Idea Transcript
~IEEE
TRAN SACTI 0 NS
0 f\J
MICROWAVE THEORY AND TECHNIQUES JUNE 1987
VOLUME MTT-35
NUMBER 6
(ISSN 0018-9480)
A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
-PAPERS
Quasi-TEM Analysis of "Slow-Wave" Mode Propagation on Coplanar Microstructure MIS Transmission Lines . : ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ·..... Y. R. Kwon, V. M. Hietala, and K. S. Champlin Field Theory Design of Ferrite-Loaded Waveguide Nonreciprocal Phase Shifters with Multisection· Ferrite or Dielectric . Slab Impedance Transformers ......................................... J. Uher, F. Arndt, and J. Bornemann Modal-S-Matrix Design of Optimum Stepped Ridged and Finned Waveguide Transformers .. J. Bornemann and F. Arndt The Method of Lines Applied to a Finline/Strip Configuration on an Anisotropic Substrate ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. M. Sherrill and N. G. Alexopoulos Uniplanar MMIC Hybrids-A Proposed New MMIC Structure .............. T. Hirota, Y. Tarusawa, and H. Ogawa Unconditional Stability of a Three-Port Network Characterized with S-Parameters ..... J. F. Boehm and W. G. Albright
545 552 561 568 576 582
SHORT PAPERS
Guided Mode Characteristics of Metal-Clad Planar Optical Waveguides Produced by Diffussion .................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. J. Al-Bader and H. A. Jamid A Novel Low-Noise Downconverter System Using a Microstrip Coupled Transmission-Mode Dielectric Resonator ..... . . . . . . . . . . . . -........................................................... M. P. Mitchell and G . R. Branner Dosimetry of Occupational Exposure to RF Radiation: Measurements and Me.thods ......... S. Tofani and G. Agnesod Decade Bandwidth Bias T's for MIC Applications up to 50 GHz .................................... .B. J. Minnis
587 591 594 597
LETTERS
Comments on "The Effect of Fringing Fields on the Resistance of a Co.nducting Film" .................. M. S. Leong 601 Daly
602
Special Issue on Quasi-Planar Millimeter-Wave Components and Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
605
PATENT ABSTRACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . • . . . . . .: . .......... : .•.. J. J. ANNOUNCEMENT
o
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TRANSACTIONS
ON MICROWAVS
THEORY
AND
TECHNIQUES,
MTT-35, NO 6, JUNE 1987
VOL
545
Quasi-TEM Analysis of “Slow-Wave” Mode Propagation on Coplanar Microstructure MIS Transmission Lines YOUNG
RACK
KWON,
VINCENT
KEITH
Abstract mode
— We
present
propagation
heavily
doped
semiconductors
surements
on four
Excellent
agreement
propagating
low-loss
is observed. ignored
mode propagation, frequencies
wfdeh
prop~ation
analysis MIS
theoretical from
shows that
afong with significant
losses
of
are included
at frequencies
the
metal,
“fsdl-wave” in the theory
below 25 GHz
results
with
a quasi-TEM wavelength
which
This on
mea-
1.0 to 12.4 GHz.
the “slow-wave”
fines is, in fact,
published
lines
have
treatments
mode mode.
reduction
been
STUDENT
MEMBER,
of “slow-wave”
transmission
at frequencies
transmission
Conduction
the attenuation
and compare
is found,
in previously
quasi-TEM coplanar
such stroctnres
on these
Relatively
a simple
on micron-sixe
M. HIETALA,
S. CHAMPLIN,
tacitly
of “slow-wave”
and are shown to dominate and to still be significant
at
paper
cently
c
OPLANAR
INTRODUCTION
been reported
circuit
model
analyze
(MIS)
that
Schottky
[10] -[12]
to
metal-insulator-semiconductor
presents
AND
a simple
quasi-TEM
[9]. The theory
as well as semiconductor
and
analysis
12.4
is similar
of
GHz
using
transmission
lines
rnicrostrip
coplanar from
quantities four
metal losses an equivalent
to those used by others
and MIS derived
with
includes
losses and employs
Schottky
Quantities
compared I.
IEEE,
coplanar microstructure MIS transmission lines on heavily doped semiconductors propagating the” slow-wave” mode. Preliminary measurements on such a structure have re-
[14].
up to at least 100 GHz.
MEMBER,
IEEE
transmission
transmission
lines
this quasi-TEM
measured on
[13],
theory
are
in the range from
1.0
coplanar
fabricated
to
lines
MIS N+
microstructure
silicon.
Excellent
transmission lines on true (i.e., not semi-insulating) serhiconductor substrates have applications in both mono-
agreement between theory and experiment is observed. Such close agreement corroborates the assumption that the
lithic
“slow-wave” mode propagating on these microstructure MIS transmission lines is, in fact, a qumi-TEM mode and
microwave
high
integrated
speed digital
structures
have been studied
and coworkers, factor”
circuits
integrated
and very
(VHSDIC’S).
experimentally
who reported
and characteristic
(MMIC’S)
circuits
by Hasegawa
measurements
impedance
Such
of “slowing
magnitude
[1] and of
attenuation coefficient [2] as functions of frequency over the range from 700 MHz to 4 GHz. Their papers refer to a quasi-TEM analysis of their coplanar MIS structure. However, their analysis does not include the effects of losses; nor does it explain the observed frequency dependence of the experimentally More extensive transmission by
other
treatments
determined numerical
quantities. analyses of
lines on semiconductors
investigators have
mode-matching
been (MM)
[3]-[8]. based method
upon
so-called either
[3]–[6],
Relatively significant
MIS
the
full-wave classical
the spectral-domain
analysis (SDA) method [4], [7], [8], or the finite-element method (FEM) [8]. Although such computational techniques have generally included semiconductor losses, they have tacitly assumed perfect metallic conductors. Accordingly, metal losses have been systematically ignored.
be treated by fairly
elementary
low-loss
propagation
wavelength
microwave reduction
techniques.
is predicted
and confirmed
by the experiments.
that coplanar heavily doped
microstructure semiconductors
along
with
by the theory
Such properties
suggest
MIS transmission lines on may be useful as transmis-
sion media for fabricating distributed components of MMIC’S. The theory shows that metal losses of the experimental
coplanar
have been described
These
can therefore
transmission
cies below
loss mechanism the previously [3]-[8], are
lines are very significant
100 GHz
and, in fact, constitute
at frequencies published
below about 25 GHz. Thus,
“full-wave”
because of their systematic
deemed
microstructure
inadequate
for
transmission II.
at frequenthe dominant
analysis
omission
accurate
treatments
of metal losses,
analysis
of
these
iines.
EXPERIMENTAL
RESIJLTS
The geometry of the experimental microstructure transmission lines is shown in Fig. 1. These structures employed coplanar
aluminum
Manuscript received November 8, 1986; revised January 24,1987. This work was supported in part by the U.S. Army Research Office and the
substrate
by a thin layer of Si02. The SiOz was grown
National Science Foundation under Grant ECS-83-16246 and by a grant from the Microelectronic and Information Sciences Center of the University of Minnesota. The authors are with the Electrical Engineering Department, University of Minnesota, Minneapolis, MN 55455. IEEE Log Number 8714119.
cm – 3) by wet oxidation at 1000”C. The center conductor and ground planes were fabricated by evaporating aluminum onto the SiOz and defining the conducting structure using standard photolithographic and etching
0018 -9480/87/0600-0545
antimony-doped,
strips N+
$01.00 01987 IEEE
separated
silicon
from
(80 ($2 ocm)-l,
an N+
silicon on
Nd = 3X1018
546
IEEE
TRANSACTIONS
ON MICROWAVE
THEORY
~
2.0-
& & ~
1.5-
AND
TECHNIQUES,
VOL.
MTT-35, NO. 6, JUNE 1987
[1
g
my
-Almnirmrn
%
1.o-
2 g
-
4
N+ Silicon
(a)
(),5 0.0
1 1.0
5.0 7.0 9.0 Frequency (GHz)
3.0
11.0
(a)
Contact
Pads
Contact
Pads
(b) Fig.
1.
(a)
Cross-sectional coplanar
view and (b) plan view MIS transmission line.
of
00 ~
micron-sized
1.0
3,0
5.0
7.0
9.0
11.0
Frequency (GHz) (b)
TABLE I DIMENSIONS (S, W, h ) AND CAPACITANCE
SCALING
(K)
FACTOR
LINES
OF THE EXPERIMENTAL
J
s
Line
w
h
K
1
4.2
6.0
0.53
1.3
2
4.2
14,0
0.53
1.3
3
8.7
9.5
0.28
1.1
4
4.7
13,5
0.28
1.2
All
dimensions
are in micrometers. 1.0
techniques.
The resulting
1) are summarized lines,
the wafer
line dimensions
in Table
thickness
(d)
I. For
(defined
all four
3.0
5.0
in Fig.
9.0
11.0
9.0
11.0
(c)
transmission
was 530 pm, the length
7.0
Frequency (GHz)
(1)
was 2500 pm, and the metal thickness (t) was 1 pm. S-parameter measurements were performed over the range from 1.0 GHz to 12.4 GHz with a computer-assisted HP 841OB test set employing a 12-term error-correction procedure. Contacts to the lines were made with a pair of Cascade Microtechl microwave probes which provided rapid,
reproducible
measurements
free of packaging
siderations. The complex characteristic impedance complex propagation coefficient were derived from
conand the
measured S parameters after first subtracting the capacitive effects of the contact pads. As seen in Fig. 2, the attenuation of each of the lines is quite small. The maximum observed attenuation, approximately 2 dB/mm (line 3) at 12.4 GHz, is, to the best of our knowledge, considerably less than any microwave attenuation
value reported
on room-temperature imaginary
parts
by others for transmission
silicon.
Fig. 3 displays
of the characteristic
lines
the real and
impedance
as func-
o.o~ 1.0
Microtech,
Inc., P.O. Box 1589, Beaverton,
OR 97075-1589.
7.0
Frequency (GHz) (d) Fig.
2. Attenuation versus frequency. (d) Line 4. Solid lines are theoretical.
tions
of frequency.
acteristic independent impedance mately
440.
which
of frequency, commonly
(a) Line 1. (b) Line 2. (c) Line Symbols are measured values.
3.
One sees that all four lines have char-
impedances
in particular, 1Cascade
5.0
3.0
are
nearly
real,
and of the order
used in microwave
has a characteristic
circuits.
impedance
relatively
of the 50-LI Line
of approxi-
2,
KWON
et al.:
ANALYSIS
OF SLOW-WAVE
MODE
PROPAGATION
547
60-
8-
1 . . . . . . . . . ,. .,,.,.
40-
.=
,,.
8 ~ L
,.
F
67
Slow-Wave Factor
;
a20~“ -
d
> D 42 m * 28 0 >4
‘0’ 0-
& -20-
,. .,. “.. ”ll” ...
‘“”’””””” ISm
u~ and i3~ are the conductivity and of the aluminum. Since current
the ground
center
plane contribution to R ~ is ignored. current, which parallel the current of the
conductor,
mediately
skin depth, densities in
planes are much less than in the center conduc-
under
flows
in
the
N + semiconductor
the center conductor
to loss [11], [12]. This current
R ~ in
current Fig.
a correction
are represented
the longitudinal
semiconductor
current flows in addition to the longitudinal metal, a parallel connection was employed of Seguinot
to
between the magnetic Losses associated with
in the semiconductor
5. Since
im-
and also contributes
is essentially
the assumption of no interaction field and the N+ semiconductor.
series connection
which can be approximated
(6)
1
respectively,
termined
1 L=—————— 4C2COF
by
for t < ?lm
Um?lms
by
This leads to [15]
model
of the center conductor:
omts
longitudinal
mapping.
expressed in ohms
1
transmission line. Since the magnetic field of the present structure is nearly that of normal CPW, C‘ can be deby conformal
t
in our simple
r
resistance
tor, the ground Longitudinal
can be expressed as
1987
0
loss resistance,
is approximated
and
L=— where
Fig. 5. This longitudinal per unit
to first-order
and
6, JUNE
“Slow-wave” mode equivalent circuit of coplarm microstructure MIS tr~smission line used in quasi-TEM analysis.
at
on an insulating
R
1
field is much
(tl~ = 56 pm
separation
results in slow-wave quasi-TEM
ductance
(
field does not “see” the N+ layer at all. the magnetic field distribution is nearly that coplanar
substrate
of the line are so
since the skin effect is unimportant
the semiconductor the magnetic Accordingly,
the center confreely penetrates
range of magnetic
the semiconductor’s
10.0 GHz).
field
because the dimensions
that the quasi-static
less than
below
the magnetic
NO.
Ct
is based upon
cause of the low impedance of the N+ semiconductor, most of the electrical energy is thereby confined to the ductor
MIT-35,
~
that transverse dimensions are so small fields are, to first-order, quasi-static. Be-
lossless insulating
VOL.
L
m
ANALYSIS
of our quasi-TEM
the assumption that transverse
TECHNIQUES,
r----+k
and 0.9 at 2 GHz
obtained by Hasegawa and coworkers [1], [2] using a much larger MIS coplanar structure on GaAs. III.
AND
current in the rather than the
et al. [13], [14]. In our model,
R= is given by
by [15] (7)
‘( In
)
2(1+JZ) (1 -w) T
where us and 6S are the conductivity respectively, of the N+ semiconductor.
for O.7071, and K3(rq) >1, respectively. Table I summarizes the possible combinations
ITI 1, then for those terminations lS{~l 1, all passive loads on port 2 will result in stability at port 1. It is not necessarily true that all r2 terminations result in stability
~=lJ(r3)l
- [(s22 – sfiAsJ)+(A;zA33
OF THE CONDITIONS
However,
F21
cl(r~)
SUMMARY
and
cl(rJ
T=
IV.
in the S;j’-plane.
+2Re
[S,:AJjrk]
= (ALZAjj–Sk~Aj)I’/ + (As+ + (Slisjj
SkkAkk – S,, A,, – Sj,Ajj)rk ‘Akk).
ANALYSIS
IAjjl’lrkl’
OF MEASURED
THREE-PORT
S-PARAMETERS
The three-port
S-parameters
of an NE46734
transistor
biased at VCB =10 V and Ic = 50 mA were measured with an automatic network analyzer at 2.4 GHz. An error-correcting
scheme [10] using the indefinite
property
for this
three-terminal device (with base port 1, emitter port 2, and collector port 3) was used to obtain the following S-
BOEHM AND
ALBRIGHT
: UNCONDITIONAL
STABILITY
OF THREE-PORT
NETWORK
585
TABLE I COMBINATIONSOF i, j, AND k FOR EQUATIONS (23), (29), AND (30) I
i
j
k
2
3
1
3
2
1
1
3
2,
3
1
2
1
2
3
2
1
3
TABLE II PREDICTEDCOMMON-COLLECTORS-PARAMETERSFROMTHE THREE-PORTMATRIX AND MSASURSD COMMON-COLLECTOR
S-PARAMETERS
~
I
%2
I
I
0.773 &j.J&
I
0.790 ~
0.017
3.51. I
Fig. 3. Stability between ports 3 and 1 viewed as a function of 17z. (Crosshatched area represents values of ra where K3 (r2) >1, F31 >
lqr,)l,F1,
> I.qr,y.)
Fig. 4. Stability between ports 1 and 2 viewed as a function of rq. (Crosshatched area represents values of r~ where K3 (r3 ) >1, F12 > Fig. 2. Stability between ports 2 and 3 viewed as a function of rl (Crosshatched area represents values of 1’1 where K3 (rl) >1, F23 >
l~(rl)l,
l~(r,)l, fil > I.r(r,)l.)
fi, > I.qrl)l.) ters were measured
parameters:
two-port tor port
[030@X
1
0.659/
– 74.01°
1.052/14.53°
In order
S-parameters,
o’~’k
O.’”k
0.466/55.53°
0.526/10.90°
0.06i/
0.346/
– 60.24°
to check the validity the common-collector
1
– 102.28°
. 1
of these error-corrected two-port
S-parame-
and compared
with
the corresponding
S-parameters found from terminating with a short circuit (l_’3 = 0.998/176
the collec.940). The
corresponding data of Table II show that good agreement exists between the measured and calculated two-port Spararneters. I’-plane plots of rl, I’z, and r~ are shown in Figs. 2, 3, and 4, respectively. These plots show which terminations satisfy
conditions
i), ii), and iii) respectively,
in Section IV.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHN?QUSS, VOL. MT”C-35, NO. 6, JUNE 1987
586
Any
such set of rl, I’z, 173which satisfies i), ii), and iii) will
guarantee that the three-port is unconditionally stable. In addition, if one wishes to determine two-port seriesfeedback
configurations,
information. emitter ports
Figs. 2–4 give vital
For example,
(port
[5]
[6]
termination
a capacitive
terr&ation
2) results in unconditional
stability
in the between
[7] [8]
3 and 1 (see Fig. 3). In Fig. 2, one sees that feedback
bver a large area in the base lead results in unconditional stability between ports 2 and 3. The designer may choose a particular termination in these areas after considering gain or noise requirements. VI. An
analytical
lishes
conditions
CONCLUSIONS
solution
has been presented
for the unconditional
which
stability
[9]
[10]
S. Tanaka, N. Shumonura, and K. Ohtake, “Active circulators—The realization of circulators using transistors,” Proc. IEEE, vol. 53, pp. 260–267, Mar. 1965. W. H. Ku, “ Stabifity of linear active nonreciporical N-ports; J. Franklin Inst., vol. 276, pp. 207-224, Sept. 1963. ‘:.Zur Widerstandstransformation Linear 2 NH. Kleinwatcher, Polej” Arch. Elek. (,lbertragrorg, vol. 10, pp. 26-28, Jan. “1956. M. F. Abulela, “Studies of some aspects of linear amplifier design in terms of measurable two-port Wd three-port scattering parameters: Ph.D. dissertation, Manchester Univ Manchester, England, 1972. D. Woods, “Reappraisal of the unconditional stability criteria for active 2-Port networks’ in terms of S-parameters,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 73-81, Feb. 1976. J. F. Boehm, “Microwave oscillator design using two-port and three-port scattering parameters,” Master’s thesis, Univ. Illinois, Urbana, Oct. 1985.
estab-
of a net-
work described with three-port S-parameters. It has been shown that nine conditions, dependent O! the terminations,
must be satisfied to guarantee that lS;l o
,Z,() .
Manuscript received October 1, 1986; revised February 4, 1987. The authors are with the Department of Electrical Engineering, Kitig Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia. IEEE Log Number 8714116.
All
n;
complex
+ jn”2
Z< ().
refractive
where
n‘2 represents
sents the imaginary
0018-9480/87/0600-0587$01.00
01987
IEEE
(2)
indices will be written part.
the real part
in the form
n2 = n’2
of n2 and n“2
repre-
588
IEEE
TRANSACTIONS
B. Theoiy
ON MICROWAVS
where
We assume that the refractive
index
increment
f3=,k(n?&f
n; — n; — o
cylinder
functions,
(14)
and
Solution We first
develop
results
analytical
sought
be seen that with
when
scheme of Section
of the propagation
increased
of (22) maybe compared III.
solutions
with
this the
In the following for
the
real
and
constants.
equation
by substituting
n;
!V = Beez
(3) in
the metal
of the Eigenvalue
Equation
solve (15) for the variable
q as follows:
q=:{c2[(2v+l)*+l] (15)
The
accurate
with
of side
(13)
of (4) is [16]
DV are the parabolic
gives
. . . . It will
give
constants
the right-hand
and decreases rapidly
of v =1,3,
of the numerical
imaginary
V = A(R/Q)’’4DU((), where
(22)
subsection,
{ = 2~~/4@/*e-Jd4 The solution
r~<s
quantity
whose roots
mode propagation
This suggests that solution
the real part
approach 3Q4
equation
values of the guided
of (22) is a small 5Q6 —+— 2Q;
is the eigenvalue
the complex
cladding
+(2v+1)([(2v
C; -4 Q2(k2
+Cl)
).
(23)
We write
for n2 ( z) and is given by (16)
+1)2+2]
v=q+jti where
q =1,3,...
and 8 is a small parameter.
(24) We designate
the
IEEE
TRANSACTIONS
real parts
of
ON MICROWAVE
THEORY
AND
TECHNIQUES,
Q. by q, and Qo., mpmtively.
q and
and (24) and noting
that
Cl,
MIT-35, NO. 6, JUNE 1987
VOL.
using (23)
C2, and Q2 are real quantities,
we
589
The complex ne~~ is now obtained written
from
(25), (26), and (33) and
in the usual form:
obtain n.,, = (n~~, + jn~$ )1”2
Qor=~(c2[(2q+l)2+l] +(2q+l) The
real
III.
[(2q+l)2+2]
part
of
C; -4 Q2(k2
is d&ermined
n~ff
+Cl)
}.
The
(25)
from (25) and the first
NUMERICAL
wave
and
,6= knc,, .
(34)
SOLUTION OF THE WAV)? EQUATION
equation
(3)
with
numerically
by using the central
T(Z+AZ)=
2W(Z)–
V(Z–
n’(z)
given
(1)
is solved
n~f~] Y(z)(Az)2
(35)
difference
by
formula
AZ)
equation of (5) as follows: n~;~= n? —Qor
while the imaginary part of n~ff may be written in terms of Qoi, the imaginary part of Q., as n~~ = – Qo, .
(27)
In order to determine the imaginary part of n~ff, we use (8), (9), (15), and (24) by writing
1
1 q+jil=
–-+ 2
where
A z is a small
n’(z)
undergoes
applied
for z >0.
conditions region
z 0
results
the solution
are
in the
by ‘f?~., According
to
we have
j$?oi
c’
k2(Qor+jQoi)–~
2J-
-k2[n2(z)-
(26)
1–~
[
V+(o) =;w(o)
1[
(36)
1 (28)
:V+(o)=; where
flor
and
respectively.
flOi
are the real
These are seen from
and
imaginary
parts
of
(8) to be
By utilizing
Go, = k’ + Cl – k2QoTC2/Q2
:v(’o).
(37)
Q., a Taylor
series expansion,
we write
(29) Y+(O+AZ)
and From
=S?+(0)+
d’~+ (0) Az–=.
(38)
;~*-(O).
(39)
(36) and (37), we have
(30) V+(Az)=; The fact that In order
n~~ <
n$f
to determine
Utilizing
a suitable
equation
is written
–2@/4]Q2~/4
n~f,
we use the method
expansion
of the left-hand
outlined
in [7].
side of (22), this
in the form
‘(-i)
.
The right-hand
side of the above equation
the aid of (16), in which,
for convenience,
B is taken
giving
to be nl/n~,
=y+ the complex
the imaginary
ji$u.
expression
part
from (31)
for
of (31) to order
v from
(24) with
q =1,3,...,
which
Initial solutions
8 gives
for
obtained
the values
of u for the first
0.81026,
real parts
and 0.72023,
of SIO and Q,
very small the quantity
within part
five modes
respectively.
are 6,
Note
mode
the square
of the metal
brackets.
of (28), the solution
only
parts are
are dominant
By using
the in
and,
(30) and the
of n~~ is found
Ti-diffused
‘3”[1+(k2Q’’ai%2)l TM
,,2. P(n:–
Qor–nm
2,
(33)
labeling
The accuracy terms
(40).
This
This
of TM
because
for both
with the & model
series expansion
at
(26), by
in all cases
corresponding A = 0.6328
modes
pm.
are shown
in
modes in metal-clad
of the existence
is discussed
of the
in [3], and our
that used in this reference.
I that the analytical
of a simplified
of the Taylor
cladding
question
is consistent
Table
by (25), numerically
parameters
three TE and TM
be confusing wave.
as given
n; = – 10.3 -- jl.0, gold
that the labeling
are in good agreement
. Im
8 in
those obtained
The waveguide and
the first
may
plasma
It is seen from
1
constants, with
n: = 4.8469,
of /3 for
mode
III.
LiNb09
I. It is noted
surface
of
the aid of (25), (26), and (33).
of the propagation
waveguides
4( ]Q2&.)3’4
also the value
of
of the value of ne~~
NUMERICAL RESULTS AND DISCUSSION
are n; = 4.9665,
Table =
with
the scheme of Section
Values
to be
hence,
by and
is assured. The efficiency
by a good estimation
(33), and (34), are compared
to
scheme of (35) are provided to (36) and (40), respectively,
when convergence
is calculated
Values
1.18194,
that
are used since their imaginary
and that the properties
according
the scheme is facilitated each
(40)
the iterative
~+ (0) and V+ ( Az)
IV.
&
with
constant
it is seen that 1’+ (0)=1.
values
estimate
imaginary
evaluated
the arbitrary
=l+AzpO
for
0.94530,
maybe
G
.(I-V)(,+;)(I-; )(1+:)...
where
+Az;
r(’+i)
po
Using
V-(0)
has been used in (28).
and numerical
results
and ~“. in which
only the first
of the Gaussian
profile
two are
IEEE TRANSACTIONS ON M3CROWAVE TH3?.ORYAND TECHNIQUES, VOL. MTT-35, NO. 6, JUNE 1987
590
TABLE
I
VALUES OF THE COMPLEX PROPAGATION CONSTANTS OF THE METAL-CLAD
●z in
Andyticd
(vm)
valtmt
of 6’
in
Numaricd
Analytical
Nurnarkd
●
value,
Valuas
Vdum
(h)
of 9’
of e“
of 6“
WAVEGUIDE WITH GAUSSIAN INDEX PROFILE Analytical
Numadcd
Analytical
Numarical
alum
valum
valuas
dims
of 0’
of &
of &
of 6“
TEO Mods
2
22.021%
22.024S2
0.79J04(-4)
0.7s740(-4)
2
22.02126
22.01726
0.32207(-S)
0.31794(-s)
J
22.0S474
22.0%J6
0.46 120(-4)
0.44ss4-4)
5
22.05474
22.05205
0.18761(-S)
o.le92q->)
4
22.07221
22.07524
0.s0965(-4)
0.20299(-4)
4
22.0722
22.070ss
0.12604(-3)
0.12789(-s)
0.13752(-4)
22.09546
22.09459
o.os6aJ(-s)
0.05783(-s)
I
7
22.09546
22.09590
0.1S9S2(-4)
10
22.10s01
22.10527
o.oem9(-4)
O.mzl x+
10
22.10s01
22.10449
0.03J87(-3)
0.0$442(-s)
15
22.)1255
22.11269
0.04582(-4)
0.04s41(+
1s
22.1 125s
22.11226
0.01869(-3)
0.01296(-s)
O.%szq-s)
O.nfml(-s)
7
TE, WI&
TM, Mode
2
21.90 JO>
21.9159S
0.9040J(-4)
o.7171q4)
3
21.96E2S
21.972S9
o.s777q-4}
4
22.004JS
22.C0676
o.4062q-4)
7
22.03457
22.0s518
10
22.075S8
15
22.09258
2
21.90JOS
21.9U9S2
0.52999(4)
5
21.96820
21 .%m
0.2S41 1(-S)
0.21442(-s)
o. Ja692(-4)
4
22.004s5
22.00510
0.1648 s(-3)
O.1 5820(-3)
0.19401(-4)
0.19222(-4)
7
22.054S7
22.053S7
0.07891(-s)
0.07ss9(->)
22.07&M
0.1182
0.1 167J(+
10
22.07558
22.07491
0.04S1 J(-s)
O.o.w I e(-~)
22.092KI
0.066%(-4)
o.06Jal(+
15
22,W2S8
22.0921a
0.02704(-S)
0.027 lq-5)
s(-4)
TE2 Mods
TM2 Nbda
2
-
2-
0.44JOJ(-4)
0.3913J(-4)
4
21.94361
21 .9460e
0.179 JI(-J)
0. I% S9(-J)
7
22.01S76
22.0152s
0.09 124(-J)
Oaoeoq->)
10
22.04740
22.04681
0.05700(-J)
0.0%
22.07J17
22.07275
0.0J261(-J)
o.oJ24q-J)
21.90764
o.6m7q-4)
4
21.94561
21.94976
7
22.01576
22.01730
0.22470(-4)
0.21628(+)
10
22.04740
22.04s11
o.1401a(-4)
0.13726(-4)
retained parabolic
22.07S17
Owl 11(-4)
22.07SS0
is of interest.
In this case, the Gaussian
and the complex
expression
(25), (26), and (33) by making
Cl=
0.07921(4)
profile
becomes
of n~ff is obtained
21.WW4
15
from
C2 = O. We thus have
dependent enough
(2q+l)n1@ (41) a= k where
mode, given
and the diffusion
mode
similar
order
in contrast
to the mode
16(-J)
depth. when
This condition
by (33) and (43). The mode
For a,
is
any large
is also true
attenuation
increases
attenuation
in wave-
guides with linear attenuation
[5] and exponential [6] profiles. In addition, the of TM modes is approximately an order of magni-
tude greater
than
that of TE modes. ACKNOWLEDGMENT
(42) l%e
authors
Minerals
and
order
the two models become
for the mode to be well guided.
n~~
with
= n; –
on the mode
particular for
n#f (parabolic)
0.17S90(-J)
3
2 1.S9384
15
0.24260(-S)
21.90S57
0.4s970(-4)
s
for
would providing
like to thank
the University
the facilities
and
M.
of Petroleum K.
Butt
for
& the
manuscript.
n~~ (parabolic)
=;
4(y)3’21m[p(n,,f:ni,1/2]. (43)
mFERENCES
[1]
Equations
(41) and (43) are the same as those in [7] obtained
the parabolic Gaussian
model.
model
Examination
tends
to
the
for
of (25) shows that for n$~, the parabolic
model
in
a manner
[2]
S. J. A1-Bader and H. A. Jamid, “Comparison of absorption loss in meta3-clad optical waveguides,” IEEE Trans. Mzcrowaue Theo~ Tech., vol. MTT-34, pp. 310-314, Feb. 1986. E. M. Garmire and H. Stoll, “Propagation losses in metat-film-substrate optical wavegnides,” IEEEJ. Qwmtum Electron., vol. QE-S, pp. 763-766, Oct. 1972.
“---
TRANSACTIONS
ON MfCROWAVE
THEORY
AND
TECHNIQUES,
VOL.
MTT-35, NO. 6, JUNE
1987
591
A. Reisinger,
“Characteristics of optical guided modes in lossy wavegnides~ Appf. Opt., vol. 12, pp. 1015-1025, May 1973. I. P. Kmninow, W. L. Mammel and H. P. Weber, “Metal-clad waveguides: Analytical and experimental study,” Appl. Opt., vol. 13, pp. 396-405, Feb. 1974. M. Masuda, A. Tanji, Y. Ando, and J. Koyama, “Propagation losses of guided modes in an opticaJ graded-index slab waveguide with metal cladding,” IEEE Trans. ikficrowaue Theo~ Tech., vol. MIT-25, pp. 773–776, Sept. 1977. T. Findakly and C. L. Chen, ‘