Idea Transcript
~IEEE
TRAN SACTI 0 NS
ON
MICROWAVE THEORY AND TECHNIQUES APRIL 1985 '
· . VOLUME
MTT-33
NUMBER
4
(ISSN 0018-9480)
A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
EDITORIAL ........................................•............................................... T. lt,oh
295
PAPERS
Waveguide and Cavity Oscillations in the Presence of Nonlinear Media ..................................... D. Censor 296 Functional Approach to Microwave Postproduction Tuning ........................... .J. W. Bandier and A. E. Salama 302 SHORT PAPERS
Highly Stable Dielectric Resonator FET Oscillators .................................................... C. Tsironis 310 The Annular Slot Antenna in a Lossy Biological Medium .................... R. D. Nevels, C. M. Butler, and W. Yablon 314 Mathematical Expression of the Loading Characteristics of Microwave Oscillators and Injection-Locking Characteristic~ .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Fukumoto, M. Nakajima, and J. Ikenoue 319 A New Cylindrical Electron Gun for Low-Power Tunable Gyrotrons with High Magnetic Compression Ratios ... J. Y. L. Ma 323 Probe Mutual Impedance in a Rectangular Waveguide ................................... A. Ittipiboon and L. Shafai 327 Slow-Wave Propagation in Two Types of Cylindrical Waveguides Loaded with Semiconductor .............. C. M. Krowne 335 Adjustment of In-Phase Mode in Circulators Using Turnstile Junctions ........................ J. Helszajn and J. Sharp 339 LETIERS
Comments on "Optical Injection Locking of BARITT Oscillators" ........................ A. J. Seeds and J. R. Forrest Further Comments on "Integration Method of Measuring Q of the Microwave Resonators" .. . P. L. Overfelt and D. J. White Comments on "New Narrow-Band Dual-Mode Bandstop Waveguide Filters" .. R. V. Snyder, J.-R. Qian, and W.-C. Zhuang Comment on "Fast-Fourier-Transform Method for Calculation of SAR Distributions in Finely Discretized Inhomogeneous Models of Biological Bodies" ................................................. A. Taflove and K. R. Umashankar Comments on "Application of Boundary-Element Method to Electromagnetic Field Problems" ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Morita, S. Kagami, and I. Fukai Comments on "Limitations of the Cubical Block Model of Man in Calculating SAR Distributions" ..................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Hagmann, H. Massoudi, C.H. Durney, and M. F. Iskqnder Correction to "Interaction of the Near-Zone Fields of a Slot on a Conducting Sphere with a Spherical Model of Man" ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-G. Zhu and K. M. Chen Correction to "Technology Summaries for Microwave Theory and Techniques-1983" ..................... J.B. Horton
343 344 344
PATENT ABSTRACTS ................................................................... · · · · · · · · · .J. J. Daly
352
345 346 347 350 351
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TRANSACTIONS
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AND TECHNIQUES, VOL. MTT-33, NO. 4, APRIL 1985
295
Editorial
F
OUR
YEARS
AGO, Dr. Reinhard Knerr, then the TRANSACTIONS Editor, started a new section, Patent This section Abstracts, in each issue of the TRANSACTIONS. has been acclaimed by many readers. Such accomplishments are due to the effort
and dedication
R. Dietrich.
is retiring
Editor
Now
position
By means sincere thanks
Norman
from
This kind
of Mr.
of Mr. Norman this Associate
of dedicated
voluntary
service is what makes the
MTT-S a great organization. I believe that everyone reading this editorial agrees with me. Now the Patent Abstracts section is in the expert hands Daly.
I thank him for accepting
this demanding
job
and I wish him success.
and Mr. John J. Daly will take over the job. of this editorial to Norman
I would
like to express my
for the time and effort
Norman 1936.
R. Dietrich He received
TATSUO
he spent.
(S’58-M66-SM’82) the B. S.E.E.
M. S.E.E. degree from
Lehigh
He is a Distinguished
ITOH
Editor
wasborn
degree
University,
Member
from
in Berks County,
Lafayette
Bethlehem,
of the Technical
College,
PA, on November
29,
Easton,
the
PA,
and
PA. Staff
at AT&T
Bell
Laboratories,
Allentown, PA. Since joining the company in 1959, he has been involved in the development of microwave circuits and subsystems. These include traveling-wave maser amplifiers, ferrite devices, microwave integrated circuits, IMPATT and FET power amplifiers, and FET low-noise amplifiers. He has recently transmitter development. He originated the
been reassigned to high bit-rate Patent Abstracts section and
since March Associate Editor, Patent Abstracts, for the MTT TRANSACTIONS Mr. Dietrich is a member of Eta Kappa Nu and Tau Beta Pi.
John J. Daly was born
in Omaha,
NE, in 1956. He received
Systems and Terminals
Development
Department
Laboratory.
and is involved
He is currently in microwave
a member radio
1981.
the B.S. degree from
State University in 1978, and the S.M. degree from M.I.T. in 1979, both engineering. He joined AT&T Bell Laboratories in 1978 as a Member of the Technical Radio
lightwave served as
Staff in the
of the Radio
transmitter
Iowa
in electrical
design.
Systems
296
IEEE
TRANSACTIONS
ON MICROWAVE
THEORY
AND
TECHNIQUES,
MYF33,
VOL.
NO. 4, APRIL
1985
Waveguide and Cavity Oscillations in the Presence of Nonlinear Media DAN
Abstract
—This
structures
paper
containing
tered in the analysis when the constitntive
the
problem
parameters
devices operated of nonlinear
This mathematical
relations
in dispersive
The
main
effect
and the dependence
amplitudes.
These are incorporated
The development the
range
determined
by experimental and
nonlinearity drical
T
model involves of the
present
and spherical
some henristic dispersion
model
assumpequation.
will
have to be
conditions
harmonic
the question
arises,
too.
It
modes in rectangular
structures,
the geometry
of the effect of the is shown
here
structures.
that
In cylin-
affects the budget of harmon-
ear systems
OF Volterra
with
memory)
by Wiener
[16],
wave mechanics
[17],
and
analytical
results
systems (weakly in communication
nonlintheory
in 1942. For relevant
and Rice [1], and Bussgang,
Presently, applied
propagation
Recent
by Dalpe,
the fundamental
in
weakly
of guided
nonlinear
media
waves and cavity
is
oscilla-
tions. The motivation for this study is threefold. Firstly, classical results for rectangular and circular waveguides, and similar cavity configurations, are relatively easy to extend to the case of nonlinear media. At the time when relevant experimental results will be available, the present results
will
involves
serve to check
some yet unjustified
the theory
used here,
heuristic
corrections.
vices are presently
And
thirdly,
just
used to investigate
Sec-
theory only
is used for low power levels. The present method nonlinear
which
assumptions.
for some devices used to date, linearized
literature
Eherman,
and
ear acoustics Starting propagation
provides
as microwave
the linear
de-
properties
[20]. with a summary of the relevant theory, the of plane periodic waves in unbounded media
is considered.
The question
original
drical,
the
[18].
are reported
[19]. For completeness,
to problems
Graham [2]. The rapid progress in nonlinear optics attracted attention to the corresponding problems in nonlinear wave propagation (e.g., see Caspers [3]). Curiously enough, somewhere along the way the awareness of the fact that we are dealing with Volterra series has been lost (see Caspers [3], Akhmanov and Khokhlov [4], Schubert and Wilhelmi [5], and Censor [6]). The importance of Volterra’s [7] for
wave propagation
theory
used here will be summarized.
of nonphase-matched
work
scattering
and applications
and Weiner
theory
[15], solitary
nonlinear
of various materials, the present theory provides the basis for diagnosing the nonlinear properties. This kind of problem also seems to be of interest in connection with nonlin-
mode coupling.
was introduced
nonlinear
ondly,
results, when these are available.
HE ANALYSIS
see Bedrosian
of
on the field
model.
of an algebraic
and cavities in particular,
ics and produces
Weak
with the same
eqnation
into the present
of validlty
boundary
induces
of
purposes,
are the production
of the dispersion
the derivation
Therefore,
(i.e., propagate
of nonlinearity
of the present
facilitate
media. For practical
the absence of shock waves, such that afl spectral
harmonics,
In waveguides,
by
Taylor
such that the series can be truncated.
of a wave are phase matched
geometry
are investigated
tool is adequate for a description
components
which
at high power levels, or
materials
nonlinear
also denotes
tions
in metallic are encoun-
analog of the well-known
nonlinearity
velocity).
waves
Kent,
series are the functional
we deal with weak nonlinearity,
phase
of
of this kind
measurements.
series for functions. constitntive
with
media. Problems
of microwave
means of microwave The Volterra
deals
nonlinear
CENSOR
analysis
of
weakly
nonlinear
dispersive systems has been recognized recently by Franceschetti and his coworkers [8]–[14] in a series of papers dealing with the theory and application of the Volterra series, and which contain many early references. The systematic application of the Volterra series to problems of wave propagation in weakly nonlinear media is a logical approach, prescribed by the fact that these series are the functional analog of the Taylor series for functions. Recently, this tool has been applied to ray tracing [6],
tions
obtained.
of the validity
The results are applied
guide and cavity problems. The case guides and cavities is conceptually simpler. It shows the production of associated modes, as well as the effect the dispersion
equation.
and spherical
be expected
of superposition
waves is discussed and general soluto canonical
The problems
of circular,
systems shows what phenomena
(according
to the present
wave-
of rectangular waveand mathematically harmonics and the of the amplitude on
theory)
cylinshould
when curved
boundaries are present. It is shown that curved boundaries might suppress harmonic production in certain cases. Also, nonlinearity acts as a mode-coupling mechanism, as shown below. These are interesting phenomena which will have to be tested experimentally. II.
GENERAL
THEORY
OF SELF-INTERACTION
The problem is considered in the frame of electrodynamics in sourceless domains, governed by Maxwell’s equations
Manuscript The author gineering.
August 8, 1983; revised November 27, 1984. is with the Department of Electrical and Computer
Ben-Gurion
Uruversity
of the Negev,
Beer-Sheva,
En-
Israel 84105.
0018 -9480/85
/0400-0296
vxE+dB/dt=O
vxH–dD/dt=O v.D=O
$01.00 ~1985 IEEE
v-
B=().
(1)
CENSOR:
For
WAVEGUIDE
reasons
harmonic)
AND
CAVITY
explained
plane-wave
OSCILLATIONS
below,
297
periodic
solutions
(as opposed
to
nonlinear
term
ll)’)(x, E, =
~
D ‘2) upon substitution
of (2) yields
are considered
E~,el@,
6=k.
x–ut
t)=~~e ~ q’
(2)
l(q+q’)*(X’Z)c:;~(qk,
q(.o, q’k, q’ti)E~~Eq,~
(8) and similar
expression
notes Cartesian
for B,, H,, D,, where
components,
(3 is the phase of the plane wave, E~, the amplitude component vector,
of
the
of the i
and
D ‘“)(x,
t) involves
vector. Substitution
kxHq+aDq=O
(3) iyk(yk,
Dq = O,
v
ollq
= O
are identically
satisfied.
harmonic
production
due to distortion
medium.
The
arbitrary
but determined
amplitudes
relations
Eq,, Dq,, H~i, and
Bq, are not
by (3) and the complicated
to be introduced
below.
(lo)
‘Jk EYJEYk “
c (“j
in (5) (hence,
also the 4 X n-dimensional
transform in (8)) are defined as parameters of the medium, independent of field amplitudes. Consequently, (10) etc.
con-
display
that
(9)
It is noted
‘wE@k
,(2)
4>q’=Y–q
Originally
by the nonlinear
~
ycd) =
Before
going on, a few observations are in order. The periodic solution, as in (2), has been stipulated in order to include
stitutive
Dy)
where we define
kxEq–d$=O v.
[6]. If periodic
of (2) etc. in
(1) yields
and
n summations
= XYDY,ely@ is stipulated and substituted in (8), and terms satisfying y = q + q‘ are regrouped, then we have
k is the propagation
q th harmonic,
and x the position
de-
i =1,2,3,
q is the order of the harmonic,
if”)
as amplitude
dependent;
seems to be useless. What
characteristic
are dealing
this is a crucial step which has to be tested experimentally, is the following. A plane wave injected into a nonlinear
weakly
nonlinear
media,
creation of shock waves is excluded. This described by (2), because shock formation different
spectral
components
propagate
phase velocities.
Phase matching
ics are produced
in a coherent
interactions, which
although
interfere
tudes of harmonic In general,
D(E),
is adequately requires that with
manner, producing
significant
but
for
relations
nonlinearity,
are given by is
(4)
and for practical The term D(”)
n th term of a Volterra
purposes
EqJEq,k
the sum
in (4) is defined
~(n) , j.
.~
i, j,.
is a tensor
as the
EyJEyk
series
assumed).
. . . v denote
(Einstein’s
Cartesian components and summation convention is
Thus, for n =1, we obtain
on substitution
the linear
case, which
of (2) yields
(27) “/d3x1dt1cf~](x1, between
infinite
limits
of (5) are constants
of integration. characterizing
(7)
t,) e-zqo(X’”l) The parameters the system.
c~,~)
The first
wave
k 1+
L) EJ(X-X1,
and ideally
undergo
relations
B(H).
self interac-
satisfying
Maxwell’s
(4) and (5), and
The budget
of ampli-
depends on the original
of the medium. is injected
excita-
In other words, if a
into
the medium
with
AEqJ
AEq,k
E~J + ~
t –L)””
the expression
AE7J “ ~
.E,(x–xn, in parentheses
AE,k — – Ey~
(11) )
(5)
t–tn)]
in (11) equals 1.
As long as the increments in (11) are small enough to justify this approximation, (9) is valid as an approximation. According
for the q th harmonic, where e~~}(qk, qti) is the four-dimensional transform according to
harmonics
to achieve, and
exactly the right amplitudes and relative phases of harmonics, this wave will propagate in the medium without modification. The assumption implied in (9) is that the ratios of amplitudes are insensitive to incremental variation, i.e., if all the amplitudes are increased by a small factor, the ratio in (10) becomes
. . . Xn, where indices
relations
tion and the properties
a hierarchy
i.e., it will
wave is present
ampli-
plane
out that they depend
we are trying
(3) and the constitutive
the corresponding
for B. In (4), it is assumed that the leading
(4) can be truncated.
be distorted,
a periodic
tudes of various
~=D(l)+~(2)+...+D(”)+...
terms are predominant,
fields? What
waves
assumed
and similarly
will
until
periodic
constitutive weak
medium equations
i.e., local nonlinear
be weak, produce
on the variable
tion,
different
waves.
nonlinear
B(H),
the
also means that harmon-
they might
constructively,
in which
if it turns
of defining
solutions of the kind in (2) already assume phase matching, i.e., all harmonics have identical phase velocities co/ Ik 1.We with
parameters
hence, the notation
is the point
to (4) and (9), we have for each harmonic
q
(12) and if considered as an expansion of D(E) about E = O, then E3)
E1f3(E1,
E2, E3)=g3(E2,
where tion
(15)
EQh~(E~,EQ,Eg)=Zl(El,Eq)
and finally,
by dividing
treatment
(16)
in (16), one equation
of the form E.3~(E~, Ez, E3)=0. Hence,
for the nontrivial
a dispersion
relation
solution
(17)
E3 # O in (17) prescribes
u = O, which
k, U, and field
involves
amplitudes. This procedure is equivalent to writing (14) in matrix form G, = F,,A~ = O, r,s =1,. ..,6, where A = (A,) = (El, Ez, E3, Hl, Hz, H3) by imposing the condition dispersion equation
k
a six component vector, and of volubility det (F,.)= O, the
F(k, u, A)=O is obtained.
The
fact
that
(18)
(18) involves
amplitudes
is a
characteristic feature of the nonlinear problem. The corresponding equations (14) and (18) for higher harmonics are independent
(18) already
The details
for
a relation
higher
the (complex)
harmonics
amplitudes
we have sufficiently
in order
because (14) and
between
are not very important
subject. At this point, theory
systems of equations,
establish
equations
determine
eral
expressions
k and u. Hence,
can
only
E(3)(E1)3+
containing
...
serve
This
means
the higher
order
tensors.
Equation
it is understood E, related (q=
to
that
transversal
waves for which
the expression difficulty
in brackets in proceeding
is that, unlike field
(21)
is the dispersion
equa-
to analyze the present
the linear case, the representation
as a superposition
justification. In general, linear media. However,
of plane
of
waves requires
superposition is not valid in nonin weakly nonlinear systems as
discussed here, it appears plausible to assume that only phase-matched nonlinear induced harmonics will be produced with significant efficiency. This implies isotropic media as considered here, the interaction colinear
waves will be negligible.
This heuristic
that in of non-
assumption,
still requiring experimental support, salvages the linear method of superposition to the extent that new solutions may be constructed from sums of plane waves which are not phase matched. Accordingly, we take the formulas for rectangular waveguides, e.g., as given by Collin [21], recast them in terms of plane waves, and replace k, Q with qk, qti, respectively, to obtain
the harmonics. Field
Thus, the fields are given by TE
TM
Hz
CXCYe
o
EZ
o
SXSYe
Zh.n~Hv
A CXSYe BSXCYe
and reso-
the
the wave
E=0.
– Zh,n~HX
the gen-
is
equations
E+u2pt~ff(E)E=0.
[kQ-~2pCeff(E)]
the total
Maxwell’s
1)
This medium admits equation becomes
Hence,
E in c,ff (E)
the field
to D,. Manipulating
Ey
guides
(19) is
(20)
EX
to discuss metallic
Z\}) is
D=~,ff(E)E
waves.
summarized
that
in the form
to the main line of our
of the harmonic
(19)
scalar 6(”), the summa-
is inapplicable.
for
compacted
problem E3)
the equations
K and
+ ...
+ ~(2)(E1)2+
kxkx E3)
E3)=z2(El,
a simple
diagonalized by multiplying by a Kroenecker d,,, and the diagonal elements made identical, and a corresponding
(3) yields
form
the
for
convention
tion (18). The main
E2,
of using the above
of
where f and g are arbitrary functions of the arguments, and in some degenerate cases not all the arguments will be present. By elimination of the factors El on the left, in (15), we obtain two scalar equations which have the general
E2~2(&,
the relatively
and cavities,
with scalar constant
●tl)E,
=
where
Ez>Eq)=gl(Ez>Eg)
the feasibility
waveguides
there set
in the form
E1fl(E1,
1985
guides and cavities.
D, = C:;)EJ + i::~E,E~
field
not
medium
(14) define
tion (14) can be manipulated to derive expressions for HI, Hz, H3 in terms of El, Ez, Es, and these are subin the first
with
4, APRIL
isotropic constitutive relation is used. This class of problems is still general enough to display typical aspects of the nonlinear class of problems. Accordingly, we define a
waves. The six homo-
dispersion
NO.
and similar
scalar equations
previously [6]. To clarify in a somewhat primitive
stituted
rectangular
at hand. The system (14) can be used
an amplitude
explained situation
MTT-33,
in the next section,
of rectangular
In order to demonstrate
dielectric where
VOL.
RECTANGULAR WAVEGUIDES AND RESONATORS
theory -
TECHNIQUES,
displays
in (3), and for q =1,
-up$;~H,Hk
AND
We begin,
simple
to t ‘“).
this yields (k xE)l-ap:)H,
THEORY
HX
– ASXCYe
HY
– BCXSve
– EY/Z,, ‘x/ze,
nM nm
(22)
CENSOR:
WAVEGUIDE
AND
CAVITY
where in the above shorthand and
SX implies
cos ( qm ~y/b
299
OSCILLATIONS
sine of
notation
CX = cos ( qn nx/a
the same argument,
), and SY follows;
and
),
respectively,
(24)
~~ and
for
B replace
n/u
with
m/b, k~nm = (nn/a)2 +(m~/b)2, and “~~~ = k2–k~,n~. The impedances are given by Zk, ~. = kZO//3n~, -Z,, n~ = &~ZO/k,
and
ZO = tip/k,
and,
for
the present
case in
particular, ZO = p,/c.ff (E), displaying the dependence on the amplitude of the first harmonic. The following characteristics of the nonlinear problem are therefore apparent from (22): since nonlinearity produces harmonics, the fields of the fundamental mode n, m appear together with harmonic modes (i.e., at frequencies qo)
qn, qm. Since phase matching
from
the outset,
phase
and
balance
group
velocities.
Any
e.g., to filter
to add a wave at a certain change
is built
into
all these modes propagate
of harmonics,
c‘(”)
the model
with
attempt
identical
to change
the
out some harmonic
or
harmonic
of the parameters
frequency,
in (10);
results in a
hence, there is a
reshuffling of the whole spectral content of the wave. To deal with ~(n), which are susceptible to these changes, is the price we have to pay for simplifying the nonlinear model from of
(8) to (9). The second main result is the dependence
parameters
excitation
affects
therefore
possible
frequencies,
●,ff (E);
on
the amplitude impedances,
to regulate certain parameters,
usually
referred
induced
transparency.
Waveguides
hence,
the wavelength,
by changing to in
the amplitude. nonlinear
This
optics
of etc. It
waveguide
property
nonlinear
dependent.
propagating
conditions
~,ff(E)
H=
waves
will be satisfied
For small amplitudes,
IV. It will not
add (with such
that
effect is negligible.
STRUCTURES
be shown that this class of canonical
merely
a complicated
problems
extension
Bessel functions
expressions.
In fact, if we adopt
given
structures,
mathematical
case, in which
trigonometric theory
CYLINDRICAL
above,
then
as opposed
As
terms will the change
we find
that
to rectangular
is
of the
replace
the
the general
curved
metallic
geometries,
will
usu-
ally suppress phase matching, and consequently also suppress coherent nonlinear interaction. The general treatment of linear vector waves is given by Stratton
[22],
completeness,
who
cites
original
work
complicated
than the rectangular
summarized.
The general expressions
in cylindrical
coordinates
Field
Hansen.
for nonsingular
E=
fields
[22, see p. 361] is TE
ihXand+n/dr (-
For more
case, the general theory is
TM
E,
Conthem
by
and since the subject is mathematically
– pa/rXnb.#. – icopzbn 8#n/&
h/r)Dza.t.
o
A’Zan+.
H,
(k2/par)Xna.*m
H,
(ik2/p@)Xa.d$./dr
ihEbni3$n/&
is quite proper
Hz
o
A’Zbn+n
metallic
(26)
In denoting o
(27)
eln~Jn(~r)el’z-iot
– Z~, nmBCXSYSze
iA CXSYS=e
EY
Zh, ~~ASXCYSZe
iBSXCYSze
HX
iA SXCYCze
HY
iBCX SYCze
– WCycze/ze3
(23)
and A’=
k2
into
even and odd parts by defining (28)
), C, = e– ‘@” and S= = sin(qlvz/d . . . Each of the plane waves concan be explicitly obtained by recastas functions in exponentials, has k kX = + nn/a, kY = + inn/b, k== sign. Hence, we are dealing waves that can be superposed, Hence,
Bessel functions
.m
A CXSYCze/Z,, ~~
since they are not phase matched.
the nonsingular
– h 2, the summation extends on – w < n a value k ~ is computed, then the nonlinear
larger
problem
waves. However,
not
admissible. prediction
VOL
(31) can be replaced by h ~. Naturally, this means that for high-order modes, where h is getting smaller compared to
d/3.
(30)
speaking,
TECHNIQUES,
If Ah ~ is small compared to h ~, then the phase mismatch e ‘~hJI’ will be negligible for some range of z, for which h in
.—n
many
AND
Ah fl=h~-hO.
~~’”(ik x~)e’’rc”s(’-’)+znhz’’”’d~’d~
Nn=L
will
THEORY
waves
M.=
(kf
ON MICROWAVE
and
each equa-
tion yields a value k.. Since i is determined by boundary kn are conditions, (34) in general means that different associated with different h ~; hence, our assumption (31) (with one h applying to all modes n) is invalid. Still, our
SPHERICAL
STRUCTURES
At this stage, where a lot of experimentation
is needed to
check the fundamentals of the theory, there is no point in bringing in all the heavy machinery for the scalar and vector spherical wave functions. These are comprehensively covered by Stratton [22, see ch. 7]. The ideas are identical, and therefore we expect the same conclusions. The repre-
CENSOR:
WAVEGUIDE
sentation
of
proximate
similar
mode
increasingly The
follows
The problem complicated,
of nonlinear
study concentrates vide the correction
by
the
wave propagation
behavior
for
behavior
of
and
by
The present
a model
plane-wave
assuming
guides and cavities
based
dispersion
periodic
solutions.
Applications
on
relations This
to rectangular
are given. This problem
the
theory
is
wave-
of nonlinear
‘This
metallic
interaction
boundaries
and harmonic
but
[16] [17] [18]
is shown to production. [20]
plane waves, i.e., when the curvature
[21]
pronounced
of wavefronts increases. Cylindrical waves are considered in some detail, the treatment for spherical structures “is only delineated,
[15]
as the waves depart
effect is increasingly
more and more from
[14]
[19]
media, are discussed.
of curved
nonlinear
[13]
is easy because
plane waves. Once the stipulation is made that nonphasematched waves do not interact, the extension to the nonlinear case is straightforward. Results are given, and practical aspects of analyzing nonlinear devices, or measuring the The presence
[12]
are
the linear fields are given as combinations of a few (at most eight, for the fully developed case of a rectangular cavity)
suppress
[10]
on weak nonlinear effects which proterms for the leading linear results. This
recapitulated.
properties
[9]
[11]
is extremely
and mathematically.
mathematically
series,
for
[8]
an ap-
AND CONCLUSIONS
SUMMARY
physically
is described
to the
as n increases
Bessel functions
conclusions
lead
constitutes
improving
[7]
of disper-
the same lines.
V.
obtained
n, m which
of the spherical
harmonics
Volterra
to (33) and (34) will
solution,
arguments).
briefly
301
vector waves in terms of plane-wave
kfl ~ for
(on account large
OSCILLATIONS
[22, pp. 416, 417], and the application
equations
results
CAVITY
of spherical
integrals sion
AND
the above conclusions
seem to be valid
in
[22]
V. Voherra, Theory of Functionuls and of Integral and Integro-Drfferential Equations. New York: Dover, 1959. G. Franceschetti and I. Pinto, “Nonlinearly loaded antennas,” in Nonlinear Electromagnetic, P. L. E. Uslenghi, Ed. New York: Academic Press, 1980, G. Franceschetti and L Pinto, “Volterra series solutions of Maxwell equations in nonlinear media,” in A tti III Riuione di Elettromagnetismo Applicato de[ CNR, (Bari), 1980, pp. 233-237. G. Franceschetti and I. Pinto, “The functioned approach to nonlinear electromagnetic,” in A tti IV Riunione di Elettromagnetismo Applicato de[ CNR, (Firence), 1982, pp. 91-96. G. Franceschetti and I. Pinto, ‘