole can be created by either a closed loop with an electric current (Fig. 3.26a) or by a pair of poles (Fig. 3.26b). The north (N) is at the top of the magnet and the South (S) is at the bottom in Fig. 3.26b. The magnet is created from a material that produces a magnetic field. We see that the configuration of the magnetic field in both cases is the same. Magnetization (also magnetic polarization) is a vector field which presents the density of permanent or induced magnetic dipole moments in a magnetic matter. A loop of an electric current (an electron moving around a nucleus) has a magnetic moment m. The magnetic moment can be described as a vector with direction perpendicular to the current loop in the right-hand rule direction (Sect. 3.16.1). The meaning of the right-hand rule is: curl (grip) your four fingers into a half-circle around the loop (of wire) with the electric current (3.26a), then your thumb shall point to the direction of the magnetic moment m. An electron in an atom revolves around a nucleus of the atom in a counterclockwise direction.

(a)

(b)

m

B

N

I I

B

B

B

S

Fig. 3.26 Magnetic moment m created by: a a loop with an electric current, b a bar magnet

172

3 Fields and Materials

Quantitative characterization of the magnetization of the individual atom (molecule) of a magnetic material is its magnetic moment:   m ¼ IDS A m2 ;

ð3:22:1Þ

where I ¼ jIj is the module of an atomic (molecular) current, DS is the differential surface vector (Sect. 1.3.2 and Fig. 3.27). Ampere square meter is a SI unit of magnetic dipole moment. If the unit volume of matter contains N closed similar atomic (molecular) currents, the magnetization of material is equal to M ¼ Nm, the magnetization is the magnetic dipole moment per unit volume, i.e. we can write that magnetization M is: M ¼ vm H ðA=mÞ;

ð3:22:2Þ

M is measured in amperes per meter and vm is the magnetic susceptibility which is a dimensionless quantity. When mi is the magnetic dipole moment of an atom, and there are N atoms per unit volume, a magnetization vector is: Pn M ¼ lim

DV!0

i¼1

mi

DV

:

ð3:22:3Þ

The magnetization vector M to the finite volume DV is the average volume density of magnetic dipole moments. Magnetization M is the vector field that can be expressed also by the density of permanent or induced magnetic dipole moments in a magnetic material. We can write at DV ! 0: M¼

dm ðA=mÞ; dV

ð3:22:4Þ

where dm is the elementary magnetic moment and dV is the infinitesimal volume.

Fig. 3.27 Vector of the magnetic moment of an atomic current

m =I∆S

I area

I

∆S

3.22

(a)

Magnetization

173

(b) B

B Fig. 3.28 Magnetization of atomic (molecular) currents: a without and b under influence of the external magnetic field

The magnetic dipole moment m (Eq. 3.22.1) can be considered as a vector with direction perpendicular to the current loop (Fig. 3.27) in the right-hand rule (Sect. 3.16.1) direction. When the applied external magnetic field is absent, the magnetic dipoles of atoms (molecules) in a material (except for a permanent magnet) possess random orientation (Fig. 3.28a). There is no net magnetic moment. After placing a magnetic material in an external magnetic field, two phenomena arise: alignment of magnetic moments of the spinning electrons and an induced magnetic moment due to the change in the orbital motion of electrons. The magnetic moment can be related to the torque s (Fig. 3.10), and it will show itself in an applied magnetic field: s ¼ Bm ðN mÞ;

ð3:22:5Þ

where s is the torque acting on a dipole, newton meter (N m) and B is the magnetic flux density of the external magnetic field, B is measured in tesla (T), the external magnetic field H is measured in amperes per meter (A/m) and m is the magnetic moment, newton meters per tesla ðN m=T ¼ A m2 Þ. It was noted in Sect. 3.13.6 that the magnetization current is a bound current which is flowing into the material volume:   Jm ¼ r  M A=m2 ;

ð3:22:6Þ

where M is the magnetization (magnetic polarization or “magnetisation” in British English). The surface density of magnetization current is: ^; Jsurf;m ¼ M  n

ð3:22:7Þ

Jsurf;m is the surface current which is flowing on the magnetic material surface. We will return to the detailed consideration on different types of material magnetizations in Sect. 3.26.

174

3.23

3 Fields and Materials

Biot–Savart Law

This law was developed by Jean-Baptiste Biot (French physicist, astronomer, and mathematician, 1774–1862) and Félix Savart (French physicist, 1791–1841). A current I which arises due to the motion of electric charges along a wire O1 O2 is a source of a magnetic field (Fig. 3.29). The Biot–Savart law determines the magnitude and direction of the vector of magnetic flux density (magnetic induction) dB in a vacuum at an arbitrary point P1 when the magnetic field is generated by the conducting wire element (small segment) of the length d‘ through which the electric current I flows and is the following: dB ¼

l0 I l I ½d‘r ¼ 0 2 ½d‘^r ; 4pr 3 4pr

ð3:23:1Þ

where the vector d‘ is the conducting wire element, r ¼ ^rr is the radius vector from the current source Id‘ to the point P1 where we want to know the magnetic flux density dB, and l0 is a vacuum permeability (Eq. 2.5.2). The vector dB perpendicular to the plane in which vectors of d‘ and r ¼ ^rr are located. The vector dB is directed in such a way that the end of the shortest rotation vector d‘ until vector r would be counterclockwise. The determination of the magnetic flux density B at point P0 (Fig. 3.30) occurs due to the entire length of current-carrying conductor wire x1 x2 and is carried out by integrating: Z B¼

Z dB ¼

wire

length x1 x2

l0 I ½d‘^r

; 4p r 2

ð3:23:2Þ

wire

length x1 x2

where the origin of Cartesian coordinate system is in point O. The integral (3.23.2) represents a vector because there is the cross product ½d‘^r ¼ d‘  ^r in the kernel of the integral. The magnetic flux density is measured volt second ¼ weber : in Tesla ¼ meter meter meter2 Fig. 3.29 Magnetic flux density dB at point P1 due to the current-carrying element Id‘

3.24

Magnetic Flux Density Around a Straight Wire

175

Fig. 3.30 Magnetic field which is created by a current wire x1 x2 at point P0

3.24

Magnetic Flux Density Around a Straight Wire

The magnetic flux density in the point P0 when the magnetic field source is a thin straight wire with the module of a current I can be calculated. In our case, the wire is located along the x-axis and has the length x1 x2 (Fig. 3.30). We consider a differential element d‘ ¼ ^xdx0 carrying current with the magnitude I. The point P0 is located at ðx; yÞ ¼ ð0; aÞ, the radius vector (Sect. 1.2.6) from the origin O to the point P0 along the line OP0 is r p ¼ ^ya; the radius vector between the current source pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Id‘ and point P0 is r ¼ rp  r0 ¼ ^ya  ^xx0 and magnitude r ¼ jrj ¼ a2 þ x20 is a distance between the source (the wire with a current density J) and point P0 . The unit vector r y^a  x^x0 ^r ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ^y sin u  ^ x cos u; r a2 þ x20

ð3:24:1Þ

where the angle u ¼ 180  u0 (Fig. 3.30), then cos u0 ¼ cosð180  uÞ ¼  cos u and cos u0 ¼ cosð180  uÞ ¼  cos u. The cross product is equal: ½d‘^r ¼ d‘  ^r ¼ ð^xdx0 Þ  ð^y sin u  ^x cos uÞ ¼ ^zdx0 sin u:

ð3:24:2Þ

From formula (3.23.1): dB ¼

l0 I l I dx0 sin u ½d‘^r ¼ ^z 0 : 2 4pr 4p r2

ð3:24:3Þ

We see that the vector dB at point P0 has the direction of the unit vector ^z (Fig. 1.14, the coordinate system is right-handed). Since u0 and u are two supplementary angles that have a sum of 180° see Fig. 3.30, we can write cos u0 ¼ cosð180   uÞ ¼  cos u, sin u0 ¼ sinð180   uÞ ¼ sin u, cot u0 ¼ adjacent=opposite ¼ x0 =a, cot u ¼ cos u= sin u ¼ x0 =a, ðcot uÞ0 ¼ csc2 udu, x0 ¼ a cot u, cosecu ¼ cscu ¼ 1= sin u,

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3 Fields and Materials

  dx0 ¼ ðcot uÞ0 du ¼  1= sin2 u ¼ acsc2 udu, cos u ¼ x0 =r, sin u ¼ a=r. From Eq. (3.24.3): dB ¼ ^z

r ¼ a= sin u ¼ acscu,

l0 I sin u dx0 l I sin u acsc2 udu l I  ¼ ^z 0 ¼ ^z 0 sin udu: 2 2 4p 4p 4pa r ðacscuÞ

ð3:24:4Þ

From Eq. (3.24.4), the module jdBj is: jdBj ¼ dB ¼ 

l0 I sin udu: 4pa

ð3:24:5Þ

Using Eq. (3.23.2), we can write: Zx2 Zx2 ½d‘^r l0 I^z adx0 l0 I^z a dx0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ ¼   3 r2 4p 4p a þ x20 a2 þ x20 a2 þ x20 x1 x1 x1 0 1 ð3:24:6Þ l I^z B x2 x1 C ¼ 0 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 4pa a2 þ x 2 a2 þ x 2

lI B¼ 0 4p

Zx2

2

1

where x1 ¼ x1 ; x2 ¼ x2 . Integrating over all angles from u1 till u2 , we receive the following: ^zl I B¼ 0 4pa

Zu2 sin udu ¼ 

^zl0 I ðcos u1 þ cos u2 Þ: 4pa

ð3:24:7Þ

u1

Infinitely long straight wire For the infinite long wire, when u1 ¼ 0 and u2  p the magnetic flux density is in the point P0 (Fig. 3.30), we can write as follows: B¼

l0 I ; 2pa

ð3:24:8Þ

where the distance a is from the point P0 till the wire (Fig. 3.30). We receive the widely known formula for the determination of value B (T) for an infinite long wire with a current I (A) at the radial distance a (m), the vacuum permittivity is given in Eq. (2.5.2).

3.24

Magnetic Flux Density Around a Straight Wire

177

B

Fig. 3.31 Direction of magnetic flux density in an infinitely long wire

I

+

A wire possesses cylindrical symmetry, and magnetic flux density lines (or magnetic field lines) are closed circles containing the wire with the current I in their centers. The direction of magnetic flux density B and magnetic field H of the long wire with the electric current I can be found out by the right-hand-rule (Sect. 3.16.1). If the thumb of your right hand points to the direction of the current I and your four fingers curl (grip) into a half-circle around the wire with the current I, then your four fingers will show the direction of B and H (Fig. 3.31). Notes about magnetic flux density of straight wire with a current depending on a location in relation to point P0 : 1. When there is an infinite wire with current x1 ! 1 and x2 ! 1, then l0 I ; magnetic flux density equals: B ¼ 2pa 2. When there is the half-infinite wire with current x1 ¼ 0 and x2 ! 1, then l0 I ; magnetic flux density equals: B ¼ 4pa 3. When the ends of a wire with a current have some large positive values x1 ! þ 1; x2 ! þ 1 at jx2  x1 j ¼ finite value, then magnetic flux density goes to zero: B ! 0.

3.25

Magnetic Flux Density Due to Circular Current Loops

We consider now another case that involves the application of the Biot–Savart law. Figure 3.32 gives a circular loop with the radius R which carries a steady current I. The loop is located in the plane xy. This EM problem is important to understand in terms of how it is possible to produce the magnetic field by a current in a loop and in this way excite waveguides and resonators which we are going to consider in Chaps. 7 and 8. We would like to know the magnetic field at a point P0 which is located along the z-axis of the Cartesian coordinate system at the distance a. For this reason, a ¼ ^zz. The differential current element Id‘ is characterized by the radius vector: r0 ¼ Rð^x cos u0 þ ^y sin u0 Þ;

ð3:25:1Þ

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3 Fields and Materials

Fig. 3.32 Circular loop with the radius R ¼ jr 0 j and the steady current I

dr0 ¼ Rð^x sin u0 þ ^y cos u0 Þ; du0

ð3:25:2Þ

then we can depict the differential current element as: 

 dr0 Id‘ ¼ I y cos u0 Þ: du0 ¼ IRdu0 ð^x sin u0 þ ^ du0

ð3:25:3Þ

The displacement vector r (Sect. 1.2.6) between the current element Id‘ and the point P0 is r ¼ ^za  r0 , where radius vectors (position vectors) jr0 j ¼ R is the distance from the origin O. We can express the displacement as: r ¼ ^za  r0 ¼ ^za  Rð^x cos u0 þ ^ y sin u0 Þ:

ð3:25:4Þ

The magnitude of the displacement r can be determined by Eq. (1.2.25): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jrj ¼ r ¼ ðR cos u0 Þ2 þ ðR sin u0 Þ2 þ a2 ¼ R2 þ a2 ;

ð3:25:5Þ

where the distance a is just the z-coordinate of the point P0 . The unit vector of ^r by Eq. (1.2.26) is: r ^za  r0 ^r ¼ ¼ : r j^za  rj

ð3:25:6Þ

Using formula (3.23.1) from the Biot–Savart law, we express the vector dB that occurs of the current element Id‘ at point P0 :

3.25

Magnetic Flux Density Due to Circular Current Loops

dB ¼

l0 I l0 I ½d‘ð^za  r0 Þ

½ d‘r

¼ : 4pr 3 4p jð^za  r0 Þj3

179

ð3:25:7Þ

The denominator of a fraction in Eq. (3.25.7) is given in Eq. (3.25.4), while as for the numerator of the fraction, we take into account Eqs. (3.25.3) and (3.25.4) so we can write the following form: ½d‘ð^za  r0 Þ ¼ ðRdu0 ð^x sin u0 þ ^y cos u0 ÞÞ  ð^za  Rð^ x cos u0 þ ^ y sin u0 ÞÞ: ð3:25:8Þ Calculating the vector product by Eq. 1.2.51, we obtain: ½d‘ð^za  r0 Þ ¼ Rdu0 ð^xa cos u0 þ ^ ya sin u0 þ ^zRÞ;

ð3:25:9Þ

where designations are given in Fig. 3.30. Now we go back to Eq. (3.25.7) which we rewrite in the form: dB ¼

l0 RI ð^xa cos u0 þ ^ya sin u0 þ ^zRÞ 0 du :  2 3=2 4p R þ a2

ð3:25:10Þ

Performing the integration of Eq. (3.25.10), we obtain that the magnetic flux density in the point P0 is: l RI B¼ 0 4p

Z2p 0

ð^xa cos u0 þ ^ya sin u0 þ ^zRÞ 0 du :  2 3=2 R þ a2

ð3:25:11Þ

The projections of vector B ¼ ^xBx þ ^yBy þ ^zBz are: Bx ¼

By ¼

 3=2 4p R2 þ a2 l0 RI

 3=2 4p R2 þ a2

Bz ¼

Z2p

l0 RaI

l0 R2 I

cos u0 du0 ¼

0

Z2p

sin u0 du0 ¼ 

0

 3=2 4p R2 þ a2

Z2p 0

l0 RaI 0 2p  2 3=2 sin u j0 ¼ 0; 4p R þ a2

du0 ¼

l0 RI 0 2p  2 3=2 cos u j0 ¼ 0; 2 4p R þ a

2pl0 R2 I l0 R2 I  2 3=2 ¼  2 3=2 : 4p R þ a2 2 R þ a2

ð3:25:12Þ

ð3:25:13Þ

ð3:25:14Þ

We see that only projection Bz is not equal to zero and the magnetic flux density B ¼ ^zBz in the point P0 is directed along ^z and B ¼ Bz . The magnetic flux density

180

3 Fields and Materials

B in the origin O at a = 0 and the radius vector jr0 j ¼ R, i.e. in the plane xy the magnetic flux density is equal to: Bo ¼

l0 I ; 2R

ð3:25:15Þ

where R ¼ jr0 j is the radius of circular loop (Fig. 3.32). Please compare Eqs. (3.25.15) and (3.24.8).

3.26

Main Types of Magnetic Materials

When a material is placed within an external magnetic field, magnetic forces of the material’s electrons will be affected. This effect is known as Faraday’s law of magnetic induction (see further Sect. 4.2). Faraday’s law of EM induction predicts how a magnetic field will interact with an electric wire loop to produce an electromotive force (voltage). However, materials can react quite differently to the presence of an applied (external) magnetic field. This reaction is dependent on a number of factors, such as atomic and molecular structure of material and net magnetic moment associated with atoms. In most atoms, electrons occur in pairs. Electrons in a pair rotate in opposite directions because this is required by Pauli’s exclusion principle. This quantum mechanical principle tells that “no more than one electron can exist in any one quantum state.” So, when electrons are paired together, their opposite spins cause their magnetic fields cancel each other. Therefore, no net magnetic field exists. Alternately, materials with some unpaired electrons will have a net magnetic field and will react more to an external magnetic field. The magnetism phenomena are based on the orbital and spin motions of electrons and how the electrons interact with one another. The different kinds of magnetism describe how materials respond to the applied magnetic fields. All matter is magnetic but some of materials are much more magnetic than others; see Table 3.8. There are five main groups of materials by their magnetic behavior: diamagnetism, paramagnetism, ferromagnetism, ferrimagnetism, antiferromagnetism. Ferromagnetic has large positive susceptibility vm and permanent magnetization even without applied external magnetic field. Earlier in this book we described the microscopic magnetic property of linear isotropic materials by the magnetic susceptibility vm which is a coefficient of proportionality between M and H (Eq. 3.19.10). We will consider the basic characteristics of magnetic materials.

3.26

Main Types of Magnetic Materials

181

Table 3.8 Magnetic susceptibility, relative, and absolute permeabilities Type of material

Magnetic susceptibility vm value

lr ¼ 1 þ vm and l ¼ lr l0

Magnetic moment of materials possessing different type of magnetism

Diamagnetic

Very small negative number:   vm ¼ 105 till  ð109 Þ; e.g. for Copper (Cu) is vm ¼ 0:77  106 Very small positive number: vm ¼ 105 till  103 , e.g. for Platinum (Pt) is vm ¼ 6:6  105 Very large positive number, function of applied external field vm  1, e.g. for Iron (Fe) is vm  100;000 Relatively large positive number, function of applied external field, e.g. for Barium Ferrite (BaFe) is vm 3 Very small positive number, e.g. for Chromium (Cr) is vm ¼ 3:6  106

lr \1 and l\l0

Atoms have no magnetic moments

lr [ 1 and l [ l0

Atoms possess randomly oriented magnetic moments

lr  1 and l  l0

Atoms possess parallel aligned magnetic moments

lr [ 1 and l [ l0

Atoms possess parallel aligned magnetic moments

lr [ 1 and l [ l0

Atoms possess both parallel and antiparallel aligned magnetic moments

Paramagnetic

Ferromagnetic

Ferrimagnetic

Antiferromagnetic

3.26.1 Diamagnetism In diamagnetic materials, the net magnetic moments (due to orbital and spinning motions of electrons in an atom) are equal to zero in the absence of an applied (external) magnetic field. The applied magnetic field changes the orbit and orbital velocity of electrons around their nuclei, and in this way, the magnetic dipole moment of atoms is changing. The magnetic dipole moment alters in the opposite direction to the applied magnetic field. Diamagnetism is the property of a material to create an internal magnetic field inside the material in the opposite direction in comparison with an applied (external) magnetic field. The induced magnetic moment disappears when the applied magnetic field is absent. The value of susceptibility is independent from temperature. All materials possess the diamagnetic effect but often this effect is masked by greater paramagnetic or ferromagnetic effects. Figure 3.33 shows the relation between the magnetization M and the applied magnetic field H (Eq. 3.19.10) in a diamagnetic material. On the left side of picture in Fig. 3.33, we see a diamagnetic material lattice of ordered arrangement of atoms (molecules or ions) that do not have magnetic moments when the applied magnetic field H is absent because diamagnetic creates an induced magnetic field in a direction opposite to the applied external magnetic field and therefore their atoms repel the magnetic field.

182

3 Fields and Materials

Fig. 3.33 Magnetic susceptibility vm of a diamagnetic material, on the left side we see its atoms which have no magnetic moment at H ¼ 0

Most elements in the periodic table of chemical elements are diamagnetic, e.g. copper, silver, and gold.

3.26.2 Paramagnetism Paramagnetism arises mainly by the magnetic dipole moments of the spinning electrons. In some materials, magnetic moments due to the orbiting and spinning electron movements are not canceled completely in the absence of the external magnetic field. Atoms of a paramagnetic have randomly oriented magnetic moments at H = 0. The molecules (atoms, ions) have a net average magnetic moment. An applied external magnetic field aligns the molecular magnetic moments in the same direction as the applied magnetic field (Fig. 3.28). The alignment process of the magnetic moments is impeded by the force of random thermal vibrations. Diamagnetic atoms have only paired electrons, whereas paramagnetic atoms, which can be made magnetic, have at least one unpaired electron. The unpaired electrons of atoms of paramagnetic realign after applying of external magnetic fields. Paramagnetic atoms attract magnetic fields. Paramagnets do not keep magnetization in the absence of an external magnetic field because thermal energy arranges electron spin orientations in random order. The paramagnetic effect is temperature dependent. At normal temperatures and when a paramagnetic material is in moderate by applied external magnetic fields, the susceptibility is small (Table 3.8). Figure 3.34 shows the relation between the magnetization M and the magnetic field H in a paramagnetic material. On the left side of picture in Fig. 3.34, we see a

Fig. 3.34 Magnetic susceptibility vm of a paramagnetic material, on the left side are shown its atoms which have randomly oriented magnetic moments at H ¼ 0

3.26

Main Types of Magnetic Materials

183

paramagnetic material lattice of ordered arrangement of atoms (molecules or ions) with their magnetic moments when the external applied magnetic field is absent. Paramagnetic materials include magnesium, molybdenum, lithium, and tantalum.

3.26.3 Ferromagnetism Ferromagnetism describes materials which form permanent magnets. Ferromagnetism is only possible when material atoms are arranged in a lattice and the atomic magnetic moments can interact to align parallel to each other. The magnetization of ferromagnetic materials can be by many orders larger than the paramagnetic ones (Table 3.8). The atomic moments in ferromagnetics show very strong interactions. Ferromagnetic material shows the parallel alignment of magnetic moments which gives large magnetization of the material even in the absence of a magnetic field. The important characteristic of ferromagnetic materials is their spontaneous magnetization. The spontaneous magnetization is the net magnetization which exists inside a uniformly magnetized microscopic volume in the absence of an applied magnetic field. Ferromagnetism can also be described in terms of magnetized domains. A magnetic domain is a small area within a magnetic material in which the magnetization is in the same direction. Sizes of domains range from a few microns to about 1 mm. Domains usually contain about 1016 atoms which are all fully magnetized in the same direction and the magnetic moments of the atoms are parallel in each domain. After the application of an external magnetic field, the magnetization vector of each domain tries to align in the direction of the external magnetic field (Fig. 3.35). The magnetization vectors of all domains line up in the direction of an external field when one corresponds to the saturation magnetization. The saturation magnetization is the magnetic moment induced to the maximum that can be obtained in saturation magnetic field for a certain ferromagnetic. Ferromagnetic materials are iron, cobalt, nickel, gadolinium, terbium, and many of their alloys (Fig. 3.35). Fig. 3.35 Magnetic susceptibility vm of ferromagnetic materials, on the left side we see its atoms with the parallel aligned magnetic moments at H ¼ 0

184

3 Fields and Materials

3.26.4 Ferrimagnetism Ferrimagnetism is only observed in ionic compounds that have more intricate crystal structures than pure chemical elements. Within ferrimagnetic materials, the exchange interactions lead to parallel alignment of atoms in one region and antiparallel alignment in other regions of a crystal (Fig. 3.36). The material is subdivided into magnetic domains. The magnetic structure of ferrimagnetics is composed of two magnetic sublattices (called A and B) separated by oxygens. The exchange interactions happen through the oxygen anions. When this occurs the interactions are called superexchange interactions. The strongest superexchange interactions are in an antiparallel alignment of spins between the A and B sublattice. The magnetic moments of A and B sublattices in ferrimagnets are not equal and result in a net magnetic moment. Ferrimagnetism is consequently similar to ferromagnetism. In ferrimagnetic materials, there is also spontaneous magnetization as in ferromagnetics. Ferrimagnetic materials usually have lower saturation magnetization in comparison with ferromagnetics. Examples of ferrimagnetic materials: YIG (yttrium iron garnet, i.e. chemical composition Y3 Fe5 O12 ), cubic ferrites made of iron oxides and other elements such as aluminum, nickel, hexagonal ferrites such as PbFe12 O19 and BaFe12 O19 . The term “garnet” means a group of silicate minerals.

3.26.5 Antiferromagnetism When the A and B sublattice magnetic moments are exactly equal, but with opposite signs the net moment is zero. This type of magnetic ordering is named antiferromagnetism. The magnetic moments of atoms (molecules) are mostly associated with the spins of electrons (Fig. 3.28) which align in a regular structure with neighboring spins pointing to opposite directions. Antiferromagnetic materials properties are very resemblant to ferromagnetic material properties, but the exchange interaction between neighboring atoms of antiferromagnetic leads to the antiparallel alignment of the atomic magnetic moments (Fig. 3.37). For this reason, the internal magnetic field cancels out the antiferromagnetic and the materials behave in the same way as paramagnetic materials.

Fig. 3.36 Magnetic susceptibility vm of a ferrimagnetic material, on the left side we see its atoms with antiparallel aligned magnetic moments

3.26

Main Types of Magnetic Materials

185

Fig. 3.37 Magnetic susceptibility vm of an antiferromagnetic material, on the left side we see its atoms with both parallel and antiparallel aligned magnetic moments

In the periodic table, the only chemical element showing antiferromagnetism at room temperature is chromium (Cr). Examples of antiferromagnetic can be the following: hematite (mineral Fe2 O3 ), alloys such as iron manganese (FeMn) and nickel oxide (NiO). The antiferromagnetic material corresponds to a vanishing total magnetization when an external applied magnetic field is absent.

3.27

Constitutive Relations for Media

The constitutive relation is the one between physical quantities that is specific to a considered substance and depicts the response of that material to the applied electric E and/or magnetic H-fields. In linear, homogeneous, isotropic media the relative permittivity er (Sect. 3.6) is a constant. We will call the mentioned media the simple one in the next text. However, in linear anisotropic media the relative permittivity is a tensor, and in non-homogeneous media it is a function of position inside the medium. The parameters of materials such as the absolute permittivity e ¼ e0 er (Sect. 3.6) and permeability l ¼ l0 lr (Sect. 3.19) are functions of space and time variables because we have been operating with the space–time representation of Maxwell’s equations. For example, the material parameter function e is a response function which relates the electric flux density Dðr; tÞ with the electric field Eðr0 ; t0 Þ existing at all moments of previously time: Zþ 1 Dðr; tÞ ¼

dr 1

0

Zt

eðrr0 ; t  t0 ÞEðr0 ; t0 Þdt0 :

ð3:27:1Þ

1

In case, the response function is local and it has the form eðr  r0 ; t  t0 Þ ¼ eðt  t0 Þdðr  r0 Þ one obtains: Zt Dðr; tÞ ¼ 1

eðt  t0 ÞEðr; t0 Þdt0 :

ð3:27:1aÞ

186

3 Fields and Materials

Here the integral Fourier transform yields the constitutive relation in common form: Dðr; xÞ ¼ eðxÞEðr; xÞ:

ð3:27:1bÞ

Analogical expression can be written for magnetic flux density Bðr; tÞ and current density Jðr; tÞ: Zþ 1 Bðr; tÞ ¼

dr

0

1

dr 1

lðr  r0 ; t  t0 ÞHðr0 ; t0 Þdt0 ;

ð3:27:2Þ

rðr  r0 ; t  t0 ÞEðr0 ; t0 Þdt0 :

ð3:27:3Þ

1

Zþ 1 Jðr; tÞ ¼

Zt

0

Zt 1

In the case of local response functions lðr  r0 ; t  t0 Þ ¼ lðt  t0 Þ dðr  r0 Þ and rðr  r0 ; t  t0 Þ ¼ rðt  t0 Þ dðr  r0 Þ after applying Fourier transform to relations (3.27.2) and (3.27.3), we obtain other constitutive relations: Bðr; xÞ ¼ lðxÞHðr; xÞ;

ð3:27:3aÞ

Jðr; xÞ ¼ rðxÞEðr; xÞ:

ð3:27:3bÞ

Formulas (3.27.1b), (3.27.3a), and (3.27.3b) are valid for harmonic fields. Expressions (3.27.1b), (3.27.3a), and (3.27.3b) describe the frequency dispersion medium. The spatial dispersion is a phenomenon where medium parameters such as permittivity, permeability, and conductivity are dependent on the wave vector of propagating EM wave (see further Sect. 5.4). For example, this depends of the permittivity at spatial dispersion caused by the non-locality (non-local interactions) of the relationship between the electrical induction D and the electric field E. The non-local interactions between D and E lead to a number of phenomena such as rotation of the plane of polarization. Spatial dispersion represents spreading effects and is important only at microscopic length scales. Spatial dispersion and time (frequency) dispersion may occur in the same medium. Spatial dispersion usually occurs if EM wavelength becomes comparable with the characteristic internal scales of medium, characterizing the degree of influence of EM waves on its elements such as electrons, atoms, and molecules. Such scales can be of the length of the mean free path of particles, radius of rotation of the charged particle in the external magnetic field. In all the mentioned cases, in order to determine the law of dispersion it is necessary to know the structure of medium and peculiarities behavior of individual atoms or molecules in an external alternating electric field. The mean free path is the average distance travelled by a moving particle between successive collisions.

3.27

Constitutive Relations for Media

187

The physical meaning of time (frequency) dispersion is as follows. Let us assume that the elements of medium such as electrons on a shell of atoms under influence of an electric field make oscillations which phase lags behind from a phase of oscillations of an external EM wave. Then the second EM waves emitted by these particles will experience additional delay and come to the observation point later than the original EM wave (see further Sect. 5.14).

3.27.1 Isotropic Media Isotropy is uniformity in all orientations; the term “Isotropy” is derived from Greek “isos=equal” and “tropos=way”. Constitutive relations for linear, homogeneous isotropic media are as follows: D ¼ eE;

ð3:27:4Þ

B ¼ lH;

ð3:27:5Þ

J ¼ rE:

ð3:27:6Þ

The absolute permittivity e ¼ e0 er ; absolute permeability l ¼ l0 lr (see Eq. 2.5.2) and conductivity r may also have complex values. The electric flux density is a function of electric field strength D ¼ DðEÞ and analogically B ¼ BðHÞ and J ¼ JðEÞ. For isotropic medium, the electric field vector E is parallel to the electric flux density vector D (see Fig. 3.38a), and the field vector H is parallel to the vector B. Constitutive relations in a vacuum:

(a)

Dvac ¼ e0 E;

ð3:27:7Þ

Bvac ¼ l0 H;

ð3:27:8Þ

(b)

Fig. 3.38 Directions of vectors D, E, P in: a isotropic material, b anisotropic material

188

3 Fields and Materials

in the matter: D ¼ e0 E þ P ¼ e0 ð1 þ ve ÞE ¼ e0 er E ¼ eE; B ¼ l0 ðH þ MÞ ¼ l0 ð1 þ vm ÞH ¼ l0 lr H ¼ lH;

ð3:27:9Þ ð3:27:10Þ

where P ¼ e0 ve E (Eq. 3.6.6) and M ¼ vm H (Eq. 3.19.10).

3.27.2 Anisotropic Media Constitutive relations of an anisotropic medium can be written as: D ¼ e0^eE;

ð3:27:11Þ

^H; B ¼ l0 l

ð3:27:12Þ

^ is the relative permeability tensors. where ^e is the relative permittivity, and l The vector D is no longer parallel to the vector E (see Fig. 3.38b). A medium is electrically anisotropic if it is depicted by the permittivity tensor ^e as follows: 2

exx ^e ¼ 4 eyx ezx

exy eyy ezy

3 exz eyz 5: ezz

ð3:27:13Þ

For lossless (dissipationless) material, all elements of tensor ^e are real, so tensor is symmetric (Hermitian, when eij ¼ e ji ) and can be written: 2

exx ^e ¼ 4 exy exz

exy eyy eyz

3 exz eyz 5; ezz

ð3:27:14Þ

where a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose, i.e. the element in the “ith” row and “jth” column is equal to the complex conjugate of the element in the “jth” row and “ith” column. The complex conjugate of “a + ib”, where “a” and “b” are reals, is “a − ib”. For susceptibility tensor ^v relates the polarization of medium by P ¼ e0 ^ vE: 2

3 2 vxx Px 4 Py 5 ¼ e0 4 vyx vzx Pz

vxy vyy vzy

32 3 vxz Ex vyz 54 Ey 5: vzz Ez

ð3:27:15Þ

The expression of the permittivity is a non-diagonal (Hermitian) tensor for a longitudinally magnetized semiconductor plasma when the interaction of the

3.27

Constitutive Relations for Media

189

carriers with an alternating electric field at microwave frequencies is taken into account. In this case, the semiconductor sample is located into the constant longitudinal magnetic field: D ¼ e0^eE;

ð3:27:16Þ

and the electric flux density D can be written by projections in the following way: 2

3 2 exx Dx 4 Dy 5 ¼ e0 4 iexy 0 Dz

iexy exx 0

32 3 Ex 0 0 54 E y 5; ezz Ez

ð3:27:17Þ

media depicted by such permittivity tensor ^e as in Eq. (3.27.17) are electrically gyrotropic (gyroelectric). In magnetically anisotropic media, the vector B is no longer parallel to the vector H. A medium is magnetically anisotropic if it is depicted by the permittivity tensor ^: l 2

lxx ^ ¼ 4 lyx l lzx

lxy lyy lzy

3 lxz lyz 5: lzz

ð3:27:18Þ

The expression of the permeability is a non-diagonal (Hermitian) tensor for a longitudinally magnetized ferrite: ^H; B ¼ l0 l

ð3:27:19Þ

the magnetic flux density B from Eqs. (3.27.18) and (3.27.19) can be written by projections in the following way: 2

3 2 lxx Bx 4 By 5 ¼ l0 4 ilxy 0 Bz

ilxy lxx 0

32 3 0 Hx 0 54 H y 5; lzz Hz

ð3:27:20Þ

^ as in Eq. (3.27.20) are magnetically media depicted by such permeability tensor l gyrotropic (gyromagnetic). So a medium is electrically anisotropic if it is described by a permittivity tensor ^e and a scalar permeability l. A medium is magnetically anisotropic if it is described ^ and a scalar permittivity e. by a permeability tensor l A medium may be both electrically and magnetically anisotropic substance or possess one of them.

190

3 Fields and Materials

3.27.3 Permittivity Tensor of Crystals Crystals are solid materials whose atoms, molecules or ions are arranged in highly ordered crystal lattice that extends in all directions. Crystals usually possess the symmetric permittivity tensor. After using the coordinate transformation, we transform a symmetric matrix into a diagonal matrix: 2

exx ^e ¼ e0 4 0 0

0 eyy 0

3 0 0 5: ezz

ð3:27:21Þ

The crystal is biaxial when exx 6¼ eyy 6¼ ezz . Three coordinate axes are reflected to the principal axes of the crystal. Some planes and direction in crystals are defined by symmetry of the crystal lattice. In crystals with certain type of lattice, there is one unique axis (principal axis) which possesses higher rotational symmetry. For cubic, triclinic, and orthorhombic crystals, the axis designation is arbitrary and there is no principal axis. All three crystallographic axes are unequal in orthorhombic, monoclinic, and triclinic crystals. For cubic crystals exx ¼ eyy ¼ ezz tensor components are equal and the crystals are isotropic. In hexagonal, tetragonal, and rhombohedral crystals, two of three terms are equal. Such crystals are uniaxial. The permittivity tensor for a uniaxial crystal is: 2

exx ^e ¼ e0 4 0 0

0 exx 0

3 0 0 5; ezz

ð3:27:22Þ

where z-axis is the optical axis. The crystal is positive uniaxial at ezz [ exx and negative uniaxial at ezz \exx .

3.27.4 Biisotropic Media We see from Eqs. (3.27.4) and (3.27.5) that for isotropic materials, the electric field E and electric flux density D as well as the magnetic field H and magnetic flux density B are parallel to one another. For materials which are more complicated in structure, such as some metamaterials, the electric fields E and D as well as the magnetic fields H and B are not necessarily parallel to each another.

3.27

Constitutive Relations for Media

191

In biisotropic media, the electric and magnetic fields are coupled. The constitutive relations for biisotropic media are: D ¼ eE þ nH;

ð3:27:23Þ

B ¼ lH þ fE;

ð3:27:24Þ

where n and f are the magnetoelectric coupling constants (coefficients) which are the intrinsic constant of each medium. Intrinsic properties are dependent mainly on the chemical composition or structure of the material.

3.27.5 Bianisotropic Media Note that a medium can be both electrically and magnetically anisotropic. The constitutive relation for a bianisotropic medium can be written as follows: ^ D ¼ ^eE þ wH;

ð3:27:25Þ

^H; B ¼ ^mE þ l

ð3:27:26Þ

^ and ^m are tensors ^ is the permeability tensor, w where ^e is the permittivity tensor, l which are present in the material bianisotropy parameters. When the bianisotropic material is placed in an electric and/or magnetic field the material becomes both polarized and magnetized.

3.27.6 Notes About Media We would also like to note that depending on the constitutive parameters such as the relative permittivity er , relative permeability lr , the electrical conductivity r, which are associated with the microscopic response of atoms and molecules of a medium one can be subdivided into: (a) Dispersive material when constitutive parameters are functions of frequency: e ¼ eðxÞ, l ¼ lðxÞ, r ¼ rðxÞ (see Fig. 3.14). (b) Non-homogeneous or homogeneous according to whether or not constitutive parameter of interest is a function of the position (coordinate rÞ: e ¼ eðrÞ, l ¼ lðrÞ, r ¼ rðrÞ. (c) Nonlinear or linear, according to whether or not constitutive parameters depend on the magnitude of applied electric E or magnetic H fields, i.e. e ¼ eðEÞ, l ¼ lðHÞ, r ¼ rðEÞ.

192

3 Fields and Materials

At the higher intensity of an electric field E, the relation between polarization and electric field can be more specifically represented by a power Taylor series expansion of the electric polarization (dipole moment per unit volume) PðtÞ at time t (in certain point with fixed coordinate r) in terms of the electric field EðtÞ:   PðtÞ ¼ e0 ve;1 EðtÞ þ ve;2 E2 ðtÞ þ ve;3 E3 ðtÞ þ ve;4 E4 ðtÞ þ ve;5 E5 ðtÞ þ    ; ð3:27:27Þ where e0 is the vacuum permittivity (Eq. 2.5.2), t is the fixed time, ve;1 is the electric susceptibility of a given dielectric material (Sect. 3.6), the coefficients ve;n are the nth order electric susceptibilities of the medium and the presence of such term is generally referred to as an nth order nonlinearity where ve;n with n > 1 are nonlinear susceptibilities: second-order, third-order, and so on. As we remember, electric susceptibility is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to the applied electric field strength. It is known that the greater the electric susceptibility, the greater the ability of a material to polarize in response to the electric field, and thereby to reduce the total electric field inside the material. Using Eqs. (3.6.8) and (3.27.9), we can write:   DðtÞ ¼ e0 EðtÞ 1 þ ve;1 þ ve;2 EðtÞ þ ve;3 E2 ðtÞ þ ve;4 E3 ðtÞ þ ve;5 E4 ðtÞ þ    : ð3:27:28Þ There is a linear relationship between polarization P and external electric field E for the linear dielectric material, from Eqs. (3.27.27) and (3.27.28): D ¼   e0 1 þ ve;1 E and P ¼ e0 ve;1 E, i.e. these expressions coincide with Eqs. (3.6.6) and (3.6.8).

3.28

Review Questions

Q3.1. What does the term “transmission media” mean? Q3.2. What is a perfect a vacuum? Q3.3. What is the speed of the EM wave transport energy through a vacuum? Q3.4. Give examples of conductor materials. Q3.5. Give examples of dielectric materials. Q3.6. What do the terms “bound charge” and “free charge” mean? Q3.7. In which direction do the positive charges displace (shift) when an external electric field E is applied? Q3.8. What do the terms “insulators” and “dielectrics” through the polarizability and electrical conductor mean? Q3.9. What are polar and nonpolar dielectrics?

3.28

Review Questions

193

Q3.10. What do the terms “instantaneous” and “permanent” dipoles mean? Q3.11. What is dielectric polarization? Q3.12. What is electric polarizability? Q3.13. Name the main dielectric polarization types. Q3.14. Explain the polarization mechanism of electronic polarization. Q3.15. Explain the polarization mechanism of atomic polarization. Q3.16. Explain the polarization mechanism of ionic polarization. Q3.17. Explain the polarization mechanism of dipolar (orientation) polarization. Q3.18. What is the absolute and relative permittivity of a dielectric medium? Q3.19. What is the electric susceptibility of a dielectric medium? Q3.20. What is the electric flux density D? Q3.21. What is the polarization vector P? Q3.22. Which one of charges qfree and qbound participates in the value r  P ¼? Q3.23. Which one of charges qfree and qbound participates in the value r  D ¼? Q3.24. What are extraneous charges? Q3.25. What is the surface charge density? Q3.26. What is the static permittivity es ? Q3.27. What are the dielectric constant and the loss factor of the complex permittivity? Q3.28. What is the loss tangent of a dielectric? Q3.29. What are the real e0 and imaginary e00 parts of the complex permittivity of medium? Q3.30. What are the displacement and polarization currents? Q3.31. What is the polarization current? Q3.32. What is the difference between the current density J and the electric current I? Q3.33. Define Ohm’s law through the material conductivity. Q3.34. What is the extraneous current? Q3.35. Write Ampere’s circuital law r  H ¼ ? in the form of the currents. Q3.36. What is the electromotive force? Q3.37. What is the magnetomotive force? Q3.38. What are electrical conductivity and resistivity? Q3.39. What special form do the magnetic field lines have? Q3.40. What is magnetic susceptibility? Q3.41. What are the magnetic scalar and vector potentials capable of calculation with regard to a magnetic field? Q3.42. What is diamagnetic, paramagnetic, ferromagnetic, ferrimagnetic, antiferromagnetic?

194

3 Fields and Materials

Q3.43. What is the magnetic dipole moment? Q3.44. What is the magnetization M of a magnetic material? Q3.45. What are constitutive relations for isotropic medium? Q3.46. What are the biaxial and uniaxial crystals? Q3.47. What are biisotropic and bianisotropic media? Q3.48. What are boundary conditions of the steady current components at the interface between two different conductive media with different conductivities?

Chapter 4

Maxwell’s Equations and Boundary Conditions

Abstract This explores Maxwell’s equations in differential (point) and large-scale (integral) forms. The author deals with derivation of boundary conditions for tangential and normal components of electric E and magnetic H fields as well as electric flux density D and magnetic flux density B of EM fields. The boundary conditions for interfaces between two dielectrics, two conductors, and dielectric-conductor media are demonstrated.

4.1

Introduction

Maxwell’s equations are a set of partial equations. They describe how electric and magnetic fields are generated by charges, currents as well as by changes of the electric and magnetic fields and how they influence the surrounding media on EM processes. From Maxwell’s equations, we can obtain certain differential equations corresponding to a physical problem. Usually, the differential equation has not only one solution but the whole family of solutions. The boundary conditions allow us to choose one that corresponds to the real physical process or phenomenon. Boundary conditions are an addition to the differential equation, which defines its behavior at the boundary (interface) of the considered area. Boundary conditions are conditions which must satisfy the desired solution of a given differential equation at the interface between two media (or part of them) of the area where the solution is sought. Now, we will present the basic concepts which we will need in studying the questions of this chapter. Differential equations from Table 2.4 for the stationary in time electric and magnetic fields can be written: r  E ¼ 0, r  D ¼ q at the constitutive relation: D ¼ e0 er E and r  H ¼ J, r  B ¼ 0 at the constitutive relation B ¼ l0 lr H: There are useful EM effects that cannot be described by the stationary in time equations (Table 2.4). One of the important EM effects is the generation of electric fields by time-varying magnetic fields named as Faraday’s law. © Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_4

195

196

4 Maxwell’s Equations and Boundary Conditions

For studying EM effects, we remind here the basic definitions which will be necessary in this chapter. • A conservative force is the force that is characterized by the work done in moving a particle between two points, which is independent of the selected path. The conservative force is dependent only on the position of the considered object. When the force is conservative, it is then possible to designate the potential at any point. If the object moves from one point to another, then the force change the potential energy of the object by a quantity which does not depend on the path taken (for curl-less or conservative fields see Sect. 1.3.7). • If the force is not conservative, then the determination of a scalar potential is not possible because taking different paths would conduct (lead) to the disagreement in finding of potential differences between the start and terminal (end, final) points. • A closed path for an electric current is called an electric circuit. A closed path for magnetic flux is called a magnetic circuit. • EM coil is an electrical conductor wire in a shape of a coil, helix, or spiral. An electrical conductor is a type of material that allows the flow of an electric current. • A galvanometer is a device for measuring an electric current. Galvanometers are an analogue of ammeters used to measure the direct current through an electric circuit. An ammeter (from ampere meter) is also used to measure the current in a circuit. • Open circuit is an electric circuit through which current cannot flow because the path is interrupted by an opening. • Open-circuit voltage is the difference of electric potential between two terminals (ends) of a device when disconnected from any electric circuit. There are not any external electric current flows between the terminals. Open-circuit voltage is interpreted as the electromotive force (emf) which is the maximum potential difference when there is no current because the circuit is not closed. When an open-circuited loop is placed in a time-changing magnetic field the emf (see Sect. 3.14.2) is induced. Although no current flow occurs in this case, the emf is a potential difference across the terminals of the loop. • In eddy currents, there are loops (closed circuits) of electric currents induced within a conductor by a changing magnetic field (see also Sect. 3.13.4) which are described by Faraday’s law of EM induction. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. Eddy currents can be induced within stationary conductors by a time-varying magnetic field created by an AC electromagnet or by relative motion between a conductor and a magnet. • An inductor (a coil or a reactor) is a passive electrical component which stores electrical energy in the magnetic field when the electric current is flowing through the inductor. An inductor usually consists of an electric conductor, as a wire which is wrapped (curled) into coils, spirals, or loops. If an electric current of the inductor changes, the time-varying magnetic field induces

4.1 Introduction

197

(makes, stimulates) the voltage in the inductor conductor (wire), which is described by Faraday’s law of induction.

4.2

Faraday’s Law of Electromagnetic Induction

Michael Faraday was an English physicist and chemist. He lived in the period of time of 1791–1867 and was the greatest scientist of the nineteenth century. Michael Faraday discovered the principles of EM induction, diamagnetism, effect of influence of an magnetic field on light rays, special behavior of certain materials in strong magnetic fields, etc. He reported numerous experiment results on the electricity and electromagnetism in three volumes entitled “Experimental Researches in Electricity”. In 1831, Michael Faraday discovered by way of experiments one of the most basic laws of electromagnetism called the Faraday’s law of electromagnetic induction. EM induction is the process by which an electric current can be induced (stimulated) to flow due to a changing magnetic field. This law explains the working principle of the electric motors, transformers, generators, and inductors. Faraday’s law shows the relationship between an electric circuit into a conducting loop and magnetic field. M. Faraday performed an experiment with a magnet and coil. During this experiment, M. Faraday noticed that when a permanent magnet was moved into or out of an EM coil it induced an electromotive force (emf) (see Sects. 3.14.2 and 4.1), in other words, a voltage, and therefore a current was produced (Fig. 4.1). M. Faraday included in his experiment a magnet and an EM coil connecting with a galvanometer. When a magnet is moved into or out an EM coil then it changes the magnetic flux through the coil and a voltage will be generated in this coil. The experiment can be seen in stages: (1) A magnet with two poles “N” (North) and “S” (South) creates the magnetic flux density B. In Fig. 4.1, the “N” pole is on the left side of the magnet, i.e. the “N”

(a)

South pole of coil

North pole of coil

(b)

EM coil

EM coil Magnet

Magnet

B B

B J

B

N

S

Magnetic field B of magnet

Binduced

J

B

N

B

Binduced

S

Magnetic field B of magnet

in coil

in coil

Direction of movement of the magnet

0

J

into EM coil

Direction of movement of the magnet

0

J

out of EM coil

Fig. 4.1 Magnet and EM coil which is connected with a galvanometer. a The magnet is moved toward the coil. b The magnet is moved out of the coil

198

4 Maxwell’s Equations and Boundary Conditions

pole is closer to the coil. When the pole “N” is moved toward the coil (Fig. 4.1a) there is an increase in the magnetic flux density B on the magnitude DB compared to its initial value and when the magnet is moved out of the coil (Fig. 4.1b) there is a decrease of B. (2) We designate increase (Fig. 4.1a) or decrease (Fig. 4.1b) of the magnetic flux density by the increment DB with the sign plus or minus. (3) The needle of a galvanometer deflects in the opposite directions for each case (Fig. 4.1) because the induced magnetic flux density Binduced and the current in the coil are opposite for (a) and (b) cases. The direction of Binduced is always opposite to the change DB: So when we move the magnet back and forth toward the coil, the galvanometer needle deflects to the left or to the right relative to the directional motion of the magnet as well as the location of the magnet poles in relation to the coil. If the magnet is held stationary (at rest) in or near the coil, the needle of a galvanometer returns back to zero position, no voltage is then observed. Similarly, if magnet is held stationary and the coil is moved away and toward the magnet, the galvanometer shows deflection of the needle in the same manner. We can also see that the faster the change in the magnetic field, the greater the induced emf (or generated voltage) in the EM coil. So M. Faraday discovered the way of producing an electric current in EM coil by using only the force of a magnetic field.

4.2.1

Fundamental Postulate for Electromagnetic Induction

The electric field intensity in an area of time-varying magnetic flux density is not conservative (see Sect. 4.1) and cannot be expressed as the gradient of a scalar potential (Sects. 2.8 and 2.9.2): rE¼

@B ; @t

ð4:2:1Þ

   ^x ^y ^z   where r  E ¼  @=@x @=@y @=@z  (see Sect. 1.3.7).  Ex Ey Ez  Taking the surface integral of both sides of Eq. (4.2.1) over an open surface and applying Stokes’s theorem (Sect. 1.3.7) we can write: I

Z E  d‘ ¼

c

Z ðr  EÞ  ds ¼ 

S

S

@B  ds ¼  @t

Z S

@B ^ds: n @t

ð4:2:2Þ

For a stationary circuit with a contour “c” and surface “S” in a time-varying magnetic field (Fig. 4.2), we can write:

4.2 Faraday’s Law of Electromagnetic Induction

199

Fig. 4.2 Magnetic flux density B that penetrates a surface S

B

B S E

с B I E  d‘ ¼  c

@ @t

ds

B

d

Z B  ds:

ð4:2:3Þ

S

We accept that a generated voltage (induced emf) in a circuit “c” which is connected to an ammeter (galvanometer), by analogy with Eq. 2.7.1, is: ZP2 U emf ¼

E  d‘;

ð4:2:4Þ

P1

where the line integral is known as the electromotive force (emf), U emf measured in volts (see Sect. 3.14.2). We admit, by analogy with Eq. (2.9.3), that the magnetic flux through a Gaussian surface (closed surface) S is: Z UB ¼

B  ds;

ð4:2:5Þ

S

where B is the magnetic flux density of an applied external magnetic field, ds is an infinitesimal vector element of a surface, UB is the total magnetic flux through the surface S which is bounded by contour “c” (Fig. 4.2). SI units of UB is weber which is Wb ¼ Volt  second ¼ Henry  Ampere. We can write the induced emf from Sect. 3.14.2: U emf ¼ 

dUB ; dt

ð4:2:6Þ

dUB is a change in the magnetic flux. The large scale (integral) form of Faraday’s law (Eq. 4.2.6) states that a voltage is induced in a conducting loop that can be stationary and immersed in a time-changing magnetic field or can be moving in a static magnetic field. This voltage is proportional

200

4 Maxwell’s Equations and Boundary Conditions

Fig. 4.3 Illustration of Faraday’s law

S B

с

to the time rate of change of magnetic flux density which penetrates the surface bounded by the contour “c” (Fig. 4.3). Faraday’s law of electromagnetic induction. Equation (4.2.6) states that the induced emf (generated voltage) U emf in a closed magnetic circuit (EM coil, Sect. 4.1) which is at rest is equal to the negative rate of increase of the magnetic flux linking the circuit. The negative sign indicates that the induced emf will cause a current flow in the closed loop in such a direction as to oppose the change in the linking magnetic flux (Fig. 4.1). This assertion is known as Lenz’s law.

4.2.2

Lenz’s Law

Lenz’s law indicates that the induced voltage U emf and the change in magnetic flux dUB have opposite signs (Eq. 4.2.6). In other words, when the magnet whose pole “N” is on the left side is pulled out of the EM coil (Fig. 4.1) the galvanometer deflects to the one of sides in response to the decreasing field. The polarity of the induced emf is such that this U emf produces a current whose magnetic field Binduced opposes the change of the magnetic flux dUB which produces this U emf . The induced magnetic field Binduced inside any EM coil always acts so as to keep the magnetic flux in the coil constant. The magnitude of the current in a given EM coil (closed circuit, loop, spiral) is proportional to the strength of the magnetic field, area of the loop, rate of change of magnetic flux and is inversely proportional to the resistivity of the material.

4.3 Maxwell’s Equations

4.3

201

Maxwell’s Equations

The principal EM phenomena are described by Maxwell’s equations. They are a set of four equations. The equations were developed on the basis of experimental results of Faraday, Ampère, and other investigators. James Clerk Maxwell (1831–1879) was a Scottish scientist in the field of mathematical physics. In 1865, Maxwell demonstrated in his publication “A dynamical Theory of the EM field” that the electric and magnetic fields can travel through space as EM waves. J. C. Maxwell also made fundamental contributions to mathematics, statistical physics, astronomy, and engineering. Maxwell’s four equations may be represented in either differential (also known as point) form or large-scale (also known as integral) form.

4.3.1

Maxwell’s Equations in Differential Form

Maxwell’s equations in differential (point) form are a set of partial differential equations that, together with the Lorentz force law (Sect. 3.17), form the foundation of classical electrodynamics, classical optics, and electric circuits. Maxwell’s equations in differential (point) form: r  Eðr; tÞ ¼ 

@Bðr; tÞ ; @t

r  Hðr; tÞ ¼ Jðr; tÞ þ

Faraday's law

@Dðr; tÞ ; @t

r  Dðr; tÞ ¼ qðr; tÞ; r  Bðr; tÞ ¼ 0;

Ampere's law

Gauss's law No name

where,    ^x ^y ^z     r  E ¼  @=@x @=@y @=@z     Ex Ey Ez        @Ez @E y @E x @E z @Ey @E x ¼ ^x    þ ^y þ ^z ; @y @z @z @x @x @y

ð4:3:1Þ ð4:3:2Þ ð4:3:3Þ ð4:3:4Þ

202

4 Maxwell’s Equations and Boundary Conditions

   ^x ^y ^z     r  H ¼  @=@x @=@y @=@z     Hx Hy Hz        @H z @H y @H x @H z @H y @H x ¼ ^x    þ ^y þ ^z ; @y @z @z @x @x @y @Dx @Dy @Dz þ þ ; @x @y @z (see Sects. 1.3.6 and 1.3.7), @Bx @By @Bz þ þ rB¼ @x @y @z and sometimes in literature Eq. (4.3.3) is named as Gauss’s law for electricity and Eq. (4.3.4) is named as Gauss’s law for magnetism. Here, Eðr; tÞ is an electric field strength (intensity), in (V/m); Bðr; tÞ is a magnetic flux density (magnetic induction), in (Tesla (T) or Wb/m2); Hðr; tÞ is a magnetic field strength (intensity), in (A/m); Dðr; tÞ is an electric flux density (electric induction, electric displacement), in (C/m2); Jðr; tÞ is a total electric current density (A/m2); qðr; tÞ is an electric charge density in (C/m3); ^r ¼ ^ xx þ ^ yy þ ^zz is the radius vector in the Cartesian system coordinate (other systems of coordinates can also be used); the operator r is the del or “nabla” operator; r is the divergence operator; r is the curl operator (Chap. 1); @=@t is a partial derivative of a function with respect to a variable t. @Dðr;tÞ is the displacement current which is a @t quantity related to changing electric field (Sect. 3.12). We see that in general cases the mentioned vectors are position and time dependent. In the total current density Jðr; tÞ [Eq. (4.3.2)], besides the electric currents described in Sect. 3.13 the extraneous current density Jext caused by external sources (Sect. 3.14.1) is also included. rD¼

4.3.2

The Sense of Maxwell’s Equations @Bðr;tÞ induces an electric field. @t @Dðr;tÞ @t (a displacement current) or/and

(Equation 4.3.1): A changing magnetic field

(Equation 4.3.2): A changing electric field an electric current Jðr; tÞ (free and/or extraneous currents) generate a magnetic field. (Equation 4.3.3): A stationary source (electric charge qðr; tÞ, i.e. electrons and, etc.) produces an electric field. (Equation 4.3.4): There are not any stationary sources for magnetic flux density (T? or “t”? here is no magnetic particle like an electron).

4.3 Maxwell’s Equations

4.3.3

203

Co-dependence of Maxwell’s Equations

The displacement current is a quantity related to changing of an electric field (Sect. 3.12). The displacement current occurs in dielectric materials and also in free space. The displacement current is zero if a current is continuous and steady because the electric field flux is constant. We can write the left side of Eq. (4.3.1) in projections: r  Eðr; tÞ ¼ curl Eðr; tÞ       @Ez @E y @E x @Ez @E y @E x    ¼ ^x þ ^y þ ^z ; @y @z @z @x @x @y

ð4:3:5Þ

Then Eq. (4.3.1) in projections is as follows: @E z @Ey @Bx  ¼ ; @y @z @t

ð4:3:6Þ

@E x @E z @By  ¼ ; @z @x @t

ð4:3:7Þ

@E y @E x @Bz  ¼ : @x @y @t

ð4:3:8Þ

We can obtain the same expressions for r  Hðr; tÞ ¼ curl Hðr; tÞ in Eq. (4.3.2). We would like to remind that divergence (Sect. 1.3.6) is @Dy @Dz x r  Dðr; tÞ ¼ div Dðr; tÞ ¼ @D @x þ @y þ @z . Equations (4.3.3) and (4.3.4) depend on Eqs. (4.3.1) and (4.3.2). We will show this dependence below. Taking the divergence of the both sides of Eq. (4.3.1), we find that: r  ðr  EÞ ¼ r 

  @B : @t

ð4:3:9Þ

The divergence of the curl of any vector field with continuous derivatives (Appendix Chap. 1:1.1) is always zero r  ðr  EðrÞÞ ¼ 0. For this reason, the Eq. (4.3.9) can be written: 

@B r  @t

 ¼ 0;

ð4:3:10Þ

204

4 Maxwell’s Equations and Boundary Conditions



@ ðr  BÞ ¼ 0: @t

ð4:3:11Þ

The changing of operators’ places r and @t@ at a resting coordinate system means that r  B ¼ div B is independent on time “t”. r  B ¼ constðtÞ and r  B  0, thus we get Eq. (4.3.4) from Eq. (4.3.1). Taking the divergence of (4.3.2), we find that:   @D r  ðr  HÞ ¼ r  J þ : @t

ð4:3:12Þ

We know from mathematics that the divergence of the curl of any vector field F is always zero: r  ðr  FÞ ¼ div curl F ¼ 0. The divergence of the curl of vector H is: r  ðr  HÞ ¼ 0. So we can write Eq. (4.3.12): 

 @D r  Jþ ¼ 0 and @t   @D ¼ 0: r Jþr  @t

ð4:3:13Þ

When the sum of all charge sources equals zero in general continuity equation [Eq. (2.14.9)], the current density can be expressed: r  J ¼  @q @t . We substitute in Eq. (4.3.13) and receive: r

  @D @q ¼ 0;  @t @t

ð4:3:14Þ

@ ðr  D  qÞ ¼ 0; @t

ð4:3:15Þ

r  D ¼ q:

ð4:3:16Þ

Equation (4.3.16) coincides with Eq. (4.3.3). So we obtain Eq. (4.3.3) from Eq. (4.3.2). Equation (4.3.3) confirms a conservation law (Sects. 2.4 and 3.9). The conservation law states that the total electric charge of a closed system remains constant over time regardless of other possible changes within the system. In physics, charge conservation is the principle that an electric charge can neither be created nor destroyed. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved.

4.3 Maxwell’s Equations

4.3.4

205

Maxwell’s Equations in Large-Scale Form

When describing interactions involving finite objects with specific shapes and boundaries it is convenient to express Maxwell’s equations in large-scale (integral) form. The differential and integral forms of the equations are mathematically equivalent by using Stokes theorem in the case of Faraday’s law [Eq. (4.3.1)] and Ampère’s law [Eq. (4.3.2)] as well as by using the divergence theorem in the case of Gauss’s law for electricity [Eq. (4.3.3)] and Gauss’s law for magnetism [Eq. (4.3.4)]. Maxwell’s equations in large-scale (integral) form are: I c

I

@ E  d‘ ¼  @t Z

H  d‘ ¼ c

J  ds þ S

@ @t

I

Z B  ds;

ðFaraday's lawÞ

ð4:3:17Þ

D  ds;

ðAmpere's lawÞ

ð4:3:18Þ

S

Z S

Z D  ds ¼

qdv;

ðGauss's lawÞ

ð4:3:19Þ

V

S

I B  ds ¼ 0: ðno nameÞ

ð4:3:20Þ

S

Here

H

E  d‘ is the line integral of the electric field along the boundary curve “c”

c

of surface S (contour H “c” is always a closed curve, Fig. 4.2), measured in joules per coulomb (volts). H  d‘ is the line integral of the magnetic field over the closed c

boundary Amperian curve (loop) “c” of the surface S, tesla-meters T  m . The surface S is fixed (unchanging in time). Value d‘ is the differential vector element of path length tangential to the path (curve, in meters) equals: ^d‘ ¼ ^xdx þ ^ydy þ ^zdz: d‘ ¼ n H S

D  ds ¼

H

ð4:3:21Þ

^ds is the electric flux (surface integral of the electric flux Dn

S

density) through the closed surface S where the surface S is the boundary of the volume V (Fig. 1.29). The magnitude ds is differential vector element of surface area S with infinitesimally small magnitude and direction normal to surface S, square meters (Fig. 4.2). Here, ds ¼ ds^ n ¼ dxdy^z is the differential surface vector [see Eqs. (1.3.4) and (1.3.7)].

206

4.3.5

4 Maxwell’s Equations and Boundary Conditions

Faraday’s Law in Integral Form

Formula (4.3.17): The large-scale form of Faraday’s law states that the voltage (emf) (Sect. 3.14.2) is induced in a conducting loop that lies along a contour “c”, which is at rest and is placed in a time-changing magnetic field, or which can be moving in a static magnetic field. This voltage (emf) is proportional to the time rate of change of magnetic flux which penetrates the surface bounded by the contour “c” (Figs. 4.2 and 4.3). This line integral is equal to the emf (generated voltage) in the loop, so Faraday’s law is the basis for electric generators. The line integral, which is on the left-hand H side of Faraday’s law (Eq. 4.3.17), is known as the electromotive force emf ¼ E  d‘. The total magnetic flux through c R the surface S, which is bounded by contour “c”, is UB ¼ B  ds and the electroS

motive force can be expressed as: emf ¼ 

@UB ; @t

ð4:3:22Þ

see also Sect. 3.14.2.

4.3.6

Ampere’s Law in Integral Form

Formula (4.3.18): Ampere’s law states that the circulation of magnetic flux density around any closed contour “c” in free space is equal to the total current flowing through a surface bounded by that contour “c”. We consider the contour “c” bounding surface S (Fig. 4.4). Let the direction of contour “c” bypass in a way that the movement along the path “c”, when looking from the tip of the vector elementary area (an infinitesimal surface) ds (Sect. 1.2.2), would be observed in a counterclockwise direction (Fig. 4.4a). This indicates that a time-changing (varying, altering) electric field gives rise to a magnetic field. We see that the electric field E and electric current density J are directed in the same side (Fig. 4.4b). The vector of magnetic flux density B is perpendicular to vectors E and J. We remember that an electric field can be created by electric charges or by time-varying magnetic fields. When the electric or magnetic field has a time dependence, both fields must be considered together as a coupled electromagnetic field (Table 2.3). The line integral term on Hthe left-hand side of Eq. (4.3.18) is known as the magnetomotive force: mmf ¼ H  d‘ (Sect. 3.14.3). c

The “integral form” of the original Ampere’s circuital law is a line integral of the magnetic field around some arbitrary closed curve “c”. The curve “c” also bounds a surface “S” through which the electric current passes and encloses the current

4.3 Maxwell’s Equations

207

(a)

(b)

E J

ds

d ds

J

с

S

J с

B

n

d

Fig. 4.4 Illustration of Ampere’s law: a vector elementary area ds , b direction of electric field E and electric current density J

(Fig. 4.4). The mathematical statement of the law is a relation between the total amounts of magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral). It can be written in a number of forms. In the formula of Ampere’s law J is the total current density (in ampere per square meter, A/m2). The first term on the right side of the Ampere’s law expression (4.3.18) represents the total current flowing through the surface “S” bounded by the contour “c” due to the electric currents depicted in Sect. 3.13 which include the extraneous current Jext caused by external sources (Sect. 3.14.1). The second term on the right side of the Ampere’s law expression is the total displacement current (Sect. 3.12) flowing through the surface S (Fig. 4.4): R Jd ¼ @t@ D  ds. Please also note that a time-varying electric field induces a magS

netic field through the displacement current Jd even when conduction and extraneous currents (Sects. 3.13.1 and 3.14.1) are absent, i.e. J ¼ 0. It is interesting to note that currents Jd allows to an EM wave propagate along the vacuum. The Ampere’s law may be expressed by the magnetomotive force (mmf): mmf ¼ J c þ J d (see Sect. 3.14.3), where J c is the conduction current (also known as drift current, free current, Sect. 3.12 and 3.13). The quantity mmf is also a quantity appearing R in the equation for the magnetic flux UB through a magnetic circuit: mmf ¼ H  d‘ ¼ UB RB . The magnetic resistance RB in magnetic circuits c

is analogous to the electric resistance in electric circuits. The magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The magnetic flux UB is usually generated by permanent magnets or electromagnets. Quantitatively, the magnetic flux through a surface S is defined as the integral of the R B magnetic induction over the area of the surface UB ¼ B  ds and emf ¼  @U @t , see S

Sect. 3.14.2.

208

4 Maxwell’s Equations and Boundary Conditions

Fig. 4.5 An electric current produces a magnetic field

B

+ J

We know that an electric current produces a magnetic induction B. The magnetic field [Eq. (3.27.5)] can be visualized as a pattern of circular field lines around the wire with a current (Fig. 4.5). Electric currents can be of different kinds (Sect. 3.13), it is only important that they would be distributed continuously with density J. Magnetic field lines are always closed! Equations (4.3.17) and (4.3.18) are obtained by integrating, respectively, Eqs.H(4.3.1) andR (4.3.2) over a surface and applying Stokes’ theorem (Sect. 1.3.7), i.e. F  d‘ ¼ ðr  FÞ  ds, where F is any vector, d‘ is the differential vector c

S

element of path length tangential to the path “c”, the magnitude ds is differential vector element of surface area S, with infinitesimally small magnitude and direction normal to surface S which is the boundary of the volume V.

4.3.7

Gauss’s Law

Formula (4.3.19):

H S

D  ds ¼

R

qdv presents Gauss’s law for electricity. This law

V

establishes the relationship between the charge q and the vector electric field E this charge. Because of D ¼ e0 er E, we can write R Hwhich affects E  ds ¼ ðq=e0 er Þdv. The area integral over any closed surface S equals to the S

V

sum of charges enclosed in the surface S and is divided by the permittivity of the medium around the charges, the magnitude ds is differential vector element of surface area S, with infinitesimally small magnitude and direction normal to surface S which is the boundary of the volume V. Let a volume V be limited to a closed surface S (Fig. 4.6). If the volume contains the total electric charge q, then this charge amount is proportional to the flow of the vector electric field through the surface S. If we consider the point charges, the total n P charge can be found by the algebraic summation of charges q ¼ qj j¼1

4.3 Maxwell’s Equations

209

Fig. 4.6 Illustration of Gauss’s law

ds

E

S

V

E Section plane of volume V

E ds E [see Eq. (2.5.4)] and then Eq. (4.3.19) can be written:

H

D  ds ¼ q. If the charge is

s

distributed continuously, Rwe can define the total charge by the integration of charge density by volume: q ¼ qdv (see Sects. 2.4 and 2.5). V

4.3.8

Magnetic Source Law

The Maxwell’s equations (4.3.20) is the magnetic source law which states that the integral of the magnetic flux density over any closed surface S vanishes (Fig. 4.7). This indicates that the total magnetic flux flowing out of a closed surface is zero (magnetic field lines form closed loops). This reflects the fact that magnetic source charges have not yet been observed in nature. Equations (4.3.19) and (4.3.20) are obtained by integrating, respectively, Eqs. (4.3.3) and R (4.3.4) over H a volume and applying the divergence theorem (Sect. 1.3.6): ðr  FÞ dv ¼ F  ds, see Fig. 4.8, where the magnitude ds is difV

S

ferential vector element of surface area S, with the infinitesimally small magnitude ds ¼ jdsj and direction normal to surface S which is the boundary of the volume V, dv is an infinitesimally small volume. According to Gauss’ law for electricity, electric flux flowing through any closed surface S is directly proportional to the sum of electric charges enclosed by the surface S. We can give some analogy which exists between an electric charge and a hepothetical magnetic monopole. We would expect to be able to formulate the magnetic source law (4.3.20) which states that the magnetic flux through any closed surface is directly proportional to the number of hypothetical magnetic monopoles enclosed by that surface S. However, as we have already discussed, hypothetical magnetic monopoles do not exist. It follows that the equivalent of Gauss’ law for

210

4 Maxwell’s Equations and Boundary Conditions

Fig. 4.7 Illustration of the magnetic source law

B

B

ds V

S

Fig. 4.8 Illustration elements to the divergence and Stokes theorems

ce

a urf

S

S

loop c

^nds

Vo lu

me

d

^nds=ds

V

magnetic fields reduces to the following: The magnetic flux though any closed surface is zero. This is just another way of saying that hypothetical magnetic monopoles do not exist and that all magnetic fields are generated by flowing currents.

4.3.9

Table of Maxwell’s Equations

Maxwell’s equations in differential and integral forms are summarized in one Table 4.1. RR R In this book, we often denote the double and triple integrals as follows: ¼ , S S RRR R ¼ where S is a surface, V is a volume. The surface integrals over the boundary V V RR H surface with the loop indicating the surface S is closed are denoted as  ¼ . To solve S

S

4.3 Maxwell’s Equations

211

Table 4.1 Combined Maxwell’s equations formulation in SI units convention Maxwell’s equation

Differential form

Faraday’s law of induction

r  E ¼  @B @t

Ampère’s circuital law

r  H ¼ Jþ

Gauss’s law

rD¼q

no name, law for magnetism

rB¼0

Large-scale (integral) form H c

@D @t

H

RR E  d‘ ¼  ddt B  ds S

H  d‘ ¼

c

RR

D

S

 ds ¼

S

RR

B 

S

RR

RR J  ds þ ddt D  ds S

RRR V

ds ¼ 0

qdv

Meaning of the equation The voltage created in a closed circuit “c” is proportional to the rate of change of the magnetic flux density B enclosed by it The magnetic field created around a closed loop is proportional to the electric current density J plus the displacement current density Jd enclosed by it The electric flux density going out of a volume is proportional to the electric charge inside the volume The total magnetic flux through a closed surface is zero. This confirms that there are no magnetic monopoles (magnetic charges)

an EM field theory problem besides the certain Maxwell’s equations, it is necessary to use constitutive relations (Sect. 3.27) and the boundary conditions!

4.4

Maxwell’s Equations for the Time–Periodic Case

Many electric applications use a sinusoidal alternating current that is varying sinusoidally in time. In alternating current (AC, also ac), the movement of electric charges reverses direction periodically. We have previously noted in Sect. 3.13 that in direct current (DC, also dc) the flow of electric charges is only in one direction. It is possible to use the complex exponential function eixt to express the time dependency in varying sinusoidal current. We will use the Euler’s formula (Sect. 1.4.2) that is a mathematical formula in complex analysis which establishes the relationship between the trigonometric functions and the complex exponential function. Euler’s formula states that the exponential function is eix ¼ cos x þ i sin x.

212

4 Maxwell’s Equations and Boundary Conditions

Euler’s formula has been named after Leonhard Euler who was the eighteenth-century physicist and scholar (erudite or learned person) and developed many concepts of modern mathematics. So if we take that the field strengths and other magnitudes are dependent on the time exponentially then we can express the ones like this: Eðr; tÞ ¼ EðrÞeixt , Hðr; tÞ ¼ HðrÞeixt and so on. The set of Maxwell’s equations (4.3.1–4.3.4) in the time-periodic case are:   @BðrÞeixt r  EðrÞeixt ¼  ; @t

Faraday's law

  @DðrÞeixt r  HðrÞeixt ¼ JðrÞeixt þ ; Ampere's law @t   r  DðrÞeixt ¼ qðrÞeixt ; Gauss's law r  BðrÞeixt ¼ 0: no name

ð4:4:1Þ ð4:4:2Þ ð4:4:3Þ ð4:4:4Þ

ixt We receive the set of equations after replacing @ e@t by ixeixt :

r  EðrÞ ¼ ixBðrÞ;

Faraday's law

r  HðrÞ ¼ JðrÞ þ ixDðrÞ; r  DðrÞ ¼ qðrÞ; r  BðrÞ ¼ 0:

Ampere's law

ð4:4:5Þ ð4:4:6Þ

Gauss's law

ð4:4:7Þ

no name

ð4:4:8Þ

Field vectors that change with space coordinates (x, y, z) and sinusoidal functions of time can be represented by the vector phasor that depends on space coordinates but does not depend on time. As an example, where a time-harmonic field:   Eðr; tÞ ¼ Re EðrÞeixt ;

ð4:4:9Þ

where EðrÞ ¼ Eðx; y; zÞ is a vector phasor that contains information on magnitude, direction, and phase. Phasors are, in general, complex quantities. We can use simplifications for differentiations and integration with respect to time t on the base of Eq. (4.4.9) are: @Eðx; y; z; tÞ=@t ¼ ixEðx; y; zÞeixt ;

ð4:4:10Þ

4.4 Maxwell’s Equations for the Time–Periodic Case

213

and Z Eðx; y; z; tÞ dt ¼Eðx; y; zÞeixt =ix:

ð4:4:11Þ

We can write time-harmonic Maxwell’s equations in terms of vector field phasors Eðx; y; zÞ and Hðx; y; zÞ, sources of total electric charge density qðx; y; zÞ and total electric current density Jðx; y; zÞ in linear homogeneous isotropic medium (simple medium, Sect. 3.27) with e ¼ e0 er and l ¼ l0 lr : r  E ¼ ixlH;

Faraday's law

r  H ¼ J þ ixeE; r  E ¼ q=e;

Ampere's law Gauss's law

r  H ¼ 0: No name

ð4:4:12Þ ð4:4:13Þ ð4:4:14Þ ð4:4:15Þ

Note 1: In the engineering and EM field theory literature, the time dependency is usually expressed by the complex exponential function eixt while in physics literature and especially in optics and plasmonics this dependence is usually taken by eixt . Note 2: Equation (4.4.14) is also known as the Coulomb’s law in differential form, where r  E is the divergence of electric field and q is the total electric charge density (charge per unit volume).

4.5

Source-Free Fields in Nonconducting Media

We derive now the Maxwell’s equations to describe the EM field in nonconductor media at constitutive relations from Sect. 3.27 for linear homogeneous isotropic medium which are D ¼ eE, B ¼ lH, J ¼ rE. The determination of source-free space and nonconducting transmission media is given in Sect. 3.1. The electric charge density q, the electrical conductivity r are the sources of the EM field. The perfect nonconducting media is characterized by the absence of q ¼ 0 and r ¼ 0 as well as the free and/or extraneous current density J ¼ 0. Maxwell’s equations for source-free case from (4.3.1) to (4.3.4) in simplified form are as follows: r  E ¼ l0 lr r  H ¼ e0 er

@H ; @t

@E ; @t

Faraday's law Ampere's law

ð4:5:1Þ ð4:5:2Þ

214

4 Maxwell’s Equations and Boundary Conditions

r  E ¼ 0; Gauss's law

ð4:5:3Þ

r  H ¼ 0: No name

ð4:5:4Þ

Equations (4.5.1–4.5.4) are first-order differential equations with two unknown vector functions Hðr; tÞ and Eðr; tÞ. We take curl of Eq. (4.5.1): r  r  E ¼ l0 lr

@ ðr  HÞ; @t

ð4:5:5Þ

and put Eq. (4.5.2) into Eq. (4.5.5), then we get: r  r  E ¼ e0 er l0 lr

@2E : @t2

ð4:5:6Þ

There is a mathematical formula for the curl of the curl of a vector field (Appendix Chap. 1:1.1) and, using Eq. (4.5.3), we get: r  r  E ¼ rðr  EÞ  r2 E:

ð4:5:7Þ

We can write taking account Eq. (4.5.3): r  r  E ¼ r2 E:

ð4:5:8Þ

Using Eq. (4.5.6), we get: @2E ¼ 0: ð4:5:9Þ @t2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We introduce the notation v ¼ 1= e0 er l0 lr and Eq. (4.5.9) can be written as follows: r2 E  e0 er l0 lr

r2 E 

1 @2E ¼ 0; v2 @t2

ð4:5:10Þ

where v is a phase speed (the module of phase velocity) of an EM wave in a medium with the relative permittivity er and permeability lr . When we take the curl of a curl of a vector field H [Eq. (4.5.2)] we get: r2 H 

1 @2H ¼ 0: v2 @t2

ð4:5:11Þ

Equations (4.5.10) and (4.5.11) are homogeneous vector equations for the electric and magnetic fields in the Cartesian coordinate system. Each of these equations can be decomposed (Sect. 1.3.7) into three scalar wave equations.

4.5 Source-Free Fields in Nonconducting Media

215

The time-harmonic Maxwell’s equations for linear homogeneous isotropic nonconducting source-free medium become: r  E ¼ ixl0 lr H; r  H ¼ ixe0 er D; r  E ¼ 0;

Faraday's law Ampere's law

Gauss's law

r  H ¼ 0: no name

ð4:5:12Þ ð4:5:13Þ ð4:5:14Þ ð4:5:15Þ

We can receive the second-order partial differential equations from Eqs. (4.5.12– 4.5.15) analogically with Eqs. (4.5.10) and (4.5.11). The equations for the time-periodic case are: r2 E þ k2 E ¼ 0;

ð4:5:16Þ

r2 H þ k2 H ¼ 0;

ð4:5:17Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ x e0 er l0 lr ¼ x=v ¼ 2p=k;

ð4:5:18Þ

where k is the wavenumber:

pffiffiffiffiffiffiffiffiffi where x ¼ 2pf , k is the wavelength, for air er ¼ lr ¼ 1 and k ¼ x e0 l0 ¼¼ x=c, c is the speed of light in a vacuum Eq. (2.1.6). Equations (4.5.16) and (4.5.17) are homogeneous vector Helmholtz’s equations.

4.6

Classification of Media Based on the Conductivity

Materials contain charged particles that respond to applied electric and magnetic fields to produce secondary polarization [(Sect. 3.6)] and magnetization [(Sect. 3.19)] fields. There are three basic phenomena (phenomenon-single) resulting from interaction between charged particles and electric and magnetic fields. There are conduction (in metals, etc.), polarization (dielectrics, etc.), and magnetization (magnetic materials). Depending on the kind of material their reaction to an applied (external) electric field, they may be classified into conductor, semiconductor, and dielectric. Important criteria of materials is the electrical conductivity r (Sect. 3.14); that is, ability of materials to conduct an electric current I. An electric current is a flow of electric charge through medium. Materials are lossless (dissipationless) if the electrical conductivity r ¼ 0 (e.g. a vacuum) does not have any conductivity.

216

4 Maxwell’s Equations and Boundary Conditions

On the basis of the right part of the Ampère’s law [Eq. (4.4.6)], we can write: JðrÞ þ ixDðrÞ ¼ rEðrÞ þ ixe0 er EðrÞ ¼ ðr þ ixe0 er ÞEðrÞ:

ð4:6:1Þ

In order for material properties show themselves as a dielectric medium (with the process of polarization, Sect. 3.2) or as a conductor medium (with the conduction currents, Sect. 3.14.1), we can define this by ratio: r tan dc ¼ ; ð4:6:2Þ xe0 er where r is in the numerator and xe0 er is in the denominator of the fraction, er is the relative permittivity of a medium and the complex absolute permittivity of the medium is e_ ¼ e0 er ¼ e0  ie00 . The magnitude r=x is the conductivity loss, tan dc is the loss tangent when the conductivity loss r=x 6¼ 0 at the dielectric loss e00b ¼ 0 (see further Eq. 4.6.5) and dc is the loss angle at e00b ¼ 0 . Value tan dc is equal to or less than the total loss angle tan d (see Eq. 4.6.6). The classification of media based on the relationship Eq. (4.6.2) gives (provides) five kinds of medium: (1) Medium is a perfect insulator (this is an idialization: insulator, nonconductor, lossless (dissipationless) medium) at r ¼ 0; (2) Medium is an imperfect insulator (low-loss medium, good dielectric, low-conductivity material, poor conductor) at r 6¼ 0; r=xe0 er  1; (3) Lossy conducting media (e.g. dopid semiconductor, poor dielectric) r=xe0 er ’ 1; (4) Medium is a good conductor (metals, high-conductivity material, high-loss medium, heavily doped semiconductor) at r=xe0 er  1; (5) Medium is a perfect conductor (this is an idialization: PEC, ideal metal) at r ! 1 (infinity). There is only a displacement current in the perfect (ideal) dielectric. There is only a conductive current in the perfect (ideal) electric conductor (PEC). A dielectric is an electrical insulator that can be polarized by an applied electric field (Sect. 3.2). When a dielectric is placed in an external electric field electric charges do not flow through the material as they do in a conductor but they only slightly shift from their average equilibrium positions causing dielectric polarization. Insulator is a substance that resists electric current. An electrical insulator is a material whose internal electric charges do not flow freely and which therefore does not conduct an electric current under the influence of an electric field. A perfect insulator does not exist but some materials such as glass, paper, teflon, and most plastics with high resistivity are very good electrical insulators. A much larger class of materials, even though they may have lower bulk resistivity (Sect. 3.15), are still good enough to insulate electrical wiring and cables. All conductors contain electric charges, which will move when an electric potential difference (measured in volts) is applied across separate points on the material. This flow of charge (measured in amperes) is what is meant by electric current. Most familiar conductors are metallic (such as aluminum, iron, silver,

4.6 Classification of Media Based on the Conductivity

217

copper). Metals are classified as conductors because their outer atom electrons are not tightly bound. A conductor is a material which contains movable electric charges. Copper and aluminum materials are the most common conductors used for electrical wiring. Silver or gold is the best conductor but it is expensive. However, there are also many non-metallic conductors. Graphite, some supercooled polymers are non-metallic conductors including solutions of salts, plasmas. There are even conductive polymers. Semiconductors have a value of fraction r=xe0 er between a conductor and an insulator. The dependency of ratio r=xe0 er on frequency f for different materials (media) is given in Fig. 4.9. The electrical conductivity r and relative permittivity er of materials are given in Tables 3.1 and 3.3.

10 10 10

6 5

so aw er

at

3

10

10

8

6

10 5

10 7

10

10

9

10

2 3

bl

ar

4

e

10

4

10

M

10

1

il so

er

at w 2

10

1

10

ry D

h

2

10

10

Se

3

il

4

es

10

7

Fr

10

8

st

10

0 r

oi

10

ω

M

10

9

σ

a

ic

M Fig. 4.9 Dependency of r=xe0 er on frequency for different media

f, Hz

218

4 Maxwell’s Equations and Boundary Conditions

Lossy materials (including a dielectric loss) are often determined by relative permittivity er and loss tangent: tan d ¼ r=xe0 er ¼ r=xe0 , where designations are given in Sect. 3.11. If there is dielectric loss in media, e.g. due to damping (attenuation) of the vibrating dipole moments, we can write as follows:   r  H ¼ J þ ixD ¼ rE þ ix_eE ¼ r þ xe00b E þ ixe0 E



r r ¼ ix e0  ie00b  i E ¼ ix e0  i e00b þ E; x x

ð4:6:3Þ

here, the term e00b is the dielectric loss and r=x is the conductivity loss, from Eq. (3.11.2) we get the complex permittivity e_ r ¼ e_ =e0 ¼ ðe0  ie00 Þ=e0 ¼ e0r  ie00r . The imaginary component e00b is associated with bound charges and dipole relaxation phenomena. The conductivity loss is caused by the movement of free charges. We can write from Eq. (4.6.3):    r  H ¼ i xe0  i xe00b þ r E;

ð4:6:4Þ

  where the imaginary part x e00b þ r can be characterized as the total effective conductivity of material. From Eq. (4.6.3), we can express the complex permittivity of medium:  

r xe00 þ r ¼ e0 1  i b 0 ¼ e0 ð1  i tan dÞ: e_ ¼ e0  i e00b þ x xe

ð4:6:5Þ

Whereas the loss tangent at the microwave range is: tan d ¼

xe00b þ r : xe0

ð4:6:6Þ

The loss tangent is frequency dependent. For dielectrics with small loss, this loss angle d is  1, and, using series expansion at x ¼ 0, we can write tan d d. The complex permittivity of material from Eqs. (4.6.5) and (4.6.6) is: e_ ¼ e0 ð1  i tan dÞ ¼ e0r e0 ð1  i tan dÞ:

ð4:6:7Þ

We see that loss tangent includes conductivity loss and dielectric damping loss of a material (Sect. 3.11). The complex permittivity frequently used in the form e_ ¼ e0 ð1  ir=xe0 Þ means that it is not included in the dielectric damping loss. In the case of lossless material tan d ¼ 0 and the permittivity is a real number and equal to e ¼ e0r e0 .

4.7 Concept of Boundary Conditions

4.7

219

Concept of Boundary Conditions

A boundary value problem is differential equations unitedly additional restraints which are called the boundary conditions. A solution to the boundary value problem is the solution to the differential equation (e.g., Maxwell’s equations) which also satisfies certain boundary conditions. There are several types of boundary conditions commonly encountered in the solution of partial differential equations, i.e. the Dirichlet, Neumann, Cauchy, Robin, and different boundary value problem formulations. A partial differential equation is an equation involving functions and their partial derivatives, e.g. wave equation. Three more often used types of boundary conditions are the following: (1) The boundary condition which specifies the value of the function itself is the first-type boundary condition, also known as the Dirichlet boundary condition. (2) The boundary condition which specifies the value of the normal derivative of the function is the second-type boundary condition, also known as the Neumann boundary condition. (3) If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is the Cauchy boundary condition. In electrostatics, a problem is usually formulated like that: to find the function which describes the electric potential inside of a given area. If the area does not contain charge, the potential must be a solution to Laplace’s equation (Sect. 2.13). The boundary conditions in the mentioned case are the interface conditions for electric and magnetic fields. If there is no electric current density in the area, it is also possible to define a magnetic scalar potential (see Sect. 3.21) using the resemble procedure. The differential forms of Maxwell’s equations can be used to determine the electric and magnetic field vectors which are single valued, bounded, and continuous in every point of the field. A single-valued function is the function that for each point in the domain (set of values for which the function is defined) has a unique value in the certain range. At two media boundaries (interface), the electric and magnetic field vectors are discontinuous and their behaviors across the boundaries are governed by boundary conditions. The integral equations (Eqs. 4.3.17–4.3.20) are assumed to hold for areas (regions) containing discontinuous media. Boundary conditions can be derived by applying the Maxwell’s equations in large-scale (integral) form to small areas at the interface of the two media. Note: EM edge conditions (electric and magnetic field behavior near a rib of a waveguide structure) and the boundary conditions at the interface (surface, boundary) of a waveguide structure are electrostatic problems because we derive these conditions at very small distances from the interface or the edge of the waveguide structure. An edge is a sharp side formed by the intersection of two surfaces of a waveguide structure. For example, when solving the problem for a rectangular dielectric waveguide EM edge conditions must be satisfied. The edge

220

4 Maxwell’s Equations and Boundary Conditions

conditions which dictate the asymptotic behavior of the EM field near the edges of the structure play a very important role in solving boundary value problems when the cross section contour of a waveguide structure contains angles. In the next sections, we will derive the boundary conditions for EM field components on the horizontal interface between two media to simplify the issue. The similar result would be if we considered any arbitrary-shaped boundary between two media.

4.8

Derivation of Electromagnetic Boundary Conditions

EM problems usually deal with structures that involve media (materials) with different physical properties. In order to solve EM problems involving contiguous (adjacent, neighboring) areas with different constitutive parameters (Sect. 3.27), it necessary to know the boundary conditions for the EM field vectors which must be satisfied at a certain interface (boundary) of the waveguide structure. We may wish to determine how the electric and magnetic fields change when crossing the interface. The EM field components obey certain rules along the boundary(interface) of two different materials. EM field vectors are: electric field strength (intensity) E ðvolt=meter Þ, electric



coulomb=meter 2 ,

flux density (electric displacement, electric induction) D

magnetic field strength (intensity) H ðampere=meter Þ, and magnetic flux density (magnetic induction) BðteslaÞ. The selected Cartesian coordinate system in Fig. 4.10 is convenient for finding the solutions of EM problems related to waveguides in Chaps. 7 and 8. On the base of Sect. 1.2.4, vectors can be decomposed into these components:

e

lan

p al

c

rti Ve

y Horizontal plane

Ey,Hy Ex,Hx

Ez,Hz

Ez,Hz

^y ^x O

z

^z

x Fig. 4.10 Decomposition of the electric and magnetic vectors into components in 3D space

4.8 Derivation of Electromagnetic Boundary Conditions

221

E ¼ ^xEx þ ^yE y þ ^zEz ¼ Ex þ Ey þ Ez ;

ð4:8:1Þ

D ¼ ^xDx þ ^yDy þ ^zDz ¼ Dx þ Dy þ Dz ;

ð4:8:2Þ

H ¼ ^xH x þ ^yH y þ ^zH z ¼ Hx þ Hy þ Hz ;

ð4:8:3Þ

B ¼ ^xBx þ ^yBy þ ^zBz ¼ Bx þ By þ Bz ;

ð4:8:4Þ

along x-axis, y-axis, and z-axis. The E and H fields’ components are shown in Fig. 4.10. We see that components Ex and Ez of the vector E are in the horizontal plane and the component E y is in the vertical plane. The components that lie in a plane are tangential to the plane, so we designate them with an index “t”, e.g. Et , Dt , H t , Bt and the unit tangent vector is ^s. The components that are perpendicular to the horizontal plane are called the normal components, so we designate them with an ^. index “n”, e.g., En , Dn , H n , Bn and the unit normal vector is n In the horizontal plane (Fig. 4.10) components Ex ¼ E t and Ez ¼ E t are tangential ones, while E y ¼ En are normal components to the plane. In the vertical plane (Fig. 4.10) the component E x ¼ E n is the normal component, while E y ¼ Et and E z ¼ Et are tangential ones. Thus, E ¼ Et þ En , where Et can be E x , E y , Ez and En can be E x , Ey depending on whether the interface is horizontal or vertical plane. There is analogical situation for the vector H ¼ Ht þ Hn . EM problems often involve media with different physical properties and require the knowledge of the relations of the field quantities at an interface, i.e. a surface forming a common boundary between two media, objects, spaces, areas, etc. For example, we may wish to determine how the electric field strength E and electric flux density D vectors change in crossing an interface. Figure 4.11 presents the interface between medium 1 with the constitutive parameters e1 ¼ e0 er1 , l1 ¼ l0 lr1 and medium 2 with the constitutive parameters e 2 , l2 . To satisfy boundary conditions at an interface point “A” we must take the limit of a distance d ! 0 from each medium when we approach to the point “A” in the interface (Fig. 4.11). The solution of the Maxwell’s equations is converted to the solution of the Laplace equation (Sect. 2.13) at very small distance d.

Fig. 4.11 Interface between two media with different constitutive parameters

Medium 1

μ1

1,

Medium 2

μ2

2,

point A

δ δ

222

4 Maxwell’s Equations and Boundary Conditions

The electrodynamic problem automatically becomes an electrostatic one at d ! 0 (see Table 2.4). We now consider an interface between two media. In order to find a relation between the components of fields at a boundary, we construct an infinitive small loop (path, contour) or pillbox (a small case, Gaussian pillbox) with its top and bottom faces.

4.9

Boundary Conditions for Et and Dt Components

We consider the interface between two dissimilar (different, various, distinct) dielectric areas (regions) and Faraday’s law presented by Eq. (4.3.17): I E  d‘ ¼  c

@ @t

Z B  ds: ðFaraday's lawÞ

ð4:9:1Þ

S

To derive the boundary condition in an arbitrary point A (Fig. 4.12), we investigate the left and right parts of Eq. (4.9.1), which represents one of the Maxwell’s equations in large-scale form. The incident plane EM wave with the electric field vector E1 from the dielectric 1 incident (impinges, falls) on the interface separated by two media with different absolute permittivities e1 ¼ e0 er1 and e2 ¼ e0 er2 , respectively. In our case, we chose that the absolute permeabilities are l1 ¼ l2 ¼ l0 . The transmitted EM plane wave into the dielectric 2 is characterized by the electrical field vector E2 . In Fig. 4.12 the value H1 is the angle of incidence and H2 is the angle of transmission (refraction)

Dielectric 1

y

E1 d

1 12

2

d

1 1

z

23

loop c point A

x

δ

d

41

δ

2

3

4

d Dielectric 2

34

E2

Interface

2

Fig. 4.12 An infinitesimal loop “c” with the differential length vector d‘ ¼ d‘12 þ d‘23 þ d‘34 þ d‘41

4.9 Boundary Conditions for Et …

223

of the plane EM wave. The counter 1234 is bypassed in a counterclockwise direction. We construct an infinite small loop (path, contour) “c” with the differential length vector d‘ (Sect. 1.3.1) which is: d‘ ¼ d‘12 þ d‘23 þ d‘34 þ d‘41 ¼ ^xjd‘12 j  ^yjd‘23 j  ^ xjd‘34 j þ ^ yjd‘41 j: ð4:9:2Þ When its horizontal sides d‘12 and d‘34 are in dielectrics 1 and 2, and each of vertical sides is half in one dielectric and half in another. The bypass loop “c” is going counterclockwise (Fig. 4.12). The coordinate system is also shown in Fig. 4.10. The loop “c” vectors obey: d‘12 ¼ ^xjd‘34 j;

ð4:9:3Þ

d‘23 ¼ ^yjd‘41 j;

ð4:9:4Þ

jd‘23 j ¼ jd‘41 j ¼ 2d;

ð4:9:5Þ

and module of the differential length vector is: jd‘j ¼ 2ðjd‘12 j þ 2dÞ:

ð4:9:6Þ

We suppose that vertical sides d‘23 and d‘41 approach zero [Eq. (4.9.5)] d ! 0 when we satisfy boundary condition at the point “A” which lies on the interface (Figs. 4.11 and 4.12). : jd‘23 j ¼ jd‘41 j ¼ 2d ¼ 0 at d ! 0:

ð4:9:7Þ

The areas, which are located inside the loop “c” (Fig. 4.12), are: areað1234Þ ¼ jd‘12 j  jd‘23 j ¼ 0. The magnitude ds is differential vector element of surface area S, with infinitesimally small magnitude and direction normal to surface S. The differential surface vector (Sect. 1.3.2) ds at d ! 0 is: ds ¼ d‘12  d‘23 ¼ ðjd‘12 j  jd‘23 jÞ^z ¼ ðjd‘12 j  0Þ^z ¼ 0:

ð4:9:8Þ

Equation (4.9.1) for the point A (Fig. 4.12) at a selected estimate value n ! which is a very small but finite one and d ! 0 is: I E  d‘ ¼ 

lim

d!0 jd‘12 j!n

c

lim

d!0 jd‘12 j!n

d dt

Z B  ds:

ð4:9:9Þ

S

In the limit d ! 0 on the base Eq. (4.9.8), we have that the right part of Eq. (4.9.9) equals zero:

224

4 Maxwell’s Equations and Boundary Conditions

lim d!0 jd‘12 j ! n

d dt

Z B  ds ¼ 0:

ð4:9:10Þ

S

So from (4.9.9) and (4.9.10) at d ! 0, we have: I lim d!0 jd‘12 j ! n

E  d‘ ¼0:

ð4:9:11Þ

c

Let us now consider the left part of Eq. (4.9.11). The contributions from the vertical sides of loop “c” to the integral approach zero, because of Eq. (4.9.7) and the differential length vector Eq. (4.9.2), is: d‘ ¼ d‘12 þ d‘34 :

ð4:9:12Þ

Since parts of the loop “c”, i.e. jd‘12 j and jd‘34 j are very small values but finite values, the sum of the contributions from the horizontal sides becomes: I lim d!0 jd‘12 j ! n

E  d‘ ¼E1  ð^xjd‘12 jÞ þ E2  ð^xjd‘34 jÞ ¼ ðE1  E2 Þ  ð^ xjd‘12 jÞ; c

ð4:9:13Þ ðE1  E2 Þ  ð^xjd‘12 jÞ ¼ ðE t1  E t2 Þjd‘12 j; I lim E  d‘ ¼ ðE t1  E t2 Þjd‘12 j; d!0 c jd‘12 j ! n

ð4:9:14Þ ð4:9:15Þ

where E1 is the electric field in the dielectric 1 and E2 is the electric field in the dielectric 2, where d‘ is the differential length vector, n ! is a very small but finite value. Equation (4.9.15) substitutes into the Eq. (4.9.11): ðEt1  E t2 Þjd‘12 j ¼ 0;

ð4:9:16Þ

E t1 and E t2 are tangential components of electric field vector in the dielectric 1 and dielectric 2. The tangential component of the vector of electric field when crossing the boundary of two dielectrics is continuous. We can write from (4.9.16) in the scalar form:

4.9 Boundary Conditions for Et …

225

E t1 ¼ Et2 ðV=mÞ:

ð4:9:17Þ

Equation (4.9.17) states that the tangential components of the electric field are continuous across the interface between two dielectrics (Fig. 4.13). As we will see later the formed surface charges on the plane between two dielectrics change only the normal component of E, while the tangential component remains constant. Now let us write the boundary condition (4.9.17) in the vector form. We can decompose the electric field intensity E into two orthogonal components (Fig. 4.14): ^  ðE  n ^Þ þ n ^ðn ^  EÞ; E ¼ Et þ En ¼ n

ð4:9:18Þ

^  ðE  n ^Þ; Et ¼ n

ð4:9:19Þ

^ ðn ^  EÞ; En ¼ n

ð4:9:20Þ

Et is the tangential component of the electric field E, En is the normal component of the electric field E. ^1 to the interface which goes from medium 1 to medium 2, i.e. in The normal n direction ^y (Fig. 4.14) because in this way we approach the interface point “A” ^2 has the direction ^ from the medium 1. Also the normal n y because we approach the interface point “A” from the media 2 along the coordinate y. We can write Eq. (4.9.17), using Eq. (4.9.19) and Figs. 4.13 and 4.14, in the vector form: ^1  ðE1  n ^1 Þ  n ^2  ðE2  n ^2 Þ ¼ 0; n

ð4:9:21Þ

^1  ðE1  n ^1 Þ  ð^ n n1 Þ  ðE2  ð^ n1 ÞÞ ¼ 0;

ð4:9:22Þ

Fig. 4.13 Electric fields at an interface of two dielectrics in terms of their tangential and normal components

E1

Dielectric 1 1

En1 Ѳ1

point A

Interface

Et2=Et Et1=Et Ѳ2

Dielectric 2 2, 2> 1

E2

En2

226

4 Maxwell’s Equations and Boundary Conditions

Fig. 4.14 A pillbox, Evectors and unit vectors to the interface of dielectric medium 1 and dielectric medium 2

y

Medium 1 1

E1

,μ1

n2

En1 x

En2

δ Et1 Et2

point A

E2 δ

E2 n1

Interface Medium 2

,μ2

2

^1  ðE1  n ^1 Þ  n ^1  ðE2  n ^1 Þ ¼ 0; n

ð4:9:23Þ

^ 1 Þ  ðE 2  n ^1 Þ ¼ ðE1  E2 Þ  n ^1 ¼ 0: ðE1  n

ð4:9:24Þ

Equation (4.9.24) taking account Eq. (1.2.56) can be written: ^1  E2 Þ ¼ 0; ^1  E1 Þ þ ðn ðn

ð4:9:25Þ

^1  ðE2  E1 Þ ¼ 0; n

ð4:9:26Þ

^1  ðE2  E1 Þ ¼ n ^2  ðE1  E2 Þ ¼ ðE1  E2 Þ  n ^1 ¼ 0; n

ð4:9:27Þ

^2 ¼ ^y are the unit vectors that is shown in Fig. 4.14, the electric ^1 ¼ ^y, n here n field vector E1 is in media 1 and E2 is in media 2. Directions of the E-field vectors from Eqs. (4.9.18–4.9.20) are shown in Fig. 4.15. The electric flux density D is related to the applied electric field E in a simple (linear homogeneous isotropic) dielectric medium by Eq. (3.6.4) and for two dielectric media we can write as follows: D1 ¼ e0 er1 E1 ¼ e1 E1 ¼ e1 ðEt1 þ En1 Þ;

ð4:9:28Þ

D1 ¼ Dt1 þ Dn1 ;

ð4:9:29Þ

D2 ¼ e0 er2 E2 ¼ e2 E2 ¼ e2 ðEt2 þ En2 Þ;

ð4:9:30Þ

4.9 Boundary Conditions for Et … Fig. 4.15 Field E1 , Et1 , En1 ^1 , directions and unit vectors n ^2 n

227

^ 1(n ^ 1•E1) En1=n ^1 E1 n

^ 1 E1 n ^1 Et1=n ^1 n

E1

^1 n

^n2 ^1 n

Interface

the back wall of the cube

Medium 1

,μ1

1

Medium 2

,μ2

2

D2 ¼ Dt2 þ Dn2 ;

ð4:9:31Þ

Where the absolute dielectric permittivity [Eq. (3.6.5)] of the first medium e1 ¼ e0 er1 and the second medium e2 ¼ e0 er2 . From Eqs. (4.9.28) and (4.9.30), we can write: E1 ¼

D1 ; e1

E2 ¼

D2 : e2

ð4:9:32Þ

On the basis of the boundary condition Eqs. (4.9.17) and (4.9.32), we can write: Dt1 Dt2 ¼ : er1 er2

ð4:9:33Þ

The relative permittivity er1 and er2 or the absolute permittivity e1 and e2 can be used in the denominator of Eq. (4.9.33), and we can write: Dt1 e1 ¼ : Dt2 e2

ð4:9:34Þ

Using the vector form Eq. (4.9.27), we can write:  ^1  n

D2 D1  e2 e1

 ^2  ¼n

  D1 D2  ¼ 0; e1 e2

^2 ¼ ^y is the unit vector in Fig. 4.15. ^1 ¼ ^y, n where n

ð4:9:35Þ

228

4 Maxwell’s Equations and Boundary Conditions

From Figs. 4.14 and 4.15, we can write: tan h1 ¼

E t1 ; En1

ð4:9:36Þ

tan h2 ¼

E t2 : En2

ð4:9:37Þ

We would like to remind you that in a simple dielectric media the directions of vectors E and D coincide (Fig. 3.38).

4.10

Boundary Conditions for Ht and Bt Components

Applying Ampere’s circular law described by Eq. (4.3.18) of the Maxwell’s equations in Large-Scale (integral) Form: I

Z H  d‘ ¼

c

J  ds þ S

@ @t

Z D  ds:

ð4:10:1Þ

S

Let us take a look at the left side of Eq. (4.10.1). Since parts of the loop “c” (Fig. 4.12), i.e. modules of differential length vectors (jd‘12 j and jd‘34 j are very small values, a selected estimate value n ! is a very small but finite one, the sum of the contributions from the horizontal sides, taking into account Eqs. (4.9.3) and (4.9.4), the left term of Eq. (4.10.1) becomes: I lim H  d‘ ¼ H1  ð^xjd‘12 jÞ þ H2  ð^ xjd‘34 jÞ d!0 jd‘12 j ! n

c

ð4:10:2Þ ¼ ðH1  H2 Þ  ð^xjd‘12 jÞ ¼ ðH t1  H t2 Þjd‘12 j:

So we can write from Eq. (4.10.2): I lim H  d‘ ¼ ðH t1  H t2 Þjd‘12 j: d!0 c jd‘12 j ! n

ð4:10:3Þ

The first term on the right-hand side of Eq. (4.10.1) does not equal zero. The first term is a surface current Js which is a finite one despite the fact that S is infinitely small area. This means that the current Js can exist on the interface (boundary surface) (Fig. 4.12), The surface current Js is distributed on the area S ¼ jd‘12 jjd‘23 j ¼ 2djd‘12 j.

4.10

Boundary Conditions for Ht and

229

The surface current can be written: Js ¼ J  d

has finite value at

d ! 0; jd‘12 j ! n J!1

ð4:10:4Þ

The closed loop d‘ ¼ d‘12 þ d‘23 þ d‘34 þ d‘41 [Eq. (4.9.2)] at d ! 0 (here d is a very small value and it is significantly smaller compared to the sum of the horizontal sides), d‘ ¼ d‘12 þ d‘34 , a selected estimate value n is a very small but finite one, J s is the surface current density between the two media [due to the flow of free electric charges only (Sect. 3.13 and Eq. (3.14.1)], which is not coming from polarization of the media). We can write: Z lim ð4:10:5Þ J  ds ¼ J s jd‘12 j: d!0 S jd‘12 j ! n The second term on the right-hand side of Eq. (4.10.1), which is the displacement current Jd (Sect. 3.12) decreases to zero, since the area S is infinitely small and Jd is a volume current. lim d!0 jd‘12 j ! n

@ @t

Z D  ds¼ 0:

ð4:10:6Þ

S

The volume current Jd equals zero because in this problem regarding the boundary conditions the volume reaches zero. From Eq. (4.10.1): I lim

d!0 jd‘12 j!n c

Z H  d‘ ¼

lim

d!0 jd‘12 j!n S

J  ds;

ð4:10:7Þ

Where d‘ is the differential length vector, a selected estimate value n ! is a very small but finite one, ds is an infinitesimal vector element of a surface S (see Fig. 4.2). We put Eqs. (4.10.3) and (4.10.5) in Eq. (4.10.7) and write in the scalar form: ðH t1  H t2 Þjd‘12 j ¼ J s jd‘12 j;

ð4:10:8Þ

H t1  H t2 ¼ J s :

ð4:10:9Þ

The tangential components of the magnetic field and the surface current are depicted in Fig. 4.16. Magnetic field components Ht1 , Ht2 are tangential to the interface (boundary) separating two environments (media). Statement: At any point on the interface the components of H1 and H2 tangential to the interface are discontinuous on the amount equal to the surface current density at that point.

230

4 Maxwell’s Equations and Boundary Conditions

(a) ^ 1(n ^ 1•H1) Hn1=n ^1 H1 n

Js

^ 1 H1 n ^1 Ht1=n

n^1

the back wall of the cube

H1

(b)

Js ^1 n

Medium 1

,μ1

^2 n

1

^1 n

Medium 2

,μ2

Interface

δ δ

2

|d 12| Js

2

Ht1 3

Interface

^1 n

1

4

Medium 1

,μ1

1

Medium 2

,μ2

2

^1 , n ^2 . b Direction of surface current Fig. 4.16 a Vectors H1 , Ht1 , Hn1 and unit vectors n ^1 Ht1

Js ¼ ½n

Magnetic field expression is: ^  ðH  n ^Þ þ n ^ ðn ^  HÞ; H ¼ Ht þ Hn ¼ n

ð4:10:10Þ

Where the tangential and normal magnetic field components are: ^  ðH  n ^Þ; Ht ¼ n

ð4:10:11Þ

^ ðn ^  HÞ; Hn ¼ n

ð4:10:12Þ

Ht is the tangential component of the magnetic field H, Hn is the normal (perpendicular) component of the magnetic field H. We can write Eq. (4.10.9) in the vector form: ^1  ðH1  n ^1 Þ  n ^ 2  ðH 2  n ^ 2 Þ ¼ Js  n ^2 ; n ð4:10:13Þ ^ 1  ðH 1  n ^1 Þ  n ^ 1  ðH 2  n ^1 Þ ¼ Js  n ^1 ; n

ð4:10:14Þ

^1 Þ  n ^ 1 þ ðH 2  n ^1 Þ  n ^1 ¼ Js  n ^1 ;  ðH 1  n

ð4:10:15Þ

^ 1 Þ þ ðH 2  n ^1 Þ ¼ Js ; ðH1  n

ð4:10:16Þ

^1  H1 Þ  ðn ^1  H2 Þ ¼ Js ; ðn

ð4:10:17Þ

^1  ðH1  H2 Þ ¼ Js ; n

ð4:10:18Þ

^ 1  ð H 2  H 1 Þ ¼ Js : n

ð4:10:19Þ

4.10

Boundary Conditions for Ht and

231

We can write from Eq. (4.10.13): ^2 ; Ht1  Ht2 ¼ Js  n

ð4:10:20Þ

^1 ; Ht1  Ht2 ¼ Js  n

ð4:10:21Þ

or we can write also:

^1 is directed from media 1 to media 2, n ^2 is directed from Where the unit vector n ^1 ¼ ^ media 2 to media 1, and n n2 ¼ ^y vectors are shown in Fig. 4.16a.

4.11

Boundary Conditions for Dn and En Components

The value of D is an electric flux density (electric displacement, electric induction, Sect. 3.6). We consider the interface between two dissimilar (different, unlike) dielectrics with the absolute permittivities e1 ¼ e0 er1 and e2 . There is a rectangular pillbox at the interface of these dielectrics (Fig. 4.17). We will use the Gauss’ law described by Eq. (4.3.19) of the Maxwell’s equations in large-scale form: I

Z D  ds ¼ qfree ¼

ðGauss's lawÞ:

qfree dv

ð4:11:1Þ

V

S

We can write on the base of Eq. (3.6.2) q ¼ qfree þ qbound ¼ qfree  r  P. The volume density qfree can contain the extraneous charges from the external sources and/or free charges arising from the conductivity of medium (Sect. 3.14.1). Value Fig. 4.17 Notations to derive the boundary conditions for normal components of the electric flux density

y d

41

n=n

4

d 2

3

Medium 1

,μ1

1

ΔS1234

x

1

8

d

n 5

Medium 2

,μ2

2

V

n

δ

2

12

δ n=n

1

6

ΔS5678

7

n

232

4 Maxwell’s Equations and Boundary Conditions

qbound is the volume density of bound (polarization) charges (Sect. 3.6). We remind here that r  D ¼qfree , r  E ¼ ðqbound þ qext Þ=e0 : Figure 4.17 shows a loop d‘ ¼ d‘12 þ d‘23 þ d‘34 þ d‘41 with its module of differential length vectors jd‘j ¼ 2ðjd‘12 j þ jd‘41 jÞ. The top and bottom pillbox surfaces are Ds1234 ¼ jd‘12 jjd‘41 j. The designations used here are: the volume between points 12345678 = 1–8, surface 1234 = 1–4, and so on. The entire surface of the pillbox is S ¼ Ds14 þ Ds58 þ 2  ð2dÞjd‘12 j þ 2  ð2dÞjd‘41 j, the entire volume of the pillbox 1 – 8 is V ¼ d‘12  d‘41  2d, at d ! 0 volume V is an infinite small and S is infinitesimal surface. Applying Gauss’s theorem (Eq. 1.3.29) we have: H R ^ds. ðr  DÞdv ¼ D  n V

S

At the limit d ! 0, the lateral surfaces of the pillbox such as Ds2673 (Fig. 4.17) tends to reach zero and the Gauss’ law can be written: I Z ^ds ¼ lim Dn qfree dv: ð4:11:2Þ lim d!0

d!0

V

S

^¼n ^2 to the top Ds14 surface of the pillbox is the direction ^ y and The normal n ^ n¼^ n1 to the bottom Ds58 surface is ^y (Fig. 4.17). The surface of the pillbox at d ! 0 is S ¼ Ds14 þ Ds58 . The surface density of free charges is lim qfree d ¼ rs;free . Surface charge density is the quantity of charge per unit area d!0

[see Eq. (2.6.8)]. We can write at d ! 0: Dn1 Ds14  Dn2 Ds58 ¼ rs;free Ds14 ;

ð4:11:3Þ

Dn1  Dn2 ¼ rs;free :

ð4:11:4Þ

We can also write as follows: e1 En1  e2 En2 ¼ rs;free :

ð4:11:5Þ

Where rs;free is the surface charge between the media (free and extraneous charges only, these charges are unbounded ones, i.e. not coming from polarization of the media). Value rs;total ¼ rs;bound þ rs;free is the total surface charge. We can also write that the normal component of E has a step of total surface charge (a discontinuous jump) on the interface surface and E n2  E n1 ¼ rs;total . At any point A on the boundary, the components of D1 and D2 normal to the boundary are discontinuous by the amount of the surface charge density of the point A. The normal component of D has a step of surface charge on the interface surface. In the vector form:

4.11

Boundary Conditions for Dn and …

233

^2  ðD1  D2 Þ ¼ rs;free n ^1  ðD2  D1 Þ ¼ rs;free n

ðC=m2 Þ; ðC=m2 Þ:

ð4:11:6Þ

We would like to remind you [Eqs. (3.6.17–3.6.19)] that the polarization vector P depends on bound (polarization) charges rs ¼ rs;bound , electric flux density vector D depends on free rs;free (or/and extraneous rs;ext ) charges, whereas the electric field E depends on all charges, i.e. rs;total . We assume that the electric field E1 incident at the interface of the dielectrics from medium 1 with the permittivity e1 , and we choose the direction of normal to ^¼n ^1 against the ordinate y, i.e. from medium 1 to medium 2. The the interface n ^¼n ^2 directed along the coordinate y, i.e. from medium 2 normal to the interface n to medium 1 (Fig. 4.17). If there are no free surface charges on the interface, the normal component of D is continuous: Dn2  Dn1 ¼ 0

at

rs ¼ 0;

ð4:11:7Þ

and Dn1 ¼ Dn2

at

rs ¼ 0:

ð4:11:8Þ

For the normal component of E: e0 er2 E n2  e0 er1 E n1 ¼ 0

at

rs ¼ 0:

ð4:11:9Þ

4.11.1 Explanation About Surface Charges on Interface of Dielectrics We consider now what happened at the interface between two dielectrics with different permittivities e1 ¼ e0 er1 and e2 . We can look at two dielectric plates touched each other as a dielectric structure immersed in the external static electric field Eexternal (Fig. 4.18). We can assume that the external electric field is created between two conductive (metal) plates of a capacitor as we considered in Sect. 3.9. We see in Fig. 3.11 that positive bounded (polarization) charges are shifted in the direction of the field Eexternal and negative bounded charges are shifted in the opposite direction. Hence, the polarization of dielectric plates occurs. According to Eq. (3.6.6), the polarization will be stronger in case of greater the permittivity of medium. If the permittivity of medium 2 is larger than the permittivity of medium 1, polarization P2 and the bound surface charge rs2;bound ¼ rs2 from medium 2 are larger as well. Since positive bounded charges of dielectric media shift in the direction of the electric field, there are positive bound surface charges rs1 at the interface from the side of medium 1 and the negative bound surface charges rs2 at the same interface

234

4 Maxwell’s Equations and Boundary Conditions

Fig. 4.18 Surface charge rs \0 at the interface of two dielectrics caused by the polarization of media

2

>

1

Medium 1

P1

1

Ei

Eexternal

Interface

τ^

^ n Medium 2

│Ϭs1│

E bound1

E bound2

P2

Ei

2

│Ϭs1│ │Ϭs2│ │Ϭs2│

Ϭs = (│Ϭs1│ │Ϭs2│) < 0 from medium 2. We can write at e2 [ e1 that P2 [ P1 , jrs2 j [ jrs1 j and the total surface charge on the interface of two media is rs ¼ ðrs1 þ rs2 Þ\0. We speak now only about bound (polarization) charges at the interface. It is important to underline that the directions vectors Eexternal , P2 and resulting electric field inside of dielectric Eres coincide (see Sect. 3.9). The electric field Ebound which is created by bound (polarization) charges is directed opposite to the field Eexternal . The direction of an electric field at the interface Ei is dependent on the sign of charges at the interface. In the case e2 [ e1 it is shown in Fig. 4.18. When e2 \e1 , P2 \P1 , jrs2 j\jrs1 j, the surface charge is positive rs ¼ ðrs1 þ rs2 Þ ¼ jrs1 j  jrs2 j [ 0 at the interface of two dielectrics (see Fig. 4.19b). In the case when e2 \e1 on the interface noncompensated charges remain positive. Figure 4.19 shows the sign of surface charges depending on the ratio of values e1 and e2 . We see that the direction of electric field Ei at the interface is different depending on the ratio of media permittivities e2 [ e1 or e2 \e1 .

Interface with the surface density charge Ϭs (b)

Medium 1

Ei

2

1

Medium 2 2

>

1

2

<

Ϭs ^ n

Ei

Interface

Medium 1

1

Ei

1

Eexternal

Eexternal

(a)

Ei Medium 2 2

Fig. 4.19 Surface charge rs ¼ rs1 þ rs2 at the interface of two dielectrics when the absolute permittivities a e2 [ e1 and b e2 \e1

4.11

Boundary Conditions for Dn and …

235

The resulting electric field Eres into a dielectric is always directed similarly to Eexternal , the polarization field Ebound (also known as Epolariz ) is always directed opposite to the field Eexternal (Sect. 3.9), and Eres ¼ Eext  E bound , the direction of electric field Ei at the interface, depends of the surface charge sign (Fig. 4.19). We would like to draw your attention to the fact that we considered the bound surface charge rs ¼ rs;bound , which occurs only by the polarization effect of medium. At the interface there can be also extraneous charges rs;ext arising from the external sources and/or free charges rs;free arising from the conductivity of medium (Sect. 3.6). These additional electric charges rs;ext or/and rs;free can create a charged interface of any sign. We would like to remind [Eqs. (3.6.17 – 3.6.19)] you that the polarization vector P depends on rs ¼ rs;bound , electric flux density vector D depends on rs;ext or/and rs;free , and electric field E depends on all charges.

4.11.2 Boundary Conditions for Polarization Vectors Now, we get the boundary conditions for the polarization vector. The flow of polarization vector P through an arbitrary closed surface S is equal to the excess bounded charge of volume dielectric V covered by the surface S and taken with the reversed sign: I

I P  ds¼ qbound ¼

S

ðqbound Þ dv:

ð4:11:10Þ

V

As follows from Eq. (3.6.17), the polarization of the dielectric medium produces an effective charge which can be interpreted as a macroscopic bound charge r  P ¼ qbound . There are two contributions to the bound charge: volume (bulk) and surface charges. Since we satisfy boundary conditions at d ! 0, volume charge is equal to zero and only rs surface charge is left. In the electrostatic situation which is at the satisfaction of boundary condition E has a curl of zero r  E¼ 0 (Sect. 2.9.2 and Table 2.4), and from Eq. (3.6.4) it follows that: r  D ¼ r  P:

ð4:11:11Þ

We see that the electric flux density D is expressed through the polarization vector P. The surface charge density of bounded charges rs;bound on the interface between two dielectrics is:

236

4 Maxwell’s Equations and Boundary Conditions

^  ðP2  P1 Þ ¼ rs;bound n



 C=m2 ;

ð4:11:12Þ

^ is the unit normal vector from medium 1 to medium 2, P is the polarization n vector. The normal component of vector P experiences a jump which equals to the density of the surface bounded charge: Pn1  Pn2 ¼ rs;bound : ð4:11:13Þ On the border (interface) of two dielectrics with different dielectric permittivity e1 , e2 and in the presence of an external electric field E polarizing charges of the different signs and different surface charge densities þ rbound1 and rbound2 arise.

4.12

Boundary Conditions for Bn and Hn Components

Using Eq. (4.3.20), we can receive the boundary conditions for the magnetic field: I B  ds ¼ 0:

ð4:12:1Þ

S

In the limit, when the side surface 2d (Fig. 4.17) tends to zero: I B  nds ¼ 0:

lim

d!0

ð4:12:2Þ

S

^ to the top Ds1234 and bottom Ds5678 surfaces of the box have The normal n opposite signs. The whole surface now is S ¼ Ds1234 þ Ds5678 at d ! 0 and: Bn1 Ds1234  Bn2 Ds5678 ¼ 0; Bn1  Bn2 ¼ 0

ðTÞ:

ð4:12:3Þ ð4:12:4Þ

At any point on the boundary, the components of B1 and B2 normal to the boundary are equal: ^  ðB2  B1 Þ ¼ 0; n ^ is the unit normal vector from medium 1 to medium 2. n

ð4:12:5Þ

4.12

Boundary Conditions for Bn and

237

The same as normal components of magnetic flux density is here: Bn1 ¼ Bn2 :

ð4:12:6Þ

For the magnetic field, using Eq. (3.27.5), we can write: ^  ðl2 H2  l1 H1 Þ ¼ 0; n

ð4:12:7Þ

l1 Hn1 ¼ l2 Hn2 ;

ð4:12:8Þ

For scalar components of magnetic field strength: H n1 ¼

l2 H n2 ; l1

ð4:12:9Þ

where l1 and l2 are the absolute permeabilities of media (Eq. 3.19.12).

4.13

Deformation of E and D Vectors at a Dielectric Sphere Boundary

The illustration of importance of boundary conditions can be the electric field and electric flux density perturbation by a dielectric sphere of radius R placed in some dielectric liquid with the absolute permittivity e2 ¼ e0 er2 (Fig. 4.20), er2 is the relative permittivity. A sphere is made of a dielectric material with the absolute permittivity e1 ¼ e0 er1 , er1 is the relative permittivity of material of sphere. The sphere is immersed into an applied (external) uniform (always the same, constant, not changing or varying) field E0 ¼ Eexternal . The origin of the spherical coordinate system is taken at the center of the dielectric sphere. The electric field inside the sphere is aligned along the applied field E0 . We want to know how the electric field is modified by the dielectric sphere. The task can be formulated in this way: determine the electric fields inside and outside of the dielectric sphere and the density of surface charges on the sphere (Fig. 4.20).

r

+ +

Ѳ

A

E0 R

+

2

+ +

1

x

+ + +

y

+

Fig. 4.20 Dielectric sphere is immersed into the external electrostatic field

z

Ϭs

238

4 Maxwell’s Equations and Boundary Conditions

In Fig. 4.20, the point “A” is arbitrary point, r is the radial distance, H is the polar angle, R is the radius of the dielectric sphere, rs ¼ rs;bound are the surface bound charges. The electrostatic problem requires the solving of the Laplace equation under certain boundary conditions [Eqs. (4.9.17) and (4.11.4)] at interference with radial distance r ¼ R. In general boundary, conditions for sphere are: Et1 ¼ Et2 ;

ð4:13:1Þ

Dn2  Dn1 ¼ rs;free ;

ð4:13:2Þ

where Dn is component of electric flux density perpendicular to the sphere surface. Discontinuity of components Dn at the interface occurs on interface due to extraneous and/or free charges (see further Sect. 3.14.1). The term “discontinuity” means “a break in continuity”. E t is the tangential component of the electric field which is parallel to the surface. Tangential component E t is always continuous. We assume in the problem under consideration that there are no extraneous or free charges inside and outside of the sphere. In this case, we can replace Eq. (4.13.2) with Eq. (4.11.9) and, taking account of D ¼ e0 er E, we can write: er2 E n2  er1 E n1 ¼ 0:

ð4:13:2aÞ

The external uniform electric field E0 induces the bounded (polarization) charges rs;bound at the dielectric sphere interface. Therefore, there are only bounded electric charges because our dielectric material of sphere has no free charges and we have not placed free electric charges on the dielectric sphere, so for this reason the surface density of free charges rs;free ¼ 0 in Eq. (4.13.2). For the electrostatic problem (Sect. 2.9.2 and Table 2.4), we know that r  E¼ 0, the electrostatic potential u can be defined by E ¼ ru that in our problem obeys the Laplace equation. This task is easier to solve through the electrostatic potential (Sect. 2.7). The solution is different for the area inside and outside the dielectric sphere. From the axial (azimuthal) symmetry of the geometry, we can take the solution of Laplace equation in the form of Legendre polynomial expansion. The potential of an external uniform (constant) electric field E0 at any point in space outside the dielectric sphere is as follows: u0 ¼ E0 r ¼ E0 r cos H ¼ E 0 rP1 ðcos HÞ:

ð4:13:3Þ

The Legendre polynomials are: P0 ðcos hÞ ¼ 1;

P1 ðcos HÞ ¼ cos H;

  P2 ðcos HÞ ¼ 3 cos2 h  1 =2 ; etc:

ð4:13:4Þ ð4:13:5Þ

4.13

Deformation of E and D Vectors …

239

Each Legendre polynomial Pn ðcos hÞ is nth degree polynomial of his argument. It may be expressed using Rodrigues’ formula: n

Pn ðcos hÞ ¼

1 dn ðcos2 h  1Þ : n!2n dðcos hÞn

ð4:13:6Þ

We will consider the solution of this problem in detail. Laplace’s equation in spherical coordinates The Laplace equation (Sect. 2.13): r2 u ¼ 0;

ð4:13:7Þ

where u is the electrostatic potential (see Sect. 2.7) which is a scalar [Eq. (2.7.3)]. The potential inside and outside the sphere is described by the Laplace equation. The Laplace operator (Laplacian) in spherical coordinates is: D ¼ r2 ¼

    1 @ 1 @ @ 1 @2 2 @ r sin H ; ð4:13:8Þ þ þ 2 2 2 r @r @r r sin H @H @H r sin H @/2

where r is the radial distance, H is the polar angle, and / is the azimuthal angle, respectively. In order to solve Laplace’s equation in spherical coordinates may be used separation of variables as follows: uðr; H; /Þ ¼ Rðr ÞWðHÞUð/Þ:

ð4:13:9Þ

Since the sphere has azimuthal symmetry (symmetric about the z-axis) this means that the solution Eq. (4.13.8) does not vary in the azimuthal angle / direction and @u=@/ ¼ 0, i.e. the electric field intensity and potential u [Eq. (4.13.7)] will depend only on two coordinates R and H. uðr; HÞ ¼ u1 ðr Þu2 ðHÞ ¼ Rðr ÞWðHÞ:

ð4:13:10Þ

Where Rðr Þ is the function only on r, Rðr Þ ¼ u1 ðr Þ is a part of potential uðr; HÞ, WðHÞ is function only of angle H, WðHÞ ¼ u2 ðHÞ a part of potential uðr; HÞ and their derivatives are R0 ðr Þ ¼ @Rðr Þ=@r, W0 ðHÞ ¼ @WðHÞ=@H . This means that @u ¼ R0 ðr ÞWðHÞ; @r

ð4:13:11Þ

@u ¼ Rðr ÞW0 ðHÞ: @H

ð4:13:12Þ

240

4 Maxwell’s Equations and Boundary Conditions

In this case, the Laplace’s equation for our problem will be simplified:     1 @ 1 @ @uðr; HÞ 2 @uðr; HÞ r sin H þ ¼ 0; r 2 @r @r r 2 sin H @H @H      1 @ 1 @ @ 2 @ r sin H þ uðr; HÞ ¼ 0; r 2 @r @r r 2 sin H @H @H

ð4:13:13Þ ð4:13:13aÞ

where uðr; HÞ ¼ u1 ðr Þu2 ðHÞ from Eq. (4.13.10). From substitution of Eqs. (4.13.11) and (4.13.12) in Eq. (4.13.13), we get: r2 u ¼

WðHÞ @  2 0  Rðr Þ @ r R ðr Þ þ 2 ðsin HW0 ðHÞÞ ¼ 0: 2 r @r r sin H @H

ð4:13:14Þ

If we multiply each term in Eq. (4.13.14) by r 2 and then divide each term by terms of Eq. (4.13.9), we obtain: 1 d 2 0  1 d r R ðr Þ þ ðsin HW0 ðHÞÞ ¼ 0: Rðr Þ dr WðHÞ sin H dH

ð4:13:15Þ

Please pay attention that derivatives in Eq. (4.13.15) are no longer partial derivatives. This is because the method of separation variables has produced two terms one of which is solely a function Rðr Þ of variable coordinate r and the other is solely a function WðHÞ of variable angle H (see Fig. 4.20). Equation (4.13.15) can be written as follows: 1 d 2 0  1 d r R ðr Þ ¼  ðsin HW0 ðHÞÞ: Rðr Þ dr WðHÞ sin H dH

ð4:13:15aÞ

This equality (4.13.15a) should be true for any r and H values, which is only possible if both left and right parts of the equation are equal to a certain constant C, that is: 1 d 2 0  r R ðr Þ ¼ C; ð4:13:15bÞ Rðr Þ dr 1 d ðsin HW0 ðHÞÞ ¼ C: WðHÞ sin H dH

ð4:13:15cÞ

Separating Variables Equation (4.13.15) allows us to separate Laplace’s equation into two separate ordinary differential equations; one is a function of variable r and the other a function of variable H. We choose C ¼ k2L and Eqs. (4.13.15b) and (4.13.15c) can be written:

4.13

Deformation of E and D Vectors …

241

1 d 2 0  r R ðr Þ ¼ k2L ; Rðr Þ dr

ð4:13:16Þ

1 d ðsin HW0 ðHÞÞ ¼ k2L ; WðHÞ sin H dH

ð4:13:17Þ

where a separation constant is k2L ¼ LðL þ 1Þ and index L ¼ 0; 1; 2. . .1 is an integer number. We have now dissimilar differential equations which we are going to solve. Solving Eq. (4.13.17) by the Frobenius method (series expansion), we get that the regular solution in the vicinity of the coordinate origin point converges at k2L ¼ LðL þ 1Þ. The separation constant k2L in the form “LðL þ 1Þ” give us a well-known differential equation whose solution we already know. Notice that there is a plus sign before the separation constant k2L in Eq. (4.13.16) and there is a minus sign before the separation constant in Eq. (4.13.17). The opposite signs must be accepted so that the sum of Eqs. (4.13.16) and (4.13.17) would be equal to zero as required by Laplace’s equation.

4.13.1 Solution for the Radial Part of Differential Equation We consider the radial Eq. (4.13.16). We multiply through by Rðr Þ and expand the derivate to find the equation: r2

d2 R dR þ 2r  LðL þ 1ÞR ¼ 0: dr dr

ð4:13:18Þ

The last equation is a second-order ordinary differential equation. The solution of Eq. (4.13.18) becomes as follows: R ¼ ArL þ Br ðL þ 1Þ ;

ð4:13:19Þ

where A and B are constants which will be determined once we apply specific boundary conditions, Rðr Þ ¼ u1 ðr Þ from Eq. (4.13.10).

4.13.2 Solution for the Angular Part of Differential Equation We solve Eq. (4.13.17) by multiplying through by WðHÞ and expanding the derivative to obtain:

242

4 Maxwell’s Equations and Boundary Conditions

d2 WðHÞ cos H dWðHÞ þ LðL þ 1ÞWðHÞ ¼ 0: þ sin H dH dH2

ð4:13:20Þ

Equation (4.13.20) is the Legendre differential equation which presents the second-order ordinary differential equation. The Legendre differential equation has two linearly independent solutions. We will introduce a new symbol for the variable f ¼ cos H. The first solution PL ðfÞ is regular at 1 f 1 it is called a Legendre function of the first kind, while the second solution QL ðfÞ is singular at f ¼ 1 and is called a Legendre function of the second kind. If L is an integer number, the function of the first kind reduces to a polynomial known as the Legendre polynomial. The Legendre differential equation can be solved by using the Frobenius method. The general solution for an integer L is then given by the Legendre polynomials. In mathematics, Legendre functions are solutions to Legendre’s differential equation. We see that the solution of Legendre differential Eq. (4.13.20) is just well-known Legendre polynomials and solution presents as follows: WðHÞ ¼ PL ðfÞ ¼ PL ðcos HÞ;

ð4:13:21Þ

where WðHÞ ¼ u2 ðHÞ from (4.13.10), see also Eqs. (4.13.3 – 4.13.6). Legendre polynomials create a complete system of orthogonal functions. Orthogonality of functions is important in solving EM problems. In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra. If you have an orthonormal system you can find the coefficients in the decomposition of function. An important property of the Legendre polynomials is that they are orthogonal on the interval 1 f 1. The orthogonality condition for Legendre polynomials is: Zp

Zþ 1 PL ðcos HÞ PN ðcos HÞ sin H dH ¼

PL ðfÞ PN ðfÞdf 1

0

¼

2 dLN ; 2N þ 1

ð4:13:22Þ

here dLN is the Kronecker delta, which is a function of two positive integer variables L and N. The function dLN equals 1 if the variables are equal to each other, i.e. L ¼ N ¼ 1, and equals 0 otherwise, i.e. L 6¼ N. For example, d11 ¼ 1 and d12 ¼ 0.

4.13.3 Constructing of the General Solution Previously, we separated Laplace’s equation into two ordinary differential Eqs. (4.13.16) and (4.13.17). The general solution to Laplace’s equation in

4.13

Deformation of E and D Vectors …

243

spherical coordinates will be constructed as a product of solutions [Eqs. (4.13.19) and (4.13.21)] in the form: uðr; HÞ ¼ u1 ðr Þu2 ðHÞ ¼ Rðr ÞWðHÞ:

ð4:13:23Þ

Using our experience with Laplace’s equation in Cartesian coordinates, we know that the full solution will be constructed as a sum of solutions (Eq. 4.13.23), i.e. the general solution of Laplace’s equation in spherical coordinates is: uðr; HÞ ¼

1

X

AL r L þ BL r ðL þ 1Þ PL ðcos HÞ:

ð4:13:24Þ

L¼0

Next step in the solution is using certain boundary conditions.

4.13.4 Satisfying of Boundary Condition Solutions in areas r\R and r [ R should be finite. Here the Cauchy boundary condition (Sect. 4.7) has to be satisfied, i.e. electrostatic potential u should be continuous on the surface. The boundary condition according to Eq. (4.13.1) is: uin jr¼R ¼ ujr¼R ;

ð4:13:25Þ

where uin and u ¼ u0 þ uout are the electrostatic potential inside and outside the dielectric sphere, respectively. uin gives the solution of Eq. (4.13.13a) inside the sphere where the singular point r ¼ 0 of the coordinate system center is and uout gives the solution of Eq. (4.13.13a) outside the sphere where r ! 1, u0 is described in Eq. (4.13.3) . uin and u are total potential uðr; HÞ inside and outside the sphere, respectively. In additional the normal component of the electric flux density should also be continuous on the boundary surface (interface) by Eq. (4.13.2a) :   @uin  @u er1 ¼ er2  : ð4:13:26Þ @r r¼R @r r¼R We denote the solution inside the sphere with the lower index “in”. In the center of coordinate, i.e. inside the sphere there is the point with coordinate r ¼ 0. We see from Eq. (4.13.24) that solution converges at r ¼ 0 when we set all coefficients BL ¼ 0. The application of this requirement is caused by the physical meaning of the solution to this problem because physical fields must be finite. The solution of Laplace Eq. (4.13.15) simplifies to:

244

4 Maxwell’s Equations and Boundary Conditions

uin ¼

1 X

AL r L PL ðcos HÞ:

ð4:13:27Þ

L¼0

We will consider the potential uðr; HÞ at the point A located from the outside of the sphere (Fig. 4.20). We denote the solution of Eq. (4.13.24) outside the sphere with the lower index “out”. We see from Eq. (4.13.27) that the solution converges at very large r (i.e. r ! 1) only if we set all coefficients AL ¼ 0. The solution of Laplace Eq. (4.13.13) for space around the sphere is: uout ¼

1 X

BL r L1 PL ðcos HÞ:

ð4:13:28Þ

L¼0

We find the amplitude coefficients AL and BL of the boundary conditions at r ¼ R: uin ¼ u0 þ uout jr¼R ; er1

ð4:13:29Þ

@ @ ðu0 þ uout Þ u ¼ er2 jr¼R : @r in @r

ð4:13:30Þ

The substitution of relations (4.13.3), (4.13.27) and (4.13.28) into Eq. (4.13.29) gives: 1 X

AL RL PL ðcos HÞ ¼ E 0 R P1 ðcos HÞ þ

L¼0

1 X

BL RL1 PL ðcos HÞ:

ð4:13:31Þ

L¼0

The substitution of relations (4.13.3), (4.13.27) and (4.13.28) into Eq. (4.13.30) leads to: er1

1 X

L AL R

L1

L¼0

PL ðcos HÞ ¼ er2 E0 P1 ðcos HÞ þ

1 X

! ðL þ 1ÞBL R

L2

PL ðcos HÞ :

L¼0

ð4:13:32Þ Equations (4.13.31) and (4.13.32) are multiplied by PN ðcos HÞ sin H dH and then integrated over the variable H interval from 0 to p. After completing the summation in Eq. (4.13.31) and using the orthogonality condition for Legendre polynomials Eq. (4.13.22) we get: AL RL

2 2 2 ¼ E0 R dL1 þ BL RL1 ; 2L þ 1 2L þ 1 2L þ 1

at L ¼ 0; 1; 2. . .1, dLN ¼ dL1 because here N ¼ 1.

ð4:13:33Þ

4.13

Deformation of E and D Vectors …

245

The summation in Eq. (4.13.32) and the use of the orthogonality condition for Legendre polynomials Eq. (4.13.22) leads to: er1 L AL RL1

  2 2 2 ¼ er2 E 0 dL1 þ ðL þ 1ÞBL RL2 ð4:13:34Þ 2L þ 1 2L þ 1 2L þ 1

at L ¼ 0; 1; 2. . .1. From the analysis of the equation system (4.13.33) and (4.13.34) we come to the conclusion that a non-trivial solution can only be provided at L ¼ 1, and then the numerical multiplier becomes 2=2L þ 1 ¼ 2=3. Hence, the system of Eqs. (4.13.33) and (4.13.34) is simplified and presented in the form: 2 2 2 A1 R ¼ E 0 R þ B1 R2 ; 3 3 3 2 2 2 er1 A1 ¼ er2 E0  2er2 B1 R3 : 3 3 3

ð4:13:35Þ

After reducing the numerical multiplier 2=3 we get: A1 R  B1 R2 ¼ E0 R; er1 A1 þ 2er2 B1 R3 ¼ er2 E 0 :

ð4:13:36Þ

We receive the system of linear equations (or linear system) which is a collection of two linear equation involving two variables. The solution of system Eqs. (4.13.36) is: Dsph

Dsph1

 R ¼  er1

  RE 0 ¼  er2 E0

Dsph2

  R ¼  er1

 ðer1 þ 2er2 Þ R2  2 þ er1 R2 ¼ ; 3  ¼ 2er2 R 2er2 R R2

ð4:13:37Þ

 3er2 E 0 R2  2  er2 E0 R2 ¼  ; ð4:13:38Þ 3  ¼ 2er2 E 0 R 2er2 R R2  RE 0  ¼ er2 E0 R þ er1 E0 R ¼ ðer1  er2 ÞE 0 R: er2 E0 

ð4:13:39Þ

For each variable A1 and B1 , the denominator is the determinant of the matrix of coefficients Dsph , while the numerator is the determinant of a matrix in which one column has been replaced with the vector of constant terms, i.e. Dsph1 and Dsph2 . The coefficient A1 is: A1 ¼

Dsph1 3er2 E 0 ¼ ; Dsph er1 þ 2er2

ð4:13:40Þ

246

4 Maxwell’s Equations and Boundary Conditions

B1 ¼

Dsph2 ðer1  er2 ÞE 0 R3 ¼ : Dsph er1 þ 2er2

ð4:13:41Þ

4.13.5 Potentials Inside and Outside the Sphere The potentials inside uin and outside uout the sphere that satisfy the Laplace’s Eq. (4.13.13) are: uin ¼ A1 rP1 ðcos HÞ ¼  uout ¼ B1 r 2 P1 ðcos HÞ ¼

3er2 E0 rP1 ðcos HÞ er1 þ 2er2

er1  er2 R3 E0 2 P1 ðcos HÞ er1 þ 2er2 r

at r R; at r R:

ð4:13:42Þ ð4:13:43Þ

The potential uðr; HÞ outside the sphere, adding the potential uout (solution of Eq. (4.13.18) outside the sphere) and the potential u0 of the external field E 0 (Fig. 4.20) we get: er1  er2 R3 E 0 2 P1 ðcos HÞ uðr; HÞ ¼ u ¼ uout þ u0 ¼ E0 rP1 ðcos HÞ þ er1 þ 2er2 r "  3 # er1  er2 R ¼ P1 ðcos HÞE0 r 1 er1 þ 2er2 r "   # er1  er2 R 3 ¼ P1 ðcos HÞE 0 r 1  at r R: er1 þ 2er2 r ð4:13:44Þ Because P1 ðcos HÞ ¼ cos H [Eq. (4.13.4)] we can express Eqs. (4.13.42) and (4.13.44) as follows: 3er2 E0 r cos H at r R; er1 þ 2er2 "   # er1  er2 R 3 u ¼ uout þ u0 ¼ E 0 r cos H 1  at r R: er1 þ 2er2 r uin ¼ 

ð4:13:45Þ ð4:13:46Þ

4.13

Deformation of E and D Vectors …

247

4.13.6 Electric Field Inside and Outside the Sphere and Bounded Surface Charges Intensity of the electric field inside the dielectric sphere is:   @ 1 @ ^ E1 ¼  ^r þ H u ; @r r @H in

ð4:13:47Þ

^ is the unit vectors in the spherical coordinate system. where ^r and H The substitution of Eq. (4.13.45) into Eq. (4.13.47) allows us to get: E1 ¼ ^r

3er2 ^ 3er2 E0 sin H E0 cos H  H er1 þ 2er2 er1 þ 2er2

at r R:

ð4:13:48Þ

The electric field outside the sphere at u ¼ uout þ u0 [Eq. (4.13.46)] can be expressed: " #    3 @ 1 @ er1  er2 R ^ E2 ¼  ^r þ H E0 cos H þ E 0 cos H u ¼ ^r 2 @r r @H r er1 þ 2er2 "  3 # e  e R r1 r2 ^ 1 þH E0 sin H er1 þ 2er2 r " # "  3  # er1  er2 R er1  er2 R 3 ^ E0 þ 1 E0 cos H þ H 1  ¼ ^r 2 E 0 sin H: r er1 þ 2er2 er1 þ 2er2 r ð4:13:49Þ ^ can be presented in terms of Cartesian units Spherical unit vectors ^r and H vectors ^x; ^y; ^z using the following transformations: ^r ¼ ^x sin H cos / þ y^ sin H sin / þ ^z cos H, ^ H ¼ ^x cos H cos / þ ^y cos H sin /  ^z sin H, then we can write the expressions: ^ sin H ¼ ð^x sin H cos / þ ^y sin H sin / þ ^z cos HÞ cos H ^r cos H  ð^x cos H cos / þ ^y cos H sin /  ^z sin /Þ sin H ¼ ^z: ð4:13:50Þ From Eq. (4.13.48) the electric field inside the dielectric sphere with the relative permittivity e1 is: E1 ¼

3er2 E0^z: er1 þ 2er2

ð4:13:51Þ

248

4 Maxwell’s Equations and Boundary Conditions

This electric field inside the dielectric sphere is parallel to the external field Eexternal ¼ E0 . The polarization field Ebound is opposite to the external field E0 . As we remember from Sect. 2.4, the positive charges shift in the direction of external electric field E0 and negative charges move in the opposite direction to E0 (Fig. 4.20). We can write the polarization field created by bounded charges of the sphere as follows:  Ebound ¼ E1  E0 ¼

     3er2 er2  er1 er2  er1  1 E 0^z ¼ E 0^z ¼ E0 : er1 þ 2er2 2er2 þ er1 2er2 þ er1 ð4:13:52Þ

The bounded charges are producing an internal electric field in sphere Ebound which is directed oppositely to the applied (external) field E0 . The field Ebound is reducing the field inside the sphere to its value. The bounded (polarization) surface charge is: ^ ¼ Pn1  Pn2 ¼ ½e0 ðer1  1ÞE1  e0 ðer2  1ÞE2 jr¼R rs;bound ¼ P  n

  ; 3er2 E0 ð^z ^rÞ  ðer2  1ÞðE2 rÞ  ¼ e0 ðer1  1Þ ðer1 þ 2er2 Þ r¼R

ð4:13:53Þ

where ð^z ^rÞ ¼ cos H and E0 ¼ Eexternal ¼ Ez . Finally the bounded (polarization) surface charge is: ^ rs;bound ¼ P  n

¼ e0 ðer1  1Þ

   3er2 er1  er2 E 0 cos H  ðer2  1Þ 2 þ 1 E0 cos H ðer1 þ 2er2 Þ er1 þ 2er2 1 ¼ e0 E0 cos H ½3er2 ðer1  1Þ  ðer2  1Þð2er1  2er2 þ er1 þ 2er2 Þ : er1 þ 2er2 ð4:13:54Þ rs;bound ¼ 3

e0 E 0 cos H : er1 þ 2er2

ð4:13:55Þ

The bound charges are producing an internal field directed oppositely to the external field E0 which is reducing the field inside the sphere to its value Eq. (4.13.51). The polarization vector into the dielectric sphere is: P¼3

e0 E 0 cos H E0 : er1 þ 2er2

ð4:13:56Þ

The dielectric sphere possesses a uniform polarization which is aimed along the direction of the applied field E0 ¼ Eexternal . The polarization of the dielectric

4.13

Deformation of E and D Vectors …

249

generates the bounded (polarization) charges (Sect. 3.9), In Fig. 4.20 surface bounded charges are indicated by rs . The directions of polarization vector P and electric field in sphere E1 coincide with E0 . Thus, one half of the sphere gets a positive charge from one side, and the other is negative from another side from polarization effect (Fig. 4.20).

4.13.7 Force Lines of Electric Field and Electric Flux Density Schematically behavior of force lines of strength of electric field and electric flux density outside and inside a dielectric sphere is shown in Figs. 4.21 and 4.22. The vectors DðrÞ and EðrÞ are different dependent on the position vector r (Sect. 1.2.6) and the original of the system coordinate is in the center of the sphere (Fig. 4.20). The dielectric sphere with the permittivity e1 is placed into the external uniform electrostatic field Eexternal ¼ E0 . The surrounding medium has the permittivity e2 . Since there are no extraneous or free charges inside and outside the sphere, the lines of D neither originate nor terminate at the interface (Fig. 4.21), and D—lines are continuous across the interface. The electric field lines are partially interrupted at the interface of sphere on the bound charges (Fig. 4.22). It is important to note that the electric field outside and inside the sphere obeys the correct boundary conditions, e.g. Equations (4.9.17) and (4.11.5).

(a) Eexternal

(b) Eexternal

> 2; D1>D2

< 2; D1 2; E1E2

2

2

+ + + +

1

1

+ + + + 2

2

Fig. 4.22 Schematic image of electric field intensity E1 inside the dielectric sphere with the permittivity e1 is as follows: a e1 [ e2 and b e2 [ e1 when an external uniform electric field Eexternal is distorted by the sphere

An influence of the dielectric sphere on the electric field is the largest in that dielectrics weaken (reduce the level, attenuate) the electric field strength in er times. According to the Coulomb’s law in a homogeneous dielectric material having the permittivity er the electric charges interact with each another er times less than in a vacuum [see Eq. (3.8.2)]. Thus, the electric field E in a homogeneous dielectric medium is in er times less than in a vacuum while the number of power lines of the electric flux density D remains constant and the same as in a vacuum. The electric field Eshape in the dielectric sphere becomes weaker by not er times times and this weakens slightly less because there is an influence of a but by eshape r   \er and Eshape ¼ E0 =eshape . As a result of the shape of a dielectric sample eshape r r influence of the shape, the electric flux density D inside the dielectric sphere increases in comparison to the value that would be in the same place in the absence of the sphere here. As we remember, the electric flux density without taking account of the shape of medium boundary is D ¼ e0 er E [Eq. (3.6.4)] and E ¼ E0 =er (Eq. 3.8.2), where E0 is the electric field in a vacuum. In our case E0 ¼ Eexternal and Eshape [ E, so in the sphere D ¼ e0 er Eshape . On the dielectric sphere placed in a vacuum this can be viewed as on the three-layer heterogeneous medium in which the permittivity changes abruptly “medium with e0 —medium with e0 er —medium with e0 ”.

4.13

Deformation of E and D Vectors …

251

In contrast to the homogeneous dielectric medium that does not affect the distribution of D-lines, the heterogeneous dielectric medium causes the redistribution of D-lines. In the case when a dielectric body with the permittivity er [ 1 is placed in an external electric field E0 formed in a vacuum (er;vacuum ¼ 1) or in a medium with a lower permittivity compared to the dielectric body permittivity, this makes the D lines in the body denser (Fig. 4.21a). In general, the magnitude and nature of the deformation E and D fields depend on the electric properties of medium, shape, size and location of dielectric bodies in an applied external electric field Eexternal because the location of charges on the body also depends on the structure of the applied electric field. The deformation of electric field inside and outside the dielectric body occurs only due to electric polarization in the dielectric. In Figs. 4.21 and 4.22 the applied electric field Eexternal is uniform. When e1 [ e2 then D1 [ D2 and E1 \E2 . When e1 \e2 then D1 \D2 and E1 [ E2 . The lines of electric flux density DðrÞ are denser in the medium with the larger permittivity since the electric field EðrÞ is lower in this case.

4.14

Boundary Conditions for E and D in Pictures

On the base of boundary conditions Sects. 4.9 and 4.11 we can write: E¼

D D ¼ ; er1 e0 e1

ð4:14:1Þ

Et1 ¼ Et2 ;

ð4:14:2Þ

Et1 ¼

Dt1 D1 sin H1 ¼ ; e1 e1

ð4:14:3Þ

Et2 ¼

Dt2 D2 sin H2 ¼ ; e2 e2

ð4:14:4Þ

The tangential component of the electric flux density vector D, when crossing the boundary of the two dielectrics, is breaking as follows: Dt1 Dt2 ¼ ; e1 e2

ð4:14:5Þ

D1 sin H1 D2 sin H2 ¼ : e1 e2

ð4:14:6Þ

252

4 Maxwell’s Equations and Boundary Conditions

If there are no surface charges on the interface of the two dielectrics, then the normal component of the electric flux density vector D [Eq. (4.11.6)], when crossing the boundary, is continuous: Dn1 ¼ Dn2 ;

ð4:14:7Þ

Dn1 ¼ D1 cos H1 ;

ð4:14:8Þ

Dn2 ¼ D2 cos H2 :

ð4:14:9Þ

The normal component of the electric field vector E [with no surface charges, Eq. (4.11.10)], when crossing the boundary of the two dielectrics, is breaking as follows: e2 En2 ¼ e1 E n1 :

ð4:14:10Þ

We can write, using Figs. 4.12, 4.13 and 4.14, expressions: tan h1 ¼

Et1 Dt1 ¼ ; En1 Dn1

ð4:14:11Þ

tan h2 ¼

Et2 Dt2 ¼ : En2 Dn2

ð4:14:12Þ

As we remember tan h is defined as the ratio of the opposite side of the angle theta to the adjacent side, see Fig. 4.23. Taking into account Eqs. (4.14.1 and 4.14.2):

Fig. 4.23 Vectors E and D behavior when they are crossing the interface of two dielectrics

4.14

Boundary Conditions for E and …

253

tan h1 Et1 =En1 Et1  E n2 E n2 ¼ ¼ ¼ : tan h2 Et2 =En2 Et2  E n1 E n1 We receive from (4.14.13), E n2 ¼ e1 E n1 =e2 ¼ er1 E n1 =er2 :

taking

account

tan h1 E n2 e1 E n1 e1 ¼ ¼ ¼ : tan h2 E n1 e2 E n1 e2

ð4:14:13Þ from

(4.14.10),

ð4:14:14Þ

The ratio of the tangents of certain angles H1 and H2 , created by the electric field E1 and E2 rays with the normal to the interface, is equal to the ratio of their respective permittivity e1 and e2 . This is also known as the law of electric flux refraction at interface (boundary). Notice that when e1 [ e2 then tan H1 [ tan H2 and when e1 \e2 then tan H1 \ tan H2 . 

 Et1 H1 ¼ arctan ; En1



 E t2 H2 ¼ arctan ; E n2

ð4:14:15Þ

h1 is an angle of incidence of the electric field in a dielectric medium 1, H2 is an angle of refraction in dielectric media 2. It can be written on the base of Eqs. (4.14.2) and (4.14.10): E t2 e2 Et1 e2 tan H2 ¼ ¼ ¼ tan H1 ; ð4:14:16Þ E n2 e1 En1 e1 tan H1 ¼

e1 tan H2 ; e2

e1 tan H2 ¼ e2 tan H1 ;   e2 H2 ¼ tan1 tan H1 : e1

ð4:14:17Þ ð4:14:18Þ ð4:14:19Þ

From Eq. (4.14.2): E 1 sin H1 ¼ E2 sin H2 :

ð4:14:20Þ

D1 cos H1 ¼ D2 cos H2 :

ð4:14:21Þ

From Eq. (4.14.7–4.14.9):

Figure 4.24 shows the distribution of E- and D-vectors at the interface of two media with the permittivity e1 and e2 when e2 [ e1 . We see in Fig. 4.24a that

254

4 Maxwell’s Equations and Boundary Conditions

(a) Et1 En1

(b) Dt1

E1

Medium 1 2

Ѳ1 Ѳ2 Et2

En2

>

1

D1

Dn1

1

Medium 1 2

Ѳ1

Interface

Ѳ2 Dt2

Medium 2 2

E2

Dn2

>

Interface

1

1

Medium 2 2

D2

Fig. 4.24 Refraction of vectors: a E and b D at the interface between two dielectrics when e2 [ e1

E vectors are discontinued (terminated) on the surface charges. Figure 4.24b shows D-vectors continuous across the interface. A larger dielectric constant (relative permittivity) e means a larger tan H and thus a larger angle H with the normal to the interface (boundary) two dielectrics, so the field tends to spread out when entering a dielectric with a higher dielectric constant. Now we will consider the electrostatic problem when the electric field vector E is perpendicular (normal) to the interface of two dielectrics. The distribution of vectors E and D in such a layered medium is shown in Fig. 4.25. We see that the electric field lines are finished (terminated) on the surface charges (Fig. 4.25a). As we known from Eq. (4.11.5), E n2 ¼ ee12 E n1 and at e1 \e2 field vectors E n2 \En1 . Since the electric field is perpendicular to the interface, the incidence angle H1 is zero and the electric flux density vector D ¼ Dn is continuous in both media (see Eq. 4.14.7). Suppose the medium 1 is a vacuum with er1 ¼ 1 then there are no electric charges in a vacuum. On the surface of the interface from the side of medium 2 polarization charges rbound (Sect. 3.9) will appear. The polarization electric field Ebound , which occurs within the medium 2, is created by rbound and is directed against the external electric field E1 ¼ Eexternal . The electric field into medium 2 is E2 ¼ Eexternal þ Ebound [Eq. (3.9.11–3.9.13)] and E 2 ¼ E1  E bound so E 2 \E 1 .

Fig. 4.25 Distributions of a electric field E and b electric flux density D lines at the interface

(a) 1<

2

Interface

(b) 1<

2

Interface

D1

D2

E2 E1

1

2 1

2

4.14

Boundary Conditions for E and …

255

Fig. 4.26 Distribution of electric field lines at two interfaces of media with different permittivities

E1

Ѳ1

Medium 1 1

Interface 1 Ѳ2 Interface 2

Medium 2

E2

2> 1 Ѳ2>Ѳ1

2

E1 Ѳ1

Medium 1 1

Figure 4.26 shows the distribution of E-lines at the interface of layered medium with the permittivity which passes abruptly “e1  e2  e1 ” when e2 [ e1 . For the simple dielectric media (Sect. 3.27) the directions of vectors E and D coincide (Figs. 3.38a and 4.24), we can write: tan H1 Dt1 =Dn1 Dt1 Dn2 ¼ ¼ : tan H2 Dt2 =Dn2 Dt2 Dn1

ð4:14:22Þ

Using Eqs. (3.6.4) and (4.14.1): P ¼ D  e0 E ¼ e0 er E  e0 E ¼ e0 ðer  1ÞE ¼ e0 ðer  1Þ

D er  1 ¼ D: e0 er er ð4:14:23Þ

As we can see from Eq. (4.14.23) the polarization of a vacuum P¼ 0 and the polarization inside the dielectric is P ¼ ere1 D. r From Eqs. (4.14.5) and (4.14.23): Pt1 ¼

er1  1 er1  1 Dt1 ¼ Dt2 ; er1 er2

ð4:14:24Þ

er2  1 Dt2 ; er2

ð4:14:25Þ

Pt2 ¼

using last two equations an Eq. (4.14.5) we can write: Pt1 Pt2 ¼ : er1  1 er2  1

ð4:14:26Þ

256

4 Maxwell’s Equations and Boundary Conditions

=

1

0

Et1

Ϭs

1

Pn2

Fig. 4.27 Components of the electric field, electric flux density and polarization at the interface of two dielectrics

Figure 4.27 shows the boundary conditions for components of the electric field E, electric flux density D and polarization P at e2 [ e1 . All extra surface charges on the interface of dielectrics are shown in rectangular boxes (Fig. 4.27).

4.15

Boundary Conditions for H and B in Pictures

We consider the interface between two different magnetic materials with dissimilar permeabilities l1 and l2 . We will consider magnetostatic interface conditions for magnetic field H and magnetic flux density B vectors. On the base of the derived equations in Sect. 4.10 and 4.12 we can establish the laws of behavior for H and B vectors (Fig. 4.28). From Eq. (3.27.5) we can write: Bt1 ¼ l1 Ht1 ;

ð4:15:1Þ

Bt2 ¼ l2 Ht2 :

ð4:15:2Þ

^1  From Eqs. (4.10.9) and (4.10.19) we have obtained H t1  H t2 ¼ 0 and n ^1 is directed ðH2  H1 Þ¼ 0 at the surface current Js ¼ 0, where the unit vector n from media 1 to media 2. The tangent component of H is continuous across the surface if there is no surface current Js present:

4.15

Boundary Conditions for H and …

257

Fig. 4.28 Vectors H and B refraction when they are crossing the interface of two media at Js ¼ 0

B t1

Medium 1

B1

μ1

Bn1

Ht1

α

Hn1

μ >μ 2

H1

1

Interface

Medium 2

μ2

β

tan α μ1 = tan β μ2

Bt1 Hn2 μ1 = = Bt2 Hn1 μ2

H2

Hn2 Bn2

Ht2 B2 Bt2

Ht1 ¼ Ht2

at

Js ¼ 0;

ð4:15:3Þ

Bt1 l1 ¼ Bt2 l2

at J s ¼ 0:

ð4:15:4Þ

Since we get from boundary conditions Eq. (4.12.6) Bn1 ¼ Bn2 anytime, we can write: l1 H n1 ¼ l2 H n2 ;

ð4:15:5Þ

H n2 l1 ¼ : H n1 l2

ð4:15:6Þ

We can write using triangulars in Fig. 4.28: tan a ¼

H t1 ; H n1

ð4:15:7Þ

tan b ¼

H t2 : H n2

ð4:15:8Þ

For isotropic magnetic medium the magnetic field vector H is parallel to the magnetic flux density vector B (see Fig. 4.29) and magnetization vector M.

258

4 Maxwell’s Equations and Boundary Conditions

Diamagnetic μr 1 M

H

Ferromagneric μr >>1 M

H

B/μ0

B/μ0

Fig. 4.29 Relations between vectors B, H and M

From Fig. 4.28 taking account relations between vectors we can write: tan a ¼

Bt1 ; Bn1

ð4:15:9Þ

tan b ¼

Bt2 ; Bn2

ð4:15:10Þ

tan a Bt1 Bn2 Bt1 ¼ ¼ ; tan b Bn1 Bt2 Bt2

ð4:15:11Þ

because from boundary conditions Eq. (4.12.6) Bn1 ¼ Bn2 is true anytime. Taking account of Eqs. (4.15.1) and (4.15.2) we can write: tan a Bt1 l1 ¼ ¼ at J s ¼ 0 because H t1 ¼ H t2 : tan b Bt2 l2

ð4:15:12Þ

The same for the magnetic fields we can write from Fig. 4.28: tan a H t1 H n2 H n2 l1 ¼ ¼ ¼ : tan b H t2 H n1 H n1 l2

ð4:15:13Þ

tan a l1 H t1 ¼ ; tan b l2 H t2

ð4:15:14Þ

From Eq. (4.15.11):

when J s ¼ 0 and H t1 ¼ H t2 then: tan a l1 ¼ : tan b l2

ð4:15:15Þ

Figure 4.30 presents the distribution of B-vector at the interface of two media with the absolute permeability l1 ¼ l0 lr1 and l2 ¼ l0 lr2 , when l2 [ l1 . Figure 4.30a shows B-vectors continuous across the interface. We see in Fig. 4.30b that H-vectors are discontinued (terminated) on the surface charges.

4.15

Boundary Conditions for H and …

259

(a) Bt1

(b) B1

Bn1

μ >μ 2

α

1

Ht1

Medium 1

μ1

H1

Hn1

μ >μ 2

α

Js β

Bn2

Bt2

Interface

B2

Medium 2

μ2

β

Hn2

Ht2

μ1 Js

Interface

H2

Medium 1 1

Medium 2

μ2

Fig. 4.30 Distribution of a the magnetic flux density B and b the magnetic field H lines at the interfaces of media with different permeability when l2 [ l1

Js is the surface current density between the two media (free current only, not coming from polarization of materials). The magnetic flux density lines are continuous in both media (Fig. 4.30a). The magnetic field H is discontinuous across the interface if there is a conduction surface current density Js (see Sect. 3.13 and 3.14). In the simple magnetic media the vectors’ B and H directions coincide (Fig. 4.29), Eq. (4.15.15) gives the law of refraction field force lines for both vectors. It follows from Eq. (4.15.15) that if the field force lines pass from medium 1 with the lower permeability l1 to the medium 2 with the greater permeable l2 then the lines are removed from the normal to the interface in medium 2, i.e. b [ a. The component of the magnetic flux density that is perpendicular to the interface is continuous across the boundary. The tangential component of the magnetic field H is continuous across the interface at Js ¼ 0. Figure 4.31 shows the behavior of all components. The normal component of B is continuous across the interface, and always Bn1 ¼ Bn2 , l1 Hn1 ¼ l2 Hn2 and Hn2 ¼ ðl1 =l2 ÞHn1 . Since B and H are related by the permeability l, we know how the normal component of the magnetic field Hn changes across the boundary. For example, when l1 =l2 \1 then Hn1 [ Hn2 .

Fig. 4.31 Components of the magnetic field H and magnetic flux densities B at the interface of two isotropic media

260

4 Maxwell’s Equations and Boundary Conditions

Note that it does not matter what the value of the conductivity r or the permittivity e is in the two contact materials because these parameters do not affect the magnetic field’s boundary conditions. ^, Blt1  Blt2 ¼ n ^  Js , Js ¼ ½ n ^ H , We know from Eq. (4.10.21) Ht2  Ht1 ¼ Js  n 1 2 ^¼n ^1 are directed from medium 1 to medium 2 see where the unit vectors n Fig. 4.31. We can express from Eq. (4.15.12) Bt2 ¼ ll2 Bt1 at Js ¼ 0. When there are sur1 face currents at the interface then the tangential component of H is not continuous (Fig. 4.31). To summarize, the component of H tangential to the interface is continuous across the interface unless there is a conduction surface current density Js . When a surface charge density rs moves with velocity v over a surface we have a surface current Is and surface current density Js ¼ rs v, where a charge per unit area from Eq. (2.5.7) rs ¼ lim Dq Ds , Ds is an area of charge distribution on the Ds!0

surface. When there are surface currents at the interface then the tangential component of H is not continuous.

4.16

Summarizing Boundary Conditions Between Lossless Dielectrics

Boundary conditions for two perfect dielectrics with conductivities r ¼ 0. The index “t” means the tangential component of the electric and magnetic fields. The index “n” means the normal components. A lossless (dissipationless) linear medium can be specified by the permittivity e, the permeability l with conductivity r ¼ 0. Boundary conditions at dielectric surfaces are given in Table 4.2. They state how vectors E and D change after intersection of the interface between two different ^ is the unit normal vector from medium 1 to medium 2 (^ media. n n¼^ n1 , Fig. 4.17), rs;bound is the bounded surface charge between the media, rs;free is the free (unbounded) surface charge between the media only, not coming from polarization of the media, rs;total ¼ rs;free þ rs;ext þ rs;bound (Sects. 3.8 and 3.14.1). Table 4.3 gives the most important formulae for dielectric medium 1 and medium 2 (Fig. 4.27). Table 4.4 gives the boundary conditions at the interface of two different magnetic media. The boundary conditions state how vectors H and B change after intersection of the interface. ^ is the unit normal vector from medium 1 to medium 2 (Fig. 4.14), Js is the n surface free current density between the two media. This current is free (unbounded, non-displacement) only, not coming from polarization of the media (see Sect. 3.14.1).

4.16

Summarizing Boundary Conditions Between Lossless Dielectrics

261

Table 4.2 Boundary conditions at any arbitrarily shaped interface between two dielectrics Vector form ^  ðE2  E1 Þ ¼ 0 n

Scalar form

Description

E t2  E t1 ¼ 0, Dt2 =Dt1 ¼ e2 =e1

The tangential component of E is continuous across the interface surface between two dielectrics ^  ðE2  E1 Þ ¼ rs;total E n2  E n1 ¼ rs;total The normal component of E has a step n of total surface charge (a discontinuous jump) on the interface surface ^  ðD2  D1 Þ ¼ rs;free Dn2  Dn1 ¼ rs;free The normal component of D has a step n of (free and extraneous) surface charge (a discontinuous jump) on the interface surface ^ Dn2  Dn1 ¼ 0; If there is no (free and extraneous) n  ðD2  D1 Þ ¼ 0 at rs ¼ 0 e2 E n2 ¼ e1 E n1 surface charge on the interface, the normal component of D is continuous ^  ðP2  P1 Þ ¼ rs;bound Pn2  Pn1 ¼ rs;bound The normal component of P has a step n of surface charge on the interface surface ^  ðE  n ^Þ þ n ^ðn ^  EÞ, Et ¼ n ^  ðE  n ^Þ, En ¼ n ^ ðn ^  EÞ E ¼ Et þ En ¼ n

Table 4.3 Formulae for media Formulae for dielectric medium 1

Formulae for dielectric medium 2

D1 ¼ e0 er1 E1 ¼ e1 E1 ¼ e0 E1 þ P1 D1 ¼ Dn1 þ Dt1 P1n ¼ e0 ðer1  1ÞEn1

D2 ¼ e0 er2 E2 ¼ e2 E2 ¼ e0 E2 þ P2 D2 ¼ Dn2 þ Dt2

Þ Pt1 ¼ ððee12 1 1Þ Pt2

Þ Pt2 ¼ ððee21 1 1Þ Pt1

Dt1 ¼ eer1r2 Dt2

Dt2 ¼ eer2r1 Dt1

En1 ¼

En2 ¼ eer1r2 En1

er2 er1

En2

Þe1 P2n ¼ e0 ðer2  1ÞEn2 ¼ e0 ðer2 1 En1 e2

We summarize the statements: 1) If there are surface charges and surface currents at the interface of a conductor, and tangential component of H and the normal component of D are not continuous. There are neither surface charges nor surface currents at the interface so the tangential component of H and the normal component of D are continuous. Ht1 and Ht2 have the same direction. Ht1 and Ht2 are perpendicular to the current Js . Table 4.5 gives the main formulae for magnetic materials, see Fig. 4.31. In Table 4.5 the value vm is called the volume magnetic susceptibility, Jm is the magnetization current which contributes to the total current density. The total current density for magnetic media is given in Eq. (3.13.19).

262

4 Maxwell’s Equations and Boundary Conditions

Table 4.4 Boundary conditions at any arbitrary- shaped interface between two magnetics Vector form

Scalar form

^ or n ^ Ht2  Ht1 ¼ Js  n ðH2  H1 Þ ¼ Js or ^ H2  ½n ^ H1 ¼ Js ½n

Bt2 ^  l  Blt1 ¼ Js n

H t2  H t1 ¼ J s ,

Description

The tangential component of H is discontinuous across the interface surface Bt2 Bt1 The tangential component of B is l2  l1 ¼ J s 2 1 discontinuous across the interface surface ^  ðH2  H1 Þ ¼ 0 at H t2  H t1 ¼ 0, The tangential component of H is n continuous across the interface surface Js ¼ 0 at J s ¼ 0 when there is no surface current present

B2 B1 The tangential component of B is B =B ¼ l =l t t 2 1 2 1 ^  l  l ¼ 0 at n 2 1 discontinuous across the interface and at J s ¼ 0 Js ¼ 0 changes its magnitude across the interface surface Bt2 ¼ ðl2 =l1 ÞBt1 ^  ðB2  B1 Þ ¼ 0 Bn2  Bn1 ¼ 0 The normal component of B is n continuous across the interface ^  ðl2 H2  l1 H1 Þ ¼ 0 or l1 H n1 ¼ l2 H n2 The normal component of H is n l1 Hn1 ¼ l2 Hn2 discontinuous across the interface and changes the magnitude H n2 ¼ ðl1 =l2 ÞH n1 ^  ðH  n ^Þ þ n ^ðn ^  HÞ, Ht ¼ n ^  ðH  n ^Þ, Hn ¼ n ^ðn ^  HÞ H ¼ Ht þ Hn ¼ n ^  ðB  n ^Þ þ n ^ðn ^  BÞ, Bt ¼ n ^  ðB  n ^Þ, Bn ¼ n ^ ðn ^  BÞ B ¼ Bt þ Bn ¼ n

Table 4.5 Formulae for media Formulae for magnetic medium 1

Formulae for magnetic medium 2

l1 ¼ lr1 l0 ;

vm1 ¼ lr1  1; l lr1 ¼ ð1 þ vm1 Þ ¼ 1 l0

lr2 ¼ ð1 þ vm2 Þ

B1 ¼ l0 lr1 H1 ¼ l1 H1 ¼ l0 ðH1 þ M1 Þ ¼ l0 ð1 þ vm1 ÞH1

B2 ¼ l2 H2 ¼ lr2 H2 þ M2 ¼ l0 ðH2 þ M2 Þ ¼ l0 ð1 þ vm2 ÞH2

M1 ¼ vm1 H1 ¼ ðlr1  1ÞH1

1 H1 ¼ Bl 1  M1 ¼ lM1

M2 ¼ vm2 H2

H2 ¼ Bl 2  M2

Jm1 ¼ r  M1

Jm2 ¼ r  M2

0

4.17

r1

0

Boundary Conditions for Dielectric-Conductor

Conductors of electricity contain a large quantity of free charged particles (Table 9.3 ), e.g. the concentration of electrons in metals can be N  1028 m3 . Electric charges of a conductive body can move freely in its volume. When we place the conductor in an external electrostatic electric field Eexternal then conduction (free) electrons start to move in an orderly way, i.e. arise a drift current (Sect. 3.13.1) but afterward the

4.17

Boundary Conditions for Dielectric-Conductor

263

current reaches equilibrium. This equilibrium becomes possible when the electric force ceases (ends, finishes) to operate the conduction electrons. This means that the electric field inside the conductor becomes zero. A mechanism of disappearance of the electric field inside the conductor is as follows: when we place a conductor in the electric field Eexternal then conduction electrons are redistributed and they create their own electric field within the conductor. In order to understand better processes in conductors let us place some negative charges on one side of a metal body. Since our charges can flow freely through metals, the negative charges will repel one another and spread in metal body as far apart as possible. One might imagine that this means that the electrons would be equally distributed throughout the metal and they will all end up on the metal interface. We know from Sect. 4.6 that for a good conductor r=xe  1 and for a perfect insulator r ¼ 0. So we come to conclusion that the electric field directly outside a conductor is perpendicular to its surface. While the electric field does not have to be zero outside the metal, the field lines are perpendicular to the interface of a conductor. Electrons (or other charge carriers) located on the interface of the conductor are able to feel the external electric field Eexternal , and the electric field force Fexternal ¼ qEexternal impacts on electrons. The electrons are unable to leave the conductor, so the perpendicular component of the electric field will have no effect on them. If the electric field has a tangential component, which is parallel to the interface, it will cause the electrons to flow along the interface which means the conductor is not in electrostatic equilibrium. The electric field into any empty cavities inside a conductor (metal) also equals zero. The electric field of conductor charges fully compensates the field Eexternal within the conductor. As a result, the total field strength inside the conductor becomes zero. We can write the electrostatic equilibrium state in the vector form: Einside ¼ Eexternal þ Efree ¼ 0;

ð4:17:1Þ

E inside ¼ E external  E free ¼ 0:

ð4:17:2Þ

and in the scalar form:

It follows from Coulomb’s law in differential form [Eq. (4.4.14)] that the volume charge density q ¼ qcrg (Eq. 2.5.8) inside the conductor is zero: q ¼ 0:

ð4:17:3Þ

The electric charges of a PEC can concentrate only on the surface (interface) of the conductor (Fig. 4.32). The surface charges on the interface are denoted as rs ¼ rcrg (Eq. 2.5.7). The electric field lines always extend from a positive charged object to a negatively charged object (Sect. 2.4). The positive charges shift (move)

264

4 Maxwell’s Equations and Boundary Conditions

Eexternal

Fig. 4.32 Perfect conductor of an arbitrary shape in an external electrostatic field

Ϭs Interface Perfect conductor

in the direction of the field E external and negative charges move in the opposite direction of Eexternal . A lossless linear medium can be specified by a permittivity e and a permeability l with electrical conductivity r ¼ 0. A lossy medium possesses of the conductivity r 6¼ 0. There are also charged particles (ions, protons, etc.) within the conductor body whose electric charges are mutually compensated and the total volume density of these electric charges q ¼ qcrg (Eq. 2.5.8) becomes zero. When the medium 1 is a perfect insulator (dielectric) r ¼ 0 and medium 2 is a perfect (ideal) electric conductor (PEC) then the electrical conductivity r ! 1. ^  Ht1 ¼ ½n ^Ht1 is the conduction surface current density. Js ¼ n The tangential components of an electric field are continuous across an interface and can be written as follows: E t1 ¼ 0; Et2 ¼ 0;

ð4:17:4Þ

Dt1 ¼ 0; Dt2 ¼ 0;

ð4:17:5Þ

rs ; E n2 ¼ 0; e1

ð4:17:6Þ

Dn1 ¼ rs ; Dn2 ¼ 0:

ð4:17:7Þ

E n1 ¼

The boundary conditions by vectors of electric and magnetic fields for the perfect dielectric–conductor interface are shown in Fig. 4.33. We remember [Eq. (4.17.1)] that the electric field inside the whole conductor body is zero. Now we will consider a metal hollow (empty) box which is placed into the electrostatic field Eexternal . We know that the conductor material of the box will be polarized and charge carriers will create a polarization field (Sect. 3.9). The polarization field has the direction which is opposite toward Eexternal and for this reason the electric field inside the air—cavity inside of the box—equals zero. This property is called the Faraday cage effect which has a number of practical applications:

4.17

Boundary Conditions for Dielectric-Conductor

Fig. 4.33 Components of the electric and magnetic fields at the interface of perfect dielectric-conductor

Ϭs

265

Ht1

En1

Interface

Perfect conductor Ϭ= E2=0, H2=0



^ n ^ t1] Js=[nH

1. Equipment which is sensitive to small electric fields is usually surrounded by Faraday cages (metal screen) to protect them from roam electric fields in the environment (random noise). 2. If you are in a car you can be relatively safe in a lightning storm because its metal frame forms a partial Faraday cage. It should be noted that Faraday cages only block fields which come from outside the cavity. The cavity will contain an electric field if a charge is placed inside. Table 4.6 presents the boundary conditions at an interface between a perfect dielectric and a perfect conductor. Note: There is no electric field inside a conductor. Any external electric field lines are perpendicular to the conductor (metal) surface. The conductor is equipotential. So the difference in potential between any two points on surface is zero. If the vector E vanishes inside a PEC then the curl E vanishes as well and the time rate of change B is zero correspondingly [see Eq. (4.3.1)].

Table 4.6 Boundary conditions for the interface perfect dielectric-perfect conductor Boundary conditions at any arbitrary- shaped interface between a perfect dielectric with conductivity r ¼ 0 and perfect electric conductor r ¼ 1 Vector form Scalar Description form ^  E ¼ 0 or The tangential component of the electric field strength E is n Et ¼ 0 ^ E ¼ 0 discontinuous across an interface, E t;outside 6¼ Et;inside , ½n E t;inside ¼ 0 ^  D ¼ rs D n ¼ rs The normal component of a flux density D is discontinuous n across an interface where a surface charge rs exists ^  H ¼ Js or Ht ¼ Js The tangential component of magnetic field strength H is n ^ H ¼ Js ½n discontinuous across an interface where a surface current exists, the amount of discontinuity ^ The normal component of a magnetic flux density B is nB¼0 Bn ¼ 0 discontinuous across an interface, Bn;outside 6¼ Bn;inside , Bn;inside ¼ 0

266

4 Maxwell’s Equations and Boundary Conditions

4.17.1 Boundary Condition at the Interface of Two Conductor Media We know from Eq. (4.6.2) that for a good conductor ratio r=xe [ [ 1 and for a perfect dielectric ratio equals zero because r ¼ 0 when a conductor is not perfect but only a good conductor (Sect. 4.6) with some electrical conductivity r which is the ability of materials to conduct an electric current I. An electric current is a flow of electric charges through conductor medium. For a good conductor, i.e. not perfectly conductive bodies, taking account that J ¼ rE from Eq. (4.14.2) we can write: Jt1 Jt2 ¼ ; r1 r2

ð4:17:8Þ

r1 and r2 are the conductivities of conductive medium 1 and medium 2, respectively. We know from Table 4.2. that e2 En2  e1 En1 ¼ rs and: e1

Jn1 Jn2  e2 ¼ rs : r1 r2

ð4:17:9Þ

The charge conservation law gives Jn1 ¼ Jn2 and  Jn1

e1 e2  r1 r2

 ¼ rs ;

ð4:17:10Þ

where here rs is surface charges on the conductor interface. Note: Symmetry is an important feature of many combinatorial search problems. To be able to solve such problems, we often need to take into account the symmetry of solutions. Table 3.3 gives the electrical conductivity r and resistivity q of some metals.

4.18

Review Questions

Q:4:1. Q:4:2. Q:4:3. Q:4:4.

What is the sense of Faraday’s law of electromagnetic induction? What is the electromotive force? What is the sense of Lenz’s law? Write Maxwell’s equations in the differential form.

4.18

Review Questions

267

Q:4:5. Explain the sense of the Maxwell’s equations. Q:4:6. What is a stationary source? Q:4:7. What does co-dependence of Maxwell’s equations mean? Q:4:8. Write Maxwell’s equations in a large-scale form? Q:4:9. Is there a magnetic monopole? Q:4:10. How can the Ampere’s law be expressed by the magnetomotive force? Q:4:11. Is the magnetic flux density vector B perpendicular or/and parallel to vectors of the electric field E and the electric current density J? Q:4:12. How are the magnetic field lines of an electric current flowing along the metal wire distributed? Q:4:13. When can you use the Euler’s formula in time dependency of the electric and magnetic field expressions? Q:4:14. Write Maxwell’s equations for the time-periodic case. Q:4:15. Please write and explain the homogeneous vector Helmholtz’s equations and explain their sense. Q:4:16. How to characterize a perfect insulator through the permittivity and the electrical conductivity? Q:4:17. How to characterize an imperfect insulator through the permittivity and the electrical conductivity? Q:4:18. How to characterize a good conductor through the permittivity and the electrical conductivity? Q:4:19. How to characterize a perfect conductor through the permittivity and the electrical conductivity? Q:4:20. How to characterize a semiconductor through the permittivity and the electrical conductivity? Q:4:21. What is the loss tangent at the microwave range? Q:4:22. What is the complex permittivity of a medium? Q:4:23. What is the total effective conductivity of material? Q:4:24. What are boundary conditions? Q:4:25. Are boundary conditions at the interface of a waveguide an electrostatic or an electrodynamic problem? Why do you think so, what are your arguments? Q:4:26. What are the tangential components of the electric field on the horizontal interface between two dielectrics in the Cartesian coordinate system? Q:4:27. What are the normal components of the electric field on the horizontal interface between two dielectrics? Q:4:28. How can the vector of the magnetic field be decomposed into three vectors in the Cartesian coordinate system? Q:4:29. Please explain how to get the boundary condition for tangential components of an electric field on the interface between two dielectric: E t1 ¼ E t2 . Q:4:30. Please explain how to get the boundary condition for the tangential components of the magnetic field across the interface between two media at the surface current Js 6¼ 0: H t1  H t2 ¼ J s .

268

4 Maxwell’s Equations and Boundary Conditions

Q:4:31. Please explain how to get the boundary condition for the normal components of the electric flux density at rs;free 6¼ 0: Dn2  Dn1 ¼ rs;free . Q:4:32. Please explain how to make sure that the normal components of polarization vectors satisfy the following: Pn1  Pn2 ¼ rs;bound . Q:4:33. How are the normal components of the electrical field E n1 and E n2 related to each other when crossing the interface of two media with the permittivities e1 and e2 ? Q:4:34. How are the tangential components of the electric flux density D related to each other when crossing the interface of two media with the permittivities e1 and e2 ? Q:4:35. How are the normal components of the electric flux density D related to each other when crossing the horizontal interface of two media with the permittivities e1 and e2 ? Q:4:36. Is the electric field E 1 in the dielectric sphere larger in comparison with the electric field E2 of a surrounding medium when the permittivity e1 of the sphere is larger than the permittivity e2 of the environment? Q:4:37. Is the electric flux density D1 in the dielectric sphere larger in comparison with the electric flux density D2 of a surrounding medium when the permittivity e1 of the sphere is larger than the permittivity of the environment e2 ? Q:4:38. The plane EM wave incident on the interface between two dielectrics with permittivities e1 and e2 . What kind of relation tan h1 =tan h2 ¼ ? is expressed through permittivities when h1 is the angle of incidence and h2 is the angle of transmission (refraction) of the plane EM wave? Q:4:39. Is there an electric field inside a perfect conductor (PEC)?

Chapter 5

Plane Electromagnetic Wave Propagation

Abstract This chapter is dedicated to the propagation of plane EM wave in different media. Here Maxwell’s equations for time-periodic plane waves are given alongside with consideration of the propagation of waves in lossless, low-loss and highly absorbing media. The phase difference between electric and magnetic fields of plane wave in good conductors is explained in detail as well. This chapter considers such questions as phase and group velocities of waves, Poynting’s theorem, the Poynting vector (instantaneous, complex, and time average), and also linear, circular, and elliptical polarizations of waves.

5.1

Introduction

What is a wave? A mechanical wave (as sound or water waves) can be imagined as a disturbance of a continuous medium that propagates with a fixed shape at a constant velocity from one point to another. This disturbance can be a regular periodic variation in value. Mechanical waves need a medium to travel. EM waves transfer energy through a medium or free space (a perfect vacuum, i.e. a space free of all matter). EM waves do not need a medium to propagate from one place to another. These waves are a result of reciprocal vibrations of electric and magnetic fields. Charges moving back and forth will produce oscillating electric and magnetic fields. EM waves can also be excited by accelerating charges. The only time-varying motion of charges leads to changing electric and magnetic fields in space. An oscillating electric field generates an oscillating magnetic field, and an oscillating magnetic field generates an oscillating electric field. A direct current circuit in which carriers’ charge is moving with a constant velocity cannot be a source of EM waves. The radiation of EM waves in modern radio equipment is achieved by using various designs of antennas, in which fast variable currents are excited. J.-B. J. Fourier (1768–1830, French mathematician and physicist) established that signals which are periodic functions of time can be presented by a linear combination of cosinusoids (cosine) and/or sinusoids (sine). We are going to analyze electric and magnetic fields in terms of their cosinusoidal (or sinusoidal) components. These components can be mathematically expressed by the real parts of the complex field. © Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_5

269

270

5

Plane Electromagnetic Wave Propagation

Harmonic vibrations (oscillations, regular periodic variations in value of about a mean) are variations in which physical value changes over time by harmonic law, i.e. cosinusoidal (or sinusoidal). As an example, an instantaneous electric field can be presented like this: Eðx; y; z; tÞ ¼ RefEðx; y; zÞeixt g; where Ref. . .g is the real part of a value in brackets, x is the angular frequency (also known as the angular speed, radial frequency, and radian frequency) of the cosinusoid (or sinusoid), and eixt is the time-dependent term. When the time variable term t is removed from the E-field expression, the field Eðx; y; zÞ only contains changes in space. The term Eðx; y; zÞ is called the vector phasor that contains information on magnitude and direction of the vector. There is no time-dependence in vector phasor Eðx; y; zÞ: A phasor (complex amplitude) is a complex number representing a cosinusoid (or sinusoid) function whose amplitude, angular frequency, and initial phase are time-invariant. The instantaneous value of a time-harmonic EM wave is the real part of the product of the phasor quantity Eðx; y; zÞ and the term eixt . We have to deal with the periodic (cyclic) behavior of oscillations. The complex term eixt is just a tool to make the math simpler because it is possible to use Euler’s formula expðixtÞ ¼ eixt ¼ cosðxtÞ  i sinðxtÞ (Sect. 1.4.2). Such representation lets us make higher-order differentiations or integrations with respect to time t replaced by the multiplication or division by higher power of ix. Note The term “instantaneous” comes from the Latin “instant” meaning “being at hand.” The term “instantaneous” means that something is happening right now, without delay. Differential equation of harmonic vibrations We will repeat in this section simple definitions concerning the amplitude, phase and phase difference of waves to deal further with questions about polarizations of EM waves which propagate in a medium (Sects. 5.15–5.21). A differential equation describing harmonic vibrations is: d2 zðtÞ þ x2 zðtÞ ¼ 0: dt2

ð5:1:1Þ

This equation of motion is also called the simple harmonic oscillator equation. Any non-trivial solution to this differential equation is the harmonic oscillation with the angular frequency x ¼ 2pf ; f is the operating frequency. A position function zðtÞ has to be such that the second time derivative position function is proportional to the negative of the position function: d2 zðtÞ=dt2 ¼ x2 zðtÞ (Eq. 5.1.1). The sine and cosine functions both satisfy this property. If we take the cosine function as an example, then: zðtÞ ¼ A  cosðð2p=T Þ  tÞ ¼ A cosðxtÞ;

ð5:1:2Þ

5.1 Introduction

271

where A is the amplitude. The sine and cosine functions are periodic with the period T ¼ 1=f . The first and second derivatives of the position function are: dzðtÞ ¼ xA sinðxtÞ; dt

ð5:1:3Þ

d2 z ð t Þ ¼ x2 A cosðxtÞ ¼ x2 zðtÞ: dt2

ð5:1:4Þ

taking account of Eq. (5.1.2):

There are three expressions for describing harmonic waves which are usually used: zðtÞ ¼ A  cosðxt þ u0 Þ ¼ A  cos u;

ð5:1:5Þ

zðtÞ ¼ A  sinðxt þ u0 Þ ¼ A  sin u;

ð5:1:6Þ

zðtÞ ¼ Aeiðxt þ u0 Þ ¼ Aeiu ;

ð5:1:7Þ

u0 is the initial (start) phase of an EM oscillation which is the value of the oscillation phase at the start point in time t ¼ 0 at the original of the coordinate system (x, y, z) = 0 for the wave process. The term phase can refer to several different things in different literature sources, e.g. (a) the phase can refer to a specified reference, such as cosðxtÞ in which the phase of zðtÞ is: u ¼ u0 ;

ð5:1:8Þ

(b) the phase of the harmonic wave is: u ¼ xt þ u0 :

ð5:1:9Þ

The value of phase u which is the argument of the cosine or sine function is referred to as the phase of vibration described by the function. The angular frequency x indicates how many radians or degrees will change the phase for 1 s, the higher the amount of x the faster the phase  u grows over  time. We remember that sinðxt þ u0 Þ ¼ cos p2  ðxt þ u0 Þ ¼ cosðxt þ u0  p=2Þ; the phase of sine function is u ¼ xt þ u0 and cosine function is u ¼ xt þ u0  p=2. When we use the cosine function for describing harmonic EM waves, the term-p=2 will be included in the initial phase u0 .

272

5

Plane Electromagnetic Wave Propagation

We see in Fig. 5.1 that the initial angle is u0 ¼ 0 of a cosinusoidal and sinusoidal function at the origin “0”. The initial angle u0 is sometimes called the phase offset or phase difference especially when we are dealing with two or more simultaneously propagating EM waves. Phase difference, also called as phase angle, phase shift in degrees is conventionally determined as a number greater than 180 and less than or equal to þ 180 . In this book, we will define the phase in each specific case. If we take that an EM wave propagates in one-dimensional space, then these expressions can be used: A cosðxt  kz þ u0 Þ ¼ A cosðuðz; tÞÞ;

ð5:1:10Þ

A sinðxt  kz þ u0 Þ ¼ A sinðuðz; tÞÞ;

ð5:1:11Þ

Aeiðxtkz þ u0 Þ ¼ A eiðuðz;tÞÞ :

ð5:1:12Þ

The argument for the cosine or sine functions is called the phase of oscillations uðz; tÞ ¼ u ¼ xt  kz þ u0 ; k is the wavenumber which can be a real or complex number depending on a medium where the EM wave propagates, z is the coordinate of the observation point of the wave process in one-dimensional space. For a wave in space of any dimension (such as 3D space): A cosðxt  k  r þ u0 Þ ¼ A cosðuðr; tÞÞ;

ð5:1:13Þ

A sinðxt  k  r þ u0 Þ ¼ A sinðuðr; tÞÞ;

ð5:1:14Þ

Aeiðxtkr þ u0 Þ ¼ Aeiðuðr;tÞÞ ;

ð5:1:15Þ

u ¼ uðr; tÞ¼xt  k  r þ u0 ;

ð5:1:16Þ

a phase u of an oscillation:

+A

sin(ωt)

1

cos(ωt) π 2

0

π 2

π

3π 2



5π 2

1

A Fig. 5.1 Cosine and sine functions with the initial angle u0 ¼ 0



φ=ωt

5.1 Introduction

273

k is the wave vector, which can be a real number or complex number; for the complex number, we use designation k ¼ ka (see Sect. 5.4), r is a radius vector of an observation point in space. A phase u of an oscillation is an argument of the function, i.e. expression written in brackets of the left part of Eqs. 5.1.13–5.1.15. The initial oscillation phase is the u0 value, which is one of the terms of the complete phase u. The phase can also be an expression of relative displacement between two waves with the same frequency. We can determine the phase as the position of a point on a waveform cycle in an instant time t. A complete cycle is determined as the interval required for the waveform to return to its initial fixed primary value. The graphic on the right of Fig. 5.2 shows how one cycle constitutes 360° of a phase. In the above expressions, the phase has a dimension of angular units (radians, degrees). The phase of the oscillating process, also expressed as cycles, that is, fractions of the period of the recurring process: 1 cycle = 2p radian = 360°. A sinusoidal (sine) or cosinusoidal (cosine) wave has unity amplitude and is repetitive after every 2p radians. As illustrated in Fig. 5.2, we can draw a circle of radius 1, also called the unit circle, and we can visualize a phasor of unity magnitude rotating around the unit circle at some fixed angular rate x. The phasor makes a complete revolution around the unit circle; it passes through 2p radians and completes one cycle. We quantify the phasor’s rotational speed as being x radians per second.

5.2

Phase Shift—Leading and Lagging Phases of Electromagnetic Waves

Figure 5.3 presents sinusoidal waves with (a) the initial (start) phase u0 ¼ 0; (b) u0 ¼ þ 30 ,(c) u0 ¼ 30 . Positive initial phase u0 in Fig. 5.3b means that a process started earlier (advance, leading ahead, previously) in comparison with

π 2

+A

φ

135°

0° 360° 0

π

φ

5π 2

A sin(φ)

45°

180°

225°

π 2

90°

180° 45°

90°

135°

270° 225°

π

3π 2

A

Fig. 5.2 Unit circle and sinusoidal wave

315°

405°

2π(rad)

315° 270°

360°

3π 2

450°

495°

φ°

274

5

(a)

(b)

Plane Electromagnetic Wave Propagation

(c)

Fig. 5.3 Illustration of a phase shift: a u0 ¼ 0; b “leading” and c “lagging” phase difference “initial”

u0 ¼ 0 (Fig. 5.3a). Negative initial phase u0 in Fig. 5.3c means that a process started later (lagging behind) in comparison with u0 ¼ 0 (Fig. 5.3a). Figure 5.3 shows how a phase of oscillations (Eqs. 5.1.10– 5.1.16) is changing with time t and at different initial phases u0 . The waveform AðtÞ ¼ Am sinðxt  u0 Þ (a) starts at the zero point (in original) of the coordinate system along the horizontal reference axis time “t”, but at the same instant of time, the waveform (c) is still negative in value and does not cross this reference axis. The term “instant of time” means a very brief (short duration) time or a particular moment or point in time. From Eq. (5.1.16) a, phase u of the oscillations is: case (a) uðaÞ ¼ xt; case (b) uðbÞ ¼ xt þ u0 , case (c) uðcÞ ¼ xt  u0 (see Fig. 5.3). The phase shift between cases (a) and (b) is uðbÞ  uðaÞ ¼ u0 , where u0 is the initial phase of case (b). The phase shift between cases (a) and (c) is uðcÞ  uðaÞ ¼ u0 , where u0 is the initial phase of case (c). Then there exists a phase difference between the waveforms. The waveform (c) can be also said as “lagging” behind the waveform (a) by the phase angle u0 . Our examples above reflect the waveforms that have the leading and lagging phase differences. Leading phase refers to a wave with the frequency f that occurs “ahead” of another wave of the same f . Lagging phase refers to a wave with frequency f that occurs “behind” another wave of the same f . When two EM waves differ in phase by –90° or +90°, they are said to be in phase quadrature. When two waves differ in phase by 180° (or −180), the waves are said to be in phase opposition (antiphase). Note An EM wave can be equivalently described through the cosine or sine functions (Eqs. 5.1.10 and 5.1.11). Figure 5.3 shows the phase shift between waves which is depicted by sine functions for simplification of the explanation, while the phase shift between waves depicted by cosine functions is shown in Fig. 5.27.

5.2 Phase Shift—Leading and Lagging …

275

Phase can also be an expression of relative displacement between two corresponding points on the waveform with the same frequency, e.g. peaks (antinodes) or zero crossings the horizontal axis (nodes). Phase difference is the difference expressed in degrees or time between two waves with the same frequency and corresponds to the same point in time. Two waves with the same frequency and no phase difference are in phase. Two waves with the same frequency and different phases have a phase difference (phase angle, phase shift), and the waves are out-of-phase with each other. When the phase difference is 180° (p radians), then these two waves are in antiphase. The phase is sometimes expressed in radians rather than in degrees. One radian of phase corresponds to approximately 57.3°. Engineers and technicians generally use degrees, whereas physicists use radians more often. The time interval for one degree of phase is inversely proportional to the frequency. If the frequency of a signal (in Hertz) is given by f ; the time t (in seconds) corresponding to one degree of phase is: t ¼ 1=ð360  f Þ:

ð5:2:1Þ

The time t (in seconds) corresponding to one radian of phase is approximately: trad ¼ 1=ð6:28  f Þ:

ð5:2:2Þ

The EM waves consist of periodic synchronized interrelated oscillations (vibrations) of electric E and magnetic H fields and can travel through a vacuum. EM waves propagate at the speed of light in a vacuum (free space); i.e. EM wave energy transports through a vacuum at the speed given in Eq. 2.1.6. Momentum pEM ¼ mEM c (product of the mass mEM , and velocity cÞ is a vector quantity possessing direction and magnitude. EM waves carry energy W EM away from their source particle, momentum pEM , and angular momentum which is the cross-product of the position vector r and momentum pEM [expresses the amount of dynamical rotation present in the EM field, Eq. (2.1.7)]. Quanta of EM waves are called photons (Chap. 9), and the protons rest mass equals zero but its energy and equivalent total (relativistic) mass is not zero. A wavefront is the location of EM wave points having the same phase: for 2D EM case a wavefront is a curve, and for 3D case a wavefront of EM wave is a surface. The electric field and magnetic field of an EM wave are perpendicular to each other. They are also perpendicular (orthogonal, normal) to the direction of the EM wave propagation (Fig. 5.4). EM radiation is relevant to the waves (or waves’ quanta, i.e. photons) of the EM fields, propagating through space carrying EM radiation energy. It includes all waves of entire EM spectrum, part whereof is given in Appendix Chap. 5:5.1. The electric E and magnetic H field components of a linearly polarized (the same as a plane polarized) EM wave oscillate in such a way that their peaks and nodes occur at the same time. In a general sense, the term “peak” is the maximum value of oscillations, either positive or negative that a waveform attains. In a more

276

5

Plane Electromagnetic Wave Propagation

λ/2

z

λ/2 λ/4

y

H

E

k node

x

Amplitude of E

node

E 0

H

n n tio atio rec opag i D pr of

anti-node

anti-node

Fig. 5.4 Snapshot of propagating transverse oscillating EM wave in a lossless medium at time t¼0

specific sense, the term “peak” is used for the highest point of the wave, and the term “trough” is used for the lowest point of the wave. The term “antipeak” is negative—going peak. The distance between any node and the antinode next to it is k/4. The term “antinode” can be used for a peak or an antipeak. Phase difference D u refers to time difference. There is no time difference between the peaks of the electric and magnetic oscillations of E- and H-field vectors of a linearly polarized EM wave propagating in a lossless (dissipationless) medium; i.e. D u is zero. That means that the electric and magnetic field components of a linearly polarized EM wave oscillate in such a way that they peak at the same time and they become zero also at the same time but they point at different directions in space, i.e. separated by an angle of 90° [see Fig. 5.4 and Eqs. (5.4.30–5.4.36)]. Wavelength k is the distance between successive positive peaks (in meters). Frequency f is the number of wavelengths per unit of time (in hertz, symbol: Hz, Hz = 1/s). The wave vector k of a plane EM wave is a vector which points to the direction in which the wave propagates. The direction of propagation of plane EM wave is always perpendicular to the wavefronts. A magnitude of the wave vector is called the wavenumber (or propagation constant, phase constant in lossless media). k ¼ 2pk ¼ x=v; v is the speed of EM wave in a simple media with er 6¼ 1 or/ and lr 6¼ 1. Vectors E; H and k form the right-screw system (Fig. 5.4). Vectors E and H always oscillate in the same phase, and instantaneous values of E and H at any pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi point are e0 er E ¼ l0 lr H in simple isotropic medium. The instantaneous value of electric (or magnetic) field quantity is the value of the quantity at a particular instant of time in the entire cycle of quantity under consideration. When the EM wave propagates into free space, then er ¼ 1; lr ¼ 1; k ¼ k 0 ¼ x=c; where c is the speed of EM wave in free space (Eq. 2.1.6).

5.2 Phase Shift—Leading and Lagging …

277

We can write the homogeneous wave equations for electric and magnetic fields given in Eqs. (4.5.10) and (4.5.11) in the scalar form: @ 2 Ey 1 @ 2 Ey ¼ 2 2 ; 2 v @t @x

ð5:2:3Þ

@2Hx 1 @2Hx ¼ : v2 @t2 @z2

ð5:2:4Þ

Equations (5.2.3) and (5.2.4) can be satisfied by plane monochromatic waves which are described by the following equations: E y ¼ E0 cosðxt  kz þ u0 Þ;

ð5:2:5Þ

H x ¼ H 0 cosðxt  kz þ u0 Þ;

ð5:2:6Þ

where E 0 and H 0 are the amplitudes of the electric and magnetic fields, u0 is the initial (start) phase. The initial phase u0 is the same because the oscillations of E and H occur with the same phase. Figure 5.4 shows a plane linearly polarized EM wave propagating from the origin along z-axis. The electric field E is in a vertical plane along the y-axis and the magnetic field H in a horizontal plane along the x-axis. The electric and magnetic fields in EM waves are in phase when the surrounding medium (environment) is lossless (dissipationless) and the polarization planes are at 90° to each other.

5.3

Spherical and Cylindrical Electromagnetic Waves

When a point source of EM energy is radiating and producing a spherical wave in all directions (Fig. 5.5) and no energy is absorbed or scattered by the surrounding medium, the intensity of the EM energy (passing through the same area) diminishes in proportion to distance R0 from the source squared. The last operates the inverse-square law of the EM energy.

Intensity WEM I= S

(a) Source power

(b) 9

I

4

WEM

Direction of propagation of EM wave

R 2R Imaginary sphere area S=4πR0, R0=R,2R,3R...

wavefronts

I

3R

Fig. 5.5 Flattening of spherical waves with the distance

A point source

Direction of propagation

278

5

Plane Electromagnetic Wave Propagation

Point sources of electric field, light, sound, EM wave radiation or gravitational force obey the inverse-square law. The inverse-square law is the physical law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of the physical quantity. The radiated power W EM is: W EM ¼ jIj  S ¼ jIj  4pR20 ;

ð5:3:1Þ

where I is the intensity of EM energy, a position R0 ¼ R; 2R; 3R. . ., which is the distance from a point source till an observation point (see Fig. 5.5) and S is a closed surface that contains the EM field source. The module of the intensity of EM energy is: jIj ¼

W EM W EM ¼ : S 4pR20

ð5:3:2Þ

Figure 5.5 shows an idealized point source of EM wave. The radiation emanating from this isotropic source streams out radially in all directions. The resulting wavefronts are concentric spheres that increase in diameter as they expand out in the environment space. The spherical wavefront suggests that it can be more convenient to determine in terms of spherical polar coordinates. The Laplacian operator in spherical coordinates is:     1 @ 1 @ @w 1 @2w 2@w r w¼ 2 r sin h ; þ 2 þ 2 2 r @r @r r sin h @ h @h r sin h @u2

ð5:3:3Þ

2

where r; h; u are the spherical coordinates, x ¼ r sin h cos u; y ¼ r sin h sin u; z ¼ r cos h. wðr; tÞ is a function that satisfies a wave equation Dw  er e0 lr l0 @@tW2 ¼ 0 and describes properties of a wave. The spherical waves are spherically symmetrical, i.e. ones that do not depend on h and u and wðrÞ ¼ wðr; h; uÞ ¼ wðr Þ. The Laplacian of wðr Þ is a simple expression: 2

  1 @ 2 @ wðr Þ r wðr Þ ¼ 2 r : r @r @r 2

ð5:3:4Þ

The wave function of a harmonic spherical wave which is an exploring function can be expressed in different ways: wðr; tÞ ¼

  A cos kðvt  r Þ; r

ð5:3:5Þ

5.3 Spherical and Cylindrical Electromagnetic Waves

279

  A ikðvtrÞ wðr; tÞ ¼ ; e r

ð5:3:6Þ

where a constant A is an amplitude of the source strength, k ¼ x=v is the wavenumber (it can be a real number in a lossless medium or a complex number in a lossy medium), the speed v ¼ c in a vacuum; see Eq. (2.1.6). This solution gives a cluster of concentric spheres filling all space at any fixed time t. Each wavefront which is a surface of constant (the same) phase is given by expression: k ðvt  r Þ ¼ const;

ð5:3:7Þ

where kðvt  r Þ is the argument of functions Eqs. (5.3.5) and (5.3.6) at the fixed time t; the expression (5.3.7) gives a surface of constant phase. The amplitude of any spherical wave is a function of radius r ¼ R0 , where R0 ¼ R; 2R; 3R. . . (Fig. 5.5a). A spherical wave decreases in amplitude, thereby changing its profile, as the wave expands and moves out from the origin (source). The wavefront of the point source becomes the plane one at a larger distance (Fig. 5.5b). We will now briefly examine the cylindrical waveform. In Fig. 5.6, we see how cylindrical waves may occur. When a plane wave incidences (falls) on a flat screen containing a long thin slit (a long narrow opening, hole, slot), then the radiation from the slit is in the form of a cylindrical wave. The Laplacian in the cylindrical system coordinate is:   1@ @w 1 @2w @2w r w¼ r þ 2 2þ 2; r @r @r r @h @z

ð5:3:8Þ

2

The wavefronts of a cylindrical wave

on of Directi tion a g a prop wave M E f o

The wavefronts of a plane wave

Direction of propagation of EM wave

Long narrow slit

Fig. 5.6 A plane wave after a long narrow slit in the screen becomes a cylindrical wave

280

5

Plane Electromagnetic Wave Propagation

where r; h; z are coordinates of the cylindrical coordinate system, x ¼ r cosðhÞ; y ¼ r sin h; z ¼ z. For the simplest case, when there is no dependence on variables h and z;, the wave function of Eq. (5.3.8) can be written as wðrÞ ¼ wðr; h; zÞ ¼ wðr Þ. When the distance r is sufficiently large, then we get: 

 A pffiffi cos kðvt  r Þ; r   A wðr; tÞ ¼ pffiffi eikðvtrÞ ; r

wðr; tÞ ¼

ð5:3:9Þ ð5:3:10Þ

where A is the constant amplitude of the source strength. Constant phase surfaces are a set of coaxial circular cylinders filling all surrounding space which propagate away from an infinite line source. A very small area of an unusually large sphere is very near a plane (Fig. 5.5b).

5.4

Plane Waves

The plane wave is one of the simplest examples of a three-dimensional wave. Figure 5.5b shows how a spherical EM wave becomes flat at distances large enough from the point source. A plane wave is a wave with a constant frequency whose wavefronts (surfaces of constant phase) are infinite ð1Þ parallel planes of the constant amplitude which are normal to the phase velocity vector (to the direction of propagation of the wave). The amplitude and phase difference (phase angle, phase shift) of the electric and magnetic fields of a plane wave remain constant on each of wavefronts (Fig. 5.7). A wave is a function of both space and time. A wave equation is a partial differential equation of the second order. In free space, the source-free ðq ¼ 0; J ¼ 0Þ 3D wave equation for a transversal EM wave is:

Fig. 5.7 Wavefronts for a harmonic plane wave

y

E

k

O

z x

H

5.4 Plane Waves

281

r2 Aðr; tÞ 

1 @ 2 Aðr; tÞ ¼ 0; c2 @t2

ð5:4:1Þ

where the equation is a hyperbolic partial differential equation, vector Aðr; tÞ is an exploring function which determines the disturbance at the certain point r at the time t (Note The same function for the spherical and cylindrical waves was designated wðr; tÞ in Eqs. 5.3.4–5.3.6 and 5.3.8–5.3.10), r is the radius vector of the point of observation, c is the speed of light in a vacuum (Eq. 2.1.6). Equation (5.4.1) can be also written in the form r2 Aðr; tÞ  ð1=c2 Þ  2  @ Aðr; tÞ=@t2 ¼ 0. As mentioned before, the EM wave is a disturbance that transfers energy through medium or free space. The wave motion occurs when a disturbance at time t1 is related to an event at time t2 when t1 \t2 . The solution of the partial differential Eq. (5.4.1) for a vacuum (free space) is: Aðr; tÞ ¼ A0 cosðxt  ðk0  rÞ þ u0 Þ;

ð5:4:2Þ

where Aðr; tÞ is the magnitude or disturbance of the wave at a given point in space and time, A0 is a constant amplitude (a peak magnitude of the oscillating quantities) of the source strength, k0 ¼ ^xkx þ ^yk y þ ^zkz is the wave vector, r is the radius vector of the point, ðk0  rÞ¼kx x þ ky y þ k z z is the scalar product, k 0 ¼ jk0 j ¼ 2p=k0 ¼ x=c ðrad=mÞ is the free-space wavenumber, either in cycles per unit distance or radians per unit distance, k0 ðmÞ is the wavelength of EM wave in a vacuum, u0 is the initial phase of an oscillation which can have “plus” or “minus” sign. Thus, the points of equal field value of Aðr; tÞ always form a plane in space. This plane then shifts with time t along the direction of propagation r with speed c in free space or with speed v in other medium. Under a monochromatic plane wave, we mean single-frequency plane wave whose amplitude is a cosine (cosinusoidal) wave or a sine (sinusoidal) wave function of r and t. Under a homogeneous plane wave, we mean the one in which the planes (surfaces) of constant phase and constant amplitudes coincide; i.e. wave amplitudes are also constant on all plane of constant phase. These planes are perpendicular to the direction of wave propagation. The scalar wave equation for a transversal plane homogeneous wave Eq. (5.4.1) in simple media (Sect. 3.27.1) as a lossless (dissipationless) isotropic dielectric, propagating in z-direction, is: @ 2 Aðz; tÞ 1 @ 2 Aðz; tÞ  2 ¼ 0: @z2 v @t2

ð5:4:3Þ

The function Aðz; tÞ depends only on one coordinate z and time. The solution of the last equation, using Eq. (5.4.2), is:

282

5

Plane Electromagnetic Wave Propagation

Aðz; tÞ ¼ A0 cosðxt  kz z þ u0 Þ;

ð5:4:4Þ

kz ¼ k ¼ 2p=k ¼ x=v is the wavenumber in a simple medium, v is the speed of EM wave in a simple media with er 6¼ 1 or/and lr 6¼ 1; k is the wavelength of EM wave in the media, A0 is a constant amplitude as in Eq. (5.4.2). The speed of EM wave in a medium is the wavelength (distance) divided by the time period: v¼

k pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k  f ¼ c= er lr ¼ 1= e0 er l0 lr ðm=sÞ; T

ð5:4:5Þ

where c is given in Eq. (2.1.6), e0 and l0 in Eq. (2.5.2). Speed is the distance which can be traveled by a wave per unit of time (in m/s). A time period T is the time needed for one complete cycle of vibration to pass a given point. Time period and frequency are in a reciprocal relationship T = 1/f. Consider this wave as traveling (propagating, moving) in the z-direction in space. We assume that the positive z-direction is on the right side of original in Fig. 5.7, and the negative z-direction is on the left. The wave has constant waveform (shape). The general solution of the wave Eq. (5.4.3) can be represented as a superposition to the forward and reverse waves: Aðz; tÞ ¼ A þ ðt  z=vÞ þ A ðt þ z=vÞ;

ð5:4:6Þ

where A þ ðnÞ and A ðnÞ are arbitrary twice differentiable functions with an argument n ¼ t  z=v. Note EM wave A þ ðnÞ ! Eyþ ðzÞ ¼ E 0þ eikz propagates in the positive direction  þ ikz propagates in the negative direction of z-axis, and A ðnÞ ! E y ¼ E0 e þ of z-axis. A ðnÞ is called a forward wave, and A ðnÞ is called a backward (reverse) wave. Plane harmonic (periodic oscillation) wave is defined by Eq. (5.4.4), and for the one-dimensional case, it can be written in simplified form: Aðz; tÞ ¼ A0 cosðxt  kzÞ;

ð5:4:7Þ

Aðz; tÞ is the disturbance at the point z and time t. The angular frequency x of a wave is the number of radians per unit time at a fixed position, whereas the wavenumber k ¼ 2p=k ¼ x=v is the number of radians per unit distance at a fixed

5.4 Plane Waves

283

time, and terms k and v are the wavelength and speed of EM wave in a media, respectively, in a vacuum k ! k0 and v ! c and ðk  rÞ ! ðk0  rÞ. The solution Eq. (5.4.4) can be written also in the form:   t z     z Aðz; tÞ ¼ A0 cos 2p  þ u0 ¼ A0 cos 2p f t  þ u0 ; T k k

ð5:4:8Þ

and also using Eq. 5.4.5:   2p ðv t  zÞ þ u0 ¼ A0 cosðk ðv t  zÞ þ u0 Þ: Aðz; tÞ ¼ A0 cos k

ð5:4:9Þ

From Eq. (5.4.9): Aðz; tÞ ¼ A0 cos

x     z ðv t  zÞ þ u0 ¼ A0 cos x t  þ u0 v v    x ¼ A0 cos xt  z þ u0 : v

ð5:4:10Þ

From Eq. (5.4.10), we receive commonly used solution record: Aðz; tÞ ¼ A0 cosððxt  kzÞ þ u0 Þ:

ð5:4:11Þ

Complex exponential form of solution is: Aðr; tÞ ¼ A0 eiðxtðkrÞ þ u0 Þ ¼ A0 eiðxtkrÞ eiu0 ¼ W0 eiðxtkrÞ ¼ Am eixt ;

ð5:4:12Þ

where the constant amplitude W0 ¼ A0 eiu0 and a wavenumber k can be real or complex numbers. In the form of complex amplitudes Am ¼ A0 eiu0 eiðkrÞ ¼ W0 eiðkrÞ , where A0 is the peak magnitude of the oscillating quantities of the source strength, u0 is the initial phase of an oscillation which can have “plus” or “minus” sign (see Sect. 5.1). We can write Eq. (5.4.3), for the electric field of EM wave propagating in zdirection in a vacuum, the partial differential equation of the second order: r2 Eðz; tÞ 

1 @ 2 Eðz; tÞ ¼0; c2 @ t 2

ð5:4:13Þ

where the speed of light in free space (Eq. 2.5.2) can be written as c ¼ dz=dt ¼ x=k0 . For the time-harmonic electric field: Eðz; tÞ ¼ EðzÞeixt :

ð5:4:14Þ

Substituting Eq. 5.4.14 in Eq. 5.4.13, we get the homogeneous vector Helmholtz’s equation:

284

5

Plane Electromagnetic Wave Propagation

r2 EðzÞ þ k20 EðzÞ ¼ 0;

ð5:4:15Þ

where the wavenumber k0 refers to the number of complete wave cycles of an EM field that exist in one meter of a linear space (a vacuum). The wavenumber k0 ¼ 2p=k0 can also be expressed in reciprocal meters ðm1 Þ. The wavenumber in free space can be expressed as: k0 ¼

2p x pffiffiffiffiffiffiffiffiffi ¼ ¼ x e0 l0 k0 c

 1  m :

ð5:4:16Þ

The source-free (because in the right part of the equation there is zero) wave Eq. (5.4.3) for a medium with the permittivity er and permeability lr becomes the homogeneous vector Helmholtz’s equation, which is a partial differential equation of the second order: r2 Eðz; tÞ 

1 @ 2 Eðz; tÞ ¼ 0: v2 @t2

ð5:4:17Þ

The source-free wave equation Eq. (5.4.17) for a simple medium becomes a homogeneous vector Helmholtz’s equation: r2 EðzÞ þ k2 EðzÞ ¼ 0;

ð5:4:18Þ

where r2 is the Laplacian, a wavenumber k refers to the number of complete wave cycles of an EM field that exist per unit distance, e.g. per one meter of linear space: k¼

 2p x pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ x e0 er l0 lr ¼ k0 er lr m1 : k v

ð5:4:19Þ

The wavelength k of a sinusoidal waveform traveling at the constant phase speed v is given by: pffiffiffiffiffiffiffiffi v c= er lr pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 k¼ ¼ ¼ k0 = er lr ¼ f e0 er l0 lr f f

ðmÞ;

ð5:4:20Þ

where f is the wave’s operating frequency. We can define frequency f as the number of repeating events (cycles) per unit time of an oscillation. The term v is called the phase speed of the EM wave: v¼

x x 1 1 1 c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffi ¼ k x e0 er l0 lr e0 l0 er lr el n

ðm=sÞ:

ð5:4:21Þ

The phase speed v usually is denoted as vp when we talk about phase velocity vp and group velocity vg .

5.4 Plane Waves

285

The index of refraction of a medium is: n¼

c pffiffiffiffiffiffiffiffi ¼ er lr : vp

ð5:4:22Þ

The magnitude of the phase velocity vp is independent from wavelength k when there is no dispersion. The angular (radial or circular) frequency means angular displacement per unit time: x ¼ 2pf :

ð5:4:23Þ

Angular frequency units are therefore degrees (or radians) per second. The period T is the duration of time of one complete cycle in a repeating event (oscillation of a wave), so the period is the reciprocal term of the frequency. The frequency f is the number of periods per unit time and is typically measured in hertz (Hz). The hertz is the unit of frequency in SI units and is defined as one cycle per second. Figure 5.8 shows a snapshot picture of an EM wave. On the left part of the figure, there is an elementary electric dipole antenna which produces the wave. The wave oscillates about the median line. A transverse wave propagating in a material medium is a wave in which particles of a medium are displaced in a direction perpendicular to the direction of energy transport. A snapshot graph is a figure of the wave’s displacement at this instant (at the same time t) along the distance (position) z. A graph that shows the wave’s displacement as a function of time t at a single position in space is called a vibration (history) graph. The nodes are located on the median line (equilibrium, rest position). A node is a point of no displacement. A wave crest (peak, extremum, antinode) is the maximum of displacement, and a wave trough (extremum, antinode) is the minimum. A crest and a trough are also called antinodes or extremums.

Snapshot Graph

A(z,t)

1 cycle

λ λ crest (anti-node)

A

~

Median line

node 0

λ

4

A

Propagation

λ

2

3 4λ

λ

5 3 4λ 2λ trough (anti-node)

7 4λ

Fig. 5.8 Snapshot graph of wave’s displacement Aðz; tÞ along the distance z

2λ z

286

5 Amplitude

λ

Higher frequency

Lower frequency Distance

crest

T

Distance

ion act

ntr trough

z

T t3

Exp ans ion

t1

z

wave period

Co

Amplitude

λ

wave length

0

0

Plane Electromagnetic Wave Propagation

Time

Higher frequency t

Time

t

t2

Fig. 5.9 Snapshot and vibration graphs at different wavelengths k ¼ v=f and periods T ¼ 1=f

The wavelength k is the distance from crest to crest, trough to trough, or from a point on one wave cycle to the corresponding point on the next adjacent wave cycle (see Figs. 5.8 and also 5.9). The contraction is the process of becoming smaller from the point t1 till t2 . The expansion is the action of becoming larger from the point t2 till t3 (Fig. 5.9). A time period which is denoted by T is the time needed for one complete cycle of vibration from point t1 to pass a given point t3 . As the frequency f of a wave increases, the time period of the wave T ¼ 1=f decreases. An essential aspect of a traveling wave in a material medium is that the wave it is a self-sustaining disturbance of the medium through which it propagates. Since the disturbance is moving, it must be a function of both position and time and can therefore be written as Aðz; tÞ. We can write Helmholtz’s equation for an EM wave with the only components E y and H x : 

 @2 @2 @2 þ þ E y þ k2 Ey ¼ 0: @x2 @y2 @z2

ð5:4:24Þ

A uniform plane wave is characterized by a uniform E y component (uniform magnitude and constant phase) over plane surfaces perpendicular to the direction of propagation of the wave: @ 2 Ey @ 2 Ey ¼ 0 and ¼ 0: @x2 @y2

ð5:4:25Þ

5.4 Plane Waves

287

Then from Eq. (5.4.15), we can write: @ 2 Ey þ k 2 E y ¼ 0: @z2

ð5:4:26Þ

The last equation is as an ordinary differential equation because E y depends only on the z-coordinate. The solution of Eq. (5.4.26) can be expressed as: þ ikz ikz E y ðzÞ ¼ E yþ ðzÞ þ E þ E y ðzÞ ¼ E 0 e 0e :

ð5:4:27Þ

In the last expression, EM wave Eyþ ðzÞ ¼ E 0þ eikz propagates in the positive  þ ik 0 z propagates in the negative direction of z-axis direction of z-axis and E  y ¼ E0 e where k is the wavenumber (propagation constant or phase constant in a lossless media). E0þ and E 0 are arbitrary constants (in general, they are complex numbers). E yþ is called a forward wave, and E  y ðzÞ is called a backward wave (see Eq. (5.4.6)). Note: Pay attention that in optics the forward wave in form Eyþ ¼ E 0þ e þ ikz is usually used and in electrodynamics (EM field theory) the opposite designation E yþ ðzÞ ¼ E 0þ eikz is used. When the waves propagate in a lossy medium, we change the term k to the term pffiffiffiffiffiffiffiffi ka ¼ k0 er lr in order to underline that the permittivity and/or the permeability may have complex values where the index a means an attenuation (loss). A lossy material is a medium in which a significant amount of the energy of a propagating EM wave is absorbed (dissipated, attenuated) per unit distance traveled by the EM wave. In topics about waveguides and resonators (Chaps. 7 and 8), the complex propagation constant will denoted traditionally for microwave electrodinamics as h ¼ h0  ih00 , where h ¼ ka . We investigate the wave propagating in the positive direction of free space:     E yþ ðz; tÞ ¼ Re E yþ ðzÞeixt ¼ Re E 0þ ðzÞeiðxtkzÞ Eyþ ðz; tÞ ¼ E 0þ cosðxt  kzÞ

ðV=mÞ;

ðV=mÞ:

ð5:4:28Þ ð5:4:29Þ

We accept that the second term on the right in Eq. (5.4.29) is a constant, cosðxt  kzÞ ¼ constant; xt  kz is a constant phase, and also this term xt  kz describes the wavefront extending toward the positive z-axis. Plane EM waves are strictly transverse when they propagate in lossless media.

288

5.4.1

5

Plane Electromagnetic Wave Propagation

Relations Between Electric and Magnetic Fields of a Plane Wave in a Vacuum

The solution, representing a plane wave, which is propagating in the direction of the wave vector k0 in a vacuum (which, similarly, will be the case for another lossless substance) from Eqs. (5.2.5) and (5.2.6) may be written in the vector form: E ¼ E0 cosðxt  k0  r þ u0 Þ;

ð5:4:30Þ

H ¼ H0 cosðxt  k0  r þ u0 Þ;

ð5:4:31Þ

where E0 and H0 are amplitude of the electric and magnetic field vectors. For the description of EM wave processes, it is convenient to use the complex form (see   Eq. 1.4.15) E ¼ E0 cosðxt  k0  r þ u0 Þ ¼ 0:5E0 eiu0 eðxtk0 rÞ þ eðxtk0 rÞ . We omit the complex conjugate expression here and accept the amplitude E0 is 0:5 E0 eiu0 , and then on the base of Eq. (5.4.30), we can write a simplified xk x þ ^ yky þ ^zk z is the expression in the complex form E ¼ E0 eðxtk0 rÞ , where k0 ¼ ^ wave vector from Eq. (5.4.2), r ¼ ^xx þ ^yy þ ^zz is radius vector from Eq. (1.2.23) and ðk0  rÞ¼kx x þ ky y þ kz z. In the last form of recording E; the derivatives of time and coordinate can be written as rE ¼ i k0 E (see Chap.1, Ex. 7), r  E ¼ i k0  E and @E=@t¼  ixE. When we substitute the expressions for electric and magnetic fields Eqs. (5.4.30) and (5.4.31) into Maxwell’s Eqs. (4.5.3) and (4.5.4), respectively, we obtain: k0  E0 ¼ 0;

ð5:4:32Þ

k0  H0 ¼ 0:

ð5:4:33Þ

We remember that if the dot product is equal to zero then the vectors are perpendicular to each other; i.e. the angle between them is 90° (Eq. (1.2.44)). Since k0 represents the direction of propagation of the EM wave, the electric and magnetic fields must be perpendicular to the direction to which the wave travels at all places and times. When we substitute the expressions for electric and magnetic fields (5.4.30) and (5.4.31) into Maxwell’ Eqs. (4.5.1) and (4.5.2), respectively, we get: k0  E0 ¼ l0 lr xH0 ;

ð5:4:34Þ

k0  H0 ¼ e0 er xE0 :

ð5:4:35Þ

According to Eq. (1.2.60), the vector product E  H ¼½EH is in the same direction as the direction of propagation of the EM wave k0 and vectors E ? H ? k0 . Figure 5.4 shows that the electric and magnetic fields are perpendicular to the wave vector.

5.4 Plane Waves

289

We will consider the special case in the next paragraph when an EM wave propagates in imperfect (lossy) dielectrics or imperfect conductor media.

5.5

Complex Amplitudes and Attenuation Constant

Considering the harmonic oscillations, we introduce the method of complex amplitudes; i.e. we take the time-dependence expðixt þ u0 Þ and @ 2 =@t2 ! x2 (see Eq. 5.1.4). The complex exponential form of solution for this case is Aðr; tÞ ¼ Am eixt (Eq. 5.4.12) and the homogeneous vector Helmholtz’s Eq. (5.4.18) takes, for the complex amplitudes of EM waves, the following form: r2 Am þ k2 Am ¼ 0;

ð5:5:1Þ

where Am ¼ A0 eiu0 eiðkrÞ is the complex amplitude, which corresponds to Em and Hm at the complex representations E ¼ Em expðixtÞ and H ¼ Hm expðixtÞ; A0 is the amplitude of the wave, which is the peak magnitude of the oscillation, u0 is the initial phase of an EM oscillation, the differential operator r2 is the Laplacian, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ka ¼ jka j ¼ ðx=cÞ eðxÞlðxÞ ¼ k0 eðxÞlðxÞ is the wavenumber, the absolute permittivity eðxÞ and/or permeability lðxÞ can be complex or real numbers. When the propagation of EM waves in media with losses is considered and the wavenumber becomes the complex number, we designate ka instead k as before (Eqs. 5.1.16 and 5.4.19). Note Here and in subsequent text, we will not always mark the complexity by the _ assuming that the magnitude _ k_ a ; h, dot (diacritical mark) above the letter, e.g. e_ ; l; can be complex or real. Equation (5.5.1) is the homogeneous Helmholtz’s equation (partial differential equation) whose solution can be obtained for simple geometries by using the method of separation of variables (also known as the Fourier method, or Fourier’s method of separation of variables). Helmholtz’s Eq. (5.5.1) for one-dimensional case dependent on the coordinate z is: d2 A m ð z Þ þ k2a Am ðzÞ ¼ 0: dz2

ð5:5:2Þ

The ordinary differential Eq. (5.5.2) corresponds to the wave Eq. (5.4.3). The general solution, based on Eq. (5.4.12), can be represented as: Am ðz; tÞ ¼ A0 eiðxtka z þ u0 Þ ¼ Am ðzÞeixt ;

ð5:5:3Þ

Am ðzÞ ¼ A0 eiu0 eika z ¼ Am0 eika z ;

ð5:5:4Þ

where Am0 ¼ A0 eiu0 and Am0 ¼ Am at the origin z ¼ 0:

290

5

Plane Electromagnetic Wave Propagation

The complex wavenumber is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k a ¼ k 0  ik 00 ¼ x lðe0  ie00 Þ;

ð5:5:5Þ

where k0 is the propagation constant (or phase constant) and the real part of the complex wavenumber ka ¼ b  ia; k00 is the attenuation constant and the imaginary part of the complex wavenumber k a which can be separated into real and imaginary parts. We designate the wavenumber k by k a to emphasize that this term is the complex one. Because we take the solution in the form Eq. (5.5.4) which is proportional to expðik a zÞ, we will operate with the magnitude ik a for EM waves propagating in a lossy medium (for lossless medium k ¼ ka ¼ k0 is a real number). The exponential propagation factor for EM waves traveling through a lossy medium eika z is described by the expression: eika z ¼ eaz eibz ;

ð5:5:6Þ

the same notations will be used in subsequent paragraphs, e.g. Sects. 5.9–5.11. We can write the correlation between the values: ik a ¼ iðk0  ik 00 Þ ¼ k00 þ ik 0 ¼ a þ ib;

ð5:5:7Þ

therefore, a ¼ k00 is the attenuation constant and b ¼ k0 is the phase constant when the EM wave propagates through a lossy medium. Substituting Eqs. (5.5.5) and (5.5.7) in Eq. (5.5.3), we will get a convenient expression: 00

0

Am ðz; tÞ ¼ A0 eaz eiðxtbz þ u0 Þ ¼ A0 ek z eiðxtk z þ u0 Þ 00

¼ A0 ek z ðcosðxt  k0 z þ u0 Þ þ i sinðxt  k0 z þ u0 ÞÞ:

ð5:5:8Þ

The instantaneous field can be presented as: 00

Aðz; tÞ ¼ ReðAm ðz; tÞÞ ¼ A0 eaz cosðxt  bz þ u0 Þ ¼ A0 ek z cosðxt  k0 z þ u0 Þ: ð5:5:9Þ From Eqs. (5.5.7) and (5.4.10) when the attenuation constant equals zero, i.e. k00 ¼ a ¼ 0 can be written as: Aðz; tÞ ¼ A0 cosðxt  k 0 z þ u0 Þ ¼ A0 cosðxðt  z=vÞ þ u0 Þ:

ð5:5:10Þ

The last equation coincides with Eqs. (5.4.10) and (5.4.11), which were obtained in the previous paragraph. The EM wave is attenuated when k00 [ 0; then we have to use Eq. (5.5.9). The ratio of two values AðzÞ=Aðz þ ‘Þ indicates how many times the decaying wave amplitude decreased on the length ‘:

5.5 Complex Amplitudes and Attenuation Constant

291

AðzÞ=Aðz þ ‘Þ ¼ expðk00 ‘Þ;

ð5:5:11Þ

where the length ‘ is a distance of the EM wave propagation. The logarithm of the ratio of two values is called an attenuation which can be measured in nepers (Np) and/or decibels (dB):   AðzÞ LNp ¼ ln ¼ lnðexpðk00 ‘ÞÞ ¼ k 00 ‘ðNpÞ; Aðz þ ‘Þ

ð5:5:12Þ

where AðzÞ and Aðz þ ‘Þ are the values of amplitudes at points z and z þ ‘ , ln is the natural logarithm. As we remember the natural logarithm function is the inverse function of the   exponential function elnðbÞ ¼ b at b [ 0 and ln eb ¼ b: The neper is defined in terms of ratios of field quantities, whereas the decibel was originally defined in terms of power ratios.  00  LdB ¼ 20 log ek ‘ 8:69 dB:

ð5:5:13Þ

The decibel (dB) and the neper (Np) have a fixed ratio to each other: 1 dB 0:115129 Np and 1 Np 8:685889 dB:

5.6

Presentation of Maxwell’s Equations for Plane Waves

For uniform plane waves, propagating along the z-axis, variations of the amplitude @ @ on the transverse coordinates are @x ¼ 0 and @y ¼ 0; i.e. the wave has only the amplitude variations in the z-direction. Faraday’s and Ampère’s laws (Eqs. 4.3.1 and 4.3.2) are: r  E ¼ l rH¼e

@H ; @t

@E ; @t

ð5:6:1Þ ð5:6:2Þ

where for the uniform plane wave are E ¼ Ex þ Ey and H ¼ Hx þ Hy because Ez ¼ 0 and Hz ¼ 0: Similarly as in Eqs. (4.3.6–4.3.8), the last two equations can be written for the uniform plane wave: @Ey ðz; tÞ @H x ðz; tÞ ¼l ; @z @t

ð5:6:3Þ

292

5

Plane Electromagnetic Wave Propagation

@Ex ðz; tÞ @H y ðz; tÞ ¼ l ; @z @t 0¼l

@H z ðz; tÞ ; @t

ð5:6:4Þ ð5:6:5Þ

and by analogy: @H y ðz; tÞ @E x ðz; tÞ ¼ e ; @z @t

ð5:6:6Þ

@H x ðz; tÞ @E y ðz; tÞ ¼e ; @z @t

ð5:6:7Þ

0¼e

@E z ðz; tÞ ; @t

ð5:6:8Þ

where e ¼ e0 er and l ¼ l0 lr (see Eqs. 3.27.4 and 3.27.5). Eqs. (5.6.5) and (5.6.8) show that both longitudinal components E z and H z of the plane wave equal zero, i.e. E z ¼ 0 and H z ¼ 0: Hence, the electric and magnetic fields of the plane wave are transverse to the direction of the propagation of an EM wave.

5.7

Intrinsic Impedance

The combination of Eqs. (5.6.3) and (5.6.7) leads to the one-dimensional wave equation in E y : @ 2 E y ðz; tÞ @ 2 E y ðz; tÞ ¼ el ; 2 @z @t2

ð5:7:1Þ

where Ey ¼ ^yEy (see Eq. 5.4.27). The solution of Eq. (5.7.1) can be expressed as a linear superposition to the forward and backward (reverse) waves. The of EM wave extending  wavefront  z toward the positive axis z is xt  kz ¼ x t  v : The solution of Eq. (5.7.1) is composed of two waves Eq. (5.4.6) traveling along the transmission line or medium in forward (along the positive z-axis) Eyþ ðz; tÞ and backward (along the negative z-axis) E y ðz; tÞ directions, respectively:   z z Ey ðz; tÞ ¼ f 1 t  þ f 2 t þ ; v v

ð5:7:2Þ

5.7 Intrinsic Impedance

293

where  z E yþ ðz; tÞ ¼ f 1 t  ; v

 z E : y ðz; tÞ ¼ f 2 t þ v

ð5:7:3Þ

On the base of Eqs. (5.4.27) and (5.5.3), we can write:   þ  ðz; tÞ ¼ E ðzÞ þ E ðzÞ eixt : E y ðz; tÞ ¼ Eyþ ðz; tÞ þ E  y y y

ð5:7:4Þ

  The term f 1 t  vz A0 eu0 eiðxtkz zÞ ¼ W0 eiðxtkz zÞ is the solution for the forward wave (see Eqs. 5.4.12 and 5.5.3). The formula for derivatives of exponential functions is: dðeu Þ=dz ¼ eu ðdu=dzÞ:   0   0 So, for example, the derivative of f 1 t  vz ¼ f 1 t  vz t  vz . The derivative of function E y ðz; tÞ with respect to the variable z, using Eq. (5.7.2), is:   @ E yþ ðz; tÞ þ E  y ðz; tÞ

 z 0  z  z 0  z  t f1 t  þ tþ f2 tþ @z v v v v   1  z z  1 þ  ¼  f1 t   f2 tþ ¼  Ey  Ey : v v v v ð5:7:5Þ

@Ey ðz; tÞ ¼ @z

¼

We can write for the magnetic field Hx ¼ ^xH x of the plane EM wave:   z z H x ðz; tÞ ¼ f 3 t  þ f 4 t þ ; v v    z z ðz; tÞ ¼ f t þ ; H xþ ðz; tÞ ¼ f 3 t  ; H  4 x v v  þ  ixt  H x ðz; tÞ ¼ H xþ ðz; tÞ þ H  x ðz; tÞ ¼ H x ðzÞ þ H x ðzÞ e :

ð5:7:6Þ ð5:7:7Þ ð5:7:8Þ

The derivative of function H x ðz; tÞ with respect to time t, using Eq. (5.7.6), is:   @H x ðz; tÞ z z ¼ f3 t  þf4 tþ ¼ H xþ ðz; tÞ þ H  x ðz; tÞ; @t v v

ð5:7:9Þ

substituting Eqs. (5.7.5) and (5.7.9) into Eq. (5.6.3): 

   1 þ Ey  E ¼ l H xþ þ H  y x : v

ð5:7:10Þ

294

5

Plane Electromagnetic Wave Propagation

From Eq. (5.7.7) according to Eq. (5.4.21):    l  E yþ  E  ¼  pffiffiffiffiffiffi H xþ þ H  y x ; el

ð5:7:11Þ

    E yþ  E  ¼ g H xþ þ H  y x ;

ð5:7:12Þ

where

g¼ g0 ¼

rffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffiffiffi l l0 lr lr ¼ g0 ; ¼ e e0 er er

rffiffiffiffiffi l0 ¼ 120p 376:73 e0

ðXÞ;

ð5:7:13Þ ð5:7:14Þ

where g0 is the intrinsic impedance (or wave impedance) of plane waves in free space, g0 is a real number, e0 and l0 are given in Eq. (2.5.2). The intrinsic impedance has dimensions of ohms. We can write from Eq. (5.7.12): H x ¼ H xþ þ H  x ¼ 

E yþ E y þ : g g

ð5:7:15Þ

 The term g is the ratio of E  y and H x in a traveling (propagating) in positive or negative direction plane wave. The term g can be considered as a characteristic of a medium, and it is known as the intrinsic impedance of the medium. Note The word “intrinsic” means “being part of a thing by its very nature, inherent,” synonyms: native, natural, true, real, essential. The wave impedance of an EM wave is the ratio of the transverse components of the electric and magnetic fields of the wave. For a transverse-electric-magnetic (TEM) plane wave propagating through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. Do not confuse this with the characteristic impedance of an infinitely long uniform transmission line (waveguide). The characteristic impedance in simplified form is the ratio of the amplitudes of voltage and current of a single wave propagating along the line, i.e. a wave traveling in one direction in the absence of reflections from the load in the other direction. The characteristic impedance of the transmission line is then: Z 0 ¼ V ðtÞ=I ðtÞ; where V ðtÞ is an alternating voltage and I ðtÞ is an alternating current. Characteristic impedance Z 0 is determined by the geometry and materials of the transmission line and, for a uniform line, does not depend on its length. The combination of Eqs. (5.6.4) and (5.6.6) leads to the one-dimensional wave equation in E x :

5.7 Intrinsic Impedance

295

@ 2 E x ðz; tÞ @ 2 E x ðz; tÞ ¼ e l ; @z2 @t2   z z Ex ðz; tÞ ¼ f 5 t  þ f 6 t þ ; v v    z z ðz; tÞ ¼ f t þ ; E xþ ðz; tÞ ¼ f 5 t  ; E  6 x v v Ex ðz; tÞ ¼ E xþ ðz; tÞ þ E x ðz; tÞ:

ð5:7:16Þ ð5:7:17Þ ð5:7:18Þ ð5:7:19Þ

The derivative of E x ðz; tÞ with respect to the variable z, using Eq. (5.7.17), can be expressed as:    @Ex ðz; tÞ @ Exþ þ E  z 0  z  z 0  z  x ¼ ¼ t f5 t  þ tþ f6 tþ @z v v v v @z ð5:7:20Þ   1  z z  1 þ  f t   f6 tþ ¼ ¼  Ex  Ex ; v 5 v v v    z z H y ðz; tÞ ¼ f 7 t  þ f 8 t þ ; ð5:7:21Þ v v   z z H yþ ðz; tÞ ¼ f 7 t  ; H  ; ð5:7:22Þ y ðz; tÞ ¼ f 8 t þ v v H y ðz; tÞ ¼ H yþ ðz; tÞ þ H  y ðz; tÞ;   @H y ðz; tÞ z z ¼ f7 t  þf8 tþ ¼ H yþ ðz; tÞ þ H  y ðz; tÞ: @t v v

ð5:7:23Þ ð5:7:24Þ

After placing Eqs. (5.7.20) and (5.7.24) into (5.6.4), we obtain: 

   1 þ þ  Ex  E þ H ¼ l H x y y ; v  þ    Ex  E x ¼ H yþ þ H  y ; g H y ¼ H yþ þ H  y ¼

E xþ E   x : g g

ð5:7:25Þ ð5:7:26Þ ð5:7:27Þ

We can rewrite the intrinsic impedance from Eqs. (5.7.15) and (5.7.27) to compare them to each other as: Eyþ E xþ ¼ g; þ ¼  Hy H xþ

ð5:7:28Þ

296

5

Plane Electromagnetic Wave Propagation

Fig. 5.10 Relationship between the electric and magnetic fields for a plane wave propagating in z-direction

y

ɳH ɳHy

E

ɳHx

Ey

Ex 0

E E y x ¼  ¼ g: H H y x

x

ð5:7:29Þ

Figure 5.10 shows the location of the electric and magnetic field components. Figure 5.10 shows the mutual arrangement of the electric field E ¼ Ex þ Ey ¼ ^xE x þ ^yEy and the magnetic field H ¼ Hx þ Hy ¼ ^ xH x þ ^ yH y . The electric E and magnetic H fields are perpendicular to each other in every traveling wave (see Eqs. 5.4.32 and 5.4.33).

5.8

Time-Periodic Case for a Plane Wave

Periodic EM waves repeat regularly over a certain period of time. For the time-periodic case, the electric and magnetic field strengths (Fig. 5.11) and other magnitudes exponentially depend on time. For this reason, we can express magnitudes like this: Eðr;tÞ ¼ EðrÞeixt ; Hðr; tÞ ¼ HðrÞeixt ; @Eðx; y; z; tÞ=@t ¼ ix Eðx; y; zÞ. In case where there is only one component Ey ¼ ^ yE y of the electric field E ¼ ^xE x þ ^yEy þ ^zE z ; E x ¼ Ez ¼ 0; is the only one component of the magnetic field Hx ¼ ^xH x , and we can write from Faraday’s law (Sect. 4.2.1) for the forward wave (Eq. 5.7.2):

y

Fig. 5.11 Plane monochromatic EM wave and the direction of it propagation

H

E 0

λ λ

4

H

x

2

3 2λ

λ

E

z P=[EH]

5.8 Time-Periodic Case for a Plane Wave

^x r  E¼ 0 0

297

^z @Eyþ @=@z ¼ ^ x ; @z 0

^y 0 Eyþ

ð5:8:1Þ

by using the time-periodic case Eq. (4.4.5), we can write for a time-periodic EM wave propagating in a vacuum:   r  E¼  ix l0 x^H xþ þ ^yH yþ þ ^zH zþ ¼ ix l0 ^ xH xþ ;

ð5:8:2Þ

because for the considered plane EM wave only one component of electric field Ey and one component of magnetic field H x exist, we can write from Eq. (4.3.6): H xþ ¼ 

þ

1 @E y ; ix l0 @z

H yþ ¼ 0; H zþ ¼ 0:

ð5:8:3Þ ð5:8:4Þ

Since the solution of Helmholtz’s Eq. (5.4.26) is Eq. (5.4.27), we can write for the forward wave:   @Eyþ ðzÞ @ E 0þ eik0 z ¼ ¼ ik 0 E yþ ðzÞ; @z @z

ð5:8:5Þ

where the electric field of EM wave propagating in a vacuum is E yþ ðzÞ ¼ E 0þ eik0 z . Upon combining Eqs. (5.8.3) and (5.8.5), we get: H xþ ¼

k0 þ E ðzÞ: x l0 y

ð5:8:6Þ

From Eq. (5.8.6) using Eq. (5.4.19), we get: H xþ

rffiffiffiffiffi pffiffiffiffiffiffiffiffiffi x e0 l0 þ k0 þ e0 þ E ðzÞ: ¼ E ðzÞ ¼ E y ðzÞ ¼ xl0 x l0 y l0 y

ð5:8:7Þ

We can write for the EM wave propagating in a vacuum by using Eq. (5.7.14): H xþ ¼

E yþ ðzÞ ; g0

ð5:8:8Þ

H xþ is in the phase with E yþ ðzÞ ¼ E0þ eik0 z . We can write the instantaneous expression for the magnetic field:

298

5

Plane Electromagnetic Wave Propagation

  Eþ Hðz; tÞ ¼ ^yH xþ ðz; tÞ ¼ ^yRe H xþ ðzÞeixt ¼ ^y 0 cosðxt  k 0 zÞ ðA=mÞ: ð5:8:9Þ g0 From Eq. (5.8.9), we can also receive the same expression as Eq. (5.8.8): H xþ ðzÞ ¼

E yþ ðzÞ : g0

ð5:8:10Þ

The magnetic and electric fields of the time-periodic plane wave are in the phase. We can write from Sect. 4.3.1: ^x ^y ^z @H xþ 0 @=@z ¼ ^ y r  H¼ 0 ; ð5:8:11Þ @z Hþ 0 0 x

  @H xþ ðzÞ @ H 0þ eik0 z ¼ ¼ ik 0 H xþ ðzÞ; @z @z

ð5:8:12Þ

where H xþ ðzÞ ¼ H 0þ eik0 z . Taking into account Eq. (5.6.7) and Eðz; tÞ ¼ E y ðzÞeixt :   @ E y ðzÞeixt @H xþ ðzÞ ¼ e0 ¼ i x e0 E yþ ðzÞ: @z @t

ð5:8:13Þ

After combining Eqs. (5.8.12) and (5.8.13): rffiffiffiffiffi pffiffiffiffiffiffiffiffiffi E yþ ðzÞ x e0 l0 k0 l0 ¼  ¼ g: ¼  ¼  þ x e0 H x ðzÞ x e0 e0

ð5:8:14Þ

The last equation coincides with Eq. (5.7.28).

5.9

Plane Waves in Lossy Media

The homogeneous vector Helmholtz’s equation given in Eq. (5.5.2) can be written for a lossy medium as follows: r2 E þ k2a E ¼ 0;

ð5:9:1Þ

where the complex number k a is in a lossy medium (see Sect. 5.5). The ratio e00 =e0 characterizes a loss tangent, which is a measure of power loss in the lossy medium (Eq. 3.11.3). Since we take the solution in the form Eq. (5.5.4) which is

5.9 Plane Waves in Lossy Media

299

proportional to eika z , we will operate with magnitude ik a in a lossy medium. The exponential propagation factor for phasor waves can be expressed as eika z ¼ eaz eibz (Eq. 5.5.6), and the EM wave attenuates as it propagates through the lossy medium and the attenuation depends upon the dielectric and conductor losses of the medium. The displacement current density resulting from the motion of bound electrons in atoms and ions in molecules (Eqs. 3.11.4 and 3.12.1) is:   Jd ¼ ix eE ¼ ix e0  ie00b E;

ð5:9:2Þ

where e00b is the dielectric loss and it can associate itself with bound charges and dipole relaxation phenomena (Eq. 4.6.5 and Sect. 3.5). The conductivity loss r=x is caused by the movement of free charges (Eq. 4.6.3). There is a conduction current density of a material which contains free charge currents (electrons or holes): J ¼ rE:

ð5:9:3Þ

The total current density is:    r r  J ¼ ix e0  ie00b  i E ¼ixðe0  ie00 ÞE ¼ ix e0  i e00b þ E; x x

ð5:9:4Þ

where e00 ¼ e00b þ xr ; e0 and e00 can be functions of frequency. There is typically a small number of free electrical carrier charges and r 0 in some insulators (Eq. 4.6.2). The conduction current dominates and the phenomenon of the bound electrons e00b is included in the conductivity in semiconductors and some metals. For the dielectric material with a real permeability and a complex permittivity, we can write: r  H ¼ ix e E ¼ ixðe0  ie00 ÞE;

ð5:9:5Þ

where e00 is the loss factor. The complex wavenumber (or complex propagation constant) in a lossy dielectric medium is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi k a ¼ x e l ¼ x ðe0  ie00 Þl;

ð5:9:6Þ

e ¼ e0 er ¼ e0  ie00 is the complex absolute permittivity. Similar case can also be with l ¼ l0  il00 , a magnetic loss tangent tan dm ¼ l0 =l00 , but here we take only the real value of l ¼ l0 .

300

5

Plane Electromagnetic Wave Propagation

The ratio e00 =e0 is called a loss tangent which is the measure of the power loss in 00 the medium, and the term tan d ¼ ee0 xre0 ; d is the loss angle. We separate the wavenumber on the real and imaginary parts: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   00    00 ffi pffiffiffiffiffiffi e e 0 0 1i 0 ¼ i x le ; ik a ¼ a þ ib ¼ ix l e 1  i 0 e e

ð5:9:7Þ

a is the attenuation constant and b is the phase constant (Sect. 5.5). The term of complex wavenumber multiplied by the imaginary unit can be expressed as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi r 1=2 ik a ¼ ix e l ¼ ix ðe0  ie00 Þl ¼ ix e0 l 1 þ ix e0



 m1 :

ð5:9:8Þ

We denoted the term of Eq. (5.9.7): s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  00 ffi e : Root1 ¼ 1i 0 e We know from mathematics that the square root z ¼ a þ ib is:

ð5:9:9Þ pffiffi z of a complex number

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0sp 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 þ b2  a p ffiffiffiffiffiffiffiffiffiffiffi ffi pffiffi a þ a a þ b A: þ i  sgn b z ¼ a þ ib ¼ @ 2 2

ð5:9:10Þ

We can write Eq. (5.9.9) using Eq. (5.9.10) as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  00 ffi  00   00  e e e Root1 ¼ 1i 0 ; where z ¼ 1  i 0 ; a ¼ 1; b ¼  0 ; ð5:9:11Þ e e e then ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 vq 1 u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e00 2 e00 2 u u Bt 1 þ e0 þ 1 t 1 þ e0  1C C; i Root1 ¼ B @ A 2 2

ð5:9:12Þ

5.9 Plane Waves in Lossy Media

301

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ik a ¼ a þ ib ¼ ix l e0  Root1 ¼ ix l e0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 vq 1 u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e00 2 e00 2 u u B t 1 þ e 0 þ 1 t 1 þ e 0  1C C; i B @ A 2 2

ð5:9:13Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ffivq 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e00 2  00 2 u u pffiffiffiffiffiffiffi B t 1 þ e0 þ 1 t 1 þ ee0  1C C; ik a ¼ h ¼ a þ ib ¼ x l e0 B þ @i A 2 2 ð5:9:14Þ where h is the propagation constant in waveguides (Chap. 7). We can write the attenuation constant: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffi u     u 2 2 u l e0 t e00 e00 ul e 0 @ a¼x  1 ¼ xt 1þ 0 1 þ 0  1A ;  2 e 2 e

ð5:9:15Þ

and the phase constant: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffi u    00 2 u 0 2 u 0 00 le t e e ul e @ b¼x 1 þ 0 þ 1 ¼ xt 1 þ 0 þ 1A :  2 e 2 e

ð5:9:16Þ

We see that the low increase of the phase constant b exists in the dielectric media with some losses because of the term ðe00 =e0 Þ2 . The exponential propagation factor for EM waves (Sect. 5.5) is: eika z ¼ eaz eibz :

ð5:9:17Þ

An EM wave propagating through a lossy medium attenuates. The attenuation depends on the dielectric and conductor losses (Eq. 5.9.4). For the intrinsic impedance for lossy media, when the uniform plane wave propagates along it, using Eq. (5.7.13), we can write:  00 1=2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi l l e g¼ ðXÞ: 1i 0 ¼ 0 00 0 0 e ð1  iðe =e ÞÞ e e

ð5:9:18Þ

Using Eq. (5.4.19) when l is a real term and e0 is the real term of e ¼ e0  ie00 , pffiffiffiffiffiffiffi we can express that real value of the wavenumber k ¼ x l e0 (Eq. 5.4.19), then the attenuation constant is:

302

5

Plane Electromagnetic Wave Propagation

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi usffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  00 2 k u e t a ¼ pffiffiffi  1 þ 0  1; e 2

ð5:9:19Þ

and the phase constant is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi usffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  00 2 u k t e 1 þ 0 þ 1: b ¼ pffiffiffi  e 2

ð5:9:20Þ

For low-loss media when the ratio e00 =e0 1, we can use the binominal theorem: ða þ bÞn ¼ an þ nan1 b þ

nðn  1Þ n2 2 nðn  1Þðn  2Þ n3 3 a b þ a b þ . . .; 12 123 ð5:9:21Þ

which for our case is:   ðe00 =e0 Þ2 ðe00 =e0 Þ4 ðe00 =e0 Þ6 2 1=2  þ  . . .: 1 þ ðe00 =e0 Þ ¼ 1þ 2 8 16

ð5:9:22Þ

Equation (5.9.19), after using Eq. (5.9.21), gives the following attenuation constant: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðe00 =e0 Þ2 ðe00 =e0 Þ4 ðe00 =e0 Þ6 a ¼ pffiffiffi  1 þ  þ . . .  1: 2 8 16 2

ð5:9:23Þ

Since for low-loss media the ratio is e00 =e0 1; we can take only the second term of the binominal series (the first term is reduced with the last term): k a pffiffiffi  2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe00 =e0 Þ2 k e00 0: 2e 2

ð5:9:24Þ

The phase constant b from Eq. (5.9.20) and after using Eq. (5.9.21) is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðe00 =e0 Þ2 ðe00 =e0 Þ4 ðe00 =e0 Þ6  þ þ    þ 1; b pffiffiffi  1 þ 2 8 16 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðe00 =e0 Þ2 k ðe00 =e0 Þ2 2 : b pffiffiffi 2 þ 4 þ ðe00 =e0 Þ k 1 þ 2 2 4 2

ð5:9:25Þ

ð5:9:26Þ

5.9 Plane Waves in Lossy Media

303

Upon using Eq. (5.9.21) to Eq. (5.9.26), we can write: ! ðe00 =e0 Þ2 b k 1þ : 8

ð5:9:27Þ

For a medium as a semiconductor when the losses are mainly conductive:  r  r  H ¼ ixðe0  ie00 ÞE ¼ ðix e0 þ rÞE ¼ ix e0 1  i 0 E: xe

5.10

ð5:9:28Þ

Plane Waves in Good Conductors

For conductors when xre0 1 where the real part of the permittivity e0 ¼ Reðe0 er Þ , r is the electrical conductivity (Sect. 4.6):   r  r  r  H ¼ ixðe0  ie00 ÞE ¼ix e0 1  i 0 E ix e0 i 0 E ¼ rE: ð5:10:1Þ xe xe Using Eq. (5.9.6), where e00 ¼ e00b þ xr ; e00b reflects losses caused by movement of bounded charges and r=x is the conductivity loss, which is caused by movement of free charges: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   r  ka ¼ x l e0  i e00b þ ; x

ð5:10:2Þ

when the loss due to free electrons is significantly higher than the loss of bounded charges, then e00b xr and we can take: e00 Since for good conductors

r xe0

r : x

ð5:10:3Þ

1; we can write from Eq. (5.10.2):

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi   00    r  e lr ix i ik a ¼ a þ ib ¼ ix l e0 1  i 0 ¼ ix l e0 1  i x e0 x e rffiffiffiffiffiffiffi lr ix ; ix ð5:10:4Þ

304

5

Plane Electromagnetic Wave Propagation

we can express the term as: rffiffiffiffiffiffiffi l r pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ix ¼ il r x ¼ 2ipf l r: ix

ð5:10:5Þ

pffiffiffi pffiffiffi pffi Because i ¼ eip=2 and i ¼ eip=4 ¼ cosðp=4Þ þ i sinðp=4Þ ¼ 2=2 þ i 2=2 ¼ pffiffiffi ð1 þ iÞ= 2 and 2i ¼ ð1 þ iÞ2 , we can write: ik a ¼ a þ ib ¼ ð1 þ iÞ a¼b¼

pffiffiffiffiffiffiffiffiffiffiffiffi p l rf ;

ð5:10:6Þ

pffiffiffiffiffiffiffiffiffiffiffiffi p l rf ;

ð5:10:7Þ

when the term: d ¼ 1=

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p l rf ¼ 1= p lr l0 r f ðmÞ;

ð5:10:8Þ

d is the depth of EM wave penetration (skin depth) (Fig. 5.12). The depth of EM wave penetration, i.e. skin depth, is a measure of the depth at which EM wave can penetrate into a material. It can be defined as the depth at which the intensity of the electric field inside the material falls to 1/e of its original value at the surface. We can write Eq. (5.10.6) using Eq. (5.10.8) in the form: ik a ¼ a þ i b ¼

1þi ; d

ð5:10:9Þ

where the attenuation constant is a ¼ 1=d; the phase constant is b ¼ 1=d: The exponential propagation factor eika z from Eq. (5.9.17) gives that the magnitude of the wave decreases exponentially and has decreased to 1/e (1/e =1/ 2.711828 = 0.3678796–0.368) of its original value after propagating at a distance pffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 1= p l rf of penetration into a good conductor (Fig. 5.12).

Fig. 5.12 Skin depth d of EM wave

y E0

e

z/δ

0.368· E0

0

z

δ

5.10

Plane Waves in Good Conductors

305

Table 5.1 Skin depth for different media Medium

Skin depth d f ¼ 103 Hz

f ¼ 106 Hz

f ¼ 109 Hz

Electrical conductivity r (S/m)

Copper

2:06 mm

65:20 lm

2:06 lm

58:5  106

0.999991

Gold

2:38 mm

75:32 lm

2:38 lm

44:2  106

1

Aluminum

2:593 mm

82 lm

2:59 lm

36:9  10

1.000022

14:3  10 4 0:01

Nickel

169:93 lm

5:37 lm

0:17 lm

Seawater Wet soil

7:96 m 159 m

0:25 m 5:03 m

7:97 mm 0:16 m

6 6

Relative permeabilitylr ¼ l=l0

600 1 1

Skin depth for different media at three different frequencies is given in Table 5.1. We can see how strongly the magnitude d depends on the frequency and electrical parameters of a material. The permeability of free space l0 is given in Eq. (2.5.2). The phase speed in the media is: vp ¼

  x 2p ¼ xd ¼ c k 0 d ¼ c  d b k0 k0 ¼

2p b

ðm=sÞ;

ðmÞ;

ð5:10:10Þ ð5:10:11Þ

where the wavenumber k0 ¼ x=c ¼ 2p=k0 ; k0 is free-space wavelength,   c is the 2p velocity of light in free space. We see from Eq. (5.10.10) when k0  d is very small, the phase speed is usually much less than the speed of light c (Eq. 2.1.6), l ¼ l0 lr . For a good conductor, the intrinsic impedance, when the uniform plane wave propagates along it, e00 =e0 1 and from Eq. (5.9.18): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi l0 lr l0 lr il0 lr x g¼ 0 00 0 00 r e ð1  iðe =e ÞÞ ie rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi ð1 þ iÞ l2pf lpf pffiffiffi ð1 þ i Þ ; r r 2 pffiffiffi eip=4 2 1 þ i ¼ ¼ ð1 þ iÞRs ; g¼ rd rd

ð5:10:12Þ

ð5:10:13Þ

306

5

Plane Electromagnetic Wave Propagation

where the surface electrical resistivity (also called specific electrical resistance) is: 1 ¼ Rs ¼ rd

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p l0 lr f : r

ð5:10:14Þ

This is the resistance of a conductor slab with a unit width ðw ¼ 1Þ and unit length ð‘ ¼ 1Þ: Value Rs is essentially equal to the dc resistance Rdc (Sects. 3.13 and 3.14) for a unit width and length of the conductor with cross-sectional area S ¼ d  w: The dc resistance: Rdc ¼

‘ : rS

ð5:10:15Þ

The dc resistance of a given conductor grows with the length of the conductor and decreases for larger cross-sectional area and conductivity of the material. In the term S ¼ d  w (Eq. 5.10.15), d is the thickness of a conductor slab. For any width w and length ‘, the ac (alternating current) resistance Rac can be calculated using the dc (direct current) resistance: Rac ¼

‘ Rs ‘ ¼ : r dw w

ð5:10:16Þ

A carrying alternating current flowing on the conductor slab has a reduced effective cross-sectional area because of the skin effect. In Eq. (5.10.16), the ac resistance Rac is expressed through the term d which is the skin depth (Fig. 5.13). The skin depth can serve us by determining the ac resistance Rac due to skin effect. Figure 5.13 shows a conductive slab (width “w ”, length “l” and thickness exceeds depth d) with a current density J flowing through a cross-section (gray plane) with d—depth and w—width. The surface electrical resistivity Rs of this slab is calculated by Eq. (5.10.14). Figure 5.13 shows the direction of electric flux current density J; as well as directions of the electric and magnetic fields. We assumed that the high-frequency current is flowing uniformly over the cross-section determined by the skin depth d and the width w of the conductor slab. The resistance of the slab is:

Fig. 5.13 Conducting medium with a sinusoidal current and skin depth

^n

w J

δ

J

H

E

5.10

Plane Waves in Good Conductors

307

rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi p lf xl : Rs ¼ ¼ r 2r

ð5:10:17Þ

^  H; where Js is the surface current per unit width, H is the Vectors Js ¼ n ^ is a unit vector perpendicular to the conductor magnetic field at the surface, n surface at any given points and pointing into adjoining dielectric region. Skin effect is the property of an alternating electric current I (Sects. 3.13 and 4.4) to become distributed inside conductors (metals, semiconductors, etc.), and it is carried out in such a way that the current density J (the current per unit area, see Sect. 3.14) is at its greatest near the surface of the conductor material and it quickly decreases with increasing depth inside the conductor. The electric current flows primarily at the “skin” layer of the conductor, between the external surface (or interface) and a level, called the skin depth. The skin effect is due to opposing eddy currents (Sect. 3.13.4) induced by the changing magnetic field resulting from the alternating current. The skin effect causes the effective resistance of the conductor medium which increases at higher frequencies (Eq. 5.10.17). The skin depth is a measure of the depth at which the current density J decreases to 1=e of its value near the surface. At high frequencies, the skin depth d becomes much smaller. So for good conductors, all currents and fields concentrate in a thin region or skin layer near the surface for time-varying fields and this region approaches zero thickness as the conductivity approaches infinity. For comparison, in perfect electric conductors (PEC) with r ! 1 (Sect. 4.6 and Fig. 4.32) the electric field is zero inside the conductor and any current flow must be only on the surface. Because the electric field equals to zero within the perfect (ideal) conductor, the continuity of the tangential electric field at an interface means that the surface tangential electric ^  E ¼ 0; the normal field Et exists only outside of the conductor E t ¼ 0; i.e. n electric flux density is Dn ¼ qs ; qs is a surface charge density, existing on the ^  D: The magnetic field also vanishes inside the coninterface (boundary), qs ¼ n ^  B ¼ 0: For the perfect conductor, the ductor Bn ¼ 0 at the conductor surface, i.e. n finite current density J per unit width is assumed to flow on the surface as a current layer with a thickness ! 0; so the current density ! 1: The current enclosed by path is the current per unit width J ¼ Js flowing on the surface of the conductor perpendicular to the direction of the tangential magnetic field at the surface. We can write for conducting medium on the base of Eqs. (4.10.3–4.10.9) using Fig. 4.16: I Z H  d‘ ¼ lim J  ds ! H t d‘¼J s d‘; ð5:10:18Þ lim d!0 jd‘12 j!n c

d!0 jd‘12 j!n

S

where d‘ is the differential vector element of path length tangential to the path “c”, H t is the tangential component of the magnetic field to the interface, and the current per unit width, i.e. a surface current density J s ¼ jJs j, is:

308

5

Js ¼ Ht

Plane Electromagnetic Wave Propagation

ðA=mÞ;

ð5:10:19Þ

the surface current density in the vector form is: ^  H: Js ¼ n

ð5:10:20Þ

Note about the term “ohmic conductor”: A conductor medium can be ohmic or not. The main difference between an ohmic and a non-ohmic conductor is whether they follow ohmic law. An ohmic conductor would have a linear relationship between the current density and the electric field (or the current and the voltage). Ohmic one is a medium that satisfies Ohm’s law, which can be written as J ¼rE; where J is the current density and r is the electrical conductivity of the medium. Non-ohmic conductors do not follow Ohm’s law and have their own electrical characteristics. Simple examples of non-ohmic conductors are bulb filaments and semiconductors like diodes and transistors.

5.10.1 Phase Difference Between E and H Fields in Good Conductors Let the EM wave propagates in a good conductor to the direction of the z-axis. The phase difference between the electric and magnetic fields in a good conductor is given by the tangent of an angle: tan u ¼

b 1; a

ð5:10:21Þ

where a is the attenuation constant and b is the phase constant. So the phase difference (phase angle, phase shift) is approximately u ¼ 45 at tan u ¼ 1 (Eq. 5.10.7). The electric field E leads (is ahead of) the magnetic field H by u ¼ 45 ; i.e. in other words, the magnetic field H is lagging behind the electric field (see Sect. 5.2). E¼^yE0 eaz cosðxt  bzÞ ¼ ^yE0 ez=d cosðxt  z=dÞ:

ð5:10:22Þ

This EM wave is traveling along the positive z-axis and is called the forward wave (see Eq. 5.4.27). Taking into account Eqs. (5.8.14) and (5.10.12) when H xþ ðzÞ ¼ Eyþ ðzÞ=g; the magnetic field is determined by: E0 E0 H ¼ ^x pffiffiffiffiffiffiffiffiffiffiffiffi eaz cosðxt  bz  uÞ ¼ ^x pffiffiffiffiffiffiffiffiffiffiffiffi eaz cosðxt  bz  p=4Þ; xl=r x l=r ð5:10:23Þ

5.10

Plane Waves in Good Conductors

309

y

Fig. 5.14 Plane monochromatic wave H- field lags behind E- field by 45 in a good conductor

e

z/δ

E 0

z H

P=[EH]

x

E0 H ¼ ^x pffiffiffiffiffiffiffiffiffiffiffiffi ez=d cosðxt  z=d  p=4Þ: x l=r

ð5:10:24Þ

When an EM wave propagates in a conductor medium, its amplitude is attenuated by the factor eaz ¼ ez=d (Eq. 5.10.9; Fig. 5.14). Skin depth d measures the exponential damping of the wave as it travels through the conductor. The effect whereby EM field in a conductor quickly decreases is known as the skin effect. The electric current and EM field are confined to a very thin layer d of the conductor surface. The phase difference (phase angle, phase shift) between the electric and magnetic fields in a good conductor is approximately p=4:

5.11

Plane Waves in Low-Loss Dielectrics

Low-loss dielectric is an imperfect insulator with a low electrical conductivity r (Sect. 4.6). For an imperfect insulator e00 e0 ; r 6¼ 0 and r=x e 1: The expression of Eq. (5.9.7) can be approximated by using the binomial 1 3 x þ    , when x ¼ iðe00 =e0 Þ; we can expansion ð1  xÞ1=2 ¼ 1  12 x þ 18 x2  16 write:   00 1=2 pffiffiffiffiffiffiffi e ik a ¼ h ¼ a þ ib ¼ ix l e0 1  i 0 e "  00   00 2 # pffiffiffiffiffiffiffi 1 e 1 e ¼ ix l e0 1  i þ : 2 e0 8 e0

ð5:11:1Þ

310

5

Plane Electromagnetic Wave Propagation

The phase constant b from Eq. (5.11.1) is: "   # pffiffiffiffiffiffiffi 1 e00 2 0 b ¼ x le 1þ 8 e0

ðrad=mÞ:

ð5:11:2Þ

The phase constant is taken in Eq. (5.11.2) but the magnitude of b changes very pffiffiffiffiffiffi slightly from the value x l e because e00 e0 . The phase velocity is: "   # x 1 1 e00 2 vp ¼ ffi pffiffiffiffiffiffi0ffi 1  b 8 e0 le

ðm=sÞ:

ð5:11:3Þ

From Eq. (5.11.1), the attenuation constant is: a¼

x e00 2

rffiffiffi l e0

ðNp=mÞ:

ð5:11:4Þ

We see that the attenuation constant a of a low-loss dielectric is a positive term and is directly proportional to the frequency x and the imaginary part of the permittivity e00 (Eq. 5.11.4). The intrinsic impedance of a low-loss dielectric from Eq. (5.9.18) is: rffiffiffi  00 1=2 rffiffiffi  l e l 1 e00 g¼ ffi 1i 0 1þi 0 e0 e0 2e e

ðXÞ:

ð5:11:5Þ

The intrinsic impedance Eq. (5.7.28) is the ratio of the electric and magnetic components for a uniform plane wave.

5.12

Flow of Electromagnetic Power and the Poynting Vector

Poynting vector is named after Prof. John Henry Poynting (1852–1914). He was an English physicist. From 1872 to 1876, he was a student at Cambridge University where he attained high honors in mathematics. In 1884, J. H. Poynting published a theorem for the conservation of energy in an EM field and an expression for the flow of EM energy, which are known, respectively, as the Poynting theorem and the Poynting vector. Nikolay Umov (1846–1915) and Oliver Heaviside (1850–1925) also independently discovered the Poynting vector. EM wave carries energy, and as the wave propagates through space, the wave can transfer energy to physical bodies placed in its path. The magnitude of the Poynting vector represents the rate at which energy flows through a unit surface

5.12

Flow of Electromagnetic Power …

311

area perpendicular to the direction of wave propagation. A direction of the Poynting vector is along the direction of wave propagation. The curl Maxwell Eqs. (4.3.1) and (4.3.2) are: rE¼

@B ; @t

r  H ¼ Jþ

@D ; @t

ð5:12:1Þ ð5:12:2Þ

where J is determined in Sect. 3.14. The verification of the following identity of vector operations is: r  ðEHÞ ¼ H  ðr  EÞ  E  ðr  HÞ;

ð5:12:3Þ

it is the same as divðEHÞ ¼ H  curlE  E  curlH: The last equation is well known from the vector differential operators of mathematics where the vector identity div ðABÞ ¼ r  ðABÞ ¼ B  ðr  AÞ  A  ðr  BÞ; in our case A ¼ E and B ¼ H: The substitution of Eqs. (5.12.1) and (5.12.2) in Eq. (5.12.3) yields: r  ðE  HÞ ¼ H 

@B @D E  E  J: @t @t

ð5:12:4Þ

In a simple medium (Sect. 3.27.1), whose constitutive parameters e; l and r do not change over time, we have:   @B @ðlHÞ 1 @ðlH  HÞ @ 1 2 H ¼H ¼ ¼ lH ; @t @t 2 @t @t 2   @D @ ðeEÞ 1 @ ðeE  EÞ @ 1 2 ¼E ¼ ¼ eE ; E @t @t 2 @t @t 2 E  J ¼ E  ðrEÞ ¼ rE 2 :

ð5:12:5Þ ð5:12:6Þ ð5:12:7Þ

Equation (5.12.4) can then be written as: r  ðE  HÞ ¼ 

@ 1 2 1 2 ð eE þ lH Þ  rE2 : @t 2 2

ð5:12:8Þ

On the left side of Eq. (3.12.8), there is the divergence of vector cross-product (Sects. 1.2.9 and 1.3.6). An integral form of Eq. (5.12.8) is obtained by integrating both sides over volume of concern:

312

5

I S

@ ðE  HÞ  ds ¼  @t

Total power leaving the volume

Plane Electromagnetic Wave Propagation

 Z  Z 1 2 1 2 eE þ lH dv  rE2 dv; 2 2

V Rate of decrease in energy stored in the electric and magnetic fields

ð5:12:9Þ

V Ohmicpower dissipated

where the divergence theorem (Sect. 1.3.6) has been applied to convert the volume integral of r  ðEHÞ to the closed surface integral of E  H: The absolute permittivity e and permeability l are given in Sect. 3.6 and Eq. (3.19.12). We recognize that the first and second terms on the right side of Eq. (5.12.9) represent the time rate of change of energy stored in the electric and magnetic fields, respectively. The last term is the ohmic power dissipated in the volume as a result of the flow of conduction current density rE in the presence of the electric field. Therefore, we may interpret the right side of Eq. (5.12.9) as the rate of decrease of the electric and magnetic energies stored and subtracted by the ohmic power dissipated as heat in the volume V: To be consistent with the law of conservation of energy, this must be equal to the power (rate of energy) leaving the volume through its surface. Thus the quantity EH is a vector representing the power flow per unit area: P ¼ ½EH



 W=m2 :

ð5:12:10Þ

Quantity P is the cross-product of the electric E and magnetic H fields and is known as the Poynting vector which is a power density vector associated with an EM field. Vector P describes the rate of EM energy transfer per unit area (per unit time). So we can say that the Poynting vector expresses the directional energy flux density of a traveling EM wave (Figs. 5.11 and 5.14). In SI units, the Poynting vector has the dimensions of power per area W/m2 (watt per square meter). [Note Power as a function of time, PðtÞ; is the rate of doing work and it is the amount of energy W used per unit time PðtÞ ¼ W=t: Energy flux is the rate of transfer (move from one place to another) energy through a surface. Energy is the property (attribute) that must be transferred to a physical body in order to execute work on or to heat the physical body. Energy cannot be created or destroyed.] Equation (5.12.9) can be written in another form: I  S

@ P  ds ¼ @t

Z

Z ðW E þ W M Þdv þ

V

qr dv;

ð5:12:11Þ

V

 1 energy e0 er  2 ¼ E x þ E2y ¼ eE2 ¼ Electric energy density; ð5:12:12Þ volume 2 2   energy l0 lr 1 ¼ H 2x þ H 2y ¼ lH 2 ¼ Magnetic energy density: WM ¼ volume 2 2 ð5:12:13Þ

WE ¼

5.12

Flow of Electromagnetic Power …

313

The electric and magnetic vectors are allowed to take on complex values. The energy density can be expressed in this case as W E ¼ 12 eE  E and W M ¼ 12 lH  H  , where E  is the complex conjugate to E; H  is the complex conjugate H (Sect. 1.4). We know that the ohmic heating (also known as resistive heating and Joule heating) is the process by which the passage of an electric current through a conductor material produces heat. The ohmic power density can be expressed as: qr ¼ rE 2 ¼ J 2 =r ¼ rE  E  ¼ J  J  =r ¼ Ohmic power density:

ð5:12:14Þ

The net energy density is the sum of the energy density due to the electric field and the energy density due to the magnetic field: W ¼ WE þ WM:

ð5:12:15Þ

In other words, Eq. (5.12.11) states that the total power flowing into a closed surface at any instant equals the sum of rates of increase of the stored electric and magnetic energies and the ohmic power dissipated within the enclosed volume. Two comments concerning the Poynting vector are as follows: First, the power relations given in Eqs. (5.12.9) and (5.12.11) relate to the total power flow across a closed surface obtained by the surface integral of ðE  HÞ; and second, the Poynting vector P has a direction normal to both E and H (see Fig. 5.11 and Sect. 5.4.1). If the region of an EM wave propagation is lossless ðr ¼ 0Þ; the last term in Eq. (5.12.11) disappears, and the total power flowing into a closed surface is equal to the rate of increase of stored electric and magnetic energies in the enclosed volume. In a static situation, the first two terms on the right side of Eq. (5.12.11) vanish and the total power flowing into a closed surface is equal to the ohmic power dissipated in the enclosed volume. Continuity equation for W ðr; tÞ and Pðr; tÞ: If the net energy density W (Eq. 5.12.15) and the Poynting vector P (Eq. 5.12.10) are the energy and energy– current (the energy flow, energy flux) densities, in this case, since the energy cannot be created or destroyed, we can write the continuity equation at r ¼ 0: @W ðr; tÞ þ r  Pðr; tÞ¼ 0: @t

ð5:12:16Þ

We can write the total energy density stored in EM field on the base of Eq. (5.12.15): 1 W ¼ ðD  E þ B  HÞ; 2

ð5:12:17Þ

314

5

Plane Electromagnetic Wave Propagation

or in linear media, the modules of the magnetic flux density jBj and electric field strength jEj; using Eq. (5.4.22), associated with this ratio B¼E ðn=cÞ ¼ pffiffiffiffiffiffiffiffi Eð er lr =cÞ, and for monochromatic EM wave, we can express them as:    1 1 2 n2 2 e þ 2 E2 : W ¼ eE þ B =l ¼ 2 2 lc

ð5:12:18Þ

The assertion (statement) that the surface integral of P over a closed surface, as given on the left side of Eq. (5.12.9), equals the power leaving the enclosed volume is referred to as the Poynting theorem.

5.12.1 Poynting’s Theorem Poynting’s theorem is a statement of conservation of energy for the EM field. The theorem is about the energy balance. The Poynting theorem states that the work done on the charges by the EM force is equal to the decrease in energy stored in the field and is less than the energy that flows out through the surface. The decrease in the EM energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net flux going out of the volume per unit time. These statements in the differential form can be expressed as: @W ¼ r  P  J  E: @t

ð5:12:19Þ

where r  P is the divergence of Poynting’s vector (energy flow), J  E is the dot product and is the rate at which the fields do work on a charged physical body, J is the current density corresponding to the motion of charges, E is the electric field. The energy density of EM wave W ¼ W EM (Sect. 2.1) propagating into non-dispersive non-absorbing (non-attenuating) linear media, using Eq. (5.12.17), is given by the formula:  1 1 W ¼ ðeE  E þ lH  HÞ ¼ eE 2 þ lH 2 : 2 2

ð5:12:20Þ

When EM wave propagates in the ^z-direction, the total energy in the volume V ¼ S  dz with the area S and thickness dz during the amount of time dt ¼ dz=c is given by the formula: dW ¼ WS dz ¼ ðW E þ W M ÞS dz ¼

e e  l l 0 r 2 E þ 0 r H 2 S dz: 2 2

ð5:12:21Þ

The Poynting theorem can be rewritten by using the divergence theorem in the integral form:

5.12

Flow of Electromagnetic Power …

@  @t

Z

315

Z W dv ¼

V

Z P  ds þ

S

ðJ  EÞ dv;

ð5:12:22Þ

V

where an area S is the boundary of a volume V [see also Eq. (5.12.11)]. Because W is the energy density for the traveling EM wave which, by integrating over the volume of the region, gives the total energy W stored in the volume V; and then by taking the (partial) time derivative, it gives the rate of change of energy.

5.13

Time-Averaged Poynting Vector

We will consider a time-harmonic plane EM wave which travels in the positive zaxis direction. An EM field for such a wave can be given in the phasor form (Sect. 5.1) by the complex amplitude vector Eðz; tÞ ¼ RefEm ðzÞeixt g; where Em ðzÞ is a phasor vector of the electric field (time-independent term) and the complex amplitude vector Hðz; tÞ ¼ RefHm ðzÞeixt g; where Hm ðzÞ is the phasor vector of the magnetic field, x is the angular frequency of the cosinusoidal (sinusoidal) wave. Phasors as Em and Hm denote a cosinusoidally varying field whose instantaneous amplitude EðtÞ follows the real part of E ¼ fEm ðzÞeixt g and H ¼ fHm ðzÞeixt g: The instantaneous Poynting vector, using Eq. (5.12.10), is:



Pinst ðtÞ¼ReðEÞ  ReðHÞ ¼ Re Em ðzÞeixt  Re Hm ðzÞeixt :

ð5:13:1Þ

We designate the complex amplitude vector Eðz; tÞ ¼ RefEg to show mathematical conversions of expressions and analogically for complex term Hðz; tÞ ¼ RefHg: Compact presentation of expressions due to using complex number properties: We use expressions for complex numbers (see Sect. 1.4.3) and write as follows: RefEg ¼ ðE þ E Þ=2 and RefHg ¼ ðH þ H Þ=2;

ð5:13:2Þ

where E; H are vectors with complex amplitudes and E ; H are the complex conjugate numbers (Sect. 1.4). We reorganize the expression of the instantaneous Poynting vector (Eq. (5.13.1)) by using Eq. (5.13.2). 1 1 1 ReðEÞ  ReðHÞ ¼ ðE þ E Þ  ðH þ H Þ ¼ ðE  H þ E  H þ E  H þ E  H Þ 2 2 4 1   ¼ ððE  H þ E  HÞ þ ðE  H þ E  H ÞÞ: 4

ð5:13:3Þ

316

5

Plane Electromagnetic Wave Propagation

We will examine and transform the first bracket in Eq. (5.13.3). We denote A1 ¼ E  H and then its conjugate complex number (Eq. 1.4.31)  is A1 ¼ðE  H Þ ¼ E  H: We have on the base of Eq. (5.13.2) that A1 þ A1 ¼ 2Re½A1  and this means that the first bracket in Eq. (5.13.3) is: ðE  H þ E  HÞ ¼ 2ReðE  H Þ:

ð5:13:4Þ

The same transformation has to be done with the next bracket containing the cross-product of two other vectors (Eq. 5.13.3). We denote B1 ¼ E  H; and then its conjugate complex number is B1 ¼ E  H after using Eq. (5.13.2), we can   write B1 þ B1 ¼ 2RefBg and this means that the second bracket in Eq. (5.13.3) is: ðE  H þ E  H Þ ¼ 2ReðE  HÞ:

ð5:13:5Þ

Using Eqs. (5.13.4) and (5.13.5), we can write Eq. (5.13.3) as: 1 1 Pinst ðtÞ ¼ ðReðE  H Þ þ ReðE  HÞÞ ¼ ReðE  H þ E  HÞ: 2 2

ð5:13:6Þ

So Eq. (5.13.6) is some compact form of instantaneous Poynting vector from Eq. (5.13.1). Derivation of Instantaneous Poynting Vector in Expanded Form As we known from Eq. (5.13.1), the instantaneous value of the Poynting vector can be expressed by real parts of the electric RefEg and magnetic RefHg fields. We can write the real part of electric field in a convenient form:   Em eixt ¼ E0m þ iE00m ðcos xt þ i sin xtÞ   ¼ E0m cos xt  E00m sin xt þ i E00m cos xt þ E0m sin xt ;

ð5:13:7Þ

ðeixt þ eixt Þ ðeixt  eixt Þ  E00m Re Em eixt ¼ E0m cos xt  E00m sin xt ¼ E0m 2 2i        1  0 1 Em þ iE00m eixt þ E0m  iE00m eixt ¼ Em eixt þ Em eixt ¼ 2 2  1 ¼ Em eixt þ Em eixt : 2 ð5:13:8Þ We can also write the real part of the magnetic field in a convenient form:

1  Re Hm eixt ¼ Hm eixt þ Hm eixt : 2

ð5:13:9Þ

5.13

Time-Averaged Poynting Vector

317

The instantaneous value of the Poynting vector by using the phasor presentations Eqs. (5.13.8) and (5.13.9):



Pinst ðtÞ ¼ Re Em ðzÞeixt  Re Hm ðzÞeixt ð5:13:10Þ  1  1 ¼ Em eixt þ Em eixt  Hm eixt þ Hm eixt ; 2 2 by multiplying each term on each other of Eq. (5.13.10), we get:  1 Em  Hm þ Em  Hm þ Em  Hm e2ixt þ Em  Hm e2ixt 4  1  1 ¼ Em  Hm þ Em  Hm þ Em  Hm e2ixt þ Em  Hm e2ixt ; 4 4 ð5:13:11Þ

Pinst ðtÞ ¼

We see that the instantaneous power flow will be fluctuating at a frequency of 2x. We can express terms in the first bracket of Eq. (5.13.11) using Eq. (5.13.4): 

   Em  Hm þ Em  Hm ¼ 2Re Em  Hm ;

ð5:13:12Þ

and in second bracket of Eq. (5.13.11): 

     Em  Hm e2ixt þ Em  Hm e2ixt ¼ Em  Hm e2ixt þ Em  Hm e2ixt   ¼ 2Re Em  Hm e2ixt : ð5:13:13Þ

Instantaneous Poynting Vector by using the phasor presentation  Pinst ðtÞ ¼

  1   1   2ixt Re Em  Hm þ Re Em  Hm e : 2 2

ð5:13:14Þ

The first term in the right side of Eq. (5.13.14) is independent of time. The next term oscillates with twice frequency EM field. Average value of Instantaneous Poynting Vector The average value of the instantaneous Poynting vector Pinst over time is determined as:

318

Pav ¼ hPi ¼

5

1 T

ZT Pinst ðtÞ dt ¼ 0

1 T

Plane Electromagnetic Wave Propagation

 ZT   1   1  Re Em  Hm þ Re Em  Hm e2ixt dt; 2 2 0

ð5:13:15Þ where the integration is accomplished over a full cycle T ¼ 2p=x: The second term is the double-frequency component e2ixt having an average value of zero. We can finally write:  1  Pav ¼ hPi ¼ Re Em  Hm : 2

ð5:13:16Þ

Equation (5.13.16) presents the time-average Poynting vector, which indicates the average real power density of a time-harmonic wave. According to some conventions, the factor of 1/2 may be left out. Multiplication by 1/2 is required to properly describe the power flow since the magnitudes Em and Hm refer to the peak fields of the oscillating quantities. The Poynting vector can be expressed directly in terms of the phasors as: 1 Pm ¼ Em  Hm : 2

ð5:13:17Þ

The time-averaged power flow Pav is given by ReðPm Þ according to the instantaneous Poynting vector averaged over a full cycle. The imaginary part is usually disregarded. It means “reactive power,” i.e. as interference, due to standing waves, which occur by waves propagating in opposite directions. In a single EM plane wave, which propagates into a simple lossless medium, E and H fields are exactly in the phase, so Pm is simply a real number according to the definition. For plane waves: In a propagating cosinusoidal (sinusoidal) linearly polarized EM plane wave of a fixed frequency, the Poynting vector always points to the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is found as: hPi ¼

1 jEm j2 ; 2g

ð5:13:18Þ

where Em is the complex amplitude of the electric field and g is the intrinsic impedance of a medium (Sect. 5.7). The expression of the average Poynting vector indicates the fact that in a plane wave the magnetic field Hm and the electric field Em are exactly in the phase.

5.13

Time-Averaged Poynting Vector

319

Complex Poynting vector SðtÞ: The complex Poynting vector of a plane wave can be expressed in following form: 1 SðtÞ ¼ E  H : 2

ð5:13:19Þ

According to some conventions, the factor of 1/2 may be left out. You can meet expression of complex Poynting vector in literature as follows: SðtÞ ¼ E  H :

ð5:13:20Þ

The real part of the complex Poynting vector SðtÞ presents the power flowing through a unit area orientated normally to vector (cross)-product E  H . The real part of vector SðtÞ is the time-average of the Poynting vector during one cycle. The imaginary part of the complex Poynting vector with some reservations can be correlated with a vector responsible for the reactive energy propagation (see also further Sects. 6.12 and 7.6.5). Summarizing of uniform plane wave properties: 1. There are no electric and magnetic field components in the direction of the wave propagation, i.e. E z ¼ 0;H z ¼ 0: 2. Direction of propagation of an EM wave is given by the direction of the Poynting vector. 3. Electric and magnetic fields are perpendicular to each other. 4. The intrinsic impedance of media in which the plane wave propagates is g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l0 lr =e0 er , and the intrinsic impedance in free space is pffiffiffiffiffiffiffiffiffiffiffi g0 ¼ l0 =e0 377ðxÞ. 5. The amplitude of electric field is g times that of magnetic field amplitude at each instant of time where g is the intrinsic impedance of a medium. 6. The electric and magnetic fields of a plane EM wave propagating in lossless (dissipationless) media are exactly in the phase. 7. The phase difference (phase angle, phase shift) between the electric and magnetic fields of a plane EM wave propagating in a good conductor is approximately 45 . 8. EM waves considered here obey the linear superposition principle; i.e. if two or more propagating waves are moving through a medium, the resultant wave function at any point is the algebraic sum of the wave functions of individual waves.

5.14

Phase and Group Velocity

The phase and group velocity are two important and interrelated concepts of the propagation of an EM wave.

320

5

Plane Electromagnetic Wave Propagation

The refractive index (or index of refraction) n of a material is a dimensionless number which depicts how the light propagates through that medium with the certain absolute permittivity e ¼ e0 er and permeability l ¼ l0 lr : n¼

rffiffiffiffiffiffiffiffiffi el ; e0 l0

c ¼ vp

ð5:14:1Þ

pffiffiffiffiffiffiffiffiffi where c ¼ 1= e0 l0 is the speed of light in a free space (Eq. 2.1.6) and vp ¼ pffiffiffiffiffiffi 1= e l is the phase velocity of light in a medium (Eq. 5.4.21). For example, refractive index nH2 O of water is 1.3311 at 20 °C and wavelength k 656.3 nm which means that light travels 1.3311 times faster in a vacuum than it does in water. The index of refraction n determines how much light is refracted or bent when entering a medium (material). The value n can be seen as the factor by which the speed vp and the wavelength k of the EM wave are reduced with respect to their a vacuum values. The refractive index of a vacuum is n ¼ 1: The phase speed vp of an EM wave in a medium determines the speed of wave crest which moves in the medium (Fig. 5.15). We want to draw your attention to a special case. The speed vp can be faster than the speed of light in a vacuum c; i.e. these crests give a refractive index below 1. This can occur close to resonance frequencies of a material (medium), e.g. for absorbing media as plasmas and for X-rays. An example of plasma with an index of refraction less than unity is the Earth’s ionosphere. In the X-ray range ðk 0:01  10 ðnmÞ; corresponding to f 3  1016  3 1019 HzÞ, the refractive indices are lower than 1 unit but very close to 1 unit (exceptions are observed at f close to resonant frequencies). As an example, water has a refractive index of 0.99999974 for X-ray at k 0.04 nm. Modern research has demonstrated the existence of media with a negative refractive index which can occur if the permittivity and permeability have simultaneous negative values. This can be reached with periodically man-made (opposite to natural) materials which are called metamaterials.

(a)

(b)

E(z,t)

Envelope of an oscillating signal

E(z,t) Vg

ω

ω

dω 0

0

0

z

0

z

ω

0

Vp

Fig. 5.15 a A single time-harmonic wave. b Sum of two waves with slightly different frequencies at time t

5.14

Phase and Group Velocity

321

A refractive index of a medium changes with the wavelength k of the EM wave alterations. This is named as dispersion which is the cause of the splitting of white light into its constituent colors in prisms. EM wave propagation in absorbing (lossy) materials (media) can be described by using the complex refractive index. The imaginary part ImðnÞ is responsible for the attenuation (loss), while the real part ReðnÞ accounts for refraction. Notice that “speed” is a scalar value, and “velocity” is a vector value. In this section we will use the terminology common used in technical literature where is usually named phase and group velosity despite the fact that the values can be scalars. From Eqs. (5.10.10) and (5.14.1): vp ¼

x k0 f ¼ pffiffiffiffiffiffiffiffi ¼ kf b er lr

ðm=sÞ;

ð5:14:2Þ

Here the phase velocity vp of EM wave (light) in the medium with the certain relative permeability and permittivity is er ¼ e=e0 ; lr ¼ l=l0 , and the wavelength pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is k ¼ k0 = er lr . The phase constant b ¼ x e0 er l0 lr in a lossless medium (Eq. 5.11.2) is a linear function of x: We have come to the same expression of the phase velocity from Eq. (5.14.2) as in Eq. (5.4.21): 1 1 vp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffi : e0 er l0 lr el

ð5:14:3Þ

We see that vp in lossless media is independent of frequency. When the wave speed of EM waves depends only on the physical properties of a medium, e.g. relative permittivity and relative permeability then the wave speed is a constant and independent of frequency. Such a medium is called a non-dispersive medium, and waves traveling through this medium will maintain a constant shape. Dispersion is the phenomenon in which the phase velocity vp of a wave depends on its frequency, and a media with this property can be named the dispersive media. The phase constant bðxÞ can not be linear function of x; e.g. in lossy media or transmission lines. The propagation of an EM wave through a physical medium is an intricate phenomenon of displacement of particles (as atoms or molecules) of the medium under the influence of the wave and the appearance of secondary waves. The superposition of these secondary waves constitutes the EM wave in the medium. The interaction of particles with the EM wave can depend on various factors such as the frequency (for dispersive media), on the direction of the wave propagation (for anisotropic media), on the sizes of media, the shape of media boundaries, on EM wave polarization. The EM waves with dissimilar frequencies propagate with different phase velocities. For this reason, the wave front shape changes. For example, the information-bearing signal consists of a set of EM waves with dissimilar frequencies which propagate with different phase velocities, causing a distortion in the signal shape.

322

5

Plane Electromagnetic Wave Propagation

Dispersive media are media in which EM waves of different frequencies travel at different velocities. In Fig. 5.15b, we can see the group velocity vg ¼ dx=db of an envelope of two waves which is designated by the dashed lines. The group velocity corresponds to a slow envelope with a frequency dx: The group velocity of an EM wave can be described as the velocity of the overall shape of waves’ amplitudes (red and dark blue dashed envelope (modulation) lines in Fig. 5.15). The phase velocity vp ¼ x0 =b0 is marked with the aquamarine color. The phase velocity corresponds to a fast oscillating wave, having an angular frequency x0 , which is inside the envelope. The phase constant of a fast oscillating wave is pffiffiffiffiffiffi b0 ¼ x0 e l. Basically, for simple materials the dispersion is such that EM waves at lower frequencies travel faster than at higher ones. As a result, the wave packet spreads out (expands, moves outward, moves away) with longer wavelengths moving faster and shorter wavelengths lagging behind. This wave packet can be considered to be a superposition of a number of harmonic waves, in other words, a Fourier series or integral. The envelope of an oscillating signal (dashed lines in Fig. 5.15b) is a smooth curve outlining its extremes. The figure illustrates a modulated cosine wave varying between upper and lower envelope curves. The envelope function may be a function of time, space, angle, or indeed of any variable. Dx xZ 0þ 2

Eðy; tÞ ¼

  Ax cos xt  bx z þ ðu0 Þx dx;

ð5:14:4Þ

x0 Dx 2

where ðu0 Þx is the initial (start) phase of an EM harmonic oscillation, which depends on frequency x; which can be ðu0 Þx = 0 (see Sect. 5.1). A group velocity vg is the velocity of propagation of the envelope (packet, collection, package, or bundle) of EM waves with different frequencies. Let’s say we have a collection of two traveling waves with equal amplitudes and marginally (a little bit, slightly) different angular frequencies x1 ¼ x0 þ Dx; x2 ¼ x0  Dx; where Dx x0 ; Dx ¼ x2  x1 . The wave’s angular frequency x0 , as mentioned before, determined the fast oscillations in the envilope (Fig. 5.15b) also means some central frequency. Note that the EM wave source has produced xt=2p ¼ f t oscillations after time t. After this time t, the initial wavefront has propagated away from the EM wave source through space to the distance x to fit the same number of oscillations bx z ¼ xt: The phase constant of the first wave b1 ¼ b0 þ Db and the second wave b2 ¼ b0  Db will be marginally different Db ¼ b2  b1 . Then using Eq. (5.4.29) and the linear superposition principle of two EM waves with frequencies x1 and x2 close to each other, we obtain:

5.14

Phase and Group Velocity

323

E ðz; tÞ ¼ E 0 cosððx0 þ DxÞt  ðb0 þ DbÞzÞ þ E 0 cosððx0  DxÞt  ðb0  DbÞzÞ; ð5:14:5Þ E ðz; tÞ ¼ E 0 cosððx0 t  b0 zÞ þ ðDxt  DbzÞÞ þ E0 cosððx0 t  b0 zÞ  ðDxt  DbzÞÞ;

ð5:14:6Þ

after entering significations: #1 ¼ ðx0 t  b0 zÞ and #2 ¼ ðDxt  DbzÞ;

ð5:14:7Þ

Equation (5.14.6) can be written as: E ðz; tÞ ¼ E 0 ðcosð#1 þ #2 Þ þ cosð#1  #2 ÞÞ:

ð5:14:8Þ

According to the trigonometric addition formulae: cosð#1 þ #2 Þ ¼ cos #1 cos #2  sin #1 sin #2 ; cosð#1  #2 Þ ¼ cos #1 cos #2 þ sin #1 sin #2 :

ð5:14:9Þ ð5:14:10Þ

Summing up the last two Eqs. (5.14.9) and (5.14.10), we get: cosð#1 þ #2 Þ þ cosð#1  #2 Þ ¼ 2 cos #1 cos #2 :

ð5:14:11Þ

Equation (5.14.8) can be written as: E ðz; tÞ ¼ 2E 0 cos #1 cos #2 :

ð5:14:12Þ

Expressions of Eq. (5.14.7) are substituted in Eq. (5.14.12): Eðz; tÞ ¼ 2E0 cosðx0 t  b0 zÞ cosðDxt  DbzÞ:

ð5:14:13Þ

Equation (5.14.13) demonstrates a fast oscillating wave having an angular frequency x0 (the second term) of the expression on the right side of the equation and the amplitude E0 (the first term) which deviates (changes, varies) slowly with the angular frequency Dx (the third term) at Dx x0 (Fig. 5.15b). The wave inside the envelope propagates with a phase velocity (speed), which is determined as: #1 ¼ ðx0 t  b0 zÞ ¼ constant; vp ¼

dz x0 ¼ : dt b0

ð5:14:14Þ ð5:14:15Þ

Equation (5.14.14) describes a wavefront of EM wave which is a surface of constant phase (see Eq. 5.4.29). Figure 5.16 shows the interpretation of phase and group speeds.

324

5

Fig. 5.16 Dispersion relations: vg ¼ dx=db and vp ¼ x0 =b0

Plane Electromagnetic Wave Propagation

ω

dω Slope dβ =Vg

ω0

A

dω dβ

ω0 Slope β0 =Vp 0

β0

β

The group velocity (speed), i.e. the velocity of the envelope, can be found out as: #2 ¼ ðDxt  DbzÞ ¼ constant:

ð5:14:16Þ

We have: vg ¼

dz Dx 1 ¼ ¼ : dt Db Db=Dx

ð5:14:17Þ

In a dispersive medium, the group velocity at Dx ! 0 is: vg ¼

1 dx ¼ db=dx db

ðm=sÞ:

ð5:14:18Þ

Taking Eqs. (5.14.2) and (5.14.18), we receive relation between the phase and group velocities:     db d x 1 x dvp 1 x dvp ¼ ¼ 1 ¼  2 : dx dx vp vp vp dx vp vp dx

ð5:14:19Þ

The phase and group velocity are related through Rayleigh’s formula: vg ¼

  dvp 1 x dvp 1 ¼ vp 1  : ¼ vp  k db=dx vp dx dk

ð5:14:20Þ

If the derivative dvp =dk [in the second term of Eq. (5.14.20)] is zero, the group velocity equals phase velocity vg ¼ vp . In this case there is no dispersion. Dispersion exists when different phase velocities of the components of the envelope cause the wave packet to extend over time.

5.14

Phase and Group Velocity

325

There can be three cases of dispersion from Eq. (5.14.20), namely: dv dv 1. No dispersion dxp ¼ dkp ¼ 0; vp is independent of x; b is a linear function of x; vp ¼ vg :

ð5:14:21Þ

dv dv 2. Normal dispersion dxp \0; dkp [ 0; vp decreasing with x vp [ v g :

ð5:14:22Þ

dv dv 3. Anomalous dispersion dxp [ 0; dkp \0; vp increasing with x vp \vg :

ð5:14:23Þ

The illustration of normal and anomalous dispersions is shown in Fig. 5.17. The refractive index given in Eq. 5.14.1 can be the complex number n ¼ n0  in00 , the real part n0 ¼ ReðnÞ is the refractive index which indicates the phase velocity, and the imaginary part n00 ¼ ImðnÞ is called the extinction coefficient. n¼

rffiffiffiffiffiffiffiffiffi el pffiffiffiffiffiffiffiffi ¼ er lr ; e0 l0

ð5:14:24Þ

where the complex relative permittivity is er ¼ e0r  ie00r ¼ n2 ¼ ðn0  in00 Þ2 . We can express the real and imaginary components as:

Fig. 5.17 Normal and anomalous dispersion

Vp(ω)=c/n(ω)

Absorption line

n(ω) 1

al

rm No

c

∆ω

∆Vp

An

ω0

om

ω

alo

us

∆Vp

ω0

∆ω

∆Vp

∆ω

al rm n No ersio sp Di

ω

326

5

e0r

0 2

00 2

¼ ðn Þ  ðn Þ

; e00r

Plane Electromagnetic Wave Propagation

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 e0r þ e00r ; ¼ 2n n ; andjer j ¼ 0 00

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jer j þ e0r 00 jer j  e0r ;n ¼ : n ¼ 2 2 0

ð5:14:25Þ ð5:14:26Þ

The dispersal law can be set either as a refractive indicator nðxÞ of frequency or as a function of the propagation constant (phase constant, Eq. 5.9.7) bðxÞ of frequency. The dependency bðxÞ is commonly used in EM field theory in the study of EM characteristics of waveguides. From the expression of the phase speed vp ¼ c=nðxÞ; we determine the propagation constant: bðxÞ ¼ 

x : vp ðxÞ

ð5:14:27Þ

The argument can be x or a wavelength of k in the media.

5.15

Polarization of Electromagnetic Waves

In Sect. 3.2, we considered the electric polarization P of dielectric materials placed in an external electric field. As we remember, the polarization of dielectrics is a phenomenon associated with limited shift of bounded charges in the dielectric or turning their electric dipoles under the influence of the electric field. In EM field theory, the term “polarization” is used for three different phenomena: (a) polarization P of dielectric materials (we have studied them in Chap. 3), (b) polarization of EM waves which propagate in a medium (we will start studying them in this chapter), and (c) polarization of the incident, reflected, and transmitted electric and magnetic fields (we will study them in the following chapter). Now we will investigate the polarization of EM waves which propagate in unlimited space of a vacuum or simple medium (Sect. 3.27.1). The word “polarization” comes from the Latin language “polus”, meaning “an axis” and also processes and states which are associated with splitting objects, primarily in space. Polarization of EM waves generally means “orientation” of EM field vectors. Polarization (also polarisation) of an EM wave specifies the geometrical orientation of the wave oscillations. Since the EM waves consist of an electric field and a magnetic field, which are perpendicular to each other, it is important to agree which of these fields determine the polarization. A wave is called polarized if we can specify a field vector direction at any time. Since there is a relation between the electric and magnetic fields (Eqs. 5.7.28 and 5.7.29) through the intrinsic impedance Eqs. (5.7.13 and 5.7.14), therefore, we only need to select one of vectors E or H: The wave polarization by agreement is the orientation of the electric field E in an

5.15

Polarization of Electromagnetic Waves

327

EM field. So the polarization of the EM wave indicates the plane in which the electric field E oscillates. The polarization of EM waves is an important characteristic in the design of waveguide devices, radar, lasers, wireless and optical fibers for telecommunications, seismology, etc. We can transmit more information through the same channel by the EM waves of different polarizations and separate these waves with the help of special equipment at the signal output. The eyes of the people are not capable of distinguishing different types of polarization and, in the contrary, many animals can see variations in polarization just as we see variations in color. It was investigated that such animals as cephalopods, insects, many amphibians, fish, and others similar to the listed ones can see differences in EM wave polarizations. The main target of polarization vision is the recognition, breaking camouflage, and increasing detection range. The polarization vision can be used for detecting transparent objects, like jellyfish (jellies). In addition, many crustaceans (such animals as crabs, lobsters, and shrimp) from their smooth surfaces reflect light that is strongly polarized. Some octopuses were able to detect patterns that had contrasts as small as 20° in the direction of polarization. Michelson contrast (also known as the visibility) is equal to ðI max  I min Þ=ðI max þ I min Þ; where I max and I min are the highest and lowest degrees of luminance. A possible advantage of polarization vision is that while an object color changes with depth due to the changing attenuation by seawater across the EM spectrum (Appendix Chap. 5:5.1), the object, due to changes in reflected polarization, would remain constant. In a laboratory experiment, a cuttlefish (marine animal) behavior changed significantly when a transparent polarization filter was placed between the mirror and the cuttlefish. The filter did not change the visible image but it only distorted the polarization. Polarization may provide the cuttlefish with a channel allowing it to hide from some predators.

5.15.1 Constructive and Destructive Interference The principle of a linear superposition of EM waves states that when two (or more) propagating waves with the same frequency are propagating to the same direction till the same point B, the resultant amplitude of E-field at point B is equal to the vector sum of the E-field amplitudes of the individual waves. Interference can be observed as a result of superposition of EM waves. EM wave’s interference is a phenomenon in which EM waves superpose (place upon each other) to form a resultant EM wave of lower, equal, or larger E-field amplitude. Interference usually refers to the interaction of EM waves that are

328

5

Plane Electromagnetic Wave Propagation

coherent with each other. Two wave sources are coherent when they have a constant phase difference (phase angle, phase shift) and the same or nearly the same frequency. Let us consider two waves with the same frequency traveling in the same direction. We add these two waves together point by point. If crests of both waves meet each other at the same point (waves are in phase), then the resulting E-field amplitude is larger than either of these two original ones. This is called a constructive interference. We will consider now the next situation where two waves with the same frequency are again traveling in the same direction. If a crest of one wave meets a trough of another wave, the resulting E-field amplitude is equal to the difference in the individual amplitudes. When the first wave is up, the second wave is down and their amplitudes are equal; in this case, these two waves precisely cancel each other out at all points and no wave is left. The EM waves are said to be out-of-phase. The sum of two waves can be less than either waves and can even reach zero. This is called the destructive interference.

5.16

Plane Wave and Polarization

A plane wave is a wave that spreads (propagates, travels) along a linear coordinate, e.g. the z-axis, and the wave is unchanged at each fixed time moment at the plane which is perpendicular to the z-coordinate. From Maxwell’s equations for the time-periodic case (Sect. 4.4), we can receive the ordinary differential equations of the second order with constant coefficients describing the plane waves: d2 H x þ x2 elH x ¼ 0; dz2

ð5:16:1Þ

d2 H y þ x2 elH y ¼ 0; dz2

ð5:16:2Þ

d2 E x þ x2 elE x ¼ 0; dz2

ð5:16:3Þ

d2 E y þ x2 elE y ¼ 0: dz2

ð5:16:4Þ

The Poynting vector of the planar wave is oriented along the z-axis of the Cartesian coordinate system (Sect. 5.13):

5.16

Plane Wave and Polarization

329

1 1 P¼ Re½EH  ¼ ReðE  H Þ¼ Pz : 2 2

ð5:16:5Þ

We know that for plane EM wave: E z ¼ 0; H z ¼ 0;

ð5:16:6Þ

E x 6¼ 0; H x 6¼ 0; Ey 6¼ 0; H y 6¼ 0:

ð5:16:7Þ

Because the projections of a field vector [E(x, y, z) or H(x, y, z)] must be constant in a plane perpendicular to the direction of EM wave propagation, i.e. in the xOyplane (Fig. 5.7), the derivatives of a function of variables x, y, z with respect to x and y must be zero. We can write for the plane wave: @ @ @ ¼ ¼ 0; 6¼ 0; @x @y @z

ð5:16:8Þ

On the base Eq. (5.4.27), the solution can be written in this form: Ex ¼ A1 eika z þ A2 eika z ;

ð5:16:9Þ

where A1 and A2 are arbitrary constants (in general, they are complex numbers) of a forward wave and a backward (reverse) wave (Sect. 5.5). Ey ¼ B1 eika z þ B2 eika z :

ð5:16:10Þ

The same designations are for E y components; i.e. the term B1 eika z denotes a forward wave and the term B2 eika z denotes a backward wave. For instantaneous value (Eq. 5.5.9): E x ðz; tÞ ¼ jA1 jeaz cosðxt  bz þ ux Þ;   E y ðz; tÞ ¼ jB1 jeaz cos xt  bz þ uy ;

ð5:16:11Þ ð5:16:12Þ

The components in the vector form are: Ex ðz; tÞ ¼ ^xE 0x eaz cosðxt  bz þ ux Þ;   Ey ðz; tÞ ¼ ^yE 0y eaz cos xt  bz þ uy ;

ð5:16:13Þ ð5:16:14Þ

where t is time in seconds. The term xt is the time phase of the EM wave in radians. For non-absorbent media: a ¼ 0:

330

5

Plane Electromagnetic Wave Propagation

  E ¼ Ex þ Ey ¼ x^E0x þ ^yE 0y cosðxt  bz þ DuÞ:

ð5:16:15Þ

Especially in antenna theory, we are concerned about the polarization of an EM wave in the plane xOy; which is orthogonal to the direction of propagation along the z-axis. This orthogonal plane is defined by vectors of the far field (Fig. 5.5), i.e. at a sufficiently large distance from the wave source. As we remember, the far field is a quasi-transverse EM (quasi-TEM) field. The light from an incandescent lightbulb (lamp) with a wire filament heated to such high temperature, light-emitting diode (LED), or the sun is unpolarized (random) light (EM waves) in which electric fields are oscillating (vibrating) in several (more than one) random directions. EM radiation from many sources propagates with an equal mixture of polarizations and is also an unpolarized EM wave. Figure 5.18a presents a randomly polarized EM wave (light) in which E vector can take any magnitude and angle at any time as the EM wave passes a selected point in space. Each arrow (red, blue, green, and black) on the plane xOy represents E vector at the point O in time. Polarized light can be produced by passing unpolarized light through polarizing filters which let waves have only one polarization to move through. Polarized EM waves are the waves in which the oscillations occur in a single plane, e.g. in the plane yOz as in Fig. 5.18b. The plane in which there are both electric fields E vector and Poynting P vector which point to the direction of propagation of EM wave is called the polarization plane. The plane of polarization is normal to the wavefront, which is perpendicular to the direction of wave propagation, i.e. to vector P: There is a diversity of methods of polarizing EM waves. It is possible to obtain plane-polarized beam of EM wave (light) from unpolarized one by removing all waves from the beam except those having oscillations along one specific direction. This can be achieved by various processes such as reflection from different surfaces, selective absorption, refraction through crystals, and scattering by small particles. The selective absorption method is the most common method to obtain

(a)

(b) y

y x

x

O O z Fig. 5.18 a Unpolarized (random) and b vertically polarized EM waves

z

5.16

Plane Wave and Polarization

331

plane-polarized EM wave by using certain kinds of materials. These materials transmit only those EM waves whose oscillations are parallel to a particular direction and will absorb those waves whose oscillations are in other directions. The simple isotropic optical materials (such as ordinary glass) do not affect the polarization of light moving through them while some materials which exhibit dichroism, optical activity, birefringence can change the polarization of light. Birefringence is the optical property of materials having a refractive index that depends on the polarization and propagation direction of EM wave. Some of these materials are used to make polarizing filters. EM waves (light) can also be partially polarized, e.g. after reflection from a surface. Polarization is the property of an EM wave that describes the position (location) of its E vector as the function of time. The tip of E vector traces linear, circular, or elliptical path at a fixed point in space. Polarization also means that there is a constant phase shift (Sects. 5.1 and 5.2) between the components Ex and Ey of the electrical vector E ¼ Ex þ Ey at each point and at any time (Fig. 5.19). According to the shape of the trace of vector E, there can be three types of polarization for harmonic EM waves: linear (also known as plane), circular, and elliptical (Fig. 5.20). If the position of the polarization plane remains constant in the stationary coordinate system, this polarization is called linear (or plane).

E

y x

Ey

^ y^ x O ^z

Ex

z1

y

xO

z

e

n la

-p

P=[EH]

Fig. 5.19 Behavior of the vector E of a monochromatic plane EM wave in time and space

(a)

(b) y

y

E z

y

(c) E

x

z

x

E z

Fig. 5.20 a Linear polarization. b Circular polarization. c Elliptical polarization

x

332

5

Plane Electromagnetic Wave Propagation

The polarization can be presented by two orthogonal linear polarizations: horizontal Ex ðor EH Þ and vertical Ey ðor EV Þ whose electric fields can be out of the phase by an angle DE ¼ uy  ux . Horizontal Ex or vertical Ey polarizations are also known as linear polarizations. Here the wave can be regarded as oscillations (vibrations) in one plane, e.g. side to side or up and down (Fig. 5.20a). This form of polarization is the most straightforward. In the linear polarization, the electric field can oscillate in any single direction; i.e. any-degree-inclined linear polarization is possible in this case. The type of polarization is defined by the path traced out by the tip of the electric field vector as the resulting waves propagate in fixed point of space over time t: In other words, polarization shows how the direction and magnitude of the vector E change in time. Linear polarization occurs when the orthogonal vibrations have the same initial phase (see Eqs. 5.1.7 and 5.2.5). Amplitudes of the orthogonal components Ex and Ey can be different or the same. Circular polarization occurs when the orthogonal vibrations have an exact p/2 phase difference (Sects. 5.1 and 5.2) and have the same amplitudes Ex ¼ Ey . Elliptical polarization occurs when the amplitudes of the orthogonal vibrations are different Ex 6¼ Ey and there is some sort of phase difference; or when the amplitudes are the same and the phase differences have certain values. In circular or elliptical polarization, the electric field rotates at a constant rate x in a plane xOy as the wave propagates (Figs. 5.19 and 5.20). The rotation can have two possible directions. If the electric field rotates in the right-hand sense with respect to the direction of wave propagation, it is called the right (also known as the right-handed, counterclockwise, or anticlockwise) circular polarization, and if the electric field rotates in the left-hand sense, it is called the left (also known as the clockwise or left-handed) circular polarization depending on the direction in which the electric field vector rotates. There are two agreements regarding the name of these polarizations. This issue will be discussed below. The term “counterclockwise” originated from the North American English and term “anticlockwise” originated from the Commonwealth English. The state of polarization is completely defined by four parameters: two amplitudes, the sign and value of the phase shift.

5.16.1 Common Notes About Polarization of Electromagnetic Waves Two situations may arise when the decomposition or superposition of EM waves occurs. The word “decomposition” means separation into constituent parts or elements. The word “superposition” means combination of two or more physical states, such as EM waves. We can imagine that a transverse (plane) EM wave, i.e. E ¼ Ex þ Ey ¼ ^xEx þ ^yE y , is radiated from one source (such as an antenna).

5.16

Plane Wave and Polarization

333

Any EM wave can be decomposed (or resolved) into two component waves orthogonally polarized to each other. Here Ex is the horizontally polarized component and Ey is the vertically polarized component (Fig. 5.19). We can also imagine that two EM waves E1 ¼ ^ xEx1 þ ^ yE y1 and E2 ¼ ^xE x2 þ ^yE y2 from different sources (antennas) interact with each other, so there is their superposition. When two or more waves propagate in the same space, the resulting amplitude at each point is the sum of the amplitudes of the individual waves. Common approvals: A plane wave propagating in free unlimited space can be decomposed into two mutually orthogonal linearly polarized terms or two circularly polarized terms of the opposite sense of rotation waves (see further Table 5.2). We will consider the basic properties. Linear to linear: 1. A linear polarized EM wave E ¼ ^xEx þ ^yE y can be decomposed (or resolved) into two perpendicularly (orthogonally) linear polarized waves. It is convenient to use the vertically ^yE y and horizontally ^xE x polarized waves (Figs. 5.19 and 5.20a). 2. If two linear waves E1 ¼ ^xE x and E2 ¼ ^yE y have exactly the same frequency and are perfectly in the phase, then after their superposition at all points of their propagation way, the resulting (total) electric field vector traces out a straight diagonal line. The angle u between the horizontal x-axis and the straight line of the resulting electric field depends on the ratio of components Ex and Ey , i.e.   u ¼ arctan E y ðz; tÞ=E x ðz; tÞ : Counter-rotating circularly polarized to linear. 3. The resulting electric field vector of the sum of two equal-amplitude counter-rotating circularly polarized waves has twice larger amplitude. The resulting EM wave is linear. Linear to circular: 4. A linear polarized EM wave can be decomposed into two equal counter-rotating components. These components are right-hand circular polarized when the E-vector rotates counterclockwise and left-hand circular polarized when the E-vector rotates clockwise by agreement IEEE for the engineering community. 5. In case when the two plane waves are exactly of the same frequency and amplitudes but out of the phase by p=2; the resultant output will be a circle (or an ellipse if amplitudes of the EM waves are different).

5.17

Approach to Polarization from Electrodynamics and Optics

Before we start studying polarization of an EM wave in details, we have to agree about the position (location) of our eyes in relation to the source of EM waves and to the direction of the wave propagation. There are different agreements about right-

334

5

Plane Electromagnetic Wave Propagation

and left-handed polarizations in scientific literature. Figure 5.21 shows a single transparent film which is visible from two sides in relation to the source of EM wave, i.e. antennas A1 and A2. In this illustration, we see a red circle with rotation counterclockwise (also known as the right-handed, counterclockwise, or anticlockwise), if we are looking at the circle from the side of the “red eye”. If we look with “blue eye” through the same transparency film on the same circle from the side of antennas A1 and A2, and the EM wave propagates in the same side, we will see that the direction of rotation has changed to the opposite (blue circular). We “cut and turned” the transparency film to see the image of the blue circle from the side of antennas (Fig. 5.21). Students do this work at home: Draw a circle with the direction of rotation on one side of the transparent film. Compare the directions of rotation from the two sides of the film. Different definitions of polarization There are two opposite approaches to polarization in electrodynamics (EM field theory) and optics. Therefore, there is some confusion about the rotation of the electric field vector clockwise or counterclockwise in some literary sources, in cases where the position of the observer eye is not indicated. In electrodynamics, (microwaves and antenna engineering) “an eye” of an observer is usually located in the opposite direction from the EM wave source (red eye in Fig. 5.21). In the last case, EM waves are usually propagating toward our eyes. Like in astronomy, the light from the stars in the sky comes through a telescope in “an eye” of an observer. The source of the EM wave (the star) and “an eye” of an observer are in the opposite directions. It is important to note that the same approach is also recommended by the Institute of Electrical and Electronics Engineers (IEEE) for the engineering community. We have also chosen to follow this approach in this book. In optics, an eye of an observer and the source of the EM wave are adjacent, that is, on one side (blue eye in Fig. 5.21). You can imagine an optical microscope (for Single transparency film that is visible from both sides of the wave source

12

E E

A1

* *2 A

z

z

9 y z

3

x

6

Fig. 5.21 “Blue eye” on the left side of the illustration is near EM wave sources (as in optics), and the “red eye” is located in the opposite side of the wave sources (as in electrodynamics or IEEE agreement)

5.17

Approach to Polarization from Electrodynamics and Optics

335

research of a material) with a light source which can be on the same side as an eye of an observer. In optics, the EM wave “spreads away” from the side of an observer. The authors of some technical articles do not always indicate the direction of the EM wave propagation along the z-axis nor do they indicate the position of an eye of an observer with respect to the source of the EM wave. For this reason, the electric field vector can rotate clockwise (as in optics) or anticlockwise (as in electrodynamics) in the right-handed polarized wave. The same uncertainty is with the left-handed polarized wave. IEEE has defined handedness for the engineering community as pictured above on the right part of Fig. 5.21. By IEEE agreement, EM wave (light) is right circularly polarized (RCP) when the E-vector rotates counterclockwise. In this case, we are observing the rotation of E-vector with the “red eye” and the EM wave is traveling toward the “red eye” of a viewer. Polarization of the wave in unlimited (infinite) space can be achieved by the summing of two linear plane waves which are perpendicularly polarized to each other and propagate in the same direction. The polarization of the resulting wave is determined by the difference in the initial phase DE ¼ uy  ux of both linear waves (Sect. 5.2) and their amplitudes Eym , Exm . We will assume in this book that the initial phase is always u0 ¼ ux ¼ 0 of the first wave and then only the initial phase u0 ¼ uy of the second wave changes, i.e. phase difference (phase angle, phase shift, Sects. 5.1 and 5.2), DE ¼ uy  ux ¼ uy  0 ¼ uy . The origin of coordinate system for the cosinusoidal function cosðxtÞ is shown in Fig. 5.1. Both linear (plane) waves must have the same direction of propagation, angular frequency x ¼ 2pf ; phase velocity vp , and the propagation constant (phase constant) b in medium. The attenuation coefficient (loss) of both waves has also been equal in a homogeneous isotropic (Sect. 3.27.1) medium and has no effect on the polarization of the resulting wave. In circular or elliptical polarization, the E-field rotates at a constant rate of an angular frequency x in a plane xOy as the wave travels. As we mentioned before, the rotation can have two possible directions. In Fig. 5.22, we see three-dimensional image of (a) clockwise (left-handed, anticlockwise), (b) counterclockwise (right-handed) circular polarizations. The direction of your fingers represents the positive direction of the electric field. The direction of your thumb represents the direction of travel. The fingers of one hand are squeezed (clenched) together in the direction of rotation of E-field; the thumb is pointed to the direction of propagation of the EM waves (Fig. 5.22). 3D picture of EM wave looks like rotating spirals. The handedness property determines the rotation of the electric field vector clockwise or counterclockwise regarding the direction of propagation. If the fingers of a person’s left hand curl in the same direction as the rotation of the electric field vector (clockwise) while the thumb is pointing to the propagation direction, the wave is the left-handed circularly polarized (Fig. 5.22a). A circularly polarized wave is right-handed if the fingers on a person’s right hand curl to the direction of

336

5

(a) y

Plane Electromagnetic Wave Propagation

Left-handed circular polarization E x

y

P=[EH] z

O

x

E

O o

(b) y Right-handed circular polarization x O

z

P

y

E

x o

z

E

z

P

P=[EH] Fig. 5.22 a Clockwise (left-handed), b counterclockwise (right-handed) circular polarizations

electric field rotation (counterclockwise) while the thumb points to the direction of propagation (Fig. 5.22b). Figure 5.23 determines dependences on the direction of an EM wave propagation P ¼ ½EH for the right-handed and left-handed circularly polarizations. We take that two waves, traveling along transmission media or waveguides in the forward or backward directions, are described as E yþ ðzÞ ¼ E 0þ eibz and E  y ¼  þ ibz þ (see Eq. 5.4.27). So the forward wave E y ðzÞ propagates to the positive E0 e direction of the z-axis and the backward wave E  y propagates to the negative direction of z-axis (Fig. 5.23). Circular polarization may be referred to as right-handed (counterclockwise, RHC) or left-handed (clockwise, LHC), depending on the direction in which the electric field vector rotates. Determination of the forward and backward waves is given in Eq. (5.4.27). The PHASOR FORM of E-field of a circularly polarized uniform plane wave (Fig. 5.23): Right-polarized, forward EM wave is shown by the red circle: EðzÞ ¼ E0þ eibz ¼ ð^x þ i^yÞE m eibz ;

ð5:17:1Þ

where the exponent term eibz shows that the wave propagates to the positive direction of z-axis. The wave propagates to the positive direction of z-axis when the unit vectors ^b ¼ ^z (see Sect. 1.4.2).

5.17

Approach to Polarization from Electrodynamics and Optics

^ iy) ^ E=Em(x+

^ E=Em(x^ iy) y

y

3π 4

ωt= π 2

π

0

z 7π 4

3π 2

ωt= 32π

5π 4

4

π

5π 4

337

x

7π 4

0

π

z 3π 4

x

π π

4

2

^ ^ RH for β= z

^ ^ E=Em(x ^ iy^)e iβz LH for β= z,

^ ^ LH for β= z

^ ^ E=Em(x ^ iy) ^ e+iβz RH for β= z,

Fig. 5.23 Right-handed (RH) and left-handed (LH) circular polarizations dependent on the direction of EM wave propagation

Left-polarized, forward EM wave is shown by the blue circle: EðzÞ ¼ E0þ eibz ¼ ð^x  i^yÞE m eibz :

ð5:17:2Þ

Left-polarized, backward EM wave is shown by the red circle: ibz ¼ ð^ x þ i^yÞE m eibz ; EðzÞ ¼ E 0e

ð5:17:3Þ

where the exponent term eibz is shown that the wave propagates in the negative direction of z-axis. The wave propagates in the negative direction of z-axis when the unit vectors ^b ¼ ^z: Right-polarized, backward EM wave is shown by the blue circle (Fig. 5.23): ibz EðzÞ ¼ E ¼ ð^x  i^yÞEm eibz ; 0e

ð5:17:4Þ

where ^b is the unit vector in the direction of propagation of EM wave, b is the wavenumber (propagation constant or phase constant in a lossless media), ^ x; ^ y; ^z are the unit vectors in the Cartesian coordinate system (Sect. 1.2.1), Em is the electric field amplitude (arbitrary constant) of E0þ or E 0 that are vectors for a forward wave and a backward wave; see Eq. 5.4.27, respectively. The unit vector of positive pffiffiffi helicity is ^e þ  ð^ x þ i^yÞ= 2, and the unit vector of negative helicity is pffiffiffi ^e  ð^x  i^yÞ= 2. Vector ð^x  i^yÞE m rotates on the side dependent on a sign “plus” or “minus” and on a direction of propagation of the wave.

338

5

Plane Electromagnetic Wave Propagation

In microwave and antenna engineering, the sense of rotation is defined when a thumb of the right and the left hand, respectively, points along or opposite to the direction of propagation (Figs. 5.21, 5.22 and 5.23).

5.18

Linear Polarization

In linear polarization, the E-field oscillates in a single direction. Linear (plane) polarization can be purely vertical, purely horizontal or can have any angle in between. The E-field oscillates in one direction at a fixed place in space over time or at a fixed time over distance. A plane wave is a wave whose surfaces of constant phase are infinite planes perpendicular to the direction of propagation (Fig. 5.7). The EM field cannot vary along the transverse coordinates x and y and @E=@x ¼ @E=@y ¼ 0; so EðzÞ must be only a function of z: There is only spatial variation in z-axis direction. We may replace the partial derivative @E=@z with the ordinary derivative @E=@z: For the sake of simplicity, we will now consider the electric field as having only one single component, e.g. Ey . EM wave can propagate in two directions, i.e. along and against the positive z-axis, i.e. forward and backward. We can express the components of the EM wave as: n o n o Ey ¼ A1 cosðxt  bzÞ þ A2 cosðxt þ bzÞ ¼ Re A1 eiðxtbzÞ þ Re A2 eiðxt þ bzÞ ; ð5:18:1Þ



A1 A2 A1 iðxtbzÞ A2 iðxt þ bzÞ cosðxt þ bzÞ ¼ Re e e þ Re ; H x ¼ cosðxt  bzÞ þ g g g g ð5:18:2Þ where A1 , A2 are the amplitudes of EM wave (Sects. 5.4 and 5.7), the wave A1 cosðxt  bzÞ represents the forward wave which propagates in the positive zaxis direction and A2 cosðxt þ bzÞ depicts the backward wave which propagates along the negative z axis, g is the intrinsic impedance. We would like to point out that in the following text we will not specify the upper “+” or “−” index in most cases, although we will assume that the wave spreads along the positive z-axis. When there is the component in the y-axis direction (Fig. 5.24a) and EM wave only propagates in the positive z-axis (forward) direction, the wave equation in scalar form is as follows: d2 E y þ b2 E y ¼ 0: dz2

ð5:18:3Þ

Linear Polarization

y O

x

339

(a)

(b) E

H

P=[EH]

z

x

y O

z

D po irec la tio riz n at of io n

Direction of polarization

5.18

Vertical plane of polarization

H

E

z

P=[EH] z

Horizontal plane of polarization

Fig. 5.24 Planes of polarization and the snapshot in time of vertical and horizontal linearly polarized waves

The solution is the following: þ þ iðxtbzÞ expðiðxt  bzÞÞ ¼ E ym e ; E y ¼ Eym

ð5:18:4Þ

where the phase constant, using Eq. (5.11.2) at e00 e0 , is b2 ¼ x2 le: Now we will consider another case when the wave has a horizontal polarization. As the electric field has only one single component in the x direction (Fig. 5.24b), we can write the wave equation in scalar form: d2 E x þ b2 E x ¼ 0: dz2

ð5:18:5Þ

The solution of Eq. (5.18.5) is: þ Ex ¼ E xm expðiðxt  bzÞÞ:

ð5:18:6Þ

A snapshot in time of a linearly polarized wave is shown in Fig. 5.24. If a linearly polarized wave has only vertical or horizontal electric field lines, the wave is, respectively, vertically or horizontally polarized. If ux ¼ uy ¼ u, the total solution can be written as:   E ¼ x^E x0 þ ^yEy0 expðiðxt  bz þ uÞÞ ¼ E0 expðiðxt  bz þ uÞÞ:

ð5:18:7Þ

In this solution, the direction of the electric field vector is independent from time xE x0 and and space and is defined by a new vector E0 which is the resulting sum of ^ ^yE y0 . This type of wave is known as a linearly polarized wave and the direction of the electric field vector E0 represents the direction of polarization. An EM wave is linearly polarized if the E-vector is always oriented along the same straight line at every instance in time. The field vector has only one component or two orthogonal linear components which are either in phase or out-of-phase (value p multiplied by n ¼ 0; 1; 2; 3. . . ); see further Table 5.2.

340

5.19

5

Plane Electromagnetic Wave Propagation

Circularly Polarized Wave

An EM wave is circularly polarized if the E-field vector end traces a circle as a function of time and space. The E-field vector must have two orthogonal linear components with the same magnitudes and the phase difference of odd (not divisible by two) multiples of p=2: So the circularly polarized wave can be expressed as two linearly polarized waves shifted by 90° in phase, i.e. Exþ ¼ Eyþ and uy ¼ ux 

p p ¼ : 2 2

ð5:19:1Þ

We can choose an initial phase value ux equal to zero, then uy ¼ p=2: The total electric field of the circularly polarized wave can be written as: þ þ E¼^ xEx0 expðiðxt  bzÞÞ þ ^yEy0 expðiðxt  bz  p=2ÞÞ:

ð5:19:2Þ

The rotation of the electric field vector is regarded as a function of space and time. The expression of Eq. (5.19.2) is known as a circularly polarized wave because of the locus traced of the electric field vector as a function of time (at a given point) is a circle, as shown in Fig. 5.22. Right- and left-handed circular polarizations are both possible, depending on the sign of the phase difference (phase angle, phase shift) p=2:

5.20

Elliptically Polarized Wave

If E xþ 6¼ E yþ , the locus (position, place, curve) of the electric vector end at the phase shift p=2 traces an ellipse and the wave is described as being elliptically polarized (Figs. 5.20c and 5.25). Elliptical polarization is the polarization of EM wave in such a way that the tip of E field vector traces an elliptical locus (curve) in space as a function of time t in any fixed plane intersecting and normal to the direction of propagation, i.e. to the Poynting vector P ¼ ½E H: The elliptic polarization is characterized by thephase shift (Sects. 5.1 and 5.2) DE ¼ uy  ux and the magnitude ratio E y ðz; tÞ E x ðz; tÞ which can be any at E x ðz; tÞ 6¼ 0: The sign of DE determines the direction of rotation. The E-field vector must have two perpendicular linear components. These components can have the same or different magnitudes. When these components have the same magnitudes, their phase difference must satisfy DE 6¼ ð2n þ 1Þp=2 or DE 6¼ np; n ¼ 0; 1; 3; 5. . ., the number ð2n þ 1Þ is the odd one. When these components do not have the same magnitudes, their time phase difference must not

5.20

Elliptically Polarized Wave

341

Fig. 5.25 Counterclockwise (right-handed) elliptical polarization

y

x

Ex

E (z,t)

Ey

z= co

ns t

z

P=[EH]

be equal to zero or multiples of p; for the reason that the resultant will be a linearly polarized EM wave at these phase differences (see also further Table 5.2).

5.20.1 Polarization Ellipse The E-field vector traces an ellipse as a function of time at a plane perpendicular to the Poynting vector. Elliptical polarization is the most general type of polarization. The linear and circular polarizations are special cases of the elliptical polarization. We can write that for the uniform plane wave, we have: Eðz; tÞ ¼ ER ðz; tÞ ¼ Ex ðz; tÞ þ Ey ðz; tÞ; Ex ðz; tÞ ¼ ^xE x ðzÞ cos xt and E x ðz; tÞ ¼ Ex ðzÞ cos xt:

ð5:20:1Þ ð5:20:2Þ

We can write from Eq. (5.20.2): cos xt ¼

E x ðz; tÞ ; E x ðzÞ

ð5:20:3Þ

because cos2 xt þ sin2 xt ¼ 1; and sin xt ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðEx ðz; tÞ=E x ðzÞÞ2 ;

Ey ðz; tÞ ¼ ^yE y ðzÞ cosðxt þ DE Þ and E y ðz; tÞ ¼ Ey ðzÞ cosðxt þ DE Þ;

ð5:20:4Þ ð5:20:5Þ

where DE ¼ uy  ux ¼ uy  0 ¼ uy is the phase shift. Upon using Ptolemy’s identities, the sum formulae for cosine are as follows: cosðxt þ DE Þ ¼ cos xt cos DE  sin xt sin DE ;

ð5:20:6Þ

342

5

Plane Electromagnetic Wave Propagation

we get: Ey ðz; tÞ ¼ ^yEy ðzÞðcos xt cos DE  sin xt sin DE Þ;

ð5:20:7Þ

Eðz; tÞ ¼ ER ðz; tÞ ¼ ^xEx ðzÞ cos xt þ ^yE y ðzÞ cosðxt þ DE Þ;

ð5:20:8Þ

EðzÞ ¼ ^xEx þ ^yEy eiDE ;

ð5:20:9Þ

where eiDE ¼ cos DE þ i sin DE . After placing Eqs. (5.20.3) and (5.20.4) into Eq. (5.20.7): E y ðz; tÞ ¼ Ey ðzÞ cosðxt þ DE Þ ¼ E y ðzÞðcos xt cos DE  sin xt sin DE Þ; ð5:20:10Þ 0 Ex ðz; tÞ cos DE  E y ðz; tÞ ¼ Ey ðzÞ@ E x ðzÞ

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Ex ðz; tÞ 2 sin DE A: 1 E x ðzÞ

ð5:20:11Þ

From Eq. (5.20.10):   E y ðzÞ cos xt cos DE E y ðz; tÞ 1 ; sin DE ¼   E y ðzÞ sin xt Ey ðzÞ sin xt  sin2 DE ¼

2   cos xt cos DE E y ðz; tÞ 1  :  E y ðzÞ sin xt sin xt

Using Eq. (5.20.3):  2   E x ðz; tÞ cos DE E y ðz; tÞ 1 sin2 DE ¼  ;  E x ðzÞ sin xt E y ðzÞ sin xt    Ex ðz; tÞ 2 cos2 DE Ex ðz; tÞ Ey ðz; tÞ cos DE  sin DE ¼ 2 E x ðzÞ Ex ðzÞ Ey ðzÞ sin2 xt sin2 xt   E y ðz; tÞ 2 1  2 ; þ E y ðzÞ sin xt

ð5:20:12Þ

ð5:20:13Þ

ð5:20:14Þ



2

sin2 DE ¼

ð5:20:15Þ

1 sin"2 xt       # E x ðz; tÞ 2 2 E x ðz; tÞ E y ðz; tÞ E y ðz; tÞ 2   cos DE  2 : cos DE þ E x ðzÞ E x ðzÞ E y ðzÞ E y ðzÞ ð5:20:16Þ

5.20

Elliptically Polarized Wave

343

Using Eq. (5.20.4), we can write:

  E x ðz; tÞ 2 sin xt ¼ 1  : E x ðzÞ 2

ð5:20:17Þ

We place Eq. (5.20.17) into Eq. (5.20.16):   ! E x ðz; tÞ 2 sin DE  1  ¼ E x ðzÞ "      # ð5:20:18Þ Ex ðz; tÞ 2 2 Ex ðz; tÞ Ey ðz; tÞ Ey ðz; tÞ 2  cos DE  2 : cos DE þ E x ðzÞ E x ðzÞ E y ðzÞ E y ðzÞ 2

We divide the left and right parts of Eq. (5.20.18) by the value of sin2 DE :   !     E x ðz; tÞ 2 E x ðz; tÞ 2 cos2 DE E x ðz; tÞ E y ðz; tÞ cos DE  1 2 ¼ E x ðzÞ E x ðzÞ E x ðzÞ E y ðzÞ sin2 DE sin2 DE   E y ðz; tÞ 2 1 þ : E y ðzÞ sin2 DE ð5:20:19Þ We transfer the second term from the left side of the equation to the right: 1¼

      E x ðz; tÞ 2 E x ðz; tÞ 2 cos2 DE E x ðz; tÞ E y ðz; tÞ cos DE    2 E x ðzÞ E x ðzÞ E x ðzÞ E y ðzÞ sin2 DE sin2 DE ð5:20:20Þ  2 Ey ðz; tÞ 1 þ ; Ey ðzÞ sin2 DE

and we simplify Eq. (5.20.20) by combining similar terms:       E x ðz; tÞ 2 sin2 DE þ cos2 DE E x ðz; tÞ E y ðz; tÞ cos DE  1¼ 2 E x ðzÞ E x ðzÞ E y ðzÞ sin2 DE sin2 DE   Ey ðz; tÞ 2 1 þ ; E y ðzÞ sin2 DE

ð5:20:21Þ

Using the trigonometric identities formula sin2 ðnÞ þ cos2 ðnÞ ¼ 1; we get the expression:     E x ðz; tÞ 2 1 E x ðz; tÞ E y ðz; tÞ cos DE 1¼  2 E x ðzÞ sin2 DE E x ðzÞ E y ðzÞ sin2 DE ð5:20:22Þ   E y ðz; tÞ 2 1 þ : E y ðzÞ sin2 DE

344

5

Plane Electromagnetic Wave Propagation

We enter the designations: E x ðz; tÞ cos xt ¼ ; E x ðzÞ sin DE sin DE

ð5:20:23Þ

E y ðz; tÞ cosðxt þ DE Þ ¼ : Ey ðzÞ sin DE sin DE

ð5:20:24Þ

xðtÞ ¼ yðtÞ ¼

We substitute Eqs. (5.20.23) and (5.20.24) into Eq. (5.20.22) and receive: 1 ¼ x2 ðtÞ  2xðtÞyðtÞ cos DE þ y2 ðtÞ:

ð5:20:25Þ

Equation (5.20.25) is the equation of an ellipse in the xOy-plane. It describes the trajectory of a point of coordinates xðtÞ and yðtÞ; i.e. normalized Ex ðz; tÞ and E y ðz; tÞ values, along an ellipse where the point moves with an angular frequency x: The trace of the time-dependent vector is an ellipse. The shape of the ellipse is depicted by its ellipticity, which is the ratio of the major and minor axes of ellipse. This ratio is also called axial ratio. The minor and major axes of ellipse are diameters (lines through the center) of the ellipse. The minor axis is the shortest diameter, and major axis is the longest diameter (Fig. 5.26). A wave with linear polarization has an ellipticity of infinity because the minor axis is zero. A circularly polarized wave has an ellipticity equal to unit because the major and minor axes are the same. The electric field vector of EM wave with elliptical and circular polarization rotates with angular frequency x: Similarly as in the circular polarization, the elliptical polarization can be either right-handed or left-handed, depending on the relation between the direction of propagation and direction of rotation (see Sect. 5.17). The parameters of the ellipse are presented below. Major axis ð2  OAÞ: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 OA ¼ Ex þ E 2y þ E 4x þ E4y þ 2E 2x E 2y cosð2DE Þ ; 2

Fig. 5.26 General polarization ellipse

ð5:20:26Þ

y E

xis ra no B) i M 2·O (

A

s axi jor A) a M ·O (2

ω Ey

Ex

O

B

x

5.20

Elliptically Polarized Wave

OA ¼ jEj

345

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1 þ 1  sin2 ð2hÞ sin2 ðDE Þ 2

:

ð5:20:27Þ

Minor axis ð2  OBÞ: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 E þ Ey  E 4x þ E4y þ 2E 2x E 2y cosð2DE Þ ; OB ¼ 2 x vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1  1  sin2 ð2hÞ sin2 ðDE Þ OB ¼ jEj : 2

ð5:20:28Þ

ð5:20:29Þ

The orientation of the ellipse is given by the tilt angle h between the semimajor axis with the x-axis: ! 1 2E x Ey  cosðDE Þ h ¼ arctan 2 2 E x þ E2y

1 ¼ arctan h0 ; 2

ð5:20:30Þ

2E E

when h0 ¼ E2 þx Ey2  cosðDE Þ: x

y

The principal value of the angle h is given in Eq. (5.20.30). In general, Eq. (5.20.30) presents an infinite number of angles h ¼ 12 arctan h0  np=2; n ¼ 1; 2; . . . It gives the angle which is the major axis with the x-axis. In antenna engineering, the shape of an ellipse is described by the axial ratio or ellipticity eEL , which is the ratio of the ellipse minor and major axes: eEL ¼

Minor axis OB ¼ : Major axis OA

ð5:20:31Þ

The ellipticity eEL value ranges between zero (linear polarization) and one (circular polarization). The handedness of elliptically polarized EM wave is as follows: 0\DE \p (left-handed handedness), p\DE \2p (right-handed handedness). The term “handedness” is the dominance of one hand over the other or the unequal distribution of skills between the left and right hands.

346

5.21

5

Plane Electromagnetic Wave Propagation

General Description of Polarizations

As mentioned in Sect. 5.20.1, the linear and circular polarizations are special cases of more general elliptical polarization:   (a) When DE ¼ uy  ux ¼ uy ¼ np; OB = 0 and h ¼  arctan Ey =E x ; the ellipse becomes a line.   (b) when DE ¼ uy  ux ¼ uy ¼  p2 þ 2np and OA ¼ OB ¼ Ex ¼ E y , the ellipse becomes a circular. We have taken here that the initial phase ux is equal to zero. The main conditions for all types of polarization are summarized in Table 5.2. The basic principles on the phase shift, leading and lagging phases of two EM waves, were set out in Sect. 5.2. In Fig. 5.27, cosines which correspond to the EM waves with different initial phases are placed in one coordinate system. The magnitudes E x , E y1 , and E y2 of EM waves are given as functions of time t in Fig. 5.27. This shows how a phase difference between waveforms changes over time. We see in Fig. 5.27 that E y1 ðtÞ is ahead (leading phase, positive initial phase þ uy ) of E x ðtÞ and Ey2 ðtÞ is behind (lagging phase, negative initial phase uy ) of Ex ðtÞ: We will analyze the behavior of the E-field vector components of the total (resulting) EM wave ER ðz; tÞ ¼ Ex ðz; tÞ þ Ey ðz; tÞ in space and in time (Eqs. 5.20.2 and 5.20.5). The component Ex is directed along x-axis and Ey is directed along y-axis. Table 5.2 Conditions of polarization No

Polarization type

Amplitudes E ym , E xm

Phase DE ¼ uy  ux , rad when ux ¼ 0; DE ¼ uy

1

Linearly (plane) polarization

Eym  E xm ; E ym 6¼ 0 or Eym \Exm ; E xm 6¼ 0

DE ¼ np, n ¼ 0; 1; 2. . .

2

RH and LH circular polarizations (Fig. 5.22)

E ym ¼ E xm ; E ym 6¼ 0

DE ¼ ð2n þ 1Þp=2 n ¼ 0; 1; 2. . .

3

RH and LH oblique elliptical polarizations (Fig. 5.22)

6 E xm ; Eym ¼ Eym ¼ 6 0; E xm 6¼ 0

DE is arbitrary; only DE 6¼ np; n ¼ 0; 1; 2; 3. . .

E ym ¼ E xm ; E ym 6¼ 0

DE is arbitrary; only DE ¼ 6 ð2n þ 1Þp=2 DE ¼ 6 np; n ¼ 0; 1; 3; 5. . .

E ym \E xm ; Eym 6¼ 0

DE ¼ ð2n þ 1Þp=2

E ym [ E xm ; E xm 6¼ 0

DE ¼ ð2n þ 1Þp=2

3.1 3.2

Horizontal elliptical polarization Vertical elliptical polarization

5.21

General Description of Polarizations

Ex(t),V/m Ey(t),V/m

347

Ey2=Eym2cos(ωt βz

φ y)



7π 2

Ex=Exmcos(ωt βz)

π

0

2

+φy

π

3π 2

5π 2



t

-φ y

Ey1=Eym1cos(ωt βz+φy) Fig. 5.27 Phase differences uy

The total EM wave ER ðz; tÞ propagates in the positive z-axis direction. The Poynting vector P¼EH is also directed toward the positive z-axis direction. We fix a coordinate point z ¼ z1 and follow the change of vector ER ðz; tÞ in time t in the xOy-plane (Fig. 5.19). For geometric construction of the trace of the tip of the ER ðz1 ; tÞ vector, which is the polarization diagram for the sum of two linearly polarized waves are used a vertical graph Ex ðz1 ; tÞ of time (located in the bottom of Fig. 5.28) and a horizontal graph Ey ðz1 ; tÞ of time. Since the coordinate z1 is fixed, it leaves out the dependency only on coordinate t: We placed horizontally the graph Ey ðtÞ in the left corner of the plane for each case we are considering (Fig. 5.28) . On the same plane, we must put the graph Ex ðtÞ vertically in the lower right corner of the plane as shown in Fig. 5.28. The number of points to which cosine (Fig. 5.27) is divided typically equals between 8 and 12. Division points of cosine are numbered in the ascending order. Horizontal lines are drawn straight from the cosine Ey ðtÞ points to the upper right corner of the plane. Vertical lines are drawn straight from the cosine Ex ðtÞ points to the upper right edge of the plane (Fig. 5.28). These vertical and horizontal lines intersect. The intersection points in the upper right corner of the plane must have the same numbering as horizontal and vertical lines were marked with. The numbered points in the upper right corner of the plane are joined in the ascending order. The shape of the figure that describes the end of the ER ðtÞ vector over time on the plane depends on the phase shift DE ¼ uy  ux ¼ uy and can be an ellipse, a circle, or a line (see Table 5.3; Fig. 5.28). The basic formulae for plotting the graph ER ðtÞ are shown below. General solution of the homogeneous vector Helmholtz’s equation for a monochromatic plane wave propagating in the positive direction is:   ER ðz; tÞ¼ ^ xE xm expðiðxt  bz þ ux ÞÞ þ ^yE ym exp i xt  bz þ uy ;

ð5:21:1Þ

348

5

Ey

φy=0°

1 2 0

Ey

1

φy=30° 2

11 10

3 4

Ey

Ey

1 12 1 12 11 3 2 11 3 10 4 4 π 10 2π 9 8 5 5 9 6 6 7 8

π 5

8

6 7

7

π

3

4

6

5

8 7

φy=90°

Ey

9 10 11 8 12 8

1 2

π 3

5

4

6



7

7

6

7

π

1 2

4

3

8

6 5

9 10 8 11 7 12 2π 6 1

1 3 1

11

10

12

3

4

4



6

π

1

3

2

4

5

7

8

9



5

π 1

4 2

3

6 7 8 2π

9

10

11 12

6

9 10

11

10

5

11

4

12

2

1

1

2

3

4

9



5



3 4 5 2π

2 3 4 1

12 1



2

10

1

11

4

8



4

10 3

11 2

5

12

2π 1

1 3

4

2

6

6 7

4

12 8 9 10 11

5

1

6 12 11 7 8 9 10 8

4

5

10

6



11 6 10 7 8

12 11

4 5

3π 6

7

8

9

10

Ex

t

11

12 1

10

11

12

2

3

4

11

2 1 3 12 11

4 10

5

1 4

3

Ex

Ex

1

12 10

Ex

2 1

0 6

1 12

3

4

1

4π 5 6 9 7 8

2

2

11

10

9

2

Ex

Ex Ex

π

7 10



5

1

12

3

φy=360° 3

3

11

4

9

1

12

Ex

1



8

9

9

φy=300° 5

12

7



1

3 11

5

5

Ex

2

2

10

9

12

4

7

7 8 9

1 2

9

π 8

9

3

11

φy=330° 3



12 1 2

3

0 6



4

10

8

8 1 9 10 11 12

8

6 7

2

3

1

7

9

5

6 5

7

6 7

1

5

8

8

9

3π 1

11 12

Ey

6

φy=270°

8 9 10 7 6 11 12 5 1

7

10

8

10 1 11 12

4 5 6

φy=180°

Ey

7

φy=240°

φy=150°

Ey

9

φy=210° 5 6 7

2

2

8



3

2

3

1

5

φy=120°

Ey

1 2

7

6

5

4

π

3

4

9

φy=180°

Ey

2

5

6

6

11 12

10

9

11 12 1 8 2π 7

10 9

2

1

2π 8

9

φy=60° 1

7 12

Plane Electromagnetic Wave Propagation

9

10 2π

11

t

12 1

Fig. 5.28 Behavior of the total electric field vector of monochromatic EM plane wave at the z ¼ const plane, ER ðz; tÞ ¼ Ex ðz; tÞ þ Ey ðz; tÞ; ux ¼ 0; Dy ¼ uy , E y is ahead of E x

5.21

General Description of Polarizations

Table 5.3 Polarization as a function of DE and ratio of E y =E x according to Fig. 5.22

349

350

5

Plane Electromagnetic Wave Propagation

ux and uy are arbitrary (yet constant) phase factors. We have taken to simplify that the initial phase is ux ¼ 0: The nature of the resulting wave ER ðz; tÞ depends on þ þ the values of Exm ¼ Exm , Eym ¼ Eym , uy . We will define the resultant vector as the vector sum of two or more vectors (Sect. 1.2.5). Composition of two orthogonal liner components: E1 ðz; tÞ ¼ Ex ðz; tÞ ¼ ^xE x ðz; tÞ;

ð5:21:2Þ

E2 ðz; tÞ ¼ Ey ðz; tÞ ¼ ^yE y ðz; tÞ:

ð5:21:3Þ

The resultant vector is: ER ðz; tÞ¼ Ex ðz; tÞ þ Ey ðz; tÞ ¼ ^xE x ðz; tÞ þ ^ yEy ðz; tÞ;

ð5:21:4Þ

Ex ðz; tÞ ¼ ^xExm cosðxt  bz þ ux Þ ¼ ^xE xm cosðxt  bzÞ;

ð5:21:5Þ

where E xm is the amplitude of the horizontal wave, b is the propagation constant that specifies the direction of propagation, x ¼ 2p=f is the angular (circular) frequency, f is the operating frequency, xt is the angular frequency of the waveform in rad/s, ðxt  bzÞ is the phase of horizontal polarized wave. The vertical component of the total (resulting) electric field is:   Ey ðz; tÞ ¼ ^yEym cos xt  bz þ uy ;

ð5:21:6Þ

where E ym is the amplitude, uy is the phase angle in degrees or radians that the waveform has shiftedeither to the left  or to the right from the reference point (the original) (Fig. 5.27), xt  bz þ uy is the phase of vertical polarized wave (Sects. 5.1 and 5.2). The total electric field ER ðz; tÞ which represents the vector sum can be written on the base of Eqs. (5.21.4) and (5.21.5):   ER ðz; tÞ ¼ ^xE xm cosðxt  bzÞ þ ^yE ym cos xt  bz þ uy :

ð5:21:7Þ

The module of the total electric field vector (the magnitude of the vector, Sect. 1.2.5) is: E R ðz; tÞ ¼ jER ðz; tÞj ¼ Ex ðz; tÞ þ Ey ðz; tÞ :

ð5:21:8Þ

The direction of the vector ER ðz; tÞ is characterized by the angle hðz; tÞ (Sect. 1.2.5):

5.21

General Description of Polarizations



Ey ðz; tÞ hðz; tÞ ¼ arctan Ex ðz; tÞ



351

    Eym cos xt  bz þ uy E y ðz; tÞ ; ¼ tan ¼ arctan Exm cosðxt  bz þ ux Þ E x ðz; tÞ 1

ð5:21:9Þ when the initial phase ux ¼ 0; then the phase shift DE ¼ uy  ux ¼ uy (Sects. 5.1 and 5.2) and arctanðnÞ ¼ tan1 ðnÞ is the inverse function of the tangent (or the inverse tangent function). The magnitude of the resultant vector ER ðz; tÞ in Eq. (5.21.8) is: jER ðz; tÞj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2ffi ðExm cosðxt  bzÞÞ2 þ Eym cos xt  bz þ uy :

ð5:21:10Þ

The angle hðz; tÞ giving the direction of the vector ER ðz; tÞ at certain variables t and z is: hðz; tÞ ¼ arctan

Ey ðz; tÞ E ym cosðxt  bz þ DE Þ ¼ arctan : Ex ðz; tÞ E xm cosðxt  bzÞ

ð5:21:11Þ

5.21.1 Shape of Curve Drawn by the End of E-Vector Depending on the Phase Shift The resulting electric field vector ER ðz; tÞ draws a curve (an ellipse, circle, or straight line) as a function of time at a plane perpendicular to the Poynting vector. In order to realize how the phase difference DE affects the polarization of resulting wave, we will consider in Fig. 5.28 a sequential increase in the phase DE with the uniform increment of 30 . Figure 5.28 demonstrates two columns showing the summation of two plane waves Ex ðz; tÞ and Ey ðz; tÞ with a phase shift DE ¼ uy . Seven pictures are placed in    each column with phase shift from 0  180 and 180  360 . The component Ex ðz; tÞ ¼ ^xE xm cosðxt  bzÞ remains constant in all phase shifts. The component  Ey ðz; tÞ ¼ ^yE ym cos xt  bz þ uy changes because the initial phase uy is different for any case. The amplitudes of waves E ym and Exm remain constant, and the value  uy changes with the uniform increment of 30 . Figure 5.28 shows the process of changing the polarization of the total wave ER ðz; tÞ ¼ Ex ðz; tÞ þ Ey ðz; tÞ when the phase shift uniformly changes. Now we will investigate how the polarization of the total wave ER ðz; tÞ depends on the ratio of amplitudes of summable waves Ex ðz; tÞ; Ey ðz; tÞ in a wide range of their relationship. The type of polarization as the function of the phase shift DE ¼ uy  ux ¼ uy and the ratio of E y =Ex from zero till infinity is shown in Table 5.3.

352

5

Plane Electromagnetic Wave Propagation

We see from Table 5.3 that the value of ratio E y =E x only affects the slope of the linear polarization or the inclination of the axis of the ellipse at the elliptical polarization (Eq. 5.20.30) but does not affect the type of polarization. In the following three paragraphs, we shall consider examples of addition of two orthogonal plane waves at the different value of their initial phase DuE ¼ uy and different ratio of their amplitudes E y =Ex .

5.22

Problem Related to Linear Polarization of Electromagnetic Waves

Draw: (1) the orthogonal linearly polarized plane waves Ex ðz1 ; tÞ; Ey ðz1 ; tÞ; (2) the polarization diagram for sum of these linearly polarized waves, i.e. the trace of the tip of the total vector ER ðz1 ; tÞ in the point z1 ¼ const (Fig. 5.19); (3) the resulting (total) electric field ER ðtn Þ; and 4) its module jER ðz; tÞj when the amplitudes of wave E x ¼ Exm ½V=m; Ey ¼ E ym ½V=m are given in a personal task, and the phase shift between these waves (Sects. 5.1 and 5.2) is Dy ¼ uy  ux ¼ uy ½rad because we assumed that ux ¼ 0: Some explanation has been given in Sect. 5.21 and Figs. 5.28 and 5.29.

Ey(t) Eym

y

E (0,8) E (1,7)

z

t 8 7 6 5 4 3 2 1 0

E (t) Em

Ex m

t

8 7 6 5 4 3

E (t)

2 1 0

E (3,5) z

8 7 6 5 4 3 2 1 0

|E (t)|

x

E (2,6)

E (4) |E (t)|

z

0

Ex(t) x

t Fig. 5.29 Polarization diagram for the linear polarization and Ex ðtÞ; Ey ðtÞ; jER ðtÞj and ER ðtÞ

5.22

Problem Related to Linear Polarization of Electromagnetic Waves

353

Table 5.4 Calculations of ER ðtn Þ and jER ðtn Þj Number of point

tn ; s

0 1 2 3 4 5 6 7 8

t0 t1 t2 t3 t4 t5 t6 t7 t8

xtn ; rad

¼0 ¼ 1=8 ¼ 2=8 ¼ 1=4 ¼ 3=8 ¼ 4=8 ¼ 1=2 ¼ 5=8 ¼ 6=8 ¼ 3=4 ¼ 7=8 ¼ 8=8 ¼ 1

xt0 xt1 xt2 xt3 xt4 xt5 xt6 xt6 xt7

¼0 ¼ p=4 ¼ p=2 ¼ 3p=4 ¼p ¼ 5p=4 ¼ 3p=2 ¼ 7p=4 ¼ 2p

cosðxtn Þ

ER ðz ¼ 0; tn Þ

jER ðtn Þj ¼ 4:24jcosðxtÞj

1 0.707 0 –0.707 –1 –0.707 0 0.707 1

4.24 ^r 3.0 ^r 0 –3.0 ^r –4.24 ^r –3.0 ^r 0 3.0 ^r 4.24 ^r

4.24 3.0 0 3.0 4.24 3.0 0 3.0 4.24

Then write the formulae of the total electric field intensity and its module, the phase shift of the plane EM waves and calculate their values for your task. Given : E x ¼ 3ðV=mÞ;

Ey ¼ 3ðV=mÞ;

DuE ¼ uy ¼ 0ðradÞ;

ER ðz; tÞ ¼ ^xEx cosðxt  bzÞ þ ^yE y cosðxt  bzÞ; jER ðz; tÞj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2ffi ðExm cosðxt  bz þ 0ÞÞ2 þ E ym cos xt  bz þ uy ;

ð5:22:1Þ ð5:22:2Þ ð5:22:3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 jER ðz; tÞj ¼ 3 ðcosðxt  bzÞÞ þ ðcosðxt  bzÞÞ ¼ 3 2ðcosðxt  bzÞÞ2 ; ð5:22:4Þ jER ðz; tÞj ¼ 4:24

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcosðxt  bzÞÞ2 ¼ 4:24 cosðxt  bzÞ:

ð5:22:5Þ

We can choose any convenient value for us of the coordinate z; e.g. z ¼ 0: Then we have: ER ðtÞ ¼ 3ð^x cosðxtÞ þ ^y cosðxtÞÞ ¼ 3 cosðxtÞð^ xþ^ yÞ: Vector (Sect. 1.2.6) ð^x þ ^yÞ ¼ r; r ¼h1; 1i , jrj ¼ cos h ¼ j1rj ¼ p1ffiffi ; h ¼ p4 :

ð5:22:6Þ

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1þ1 ¼ 2 ;^ r ¼ jrrj ¼ ^xpþffiffi2^y ;

2

Time t = 0, 1, 2,…8 s, x ¼ 2pf ¼ 2p ðradÞ at f = 1 Hz. ER ðz ¼ 0; tn Þ ¼ 3 cosðxtÞð^x þ ^yÞ ¼ 4:24 cosðxtÞ^r; jER ðz; tÞj ¼ 4:24

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcosðxtÞÞ2 ¼4:24jcosðxtÞj:

ð5:22:7Þ ð5:22:8Þ

354

5

Plane Electromagnetic Wave Propagation

The cosine function cosðxt  bzÞ has positive and negative values. The total (resulting) vector ER ðz; tÞ can be positive and negative, and the module jER ðz; tÞj has only positive values. We are constructing the polarization diagram (Fig. 5.29) on the basis of Table 5.4. The technique of adding vectors Ey ðtÞ and Ex ðtÞ at fixed times is shown in Fig. 5.29. Exm ðtÞ and Eym ðtÞ are amplitudes of electric field of considered EM waves, i.e. their maximum values. In the upper left corner of the Fig. 5.29, we see the complete cycle of vertical plane wave Ey ðtÞ with the initial (start) phase uy ¼ 0: In the bottom right corner, there is the complete cycle of the horizontal plane wave Ex ðtÞ with the initial phase ux ¼ 0: In Fig. 5.29, the time of one complete cycle of EM wave is divided in intervals. We arbitrarily chose eight segments. For the plane wave with the electric field Ex ðtÞ from points “0, 1, 2,…8” you need to draw, in the beginning, horizontal lines (black with arrows) till the crossing with the green curve cosðxt  bzÞ: After that from the new points on the green curve cosðxt  bzÞ, you need to draw a vertical (green) line. The numbering of all received points coincides with the initial points “0, 1, 2,…8” on the vertical t- axis. For the plane wave Ey ðtÞ from points “0, 1, 2,…8” on the horizontal t axis, you need to draw, in the beginning, vertical lines till the red curve cosðxt  bzÞ: After that from the new points on the red curve cosðxt  bzÞ, you need to draw a horizontal red line. The intersection of the red horizontal and green vertical lines from the points with the same numbering gives the polarization diagram for the sum of two plane waves (the trace of the tip of the total vector ER ðz1 ; tÞ at z1 ¼ constÞ. The resulting vector ER ðtÞ from data of Table 5.4 is given on the lower left corner of the plane. The module jER ðtÞj is represented by a dashed blue line which is also in the same picture of the plane (Fig. 5.29).

5.23

Problem Related to Circular Polarization of Electromagnetic Waves

Draw: (1) the orthogonal linearly polarized plane waves Ex ðz1 ; tÞ; Ey ðz1 ; tÞ; (2) the polarization diagram for sum of these linearly polarized waves, i.e. the trace of the tip of the total vector ER ðz1 ; tÞÞ in the point z1 ¼ const (Fig. 5.19); (3) the resulting (total) electric field ER ðtn Þ; and 4) its module jER ðz; tÞj when the amplitudes of wave E x ¼ E xm ½V=m; E y ¼ Eym ½V=m are given in a personal task, and the phase shift between these waves (Sects. 5.1 and 5.2) is Dy ¼ uy  ux ¼ uy ½rad because we assumed ux ¼ 0: Then write the formulae of the total electric field intensity and its module and the phase angles of the plane EM waves and calculate their values for your task. Determine the direction of rotation of the vector ER for the circularly polarized wave.

5.23

Problem Related to Circular Polarization of Electromagnetic Waves

355

Fig. 5.30 Polarization diagram for the counterclockwise (right-handed) circular polarization and Ex ðtÞ, Ey ðtÞ; jER ðtÞj and ER ðtÞ

Table 5.5 Calculations of ER ðtn Þ and jER ðtn Þj Number of point

tn ; s

0 1 2 3 4 5 6 7 8

t0 t1 t2 t3 t4 t5 t6 t7 t8

¼0 ¼ 1=8 ¼ 1=4 ¼ 3=8 ¼ 1=2 ¼ 5=8 ¼ 3=4 ¼ 7=8 ¼1

xtn ; rad xt0 xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8

¼0 ¼ p=4 ¼ p=2 ¼ 3p=4 ¼p ¼ 5p=4 ¼ 3p=2 ¼ 7p=8 ¼ 2p

cosðxtn Þ

sinðxtn Þ

ER ðtn Þ; V/m

jER ðtn Þj ¼ E xm

1 0.707 0 –0.707 –1 –0.707 0 0.707 1

0 0.707 1 0.707 0 –0.707 –1 –0.707 0

3 2.12 0 –2.12 –3 –2.12 0 2.12 3

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

356

5

Given : E x ¼ 3ðV=mÞ;

E y ¼ 3ðV=mÞ;

Plane Electromagnetic Wave Propagation

DuE ¼ uy ¼ p=2ðradÞ:

  ER ðz; tÞ ¼ E1 þ E2 ¼ ^xEx cosðxt  bzÞ þ ^yEy cos xt  bz þ uy ; jER ðz; tÞj ¼

ð5:23:1Þ ð5:23:2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ðE xm cosðxt  bz þ 0ÞÞ2 þ Eym cosðxt  bz  p=2Þ ; ð5:23:3Þ

when E xm ¼ E ym : jER ðz; tÞj ¼ E xm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðxt  bzÞ2 þ cosðxt  bz  p=2Þ2 ;

ð5:23:4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðxt  bzÞ2 þ sinðxt  bzÞ2 ¼ Exm ;

ð5:23:5Þ

jER ðz; tÞj ¼ E xm

We can choose any value of the coordinate z; e.g. z ¼ 0: Then we have:    ER ðtÞ ¼ 3 x^ cosðxtÞ þ ^y cos xt þ uy ¼ 3ð^x cosðxtÞ þ ^ y cosðxt  p=2ÞÞ: ð5:23:6Þ Time t = 0, 1, 2,…8 s, x ¼ 2pf ¼ 2pðradÞ at f ¼ 1 Hz: ER ðz; tÞ ¼ E xm ð^x cosðxtÞ þ ^ ysinðxtÞÞ;

ð5:23:7Þ

jER ðz; tÞj ¼ jER ðz ¼ 0; tn Þj ¼ E xm :

ð5:23:8Þ

The cosine function cosðxt  bzÞ has positive and negative values. The resulting vector ER ðz; tÞ can be positive and negative, and the module jER ðz; tÞj has only positive values. We are building the polarization diagram (Fig. 5.30) on the basis of Table 5.5. The explanation of the diagram construction in Fig. 5.30 is given in Sects. 5.21 and 5.22. We see that the module jER ðtn Þj ¼ const at the circular polarization, and its value is represented by a blue dashed straight line on the lower left corner image of the plane.

5.24

Problem Related to Elliptical Polarization of Electromagnetic Waves

Draw: (1) the orthogonal linearly polarized plane waves Ex ðz1 ; tÞ; Ey ðz1 ; tÞ; 2) the polarization diagram for the sum of these linearly polarized waves, i.e. the trace of the tip of the total vector ER ðz1 ; tÞÞ in the point z1 ¼ const (Fig. 5.19); (3) the

5.24

Problem Related to Elliptical Polarization of Electromagnetic Waves

357

resulting (total) electric field ER ðtn Þ; and 4) its module jER ðz; tÞj when the amplitudes of wave E x ¼ Exm ðV=mÞ; Ey ¼ E ym ðV=mÞ are given in a personal task, and the phase shift between these waves (Sects. 5.1 and 5.2) is Dy ¼ uy  ux ¼ uy ðradÞ because we assumed ux ¼ 0: Then write the formulae of the total electric field intensity and its module and the phase shift of the plane EM waves and calculate their values for your task. Determine the direction of rotation of the vector ER for the elliptically polarized wave. Given : E x ¼ 3ðV=mÞ;

E y ¼ 3ðV=mÞ;

DuE ¼ uy ¼ p=4ðradÞ;

  ER ðz; tÞ ¼ E1 þ E2 ¼^xEx cosðxt  bzÞ þ ^yE y cos xt  bz þ uy ; jER ðz; tÞj ¼

ð5:24:1Þ ð5:24:2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ðE xm cosðxt  bz þ 0ÞÞ2 þ Eym cosðxt  bz  p=4Þ ; ð5:24:3Þ Exm ¼ Eym ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jER ðz; tÞj ¼ E xm cosðxt  bzÞ2 þ cosðxt  bz  p=4Þ2 ;

ð5:24:4Þ ð5:24:5Þ

We can choose any value of the coordinate z, e.g. z = 0. Then we have:    y cosðxt  p=4ÞÞ: ER ðtÞ ¼ 3 x^ cosðxtÞ þ ^y cos xt þ uy ¼ 3ð^x cosðxtÞ þ ^ ð5:24:6Þ Time t = 0, 1, 2,…8 s, x ¼ 2pf ¼ 2pðradÞ at f = 1 Hz. ER ðz; tÞ ¼ jER ðz ¼ 0; tn Þj ¼ Exm ð^x cosðxtÞ þ ^ y cosðxt  p=4ÞÞ; jER ðz ¼ 0; tn Þj ¼ E xm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðxtÞ2 þ cosðxt  p=4Þ2 :

ð5:24:7Þ ð5:24:8Þ

The cosine function cosðxt  bzÞ has positive and negative values. The resulting vector ER ðz; tÞ is positive and negative, and the module jER ðz; tÞj has only positive values. We are constructing the polarization diagram (Fig. 5.31) on the basis of Table 5.6. The explanation of the diagram construction in Fig. 5.31 is given in Sects. 5.21 and 5.22. We see that the module jER ðtn Þj ¼ const at the elliptical polarization, and its value is represented by a blue line that pulsates (jumps) between two values (Table 5.6) on the lower left corner image of the plane.

358

5

Plane Electromagnetic Wave Propagation

Fig. 5.31 Counterclockwise (right-handed) elliptical polarization and Ex ðtÞ; Ey ðtÞ; jER ðtÞj

Table 5.6 Calculations of ER ðtn Þ and jER ðtn Þj Number of point

tn ; s

0 1 2 3 4 5 6 7 8

t0 t1 t2 t3 t4 t5 t6 t7 t8

5.25

¼0 ¼ 1=8 ¼ 1=4 ¼ 3=8 ¼ 1=2 ¼ 5=8 ¼ 3=4 ¼ 7=8 ¼1

xtn ; rad xt0 xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8

¼0 ¼ p=4 ¼ p=2 ¼ 3p=4 ¼p ¼ 5p=4 ¼ 3p=2 ¼ 7p=4 ¼ 2p

cosðxtn Þ

cosðxtn  p=4Þ

ER ðtn Þ; V/m

jER ðtn Þj

1 0.707 0 –0.707 –1 –0.707 0 0.707 1

0.707 1 0.707 0 –0.707 –1 –0.707 0 0.707

3ð^ x þ 0:707^ yÞ 3ð0:707^ xþ^ yÞ 2:12^ y 2:12^ x 3ð^ x  0:707^ yÞ 3ð0:707^ x^ yÞ 2:12^ y 2:12^ x 3ð^ x þ 0:707^ yÞ

3.67 3.67 2.12 2.12 3.67 3.67 2.12 2.12 3.67

Standing Electromagnetic Waves

Standing EM waves can be formed by restraining the EM waves between two perfectly conductor planes separated from each other by distance “d”. Both EM waves have the same frequency x ¼ x1 ¼ x2 , the propagation constant

5.25

Standing Electromagnetic Waves

359

b ¼ 2p=k1 ¼ 2p=k2 , the electric field amplitude E 0 ¼ E 0;1 ¼ E0;2 , and the magnetic field amplitude H 0 ¼ H 0;1 ¼ H 0;2 . There are two cosinusoidal plane EM waves (Fig. 5.11). We can write for the first wave which is traveling (propagating) in the positive direction along the z-axis (forward wave Sect. 5.4): E y1 ðz; tÞ ¼ E 0 cosðxt  bzÞ;

ð5:25:1Þ

H x1 ðz; tÞ ¼ H 0 cosðxt  bzÞ:

ð5:25:2Þ

The second EM wave traveling in the negative direction of the z-axis (backward wave Sect. 5.4): E y2 ðz; tÞ ¼ E 0 cosðxt þ bzÞ;

ð5:25:3Þ

H x2 ðz; tÞ ¼ H 0 cosðxt þ bzÞ:

ð5:25:4Þ

We will use the EM wave’s superposition principle: E y ðz; tÞ ¼ E y1 ðz; tÞ þ Ey2 ðz; tÞ ¼ E0 ðcosðxt  bzÞ þ cosðxt þ bzÞÞ;

ð5:25:5Þ

H x ðz; tÞ ¼ H x1 ðz; tÞ þ H x2 ðz; tÞ ¼ H 0 ðcosðxt  bzÞ þ cosðxt þ bzÞÞ:

ð5:25:6Þ

The angle sum and angle difference identities formula is as follows: cosðh1  h2 Þ ¼ cosðh1 Þ cosðh2 Þ  sinðh1 Þ sinðh2 Þ:

ð5:25:7Þ

We apply Eq. (5.25.7) in Eqs. (5.25.5) and (5.25.6): E y ðz; tÞ ¼ E0 ðcos bz cos xt þ sin bz sin xt  cos bz cos xt þ sin bz sin xtÞ; ð5:25:8Þ Ey ðz; tÞ ¼ 2 E 0 sin bz sin xt;

ð5:25:9Þ

H x ðz; tÞ ¼ H 0 ðcos bz cos xt þ sin bz sin xt þ cos bz cos xt  sin bz sin xtÞ; ð5:25:10Þ H x ðz; tÞ ¼ 2H 0 cos bz cos xt:

ð5:25:11Þ

The total fields Ey ðz; tÞ and H x ðz; tÞ satisfy the wave Eqs. (5.16.4) and (5.16.1), respectively. The waves described by Eqs. (5.25.9) and (5.25.11) are standing waves which cannot propagate (travel) but can oscillate in space and time between two perfect conductor planes. The total electric field (Eq. 5.25.9) is equal to zero at all times when function sin kz ¼ 0:

360

5



Plane Electromagnetic Wave Propagation

np np n ¼ ¼k ; k 2p=k 2

ð5:25:12Þ

where n = 0, 1, 2, 3…(nodal plane of EÞ. The planes which contain the points z ¼ k n2 ; n ¼ 0; 1; 2 . . . 1 are called the nodal planes of the electric field (Figs. 5.4 and 5.8). When sin kz ¼ 1 then:       1 p 1 p n 1 ¼ nþ ¼k þ z ¼ nþ ; 2 k 2 2p=k 2 4

ð5:25:13Þ

wheren ¼ 0; 1; 2 . . . 1 (antinodal plane of EÞ. The planes which contain the points z ¼ k n2 þ 14 ; n ¼ 0; 1; 2. . .1 are named as the nodal planes of the electric field (Figs. 5.4 and 5.8). We see from Eq. (5.25.9) that the amplitude of the electric field is 2E 0 . The planes that contain these points are the antimodal (crest and trough Fig. 5.8) planes of the E-field. We see from Eq. (5.25.9) for the time-dependence that the electric field equals zero in all cases where sin xt ¼ 0 then: t¼

np nT ¼ ; x 2

n ¼ 0; 1; 2; 3. . .;

ð5:25:14Þ

where the angular frequency x ¼ 2pf ¼ 2p=T; T is the period. Comparing Eqs. (5.25.9) and (5.25.11), we see that in standing EM waves, Efield and H-field are shifted by p=2 because E y ðz; tÞ is determined by the sine functions and H x ðz; tÞ is depicted by the cosine. From Eqs. (5.25.1) and (5.25.2), we see that the electric and the magnetic fields are always in phase for the traveling EM wave.

5.26

Review Questions

Q5.1. Can EM waves propagate in a space free of any matter? Q5.2. What is the instantaneous value of a time-harmonic electric Eðx; y; z; tÞ field? Q5.3. Write a differential equation describing a harmonic vibration. Q5.4. What is a phase shift (phase angle, phase difference) between two EM waves? Give their definitions. Q5.5. What is an initial phase of an EM wave? Q5.6. What is the difference between the leading and the lagging phase of a wave? Q5.7. What is a wavefront of a plane EM wave? Q5.8. Describe the terms for an EM wave: antinode, peak, antipeak, crest, through, extremum.

5.26

Review Questions

361

Q5.9. Describe the wave vector k of a plane EM wave which can propagate in vacuum and lossless materials. Q5.10. What is the phase difference between the electric and magnetic field of an EM wave when the wave propagates in a lossless material? Q5.11. What is the name of the surface of a constant phase, i.e. kr ¼ constant? Q5.12. What is the angle between the planes in which the electric and magnetic fields are located? Q5.13. What is a monochromatic plane EM wave? Q5.14. Describe the meaning of the homogeneous plane wave. Q5.15. Write the scalar wave equation for a transversal plane homogeneous wave. Q5.16. Write the homogeneous Helmholtz’s equation. Q5.17. What are the forward and backward waves? Q5.18. What is the difference between the propagation constant EM waves which propagate in lossless and lossy media? Q5.19. Write Faraday’s and Ampère’s laws for plane waves. Q5.20. What is the intrinsic impedance of plane waves in free space? Q5.21. What is the intrinsic impedance of plane waves in lossless materials? Q5.22. How can you characterize a time-periodic EM wave? Q5.23. Write the homogeneous Helmholtz’s equation for EM wave propagating into a lossy medium. Q5.24. How the ratio e00 =e0 is to be named? Q5.25. What is the attenuation constant of a plane wave which is propagating in an isotropic lossy media? Q5.26. What is the phase constant of a plane wave which is propagating in a simple lossy media? Q5.27. What is the skin depth of a plane wave propagating in a good conductor? Q5.28. What is the phase difference between E and H fields in the plane wave, propagating in a good conductor? Q5.29. How do you describe the Poynting vector of an EM wave? Q5.30. What is the instantaneous value of the Poynting vector? Q5.31. What is the time-averaged Poynting vector? Q5.32. What are phase and group velocities? Q5.33. What is the normal and anomalous dispersion? Q5.34. What are the constructive and destructive interferences? Q5.35. What are the clockwise, left-handed, counterclockwise, right-handed circular polarizations? Q5.36. Describe the linearly polarized EM wave. Q5.37. Describe the circularly polarized EM wave. Q5.38. Describe the elliptically polarized EM wave. Q5.39. What is a polarization ellipse? Q5.40. What kind of relationship is expressed by ellipticity? Q5.41. At what frequency does the electric field vector of EM wave with the elliptical or circular polarization rotate?

362

5

Plane Electromagnetic Wave Propagation

Q5.42. What conditions ðExm ; Eym ; DE Þ should be satisfied so that the summation of two orthogonal plane waves would result in a total linearly polarized wave? Q5.43. What conditions ðExm ; Eym ; DE Þ should be satisfied so that the summation of two orthogonal plane waves would result in a total circularly polarized wave? Q5.44. What conditions ðExm ; Eym ; DE Þ should be satisfied so that the summation of two orthogonal plane waves would result in a total elliptically polarized wave? Q5.45. Specify the standing EM waves between two perfect conductor planes. Q5.46. What is the difference between standing waves and propagating ones?

Chapter 6

Reflection and Transmission of Plane Electromagnetic Waves

Abstract This chapter examines oblique and normal incident of parallel and perpendicular polarized plane waves on the interface of two dielectrics. The reflected and refracted waves which arise afterward are then analyzed. The author also explains how to determine the phase of two arising waves. The chapter presents oblique and normal incident and reflection of parallel and perpendicular polarized waves on the conducting flat interface and gives distributions of total electric and magnetic fields. Other items explored in this chapter include patterns of EM fields, currents, and charge allocations.

6.1

Introduction

The EM field can be radiated away from an antenna. E-, H-field vectors together with the wave vector k (or the Poynting vector P, Sect. 5.12) must form a right-handed triad for each EM wave. E-, H-field vectors are described for time-periodic case by expressions Eðr; tÞ ¼ E0 eiðx tkrÞ , Hðr; tÞ ¼ H0 eiðx tkrÞ (Eq. 5.4.12) where x is an angular frequency, r is the radius vector (Sect. 1.2.6), E0 and H0 are amplitudes of the electric and magnetic field vectors. The EM wave is traveling (propagating) in a vacuum or other isotropic media (Sect. 3.27.1) and the wave can experience such phenomena as reflection, scattering, transmission, refraction, diffraction, interference, and other similar processes. We will describe these phenomena briefly. Reflection is phenomenon when an EM wave turns back from the direction it was propagating due to hitting a reflective object. Another definition: Reflection is the change in one direction of a wavefront at an interface of two different media so that the wavefront changes its direction back in the opposite direction into the medium from which it initiated (originated). Figure 6.1 shows the wavefronts of the incident (red color), reflected (blue), and transmitted (known also as refracted, green color) EM plane wave. The incident plane wave impinges (incident, falls) on the interface separated to media with different absolute permittivities e1 ¼ e0 er , e2 ¼ e0 er2 and absolute permeabilities l1 , l2 , respectively. We see the interference © Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_6

363

364

6

Reflection and Transmission of Plane Electromagnetic Waves

phenomenon of the incident and reflected waves in the center of the upper half-plane close to the normal (Fig. 6.1b). The term “scattering” means a change in the direction of movement of an EM wave or a particle or other objects because of a collision with some obstacles. Different from the reflection in which an EM wave is deflected in one direction in scattering the EM wave can be deflected in many directions. EM waves are one of the most objects which often undergo scattering. The most commonly used are Rayleigh, Mie, Optical, Raman, Compton, Brillouin scatterings. For example, Raleigh scattering arises when the size of a scatter is much smaller than the wavelength of the incident EM wave. Mie scattering arises when the size of a scatter is comparable to the wavelength of the incident EM wave, e.g. Mie scattering can be caused by pollen, dust, water droplets, and other particles in the lower layer of the atmosphere. The term “transmitted” means that something, such as an EM signal, passes from one place to another, e.g. the transmission or passage of an EM wave through a medium. Refraction which is also known as transmission is the change in direction of EM wave propagation due to passing from one medium to another medium. Diffraction is the phenomenon by which an EM wave (light) is spread out (dispersed) as a result of bends of obstacle edges or passing through a narrow aperture (also known as slit, slot, hole). Diffraction is typically accompanied by

(b)

Medium 1

cid

W av e

1,

en

t

μ

r vef Wa

fro

Wa vef

ron

t

φ

φ

2

Interface

x

y

φ

Wavefront

Tran Refracte smitted or d plane wave

nt

μ

Medium 1 Medium 2

W av ef

2,

Int e

Medium 2

ro

rfa c

e

W av

efr

on

t

Re fle

ct

nt

t

on

1

ed

In

normal

(a)

z

Fig. 6.1 A plane wave incident on the interface of two media, the incident wave reflected and transmitted

6.1 Introduction

365

interference between the EM waves to which the initial wave is subdivided. The diffraction behaviors are exhibited when the EM wave encounters an obstacle or a slit which is comparable in size to the wavelength of the wave. Diffraction occurs with all kinds of waves, including sound waves, water waves, and EM waves.

6.2

Main Definitions About Incident Plane Waves

Consider a flat (plane) interface between two dielectric media with the different absolute permittivities e1 ¼ e0 er , e2 ¼ e0 er2 and absolute permeabilities l1 , l2 . The interface plane (interface surface) defines the boundary between two media with the different refractive indices (known also in the plural form as the indexes and in the pffiffiffiffiffiffiffiffiffiffiffi singular form: the refractive index or index of refraction) n1 ¼ er1 lr1 and n2 ¼ pffiffiffiffiffiffiffiffiffiffiffi er2 lr2 (Eq. 5.4.22). When an EM wave incident (impinges, falls) on the interface between two media, two other waves occur, i.e. the transmitted (also known as refracted) and reflected waves. The fraction (a part of the whole) of the incident wave which is reflected is named as the reflected wave. The fraction of the incident wave which is transmitted is named as the transmitted wave. The polarization of the reflected and transmitted waves depends on the polarization of the incident wave, angle ui , and some parameters of media. In Fig. 6.2, ui is the angle of incidence, ur is the angle of reflection, and ut is the angle of transmission (also known as angle of refraction). The angle of incidence ui is equal to the angle of reflection ur ¼ ui . This is the law of reflection. The incident, reflected, and transmitted wave vectors form a plane that is called the plane of incidence which also includes the normal to the interface surface (or simple to the interface). The plane of the interface is the plane that defines the surface between the two media. In Fig. 6.2 and in the further figures, vectors Pi , Pr , Pt are the

Fig. 6.2 Interface plane, plane of incidence, and wavefronts

1,

μ

1

Pi

φ φ

Normal to interface plane

Medium 2 2,

μ

2

Plane of incidence

i

r

f to ron ave vef w Wa ected refl

Medium 1

W inc avefr ide ont tn wa of ve

Phase plane

Pr

Pt Ψt f to e ron av vef d w Wa mitte ns tra

Interface plane

366

6

Reflection and Transmission of Plane Electromagnetic Waves

Poynting vector of the incident, reflected, and transmitted plane waves, respectively. Snell’s law (also known as Snell–Descartes law and the law of refraction) is used to describe the relationship between the angles ui and ut when referring to EM waves passing through a boundary (interface) between two different simple media (Sect. 3.27.1). Snell’s law is: sin ui n1 ¼ sin ut n2 ;

ð6:2:1Þ

the last expression can be written as: sin ui n2 ¼ ; sin ut n1 sin ut ¼ sin ui

n1 ; n2

ð6:2:2Þ ð6:2:3Þ

and the angle of transmission (refraction) can be defined as:   n1 ut ¼ arcsin sin ui : n2

ð6:2:4Þ

Snell’s law was named after Dutch astronomer and mathematician W. Snell (Willebrord Snel van Royen, 1580–1626). Snell’s law states that the ratio of sines of the angles ui and ut is equivalent to the reciprocal of the ratio of the indices of refraction (Eq. 6.2.2) and is also equivalent to the ratio of phase velocities (or phase speeds because here they are scalars) in two media under consideration: sin ui v1 k1 ¼ ¼ ; sin ut v2 k2

ð6:2:5Þ

where v1 ¼ c=n1 , v2 ¼ c=n2 are the speeds of an EM wave and k1 , k2 are the wavelengths of the EM wave in media 1 and 2, respectively, c is the speed of the EM wave (light) in a vacuum, Figure 6.3a presents a case when n2 [ n1 and ui [ ut . Figure 6.3b presents a case when n2 \n1 and ui \ut . The phase of the transmitted (refracted) wave always coincides with the phase of the incident wave. Transmitted wave is usually made up of the portions of the incident wave which travels from media 1 into media 2. A phase of reflected wave can coincide or differ on p when comparing it with a phase of an incident wave. We will consider this question more thoroughly in Sect. 6.6.

6.2 Main Definitions About Incident Plane Waves

(b) Incident EM wave

Angle of incidence

φ =φ i

i

n

i

φ n

φ =φ

φφ

Reflected EM wave

r

Pi=[EiHi]

1

r

Angle of reflection

φ φ

n

Pr=[ErHr]

normal

normal

(a)

367

t

Pt

n φ 2

1

t

i

z

Fig. 6.3 Angles of incidence, reflection, transmission: a Refractive indices n2 [ n1 . b Refractive indices n2 \n1

6.2.1

Perpendicular and Parallel Polarizations

When the plane EM wave incident (falls, impinges) on the interface between two media, there can be two orthogonal linear polarizations of the incident wave that are most important for the reflection and transmission. These orthogonal linear polarizations are called the perpendicular ð?Þ and parallel ðk Þ polarizations (Fig. 6.4). These linear polarizations are usually defined by their E-field vectors relative orientation to the plane of incidence.

Fig. 6.4 Mutual location of the electric field vectors of perpendicular and parallel polarized waves

368

6

Reflection and Transmission of Plane Electromagnetic Waves

In Fig. 6.4, the electric field vectors of the incident, reflected, and transmitted waves with the perpendicular polarization Ei;? , Er;? , Et;? (blue color) are represented by blue vectors, and , the E-fields of waves with the parallel polarization Ei;k , Er;k , Et;k (red color) are marked by the red vectors. The direction of the wave vector coincides with the corresponding Poynting vector (Figs. 6.4 and 6.5); i.e. direction of ki coincides with the Poynting vector of the incident wave Pi ¼ Ei  Hi ¼ ½Ei Hi , direction of kr coincides with Pr ¼ Er  Hr ¼ ½Er Hr , and direction of kt coincides with Pt ¼ Et  Ht ¼ ½Et Ht , where ki , kr , kt are wave vectors which lie in a plane called the plane of incidence. We are now implying that media are lossless (dissipationless) and that the magnitude of wave vectors (also known as wavenumbers or the phase constant) is the real numbers, i.e. ki ¼ jki j and kr ¼ jkr j. Modules of wave vectors in a lossless medium 1 (Sect. 5.4) are equal: jki j ¼ jkr j ¼ 2p=k1 ;

ð6:2:6Þ

and the wave vector in a lossless medium 2 is: jkt j ¼ 2p=k2 :

ð6:2:7Þ

The wavelength of the propagating wave (5.4.20–5.4.22) is: vp c ¼ vp T; ¼ nf f

Fig. 6.5 Wave vectors of the incident, reflected, and transmitted EM waves

ð6:2:8Þ

^n

Pi

φ φ i

z

ki cos φi

φr=φi k cos φr

x

ki φ i

ki=kr z

r

^nr

φ

kr=ki= 2λπ1

kt= 2λπ2

t

φt kt cos φt

y

ki sin φi kr sin φr ^ni

Pr

ki

Medium 1 Medium 2

r



^nt

x

Pt kt kt sin φt

x

6.2 Main Definitions About Incident Plane Waves

369

where T ¼ 1=f is the time period of EM wave (Fig. 5.9), and n is the refractive index of medium (Eq. 5.14.1). Figure 6.5 shows designations which are going to be used in the next para^ to the interface, the unit vectors n ^i , n ^r , n ^t coincide with graphs: the unit vector n directions of propagation of the incident, reflected, and transmitted waves, ^i and ki coincides for the incident wave. The respectively. The direction of vectors n same property of similar vectors is for the reflected and transmission waves. At the bottom of Fig. 6.5, there are triangles which indicate the relationship between values expressed through angles ui , ur , and ut . Figure 6.6 shows the perpendicular polarized incident and transmitted waves when E-field vector is perpendicular to the plane of incidence. We see that electric fields Ei and Et can be directed forwards the paper, i.e. to the side away from a reader (Fig. 6.6a), or on the opposite side, i.e. to the reader (Fig. 6.6b). The incident wave, reflected, and transmitted waves are described by these expressions: Eincident ¼ Ei ¼ E0;i eiðxtki rÞ ;

ð6:2:9Þ

Ereflected ¼ Er ¼ E0;r eiðxtkr rÞ ;

ð6:2:10Þ

Etransmitted ¼ Et ¼ E0;t eiðxtkt rÞ ;

ð6:2:11Þ

where r is the radius vector (Sect. 1.2.6), E0;i ; E0;r ; E 0;t are specify the amplitudes (which are the peak magnitudes, i.e. the maximum absolute values, Sect. 5.2) of the incident, reflected, and transmitted wave fields Ei ðzÞ, Er ðzÞ, and Et ðzÞ at z ¼ 0, respectively. Value k t is the wavenumber in medium 2. We have from Eqs. (6.2.9) to (6.2.11) at t ¼ 0 that ki  r ¼ kr  r ¼ kt  r, i.e. that the phases on the interface do not change. Since the scalar product of these three wave vectors ki , kr , and kt

(b)

Hi Ei Pi

φ

Plane of incidence

i

y O

Interface

φ

Pr

t

z

Medium 1 Medium 2 x

Et Pt

normal

normal

(a)

Ei Hi Pi

φ

Plane of incidence

y O

Interface

φ

Pr

i

t

z

Medium 1 Medium 2 x

Et Pt

Fig. 6.6 Perpendicular polarization of a plane wave: a Ei of incident wave is directed away from a reader. b Ei of incident wave is directed to the opposite side (to the reader)

370

6

Reflection and Transmission of Plane Electromagnetic Waves

must be equal, the three vectors must lie in the same plane (according to Eq. 1.2.45), and this plane is the plane of incidence. The phase constants ki ¼ jki j and kr ¼ jkr j in a lossless medium 1, using Eqs. (6.2.6)–(6.2.8), are: jki j ¼ jkr j ¼

x 2p 2pn1 xn1 ; ¼ ¼ ¼ vp1 k1 k0 c

ð6:2:12Þ

and the wavenumber kt ¼ jkt j in a lossless medium 2 is: jkt j ¼

x 2p 2pn2 xn2 ; ¼ ¼ ¼ vp2 k2 k0 c

ð6:2:13Þ

where n1 , n2 are refractive indices of media and we have at the interface z ¼ 0: ki  rjz¼0 ¼ kr  rjz¼0 ¼ kt  rjz¼0 :

ð6:2:14Þ

In general case, the wave vectors (Fig. 6.5) are:  x  k i ¼ ni k i ¼ n 1 k i;x ^x þ ki;y ^ y þ ki;z^z ; c  x  k r;x ^x þ k r;y ^ y þ k r;z^z ; kr ¼ nr k r ¼ n1 c  x  k t;x ^x + k t;y ^ y þ kt;z^z ; kt ¼ nt kt ¼ n2 c

ð6:2:15Þ ð6:2:16Þ ð6:2:17Þ

where ^x, ^y, ^z are the unit vector along the x-, y- and z-axes, respectively, in Cartesian coordinates. The pairs of wave vectors must lie in the same plane which is perpendicular to the interface that ensues (Fig. 6.5): ^  ðki  kr Þ ¼ n ^  ki  n ^  kr ¼ 0; n

ð6:2:18Þ

^  ðki  kr Þ ¼ ^yjn ^jjkr j sin ur ¼ ^ ^jjki j sin ui  ^yjn n yjki j sin ui  ^ yjkr j sin ur ; ð6:2:19Þ ^j ¼ 1, and using Eqs. (6.2.12) and (6.2.19), we because the unit vector module is jn get: ^  ð ki  k r Þ ¼ n

2p 2pn1 2pn1 y y ðsin ui  sin ur Þ^y ¼ sin ui ^ sin ur ^ k1 k0 k0

ð6:2:20Þ

^ is the and here, k0 ¼ x=c is the wave length in a free space (a vacuum), and n normal to the interface.

6.2 Main Definitions About Incident Plane Waves

371

Using Eq. (6.2.13), we receive: ^  ð k i  kt Þ ¼ n

2p ðn1 sin ui  n2 sin ut Þ^ y: k0

ð6:2:21Þ

^ i k ki , n ^r k kr and n ^t k kt can be expressed as: Unit vectors n ^i ¼ ^x sin ui þ ^z cos ui ; n

ð6:2:22Þ

^r ¼ ^x sin ur  ^z cos ur ; n

ð6:2:23Þ

^t ¼ ^x sin ut þ ^z cos ut ; n

ð6:2:24Þ

where angles of the incidence, reflection, transmission and other notations are given in Fig. 6.5.

6.2.2

Perpendicular Polarization

The perpendicular polarization of EM wave is the polarization for which the E-field vector is perpendicular to the plane of incidence and H-field vector lies in the plane (Fig. 6.6). We would like to draw your attention to the fact that the electric fields Ei and Et are always directed to the same side; i.e. away from a reader, we see the red cross (Fig. 6.6a) or to the opposite side, and we see the red point (Fig. 6.6b). The phase of the vectors Ei and Et always coincides. In scientific literature, the perpendicular polarization is also known as the s-polarization (from the German senkrecht (perpendicular)), r-polarization (sigma-polarized or sagittal plane polarized), TE-polarization (transverse electric), and H-polarization (H-field vector lies in the plane of incidence).

6.2.3

Parallel Polarization

Figure 6.7 presents the parallel polarized incident and transmitted waves for which the E-field vectors lie in the plane of incidence. We see that magnetic fields Hi and Ht (blue color) are directed to the same side; i.e. into paper, away from a reader, we see the blue cross (Fig. 6.7a) or to the opposite side, and we see a blue point (Fig. 6.7b). The phase of the vectors Hi and Ht always coincides. In scientific literature, the parallel polarization is also known as the p-polarization (from the German parallel), p-polarization (pi-polarized or tangential plane polarized), TM-polarization (transverse magnetic), and E-polarization (E-field vector lies in the plane of incidence). Notes: Horizontal and vertical polarizations The orientation of E and H field vectors of EM wave is very important in radio wave communication. The direction of E field vector specifies the polarization of

372

6

Reflection and Transmission of Plane Electromagnetic Waves

Hi

φ

Ei

Plane of incidence

Pi y O

Interface

φ

Pr

i

t

z

Medium 1 Medium 2

Ht Pt

normal

(b) normal

(a)

Ei

φ

Hi

Plane of incidence

Pi y O

Interface

φ

Pr

i

t

z

Medium 1 Medium 2

Ht Pt

Fig. 6.7 Parallel polarization of a plane wave: a Hi of incident wave is directed away from a reader. b Hi of incident wave is directed to the opposite side (to the reader)

the antenna. There are two types in linear polarization for linear polarized antennas, i.e. horizontal and vertical polarizations. In order to make communication link work effectively, both transmitting and receiving antennas should be in the same polarization. Since the plane of linear polarization can be a plane parallel to the Earth’s surface and perpendicular to it, they are usually named, respectively, horizontal and vertical plane of polarization. 1. At the vertical polarization, the electric field E vector of EM wave is perpendicular to the Earth’s surface. The E vector is in the plane xOz; i.e. E lies in the plane of incidence for the vertical polarization which is analogous to the parallel polarization (Fig. 6.7). 2. At the horizontal polarization, the electric field E vector of EM wave is parallel to the Earth’s surface. The E vector is in the plane yOz; i.e. E? to the plane of incidence for the horizontal polarization which is in some way analogous to the perpendicular polarization (Fig. 6.6).

6.3

Fresnel’s Equations

The Fresnel equations describe the reflection and transmission of EM waves which occur after incident of a plane EM wave on the interface of two media with different refractive indices n1 and n2 . The Fresnel equations (or Fresnel conditions) were derived by A.-J. Fresnel (French engineer and physicist, 1781–1868). An EM wave of arbitrary polarization will need to be resolved into components (Fig. 1.3) where the electric field vector of perpendicular polarized wave

6.3 Fresnel’s Equations

373

is directed perpendicularly to the plane of incidence and the electric field vector of parallel polarized wave is located in the plane of incidence. The reflection and transmission coefficients have to be found separately. This distinction between the parallel and perpendicular polarizations arises due to the fact that the boundary conditions require matching of tangential components of electric and magnetic fields, and for the polarizations, different components are on the interface. The magnetic field is tangential to the plane of interface (Fig. 6.6) for the perpendicular polarized EM wave while for the parallel polarized wave the electric field is tangential to the plane of interface (Fig. 6.7). The reflection R and transmission T coefficients which are the functions of the incident wave polarization and the angle of incidence can be determined by EM boundary conditions. Boundary conditions on the interface are: E t1 ¼ Et2 and H t1 ¼ H t2 at z ¼ zinterface ¼ 0;

ð6:3:1Þ

where Et and H t are the tangential projections of electric E and magnetic H fields, i.e. magnitudes of vector components Et and Ht . There are two waves in medium 1: the incident wave with vectors Ei ; Hi and reflected wave Er ; Hr . There is only one wave in medium 2: the transmitted wave with vectors Et ; Ht . If we take that the amplitude of the incident wave equals 1, we can write the equation: 1  R ¼ T;

ð6:3:2Þ

where R is the reflection coefficient, and T is the transmission coefficient. Parallel polarized EM plane wave. The incident wave, reflected, and transmitted waves are described by Eqs. (6.2.9)–(6.2.11). Ignoring the term which describes the rapidly varying parts of the waves ðei x t Þ and keeping only the amplitudes E 0;i , E 0;r , and E 0;t (which are from term eikr at r ! 0 on the interface), we can write expressions for the parallel polarized wave. The boundary condition Eq. (6.3.1) for tangential components of the electric field, using Fig. 6.5, is: E 0;i cos ui  E0;r cos ui ¼ E0;t cos ut ;

ð6:3:3Þ

  cos ui : E 0;t ¼ E 0;i  E0;r cos ut

ð6:3:4Þ

The boundary condition Eq. (6.3.1) for tangential components of magnetic field, Fig. 6.7 (b) with the same phase of all magnetic field vectors, is: H 0;i þ H 0;r ¼ H 0;t :

ð6:3:5Þ

374

6

Reflection and Transmission of Plane Electromagnetic Waves

The last equation, using Sect. 5.7 and Fig. 5.10, we can write: E0;i E 0;r E 0;t þ ¼ ; g1 g1 g2 E0;t ¼

ð6:3:6Þ

 g2  E 0;i þ E 0;r : g1

ð6:3:7Þ

When Eq. (6.3.4) is substituted in (6.3.7), we get:   cos ui g2   E 0;i  E 0;r ¼ E 0;i þ E 0;r : cos ut g1

ð6:3:8Þ

Grouping terms E 0;i and E 0;r at either side of re-arranged, we receive:     g2 g2 E 0;i cos ui  cos ut ¼ E 0;r cos ui þ cos ut g1 g1

ð6:3:9Þ

The electric field amplitude reflection coefficient for the parallel polarized EM waves is:  Rk ¼

cos ui  cos ut gg2



E0;r g cos ui  g2 cos ut 1 ¼ 1 ¼ g2 E 0;i g1 cos ui þ g2 cos ut cos ui þ cos ut g

ðdimensionlessÞ;

1

ð6:3:10Þ where E0;i is the amplitude of the incident plane wave at the interface when z = 0, and E 0;r is the amplitude of the reflected plane wave at z = 0; angles ui and ut are shown in Fig. 6.3. From Eq. (6.3.6), we define: E0;r ¼

g1 E0;t  E0;i : g2

ð6:3:11Þ

Substituting Eq. (6.3.11) in Eq. (6.3.9), we get:  2E 0;i cos ui ¼ E 0;t

 g1 cos ui þ cos ut : g2

ð6:3:12Þ

From the last equation, we get the Fresnel equation for the electric field amplitude coefficient of transmission for the parallel polarized EM waves: Tk ¼

E 0;t 2g2 cos ui ¼ E0;i g2 cos ut þ g1 cos ui

ðdimensionlessÞ;

ð6:3:13Þ

6.3 Fresnel’s Equations

375

where E 0;t is the amplitude of the transmitted plane wave at z = 0, and the angle ut is shown in Fig. 6.3. The transmission coefficients are always positive. The incident and transmitted waves are always in the phase. From Eq. (5.7.1), the intrinsic impedance of medpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ium 1 and medium 2 is as follows: g1 ¼ g0 lr1 =er1 and g2 ¼ g0 lr2 =er2 , eta0  120p ðXÞ is the wave impedance of plane waves in a vacuum. Perpendicular polarized EM plane wave. Ignoring the terms which define the rapidly varying parts of the waves ðei x t Þ and keeping only the amplitudes E0;i , E0;r , and E 0;t (which are from term eikr at r ! 0 on the interface) in Eqs. (6.2.9)–(6.2.11), we can write expressions for the perpendicular polarized wave. The boundary condition Eq. (6.3.1) for tangential components of the electric field, using Fig. 6.6b with the same phase of all electric field vectors, is: E 0;i þ E0;r ¼ E0;t :

ð6:3:14Þ

The boundary condition Eq. (6.3.1) for tangential components of the magnetic field is: H 0;i cos ui  H 0;r cos ui ¼ H 0;t cos ut :

ð6:3:15Þ

Using expression from Sect. 5.7, we get: E 0;i E0;r E0;t cos ui  cos ui ¼ cos ut ; g1 g1 g2

ð6:3:16Þ

and substituting Eq. (6.3.14) in Eq. (6.3.15): 

   E0;i  E 0;r E 0;i þ E 0;r cos ui ¼ cos ut : g1 g2

ð6:3:17Þ

Grouping terms E 0;i and E 0;r in Eq. (6.3.17), we get: E 0;i ðg2 cos ui  g1 cos ut Þ ¼ E0;r ðg2 cos ui þ g1 cos ut Þ:

ð6:3:18Þ

The reflection coefficient R? for the perpendicular polarized EM wave, using Eq. (6.3.18), is: R? ¼

E0;r g2 cos ui  g1 cos ut ¼ E 0;i g2 cos ui þ g1 cos ut

ðdimensionlessÞ:

ð6:3:19Þ

We see that when we know the electric field of incident wave E 0;i which is usually taken to be unity (because the normalization to the amplitude of the incident wave is carried out) and the reflection coefficient R? of the transmission line (waveguide), then we can easily find the electric field of reflected wave E 0;r .

376

6

Reflection and Transmission of Plane Electromagnetic Waves

The transmission coefficient can be found by using Eq. (6.3.14): E0;r ¼ E0;t  E0;i :

ð6:3:20Þ

Substituting Eq. (6.3.20) in Eq. (6.3.18), we have: E 0;i ð2g2 cos ui Þ ¼ E 0;t ðg2 cos ui þ g1 cos ut Þ:

ð6:3:21Þ

The transmission coefficient T ? for the perpendicular polarized EM waves is: T? ¼

E 0;t 2g2 cos ui ¼ E 0;i g2 cos ui þ g1 cos ut

ðdimensionlessÞ:

ð6:3:22Þ

The intrinsic impedance, generally speaking, can be a complex number. The complex intrinsic impedance occurs for lossy materials. In this case, the electric and magnetic fields of the transmitted wave are not in-phase.

6.4

Total Internal Reflection and Critical Angle

When an incident plane wave impinges on the interface between two media with specific refractive indices, important physical phenomena may occur. Figure 6.8a–c shows three cases relating to an EM wave propagating from an optically dense medium 1 with n1 to a rare medium 2 with n2 when er1 [ er2 and n2 =n1 \1. For this case, the critical angle ucr and the Brewster angle uBi can be calculated. First, we will consider the concept of a critical angle. 1. The first determination of critical angle. The critical angle ucr is defined as the angle of incidence ui of an EM plane wave that provides an angle of trans^), and in other words, the mission (refraction) of 90° (relative to normal n

(a)

(c)

(b) n^

Medium 1

Pi

Pr

φi

φi1

Parallel polarization

Medium 1

Er 85°

Pr=

[ErH

]

r

y

Hi x

6.7°

Et

Ht

z

Pt

Medium 2

[ErH Hr Pr=

Ei

Pi

Hr

Medium 1

φ +φ >π/2 φ >φ i

t

i

t

n2/n1>1

85°

Er

y

x 6.7°

Et

Ht z

]

r

Medium 2

Pt

Fig. 6.10 Parallel polarized wave at ui þ ut [ p=2, ui [ ut ; n2 =n1 [ 1: a Hi of incident wave are directed away from the reader. b Hi of incident wave are directed to the opposite side (to the reader)

6

Reflection and Transmission of Plane Electromagnetic Waves

M

H

E

ne ag tic

90°

fie ld

Electric field

384

E P=[EH] H Poynting vector

P

Fig. 6.11 Determination of directions: index finger—E, middle finger—H, and thumb finger— P vectors

Right-hand rule: We use this rule to determine the direction of one of vectors. If we put two appropriate fingers in directions of the vectors for which we know their direction exactly, the third finger will show the unknown direction of the third vector. Directions of EM field vectors: The index finger shows E‒vector, the middle finger indicates H‒vector, and the thumb finger points in the direction of P‒vector. For example, if we know the directions of H‒vector and P‒vector and we locate the appropriate fingers (Fig. 6.11) along these directions then the index finger shows the direction of E‒vector. Example 1 We will consider the parallel polarized wave. We know that the phase shift between Ei and Er is 0 (left column in Table 6.2), so we have to start considering the vector Hi which is perpendicular to the plane of incidence. Vector Hi may have two directions which are different from p; i.e. with opposite directions, one direction is from us to the sheet of paper, we see the blue cross (Fig. 6.10a) and another direction of vector Hi is toward us, we see the blue point (Fig. 6.10b). We will consider the vector Hi that is represented by the blue cross as in Fig. 6.10a. We can determine the direction of vector Ei if we know the direction of Hi which is always perpendicular to the sheet of paper at the parallel polarization and we also know the direction of the Poynting vector Pi ¼ ½Ei Hi  . In order to define the direction of vector Ei we need to apply the right-hand rule (Fig. 6.11). As a result we defined the direction of Ei. Now, we can define how the magnetic field vector Hr is located. The direction of vector Hr can be to or from the reader and this vector is marked with a point or a cross. We must always remember that P = [EH] and when vector P changes it direction, then is obliged to change direction only one of vectors or E or H. We know the direction of vector Pr changes its direction as compared to vector Pi . From Table 6.2 (left column, for parallel polarized wave), the phases of vectors Ei and Er are the same. This means that since vector Pr changes its direction and vector Er does not change the direction as compared with the incident

6.6 Determination of a Phase of Reflected Wave

385

Table 6.3 Phase of reflected wave at ui þ ut \p=2 ui þ ut \p=2 ui [ ut ; n2 =n1 [ 1 Vacuum: n1 ¼ 1 Glass: n2 ¼ 2:7 Brewster’s angle  

Polarization of EM wave

uB;i ¼ arctan

ui \ut ; n2 =n1 \1 pffiffiffiffiffiffiffiffi See water n1 ¼ er lr  8:5 Vacuum n2 ¼ 1 Brewster’s angle  

n2 n

uB;i ¼ arctan

2:71 

uB;i ¼ arctan 1  69:68 ui ¼ 45 ; sinð45 Þ  0:707 Snell0 s law : n1 sin ut ¼ sin ui n2 ut  15:17 ui þ ut  60:17

n2 n

 11 

uB;i ¼ arctan 8:5  6:73 u i ¼ 2 Snell0 s law : n1 sin ut ¼ sin ui n2 ut  17:3 ui þ ut  19:28 Phase of reflected wave Er as compared to the phase Ei corresponds to the value “0” or “p” below Parallel p 0 polarization Perpendicular p 0 polarization

wave Ei , so Hr must change the direction to the opposite on the base of expression Pr ¼ ½Er Hr  in the case under consideration. When we know directions of Hr and Pr , we define the direction of Er by using Fig. 6.11. The directions of vectors of the incident, reflected, and transmitted EM waves for the considered example 1 are given in Fig. 6.10. Figure 6.12 presents directions of vectors of incident, reflected, and transmitted waves at the same basic parameters as in Fig. 6.10 but only for perpendicular polarized EM wave.

(а)

(b)

Perpendicular polarization

Medium 1

Hi

Pi

Er

85°

Ei

φ +φ >π/2 φ >φ i

t

i

t

n2/n1>1

y

Et

z

Pr

Pi

Hr

Hi

x 6.7°

Ht Medium 2

Pt

Perpendicular polarization

Ei

φ +φ >π/2 φ >φ i

t

i

t

n2/n1>1

Medium 1

Hr 85°

Pr Er

y

x 6.7°

Ht

z

Et

Medium 2

Pt

Fig. 6.12 Perpendicular polarization ui þ ut [ p=2, ui [ ut ; n2 =n1 [ 1: a Ei of incident wave are directed away from a reader. b Ei of incident wave are directed to the opposite side (to the reader)

386

6

Reflection and Transmission of Plane Electromagnetic Waves

We see from Table 6.2 that phase shift of the electric field vectors Ei and Er is p for perpendicular polarized EM wave. If vector Ei is shown by a cross (Fig. 6.12a), the reflected wave Er will be represented by a point because the phase shift between them equals p. The vectors Pr and Er change their directions (Table 6.2, left column for perpendicular polarization), and therefore, Hr must not change the phase on the basis of expression Pr ¼ ½Er Hr . The directions of vectors of the incident, reflected, and transmitted EM waves are demonstrated in Fig. 6.12. We would like to note that E, H and P vectors of incident, reflected, and transmitted waves satisfy the rule presented in Fig. 6.11. Please pay attention that we can precisely specify the direction of the vector which is perpendicular to the sheet of paper. So, we always have to consider the Poynting vector and one of vectors, either E or H, which is perpendicular to the sheet of paper. We can then define the vector that lies in the plane of the page. Example 2 Now, we will consider the second case when ui \ut ; n2 =n1 \1 at ui þ ut [ p=2, right column in Table 6.2. In this case n2 =n1 ¼ 1=8:5 ¼ 0:118, the incident angle ui ¼ 6:72 ; ut  84:1 , ui þ ut  90:73 , and for parallel polarized EM wave, the phase of the reflected wave is lagging behind in comparison with the incident wave phase by p (Fig. 6.13). Vectors Pr and Er change their directions (Table 6.2, right column, for parallel polarization) in comparison with the incident wave, and therefore, Hr must not change the phase on the base of expression Pr ¼ ½Er Hr . We see from Fig. 6.13 that directions of Hi and Hr coincide. The directions of vectors of the incident, reflected, and transmitted EM waves for considered example 2 are presented in Fig. 6.13. Figure 6.14 shows the incident of the perpendicular polarized wave. We can determine the direction of vector Hi of the incident wave if we know the direction

(a)

(b) Parallel polarization

Parallel polarization

Pr=[ErHr]

Pr=[ErHr]

Hi

Ei

Ei Er

Hr Hi

Er

6.72°

6.72°

Medium 1 Medium 2

i

t t

n2/n1π/2 φ π/2 φ 2b, d>2a

Fig. 7.62 TE101 distributions of E (red color), H (blue) , Js (black), Jd (green), qs for TE101 resonator mode with excitations by a probe, a loop, and a slot at t = 0 and t = T/4

7.11

Transverse Magnetic TMmnp Resonator Modes

The expressions for the EM field components for TMmn ðEmn Þ modes in a waveguide have been given in Eqs. 7.7.25, 7.7.28 – 7.7.31. The longitudinal variation for a propagating (traveling) wave in the positive z-direction is depicted by the factor eihz as indicated in Eq. (7.7.25). This propagating wave will be reflected by the end resonator conductor wall at z ¼ d. The reflected wave propagates in the negative z-direction which is described by a factor eihz . The superposition of waves with similar amplitudes and propagating in the opposite directions gives a standing wave sinðhzÞ or cosðhzÞ type of oscillations.

524

7 Rectangular Hollow Metallic Waveguides and Resonators

The selection type of solution sinðhzÞ or cosðhzÞ) depends on the wave field component which is being considered. Boundary conditions at the resonator conducting surfaces require that the tangential electric field components would disappear at z ¼ 0 and z ¼ d (Fig. 7.43). This means that components Ex ðx; y; zÞ and E y ðx; y; zÞ have to be equal to zero at z ¼ 0 and z ¼ d. Dependence on z is sin hz at z ¼ 0 and h ¼ pp=d. This means that we have to select a function depending on the coordinate z so that it is equal to zero at z = 0; therefore, we must choose sinðppz=d Þ which is zero at z = 0. We will consider getting the field component in the resonator in more detail. The approach to solving the problem is the same as in Sect. 7.10.2. The EM wave reflected from the resonator end wall at z ¼ d and propagated in the negative direction of z-axis has these components: Ez

Ex

Ey

Hx

Hy

ref;mn ðx; y; zÞ

ref;mn ðx; y; zÞ

ref;mn ðx; y; zÞ

ref;mn ðx; y; zÞ

ref;mn ðx; y; zÞ

npy

ei hz a b at m ¼ 1; 2; 3. . . and n ¼ 1; 2; 3. . .;

¼ E0

ref;mn

sin

ihmn @Ez;mn k 2?mn @x ih mp E0 ¼ 2 k ?mn a

mpx

sin

ð7:11:1Þ

¼

ihmn @E z;mn k2?mn @y ih np E0 ¼ 2 k?;mn b

ref;mn

cos

ref;mn

sin

mpx

a

npy

ei hz ; b

ð7:11:2Þ

npy

ð7:11:3Þ

sin

¼

ixe0 er @E z;mn k2?mn @y ixe0 er n p E0 ¼ 2 k?mn b

mpx

a

cos

b

ei hz ;

¼

ixe0 er @E z;mn k 2?mn @x ixe0 er mp E0 ¼ 2 k ?mn a

ref;mn

sin

mpx

a

npy

cos ei h z ; b

ð7:11:4Þ

¼

Hz

ref;mn

cos

ref;mn ðx; y; zÞ

mp np

x sin y ei hz ; a b

¼ 0:

ð7:11:5Þ

ð7:11:6Þ

The tangential electric field components have to be equal to zero Et ¼ 0 at z ¼ 0 and z ¼ d. These tangential components to the front and back end resonator conductor walls lying in the plane of xOy (Fig. 7.43). Because the resonator under our consideration is a linear system consequently, the superposition principle can be applied to the resonator. The total electric field at a point is just equal to the vector sum of all

7.11

Transverse Magnetic TMmnp Resonator Modes

525

fields at the point. The total field components for modes propagating in the positive and negative directions in the resonator obey the boundary conditions as follows: E xres;mn ¼ E x;mn þ Ex

ref;mn

at z ¼ 0 and z ¼ d;

ð7:11:7Þ

E yres;mn ¼ E y;mn þ Ey

ref;mn

at z ¼ 0 and z ¼ d;

ð7:11:8Þ

where indexes m ¼ 1; 2; 3. . . and n ¼ 1; 2; 3. . . can have any you need values, E xres;mn is x-component of the total electric field of resonator for mode (oscillation), E x;mn is x-component of the total electric field for a propagating wave (mode) TE in the positive z-direction described by the factor eihz , and E x ref;mn is xcomponent of the electric field for a reflected wave TE described by the factor eihz . Similar designations are accepted for other components of electrical and magnetic fields of resonator H xres;mnp ; H x;mn , H x ref;mn etc. Substituting Eqs. (7.7.28) and (7.11.2) into boundary conditions (7.11.7), we get: 

ihmn mp mp np  cos x sin y E0;mn ei hmn z  E 0 a b k2?mn a

ref;mn e

i hmn z



¼0

ð7:11:9Þ

at z ¼ 0 and z ¼ d;  at z ¼ 0, we receive E0;mn e0  E0 amplitude coefficients is: E0 

ref;mn e

ref;mn

0



¼ 0, and the correlation between

¼ E 0;mn :

  ihmn mp mp np

cos x sin y E 0;mn ei hmn z  ei hmn z ¼ 0; 2 a b k ?mn a

ð7:11:10Þ ð7:11:11Þ

  and taking into account Eq. (1.4.16), we have eihz  eihz ¼ 2i sinðhzÞ and for the component Ex we get: 2

h mp mp np

cos x sin y E 0;mn sinðhzÞ ¼ 0: a b k 2?mn a

ð7:11:12Þ

From Eq. (7.11.12) at z ¼ d it follows: sinðhdÞ ¼ 0;

hd ¼ pp;

h ¼ pp=d at p ¼ 1; 2; 3. . .

ð7:11:13Þ ð7:11:14Þ

For TMmn modes, the number p cannot be started from zero because at p ¼ 0 all components of the EM field disappear. Substituting Eqs. (7.7.29) and (7.11.3) into boundary conditions (7.11.8), we get for the component Ey presentation:

526

7 Rectangular Hollow Metallic Waveguides and Resonators

ih np E0 b

k2?;mn 

ref;mn

sin

mpx

a

cos

npy

ih np mp np  sin x cos y E 0;mn ei hz  E 0 a b k2?;mn b

b

ei hz ;

ref;mn e

i hz

ð7:11:15Þ 

¼ 0;

ð7:11:16Þ

where h ¼ pp=d at p ¼ 1; 2; 3. . . according Eq. (7.11.14) and from Eq. (7.11.10) E 0 ref;mn ¼ E 0;mn . We can write: 

  ih np mp np

sin x cos y E 0;mn ei hz  ei hz ¼ 0; 2 a b k?;mn b

ð7:11:17Þ

and using Eq. (1.4.16), we get for the component Ey presentation: 2

h np mp np

sin x cos y E0;mn sinðhzÞ ¼ 0: a b k2?;mn b

ð7:11:18Þ

On the base of Eqs. (7.7.30) and (7.11.4) for magnetic fields of incident and reflected waves and using a relation between amplitude coefficients (7.11.10), we get for component H x presentation: ixe0 er np mp np  sin x cos y E 0;mn ei hmn z þ E0 a b k2?mn b Because amplitude coefficients E 0 write Eq. (7.11.19) as follows:

ref;mn

ref;mn e

i hmn z



¼ 0:

ð7:11:19Þ

¼ E0;mn from Eq. (7.11.10), we can

  ixe0 er np mp np

sin x cos y E 0;mn ei hmn z þ ei hmn z ¼ 0; 2 a b k?mn b

ð7:11:20Þ

  taking into account Eq. (1.4.15), we have eihz þ eihz ¼ 2 cosðhzÞ, and for the component H x , we get: 2

ixe0 er np mp np

sin x cos y E 0;mn cosðhzÞ ¼ 0; a b k2?mn b

ð7:11:21Þ

where from Eq. (7.11.14) h ¼ pp=d at p ¼ 1; 2; 3. . . On the base Eqs. (7.7.31) and (7.11.5) for magnetic fields of incident and reflected waves, we get for component H y presentation: 

ixe0 er mp mp np  cos x sin y E 0;mn ei hmn z þ E 0 a b k2?mn a

ref;mn e

i hmn z



;

ð7:11:22Þ

7.11

Transverse Magnetic TMmnp Resonator Modes

527

using  ihz relation  between amplitude coefficients and taking into account Eq. (1.4.15) þ eihz ¼ 2 cosðhzÞ, we have: e 

  ixe0 er mp mp np

cos x sin y E 0;mn ei hmn z þ ei hmn z ; 2 a a b k?mn

ð7:11:23Þ

and we can write for H y  component presentation: 2

ixe0 er mp mp np

cos x sin y E 0;mn cosðhzÞ; a b k 2?mn a

ð7:11:24Þ

where from Eq. (7.11.14) h ¼ pp=d at p ¼ 1; 2; 3. . .. On the base Eqs. (7.7.25a) and (7.11.1) for electric fields of incident and reflected waves, we get for the component E z presentation: E 0;mn sin

mpx npy   sin ei hmn z þ ei hmn z ¼ 0; a b

ð7:11:25Þ

and using  relation between amplitude coefficients and taking into account  ihz ihz þe Eq. (1.4.15) e ¼ 2 cosðhzÞ, we can finally write for Ez -component presentation:: 2E 0;mn sin

mpx npy

sin cosðhzÞ ¼ 0; a b

ð7:11:26Þ

where from Eq. (7.11.14) h ¼ pp=d at p ¼ 1; 2; 3. . .. All constant multipliers (numerical coefficients) as 2 in Eqs. (7.11.12), (7.11.18), (7.11.21), (7.11.24), and (7.11.26) we include in the amplitude coefficients E 0;mn . The coefficient E0;mn is different for modes with different indexes. We denote further E 0;mn ¼ E 0 . Using Eqs. (7.11.12), (7.11.18), (7.11.21), (7.11.24),   and (7.11.26), we get components of resonator TMmnp H mnp modes (oscillations) in the phasor field presentation: mpx npy ppz

sin cos ; ð7:11:27Þ a b d mp np ppz

1 mp pp E 0;mn cos x sin y sin E xres;mn ðx; y; zÞ ¼  2 ; ð7:11:28Þ a b d k ?mn a d E zres;mnp ðx; y; zÞ ¼ E 0;mn sin

Eyres;mnp ðx; y; zÞ ¼ 

mp np ppz

1 np pp E 0;mn sin x cos y sin ; b d a b d

k 2?;mn

ð7:11:29Þ

528

7 Rectangular Hollow Metallic Waveguides and Resonators

H xres;mnp ðx; y; zÞ ¼

ppz

ixe0 er np mp np

sin x cos y E 0;mn cos ¼ 0; 2 a b d k?mn b

ð7:11:30Þ H yres;mnp ðx; y; zÞ ¼ 

mp np ppz

ixe0 er mp E x sin y cos cos ; ð7:11:31Þ 0;mn a b d k 2?mn a H zres;mnp ðx; y; zÞ ¼ 0;

ð7:11:32Þ

where m ¼ 1; 2; 3. . ., n ¼ 1; 2; 3. . . and p ¼ 1;2; 3. ... Instantaneous Field Expressions for TMmnp E mnp Oscillations The instantaneous field components for TMmnp oscillations (modes) are obtained by multiplication of the phasor field expressions (7.11.27)–(7.11.32) on ei xt which after using Euler’s formula Eq. (1.4.7) equals to cosðxtÞ þ i sinðxtÞ, and then taking the real part of the product, we have instantaneous components of TMmnp modes: mpx npy ppz

sin cos cosðxtÞ; ð7:11:33Þ a b d mp np ppz

1 mp pp E 0;mn cos x sin y sin Exres;mn ðx; y; z; tÞ ¼  2 cosðxtÞ; a b d k?mn a d E zres;mnp ðx; y; z; tÞ ¼ E 0;mn sin

ð7:11:34Þ Eyres;mnp ðx; y; z; tÞ ¼ 

mp np ppz

1 np pp E0;mn sin x cos y sin cosðxtÞ; b d a b d

k 2?;mn

ð7:11:35Þ H xres;mnp ðx; y; z; tÞ ¼ 

ppz

xe0 er np mp np

sin x cos y E cos sinðxtÞ; 0;mn a b d k2?mn b ð7:11:36Þ

Table 7.8 Resonance frequency f mnp and wavelength kmnp for TMmnp

m

n

p

f mnp (GHz)

kmnp (mm)

1 1 1 1 2 2 1 2 2 1

1 1 1 1 1 1 1 1 1 1

0 1 2 3 0 1 4 2 3 5

16.36 16.63 17.42 18.67 19.88 20.10 20.29 20.76 21.82 22.19

18.34 18.04 17.22 16.07 15.09 14.92 14.79 14.45 13.75 13.52

Transverse Magnetic TMmnp Resonator Modes

7.11

H yres;mnp ðx; y; z; tÞ ¼

529

mp np ppz

xe0 er mp E x sin y cos sinðxtÞ; cos 0;mn a b d k2?mn a ð7:11:37Þ H zres;mnp ðx; y; z; tÞ ¼ 0;

ð7:11:38Þ

where m ¼ 1; 2; 3. . ., n ¼ 1; 2; 3. . . and p ¼ 1; 2; 3. . ..

7.11.1 Resonance Frequency of TMmnp Mode Table 7.8 gives resonance frequency f mnp calculated by Eq. (7.10.11) because formulae for TEmnp and TMmnp ð¼ Emnp Þ are the same and the wavelength kmnp is calculated by Eq. (7.10.4) for oscillations with different indexes “m”, “n”, and “p” when the resonator sizes are a  b  d ¼ 0:023  0:01  0:05 m3 (Fig. 7.43). As we mentioned in Sect. 7.10.1, if we change the coordinate axis, we can rename mode TE101 as mode TM110 (Fig. 7.53), so for this reason, we will study another mode with different structure of EM fields. We see from Table 7.8 that the mode TM110 is followed by the mode TM111 with the resonance frequency f mnp ¼ 16:63:

y

b

x

O a

Fig. 7.63 Electric (red color) and magnetic (blue color) field structure of TM111 resonator mode in the xOy plane at z = d

530

7 Rectangular Hollow Metallic Waveguides and Resonators

x

a

O

z

Fig. 7.64 Electric (red color) and magnetic (blue color) field structure of TM111 resonator mode in xOz plane at y ¼ b=2

y

b

O

z

Fig. 7.65 Electric (red color) and magnetic (blue color) field structure of TM111 resonator mode in the yOz-plane at x ¼ a=2

Transverse Magnetic TMmnp Resonator Modes

7.11

531

y

b

O

x a

Fig. 7.66 Surface conduction current density Js of TM111 resonator mode in the xOy plane at z=0

x

a

O

z

Fig. 7.67 Surface conduction current density Js of TM111 resonator mode in the xOz plane at y=b

532

7 Rectangular Hollow Metallic Waveguides and Resonators

y

b

O

z

Fig. 7.68 Surface conduction current density Js of TM111 resonator mode in the plane when x=0

7.11.2 Transverse Magnetic TM111 Mode We will consider now TM111 ð¼ E 111 Þ resonator mode. Our calculation of this mode is fulfilled at t ¼ T=8. At this time moment t, the electric and magnetic fields exist simultaneously. Visual pictures are carried out by using our homemade computer programs in MATLAB. Figures 7.63–7.68 show distributions of E-, H-fields and surface conduction current Js calculations by formulae (7.11.33–7.11.38) at the resonator sizes a  b  d ¼ 0:023  0:01  0:05 m3 (see Fig. 7.43). Figure 7.63 shows the distribution of vectors E and H of TM111 mode near the end resonator side a  b at z ¼ d. We have chosen here the plane at z d for consideration of fields because the electric field concentrates near the center of conductor resonator sides, while the electric field of TE101 mode is getting stronger in plane z ¼ d=2, i.e. in the center of free space inside the resonator (see Fig. 7.53). In Fig. 7.63, we see that magnetic field lines form oval closed lines which are parallel to the end resonator wall a  b. The electric field lines are perpendicular to the end wall a  b and directed from us in the plane, so we see crosses of the electric field vectors. Figures 7.64 and 7.65 show four concentrated locations of the magnetic field in the resonator corners. We see points or crosses of vectors H because the magnetic field rings are intersected by the horizontal xOz or vertical yOz planes (Fig. 7.53). The electric and magnetic fields are stronger near the resonator walls and weaker in the resonator center for the mode TM111 . The vector field arrows protruding beyond

7.11

Transverse Magnetic TMmnp Resonator Modes

533

Fig. 7.69 Distribution of the electric field intensity of TM111 resonator mode where the intensity color scale is shown on the right

Fig. 7.70 Distribution of the magnetic field intensity of TM111 resonator mode where the intensity color scale is shown on the right

the resonator indicate that the points in which they correspond relate to the space inside the resonator which is very close to the wall. The distributions of the surface conduction current density on the resonator walls are shown in Figs. 7.66, 7.67, and 7.68. We see that the current density is weak in the centers of the conductor walls.

534

7 Rectangular Hollow Metallic Waveguides and Resonators

Figures 7.69 and 7.70 show the distribution of intensity of the electric field and the magnetic field, respectively. The intensity scale of vectors E and H is given on the right side of pictures (dark red color is the maximum intensity which is equal to 1 in conventional units). We see that the electric and magnetic fields concentrate near the resonator walls. We would like to remind you that Figs. 7.63–7.70 present distributions at t ¼ T=8 (Fig. 7.50).

7.11.3 Schematic Sectional View for TM111 Mode Now, we will analyze the schematic view of TM111 ðE111 Þ at moments which is shown in Fig. 7.50: t ¼ t2 ¼ T=4 (when the magnetic field H 6¼ 0, and the electric field E¼ 0) and t ¼ t3 ¼ T=2 (when H ¼ 0, E 6¼ 0) as shown in Fig. 7.50.

A-A

t2=T/4

1z

d=λw/2

1z - 1z

Ez

t2=T/4

B

A

Ez

Probe

b

a t2=T/4

1z 1z

B-B

A

Loop

2z

D-D

B

λw/4

t3=T/2

1z

t3=T/2

2z

Slot

2z C-C

Slot

Probe

C

t3=T/2

D

2z - 2z

H Loop

2z

E Js

H Jd

qs

C

D

TM 111 m=1, n=1, p=1 a>2b, d>2a

Fig. 7.71 Distributions of E (red color), H (blue), JS (black), Jd (green), qs for TM111 ð¼ E 111 Þ oscillation when there are excitements by a probe, a loop, and a slot at time moments t = T/4 and t = T/2

7.11

Transverse Magnetic TMmnp Resonator Modes

535

Schematic view of the distributions of E (red color), H (blue), Jd (green), Js (black), and surface charges qs for the mode TM111 of the rectangular resonator is presented in Fig. 7.71. In the resonator, the time moment t is very important (see Figs. 7.50, 7.51, and 7.52). In Fig. 7.71, we see a probe, a loop, and a slot for the excitation of the resonator. The electric field at t ¼ t2 ¼ T=4 has the maximum value E and H¼ 0. The view of processes which are happening in the resonator at this time t ¼ t2 is shown in Fig. 7.71 (top four sections). In the longitudinal cross-section “1z–1z”, we see lines of vectors E which start or end on the charges qs . In the section “A–A” field E-vectors are perpendicular to the plane. The section “A–A” demonstrates the law of E z component distribution along the waveguide wall “a” and “b”. In the plane “B–B”, the electric field lines lie in the plane, in the center E-vector lines change their directions and go in the directions perpendicular to the plane. In the center of plane “B–B”, we see crosses of the E-vectors. We can see images only of H, Js , Jd in Fig. 7.71 which corresponds to time moment t ¼ t3 ¼ T=2. In the cross-section “C–C” (bottom left section), the magnetic field is depicted by closed curves. We also see vectors of current that flows on the conducting resonator walls and which are closed through current density Jd (green points) and form together closed current circuits (see also Fig. 7.51). Section “2z–2z” (bottom rigth section in Fig. 7.71) shows points and crosses of the magnetic field in the corners of cross-section because the plane cuts the magnetic field closed curves at t=t3=T/2. Section “2z–2z” shows the probe, loop, and slot for excitation of this mode. In this section, we see the displacement current lines which are closed via the conduction current Js and the last current flows on the conducting resonator walls. Section “D–D” gives the distribution of the displacement current as well as crosses and points of field H-vector lines at t=t3=T/2.

7.12

Quality Factor

Resonators store energy in the electric and magnetic field for any mode. Quality factor Qres is a measure of the bandwidth of the resonator, and it is a dimensionless value. Here, we can express the quality factor as follows: Qmnp ¼

2p f mnp W mnp ; Pmnp

ð7:12:1Þ

where Pmnp is the time-average power dissipated in one period of the resonant frequency, magnitude W mnp ðtÞ ¼ W E;mnp ðtÞ þ W M;mnp ðtÞ (see Eq. (7.10.10)) is the total time-average energy stored in the resonator at a resonant frequency, W E;mnp ðtÞ ¼ W E calculated by Eq. (7.10.8), W M;mnp ðtÞ ¼ W M calculated by Eq. (7.10.9), and f mnp is the resonant frequency calculated by Eq. (7.10.11).

536

7 Rectangular Hollow Metallic Waveguides and Resonators

We will analyze the case d [ a [ b (Fig. 7.43) when two largest sides are “d” and “a”. For this case, the main mode with the lowest cutoff frequency is TE101 (Table 7.7). The lowest resonant frequency is: f 101

c ¼ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 2: 2 a d

ð7:12:2Þ

The TE101 mode has three nonzero EM field components Ey;101 Eq. (7.10.48), H x;101 Eq. (7.10.49), and H z;101 Eq. (7.10.50). The time-average stored electric energy from Eq. (7.10.8) is: Zd Z b Z a WE ¼ 0

¼

0

2 er e0 E y dxdydz 2

0

er e0 f mnp lr l0 p4 k 4? a2

Zd Z b Z a H 20

sin2 0

0

px pz

dxdydz; a d

ð7:12:3Þ

0

where k? ¼ ðp=aÞ from Eq. (7.10.46): W E ¼ er e0 ðlr l0 Þ2 f 2101 H 20

 3  a bd 1 ¼ er e0 ðlr l0 Þ2 a3 b d H 20 f 2101 : 2 2

ð7:12:4Þ

The time-average stored magnetic energy from Eq. (7.10.9) is:

WM ¼



Zd Zb Za l l jH x j2 þ jH z j2 r 0 2 0

WM

0

dxdydz;

ð7:12:5Þ

0

#   



p 4 1 2 2 px 2 pz

2 px 2 pz cos þ cos sin dxdydz sin k? da a d a d 0 0 0  a 2  ll ¼ r 0 H 20 1 þ abd: d 8 ll ¼ r 0 H 20 2

Z d Z b Z a "

ð7:12:6Þ Substituting the resonance frequency of the TE101 mode from Eq. (7.12.2) in Eq. (7.12.3), we see that at the resonant frequency the time-average stored electric energy and magnetic energy are equal—W E ¼ W M . The total time-average energy stored in the resonator Eq. (7.10.10) is:

7.12

Quality Factor

537

W mnp ¼ 2W E ¼ 2W M ;

ð7:12:7Þ

1 W mnp ¼ er e0 ðlr l0 Þ2 a3 b d H 20 f 2101 ; 2

ð7:12:8Þ

 a 2  lr l0 2 H0 1 þ abd: d 8

ð7:12:9Þ

or W mnp ¼

For determination of the time-average power dissipated Pmnp in one period of the resonant frequency (an amount of time which takes to complete one cycle is called a period, and the number of cycles in one second is the frequency of the oscillation), we are taking into account the fact that the power loss per unit area (see Sect. 7.8) is: Ploss=area ¼

jJ s j2 Rs jH t j2 Rs ¼ : 2 2

ð7:12:10Þ

The time-average power dissipated in one period of the resonant frequency is: I Pmnp ¼

Rs Ploss=area ds ¼ Rs 2

s

Zd Z b 2

0

ð7:12:11Þ

0

Zd Z a

jH z j dxdy þ Rs jH x j2 þ jH z j2 dxdz; 2

jH x j dxdy þ Rs 0

jH t j2 dx dy; 0

Zb Za Pmnp ¼ Rs

Zb Za

0

0

0

0

ð7:12:12Þ Pmnp ¼

     Rs H 20 a2 b 1 b 1 þ þ þd : a 2 2 d d 2

ð7:12:13Þ

The resistance of a conductor in a resonator according Eq. (5.10.14) is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pf mnp lr l0 Rs ¼ : rc

ð7:12:14Þ

After inserting (substituting) Eqs. (7.12.8) and (7.12.13) in Eq. (7.12.1), we receive the quality factor for the TE101 mode: Q101

  pf 101 lr l0 d 2 þ a2 abd 2p f 101 W 101      ; ¼ ¼ P101 Rs d a d 2 þ a2 þ 2b d 3 þ a3

ð7:12:15Þ

538

7 Rectangular Hollow Metallic Waveguides and Resonators

  ðplr l0 f 101 rc Þ1=2 d 2 þ a2 abd   3  : Q101 ¼   2 d a d þ a2 þ 2b d þ a3

ð7:12:16Þ

For the case a ¼ b ¼ d when three sides are equal, there are three lowest-order modes TM110 , TE011 , and TE101 with the quality factor: Q101 ¼ aðplr l0 f 101 rc Þ1=2

at

c f 101 ¼ pffiffiffi ; a 2

ð7:12:17Þ

where c is the speed of light (Eq. 2.1.6), “a” is the wall size of a resonator (Fig. 7.43), and rc is the electrical conductivity of a material from which the resonator is made.

7.13

Excitation of Rectangular Resonators by Probes, Loops, and Slots

The principles of the resonator excitation are the same as for waveguides in Sect. 7.9. To excite the cavity resonator, the probes, loops, and slots in conductive resonator walls are applied. Figure 7.72 shows the location of a probe, a loop, and a slot in order to excite oscillation TE101 . When to the exciting element the energy is brought from a generator which is operating at high frequencies, different types of oscillations may arise. Since the resonator has an infinite number of discrete natural frequencies, if you change the input frequency of the generator, it is possible to observe a large number of resonant oscillations with different structure of EM fields.

Fig. 7.72 Possible locations of exciting elements in the rectangular resonator

slot

H Jc

E

probe

H loop

7.14

7.14

Conclusions About Resonators

539

Conclusions About Resonators

1. Volumetric resonator has many resonance frequencies f mnp each of which corresponds to a different type of oscillations TEmnp also known as H mnp , TMmnp also known as E mnp . 2. The designation by letters TE and TM defines the existence of the longitudinal component Hz or Ez. Numeric index values of m, n, p indicate the number of half-wave of a standing wave along the relevant coordinates (sides) of the resonator: m ! a; n ! b; p ! d: 3. Oscillations TMmnp with indexes m ¼ 1; 2; 3. . ., n ¼ 1; 2; 3. . . and p = 0 are possible. The tangential components Ex and Ey have to change by the law sinðhzÞ at the resonator end conducting walls because the boundary conditions require that Et ¼ 0 while the longitudinal wavenumber automatically becomes equal to zero (h = 0) at p = 0. In this case, the boundary conditions are satisfied at any length of a resonator. 4. Oscillations TEmnp with indexes m ¼ 1; 2; 3. . ., n ¼ 1; 2; 3. . . and p = 0 cannot exist, as transverse components along the longitudinal coordinate should not change E x ðzÞ ¼ const, E y ðzÞ ¼ const. Therefore, components E x , Ey violate the boundary conditions on the lateral walls of a resonator. 5. The smallest resonance frequencies f mnp have the lowest types of vibrations (oscillations) because by reducing the value of indexes “m”, “n” and “p” f mnp decreases. 6. The increase of the geometric dimensions “a”, “b”, and “d” of the resonator cavity reduces the resonant frequency f mnp . 7. Dimensions for the lower types of resonator oscillations commensurate with a wavelength kw . 8. The main type of oscillations matches the lowest resonance frequency. The smallest types of oscillations are oscillations H 011 , H 101 , E110 . The choice which of these oscillations is the main (fundamental, dominant, principal) one depends on the ratio of sides of the resonator. It is determined by the size of the minimum side. H 011 ! min “a”, H 101 ! min “b”, E 110 ! min “d”. This can be explained by the fact that the minimum size of side gives the greatest increase in frequency f mnp . It means that this size should be deleted by taking the index on this side to be equal to zero. The main type is the lowest type of oscillations which does not have any field variation along the shortest side of a resonator. Therefore, these three types of oscillations have the same field structure and differ only in the orientation of the coordinate system relative to the resonator. 9. Changes over time of the electric and magnetic components of oscillations are shifted on the phase p=2 that determines the multiplier i ¼ eip=2 . Therefore, there are moments of time when all the energy is concentrated only in the electric field or only in the magnetic field (Figs. 7.50, 7.52, and 7.28).

540

7 Rectangular Hollow Metallic Waveguides and Resonators

In the end of this book the following data are presented (Appendixes Chap. 7:7.3 and 7:7.4): TEmnp and TMmnp distributions of the electric and magnetic fields in Section “1–1” at z = d/2, Section “2–2” at x = d/2 and Section “3–3” at y ¼ b=2 (see Fig. 7.13) for the rectangular resonator with sizes a  b  d ¼ 0:023  0:01  0:05 m3 at time t ¼ T=8.

7.15

Examples and Problems

Resonator Example 1 Find the main mode with the lowest cutoff frequency in the rectangular cavity resonator filled with air er ¼ lr ¼ 1 (a) d [ a [ b; (b) a [ b [ d; (c) a ¼ b ¼ d; (d) a [ d [ b; (e) b [ a [ d (Fig. 7.43). When z-axis is chosen for the third index “p” of resonator modes. We remember that for TMmnp mode neither m nor n can be equal to zero but the third index “p” can be zero (see Table 7.8). For TEmnp modes either m or n but not both simultaneously can be equal to zero but “p” cannot be zero (see Table 7.7). We will use Eq. (7.10.11). We see from the equation that a larger size of a resonator provides a lower cutoff frequency. (a) For the case d [ a [ b when the two largest sides are “d” and “a”, the main mode with the lowest cutoff frequency is TE101 (see Tables 7.7). The lowest resonant frequency is: f 101 ¼

c 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ : a2 d 2

Please calculate the cutoff frequency for TE101 , TE102 , TE103 when a  b  d ¼ 0:023  0:01  0:05 ðm3 Þ and compare it with Table 7.7. (b) For the case a [ b [ d when the two largest sides are “a” and “b”, the main mode with the lowest cutoff frequency is TM110 : f 110

c ¼ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ : a2 b2

(c) For the case a ¼ b ¼ d when three sides are equal, there are three lowest-order modes TM110 , TE011 and TE101 with the same EM field pattern. We name these modes as degenerated ones. These modes have the following resonant frequency: 1 c f 110 ¼ f 011 ¼ f 101 ¼ pffiffiffi : 2a

7.15

Examples and Problems

541

(d) For the case a [ d [ b when the two largest sides are “a” and “d”, the main mode with the lowest cutoff frequency is TE101 . The lowest resonant frequency is: f 101

c ¼ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ : a2 d 2

Please compare cases (a) with sizes d > a and case (c) with sizes a > d. Example 2 Calculate the size of a cubic resonator when a = b = d which is made of brass with rc;CuZn10 ¼ 2:5  107 ðS=mÞ in order to have the main resonant frequency of 10 GHz and 30 GHz. Calculate the quality factor of the TE101 mode. There are three lowest-order degenerated modes TM110 , TE011 , and TE101 . l0 a ffiffi 8 , Q101 ¼ pf 101 ¼ It is given that: f 110 ¼ f 011 ¼ f 101 ¼ p1ffiffi2 ac, a ¼ p310 3Rs 2f 101 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a pf 101 l0 rc . 3 Resonator made of brass at f 101= 10 GHz: 3  108 3  108 pffiffiffi ðHzÞ; a ¼ pffiffiffi ¼ 0:0212 ðmÞ; 2  1010 a 2 0:0212 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1010 ð4p107 Þð2:25  107 Þ 6657: ¼ 3

f 101 ¼ Q101

Resonator made of brass at f f 101 ¼ Q101 ¼

101=

30 GHz:

3  108 3  108 pffiffiffi ðHzÞ; a ¼ pffiffiffi ¼ 7:07  103 ðmÞ; 2  30  109 a 2

7:07  103 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð30  109 Þð4p  107 Þð2:5  107 Þ ¼ 4054: 3

Example 3 Resonator made of silver at 10 GHz. Calculate the size of a cubic resonator when a ¼ b ¼ d which is made of silver with rc;Ag ¼ 6:25  107 ðS=mÞ for the resonator mode TE101 at frequency f101=10 GHz. Calculate the quality factor of the mode TE101 . 3  108 3  108 pffiffiffi ðHzÞ; a ¼ pffiffiffi ¼ 0:0212 ðmÞ; 2  1010 a 2 0:0212 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1010 ð4p107 Þð6:25  107 Þ 11; 100: ¼ 3

f 101 ¼ Q101

542

7.16

7 Rectangular Hollow Metallic Waveguides and Resonators

Review Questions

Q7:1. What does the classification of EM modes as TEM, TEmn (=Hmn), TMmn (=Emn), HEmn, EHmn mean? Q7:2. Which of EM field components Ex, Ey, Ez, Hx, Hy, Hx are important for the classification considered herein? Q7:3. How to describe TEM and HEmn modes by the classification? Q7:4. What type of mode is there when E z 6¼ 0 and H z 6¼ 0? Q7:5. What type of mode is there when E z ¼ 0 and H z 6¼ 0? Q7:6. What type of mode is there when E z 6¼ 0 and H z ¼ 0? Q7:7. Which of EM field components E x , Ey , Ez , H x , H y , H z in the rectangular waveguide are tangential to the horizontal wide walls of the waveguide? Q7:8. Which of EM field components E x , Ey , Ez , H x , H y , H z in the rectangular waveguide are tangential to the vertical narrow walls of the waveguide? Q7:9. What is the instantaneous expression of the electric and magnetic field for time-harmonic EM fields, i.e. Eðx; y; z; tÞ = ? Q7:10. What are the phase and group speeds of a TEM mode in a microstrip line and the speed of light in the homogeneous medium? Q7:11. What are the phase and group speeds of TE modes in the waveguide? Compare them with the speed of light in vacuum. Q7:12. What are the phase and group speeds of TM modes in the waveguide? Compare them with the speed of light in vacuum. Q7:13. Write Helmholtz’s equations for the electric and magnetic fields. Q7:14. Write the expression of the wave impedance for TEmn modes through the components of Ey and Hx. Q7:15. Write the expression of the wave impedance for TMmn modes through the components of Ex and H y . Q7:16. What is a cutoff frequency and a wavelength of a waveguide mode? Q7:17. Give the definition of the concept of the main mode of waveguides. Q7:18. Give the definition of the concept of the higher mode of waveguides. Q7:19. How is the wave impedance TE and TM modes related to the frequency? Q7:20. What is the dependency of component Ez for TMmn mode on the coordinate “x” and “y” in the rectangular waveguides? Q7:21. What is the dependency of component H z for TEmn mode on the coordinate “x” and “y”? Q7:22. Write the dependency of solution on the coordinate “z” and time “t” for TEmn modes in the waveguides, i.e. H(x, y, z, t) = H(x, y)(function on z and t). Q7:23. How the method of the separation of variables is applied to the solution of Helmholtz’s equations for TEmn modes of rectangular waveguides? Q7:24. Write the expression of the longitudinal component H z and explain the distribution of the component on the waveguide cross-section. Q7:25. How the method of the separation of variables is used in the solution of Helmholtz’s equations for TMmn modes of rectangular waveguides?

7.16

Review Questions

543

Q7:26. Write the expression of the longitudinal component E z and explain the distribution of the component on the waveguide cross-section. Q7:27. How many electric and magnetic components (Ex, Ey, Ez, Hx, Hy, Hz) TEmn modes may contain? Q7:28. Which type of boundary conditions must be satisfied at the metallic waveguide walls for transverse electric modes TEmn ? Q7:29. What do subscripts “m” and “n” in TEmn modes indicate? Q7:30. Which type of boundary conditions must be satisfied at the metallic waveguide walls for transverse electric modes TMmn ? Q7:31. What do subscripts “m” and “n” in TMmn modes indicate? Q7:32. What does the transversal propagation constant of rectangular waveguides indicate? Q7:33. What does the longitudinal propagation constant of rectangular waveguides indicate? Q7:34. What does the wavenumber k mean? Q7:35. Please draw the dispersion characteristics of the main and higher modes of a rectangular waveguide and explain them. Q7:36. Write the cutoff frequency expression for TMmn and TEmn modes. Q7:37. Specify the three methods of excitation of mode TE10 in the waveguide. Q7:38. How do we excite the mode TE01 ? Q7:39. How do we excite the mode TE20 ? Q7:40. What is the difference between traveling (propagating) and vanishing modes? Q7:41. Write the value of the subscripts “m” and “n” for the main mode of the metallic rectangular waveguide, m = what? and n = what? Q7:42. How many components does the mode TE10 have? Q7:43. Draw a 3D distribution of the conduction current on the walls of a rectangular waveguide and the displacement current for TE10 mode. Q7:44. Draw a 3D distribution of the electric and magnetic fields in a rectangular waveguide for TE10 mode. Q7:45. Why do waveguide losses increase when approaching the cutoff frequency of propagating modes and at higher frequencies? Q7:46. What is the difference between a waveguide and a resonator? Q7:47. How does the resonance frequency in a resonator depend on the numbers of standing half-wave “m”, “n”, “p”? Q7:48. How do the values E; qs and H; J s ; J d change over time in a resonator? Q7:49. What do subscripts “m”, “n” and “p” in resonator modes indicate? Q7:50. How do we excite the TM11 resonator mode by a probe? Q7:51. Which resonance mode TE or TM and with what indexes ”m”, “n”, “p” has the lowest value of the critical frequency?

Chapter 8

Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Abstract This chapter contains solutions and calculations for circular cylindrical hollow metallic waveguides and cavity resonators. Here the solutions of Helmholtz’s equations in cylindrical coordinates are presented, Bessel functions’ and their first derivatives’ properties are discussed in detail. Other items explored in this chapter include EM characteristics of TE and TM modes, explanation of mode indexes, dispersion characteristics, and cutoff frequencies. There is also a schematic sectional view of TE11, TM01, TE111, TM011 modes in waveguides and resonators.

8.1

Introduction

EM waves can also propagate in a hollow circular cylindrical metallic (conducting, metal) tube (pipe) having a uniform circular (round) cross-section and the internal (inner) radius “a” (Fig. 8.1a). The waveguide is centered relative to the longitudinal z-axis. It is important to note that elliptical cylindrical waveguides are also frequently used for which solutions are searched in the elliptical coordinate system and other functions are applied for solving, such as Mathieu ones. The hollow circular cylindrical waveguide can be filled with air or other insulator (dielectric) medium. We assume that the waveguide has infinite length along the z-axis. We will consider the lossless tube waveguide with the electrical conductivity of metal walls rc ¼ 1 and the electrical conductivity of the dielectric filling r ¼ 0. The dielectric filling is a linear isotropic homogeneous non-conducting medium, i.e. an insulator with the EM constitutive parameters r ; lr and r ¼ 0: Usually, metallic waveguides are air filled with r ¼ 1 and lr ¼ 1: Figure 8.1b shows the arbitrary points M 1 ; M 2 ; . . .; M n where we can calculate the ^ are the unit ones, the radial coordinate r and EM field components, vectors ^r, and u azimuthal (angular) coordinate u of the chosen arbitrary point. Circular cylindrical waveguides are widely used as an EM field line in microwave communications. Segments of the waveguide can be used as elements of the rotating joints intended for transmission of EM energy from the static (non-moveable) to moving waveguides. The waveguide can be used to excite the © Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_8

545

546

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

(a)

φ=0

(b)

^r

φ=0 a

z

Mn

r

φ

M2 M1 ^ r

^ φ

^ φ

0 2a

Fig. 8.1 Circular cylindrical waveguide: a its geometry with radius a and b used designations

open (without a screen) circular cylindrical dielectric waveguides or when the circular polarization is to be transmitted to certain antennas. However, circular waveguides are used less frequently than rectangular ones. This is due to the polarization instability of the main mode TE11 in the circular cylindrical waveguide. The polarization change, i.e. EM field rotation, arises due to the perfect symmetry of a circular waveguide. When a mode has a certain polarization at the beginning of the waveguide line, then the polarization of the mode has already another direction at the waveguide end because of various random or deliberate waveguide deformations or inclusions. It is very valuable that in the circular waveguide modes can exist as the symmetric types of waves, e.g. TM01 and TE01 . The mode TM01 is interesting because it is often used in waveguide rotary joints. The attenuation of TE01 mode decreases with increasing of the frequency in contrast to other waves. The circular copper waveguide attenuation at the radius a ¼ 2:5 cm, and the length of 1 km is only several decibels in ranges SHE and EHF (Appendix Chap. 5:5.1) at the cutoff frequency ðf c ÞTE01  23 GHz. The difficulty lies in the fact that the wave TE01 is the third higher mode and therefore can propagate along the waveguide with the main TE11 and other waves of higher types which disturb and thus are referred to as parasitic modes. In order to eliminate unwanted parasitic waves you can use filters, but this will complicate the device operating on the mode TE01. An EM field can propagate along a circular waveguide in different ways. Two common modes of the circular waveguide are known as transverse electric TEmn (also known as H mn ) and transverse magnetic TMmn (also known as E mn ) Sect. 7.2. The waveguide must possess a certain minimum diameter relative to the wavelength of microwave signal. If the waveguide diameter is too small or the frequency is too low, the EM fields cannot propagate. At any frequency above the cutoff frequency, the waveguide will operate (work) well, although certain waveguide operating characteristics depend on the number of half-wave field variations on the angular and radial coordinates in the cross-section. The common principles of excitation of cylindrical waveguides are the same as for rectangular waveguides; see Sect. 7.9.

8.2 Solution of Helmholtz’s Equation in Cylindrical Coordinates

8.2

547

Solution of Helmholtz’s Equation in Cylindrical Coordinates

From the time-harmonic Maxwell’s equations analogical to Eqs. (7.3.1) and (7.3.2) for a source—free insulator medium, we get the set of equations in the cylindrical coordinate system: 1 @E z @Eu  ¼ i xl0 lr H r ; r @u @z

ð8:2:1Þ

@Er @Ez  ¼ i xl0 lr H u ; @z @r

ð8:2:2Þ

1 @H z @H u  ¼ i x0 r E r ; r @u @z

ð8:2:3Þ

@H r @H z  ¼ i x0 r E u ; @z @r   1 @ rEu 1 @Er  ¼ i xl0 lr H z ; r @r r @u   1 @ rH u 1 @H r  ¼ i x0 r E z : r @r r @u

ð8:2:4Þ ð8:2:5Þ ð8:2:6Þ

The complex amplitudes of electric E and magnetic H field intensities for EM waves which propagate along the z-axis are: Eðr; u; zÞ¼ Eðr; uÞeihz ;

ð8:2:7Þ

Hðr; u; zÞ¼ Hðr; uÞeihz :

ð8:2:8Þ

Now, in a similar way as we got Eqs. (7.4.35)–(7.4.38) we can express the transverse components of the EM field through the longitudinal ones in the cylindrical coordinates:   i @E z xl0 lr @H z þ Er ¼ 2 h ; ð8:2:9Þ @r r @u k?   i h @E z @H z  xl0 lr ; Eu ¼ 2 @r k? r @u

ð8:2:10Þ

  i x0 r @Ez @H z h ; Hr ¼ 2 r @u @r k?

ð8:2:11Þ

548

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

  i @Ez h @H z þ H u ¼ 2 x0 r : r @u @r k?

ð8:2:12Þ

We put Eqs. (8.2.9) and (8.2.10) into Eq. (8.2.5) and get:    1@ @H z 1 @2Hz  2 r þ x 0 r l0 lr  h2 H z ¼ 0: þ 2 2 r @r r @u @r

ð8:2:13Þ

We put Eqs. (8.2.11) and (8.2.12) into Eq. (8.2.6) and receive:    1@ @E z 1 @ 2 Ez  2 r þ x 0 r l0 lr  h2 E z ¼ 0: þ 2 2 r @r r @u @r

ð8:2:14Þ

The basic equations to be satisfied by the time-harmonic complex amplitude of electric E and magnetic H field intensities in the charge-free insulator area in the general form of homogeneous vector Helmholtz’s equation (7.4.19) are:     r2? E þ x2 0 r l0 lr  h2 E = r2? E þ k2  h2 E = 0,

ð8:2:15Þ

    r2? H þ x2 0 r l0 lr  h2 H = r2? H þ k 2  h2 H = 0.

ð8:2:16Þ

3D Laplacian operator (Laplacian, differential operator) can be written in the following form: r2 ¼ r2? þ

@2 ; @z2

ð8:2:17Þ

r2 is separated into two parts r2? and r2z , where r2? ¼ r2ru is the 2D  transversal Laplacian operator for the transverse r, u coordinates and r2z ¼ @ 2 @z2 is the longitudinal (axial) part for the z-coordinate. The electric field vector Eðr; u; zÞ and the magnetic field vector Hðr; u; zÞ can be written as the sum of the transversal and longitudinal components: ð8:2:18Þ E ¼ ET þ ^zEz ; H ¼ HT þ ^zH z ;

ð8:2:19Þ

where index “T” means the transverse terms and electric field components Er , E u , H r , H u are transverse components. The longitudinal components are E z and H z . 2 ihz When the dependence on the z-coordinate is eihz , then we have @ e ¼ @z2

h2 eihz and the Laplacian operator can be written by analogy with Eqs. (7.4.20) and (7.4.22) as:

8.2 Solution of Helmholtz’s Equation in Cylindrical Coordinates

549

r2ru E þ k 2? E = 0,

ð8:2:20Þ

r2ru H þ k 2? H = 0.

ð8:2:21Þ

where k? is the transverse propagation constant in the circular waveguide: k? ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2  h2 ¼ x2 0 r l0 lr  h2 ;

ð8:2:22Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the wavenumber k ¼ x 0 r l0 lr is given in Eq. (7.4.11) and the longitudinal propagation constant h ¼ h0 ¼ 2p=kw . In this chapter, we take h ¼ h0 ¼   Re h_ the same as in Eq. (7.2.2). Since each component of vectors Eðx; y; zÞ and Hðx; y; zÞ must satisfy Helmholtz’s equations (8.2.20) and (8.2.21), we can write the equation regarding the longitudinal component Ez and H z as:

 @ z r @E where r2ru E z ¼ 1r @r + @r

r2ru E z þ k 2? Ez = 0,

ð8:2:23Þ

r2ru H z þ k 2? H z = 0,

ð8:2:24Þ

1 @ 2 Ez r2 @u2 ,

 @ z and r2ru H z ¼ 1r @r r @H + @r

1 @2 Hz r 2 @u2 .

Despite the fact that Eqs. (8.2.23) and (8.2.24) are quite similar to Eqs. (7.5.3) and (7.5.19), their solutions are different. We see that the inner surface of metallic wall of a circular cylindrical waveguide coincide with the coordinate surface r ¼ a of the cylindrical coordinate system ðr; u; zÞ. We can write Eq. (8.2.23) in cylindrical coordinates: r2ru E z þ k 2? Ez =

  1@ @E z 1 @ 2 Ez r + k 2? E z ¼ 0. + 2 r @r r @u2 @r

ð8:2:25Þ

We can write Eq. (8.2.24) in the cylindrical coordinates: r2ru H z þ k 2? H z =

  1@ @H z 1 @2Hz r + k 2? H z = 0. + 2 r @r r @u2 @r

ð8:2:26Þ

We use the method of separation of variables for a solution of Eqs. (8.2.25) and (8.2.26). We will demonstrate here only a solution for the last equation. We write a partial solution to the equation in the form of two functions: H z ðr; uÞ ¼ Rðr ÞUðuÞ:

ð8:2:27Þ

where Rðr Þ is the radial distribution function of r coordinate only and UðuÞ is the angular distribution function of u coordinate only.

550

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

We substitute Eq. (8.2.27) into Eq. (8.2.26) and receive the following:   r d dRðr Þ 1 d2 UðuÞ r : þ k2? r 2 ¼  Rðr Þ dr dr UðuÞ du2

ð8:2:28Þ

We see that the left side of Eq. (8.2.28) is a function of r and the right side is a function of u only. We will denote the separation constant by number m. We will separate Eq. (8.2.28) into two ordinary differential equations, and one of the differential equations is: 

1 d2 UðuÞ ¼ m2 : UðuÞ du2

ð8:2:29Þ

Equation (8.2.29) can be rewritten as: d2 UðuÞ þ m2 UðuÞ ¼ 0: du2

ð8:2:30Þ

The appropriate solution of Eq. (8.2.30) can be: UðuÞ ¼ A1 cosðmuÞ þ A2 sinðmuÞ;

ð8:2:31Þ

where A1 and A2 are arbitrary constants. One of the functions cosðmuÞ or sinðmuÞ can be chosen because there is the symmetry of the circular waveguide by angular coordinate u. However, in order to satisfy the obvious requirement of the solutions periodicity on the angle u with a period of no more than 2p, index m must be an integer number including zero. When the index m is equal to zero, the solution can be only cosðmuÞ ¼ cosð0Þ. When we change the angular coordinates u1 ¼ u on u1 ¼ u þ 2p, the field’s value is not changed in the points for both cases, so we can record: UðuÞ ¼ Uðu þ 2pÞ:

ð8:2:32Þ

The expression for the angular distribution function UðuÞ finally has the following form: UðuÞ ¼ A cosðmuÞ:

ð8:2:33Þ

Images of the angular distribution function UðuÞ are presented in Fig. 8.2. Functions UðuÞ with different “m” are placed along the perimeter of the waveguide. The number of “m” shows how many standing waves of the field fit around the circumference (perimeter) of the waveguide. The number “m” is the first index of mode (wave) type. There are no variations of the field (longitudinal component H z ) when “m” is equal to zero; see the left

8.2 Solution of Helmholtz’s Equation in Cylindrical Coordinates

φ

m=0

551

wavequide

m=1

m=2 m=3

m=4

Fig. 8.2 Angular distribution function UðuÞ ¼ A cosðmuÞ at m ¼ 0; 1; 2; 3; 4 behavior and designations

picture in Fig. 8.2. There is one variation of the field when “m” equals to one. There are two, three, and four variations where “m” equals 2; 3; 4, respectively. The value of “m” may be arbitrary. The next differential equation from Eq. (8.2.28) corresponds to a radial distribution of the field and can be written as follows:   r d dRðr Þ r þ k2? r 2 ¼ m2 ; Rðr Þ dr dr

ð8:2:34Þ

  d2 Rðr Þ 1 dRðr Þ m2 2 þ k þ  Rðr Þ ¼ 0: ? dr 2 r dr r2

ð8:2:35Þ

Equation (8.2.35) is Bessel’s differential equation. The solution of Bessel’s equation is: Rðr Þ ¼ B1 J m ðk? r Þ þ B2 Y m ðk? r Þ;

ð8:2:36Þ

where J m ðk? r Þ is the Bessel function of the first kind of the order m, Y m ðk? r Þ is the Bessel function of the second kind of the order m which is known as the Neumann function (Ym(z)–>∞ at the function argument z–>0), k? is given in Eq. (8.2.22). By frequency conditions, we mean the cutoff frequencies which are unique for every particular waveguide mode that is supposed to be propagating in a waveguide of a given radius. The EM field of mode propagating inside the waveguide must be finite at all r\a including r ¼ 0. When B2 6¼ 0, then there is a contradiction (disagreement) because the left side of Eq. (8.2.36) must have a finite value at r ¼ 0 while the right side of Eq. (8.2.36) is drawn to infinity at r ¼ 0. We come to the conclusion that the coefficient B2 ¼ 0 and the solution of Eq. (8.2.35) is determined by the Bessel function of the first kind. The second term Y m ðk? r Þ in Eq. (8.2.36) would need to be taken into account only in cases where point r ¼ 0 in the waveguide is not included in the scope of the consideration, for example, for the layered waveguide as a coaxial cable or open (without conductor screen) dielectric waveguide for layers that do not contain point r ¼ 0.

552

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

So the solution for the circular cylindrical waveguide (Fig. 8.1) is: Rðr Þ ¼ Rm ðr Þ ¼ B J m ðk ? r Þ:

ð8:2:37Þ

The solution of Helmholtz’s equation (8.2.26) after using Eqs. (8.2.27), (8.2.33), and (8.2.37) can be written as follows: X X H z;m ðr; uÞ ¼ Rm ðr ÞUm ðuÞ ¼ Am Bm J m ðk? r Þ cosðmuÞ; ð8:2:38Þ m

m

and since the coefficients A and B have arbitrary values, we denote H 0 ¼ A B and the solution of Eq. (8.2.26) for TE modes as: H z;m ðr; uÞ ¼ H 0;m J m ðk? r Þ cosðmuÞ:

ð8:2:39Þ

Analogically, the solution of Helmholtz’s equation (8.2.25) for TM modes is: E z;m ðr; uÞ ¼ E0;m J m ðk? r Þ cosðmuÞ:

ð8:2:40Þ

The value of k? is determined from the dispersion equation which composites of boundary conditions separately (in different ways) for TE and TM modes. In order to build solutions, we will consider properties of Bessel functions.

8.3

Bessel Functions and Their Properties

We will investigate only the Bessel function J m ðk ? r Þ of the first kind of the order m since we are interested in the solutions given in Eqs. (8.2.39) and (8.2.40). Figure 8.3 presents the Bessel functions of the first kind of the order m with the argument from 0 to 10. The function J m ðk? r Þ is deferred on the ordinate axis, and the argument of the function k? r is on the abscissa axis.

Fig. 8.3 Bessel functions J m ðk ? r Þ of the first kind of the order m when m ¼ 0(red color), ¼ 1 (green), ¼ 2 (blue), ¼ 3 (black), ¼ 4 (dashed red), ¼ 5 (dashed green)

Jm(k r) 1 0.8 0.6

J0(k r) J1(k r) J2(k r)

0.4

J3(k r)

J4(k r)

6

8

J5(k r)

0.2

-0.2 -0.4

2

4

10

kr

8.3 Bessel Functions and Their Properties

553

The Bessel functions are known as the cylindrical harmonics. In mathematics, cylindrical harmonics are a set of linearly independent solutions to Laplace’s differential equation expressed in cylindrical coordinates. Bessel functions of the first kind are solutions of Bessel’s differential Eq. (8.2.39), and they are finite at the origin k ? r ¼ 0 of the abscissa axis for an integer positive index m. It is possible to define the Bessel functions by the following power series:  m þ 2s 1 X ð1Þs k? r J m ðk ? r Þ ¼ ; s!ðm þ sÞ! 2 s¼0

ð8:3:1Þ

where the symbol “ ! ” means the factorial function. The factorial of a nonnegative integer n is the product of all positive integer less than or equal to n Q n! ¼ j ¼ 1  2  3    ðn  1Þ  n. One must keep in mind that value of 0! is 1. As j¼1

it follows from Eq. (8.3.1), functions J m ð0Þ ¼ 0 for all indexes m except index m ¼ 0, for zeroth order, J 0 ð0Þ ¼ 1. Cylindrical functions J m ðk ? r Þ in the cylindrical coordinate system play the same role as sine and cosine functions in the Cartesian (rectangular) coordinate system. We see from Fig. 8.3 that there are common features between Bessel functions and the harmonic trigonometric functions. There are also differences between the Bessel functions (Fig. 8.3) and trigonometric ones such as sine and cosine: 1. Bessel functions are not periodic functions. 2. Amplitude of Bessel functions is not a constant and decreases with the growth of a function argument k? r. 3. The higher the index m, the stronger the shift of cylindrical functions on the abscissa k? r axis (Fig. 8.3). 4. When k? r becomes very large, J m ðk? r Þ approaches a cosinusoidal form because pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at J m ðk? r Þ ¼ 2=pðk? r Þ cosðk? r  mp=2  p=4Þ þ O 1=ðk ? r Þ3=2 ðk? r Þ ! 1. In the theory of cylindrical functions, we know the important recurrence relations for the Bessel functions of the first kind, namely: J m1 ðk ? r Þ þ J m þ 1 ðk? r Þ ¼ J 0m ðk ? r Þ ¼ J m1 ðk? r Þ 

2m J m ðk ? r Þ; ðk ? r Þ

ð8:3:2Þ

m m J m ðk ? r Þ ¼ J m ðk? r Þ  J m þ 1 ðk? r Þ; ð8:3:3Þ ðk ? r Þ ðk ? r Þ

@ ððk? r Þm J m ðk ? r ÞÞ ¼ ðk? r Þm J m1 ðk? r Þ; @ ðk ? r Þ

ð8:3:4Þ

554

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

@ ððk ? r Þm J m ðk? r ÞÞ ¼ ðk? r Þm J m þ 1 ðk ? r Þ; @ ðk ? r Þ

ð8:3:5Þ

m J m ðk ? r Þ  J m þ 1 ðk? r Þ; ðk ? r Þ

ð8:3:6Þ

1 J 0m ðk? r Þ ¼ ðJ m1 ðk? r Þ  J m þ 1 ðk? r ÞÞ; 2

ð8:3:7Þ

J m ðk ? r Þ ¼ ð1Þm J m ðk? r Þ;

ð8:3:8Þ

J 0m ðk ? r Þ ¼

@J 0 ðk? r Þ ¼ J 1 ðk ? r Þ; @ ðk ? r Þ

ð8:3:9Þ

ððk ? r ÞJ 1 ðk? r ÞÞ0 ¼ ðk ? r ÞJ 0 ðk? r Þ:

ð8:3:10Þ

J 00 ðk? r Þ ¼

For our research of the solutions, the greatest interest is drawn to the argument values k? r at which the Bessel functions J m ðk ? r Þ or their derivatives J 0m ðk ? r Þ ¼ @J m ðk? r Þ=@ ðk ? r Þ become equal to zero. The designations for these two cases are: 1. mmn is the root of the Bessel function when J m ðk ? r Þ ¼ 0 at the specified m and n, i.e. the Bessel function argument k ? r ¼ k ?mn r ¼ mmn at which the Bessel function is equal to zero. Below, we will see that these roots give the solutions for the transverse magnetic TMmn modes; 2. m0mn is the root of the derivative of the Bessel function J 0m ðk ? r Þ ¼ @ ðJ m ðk? r ÞÞ=@ ðk ? r Þ ¼0 at the specified m and n, i.e. the Bessel function argument k? r ¼ k?mn r ¼ m0mn at which the derivative of the Bessel function is equal to zero. The prime symbol at value m0mn is a notation (not a derivative of it, just resembles a connection with J 0m ). Below, we will see that these roots give the solutions for transverse electric TEmn modes.

The physical meanings of indexes m and n are: The first index m means the number of half-wave field variations on the angular coordinate u, and the second index n means the number of half-wave field variations on the radial coordinate r. There are infinitely many zeros of Bessel function J m ðk? r Þ as well as many zeros of the derivative J 0m ðk ? r Þ. The first few roots mmn ¼ k? r of the Bessel function when J m ðk? r Þ ¼ 0 are tabulated in Table 8.1 for nonnegative integer values of m and n [see also Fig. 8.3 for the Bessel functions J m ðk? r Þ]. The first roots m0mn ¼ k? r of the derivative of the Bessel function J 0m ðk ? r Þ are given in Table 8.2 for nonnegative integer values of m and n. Analyzing the curve of function J 0 ðk ? r Þ in Fig. 8.3, we see that the curve of the corresponding function crosses the abscissa axis for the first time at a point

8.3 Bessel Functions and Their Properties

555

Table 8.1 Zeros of Bessel functions J m ðk ? rÞ n

1 2 3 4 5

Roots mmn of Bessel function (corresponds to the TMmn modes) m ¼ 1, m ¼ 2, m ¼ 3, m ¼ 4, m ¼ 0, J 1 ðk ? r Þ J 2 ðk ? rÞ J 3 ðk ? r Þ J 4 ðk ? r Þ J 0 ðk ? r Þ

m ¼ 5, J 5 ðk ? r Þ

m01 = m02 = m03 = m04 = m05 =

m51 = m52 = m53 = m54 = m55 =

m11 = m12 = m13 = m14 = m15 =

2.405 5.520 8.654 11.792 14.931

m21 = m22 = m23 = m24 = m25 =

3.832 7.016 10.174 13.324 16.471

m31 = m32 = m33 = m34 = m35 =

5.136 8.417 11.620 14.796 17.960

6.380 9.761 13.015 16.224 19.409

m41 = m42 = m43 = m44 = m45 =

7.588 11.065 14.373 17.616 20.827

8.772 12.339 15.700 18.980 22.218

approximately equal to 2.405 (see Table 8.1). In accordance with the agreement, this point is denoted as m01 which is the first root of the Bessel function. The following roots of function J 0 ðk? r Þ in increasing order are in points of the x-axis m02 ¼ 5:520, m03 ¼ 8:654, m04 ¼ 11:792, m05 ¼ 14:931, and so on. Figure 8.3 only shows the Bessel function with arguments k? r  10. If we compare the values of the roots of Bessel functions from Table 8.1 with values of its derivatives from Table 8.2, we can note in Table 8.3 that there is true equality: m1n ¼ m00n . We see in Fig. 8.4 that the root m001 ¼ 3:832 of transverse electric TE01 mode coincides with the root m11 ¼ 3:832 of transverse electric TM11 mode. Roots m0mn correspond to the Bessel function extremum, i.e. the maximum or minimum value of the function. The first positive maximum of the Bessel functions belongs to J 1 ðk? r Þ and corresponds to the value of the Bessel function argument m011 ¼ 1:841 (Fig. 8.4). We also see that the first root of Bessel function J 1 ðk? r Þ Table 8.2 Zeros of derivatives of Bessel functions J 0m ðk ? r Þ n

Roots m0mn of derivatives of Bessel function (corresponds to the TEmn modes) m ¼ 0, m ¼ 1, m ¼ 2; m ¼ 3, m ¼ 4, m ¼ 5, J 00 ðk? r Þ J 01 ðk ? rÞ J 02 ðk ? rÞ J 03 ðk ? rÞ J 04 ðk ? r Þ J 05 ðk ? r Þ

1 2 3 4 5

m001 = m002 = m003 = m004 = m005 =

m011 = m012 = m013 = m014 = m015 =

3.832 7.016 10.174 13.324 16.471

m021 = m022 = m023 = m024 = m025 =

1.841 5.331 8.536 11.706 14.864

m031 = m032 = m033 = m034 = m035 =

3.054 6.706 9.970 13.170 16.348

4.201 8.015 11.346 14.586 17.789

m041 = m042 = m043 = m044 = m045 =

5.318 9.282 12.682 15.964 19.196

m051 = m052 = m053 = m054 = m055 =

6.416 10.520 13.987 17.313 20.576

Table 8.3 Resemblance of roots corresponding to J 1 ðk ? rÞ and J 00 ðk ? r Þ n

1

Roots m1n correspond m = 1, J 1 ðk ? r Þ m11 Roots m00n correspond m = 0, J 00 ðk ? r Þ m001

2 to the TMmn modes ¼ 3:832 m12 ¼ 7:016 to the TEmn modes ¼ 3:832 m002 ¼ 7:016

3

4

5

m13 ¼ 10:174

m14 ¼ 13:324

m15 ¼ 16:471

m003 ¼ 10:174

m004 ¼ 13:324

m005 ¼ 16:471

556

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators Jm(k r)

Fig. 8.4 Bessel functions J 0 ðk ? r Þ, J 1 ðk ? r Þ, J 2 ðk ? r Þ of the first kind

1 0.5

J0(k r)

J2(k r)

O -0.5

TE 11

J1(k r)

1

ν'

TE 21

TE 01

TM 01

2

4

11=1.841

ν

01=2.405

ν'

TM 21

TM 11 3

ν'

01=3.832

21=3.054

ν

kr

5

ν

21=5.1

11=3.832

coincides with the first root of the derivative of Bessel function J 00 ðk? r Þ and m11 ¼ m001 ¼ 3:832. It should be noted that the maximum of Bessel functions J 0 ðk? r Þ ¼ 1 at k? r ¼ 0 is not a root.

8.4

Solutions for TEmn Modes in Circular Waveguides

We need to base ourselves on Helmholtz’s differential equation for the component H z when researching the TE modes (waves). We can write Helmholtz’s equation (8.2.26) in the following form: r2ru H z þ k2? H z =

@ 2 H z 1 @H z 1 @2Hz þ þ + k2? H z ¼ 0; r @r r 2 @u2 @r 2

ð8:4:1Þ

where k? is given in Eq. (8.2.22). We will search a solution of Eq. (8.4.1) while taking into account that the longitudinal component for TE modes E z ¼ 0 is in this form: H z ðr; u; zÞ ¼ H z ðr; uÞeihz :

ð8:4:2Þ

The set of TE mode components from Eqs. (8.2.9–8.2.12) can be expressed as: Er ðr; u; zÞ ¼  Eu ðr; u; zÞ ¼

ixl0 lr @H z ðr; uÞ ihz e ; @u r k2?

ixl0 lr @H z ðr; uÞ ihz e ; @r k 2?

E z ðr; u; zÞ ¼ 0;

ð8:4:3Þ ð8:4:4Þ ð8:4:5Þ

8.4 Solutions for TEmn Modes in Circular Waveguides

ih @H z ðr; uÞ ihz e ; @r k 2?

ð8:4:6Þ

ih @H z ðr; uÞ ihz e : @u r k2?

ð8:4:7Þ

H r ðr; u; zÞ ¼  H u ðr; u; zÞ ¼ 

557

The differential Eq. (8.4.1) is solved by the method of separation of variables (also known as the Fourier method) which is described in Sect. 8.2. We use the solution of Helmholtz’s equation given in Eq. (8.2.39), and the phasor field expression is presented in the following form:   H z;mn ðr; uÞ ¼ H 0;mn J m k ?;mn r cosðmuÞeihmn z :

ð8:4:8Þ

After the insertion of Eq. (8.4.8) into Eqs. (8.4.3)–(8.4.7), we get expressions of other components for TE modes: E r;mn ðr; u; zÞ ¼

ixl0 lr m H 0;mn J m ðk?mn r Þ sinðmuÞeihmn z ; r k 2?mn

ð8:4:9Þ

E u;mn ðr; u; zÞ ¼

ixl0 lr H 0;mn J 0m ðk ?mn r Þ cosðmuÞeihmn z ; k?mn

ð8:4:10Þ

E z;mn ðr; u; z; tÞ ¼ 0; H r;mn ðr; u; zÞ ¼  H u;mn ðr; u; zÞ ¼

ihmn H 0;mn J 0m ðk ?mn r Þ cosðmuÞeihmn z ; k ?mn

ihmn m H 0;mn J m ðk?mn r Þ sinðmuÞeihmn z : r k2?mn

ð8:4:11Þ ð8:4:12Þ ð8:4:13Þ

The electric field of the TE mode types has only two transverse components E r , E u , and only one component E u is tangent to the metal walls of the circular cylindrical waveguide. The tangential component from Eq. (8.4.4) is proportional to the derivative of H z ðr; uÞ with respect to the independent variable “r”: Eu 

@H z ðr; uÞ : @r

ð8:4:14Þ

Then, the boundary condition can be written as follows: @H z ðr; uÞ jr¼a ¼ 0: @r

ð8:4:15Þ

The tangential components of the magnetic field will automatically satisfy at the border r ¼ a when the condition (8.4.15) is satisfied in the problem under analysis.

558

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

The study of TE mode propagation in a circular metallic waveguide is reduced to the solution of the homogeneous Neumann boundary value problem (Sect. 4.7) for Helmholtz’s equation: @ 2 H z 1 @H z 1 @2Hz þ þ + k 2? H z ¼ 0, r @r r 2 @u2 @r 2 at

@H z ðr; uÞ jr¼a ¼ 0: @r

ð8:4:16Þ ð8:4:17Þ

Usually, when solving Eq. (8.4.16) the boundary condition Eu jr¼a ¼ 0 is used. In order to calculate the unknown transverse propagation constant k⊥ , we have to investigate the partial derivative of function Eq. (8.2.38): @H z ðr; uÞ @J m ðk ? r Þ ¼ H 0 k? cosðmuÞ: @r @k? r

ð8:4:18Þ

The boundary conditions will be satisfied when the following condition is satisfied: @J m ðk? r Þ ¼ 0jr¼a : @k? r

ð8:4:19Þ

Equality (8.4.19) is an equation regarding the argument k? r of the Bessel function. The equality has an infinite number of roots which are identified by us as m0mn (see Table 8.2). The considered boundary value problem describing the wave propagations has an infinite number of nontrivial solutions for each of the solutions: k ?mn a ¼ m0mn ;

ð8:4:20Þ

where indexes m ¼ 0; 1; 2; . . .; 1 and n ¼ 1; 2; 3; . . .; 1. We can express the transverse propagation constant of the circular waveguide as follows: k ?mn ¼ m0mn =a:

ð8:4:21Þ

The longitudinal magnetic field component of TE mode can be finally expressed as: H z;mn ðr; u; zÞ ¼ H 0;mn J m

 0  mmn r cosðmuÞeihmn z : a

ð8:4:22Þ

8.4 Solutions for TEmn Modes in Circular Waveguides

559

We have expressions of other components for TE modes:  0  ixl0 lr m a2 m r Er;mn ðr; u; zÞ ¼  2 H 0;mn J m mn sinðmuÞeihmn z ; 0 a r mmn

ð8:4:23Þ

 0  ixl0 lr a m r 0 E u;mn ðr; u; zÞ ¼ H 0;mn J m mn cosðmuÞeihmn z ; m0mn a

ð8:4:24Þ

Ez;mn ðr; u; zÞ ¼ 0;

ð8:4:25Þ

 0  ihmn a 0 m mn r H J cosðmuÞeihmn z ; 0;mn m m0mn a

ð8:4:26Þ

 0  ihmn m a2 m r H u;mn ðr; u; zÞ ¼  2 H 0;mn J m mn sinðmuÞeihmn z : 0 a r mmn

ð8:4:27Þ

H r;mn ðr; u; zÞ ¼ 

The longitudinal propagation constant is: hmn

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k 2 r lr  k2?mn ;

ð8:4:28Þ

where the wavenumber k ¼ x=c is given in Eq. (5.4.19). After taking account of Eq. (8.4.21), we can write as follows:

hmn

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  0 2ffi 2p m ¼  mn ; k a

ð8:4:29Þ

and here hmn is the longitudinal propagation constant for a particular TE mode with specific indexes “m” and “n”.

8.5

Instantaneous Field Expressions for TEmn Modes

The instantaneous field components of the TEmn ¼ ðH mn Þ modes are obtained by multiplication of the phasor field expressions (8.4.22)–(8.4.27) on term ei xt . We will get the multiplier eiðxthzÞ which upon using Euler’s formula (Eq. 1.4.7) is equal to cosðxt  hzÞ þ i sinðxt  hzÞ, and then, taking the real part of the product, we can write as follows:

560

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

H z;mn

 0  m r ¼ H 0;mn J m mn cosðmuÞ cosðxt  hmn zÞ: a

ð8:5:1Þ

We have for TE modes the following expressions of other components: E r;mn

 0  xl0 lr m a2 m r ¼   2 H 0;mn J m mn sinðmuÞ sinðxt  hmn zÞ; 0 a r mmn

Eu;mn ¼ 

H r;mn

 0  xl0 lr a 0 mmn r H J cosðmuÞ sinðxt  hmn zÞ; 0;mn m m0mn a

ð8:5:2Þ

ð8:5:3Þ

Ez;mn ¼ 0;

ð8:5:4Þ

 0  hmn a m r 0 ¼ 0 H 0;mn J m mn cosðmuÞ sinðxt  hmn zÞ; mmn a

ð8:5:5Þ

 0  hmn m a2 m r H u;mn ¼   2 H 0;mn J m mn sinðmuÞ sinðxt  hmn zÞ; a r m0mn

ð8:5:6Þ

where m ¼ 0; 1; 2; . . .; 1 and n ¼ 1; 2; 3; . . .; 1. In Eqs. (8.5.2), (8.5.3), (8.5.5), and (8.5.6), the transverse components of TEmn modes are given. Components depend on coordinates r; u; z and time t, i.e. E r;mn ¼ Er;mn ðr; u; z; tÞ, E u;mn ¼ Eu;mn ðr; u; z; tÞ. Term hmn is given in Eq. (8.4.29), and m0mn is in Table 8.2. The integer “m” and “n” designate the number of half-wave variations in u and r directions, respectively.

8.6

Explanation of TEmn Mode Indexes

The first number of the TEmn modes index (subscript) for the circular waveguide always represents the number of half-wave field variations in the u direction, and the second number represents the number of half-wave field variations in the r direction. The same index dependencies are for TMmn modes (Sect. 8.3). Figure 8.5 shows the formula (8.4.22) in simplified form and shows how the solution depends on indexes “m” and “n”. The solution of Helmholtz’s differential equation for the component H z is shown in Fig. 8.5.

dependence on the φ coordinate

Fig. 8.5 To the explanation of TEmn mode indexes

H z,mn = H 0 J m

ν 'm n r a

cos m φ e-ih m n z dependence on the r coordinate

8.6 Explanation of TEmn Mode Indexes

561

We see that the dependence on the u coordinate is quite intricate. The dependency on the index “m” is also contained in the multiplier eihz because h ¼ hmn . Figure 8.6 presents the distribution of the longitudinal component H z in the cross-section of the circular waveguide for TE01 , TE02 ,TE03 ,TE11 ,TE22 ,TE41 modes. We see that the radial distribution repeats the segment of Bessel function of the first kind of the order “m” till the n-th function extremum where n ¼ 1; 2; 3; . . .; 1. We see that the larger indexes “m” and “n”, the more complicated distribution of the longitudinal component H z . We want to remind you that the transverse components of TE modes are expressed through the longitudinal component H z . Functions UðuÞ and Rðr Þ shown in Fig. 8.6 are defined in Sect. 8.2.

TE 01, m=0, n=1

TE 02, m=0, n=2

R(r)

R(r)

φ(φ)

φ(φ)

J0(k r)

1

0 2

6

8

kr

0 2

6

TE 22

φ(φ)

R(r)

0

kr

8

J0(k r)

2

6

m

TE 41

φ(φ)

0 8

10

kr

φ(φ)

R(r)

m 1

kr

8

n=3

R(r)

J1(k r),m=1

n=1

m

1

n=2

TE 11

2

R(r)

J0(k r)

n=1

1

φ(φ)

m

m

1

TE 03, m=0, n=3

m

J2(k r),m=2

1

0 2

4

10

n=2

kr

J4(k r),m=4

0 2

4

6

10

kr

n=1

Fig. 8.6 Distribution of the longitudinal component H z on the u and r coordinates

562

8.7

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Expressions Corresponding to Transverse Electric Modes

The general expressions of the wavenumber and longitudinal propagation constant are the same for a rectangular waveguide (Sect. 7.5.3) and a circular cylindrical waveguide. The cutoff frequency of a certain mode from Eqs. (7.5.37), (8.4.21), and also Eq. (8.4.28) at hmn ¼ 0 can be written as follows: ðf c ÞTEmn ¼

c k?mn m0mn pffiffiffiffiffiffiffiffi ¼ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p r lr 2pa 0 r l0 lr

ðHzÞ:

ð8:7:1Þ

The cutoff wavelength of a propagating mode in a waveguide, using Eq. (7.5.38), is: ðkc ÞTEmn ¼

2p 2pa ¼ 0 k?mn mmn

ðmÞ:

ð8:7:2Þ

The wavelength of a propagating mode at f [ f c , using Eq. (7.5.41), is: ðkw ÞTEmn ¼

2p k k ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2ffi hmn 1  ðf c ÞTEmn f 1  k ðkc ÞTEmn

ðmÞ: ð8:7:3Þ

The wavelength k of a plane EM wave in the unbounded insulator medium with constitutive parameters r ; lr can be expressed from Eq. (7.5.42) as: k¼

2p vl ¼ k f

ðmÞ;

ð8:7:4Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k is the wavenumber and vl ¼ 1= 0 r l0 lr is the speed of EM waves in an insulator medium from Eq. (7.5.44). The phase speed (scalar quantity) of modes, using Eq. (7.5.50), is:   vl vl vph ¼ vph mn ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . ffi 1  ðf c ÞTEmn f

2

1  k ðkc ÞTEmn

2

ðm=sÞ: ð8:7:5Þ

The longitudinal propagation constant, using Eq. (7.5.40), is: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2ffi 2 h ¼ hmn ¼ k  ðk?mn Þ ¼ k 1  ðf c ÞTEmn f ¼ 1  k ðkc ÞTEmn  1  m ; ð8:7:6Þ where m ¼ 0; 1; 2; . . .; 1 and n ¼ 1; 2; 3; . . .; 1.

8.8 Dispersion Characteristics of TEmn Modes

8.8

563

Dispersion Characteristics of TEmn Modes

It is important to know dispersion characteristics of a circular waveguide as well as the cutoff frequency of the main mode and higher modes. In this chapter, we give our calculations placed in graphs and tables for the circular cylindrical waveguide with radius a ¼ 0:01ðmÞ. Table 8.4 presents cutoff frequency f c and wavelength kc of TEmn modes calculated by formulas (8.7.1) and (8.7.2). The mode only starts to propagate from its cutoff frequency f c . We see in Fig. 8.7 and Table 8.4 that the mode TE11 is the main one because this mode has the lowest cutoff frequency f c ¼ 8:79 GHz. Figure 8.7 presents dispersion characteristics of TE modes. Table 8.4 shows tabulated values cutoff frequency ðf c ÞTEmn and wavelength ðkc ÞTEmn which is given in millimeters (SI unit symbol mm) for the first ten modes: TE11 , TE21 , TE01 , TE31 , TE41 , TE12 , TE51 , TE22 , TE02 , TE61 in the frequency range till 40 GHz. Cutoff frequency corresponds to the longitudinal propagation constant h ¼ 0.

8.9

Main Mode TE11 of Cylindrical Waveguides

The main mode of the circular cylindrical metallic waveguide is TE11 mode. We will analyze TE11 mode in more detail because many microwave devices operate on this mode, and this is an important mode. TE11 mode is called in technical literature as the main mode (also known as the dominant or fundamental or principal). The mode TE11 has the lowest cutoff frequency compared to other modes of the circular cylindrical waveguide. The modes with the cutoff frequencies higher than f c;TE11 are named as the higher modes; see Fig. 8.7 and Table 8.4. The set of instantaneous field components of TE11 mode after using Eqs. (8.5.1)–(8.5.6) at indexes m ¼ 1 and n ¼ 1 are the following: Table 8.4 Cutoff frequency ðf c ÞTEmn and wavelength ðkc ÞTEmn of TEmn modes Mode type TEmn

m

n

ðf c ÞTEmn GHz

ðkc ÞTEmn mm

Main mode, TE11 TE21 TE01 TE31 TE41 TE12 TE51 TE22 TE02 TE61

1 2 0 3 4 1 5 2 0 6

1 1 1 1 1 2 1 2 2 1

8.79 14.58 18.29 20.05 25.39 25.45 30.63 32.01 33.49 35.81

34.15 20.58 16.40 14.96 11.82 11.79 9.79 9.37 8.96 8.38

564

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Fig. 8.7 Dispersion characteristics of the main TE11 and higher modes of the circular cylindrical waveguide

 0  xl0 lr a2 m r E r;TE11 ðr; u; z; tÞ ¼   2 H 0;11 J 1 11 sinðuÞ sinðxt  h11 zÞ; 0 a r m11

ð8:9:1Þ

 0  xl0 lr a 0 m11 r E u;TE11 ðr; u; z; tÞ ¼  H 0;11 J 1 cosðuÞ sinðxt  h11 zÞ; m011 a

ð8:9:2Þ

Ez;TE11 ðr; u; z; tÞ ¼ 0; Hr;TE11 ðr; u; z; tÞ ¼

 0  h11 a 0 m11 r H J cosðuÞ sinðxt  h11 zÞ; 0;11 1 m011 a

 0  h11 a2 m r H u;TE11 ðr; u; z; tÞ ¼   2 H 0;11 J 1 11 sinðuÞ sinðxt  h11 zÞ; 0 a r m11  H z;TE11 ðr; u; z; tÞ ¼ H 0;11 J 1

 m011 r cosðuÞ cosðxt  h11 zÞ: a

ð8:9:3Þ ð8:9:4Þ ð8:9:5Þ

ð8:9:6Þ

We see from Eqs. (8.9.1)–(8.9.6) that there is only the Bessel function J 1 ðk? r Þ and its derivative in the determination of mode TE11 components. The roots of this function are shown in Fig. 8.4. The distribution of component H z variations of the main mode TE11 along the transverse coordinates r and u is given in the bottom left corner of Fig. 8.6. Because the root m011 of the Bessel function which corresponds to the TE11 mode is equal to 1.841 (Table 8.2), we can write as follows:

8.9 Main Mode TE11 of Cylindrical Waveguides

ðkc ÞTE11 ¼

2p 2pa 6:28 a ¼ 3:41 a ¼ 0 ¼ k?11 m11 1:841

565

ðmÞ;

ð8:9:7Þ

where “a” is the waveguide radius (Fig. 8.1a).

8.10

Electromagnetic Field Distributions of TE11 Mode

The electric field magnitude (Sect. 1.2.6) in a chosen arbitrary point Mn (Fig. 8.1b) is calculated in scalar form by the following expression: E¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2r þ E 2u þ E 2z :

ð8:10:1Þ

We can find the direction of the vector E in any point of waveguide: ^ Eu þ ^z Ez ; E ¼ ^r E r þ u

ð8:10:2Þ

and the magnetic field magnitude and the direction of vector H in any point of waveguide are calculated in scalar and vector forms by the following expressions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2r þ H 2u þ H 2z ;

ð8:10:3Þ

^ H u þ ^zHz : H ¼ ^rH r þ u

ð8:10:4Þ



Calculations of the electric E and magnetic H field components are performed on the base of Eqs. (8.9.1)–(8.9.6). Computing and visual pictures are made by using homemade computer programs in MATLAB. The waveguide radius is taken as a ¼ 0:01 m in calculations. The cutoff frequency is ðf c ÞTE11 ¼ 8:79 GHz and wavelength ðkc ÞTE11 ¼ 0:034 m. Figure 8.8 presents planes for which the electric and magnetic field distributions are calculated. Cutting plane lines show a plane exposed by cutting and removing an imaginary section of the object. The exposed plane is called the sectional view, and the line used to cut the object is referred to as the cutting plane line. We agree with this book that for the sections “A–A” and “B–B” (Fig. 8.8) the removing imaginary section is the upper part of the waveguide. The arrows that are next to the cutting plane and point downward indicate the plane which remains for our consideration. We also have to consider drawing of the bottom part of the waveguide from the cutting plane “A–A” because arrows next to the cutting plane point downward (and we remove the upper part of the waveguide). If the arrows next to the cut plane are pointing upward, we should consider the cross-section of the upper part of the waveguide and remove the bottom part of the waveguide.

566

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Fig. 8.8 Planes “1z  1z”, “A  A” and “B  B” in a circular cylindrical waveguide

A-A φ

1z

r

1z

z Mn(r,φ)

a

1z - 1z z

A

A

B

B r

O 2a

The direction of the arrows for the section “B–B” also corresponds to the bottom part of the waveguide from the cutting plane “B–B” because the arrows point downward (see Fig. 8.8). Figures 8.9 and 8.10 present the electric E (red color) and magnetic H(blue) field distributions in planes “1z  1z”, “A  A”, and “B  B” of Fig. 8.8. The fields are presented diagrammatically as a set of lines with arrows on a certain plane, and they are called the electric or magnetic field lines (Sects. 2.4 and 3.16). Figure 8.9 shows the longitudinal section of the circular waveguide. Here we see the electric field in the form of the red points and crosses because the electric field is perpendicular to the plane “1z  1z”. The electric and magnetic field direction changes at every kw =2 distance. We see that E is stronger in the central part of the waveguide. The magnetic field vectors are located in the plane 1z–1z. Magnetic field vectors are the longest where the H field is stronger. We see that H vectors have the largest values near the waveguide walls. The distance between the maximums of electric and magnetic fields is kw =4. Figure 8.10a,b shows the distribution of EM field in two cross-sections “A  A” and “B  B”. The first cross-section “A  A” is away from the origin “O” on the distance equal to kw =2 along the z-axis (see Figs. 8.8 and 8.9). The second cross-section “B  B” is away from the origin “O” on the distance equal to kw =4. Figure 8.10 a presents the cross-section “A  A” where we only see blue points and crosses of magnetic field line because the H-field is perpendicular to the cross-section. In cross-sections “B  B” (Fig. 8.10b), we see electric field lines (red) go in the radial directions and the magnetic field lines (blue) form concentric circles. The field arrows that protrude beyond the waveguides indicate that the

Electromagnetic Field Distributions of TE11 Mode

8.10

r

B

567

A

a

z

O

B

A

λw/2

λw/2

Fig. 8.9 Electric (red color) and magnetic (blue) field vectors of TE11 mode in the points of a grid in the plane “1z  1z”

(a)

(b) a

a

1z

z

1z 1z

z

1z

Fig. 8.10 Electric (red color) and magnetic (blue) field vectors of TE11 mode in the points of a grid in planes a “A  A” at z ¼ kw =2 and b “B  B” at z ¼ kw =4

points in which they correspond relate to the space inside the waveguides which is very close to the wall.

8.11

Schematic Sectional Views for TE11 Mode

Schematic view of distributions of the electric field E(red color), magnetic field H(blue), displacement Jd (green), surface conduction Js current (black), and surface

568

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

charges qs (red circles with plus or minus) for the main mode TE11 of the circular waveguide is presented in Fig. 8.11. The longitudinal section of waveguide is the view “1z  1z” (Fig. 8.8) demonstrated by the blue rings of the magnetic field and black lines of Js . The electric field and displacement current lines are normal to the plane “1z  1z”; for this reason, we see points and crosses of E and Jd . The right figures in Fig. 8.11 are two cross-sections of the waveguide “A  A” and “B  B”. Figure 8.11 shows the waveguide excitation by a probe, loop, and slot. The common principles of the waveguide excitations were given in the description of Sect. 7.9. In cross-sectional view “A  A” (the upper right picture), the displacement current lines (green) are presented. The magnetic field is submitted by crosses and points because the section “A  A” dissects the force lines of H-field. Notice: The direction of the arrows for the cutting plane “ A  A”in the longitudinal section “ 1z  1z”of the waveguide corresponds to the right part from the plane “ A  A” according to our agreement. So we see the right part of the waveguide in the plane “A  A”. In this cross-section, we see the slot and loop for the waveguide excitement. The section “A  A” gives the law of H z component distribution along r and u coordinates. There is one variation of the field along the u coordinate because of the index m = 1 (Fig. 8.6). The distribution of the component H z along the r coordinate repeats the curve features of the function J 1 ðk? r Þ from the coordinate origin to the first maximum of the function because of the index n = 1 (Fig. 8.4).

1z - 1z B

Probe

A

A-A 1z

Loop

H

Js

a

E

Jd

Hz(φ) B

A

Slot

P=[EH]

Slot

λw/4

TE 11

Js

H

Jd

qs m=1, n=1

Hz (r)

B-B 1z

λw E

1z

Probe

E

1z

Fig. 8.11 Schematic view of the distributions of the electric E (red color), magnetic H (blue) fields, displacement Jd (green), surface conduction Js (black) currents, and charges qs (red circles with plus or minus) for the main mode TE11 in the longitudinal section “1z  1z” and two cross-sections “A  A” and “B  B” of the circular waveguide

8.11

Schematic Sectional Views for TE11 Mode

569

We see the Poynting vector P in Fig. 8.11 showing the direction of TE11 mode propagation. It is important to know the direction of EM mode propagation in order to know the position and direction of E-field lines in comparison with current Jd lines. The electric field E lags from the displacement current Jd (Fig. 6.23). In the bottom right picture (section “B  B”), the electric field lines (red) are presented. The direction of the arrows for the section “B  B” in the longitudinal section “1z  1z” of the waveguide corresponds to the right part from the cutting plane “B  B”. In the section “B  B”, we see surface charges qs (red circles with plus or minus) and a probe for the waveguide excitation. We have chosen the section “A  A” on the distance kw =4 in the direction of EM wave propagation, i.e. in the direction of the Poynting vector P ¼ ½EH, from the section “B  B”, and for this reason, the directions of Jd - and E- vector lines coincide.

8.12

Solutions of Helmholtz’s Equation for TEmn Modes

We need to base ourselves on Helmholtz’s equation for the component E z when researching the TM modes (waves). We can write Helmholtz’s equation (8.2.25) in the following form: r2ru E z þ k 2? E z =

@ 2 E z 1 @Ez 1 @ 2 Ez þ 2 þ + k 2? E z ¼ 0, 2 r @r r @u2 @r

ð8:12:1Þ

where k? is described in Eq. (8.2.22). We know that for TM mode H z ¼ 0, and we will search a solution of Eq. (8.12.1) in this form: E z ðr; u; zÞ ¼ Ez ðr; uÞeihz :

ð8:12:2Þ

We have from Eq. (8.2.9)–(8.2.12): Er ¼

ih @Ez ; k2? @r

ð8:12:3Þ

Eu ¼

ih @Ez ; r k2? @u

ð8:12:4Þ

ix0 r @E z ; r k 2? @u

ð8:12:5Þ

Hr ¼

570

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Hu ¼

ix0 r @E z : @r k 2?

ð8:12:6Þ

The EM problem Eq. (8.12.1) is solved by the method of separation of variables which is given in Sect. 8.2. We write the solution of Helmholtz’s equation for TM modes which is presented in Eq. (8.2.40) in the form: Ez;mn ðr; uÞ ¼ E0;mn J m ðk?mn r Þ cosðmuÞ:

ð8:12:7Þ

The phasor field (Sect. 4.4) is: E z;mn ðr; u; zÞ ¼ E 0;mn J m ðk?mn r Þ cosðmuÞeihmn z :

ð8:12:8Þ

For TM modes, after substituting Eq. (8.12.8) into Eqs. (8.12.3)–(8.12.6), we have the expressions of other components, namely: ihmn E 0;mn Jm0 ðk?mn r Þ cosðmuÞeihmn z ; k ?mn

ð8:12:9Þ

ihmn m E0;mn J m ðk?mn r Þ sinðmuÞeihmn z ; r k 2?mn

ð8:12:10Þ

ix0 r m E 0;mn J m ðk?mn r Þ sinðmuÞeihmn z ; r k2?mn

ð8:12:11Þ

ix0 r E0;mn J 0m ðk ?mn r Þ cosðmuÞeihmn z ; k ?mn

ð8:12:12Þ

Er;mn ðr; u; zÞ ¼  E u;mn ðr; u; zÞ ¼

H r;mn ðr; u; zÞ ¼ 

H u;mn ðr; u; zÞ ¼ 

H z;mn ðr; u; zÞ ¼ 0:

ð8:12:13Þ

The coefficient E 0;mn depends on the field strength of the excitation and can be different for different modes. We can take for simplification of calculations E0;mn ¼ 1 when we do not have information on the mode excitation. The eigenvalues of TM modes are determined on the basis of the boundary condition which indicates that the component E z;mn must disappear at the metal wall of waveguide when the coordinate r is equal to the waveguide radius “a”: Ez;mn ¼ 0jr¼a ;

ð8:12:14Þ

8.12

Solutions of Helmholtz’s Equation for TEmn Modes

571

and taking into account Eq. (8.12.7), we have: J m ðk ? r Þ ¼ 0:

ð8:12:15Þ

The last condition can be fulfilled only in case when the number belongs to an infinite discrete sequence and satisfies the ratio: k ?mn a ¼ mmn ;

ð8:12:16Þ

where mmn is the root of the Bessel function J m ðk? r Þ from Table 8.1, and the set of transverse propagation constants is: mmn : ð8:12:17Þ k?mn ¼ a The complex amplitude of the longitudinal component E z;mn for TMmn modes is: E z;mn ðr; u; zÞ ¼ E 0;mn Jm

m r mn cosðmuÞeihmn z : a

ð8:12:18Þ

We can write the other components from Eqs. (8.12.9)–(8.12.13) in the following form: E r;mn ðr; u; zÞ ¼  E u;mn ðr; u; zÞ ¼ H r;mn ðr; u; zÞ ¼ 

m r ihmn a mn E 0;mn J 0m cosðmuÞeihmn z ; mmn a

ihmn m a2 r ðmmn Þ2

E 0;mn J m

ix0 r m a2

H u;mn ðr; u; zÞ ¼ 

2

r ðmmn Þ

m

E0;mn J m

mn

a

r

sinðmuÞeihmn z ;

m r mn sinðmuÞeihmn z ; a

m r ix0 r a mn E 0;mn J 0m cosðmuÞeihmn z ; mmn a H z;mn ðr; u; zÞ ¼ 0:

ð8:12:19Þ ð8:12:20Þ ð8:12:21Þ ð8:12:22Þ ð8:12:23Þ

So Eqs. 8.12.18– (8.12.23) is a set of electric and magnetic field components for TMmn modes at m ¼ 0; 1; 2; . . .; 1 and n ¼ 1; 2; 3; . . .; 1. The longitudinal propagation constant hmn is given in Eq. (8.4.28).

8.13

Instantaneous Field Expressions for TMmn Modes

The instantaneous field components of the TMmn ¼ ðEmn Þ modes are obtained by multiplication of the phasor field expressions (8.12.18)–(8.12.22) on term ei xt .

572

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

We will get the multiplier eiðxthzÞ which, after using Euler’s formula (Eq. 1.4.7), is equal to cosðxt  hzÞ þ i sinðxt  hzÞ, and then, taking the real part of the product [see Eqs. (7.4.2 and 7.4.3)] we can write as follows: m r mn cosðmuÞ cosðxt  hmn zÞ; a m r hmn a mn E r;mn ðr; u; z; tÞ ¼ E 0;mn J 0m cosðmuÞ sinðxt  hmn zÞ; mmn a E z;mn ðr; u; z; tÞ ¼ E 0;mn J m

E u;mn ðr; u; z; tÞ ¼  H r;mn ðr; u; z; tÞ ¼

hmn m a2 r ðmmn Þ2

x0 r m a2

H u;mn ðr; u; z; tÞ ¼

r ðmmn Þ

2

ð8:13:2Þ

m r mn sinðmuÞ sinðxt  hmn zÞ; a

ð8:13:3Þ

m r mn sinðmuÞ sinðxt  hmn zÞ; a

ð8:13:4Þ

E 0;mn J m

E 0;mn J m

ð8:13:1Þ

m r x0 r a mn E 0;mn J 0m cosðmuÞ sinðxt  hmn zÞ; mmn a H z;mn ðr; u; z; tÞ ¼ 0;

ð8:13:5Þ ð8:13:6Þ

and here indexes can be in the ranges at m ¼ 0; 1; 2; . . .; 1 and n ¼ 1; 2; 3; . . .; 1. We can take the value of root mmn for a chosen TM mode with the certain indexes “m” and “n” from Table 8.1 and hmn is given in Eq. (8.4.28).

8.14

Explanation of TMmn Mode Indexes

The first index (subscript) “m” of any transverse magnetic mode for the circular waveguide always represents the number of half-wave field variations in the u direction, and the second index “n” represents the number of half-wave field variations in the r-direction. Figure 8.12 shows a simplified formula (8.12.18) and shows how the solution depends on indexes “m” and “n”. We see that the dependence on the u coordinate is quite intricate in the formula and “m” participates in this dependence several times. We see that the dependence on the u coordinate is quite intricate. In the dependence of H z ðr; u; zÞ on z-coordinate, “m” is also present through multiplier eihz because of h ¼ hmn . dependence on the φ coordinate

Fig. 8.12 To the explanation of TMmn mode indexes

E z, mn =E 0 J m

ν 'm n r a

cos mφ e -ih m n z dependence on the r coordinate

8.14

Explanation of TMmn Mode Indexes

TM 01, m=0, n=1

TM 02, m=0, n=2

TM 03, m=0, n=3

R(r)

R(r)

R(r)

φ(φ)

1

φ(φ)

J0(k r),m=0

1

0 2

6

J0(k r) ,m=0

2

6

φ(φ)

1

0

kr

8

n=1

0

kr

8

J0(k r) ,m=0

2

6

kr

8

n=3

n=2

TM 11, m=1, n=1

TM 22, m=2, n=2

TM 41, m=4, n=1

R(r)

R(r)

R(r)

φ(φ)

1

573

J1(k r),m=1

1

0 2

8

n=1

10

kr

φ(φ)

J2(k r) ,m=2

1

0 2

4

10

n=2

kr

φ(φ)

J4(k r) ,m=4

0 2

4

6

10

n=1

kr

Fig. 8.13 Schematic distribution of the longitudinal component E z on the u and r coordinates

Figure 8.13 shows the schematic distribution of component E z for modes TM01 ,TM02 ,TM03 ,TM11 ,TM22 TM41 . The top row of the figure shows how the dependence of Ez distribution varies from the value of second index “n” when the first index “m” is the same. The left and right bottom row pictures at “n = 1” with different “m” show how the field distribution depends on “m”.

8.15

Expressions Corresponding to Transverse Magnetic Modes

The cutoff frequency of certain mode TMmn from Eq. (7.5.37) is:

574

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

ðf c ÞTMmn ¼

c k?mn mmn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2pa 0 r l0 lr

ðHzÞ;

ð8:15:1Þ

where k?mn is given in Eq. (8.12.17). The cutoff wavelength of propagating mode in the waveguide when f [ f c from Eq. (7.5.38) is: ðkc ÞTMmn ¼

2p 2pa ¼ k ?mn mmn

ðmÞ:

ð8:15:2Þ

The wavelength of a propagating mode from Eq. (7.5.41): ðkw ÞTMmn ¼

2p k k ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . 2ffi hmn 1  ðf c ÞTMmn f 1  k ðkc ÞTMmn

ðmÞ; ð8:15:3Þ

where k is a wavelength of a plane EM wave in the unbounded insulator medium with constitutive parameters r ; lr . We can write from Eq. (7.5.42): k¼

2p 1 vl ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k f 0 r l0 lr f

ðmÞ;

ð8:15:4Þ

where k is the wavenumber in an insulator (dielectric) medium and the speed vl of EM wave in the medium is presented in Eq. (7.5.44). The phase speed (scalar quantity) of modes from Eq. (7.5.50) is: 

vph

 mn

vl vl ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . ffi 1  ðf c ÞTMmn f

2

1  k ðkc ÞTMmn

2

ðm=sÞ:

ð8:15:5Þ

The longitudinal propagation constant from (7.5.40) is: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi 2 2 h ¼ hmn ¼ k  k?mn ¼ k 1  ðfc ÞTMmn = f rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   ¼ 1  k=ðkc ÞTMmn m1 :

ð8:15:6Þ

Dependence of the longitudinal propagation constant on frequency h ¼ F ðf Þ is the dispersion characteristic of a waveguide.

8.16

8.16

Dispersion Characteristics and Cutoff Frequency of TMmn Modes

575

Dispersion Characteristics and Cutoff Frequency of TMmn Modes

It is important to know dispersion characteristics and the cutoff frequency of modes of the circular waveguide to create devices or use it for measurements. In this chapter, we give our calculations which are placed in graphs and tables for the waveguide, as before, with radius a ¼ 0:01m. Table 8.5 presents cutoff frequency f c and wavelength kc of TMmn modes calculated by formulas (8.15.1) and (8.15.2). The mode only starts to propagate from its cutoff frequency f c . Figure 8.14 presents dispersion characteristics of TM modes. Table 8.5 shows values cutoff frequency and wavelength for the first ten modes: TM01 , TM11 , TM21 , TM02 , TE31 , TM12 , TM41 , TM22 , TM03 , TM51 in the frequency range from *11 to 45 GHz. As we see in Fig. 8.14, each of all presented TE modes has a different cutoff frequency. Cutoff frequency corresponds to the longitudinal propagation constant hmn ¼ 0. Since we know that the next mode after the main transverse electric TE11 mode is the transverse magnetic TM01 mode, we can calculate values f cnt , Df , and df (Eqs. 7.6.46–7.6.48). The cutoff frequencies of modes TE11 and TM01 are f c jTE11 ¼ 8:79 GHz and f c jTM01 ¼ 11:48 GHz, respectively. We take the operating frequency range from f 1 jTE11  9 GHz to f 2 jTM01  11 GHz. The central frequency of operating range calculated by Eq. (7.6.46) is: f cnt ¼ f 1 þ2 f 2  9 þ2 11 ¼ 10 GHz, the operating frequency range Df of a microwave devices Eq. (7.6.47) which operates on the main mode is: Df ¼ f 2  f 1  11  9 ¼ 2 GHz, and the broadband width df of the waveguide calculated by Eq. (7.6.48) is: 2  100% ¼ 20%. df ¼ fDf  100% ¼ 10 cnt

Table 8.5 Cutoff frequency ðf c ÞTMmn and wavelength ðkc ÞTMmn of TMmn (=Emn ) Mode type TMmn

m

n

ðf c ÞTMmn GHz

ðkc ÞTMmn mm

TM01 TM11 TM21 TM02 TM31 TM12 TM41 TM22 TM03 TM51

0 1 2 0 3 1 4 2 0 5

1 1 1 2 1 2 1 2 3 1

11.48 18.29 24.52 26.36 30.46 33.49 36.23 40.19 41.32 41.88

26.14 16.40 12.24 11.38 9.85 8.96 8.28 7.46 7.26 7.16

576

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Fig. 8.14 Dispersion characteristics of the TMmn modes of circular cylindrical waveguide

Values f cnt , Df , and df are important for microwave devices created on the base of a circular waveguide. Compare the broadband width df for the circular waveguide with the rectangular waveguide (Sect. 7.6.3). We see that the broadband width df of circular metallic waveguide is about three times less in comparison with the rectangular metallic waveguide.

8.17

Joint Dispersion Characteristics of TEmn and TEmn Modes

It is known that an excitation of the main mode TE11 can be also accompanied by an excitation of higher modes of both types. We present joint dispersion curves in Fig. 8.15 to trace the sequence of modes TEmn and TMmn . In Fig. 8.15, there are 20 modes TE11 , TM01 , TE21 , etc. which go in the order of increasing cutoff frequency. We see that the modes TM11 and TE01 cutoff frequency values (Tables 8.4 and 8.5) and curves coincide.

8.18

Formulae for TM01 Mode

We see in Fig. 8.15 that TM01 mode is the first higher mode which is next after the main TE11 mode. It is known that at an excitation of the main mode TE11 in the beginning higher (parasitic) waves which have the similar EM field structure with

8.18

Formulae for TM01 Mode

577

Fig. 8.15 The sequence of occurrence of TEmn and TMmn modes in circular cylindrical waveguides

the main mode is excited. For this reason, we investigate the EM field structure of TM01 mode. We can write formulae on the base of Eqs. (8.13.1)–(8.13.6) for TM01 mode: m r 01 cosðxt  h01 zÞ; a m r h01 a 01 E r;TM01 ðr; u; z; tÞ ¼ E0;01 J 01 sinðxt  h01 zÞ; m01 a Ez;TM01 ðr; u; z; tÞ ¼ E 0;01 J 1

H u;TM01 ðr; u; z; tÞ ¼

ð8:18:1Þ ð8:18:2Þ

E u;TM01 ðr; u; z; tÞ ¼ 0;

ð8:18:3Þ

H r;TM01 ðr; u; z; tÞ ¼ 0;

ð8:18:4Þ

m r x0 r a 01 E0;01 J 01 sinðxt  h01 zÞ; m01 a H z;TM01 ðr; u; z; tÞ ¼ 0:

ð8:18:5Þ ð8:18:6Þ

We use these formulae for the calculation of TM01 mode components. Since of the root m01 of the Bessel function which corresponds to the TM01 mode is equal to 2.405 (Table 8.1) at r ¼ 1, lr ¼ 1, we can write as follows: ðkc ÞTM01 ¼

2p 2pa 6:28 a ¼ 2:61 a ¼ ¼ k ?01 m01 2:405

ðmÞ;

ð8:18:7Þ

578

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

ðf c ÞTM01 ¼

c c ðHzÞ; ¼ ðkc ÞTM01 2:61 a

ð8:18:8Þ

where c is the speed of light in a vacuum (Eq. 2.1.6), “a” is the waveguide radius (Fig. 8.1).

8.19

Calculation of TM01 Field Distributions

Figures 8.16 and 8.17 give calculations of the EM field for TM01 mode by the homemade computer program in MATLAB. The waveguide radius is a ¼ 0:01 m. The cutoff frequency is equal to ðf c ÞTM01 = 11.48 GHz (Table 8.5). Calculations are done according to formulas (8.18.1)–(8.18.6). The magnetic field has its maximum values close to the conductor wall of the waveguide in contrast to the electric field which has its maximum values in the center of the waveguide in certain places along the waveguide (Fig. 8.16). We see that magnetic and electric fields change their directions to the opposite ones at every kw =2 distance along the z-coordinate. The distance between maximum values of the electric and magnetic fields is kw =4. Figure 8.17 presents the EM field in two cross-sections. As the arrows for the section “A–A” and “B–B” in the waveguide longitudinal section “1z–1z” (Fig. 8.16) are directed to the left from the cutting planes, we see the left part of the waveguide in Fig. 8.17. r

B

A

a

z

O

B λw/4

A λw/2

λw/2

Fig. 8.16 Electric (red color) and magnetic (blue) field vectors of TM01 mode in the points of a grid in plane “1z  1z”

8.19

Calculation of TM01 Field Distributions

(a)

579

(b) a

a

1z

z

1z

1z

z

1z

Fig. 8.17 Electric (red color) and magnetic (blue) field vectors of TM01 mode in the points of a grid in the plane a “A  A” at z ¼ kw =2 and b “B  B” at z ¼ kw =4

In the cross-section “A–A” (Fig. 8.17a), we see the electric field vector crosses which are localized in the center. This means that in this cross-section the electric field is perpendicular (normal) to the plane of “A–A”. The very weak magnetic field forms the concentric circles which consist of faintly visible blue points close to the waveguide wall in Fig. 8.17a. The magnetic field vectors are located in the plane “A–A”, and they are perpendicular to E-field vectors, and the Poynting vector P is directed along the z-axis. In Fig. 8.17b, we can see the electric field vector lines (red) which are radially directed. The E vectors are perpendicular to the conductor wall because the conductor is given here as the perfect one. The magnetic field vectors (blue) form the closed concentric circles. The H vector lengths are larger in points which are closer to the waveguide wall and shorter in places closer to the waveguide center. It means that the magnetic field H is weaker in the waveguide center. The EM field of TE01 in the cross-section “B–B” concentrates closer to the conductor wall. The field arrows that protrude beyond the waveguides indicate that the points in which they correspond relate to the space inside the waveguides very close to the wall.

8.20

Schematic Sectional Views for TM01 Mode

Schematic view of distributions of the electric field E (red color), magnetic field H (blue), displacement current density Jd (green), surface conduction current density Js (black), and surface charges qs (red circles with plus or minus) for the first higher mode TM01 of the circular waveguide is presented in Fig. 8.18. Here we see the waveguide excitation by the probe, loop, and slot. The description of the waveguide excitation is given in Sect. 7.9.

580

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

B

Probe

qs

1z - 1z

P=[EH]

Slot

A-A 1z

A

Js

H

H

Ez(φ)

Js

Ez(r)

Js z

A B

qs

Loop

Slot

λw/4

λw/2

1z

Slot

B-B 1z

λw E

Js Jd

H

qs Loop

1z

Fig. 8.18 Schematic view of the distributions of the electric field E (red color), magnetic field H (blue), displacement Jd (green), surface conduction Js (black) currents, and charges (red circles with plus or minus) qs for the mode TM01 in the longitudinal section “1z  1z” and two cross-sections “A  A” and “B  B” of circular waveguide

The longitudinal section of the waveguide is the view “1z–1z” where there are blue crosses and points of the magnetic field whose vector lines are perpendicular to the section “1z–1z”. The vector E and Jd lines are located in the plane “1z–1z”. We see that that the displacement current advances the electric field at distance kw =4 along the positive z-axis direction (see Fig. 6.28). The right figures show two cross-sections of the waveguide “A–A” and “B–B”. Note that the direction of the arrows for the section “A–A” and “B–B” in the longitudinal section “1z–1z” corresponds to the right part of the waveguide from the cutting planes. In cross-sectional view “A–A” (the upper right picture), we see the displacement current Jd lines (green) which are situated in this plane. We also see here black lines and points of the surface conduction current Js which flows on the waveguide wall. The currents Jd and Js form together closed electrical circuits (contours). The electric field is submitted by crosses because the section “A–A” dissects the force lines of E. In the section “A–A”, the law of E z component distribution along r and u coordinates, i.e., E z ðuÞ and Ez ðr Þ, is presented (see Figs. 8.2–8.4; Table 8.1). In the lower right picture (cross-sectional view “B–B”), the electric field lines (red) are demonstrated which go out of surface charges qs (red circles with plus or

8.20

Schematic Sectional Views for TM01 Mode

581

minus). In the section “B–B”, we see the slot and loop for the waveguide excitement. We have chosen the section “A–A” at the distance kw =4 in the direction of EM wave propagation from the section “B–B”, and for this reason, the directions of Jd  and E-vector lines coincide. The configuration of vectors Jd and E along the zaxis is the same; only they are shifted relative to each other along z-axis of the distance kw =4 because Jd ¼ ixr 0 Eðr; u; zÞeixt (see Fig. 6.28).

8.21

Circular Cylindrical Cavity Resonators

The hollow circular cylindrical resonator can be made by placing conducting walls on both ends of a circular waveguide (Fig. 8.1); i.e., the cavity has both ends short-circuited. The microwaves bounce back and forth between the walls of the cavity. At the resonance frequencies of the cavity, microwaves reinforce the formation of standing waves in the cavity. The cavity resonator is a device that exhibits resonant behavior, i.e., oscillates at certain resonance frequencies with greater amplitudes than others. In electronics, microwave cavity resonators consisting of hollow metal boxes are used in microwave transmitters, receivers, band-pass filters, test equipments to control frequency, etc. We have noted in Sects. 6.11.3 and 6.12.3 that the vector Jd is ahead of the vector E in space by the quarter of wavelength kw =4 along the direction of the wave propagation in waveguides, i.e., in the direction of the Poynting vector. We have noted also that the vector Jd can be ahead of the vector E in time by the value of a quarter of a period T=4. The displacement current Jd is ahead in time on T=4 compared with the electric field E in the resonator, but space distributions of vector fields Jd and E have identical form. The magnetic field H is located around the current and, for standing waves, simultaneously around the electric field in the same time moment as T=8 (Fig. 8.19).

Fig. 8.19 Changing of values E, qs , H; Js ; Jd over time in hollow cylindrical resonators

d

E, q s

H,Jd,Js a

t0 0

t1 T/8

t2 T/4

t3 T/2

t

582

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

The magnetic field in a resonator mode (oscillation) is displaced on the z-axis in comparison with the magnetic field of a traveling mode in a circular cylindrical waveguide on quarter wavelengths in the waveguide. This feature ensures the satisfaction of the boundary conditions at the end resonator walls H n ¼ 0 and E t ¼ 0. The magnetic field, displacement current, and conduction current are shifted in time on t ¼ T=4 in comparison with the electric field. The time moments when we are going to analyze the circular cylindrical resonator are shown in Fig. 8.19, i.e., t0 ¼ 0, t1 ¼ T=8, t2 ¼ T=4, t3 ¼ T=2. The electric field is maximum, and the magnetic field is equal to zero in the time moment t0 . On the base of Eqs. (7.10.3) and (8.7.2), we can write the wavelength of the TE resonator modes (oscillations): 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kTEmnp ¼ r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :  2 . 2 2 0 2 m þ ð p=d Þ pa mn 1 ð kc Þ þ ðp=2d Þ

ð8:21:1Þ

TEmn

We can express the resonance frequency for the TE modes from (7.10.4): f TEmnp ¼

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c r lr c r lr  0  2 mmn pa þ ðp=d Þ2 : ¼ kmnp 2

ð8:21:2Þ

On the base of Eqs. (7.10.3) and (8.15.2), we can write the wavelength of the TM resonator modes (oscillations): 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kTMmnp ¼ r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; . 2 ðmmn =paÞ2 þ ðp=d Þ2 1 ð kc Þ þ ðp=2d Þ2

ð8:21:3Þ

TMmn

where the resonance frequency of the TM modes is: f TMmnp ¼

pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c r lr ðmmn =paÞ2 þ ðp=d Þ2 : 2

ð8:21:4Þ

We see that the resonance frequencies of the TE and TM resonator modes with the same indexes “m”, “n” are different. The term m0mn from Table 8.2 participates in Eqs. (8.21.1), (8.21.2) for TE modes and mmn from Table 8.1 participates in Eqs. (8.21.3) and (8.21.4) TM modes. The physical meanings of indexes m and n are given in Sect. 8.3, and the third index p in TEmnp and TMmnp modes means the number of half-wave field variations along the length d in the z-axis.

8.22

Expressions for Electromagnetic Field Components of TEmnp Modes

8.22

583

Expressions for Electromagnetic Field Components of TEmnp Modes

The common reasoning for a hollow circular cylindrical resonator is the same as for a hollow rectangular resonator. The total field in the resonator is a superposition of incident and reflected waves. The radial and azimuthal electric field components are the following: E r;TEmnp ¼ Er;mn þ E r

ref;mn

E u;TEmnp ¼ E u;mn þ Eu

¼0

ref;mn

at z ¼ 0 and z ¼ d;

ð8:22:1Þ

¼ 0 at z ¼ 0 and z ¼ d;

ð8:22:2Þ

where Er;mn is the electric field radial component of TE wave propagating in the positive z-axis direction , Er ref;mn is the electric field radial component of the reflected wave propagating in the negative z-axis direction, Eu is the electric field azimuthal component of TE wave, E u ref is the electric field azimuthal component of reflected wave, and d is the resonator length. Expressions of EM field components of TEmnp mode match with formulas (8.4.22)–(8.4.27) for a circular waveguide. The EM wave reflected from the resonator end wall and propagated in the negative direction of z-axis has these components: Er

Eu

ref;mn ðr; u; zÞ ¼

ixl0 lr m a2  2 H 0 r m0mn

ixl0 lr a H0 ref;mn ðr; u; zÞ ¼ m0mn Ez

0 ref;mn J m

ref;mn ðr; u; zÞ

ih a H r ref;mn ðr; u; zÞ ¼ 0 H 0 mmn Hu

 ref;mn J m

ref;mn ðr; u; zÞ ¼ H 0

 m0mn r cosðmuÞeihz ; a

ð8:22:4Þ

¼ 0;

ð8:22:5Þ ð8:22:6Þ

 0  mmn r sinðmuÞeihz ; ref;mn J m a

ð8:22:7Þ

 Hz

ð8:22:3Þ

 0  mmn r cosðmuÞeihz ; a

0 ref;mn J m

ih m a2 2 H 0 ref;mn ðr; u; zÞ ¼   r m0mn



 m0mn r sinðmuÞeihz ; a

ref;mn J m

 m0mn r cosðmuÞeihz : a

Substituting expressions for Er;mn from (8.4.23) and E r Eq. (8.22.1) at z ¼ 0, we have:

ref;mn

ð8:22:8Þ from (8.22.3) in

584

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

 0  ixl0 lr m a2 mmn r sinðmuÞðH 0;mn þ H 0  2 J m 0 a r mmn

ref;mn Þ

From Eq. (8.22.9) following that ðH 0;mn þ H 0 written: H0

ref;mn

¼ 0 at z ¼ 0:

ref;mn Þ

ð8:22:9Þ

¼ 0, this equality can be

¼ H 0;mn :

ð8:22:10Þ

The same boundary condition (8.22.1) at z ¼ d gives the expression as follows:  0  ixl0 lr m a2 mmn r sinðmuÞðH 0  2 J m a r m0mn

ref;mn e

ihz

þ H 0;mn eihz Þ ¼ 0 at z ¼ d; ð8:22:11Þ

and after substituting Eq. (8.22.10) in Eq. (8.22.11) we get the expression:  0  ixl0 lr m a2 mmn r H J sinðmuÞðeihz  eihz Þ ¼ 0  2 0;mn m 0 a r mmn

at z ¼ d;

ð8:22:12Þ

and bearing in mind Eqs. (1.4.15) and (1.4.6), we can write: 

 eihz  eihz ¼ 2i sinðhzÞ;



ð8:22:13Þ

 eihz þ eihz ¼ 2 cosðhzÞ;

ð8:22:14Þ

and after using Eq. (8.22.13), finally we get:  0  2xl0 lr m a2 mmn r sinðmuÞ sinðh d Þ ¼ 0  2 H 0;mn J m a r m0mn

at z ¼ d:

ð8:22:15Þ

We get out from Eq. (8.22.15): sinðhd Þ ¼ 0 and hd ¼ pp;

ð8:22:16Þ

and from the last expression, the propagation constant in z-direction is: h ¼ pp=d at p ¼ 1; 2; 3; . . .; 1:

ð8:22:17Þ

Index p shows the number of variations along the z-axis of a circular cylindrical resonator. The index p has to start from the unit because at p ¼ 0 all EM field components of resonator TEmnp mode disappear.

8.22

Expressions for Electromagnetic Field Components of TEmnp Modes

585

We will use Eqs. (8.4.22–8.4.27) and (8.22.3–8.22.8) to get the expressions for the EM field components of TEmnp mode. After taking account of Eqs. (8.22.13, 8.22.14, and 8.22.17), we finally receive expressions as follows:  0 

xl0 lr m a2 m r ¼  2 H 0;mnp J m mn sinðmuÞ eihz  eihz 0 a r mmn  0  pp 2xl0 lr m a2 mmn r z ; ¼ sinðmuÞ sin  2 H 0;mnp J m a d r m0mn

ð8:22:18Þ

 0 

ixl0 lr a mmn r 0 H J cosðmuÞ eihz  eihz 0;mnp m 0 mmn a  0  pp 2xl0 lr a m r z ; ¼ H 0;mnp Jm0 mn cosðmuÞ sin 0 mmn a d

ð8:22:19Þ

E z;TEmnp ¼ 0;

ð8:22:20Þ

 0 

ipp a mmn r 0 H J cosðmuÞ eihz þ eihz 0;mnp m 0 mmn d a  0  pp i2pp a m r z ; H 0;mnp J 0m mn cosðmuÞ cos ¼ 0 mmn d a d

ð8:22:21Þ

 0 

i h m a2 m r H u;TEmnp ¼  2 H 0;mnp J m mn sinðmuÞ eihz þ eihz a r m0mn   pp i2 pp m a2 m0 r z ; ¼  2 H 0;mnp J m mn sinðmuÞ cos a d d r m0mn

ð8:22:22Þ

Er;TEmnp

Eu;TEmnp ¼

H r;TEmnp ¼ 

H z;TEmnp

 0  pp m r z : ¼ i 2H 0;mnp J m mn cosðmuÞ sin a d

ð8:22:23Þ

In Eqs. (8.22.18)–(8.22.23), we see the EM field components of TEmnp modes which are dependent on the cylindrical coordinates r; u; z and independent from time, i.e. E r;TEmnp ¼ E r;TEmnp ðr; u; zÞ,E u;TEmnp ¼ E u;TEmnp ðr; u; zÞ (about vector phasors see Sect. 5.1). In order to calculate (determine) the electric and magnetic field components of the resonator mode we can take the value of root m0mn for a chosen mode with the certain indexes “m” and “n” from Table 8.2, and the term h is given in (8.22.17). The constant multiplier “2” in equations can be entered into the arbitrary amplitude coefficient, and it is possible for this amplitude coefficient in calculations (after its normalization) to get equal to 1.

586

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

8.23

Instantaneous Field Expressions for Resonator TEmnp Modes

We accepted here as in Sect. 5.1 that the instantaneous expression of the electric and magnetic field components which are the complex magnitudes for time-harmonic EM fields are:

Eðr; u; z; tÞ ¼ Re Eðr; u; zÞeixt ;

ð8:23:1Þ

Hðr; u; z; tÞ ¼ Re Hðr; u; zÞeixt ;

ð8:23:2Þ

where Eðr; u; zÞ is a 3D vector phasor that contains information about the magnitude and phase. We remember that Euler’s formula (Eq. 1.4.7) gives the simple ratio eixt ¼ cosðxtÞ þ i sinðxtÞ. We express the instantaneous field components of TEmnp from (8.22.18)–(8.22.23) while taking into account Eqs. (8.23.1) and (8.23.2). The components are as follows: E r;TEmnp

 0  pp xl0 lr m a2 m r z cosðxtÞ; ¼  2 H 0;mnp J m mn sinðmuÞ sin a d r m0mn

E u;TEmnp ¼

 0  pp xl0 lr a mmn r 0 z cosðxtÞ; H J cosðmuÞ sin 0;mnp m 0 mmn a d E z;TEmnp ¼ 0;

H r;TEmnp ¼ H u;TEmnp ¼ 

H z;TEmnp

 0  pp pp a mmn r 0 z sinðxtÞ; H J cos ð mu Þ cos 0;mnp m m0mn d a d

 0  pp ppm a2 mmn r z sinðxtÞ; H J sin ð mu Þ cos  2 0;mnp m a d d r m0mn  0  pp m r z sinðxtÞ: ¼ H 0;mnp J m mn cosðmuÞ sin a d

ð8:23:3Þ

ð8:23:4Þ ð8:23:5Þ ð8:23:6Þ ð8:23:7Þ

ð8:23:8Þ

These expressions we use for EM field calculation of TEmnp resonator modes. Here indexes can be in the ranges at m ¼ 0; 1; 2; . . .; 1, n ¼ 1; 2; 3; . . .; 1, p ¼ 1; 2; 3. . .; 1 where it is possible to get m ¼ 0 and n 6¼0; p 6¼ 0. Constant Multiplier “2” is entered into arbitrary coefficient H 0;mnp which (after its normalization) is possible to take equal to 1 in calculations.

8.24

8.24

Resonance Frequency of TEmnp Modes

587

Resonance Frequency of TEmnp Modes

Table 8.6 shows resonance frequency values f mnp ¼ f TEmnp calculated by Eq. (8.21.2) and wavelength kmnp ¼ kTEmnp calculated by Eq. (8.21.1) for oscillations with different indexes m, n, and p when the resonator radius is a ¼ 102 m and the resonator length is d ¼ 5  102 m. We see that the main mode of hollow circular cylindrical resonator is TE111 mode with the lowest cutoff frequency f 111 ¼ 9:28 GHz.

8.25

Electromagnetic Field Calculations of TE111 Mode

We see from Table 8.6 that the lowest cutoff frequency has TE111 ð¼ H 111 Þ resonator mode (oscillation), so this mode is the main one. For this reason, we present here our calculations of this oscillation at the moments t ¼ 0; T=8; T=4; T=2. Computing and visual pictures are carried out by using our homemade computer programs in MATLAB. In Figs. 8.20, 8.21, 8.22, 8.23, 8.24, 8.25, 8.26 and 8.20, we see computations of distributions of the electric E and magnetic H fields by Eqs. (8.23.3)–(8.23.8) when the resonator radius is a ¼ 102 m and the length is d ¼ 5  102 m. Figures 8.20 and 8.21 present the electric and magnetic field distributions at the time moment t ¼ 0 (see Fig. 8.19). We see that the electric field (red color) E concentrates in the central part of the resonator and its line arrows are perpendicular to the plane “1z–1z” and aimed toward us. The magnetic field is absent at t ¼ 0. The electric field is very weak in the cross-section “B–B” when z  0 (Fig. 8.8) at t ¼ 0, so for this reason we do not show this field distribution in this particular cross-section here. Figure 8.21 presents the EM field distribution in the cross-section “A–A” at z ¼ d=2. Table 8.6 Frequency and wavelength of TEmnp resonator mode

No.

m

n

p

TEmnp

f TEmnp (GHz)

kTEmnp (mm)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1 1 1 1 2 2 2 1 0 2

1 1 1 1 1 1 1 1 1 1

1 2 3 4 1 2 3 5 1 4

TE111 TE112 TE113 TE114 TE211 TE212 TE213 TE115 TE011 TE214

9.28 10.64 12.58 14.87 14.88 15.76 17.13 17.38 18.54 18.88

32.32 28.20 23.85 20.17 20.16 19.03 17.51 17.26 16.18 15.89

588

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

A

a

z

O

A d Fig. 8.20 Electric field E structure (red color) of TE111 mode in the plane “1z  1z” at t ¼ 0

The field arrows that protrude beyond the resonator indicate that the points in which they correspond relate to the space inside the resonator which is very close to the wall. The EM field distribution in the three cross-sections at the time moment t ¼ T=8 (see Figs. 8.8 and 8.19) is given in Figs. 8.22 and 8.23. Figure 8.22 presents the longitudinal cross-section “1z  1z”. Both electric and magnetic fields exist in the Fig. 8.21 Electric field E structure (red color) of TE111 mode in plane “A  A” at z ¼ d=2 and t ¼ 0

a

1z

z

1z

8.25

Electromagnetic Field Calculations of TE111 Mode

B

589

A

a

z

O

B

A d

Fig. 8.22 Electric E (red color) and magnetic H (blue) field structures of TE111 mode in the cross-section “1z  1z” at t ¼ T=8

(a)

(b) a

a

1z

z

1z

1z

z

1z

Fig. 8.23 Electric E (red color) and magnetic H (blue) field structures of TE111 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ T=8

resonator at time t ¼ T=8. The electric field concentrates in the resonator center and is directed perpendicularly toward the plane “1z  1z”. E vector arrows are aimed toward us and we see points. Magnetic field vectors create closed ellipsoid curves (blue arrows). H vectors have the largest values close to the conductor walls in the central part along the length d. The EM field has minimal value at the end walls of resonator when z ¼ 0 and z ¼ d. Figure 8.23 presents the structure of TE111 mode in two cross-sections.

590

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

B

A

a

z

O

B

A d

Fig. 8.24 Magnetic field H (blue color) structure of TE111 mode in the plane “1z  1z” at t ¼ T=4

(a)

(b) a

a

1z

z

1z

1z

z

1z

Fig. 8.25 Magnetic field H (blue color) structure of TE111 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ T=4

We see blue points and crosses of the magnetic field vectors in the plane “A  A” at z ¼ d=2 because the plane intersects H vector curves. The electric field is radial (perpendicular) to the lateral wall, and its vectors change directions along the waveguide perimeter. The maximum values of E and H are shifted in space of cross-section at p=2 (Fig. 8.23a). The field arrows that protrude beyond the resonator indicate that the points in which they correspond relate to the space inside the resonator very close to the wall.

8.25

Electromagnetic Field Calculations of TE111 Mode

591

A

a

z

O

A d Fig. 8.26 Electric field E (red color) structure of TE111 in the plane “1z  1z” when t ¼ T=2 Fig. 8.27 Electric field E (red color) structure of TE111 in the cross-section “A  A” at z ¼ d=2 and t ¼ T=2

a

1z

z

1z

In the cross-section “B  B” (Fig. 8.23b), we only see H vectors which form concentric circles. Vector arrows are directed clockwise at the cross-section “B  B” ðz  0Þ. The explanation about our agreement how to take an image in the cross-section is set out in Sect. 8.10. The EM field structure is given in the longitudinal plane of the resonator (Fig. 8.24) and two cross-sections (Fig. 8.25) at time t ¼ T=4. We see in Figs. 8.24 and 8.25 only blue vectors because the electric field is zero. The distribution of H in

592

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

the plane “1z  1z” shows that the magnetic field lines form closed curves along the length d. Vectors of H are directed clockwise in the cross-section “1z  1z”. In the cross-section “A  A” (Fig. 8.25a), we see the blue points in the left part and the blue crosses in the right part. This is due to the fact that the plane “A  A” intersects the closed curves of H and the magnetic field lines are perpendicular to this cross-section. We see enclosed concentric circles composed of magnetic field vectors in the cross-section “B  B” at z  0. H vectors aim in a clockwise direction at the cross-section “B  B” (Fig. 8.25b). The EM field structure is given in Figs. 8.26 and 8.27 at the time t ¼ T=2. The magnetic field is zero for this period of time as shown in Fig. 8.19. In the longitudinal cross-section (Fig. 8.26), the field distribution is similar to the analogical one at t ¼ T=8 (Fig. 8.20) with the exception of the E vector direction which is changed to the opposite. Note that the red points of E vectors are depicted in Fig. 8.20, while now we see the crosses of E vectors in Fig. 8.26. We do not give here the distribution of EM field in the cross-section “B–B” at z  0 because field components are very weak near the end wall of the resonator. Figure 8.27 presents the electric field distribution of oscillation TE111 in the cross-section “A  A” at z ¼ d=2 when t ¼ T=2. The electric field vectors are directed downward at the top and bottom parts of the cross-section “A  A”. Comparing the Figs. 8.27 and 8.21, it can be noted that the general picture of E distributions is similar but the electric field lines have the opposite direction in these two cases. This is to be expected according to Fig. 8.19 because the electric fields are in opposite phases at t ¼ 0 and t ¼ T=2. The field arrows that protrude beyond the resonator indicate that the points in which they correspond relate to the space inside the resonator very close to the lateral wall.

8.26

Schematic Sectional Views of TE111 Mode

Schematic view of the distributions of the electric field E (red color), magnetic field H (blue), displacement Jd (green), surface conduction Js current (black), and surface charges qs (red circles with plus or minus) for the mode TE111 ð¼ H 111 Þ of the circular cylindrical resonator is presented in Fig. 8.28 at time periods t ¼ 0 and t ¼ T=4. Here we see the resonator excitation by the probe, loop, and slot. The explanation of the principle of excitations is given in Sect. 7.9. There are two sections of resonator “1z  1z” and “A  A” relating to the period of time t ¼ 0 in the left part of Fig. 8.28. According to Fig. 8.19, at this period of time there are only the electric field and surface charges qs . We see the red points of E vectors in the plane “1z  1z” because the electric field is perpendicular to this plane and E concentrates in the resonator center (see also Fig. 8.20). The probe, slot, and loop are shown in this longitudinal section.

8.26

Schematic Sectional Views of TE111 Mode

t=0

593

A-A

B-B

t=T/4 Hz

1z

1z

Probe

Hz(φ) Hz(r) 2z - 2 z

Js

d=λw/2

A

2z Probe

Probe

A

Hz

2z

λw/4

1z - 1z

a

Js B

B

Jd

Probe Loop

Loop

C

C Slot

E H

qs

Slot

Js Jd

C-C

Loop

2z

2z

Slot Fig. 8.28 Distributions of electric field E (red color), magnetic field H (blue), displacement Jd (green), surface conduction Js (black) currents, and charges qs (red) for TE111 mode with excitements by the probe, loop, and slot at time moments t = 0 (on the left) and t = T/4 (on the right side)

The arrows at the section “A–A” indicate the upward direction, and this corresponds to the top part of the section “1z  1z” in Fig. 8.28. We see vectors E and the charges qs in the section “A  A” at t ¼ 0. Compare the E distribution which is schematically presented in the section “A  A” with calculations in Fig. 8.21. In the longitudinal section “2z  2z” at time t = T/4, there are closed magnetic field lines which are grouped near the lateral walls of the resonator as well as the displacement Jd and surface conduction Js currents. The electric field and charges are absent at this period of time (see Fig. 8.19). We can see the green crosses of Jd current in the center of the resonator in the section “2z  2z”. The green crosses demonstrate that the Jd current flows down in the airspace from the top to the

594

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

bottom parts of the resonator wall. We see the Js current which flows at the bottom of the resonator (black lines); then, it turns and flows up (black points) on the cylindrical surface of the resonator wall. The directions of Jd vectors are opposite to E vectors according to Fig. 8.19. For explanations regarding images of the cutting planes, see Sect. 8.10. In the cross-section “B  B”, we see the displacement current (green lines), magnetic field blue crosses and points. Currents Js and Jd form a closed electrical circuit. In this section “B  B”, there are radial H z ðr Þ and azimuthal H z ðuÞ distribution functions which are given in Eq. (8.23.8). The cross-section “C  C” shows blue lines of magnetic field and black lines of the surface conduction current that flows on the end wall of resonator as well as the slot and loop for the excitation of resonator.

Transverse Magnetic TMmnp Resonator Modes

8.27

The components of TMmn ð¼ Emn Þ modes propagating in the positive z-axis direction of circular cylindrical waveguide are determined by Eqs. (8.12.18)– (8.12.23). In order to get the expression for the reflected wave, it is enough to change the sign before the propagation constant h and to replace the amplitude coefficient in Eqs. (8.12.18)–(8.12.23). We can write expressions for the components of backward wave, i.e., the wave reflected from the end resonator wall that is located at z ¼ d: Er

Eu

ref;mn ðr; u; zÞ

Ez Hr

Hu

ih a E0 mmn

ref;mn ðr; u; zÞ ¼

¼

ref;mn ðr; u; zÞ

ref;mn ðr; u; zÞ

¼

ref;mn ðr; u; zÞ

0 ref;mn J mn

ih m a2

E0 r ðmmn Þ2 ¼ E0

¼

r ðmmn Þ2

E0

ix0 r a E0 mmn Hz

ref;mn J m

ref;mn J m

ix0 r m a2

m r mn cosðmuÞeihz ; a m r mn sinðmuÞeihz ; a

m r mn cosðmuÞeihz ; a

ref;mn

Jm

0 ref;mn J mn

ref;mn ðr; u; zÞ

m r mn sinðmuÞeihz ; a

m r mn cosðmuÞeihz ; a

¼ 0:

ð8:27:1Þ ð8:27:2Þ ð8:27:3Þ ð8:27:4Þ ð8:27:5Þ ð8:27:6Þ

The total field in the cavity resonator is a superposition of the incident wave (which is described in the same manner as in a circular waveguide) and reflected wave from the resonator end walls:

8.27

Transverse Magnetic TMmnp Resonator Modes

Er;TMmnp ¼ Er;mn þ E r E u;TMmnp ¼ E u;mn þ Eu

ref;mn ref;mn

595

¼0

at z ¼ 0 and z ¼ d;

ð8:27:7Þ

¼0

at z ¼ 0 and z ¼ d;

ð8:27:8Þ

where Er;mn is the electric field radial component of TM wave propagating in the positive direction of z axis (Eq. 8.12.19), Er ref;mn is the electric field radial component of the reflected TM wave propagating in the negative direction of z axis (Eq. 8.27.1), E u;mn is the electric field azimuthal component (Eq. 8.12.20), E u ref;mn is the electric field azimuthal component of the reflected TM wave (Eq. 8.27.2), and d is the resonator length. Substituting expressions for E r from Eqs. (8.12.19) and (8.27.1) to Eq. (8.27.7), we get: ih a 0 mmn r J cosðmuÞðE0 mmn m a

ref;mn

From Eq. (8.27.9) following that ðE0 can write: E0

ref;mn

 E0;mn Þ ¼ 0 ref;mn

at z ¼ 0:

ð8:27:9Þ

 E 0;mn Þ ¼ 0 and the equality, we

¼ E 0;mn :

ð8:27:10Þ

In the formulas (8.27.1)–(8.27.6), value E 0 ref;mn is the amplitude coefficient (factor) of the reflected wave which propagates in the negative z-axis direction. This amplitude coefficient coincides with the amplitude coefficient of propagating wave in the positive z-axis direction for TM modes. Boundary condition (8.27.7) at z ¼ d taking into account Eq. (8.27.10) is:   ih a 0 mmn r Jm cosðmuÞE 0;mn eihz  eihz ¼ 0 at z ¼ d; mmn a

ð8:27:11Þ

and bearing in mind Eq. (8.22.13) we receive: 2

h a 0 mmn r J cosðmuÞE 0;mn sinðhd Þ ¼ 0 mmn m a

at z ¼ d:

ð8:27:12Þ

We have from Eq. (8.27.12): sinðhd Þ ¼ 0 and hd ¼ pp;

ð8:27:13Þ

h ¼ pp=d;

ð8:27:14Þ

and from here

p is the index which can be equal to 0; 1; 2; 3; . . .; 1. Index p shows the number of variations along the z-axis of a circular cylindrical resonator. When p ¼ 0, then

596

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

components E r;TMmn0 and E u;TMmn0 disappear but left components Ez;TMmn0 , H z;TMmn0 , H u;TMmn0 that do not depend on the z-coordinate. Note: Pay attention that for TEmnp the index p is never equal to zero, i.e. p 6¼ 0. The final expressions for the EM field components of TMmnp mode, while taking account of Eqs. (8.22.13), (8.22.14), and (8.27.14), are: m r

ih a mn Er;TMmnp ¼  E 0;mnp J 0m cosðmuÞ eihz  eihz mmn a ð8:27:15Þ pp m r 2pp a mn 0 z ; E0;mnp J m cosðmuÞ sin ¼ mmn d a d m r

mn sinðmuÞ eihz  eihz a r ðmmn Þ pp m r 2pp a2 m mn z ; E J ¼ sin ð mu Þ sin 0;mnp m a d r d ðmmn Þ2

ð8:27:16Þ

m r

mn cosðmuÞ eihz þ eihz a pp m r mn z ; ¼ 2E0;mnp J m cosðmuÞ cos a d

ð8:27:17Þ

m r

mn sinðmuÞ eihz þ eihz 2 a r ðmmn Þ 1 pp m r X i2x0 r m a2 mn z ; ¼ E J sin ð mu Þ cos 0;mnp m a d r ðmmn Þ2 M

ð8:27:18Þ

m r

i2x0 r a mn E 0;mnp J 0m cosðmuÞ eihz þ eihz mmn a pp m r i2x0 r a mn z ; E 0;mnp J 0m ¼ cosðmuÞ cos mmn a d

ð8:27:19Þ

H z;TMmnp ¼ 0;

ð8:27:20Þ

Eu;TMmnp ¼

iha2 m

2

E 0;mnp J m

E z;TMmnp ¼ 2E0;mnp J m

H r;TMmnp ¼ 

ix0 r m a2

E 0;mnp J m

H u;TMmnp ¼ 

where m ¼ 0; 1; 2; . . .; 1, n ¼ 1; 2; 3; . . .; 1, p ¼ 0; 1; 2; . . .; 1. Indexes m and p can be differently or simultaneously equal to zero, for example, TM010 mode. Index m is the number of half-wave variations along the u-coordinate and the order of the Bessel function, and index n is the number of the Bessel function root. Index n designates the number of half-wave variations in the r-direction. Index p designates the number of half-wave variations in the z-directions. Components depend on coordinate r; u; z, i.e., E r;TMmnp ¼ E r;TMmnp ðr; u; zÞ,Eu;TMmnp ¼ Eu;TMmnp ðr; u; zÞ. The multiplier “2” in Eqs. (8.27.15)–(8.27.19) can be entered into arbitrary coefficient E0;mnp which (after its normalization) is possible to take equal to 1 in calculations.

8.28

Instantaneous Field Expressions for Resonator TMmnp Modes

8.28

597

Instantaneous Field Expressions for Resonator TMmnp Modes

We can write from formulas (8.27.15)–(8.27.20) by using Eqs. (8.23.1) and (8.23.2) and Euler’s formula (Eq. 1.4.7) eixt ¼ cosðxtÞ þ i sinðxtÞ. The following instantaneous field components: E r;TMmnp ¼  Eu;TMmnp ¼

pp m r ph a mn z cosðxtÞ; E 0;mnp J 0m cosðmuÞ sin mmn d a d

pp a2 m r d ðmmn Þ

E J 2 0;mnp m

E z;TMmnp ¼ E0;mnp J m H r;TMmnp ¼

x0 r m a2

H u;TMmnp ¼

r ðmmn Þ2

pp m r mn z cosðxtÞ; sinðmuÞ sin a d

pp m r mn z cosðxtÞ; cosðmuÞ cos a d

E0;mnp J m

pp m r mn z sinðxtÞ; sinðmuÞ cos a d

pp m r x0 r a mn z sinðxtÞ; E 0;mnp J 0m cosðmuÞ cos mmn a d H z;TMmnp ¼ 0:

ð8:28:1Þ ð8:28:2Þ ð8:28:3Þ ð8:28:4Þ ð8:28:5Þ ð8:28:6Þ

TMmnp can have five or less components which are not equal to zero. Indexes m ¼ 0; 1; 2; . . .; 1, n ¼ 1; 2; 3; . . .; 1, p ¼ 0; 1; 2; . . .; 1. Indexes m and p can be differently or simultaneously equal to zero, for example, TM010 mode. Here components depend on coordinates r; u; z and time t, i.e., Er;TMmnp ¼ E r;TMmnp ðr; u; z; tÞ,E u;TMmnp ¼ E u;TMmnp ðr; u; z; tÞ. Constant multiplier “2” is entered into arbitrary coefficient E0;mnp which may get equal to 1 in calculations.

8.29

Resonance Frequency of TMmnp Modes

Table 8.7 presents the resonance frequency f TMmnp values calculated by Eq. (8.21.4) and wavelength kTMmnp calculated by Eq. (8.21.3) for oscillations with different indexes m, n, and p when the resonator radius is a ¼ 102 m and the resonator length is d ¼ 5  102 m. We see that the lowest cutoff frequency has the oscillation TM010 , i.e. f TM010 ¼ 11:48 GHz.

598

8.30

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Resonator Mode TM010

The first mode of the transverse magnetic ones TMmnp according to Table 8.7 is TM010 mode. The term TM indicates that H vectors are perpendicular to the plane “1z  1z”. We see that there are no variations on the first “m” and third “p” indexes, i.e. along the coordinates u and z. The first number in the subscript (index) is the azimuthal mode number; when m = 0, then the mode is the symmetrical one. The second index is the radial mode number. The radial mode number minus one ðm  1Þ is the number of nodes in a radial variation of the E z component. EM field components using Eqs. (8.28.1)–(8.28.6) are: Er;TM010 ðr; u; z; tÞ ¼ 0;

ð8:30:1Þ

E u;TM010 ðr; u; z; tÞ ¼ 0;

ð8:30:2Þ

m r 01 E z;TM010 ðr; u; z; tÞ ¼ E 0;TM010 J 0 cosðxtÞ; a

ð8:30:3Þ

H r;TM010 ðr; u; z; tÞ ¼ 0;

ð8:30:4Þ

H u;TM010 ðr; u; z; tÞ ¼

m r x0 r a 01 E 0;TM010 J 00 sinðxtÞ; m01 a

H z;TM010 ðr; u; z; tÞ ¼ 0;

ð8:30:5Þ ð8:30:6Þ

The term m01 ¼ 2:405 can be taken from Table 8.1. So this resonator mode has only two components E z;TM010 and H u;TM010 . The EM structure of this mode is very simple. The calculations of EM field at the period of time t1 ¼ T=8 in the longitudinal cross-section “1z  1z” and the cross-section “A  A” are given in Fig. 8.29. The electric and magnetic fields simultaneously exist at the time moment t1 ¼ T=8 as shown in Fig. 8.19.

Table 8.7 Resonator TMmnp ð¼ EmnpÞ

No.

m

n

p

TMmnp

f TMmnp (GHz)

kTMmnp (mm)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

0 0 0 0 0 1 1 0 1 1

1 1 1 1 1 1 1 1 1 1

0 1 2 3 4 0 1 5 2 3

TM010 TM011 TM012 TM013 TM014 TM110 TM111 TM015 TM112 TM113

11.48 11.86 12.95 14.59 16.61 18.29 18.54 18.89 19.25 20.39

26.14 25.29 23.16 20.57 18.07 16.40 16.18 15.88 15.58 14.72

8.30

599

Resonator Mode TM010

The resonator radius is a ¼ 102 m, and the resonator length is d ¼ 5  102 m in calculations of Fig. 8.29. The magnetic field is zero along the z-axis. The transverse propagation constant: k ?010 ¼

  m01 2:405 ¼ 240:5 m1 : ¼ a a

ð8:30:7Þ

We see that k?010 is inversely proportional to the radius “a”. The wavelength of the TM010 resonator modes according to Eq. (8.21.3) is: kTM010 ¼

2pa 2p a  0:026ðmÞ; ¼ m01 2:405

ð8:30:8Þ

where kTM010 is directly proportional to the radius “a”. And the resonance frequency of the TM010 mode on the base Eq. (8.21.4) is: f TM010

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi m01 c r lr 2:405 c r lr 1 ¼   11:5 ðGHzÞ; ¼ a 2p a 2p

ð8:30:9Þ

when r ¼ 1 and lr ¼ 1. We pay your attention that transverse propagation constant k?0n0 and wavelength kTM0n0 for all TM0n0 modes with n ¼ 1; 2; 3; . . .; 1 depend only on the radius “a” at fixed indexes. TM0n0 modes may be used for particle acceleration. The longitudinal electric field is at its maximum in the center and uniform along the z-axis, i.e., along the propagation of a charged particle (electron, ion, etc.) beam. The absence of the magnetic field on the z-axis is very important because the transverse H could deflect the electron beam. The TM010 ð¼ E 010 Þ mode of cylindrical cavity resonator is often used for measuring complex permittivity of liquid materials.

(a)

A

(b) a

a z

O

1z

z

1z

A d

Fig. 8.29 Electric (red color) and magnetic (blue) field structures of TM010 in: a plane “1z  1z” and b cross-section “A  A” at z ¼ d=2 when t ¼ T=8

600

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Fig. 8.30 Schematic distributions of E and H for TM010 at the time moment t ¼ T=8

A-A

TM 010 t=T/8 1z

a

1z

1z - 1z H

A

A E

r

a

O

Schematic view of the distributions of the electric field E (red color), magnetic field H (blue) for the higher mode TM010 of the circular resonator is presented in Fig. 8.30 at the period of time t ¼ T=8 when E and H fields exist simultaneously. We would like to pay your attention to the fact that the direction of the magnetic field determines here the displacement current Jd which is in antiphase with E-field (Fig. 8.19). We see in Figs. 8.29 and 8.30 that the distribution of EM field of TM010 is very simple, so for this reason we will not explore this mode in more detail.

8.31

Calculations of TM011 Field Distributions

In order to better understand the processes occurring in the resonator over time, we present here our calculations of TM011 mode at the four periods of time t ¼ 0; T=8; T=4; T=2. Computing and visual pictures are fulfilled by using our homemade computer programs in MATLAB. Figures 8.31, 8.32, 8.33, 8.34, 8.35, 8.36, 8.37 and 8.38 show our computations of distributions of the electric E and magnetic H fields by formulas (8.28.1)–(8.28.6) when the resonator radius is a ¼ 102 m ¼ 1 cm and the length is d ¼ 5  102 m ¼ 5 cm. We see from Eqs. (8.28.1)–(8.28.6) that for this mode with “m ¼ 0” three components E u , H r , and H z equal zero. Figures 8.31 and 8.32 present the electric field distributions at the time moment t ¼ 0. The electric field lines are situated in the plane “1z  1z” and concentrate in the end parts of the resonator. Their line arrows are aimed in the resonator center

8.31

Calculations of TM011 Field Distributions

B

601

A

a

z

O

B

A d

Fig. 8.31 Electric field (red color) distribution of TM011 mode in cross-sections “1z  1z” at t¼0

from two resonator ends. The electric field is very low and distributed uniformly throughout the cross-section area in plane “A  A” at z ¼ d=2 when t ¼ 0 (Figs. 8.31 and 8.32a). The magnetic field is absent at t ¼ 0 (see Fig. 8.19). We observe densely located red points of E in the center of plane “B  B” at z  0; i.e., the electric field is perpendicular to this cross-section (Fig. 8.32b). The EM field structure is given in the longitudinal plane “1z  1z” (Fig. 8.33) and two cross-sections (Fig. 8.34) at the period of time t ¼ T=8. The electric and magnetic fields exist simultaneously at this period of time. The electric field lines are shorter, i.e. E is weaker, at t ¼ T=8 comparing with the electric field at t ¼ 0 (Figs. 8.31 and 8.33). We see in Fig. 8.33 blue points and crosses of H vector lines that are perpendicular to the plane. The magnetic field is localized in the resonator corners. The distribution of H in the plane “1z  1z” demonstrates that the magnetic field lines form the closed curves in 3D view. The H vectors directed from points which are located in the bottom left part of plane “1z  1z” till the crosses that are in the top left part of the resonator at z ¼ 0 d=2. The H vectors are directed at the opposite way in the resonator right area when z ¼ d=2 d. The electric field is directed in a special manner. E vectors are perpendicular to the resonator end walls. E-field lines describe by the hyperbolic-type lines which start on the end wall with z = 0 and finish on the top or bottom lateral part of cylindrical wall of resonator. The analogical picture of E-field lines is in the right part of resonator. E field starts from the end wall with z = d and finishes on the top or bottom lateral part of cylindrical wall with coordinates z ¼ d=2 d.

602

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Fig. 8.32 Electric field (red color) distribution of TM011 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ 0

B

A

a

z

O

A

B d

Fig. 8.33 Electric (red color) and magnetic (blue) field vector distributions of TM011 mode in the sections “1z  1z” at t ¼ T=8

We only see weak vectors of E in the cross-section “A  A” at z ¼ d=2 when t ¼ T=8 (Fig. 8.34a). The electric field concentrates in the center of the cross-section “B–B” at z  0, and the magnetic field tangentially localizes near the wall (Fig. 8.34b). The EM field structure is given in the longitudinal plane “1z  1z” (Fig. 8.35) and two cross-sections “A  A” and “B  B” (Fig. 8.36) at the time moment

8.31

Calculations of TM011 Field Distributions

603

(a)

(b) a

a

1z

z

1z

1z

z

1z

Fig. 8.34 Electric (red color) and magnetic (blue) field vector distributions of TM011 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ T=8

A

B

a

z

O

A

B d

Fig. 8.35 Magnetic field (blue color) structure of TM011 mode in the longitudinal section “1z  1z” at t ¼ T=4

t ¼ T=4. There are only blue H vectors in the figures because the electric field is zero according to Fig. 8.19. The magnetic field is localized in the resonator corners. Vectors of H change their directions in the middle of the resonator at z ¼ d=2 (Fig. 8.35). The distribution of H vector lines in the plane “1z  1z” shows that the field lines are perpendicular to this plane and form closed curves in the 3D view. The H vector lines are directed from points which are located in the bottom part of the plane “1z  1z” till the crosses that are in the top left part of the resonator at z ¼ 0 d=2:

604

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

(a)

(b) a

a

1z

z

1z

1z

z

1z

Fig. 8.36 Magnetic field (blue color) structure of TM011 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ T=4

The H vector lines directed in the opposite way in the right area with z ¼ d=2 d. In Fig. 8.36, there are two cross-sections. In the cross-section “A  A” at z ¼ d=2, there are the calculated H vectors. They are so small that they are difficult to distinguish. We usually give the reference in the text and do not give an image when the fields are weak as in Fig. 8.36a. Here we have placed the section “A  A” only as an example of the smallness of field values. We see enclosed concentric circles which are composed by the magnetic field vectors in the cross-sections “B  B” at z  0. Figures 8.37 and 8.38 present the electric field distributions at the time moment t ¼ T=2. We see that the electric field lines are situated in the plane “1z  1z” and concentrate in the end parts of the resonator. Its line arrows are aimed from the resonator center to resonator ends; i.e. vectors have the opposite direction than in Figs. 8.31 and 8.33. The magnetic field is absent at t ¼ T=2 (Fig. 8.19). The electric field is very low and distributed uniformly throughout the cross-section area in plane “A  A” at z ¼ d=2 (Fig. 8.38a). The electric field is perpendicular to cross-section “B  B” at z  0. We see that the largest density of red points is located in the center of plane “B  B” (Fig. 8.38b).

8.32

Schematic Sectional View of TM011

Schematic view of distributions of the electric field E (red color), magnetic field H (blue), displacement Jd (green), surface conduction Js current (black), and surface charges qs (red circles with plus or minus) for the mode TM011 (=E 011 ) of the resonator is presented in Figs. 8.39 at the periods of time t ¼ 0 and t ¼ T=4. Here

8.32

605

Schematic Sectional View of TM011

B

A

a

z

O

B

A d

Fig. 8.37 Electric field (red color) structure of TM011 mode in the longitudinal section “1z  1z” at t ¼ T=2

Fig. 8.38 Electric field (red color) structure of TM011 mode in cross-sections a “A  A” at z ¼ d=2 and b “B  B” at z  0 when t ¼ T=2

we give the resonator excitation by the probe, loop, and slot. Explanation of the principle of excitations is given in Sect. 7.9. The longitudinal section of waveguide in the period of time t = 0 is given at the top three sections “1z  1z”, “A  A”, “B  B”. The hyperbolic-type curves of electric field vectors are shown in the view “1z  1z”. The electric field lines are situated in the plane “1z  1z” and concentrated in the end parts of the resonator (see also Fig. 8.31). The field line arrows are aimed in the center from two resonator

606

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

(a)

A

Loop

1z - 1z

B

A-A

1z

Loop

Probe

Ez(φ) Ez(r) Slot

A

z

B

1z

Slot

B-B

d=λw/2

1z

E Js

qs 1z

(b) Loop

2z - 2z

С

C-C Loop

D Probe

2z

H

Probe

Jd

Js

Js Slot

С

z

D

Slot

Jd

d=λw/2

2z

D-D

2z

Probe

Js Jd

Js

H 2z

Fig. 8.39 Distributions of electric field E (red color), magnetic field H (blue), displacement Jd (green), surface conduction Js (black) currents, and charges qs (red) for TM011 mode at the time moments a t = 0 and b t = T/4 (three bottom cross-sections: “2z  2z”, “C  C”, “D  D”)

8.32

Schematic Sectional View of TM011

607

end walls. The magnetic field is absent at t ¼ 0 (see also Fig. 8.19). The probe, slot, and loop are shown in the section “1z  1z”. We see the red crosses of E in the center of the section “A  A”. The electric field lines are perpendicular to this section. This section gives the law of H z component distribution along r and u coordinates. The E vector lines are radially directed to the lateral cylindrical wall in the cross-section “B  B” at t ¼ 0. The explanation of our agreement how to take an image in the cross-section is set out in Sect. 8.10. The longitudinal section of resonator is presented in the view “2z  2z” with distributions of Jd , Js currents and H-field at t ¼ T=4. Here the blue crosses and points of the magnetic field in the resonator corners are shown (compare with Fig. 8.35). H vector lines are perpendicular to the plane “2z  2z”. The distribution of H in this plane shows that the magnetic field lines form closed curves in 3D view. The magnetic field vectors are directed from points which are located on the bottom left corner of the plane “2z  2z” till the crosses that are located in the left top part of the resonator. The magnetic field vectors are directed in the opposite way in the right resonator part. The conduction current Js flows on the resonator wall and creates together with Jd a close electrical circuit. The cross-section “C  C” shows closed magnetic field lines which are grouped near the walls of the resonator. The probe, slot, and loop for the resonator excitation are shown in this section. Magnetic field H vectors are directed clockwise in the section “C  C”. The vector lines of Jd in the cross-section “D-D” of Fig. 8.39b are directed opposite to the vector lines of E in section “B-B” of Fig. 8.39a because they are out of the phase. The electric field and charges are absent at this period of time t = T/4 (see Fig. 8.19).

8.33

Review Questions

Q8:1. What is the difference between circular cylindrical and rectangular waveguides? Q8:2. What types of modes can propagate in circular cylindrical waveguides? Q8:3. What do the main mode and the higher-order modes of circular waveguide mean? Q8:4. What do TEmn modes mean? Which component of EM field is always equal to zero? Q8:5. What do TMmn modes mean? Which component of EM field is always equal to zero? Q8:6. What does the first index “m” indicate and which coordinate is it associated with?

608

8 Cylindrical Hollow Metallic Waveguides and Cavity Resonators

Q8:7. What does the second index “n” indicate and which coordinate is it associated with? Q8:8. Write the value of indexes “m” and “n” for the main mode of circular waveguide, m = what? and n = what? Q8:9. What do the cutoff frequencies of TEmn and TMmn modes mean? Q8:10. Write the expression of cutoff frequency for TEmn modes. Q8:11. Write the expression of cutoff frequency for TM  mn modes.  Q8:12. How many electric and magnetic components Er ; E u ; E z ; H r ; H u ; H z can TEmn modes contain in the circular waveguide?   Q8:13. How many electric and magnetic components Er ; E u ; E z ; H r ; H u ; H z can TMmn modes contain? Q8:14. Write the expressions of H z and E z components for TEmn modes of the waveguide. Q8:15. What values can indexes “m” and “n” take for TEmn modes? Q8:16. Write the expressions of H z and E z components for TMmn modes of the waveguide Q8:17. What values can indexes “m” and “n” take for TMmn modes? Q8:18. Draw the Bessel functions of zero order and show the root of TM01 mode. Q8:19. Draw the Bessel functions of the first order and show the root of TE11 and TM11 modes. Q8:20. Draw the Bessel functions of the second order and show the root of TE21 and TM21 modes. Q8:21. What does the transversal propagation constant mean? Q8:22. What does the longitudinal propagation constant mean? Q8:23. What are instantaneous field components and what the multiplier should be used? Q8:24. What are the dispersion characteristics of circular cylindrical waveguides? Q8:25. What is the difference between cylindrical cavity resonators and cylindrical waveguides? Q8:26. What additional boundary conditions appear in the resonator in comparison with the waveguide? Q8:27. What is the difference between the expressions of resonance frequencies for TEmnp and TMmnp modes through roots of the Bessel function and its derivative? Q8:28. What types of modes (oscillations) can occur in circular cylindrical resonators? Q8:29. What does the first index “m” indicate in the resonator mode and which coordinate is it associated with? Q8:30. What does the second index “n” indicate in the resonator mode and which coordinate is it associated with? Q8:31. What does the third index “p” indicate in the resonator mode and which coordinate is it associated with?

8.33

Review Questions

609

Q8:32. Write the value of indexes “m”, “n”, and “p” for the first mode of circular resonator “m ¼ what?”, “n ¼ what?”, and “p ¼ what?” Q8:33. What initial value can the index “m” (0 or 1?), the index “n”, and the index “p” take for resonator TEmn modes? Q8:34. What initial value can the index “m” (0 or 1?), the index “n”, and the index “p” take for resonator TMmn modes? Q8:35. How many components does the TM010 mode contain? Q8:36. How many components does the TE111 mode contain?

Chapter 9

Plasmonics

Abstract This chapter describes the plasmonic phenomena. Here, the main models for calculations of complex permittivity of conductors such as Drude, Drude– Sommerfeld, Drude–Lorentz Debye, Brendel–Borman are presented. The author also deals with Helmholtz’s equations for surface plasmon polariton (SPP) waves, their solutions and dispersion equations on the flat and cylindrical interface insulator (or semiconductor)–conductor. This chapter lists calculations of dispersion characteristics made by using formulae of the textbook. The dispersion characteristics of EM waves propagating along the flat gold/silver–air, gold/silver– semiconductor interfaces and also dispersion characteristics of TE0n and TE0n modes for gold cylinders with different radii are thus calculated, and then, the author compares the calculations with analogical results from literature finding that they also coincide.

9.1

Introduction

The rapid development of micro- and nanofabrication technology stimulated the studying of nanophotonics and its branches such as plasmonics, metamaterials (composite materials engineered to have a property that is not found in nature) for high-speed data transmission, sensitive optical detection, manipulation of ultrasmall objects, and visualization of nanoscale patterns. Nanophotonics studies the interaction of nanometer-scale objects such as nanometal particles, nanocrystals, semiconductor nanodots (very small semiconductor particles) with light. Plasmonics considers the interaction between EM waves (light) and free (conduction) electrons (Sects. 3.8 and 3.15) in conductors (metals, semimetals, semiconductors). Free electrons in conductors excited by electric field components of EM wave can have collective oscillations with a frequency close to the frequency of acting EM wave. Plasma oscillations are fast oscillations of the electron density in conducting media such as metals or plasmas (Sect. 3.1).

© Springer Nature Singapore Pte Ltd. 2019 L. Nickelson, Electromagnetic Theory and Plasmonics for Engineers, https://doi.org/10.1007/978-981-13-2352-2_9

611

612

9 Plasmonics

Plasmonics is based on a physical phenomenon called plasmons. Plasmons by their physical nature are a composition of free electron plasma and EM waves. Plasmons are similar to plasma inside the conductor and outside it plasmons are like an EM wave. The nature of the surface of medium on which this plasmon is created determines the frequency of its oscillations. The conductor body surface can be a smooth flat, cylinder, sphere, corrugated; it can contain different shape grooves as well as the surface of nanoparticles of different shapes and sizes can be considered. In a piece of conductor or on its surface different types, volume (also known as bulk) and surface plasmons may be excited. There are also the localized plasmons which are associated with oscillations of conduction electrons on nanoparticles. For general review, we will shortly mention here different types of plasmonic waves which can occur due to conducting nanoparticles and under other conditions show the variety of phenomena existing in plasmonics. Here, we are basically going to consider surface plasmon polariton (SPP) modes which are inherently surface charge density waves propagating along the conductive flat or cylindrical interfaces. SPP, which we are going to consider, can propagate on the conductor–insulator or conductor–semiconductor interface. This chapter presents plasmonics from the point of view of EM field theory and microwave waveguide technology with dependence from the longitudinal coordinate and time eiðxthzÞ as it is accepted in microwave range (Chaps. 7 and 8).

9.2

Chronological Development of Plasmonics

Plasmonics has been known from the old times, and metals with higher conductivity such as silver and gold particles (powder) were implanted into glass substance. Silver and gold particles give new colors and shades of colors (stained glass) dependent on the metal quantity, particle size and shape. The term “stained glass” can refer to colored glass. The optical properties of composition made of glass with metal nanoparticle inclusions were widely utilized through the centuries to decorate windows and art painting (Fig. 9.1a, b) and also for creation of three-dimensional structures (Fig. 9.1c). The synthesis of colloidal gold of the fourth-century Lycurgus Cup is well known (Fig. 9.1c). The Cup is made of a dichroic glass. Dichroic glass is the one which displays different colors by undergoing a color change in certain lighting conditions. The Lycurgus Cup exhibits a different color depending on whether or not light is passing through it: red when light comes from behind of a subject (the cup), green when light comes from in front of the cup and when light incidentally falls on the other angle it can be of other shades of color. Figure 9.1c shows the same Lycurgus Cup lighted which is illuminated by light at different angles. In the last color phenomena (Fig. 9.1), localized surface plasmons (LSPs) are used. LSPs are a result of localization of surface plasmons on metal particles.

9.2 Chronological Development of Plasmonics

613

Fig. 9.1 a, b Stained glass windows. c Fourth-century Lycurgus Cup under different lighting conditions

In the sixteenth century, the Paracelsus claimed to have created a potion called Aurum Potable (Latin: potable means gold). In the seventeenth century, the glass-coloring process was refined by Andreus Cassous and Johann Kunckel, allowing them to produce a striking deep ruby-colored form of glass. However, the secret of stained glass was not cracked until a German physicist Gustav Mie (1868– 1957) presented a theory that explained the optical properties of spheres in 1908. The Mie theory (mentioned in Sect. 6.1) showed that the size and the refractive index (Sect. 5.14) of sphere material as well as EM properties of environment (surrounding media) determine a color of a composite material. American physicist Robert Williams Wood (1868–1955) discovered plasmon oscillations in nanoparticles. In 1902, R. W. Wood, observing the reflection spectrum of continuous light from a metallic diffraction grating, found a new phenomenon which is now named Wood’s anomalies. At approximately the same time, the Maxwell–Garnett model which depicts the theoretical modeling of composite materials originated. Composite materials are made of two or more constituent materials with significantly different electrophysical properties. The model explained the bright color in the glass with metal nanoparticle inclusions. The model used Rayleigh approaches (1871) to the EM properties of spherical nanoparticles.

614

9 Plasmonics

The Drude model of conductors (metals, etc., Sects. 3.15 and 4.6) was proposed by Paul Drude to describe transport properties of electrons in conductors in 1900. The Drude model was complemented by Dutch physicist Henrik Lorentz (1853– 1928) in 1905. This last classical model is known as the Drude–Lorentz model. In 1933, the Drude theory was further developed by using knowledge of quantum mechanics by Arnold Sommerfeld (1868–1951) and this led to the emergence of the Drude–Sommerfeld model. From a formal point of view as light is EM waves, the Maxwell’s equations with the corresponding boundary conditions can completely describe optical phenomena at the conductor–insulator (or semiconductor) interface and nanoparticles. The solutions of the Maxwell’s equations found by Mie (1908), Sommerfeld (1909) and other scientists in the beginning of the twentieth century are completely applicable to the plasmon phenomena on the metal–insulator interface as well as on noble metal (silver (Ag), gold (Au), etc.) nanoparticles. The existence of surface plasmons was probably first predicted by Rufus Ritchie (1957). R. Ritchie published the theoretical work on collective electron modes, surface EM waves in solids (solid-state medium, solid is one of the five fundamental states of matter, Sect. 3.1). Solid examples are dielectrics, semiconductors, semimetals, metals (Sect. 4.6). In the following several decades, surface plasmons were extensively studied by many scientists: T. Turbadar in the fifties–sixties of twentieth century, Heinz Raether, Erwin Kretschmann, Andreas Otto and Vladimir Agranovich in the sixties and seventies of twentieth century. In the early twenty-first century, great interest in the surface and localized plasmons arose. This interest can be attributed to the new modern nanotechnology to produce nanoparticles of almost any shape, size, and composition. Now, the computer calculation methods allow optimizing the properties of nanoparticles and nanodevices. Scanning microscopes allow anyone to thoroughly diagnose options with regard to each nanoparticle and different nanodevices. Notes to explain terminology Purposeful studies of surface plasmons began in the 1980s when the chemists have begun to study this phenomenon by means of Raman spectroscopy, and then, they used the laser light scattering to define the structure of a sample by the molecular oscillations. The word “plasmon” was adapted (originated) from plasma physics taken to the limit of the degenerate electron gas. It should be noted that literature on plasmonics often uses three different names for the same modes, i.e. “surface plasmon polaritons,” “surface plasmons,” or just “plasmons.” We use here abbreviations for surface plasmon polaritons in plural: SPPs and in singular: SPP. In physics, a plasmon is a quantum (portion) of plasma oscillation. A quantum is the minimum amount of any physical entity. For example, a photon is a single quantum of light (a light quantum or a light particle). There is also such approach: when an SP (surface plasmon) couples with a photon, the resulting hybrid excitation is called a surface plasmon polariton (SPP). SP refers to charge

9.2 Chronological Development of Plasmonics

615

oscillations alone, while SPP refers to the entire excitation of the charge oscillations and the EM wave. SPPs are surface waves with a longitudinal electric field component (TM (transverse magnetic) wave in nature for usual nonmagnetic media) and are slow propagating waves. The surface plasmon polariton is a surface wave with a longitudinal field component that propagates at the interface between a conductor and an insulator (dielectric, semiconductor) usually at PHz frequency when the conditions are right and can propagate along the interface at a certain wavelength shorter than that of incident EM wave (light) until its energy is lost by absorption in the conductor, radiation in free space, etc. A coupled photon and plasmon is called a plasmon polariton. Propagating usual EM waves are forbidden in materials where the real part of the material relative permittivity e0r (Eq. (3.11.2)) is negative as it happens with metals below the plasma frequency. In the last case, there may exist an evanescent wave which is an exponentially damped wave. But the EM wave propagation is possible on the surface of separation of a conductor (with the negative e0 also index “r” at epsilon) and an electric insulator or semiconductor (with the positive e0r ). As we remember from Sect. 3.2, the electrical insulator material (also known as the dielectric or nonconductor) can be polarized by an applied electric field. The electric charges do not flow through the insulator while the charges flow in conductors (Sect. 3.13). The term “surface plasmon polariton” denotes that the wave involves both conduction electron motion in a metal, i.e. surface plasmon, and EM waves in an insulator like air. The exciting EM wave is usually light in air (see Sect. 9.14). SPPs are EM surface waves confined to the interface of two materials with the relative permittivity (dielectric function, Sect. 3.10) e0 also index “r” at epsilon of opposite signs. SPPs can occur as a result of an interaction between an EM wave (light) and a collective surface electron density oscillation of free electrons of a conductor. Here, we are going to study SPP modes propagating only on the flat (plane) and cylindrical interfaces (surfaces). SPPs usually occur in infrared (IR) frequency range with f ¼ ð0:3  430Þ THz and wavelengths k ¼ 1 mm  700 nm; visible (VIS) frequency range with f ¼ ð0:43  0:79Þ PHz and k ¼ 700 nm  400 nm as well as ultraviolet (UV) frequency range with f ¼ ð0:79  30Þ PHz and wavelengths k ¼ 400 mm  10 nm and they can propagate along a metal–insulator (metal–air) or metal–semiconductor interface. EM waves interact with media in different ways across the ranges of spectra. One can observe semiconductor plasma oscillations and molecular rotation in microwave range (Appendix Chap. 5:5.1) through far IR (about 0.3–30 THz) frequency range. There are molecular vibration and plasma resonant oscillations in metals in near IR (about 120–400 THz) frequency range. There are molecular electron excitations and plasma resonant oscillations in metals in VIS frequency range. Excitation of molecular and atomic valence electrons including ejection of the electrons (photoelectric effect) can be observed in UV frequency range. Metals have their own atom resonance oscillations with frequencies in the optical range. The optical range radiation accepts frequencies from 0.3 THz till 3 PHz ðk ¼ 1 mm  100 nmÞ by DIN 5031 standard. DIN means “Deutsches Institut für Normung” (Engl. “German Institute for Standardization”).

616

9 Plasmonics

The elementary particle (portion, quantum) of light is a photon. The photon always moves at the speed of light in a vacuum. As the photon energy grows and becomes comparable to the energy bandgap (forbidden band, energy gap, bands of energy) of solids, a new conduction process may occur. The photons can excite an electron from an occupied state in the valence band to an unoccupied state in the conduction band. This process is called the interband (also known as interzone) transition (see Figs. 9.3 and 9.4). In other words, the interband transitions are transmissions between the energy states available to the electrons. It should be noted that there is also the intraband transmission within the conduction band which applies to free carriers (see also Sect. 9.5). We see that there are prefixes “inter” and “intra” which give a different meaning to the word. An interband transition is an electronic transition between different bands, and an intraband transition is a transition between electronic states within the same band. Notes about Sommerfeld–Zenneck surface wave on conductor surface The SPP wave is sometimes wrongly associated with a Sommerfeld–Zenneck (or simple Zenneck, ground) wave. The Zenneck surface wave is non-uniform EM plane wave incident at the complex Brewster angle (Sects. 6.3–6.5) on a conductor interface between two homogeneous media having different signs of the permittivity. The same requirement to distinguish in the signs of the permittivity of bordering materials is for the propagating SPP waves along their interface. Zenneck surface wave is an EM wave that uses the lossy earth or sea surfaces as a waveguide enabling the transmission of EM waves along larger distances. These waves correspond to the propagation of radio waves (f * 20 kHz–300 GHz) along the surface of the Earth and they decay exponentially vertical to the conductor surface. Both the Zenneck wave and the SPP wave are TM waves in non-magnetic media (Sect. 9.12) and are non-radiating as they have, in general, exponentially descending fields with increased distance. The Zenneck waves are fast propagating waves and SPPs are slow propagating waves. This electron wave produces its own EM wave, and this plasmonic wave is confined to a very small region near the interface. TM SPP wave is a surface wave with a longitudinal electric field component that propagates at the interface between a metal and a dielectric at petahertz frequency when the SPP can propagate along the metal– dielectric interface at a wavelength shorter than that of incident EM wave (light).

9.3

Definitions Used in Plasmonics

In previous Chaps. 7 and 8 on hollow metallic waveguides, we have dealt with frequencies measured in GHz, i.e. frequency ranges SHF, EHF, THF (Appendix Chap. 5:5.1) while operating frequencies in plasmonics can be several orders higher. For this reason, there is a specific terminology in plasmonics which is not used in EM field theory for microwaves.

9.3 Definitions Used in Plasmonics

617

We are going to employ very high operating frequencies and very small sizes of a waveguide; for this reason, dimensions which are usually used in plasmonics are as follows: 1 petahertz ð1 PHzÞ ¼ 1015 Hz ¼ 106 GHz ¼ 103 THz: Notation “pm” is picometre (International spelling) also known as picometer (American spelling), 1 pm ¼ 1012 m: For comparison purposes, the sizes of atoms range between 62 and 520 pm. Another notation “nm” is nanometer, i.e. 1 nm ¼ 109 m; “lm” is micrometer, i.e. 1 lm ¼ 106 m; and “mm” is millimeter, i.e. 1 mm ¼ 103 m: There are peculiarities in the use of units of the operating frequency f measurement in plasmonics and optics. The frequency of light is measured in several different units in the scientific literature, i.e. Hz (hertz), eV (electron volt), cm−1 (centimeter to the power minus 1), K (kelvin). The unit kelvin is used because the energy of a photon E is expressed as E ¼ h f ¼ hx ¼ h kc and E ¼ k T where the product of the Boltzmann constant k and the temperature T is sometimes used as a unit of energy, f k ¼ c (Eq. 5.14.2), c is the speed of light in a vacuum (m/sec) (Eq. 2.1.6), k is the wavelength of EM wave (light) (m), and x ¼ 2pf is the angular frequency (radians per second). In expression of energy of a photon E, term h is the Planck’s constant and the reduced Planck constant is h ¼ h=2p and they are the following: h  4:13567  1015 eV s ¼ 6:62607  1034 J s and  h  6:58212  1016 eV s 34  1:05457  10 J s: Please pay attention that symbols of the Planck’s constants h,  h, and the Boltzmann constant k looks like the propagation constant h (which is the real part _ and the wavenumber k (Sects. 4.5 and 5.4). of the complex propagation constant h) Here, we give several relationships between the most commonly used in plasmonics units: 1 THz ¼ 1012 Hz ¼ 0:00414 eV; 1 THz corresponds to the wavelength of 300 lm: 1 eV ¼ 241:8 THz: 1 eV corresponds to the wavelength of 1:2398 lm and also 2 eV  0:620 lm; 6 eV  0:207 lm, etc. (see Sect. 2.12). Values in cm1 correspond to the so-called wavenumber which is  1=k; 1 cm1 ¼ 123:98 leV ¼ 0:0299795 THz: The relations between the various units which are used in the scientific literature on plasmonics are given in Table 9.1.

Table 9.1 Conversion of values eV, Hz, cm−1, K eV (electron volt)

Hz (hertz)

eV

1

Hz

4:13558  1015

2:41804  10 1

1:23981  104 4

cm K

−1

14

8:61705  10

cm1

K (kelvin)

8065.73

11604.9

2:99793  1010

3:33565  1011 1

4:79930  1011 1.42879

2:08364  1010

0.695028

1

618

9 Plasmonics

We use definitions and quantities in this chapter that are usually applied in quantum mechanics and optics. Quantum mechanics is a branch of physics which deals with physical phenomena that refer to structures with sizes of nanoscales (1– 100 nm). Optics is the branch of physics which involves the behavior and properties of light and its interactions with media (mediums). In physics, the electronvolt (symbol eV; also written as electron volt) is a unit of energy. By definition, it is the amount of energy gained (or lost) by the charge of a single electron moved across an electric potential difference of one volt. Thus, it is 1 V (1 J per coulomb, 1 J/C) multiplied by the elementary charge ðe  1:60218  1019 CÞ; and see Sect. 2.4. Therefore, one electron volt is approximately equal to 1:60218  1019 J where symbol J means “joules.” The relationship between frequency f and energy of photons E are shown in Table 9.2. Difference between media through the available energies for electrons The difference between insulators, semiconductors, and conductors can be depicted by the available energies for electrons in materials. The energy of an electron in an atom may have only permissible values. Free atoms have discrete energies while atoms in solids due to influence of atoms on each other have available energy in the form of bands. So, the energy spectrum of electrons in solids has band structure. In solid media (materials), the electronic band structure (band structure) of material is described by ranges of allowed energy bands (simply bands) that electrons may have within the solid and energy intervals named as band gaps (forbidden gaps, energy bandgaps, energy gaps, bands of energy) Eg which point out that electrons cannot have such energy. The term forbidden band determines the electron energy difference between the top of the valence band and the bottom of the empty conduction band. In insulators (dielectrics), the electrons in the valence

Table 9.2 Relation between frequency, wavelength, and energy of photons

No

Frequency, f

Wavelength, k

E¼ hx

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

3 GHz 30 GHz 300 GHz 3 THz 30 THz 300 THz 3 PHz 30 PHz 300 PHz 3 EHz 30 EHz 300 EHz

1 dm 1 cm 1 mm 100 lm 10 lm 1 lm 100 nm 10 nm 1 nm 100 pm 10 pm 1 pm

12.4 leV 124 leV 1.24 meV 12.4 meV 124 meV 1.24 eV 12.4 eV 124 eV 1.24 keV 12.0 keV 124 keV 1.24 meV

9.3 Definitions Used in Plasmonics

619

band are separated by a large band gap E g from the conduction band; in semiconductors, there is a sufficiently small width (narrow) gap E g , and in conductors (like metals), the valence band overlaps the conduction band. The value of the bandgap of insulators is usually greater (e.g. >4 eV) than the one of a semiconductor (e.g. 0, insulator , index ''i''

y

E

Flat Interface

i

i z

H x

i

Ey

O

i

εr m

εr

m

m

Ez

Hx

z

m

Ey

area I, y0,area II, z>0 Re(h SPP)=

SPP

2π λ SPP

z

x m

εr(ω)

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