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INTEGRAL-OPERATOR ANALYSIS FOR SCATTERING AND

COUPLING IN OPEN-BOUNDARY DIELECTRIC WAVEGUIDES

BY

Shuhui Victor Hsu

A DISSERTATION

Submitted‘to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering and Systems Science

1984

ABSTRACT

INTEGRAL-OPERATOR ANALYSIS FOR SCATTERING AND
COUPLING IN OPEN-BOUNDARY DIELECTRIC WAVEGUIDES
BY

Shuhui Victor Hsu

Integral-operator analysis is employed to study two
classes of commonly encountered problems in Open-boundary
dielectric waveguides. They are the scattering by obstacles
along the waveguide and the coupling between waveguides in a
multi-guide system. First, in the scattering treatment, an
equivalent polarization current is identified from the
contrast of refractive indices between the discontinuity
region and the unperturbed background. Exploitation of this
current establishes an electric-field integral equation
(EPIE) describing the unknown discontinuity field, which,
leads to the formulation of scattering coefficients.
various solutions to the EFIE are discussed, including the
Fourier transform method, Method of Moments, and iterative
sloutions etc. In the treatment of waveguide coupling, a
similar procedure yields a system of simultaneous EFIEs
describing the coupled system-mode field for each waveguide.

Subsequent coupled-mode perturbation approximation yields

modal amplitude coefficients and the coupling coefficients.
Applications of the above analysis and solutions are
demonstrated via one-dimensional slab waveguides. Merits of
various solution approaches are evaluated. Moreover, the
correctness of the obtained results are verified; this
consequently confirms that the integral-operator analysis
provides an alternative to the conventional boundary-

value analysis.

dedicated to

my father, Chin-lin Hsu

my mother, Su-ying Kuo

and my wife, Sufen Susan Hsu

for her patience and years of
tolerance. '

ii

ACKNOWLEDGEMENT

The author wishes to gratefully acknowledge several
persons, without whom this dissertation could not have been
completed. My achievements, great or little, were possible
through their participation.

My father taught me a Chinese proverbr "When you drink
water, think of its source.” Dr. K. M. Chen, a member of my
guidance committee, has been fundamental in my achieving the
Ph. D. degree at Michigan State. He was instrumental in
first bringing me to this university and he has provided
many years of caring friendship. I want to give him my
special thanks.

Dr. Dennis P. Nyquist has been my academic advisor and
the chairman of my guidance committee. Be inspired and
motivated me through many years of graduate studies,
especially in the formulation and development of my research
subject. He was extraordinarily patient and persistent in
his neverending encouragements. His very positive influence
on my personal and technical development will carry forward
into my future endeavors. I am forever grateful to him.

Dr. Bong Ho served on my guidance committee and
provided valuable assistance in many ways beyond the

academic realm. I thank him and the other members of my

iii

committee, Dr. Jes Asmussen and Dr. Byron Drachman.
Moreover, I am appreciative of the support and kind
assistance received from Dr. John Kreer, Chairman of the
Electrical Engineering and Systems Science Department, and
from the Department staff: Enid Maithand, Pauline Van Dyke,
and Ginny Mrazek. Special thanks go to Dr. S. Mao for his
friendship and encouragement, to Joanne Weiss and her family
for being my first hosts in the United States, and to Jane
Chen and her husband Peter for her typing of the manuscript
and their kind hospitality.

These generous people and many others like them whom I
haven't mentioned have given very much to me; Being from
another part of the world, I can truly recognize that it is
they who make this University and this country so great. In
meeting the many challenges of the past few years, their
help was essential to me. I gratefully thank every one of

them.

iv

TABLE OF CONTENTS
Page
LIST OF TABLES ....................................... vii
LIST OF FIGURES . ...... ............................... viii
Chapter
I. Introduction ...................................... 1
PART I

II. Integral-Equation Formulation for Scattering by
Dielectric Discontinuities along Open-Bounday
waveguide ...................................... 8

2.1 Equivalent Current Description for the
Discontinuity Region ....................... 9
2.2 Formulation of EFIE for Unknown Field in the
Discontinuity Region ....................... 13
2.3 Depolarizing Dyad for Electric Dyadic Green's
Function ................................... 19
2.4 Scattering Coefficients and Mode Conversion .. 25
2.5 Solutions to Electric Field Integral Equation. 30
2.5.1 Approximate Radiationless Solution to
EFIE ................................. 31
2.5.2 Approximate Solution to EFIE for
Radiating Discontinuity .............. 36
2.5.3 Moment-Method Numerical Solution for
Discontinuity Field .................. 38
2.5.4 Iterative Solution to EFIE ............. 41

III. Application of Integral-Operator Analysis to
Scattering by Slice Gap Discontinuity in a
Dielectric Slab wavegUide OOOOOOOIOOOOOOOOOOOOOO 43

3.1 Introduction ................................. 43
3.1.1 TE Propagation Modes for Slab Waveguide
and Their Normalization .............. 47
3.1.2 Scalar EFIE for TE Mode Scattering
along Slab Wavegudie ................. 49
3.2 Approximate Treatment for scattering along
Slab Waveguide ............................. 53
3.2.1 Analytical Solution Without Radiation
Contribution ......................... 53
3.2.2 Approximate Solution for Radiating
Discontinuity ........................ 59

Chapter Page

3.3 Method of Moments Numerical Solution ......... 61
3.3.1 Discretization of Scalar 2-D EFIE to

MoM Matrix Equation .................. 61

3.3.2 Numerical Results ...................... 70

3.4 Iterative Solution ........................... 79

PART II

IV. Integral-Operator Formulation of Coupled
Dielectric Waveguide System ................... 82
4.1 Equivalent Polarization Description of
Heterogeneous Waveguide Core ............... 83
4.2 Electric Field Integral Equation Description
for Guided Waves Supported by Open-Boundary
Dielectric Waveguide System ................ 85
4.3 Homogeneous EFIE's for Natural Surface-Wave
Modes along Coupled Waveguide System ....... 91
4.4 Integral-Operator Based Coupled-Mode
Perturbation Approximation ................. 99

V. Application of Integral—Operator Analysis to
Coupled Slab Waveguide System ................. 110
5.1 Introduction ................................. 110
5.2 Specialization of EFIE for Coupled Slab-
Waveguide System ........................... 111
5.3 Fourier-Exponential Transform Solution for
Step-lndex Slabs ........................... 116
5.3.1 Coupled TE Modes ....................... 116
5.3.2 Coupled TM Modes ....................... 124
5.4 Perturbation Approximation ................... 126
5.4.1 Specialization For Coupled TE Modes .... 126
5.4.2 Degenerate Coupled TE Modes between
two Slab Waveguides .................. 130
5.5 Numerical Results ............................ 131

VI. Conclusion ...................................... 146
LIST 0? REFERENCES ................................... 151
APPENDIX A SLABZ SOURCE LISTING ...................... 156
APPENDIX B SLABZ OUTPUT SAMPLE ....................... 171

APPENDIX C OSWDSC SOURCE LISTING I O O O O O O O O O C O O O O O O O O O O 175

APPENIDXDOSWDSC OUTPUT SMPLE OOOOOOOOOOOOOOOOCOOOOO 180

vi

Table

3.1

3.2

LIST OF TABLES .
Page

Reflection and transmission coefficients for
TEO slab-waveguide mode incident upon

dielectric-slice discontinuity of various
configurations, as calculated by several

methods; resulting reflected, transmitted,

and radiated powers included. (2t/lo=0.3,

“131.6, “231.0) ...........o..................... 72

Mode conversion coefficients (reflection and
transmission) for TEO slab-waveguide mode

incident upon dielectric-slice discontinuity

of various configurations; slab width is such

that it supports the propagation of TEO and

TEz modes. ...................................... 78

vii

LIST OF FIGURES
Figure Page

2.1 Scattering (reflection and radiation) of an
incident surface-wave mode by a heterogeneous
device-discontinuity region along an open-
boundary dielectric waveguide of arbitrary
cross-section shape. ............................ 10

2.2 Surface charge layer created by the interrup-
tion of equivalent polarization current due to
the exclusion of principal volume 6V around the

source-point singularity at f a F'. ............. 20

2.3 Cylindrical principal volume 6V with a<<b,
centered at field point (the origin), having
its axis parallel with the principal axis of
prepagation z. .................................. 22

2.4 Rectangular pillbox as principal volume with
c<<a, c<<b, centered at field point (the origin),
having its axis parallel with the principal axis
of prOpagation z. ............................... 24

2.5 Locations of input and output terminal planes
at 21 and 22 for the definition of reflection

and transmission coefficients appropriate for
the incident m'th surface-wave mode and the
scattered n'th surface-wave mode. ............... 26

3.1 Slice discontinuity region Vd of length 220:

width 2d along an one-dimensional slab having
refractive index n3(x,z). ....................... 44

3.2 Graphical solutions to eigenvalue equation for
even and odd symmetric-slab surface-wave modes
for case of n1=1.6, n2=l.0, thickness of slab

t:zd. 00.0.0.0....0...OOOOOOOOOOOOOOOOOOOOOOOOO. 46

3.3 Normalized phase constant as a function of index
contrast for the axial propagation in the slice-
discontinuity region of a slab waveguide. ....... 55

viii

Figure(cont'd)

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

Normalized phase constant as a function of
refractive index n3 of the slice region for

the axial propagation in the slice-discontinuity

region. ......OOOOOOO..........OOOOOOOOOOOOOOOOOO

Magnitude of reflection coefficient as a function
of its refractive index n3 for three values of

the normalized lengths of the slice-discontinuity

region. 0.000.000.0000.0.........OOOOOOOOOOOOOCOO

Magnitude of transmission coefficient as a
function of its refractive index n3 for three

values of the normalized lengths of the slice
discontinuity region. 0....OOOOOOOOOOOOODOOOOOOOO

Comparison of reflection and transmission coeffi-
cients obtained for slice discontinuity in a slab

waveQUide With Zd/Aoa 0.3. ......OOOOOOCOOOOOOOOO

Comparison of reflection and transmission coeffi-
cients obtained for slice discontinuity in a slab

waveguide with 2d/lo= 0.3. ......................
Partitions of slice discontinuity region Vd for
the application of Method of Moment solution. ...
Distribution of field Ey(x,z) excited in

dielectric slice-discontinuity region of slab
waveguide by T20 incident-mode wave. ............

Relative amplitudes and phases of fields in the
slice-discontinuity region as a function of
normalized axial locations. The slab supports
dual modes with a TEO mode incident. ............

Relative amplitudes and phases of fields in the
slice-discontinuity region as a function of
normalized axial locations. The slab supports
dual modes with a TEO mode incident. ............

Relative amplitudes and phases in the slice-
discontinuity region with various indices for

the slice region, as a function of normalized
axial locations. The slab supports dual-mode
proPagation. ....................................

ix

Page

56

57

58

60

62

63

71

74

75

76

Figure(cont'd)

4.1

4.2

4.3

5.2

5.3

5.5

5.6

5.7

5.8

Three-dimesional configuration of principal
volume which contributes to the depolarizing
dyad of the 3-d EFIE for a Open-boundary~
waveguide. ......................................

Two-dimensional configuration of principal volume
shown as enclosed by principal contour C5. ......

Configuration of N-coupled, Open-boundary
dielectric waveguides. ..........................

Contributions of surface charges which arise
from the index discontinuity between each slab

and its surround Cladding. ......OOOOOOOOOOOOOOOO

Configuration apprOpriate for study of non-
degenerate TE surface—wave mode coupling between
two slab waveguides. ......OOOOOOOOO0.00.00.00.00

Configuration appropriate for study of
non-degenerate TE surface-wave mode coupling
between the m'th and the n'th guides in a
N-coupled slab waveguide system. ................

Configuration apprOpriate for study of degenerate
TE surface-wave mode coupling between two
identical Slabs. .....OOOOOOOOOO0.0000000000...OI

Normalized phase constant shift for two
propagating modes of a degenerately coupled
two-Slab SYStem. (“131.6, “28100) cocoon-00......

Comparison of resulting values for normalized
phase constant shifts from various solutions

in a degenerately coupled two-slab system.
(“1.106, “2.100) coco-000.000000000000000.0000...

Comparison of results from integral-operator-
based, coupled-mode perturbation solution

with numerical solutions to the exact eigen-
value equation for phase constant shift (ABd)
due to degenerate-mode coupling between
identical slab waveguides with variable Spacing

s/d. ............................................

Comparison of results from integral-Operator-
based, coupled-mode perturbation solution with
numerical solutions to the exact eigenvalue
equation for phase-constant shifts (AB)d due to
degenerate-mode coupling between identical slab
waveguides with variable spacing s/d. ...........

page

90

94

96

113

117

129

132

133

134

136

137

Figure(cont'd)

5.9

5.10

5.11

5.12

5.13

5.14

Comparison of results from integral-Operator-
based, coupled-mode perturbation solution with
numerical solutions to the exact eigenvalue
equation for phase-constant shift (A8)d due to
degenerate-mode coupling between identical

slab waveguides with variable Spacing s/d. ......

Comparison of results from integral-operator-
based, coupled-mode perturbation solution with
the exact solution of Wilson and Reinhart for
phase-constant shifts (A8)d1 due to degenerate-

mode coupling between different slab wave-guides
With variable SPECIDQ SZI/dlo 0.0000000000000000.

Results of integral-Operator-based, coupled-

mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; phase-constant shifts (exact and coupled-
mode) for variable spacing s/dl. ................

Result of integral-operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling beteen differing slab wave-

guides; phase-constant shifts for variable
spacing 8/61. .00....0............OOOOOOOOOOOOOOO

Results of integral-operator-based, coupled mode
perturbation solution for non-degenerate TEO-

mode coupling between differing slab wave-guides;
amplitude ratios for variable spacing s/dl, .....

Results of integral-Operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; phase-constant shifts (exact and coupled-
mode) for variable thickness dz/d1° .............

Results of integral-operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; modal amplitude ratio for variable
thiCKness dZ/dlo .........................0......

xi

Page

138

139

141

142

143

144

145

Figure(cont'd)

6.1

Page

Configuration of integrated, open-boundary,
dielectric-waveguide system consisting of
arbitrary-shaped, graded-index core regions

adjacent to the film/overlay interface depo-

sited upon a uniform substrate. ................. 150

xii

CHAPTER I
INTRODUCTION

Open-boundary dielectric waveguides, as Opposed to the
conventional closed-boundary metallic wavegbides, are
dielectric structures capable of guiding prOpagating
discrete TE, TM or hybrid modes and radiation modes having a
continuous eigenSpectrum. In its most elementary form, it
consists of a dielectric guiding core which provides a
positive contrast of refractive index relative to that of
the surrounding medium (cladding) within which the core is
immersed. Confined electromagnetic (EM) fields possess the
usual complex-exponential propagation dependence along the
waveguiding axis, but are characterized by a real
exponential decay along the direction normal to and away
from the guiding structure. Field confinment in the core is
essentially a consequence of the phenomenon of total
internal reflection at the core-cladding interface. Such
structures therefore also geneally known as ”surface
waveguides”.

Interest in EM propagation along open-boundary
dielectric stuctures has existed since the early part of
this century [1] and has progressed with a varying degree of

intensity from that time [2]. Recent develOpment and

2
applications of semiconductor lasers in the communication
area has stimulated widespread interest in certain classes
of dielectric waveguide for guiding light waves, e4L, the
”optical fiber” [3]. Together with the expansion of
activities from the microwave spectrum into the millimeter
wavelength region in the past decade these studies have
culminated in a large store of information characterizing
such surface waveguides as transmission and circuit system.
Yet, according to Kogelnik [4], integrated optics though
intriguing, remains in its infancy at the research stage; a
similar review for the dielectric waveguide microwave
integrated circuits was given by Knox [5].

Taylor and Yar iv [6] point out in their review paper
that virtually all integrated-optics decvices, iue.,
couplers, modulators, switches and filters, depend
critically for their operation upon the characteristics of
low order surface-wave eigenmodes supported by isolated or
coupled systems of integrated dielectric waveguides. Among
the class of uniformly-clad, isolated waveguides, exact
solutions exist [7,8] only for planar-slab structures or
fibers having circular or elliptical cross-section shape,
while the only coupled system which permits an exact
solution is composed of parallel slabs. Since boundary
conditions at the core/surround interface are inseparable
for more general core geometries, conventional differential-
operator based methods [9,10,11] become ineffective for

such guides. They have, however, provided approximate

3

solutions to these problems as demonstrated by Marcatile
[12] and Goell [13]. The integral-operator description [14-
16], related to Katsenelenbaum's [l7] polarization integral
equation, for uniformly-clad, open-boundary dielectric
waveguides, provides a conceptually-exact formulation for
propagation modes supported by the waveguiding system having
any number of graded-index cores with arbitrary cross-
section shape.

This dissertation, consisting of two parts, describes
the integral equation formulation as an alternative to the
conventional boundary-value analyses in the areas of surface
wave research where knowledge of basic phenomena and
accurate solutions remain relatively incomplete. 'This
research includes the scattering of surface waves by
obstacles along the cladded dielectric waveguide [18,19] in
Part I, and the modal coupling phenomena in a multi-
waveguide system [15] in Part II.

The most comprehensive available treatments for
discontinuities along open dielectric waveguides are those
by Marcuse [20] which deal with the abrupt junction between
two dissimilar guides and the interaction of surface waves
'with small, distributed surface irregularities. .Among all
discontinuities which have been studied, approximate
analyses [20-23] of the abrupt junction between dissimiliar
(primarily planar slab) waveguide sections have
predominated. Rigorous treatments [24,25] of output

coupling from a planar, solid state hetero-junction laser

4
have been advanced, as well as a Greenfs function approach
to scattering from periodic discontinuities in a planar DFB
laser [26]. The only rigorous analysis for ensembles of
step discontinuities along planar-slab waveguides is
evidently Rozzi‘s [27,28] investigation based on a two-
dimensional integral-equation formulation for the fields in
transverse discontinuity planes. Vassalo [29] has provided
a large-scale, scattering-coefficient, circuit formulation
for discontinuities along open waveguides. Mode conversion
as a result of scattering was investigated by Lewin [24] for
a heterojunction laser.

Surface-wave-mode coupling between adjacent, uniformly-
clad dielectric waveguides has been studied by a number of
investigators [31-33]. The most complete study of coupled
mode theory was given by Miller [34] for conducting
waveguides and, for integrated configurations, Marcatili's
[12] approximate analysis of coupling between rectangular
waveguides remains the primary work. Most conventional
differential-operator based treatments [31.35.36] are
approximate in nature. 'Their application is limited to
weakly guided or degenerately coupled systems where accurate
solutions are possible. To obtain more accurate results for
non-degenerate coupling, Kuester and Chang [33] presented a
variational approach [38], assuming that coupled guides are
well separated. The only exact treatment was given by Jones
[39] for coupling of parallel fibers. There, a surface

dyadic Green's function including contributions by the

5
continuous spectrum was considered. iHowever, subsequent
treatment was based upon a set of coupled differential
equations converted from integral equations initially
formulated for the transverse cross-sectional plane of a
coupled waveguide configuration.

In Chapter II of Part I, the discussion on scattering
of surface waves along dielectric waveguides begins with the
recognition of index contrast between the discontinuity
region (or obstacle) and the waveguiding region. This leads
to the identification of equivalent induced polarization
current in the discontinuity region. The fundamental
electric-field integral equation (EFIE) for the unknown
electric field in the discontinuity region is developed in
terms of the electric dyadic Green's function. The nature
of the imbedded source point singularity in this EFIE is
pointed out as well as the effects of the depolarizing dyad
which results from the associated principal-value
integration. Scattering coefficients (both reflection and
transmissions) are subsequently formulated in terms of the
solution for that unknown discontinuity field. Mode conver-
sion from an incident principal mode to higher propagation
modes in the scattering process are discussed. Applications
of the above EFIE are presented in Chapter III with
specialization to a slice gap discontinuity along a one-
dimensional slab waveguide» Several solution technigues are
discussed, including: i) approximate closed-form

radiationless solution, ii) approximate solution including

6

the radiation contribution, iii) Moment-Method numerical
solution and iv) iterative solution in the spatial domain.

Part II discusses the coupling between multiple
dielectric waveguide systems. In Chapter IV, following the
identification of an equivalent polarization current which
arises due to the index contrast between the guiding core
and its surround cladding, the formulation of an integral-
operator description for a single dielectric waveguide is
first presented. This EFIE is subsequently generalized to
describe the EM field supported by a coupled system of N
waveguides. These EFIEs are subsequently specialized to
describe the natural, guided, axially-propagating, coupled
surface-wave modes supported by the systems. A perturba-
tion analysis based upon the isolated guide‘s modal field,
and the EFIE which it satisfies, is developed and
subsequently applied to a system of two coupled waveguides.
The system mode propagation constant, which depends upon the
degree of coupled-guide interaction, is obtained for a
weakly coupled system. ZFurthermore, this conceptually-exact
formulation with perturbation approximation is shown to
recover the results of the standard differential-operator
based coupled mode theory. Application of this integral-
Operator-based coupled mode theory is demonstrated in
Chapter V. When a coupled slab-waveguide system is
considered, Fourier transform solution to the coupled EFIEHs
are shown to recover the well known characteristic equation.

for a two guide system. This confirms the correctness of

7
the integral-operator approach. Numerical calculations for
phase constant shift AB, due to coupling, are studied using
the perturbation approximation for two coupled slab
waveguides, both degenerate and non-degenerate cases.
Results are compared with the exact solutions obtained by

other researchers.

CHAPTER II

p INTEGRAL-EQUATION FORMULATION FOR SCATTERING BY DIELECTRIC

DISCONTINUITIES ALONG OPEN-BOUNDARY WAVEGUIDE

In practice, a surface waveguide is only the intercon-
' necting component of a complex network which consists of
both active and passive wave processing devices such as
modulator, amplifier, and directional coupler etc. It is
therefore commonplace to encounter some kind of discontinui-
ties at these device interfaces in addition to any imperfec-
tions which arise from wall irregularities, inadverdent
bends, etc. of the guide sturcture itself. And, when a
surface wave is incident upon these discontinuities, it is
subsequently scattered, iae., reflected, transmitted and
radiated.

There have been several treatments on the subject of
the scattering of surface-wave modes in a dielectric
waveguide by obstacles such as step discontinuity [21,28],
or the losses due to waveguide tapers and random wall
perturbations [44]. Most of the approaches are either
variation method or mode matching technique. This chapter
presents an analytical formulation [18,19] to calculate the
amplitude of scattered waves through the application of a

polarization integral equation [45,46]. Such that, the

9
advantage of digital computation could then be utilized for
any arbitrarily-shaped discontinuity as often the case of
pratical concern.

Consider an arbitrarily-shaped discontinuity region Vd
of permittivity 6(E) along an open-boundary dielectric
waveguide of permittivity 5g imbedded in a surrounding
cladding medium of permittivity 5c (Figure ZJJ. We can
immediately identify a contrast of permittivity between the
discontinuity region and the unperturbed waveguide system.
This contrast gives rise to an equivalent polarization
current, which in turn maintains the scattered field. An
integral equation is formulated for the unknown electric
field, which is proportional to the equivalent polarization
current within the discontinuity region. By solving for
this unknown field, the amplitudes of the reflected, trans-
mitted and radiated fields are readily calculated. Also to
be discussed in this chapter is the conversion of modal
fields due to scattering by the discontinuity in a dielec-

tric waveguide capable of supporting multi-mode propagation.

2.1 Equivalent Current Description for the Discontinuity

 

Region
Referring to Figure 2.1, let eu(E) be the permittivity

profile of the unperturbed, axially-uniform (5= fix + 22 =
2-d position vector) dielectric waveguide with the following

decomposition

10

0”er SOUI’CB

(7)39 = jw'P’ @1455

/

 
    
 

«a 6

 

 

 

 

 
 
 

.— 5 SURROUND _..°°
‘4\/\/\"§i I ES
.... .. 6' ""
65W 9 g ‘9)
dielectric e SURROUND -——-av
waveguide c (6u = 6c)

discontinuit
(device) region

Figure 2.1 Scattering (reflection and radiation) of
an incident surface-wave mode by a hetero-

geneous device-discontinuity region along
an open-boundary dielectric waveguide of
arbitrary cross-section shape.

11

89(5), at points in the graded-index
waveguide core
cum) =

€c(5)' at points in the surround.

Note that in general the surround need not be homogeneous;
in the case of integrated-optics system, it could be a
layered dielectric system with substrate covered by film and
overlay dielectric regions. The discontinuity region Vd has
a complex permittivity of €(f), where e(?) differs from
59(5). Incident wave E1 induces an equivalent polarization
distribution in Vd, and the latter polarization excites the
scattered field ES. It is the sum of the impressed field
E1, due to remote sources with the discontinuity absent, and
E5, the scattered field excited by the discontinuity,
results in the total field E anywhere inside the disconti-

nuity region as

(2.1.1)

We can identify the equivalent polarization current

from the Ampherefls Law of the Maxwell equations by adding
and substracting to it the displacement current of the

unperturbed waveguide jweuh(5)E (harmonic time dependence

ejmt implied but suppressed throughout) to obtain

where'

Lil

eq

eq

31:

The induced current 3

12
3e<£> + jw[e(E) - eu(5) §(E)]+»jweu(6)fi(f)
-e _ _ _ , _ - _
J (r) + Jeq(r) + jweu(p)E(r)

3t(E) + jweu(E)E(E)
(2.1.2)

impressed electric current which maintains

impressed incident field Ei,

ijeq (2.1.3)
equivalent induced polarization current which
describes discontinuity region Vd and

maintains scattered field Es, with

[e(E) -eu(E)] E(E)
so [n2(E) - n:(E)]E(E)
e 5n2(E)§(E)

0 (2.1.4)

a polarization density in terms of the

contrast of refractive index 6n2(?), and

-e —
J + Je (2.1.5)

9

= the total effective current

eq' which is proportional to the con-

trast of refractive index nonvanishing only in discontinui—

ty region Va, is now expressed in terms of total field

in that region as

l3

3eq(E) = jweoan2(E)E(E) .

(2.1.6)

2.2 Formulation of EFIE for the Unknown Field in the

Discontinuity Region

 

Since the scattered field ES(E) is induced by polariza-
tion current Seq of (2.1.6), which is proportional to the

unknown field 6(2), equation (ZJLJ) is rearranged to

(2.2.1)
Such that both terms on the left-hand side of the above
equation depend upon unknown total field E(?). When E5 is
expressed as an integral operation on EG), equation (2.2.1)
subsequently leads to the fundamental integral equation
which describes unknown discontinuity field E(?).

We proceed to expand scattered field E3 in the
complete set of eigenfunctions (both discrete and
continuous) of the unperturbed dielectric waveguide.
Solving for the amplitude spectrum of these spectral
components (eigenfunctions) will then yield a complete
description of the scattered field. Let En1(?) be the
n'th discrete surface-wave mode and Ecifij) be a
spectral component of the continuous eigen-spectrum having
a two-dimensional spectral frequency E = §£x + 95y. The

upper and lower signs of the superscrip 33 represent the

14
wave traVelling in +2 and -z directions respectively. Then

on

ESE) =23: if: (E) +1] A (fifiiéfémz a

n

--m

for z .2 2' (2.2.2)

where z' locates an element of polarization current with
g=‘/g§+g§ while afi and A¢(p) are, respectively, the
amplitude coefficients of the discrete eigenmodes and conti-
nuous eigen-spectrum. The modal eigenfields propagating in

the :2 directions are

Ijan

that) = aim 3.538%”- [ét(5.§) . 2ez(5,:)]e¥i8<€>z

(2.2.3)
where 3n is the phase constant of the discrete n'th surface-
wave mode while 3(E) = [kg-521112 is the phase constant
of the continuous spectral component with spatial frequency
'5.

Amplitudes of discrete surface-wave modes and their
orthogonality properties are well known [9,41,42]. Through
the application of Lorentz Reciprocity Theorem, these
properties have also been established in a general manner
for the continuous radiation—mode spectral components [43].

These normalization and orthogonality relations for the

transverse field components Et(5) and ht(5), which apply

15

over the infinite transverse cross—sectional plane of the

dielectric waveguide, are

6
- 5 (5H5 (6) ds = —m-5‘-
.ICS [ tm. tn ] 2

N)

(2.2.4)

for discrete surface-wave modes, and

N)
r—-fi
CD]
A
“OI
‘
ml
V
X
5’!
r.-
A
‘OI
‘
ml
.3
t__J
D;
(D
II
N I

1..

(2.2.5)
for spectral components of continuous

radiation mode.

With the above normalizations, the amplitude coefficients are

then obtained [42,43] as

(2.2.6)

(2.2.7)

Substitution of'a: and AWE) into (2.2.2) for

scattered field Es leads to

-¥-.-.- -- ,-¢_-2
..ff [L Ec(r ,g) Jeq(r )dV]Ec(r.€)d E
d

—00

16

(2.2.8)
Scattered field E5 can therefore be represented by the

following integral operator

-3— - “-‘. .- -I U
E (r) —.l; G(r|r ) Jeq(r )dV ,
d (2.2.9)
iae., Es(?) maintained by equivalent volume polarization
current seq is expressed in terms of electric dyadic Green's

function 6(EIE'). This Green's function has been

constructed from (2.2.8) as

5(EIE') = Ed(E|E') + E (EIE') ' with

(2.2.18)
3 contribution by discrete surface—wave modes,
and
=-- -t---*‘-.-2 =- -.
G (rIr') = -pv Ec(r,€)EC(r ,g)a g + L6(r - r )

r

(2.2.11)
= contribution by continuous radiation-mode

spectrum.

17
To obtain the component forms of Gd(E|F') and Gr(?|?'). we
substitute expressions (2.2.3) into (2.2.18) and (2.2.11)
such that
-- _ .
Edtflf') =-ZN:[5tn(B) .+. Eeznm] [5tn('5') $282.15") e+JBn(Z z ).
n I (2.2.12)

It is noted that exp [ern(z-z')]= exp [-j3n|z-z'|]
because, for an element of polarization current at 22.2.22'
for forward scattered waves and z 32' for backward scattered
waves, resulting in

-' .. '
EdGlE') =-i[étn(5) 1 2e2n(5)] [5131wa 2e2n(5,)] e JBnlz z I.
n (2.2.13)

Similarly

Gill

rm?) “FY/:1. [Std/é) : £ez(6.'é)] [ét(6'.E) ; 2ez(6'.2)]

x e'jB‘E’Iz‘z'laZg + i 6(E - E')
(2.2.14)

The principal-value notation PV in Gr(?|?U indicates
that the integration over the discontinuity region Vd should
be taken in a manner which excludes the source-point
singularity when F' passes through field point E, i.e.,
F-E'aO. Furthermore, a depolarizing dyadic quantity
is found necessary in Gr(?IFU to evaluate the contribution
from this source-point singularity [40,47,48]. Depolarizing

dyad I is identified and evaluated in the following section

18
as appropriate for the case of dielectric waveguides.
From equation (2.2.9), the scattered field E5 can be
written in terms of its source polarization current

Jequmeoan2(f)fi(f) as

= jwe 0.].6 n2(r' )G(r|r' ) ° E(r' )dV' -
Vd

(2.2.15)
With the above integral operator for E55), relation (21.2.1)
becomes an electric field integral equation (EFIE) for the

unknown total field E(E) within Vd

172(5) - —z—f 6n2(E')c=;(E|i-') ~E:(E')dv' = aim
V
d

for all I EVd
(2.2.15)

Where kosw poeois the free-space wave number and 20: 0/50
is the associated intrinsic impedance. EFIE (2.2.16) con-
stitutes the fundamental mathematical model which charaete-
rizes the fields of discontinuity region in a dielectric
waveguide. It is normally assumed in the following discus-
sions that a remote source 3e which maintains an impressed

field E1 consisting of single surface-wave mode in the

19
region of interest, such that solutions to the EFIE for E1
excited in Vd by E1 lead subsequently to the scattered

field interior to Vd through expression (2.2.15).

2.3 Depolarizing Dyad for Electric Dyadic Green's Function
The dyadic Greends function Er(?|ff) of (2.2.14),
which is the contribution due to the continuous radiation

spectrum, has a ———£———— singularity at E = E', the source

E - 5'

point. ThereforL, exprLssion (ZJLll) for the scattered
field E3 possesses a non-integrtable singularity and the
integral does not exist unless an infinitesmal volume 6V,
the principal volume, surrounding F = f' is excluded as
shown in Figure 2.2. Mathematically, the integral of
5r(? [EU is carried out in this principal value sense such
that the spatial frequency integral in Er is rendered
convergent; however, physically the exclusion of 6V inter-
rupts the equivalent current (proportional to electric
field) of the discontinuity region. As a result, a non-
physical polarization charge layer is created on the surface
of 6V. Consequently, the effect of the surface charge due
to the exclusion of 6V, which is built into the principal-
value integral, should be substracted in order to obtain a
correct result [49]. It is the purpose of this section to
demonstrate that the charge density on the principal volume
does maintain a finite value of electric field at its center
as 6V approaches zero in the limit.

Consider a principal volume 6V which is cylindrical

in shape with height 2a and radius 2b such that a/b<<1 as

20

DISCONTINUITY ES

REGION
/,”~.__./ ‘\

 

 

 

 

 

CLADDING

 

 

Figure 2.2

Surface charge layer created by the inter-
ruption of equivalent polarization current
due to the exclusion of principal volume QV
around the source-point singularity at r=r'.

21
shown in Figure 2.3. A coordinate system is chosen so that
its origin coincides with the field point; 6V is oriented
with its axis parallel with the principal axis of propaga-
tion 2. Since 6V is a small volume, with quasi-static
approximation, the electric field E at the center of 6V

can be expressed in terms of scalar and vector potentials

as
E = - V¢ - ij
.__~_ _1_ 13 ESE
4“ As R3 (2.3.1)

where e is the local value of permittivity and R is the
distance between the source point and field point. Although
not shown in (2.3JJ, the volume integral involving vector
potential A vanishes in the limit as 6V approaches zero.
n ,the surface charge density over AS, is equal to
-(fi-3)/jm from the equation of continuity, n is the surface
normal of AS as shown in Figure 2.3.

In the limit as 6V approaches zero, the surface
integral (2.3JJ over AS=ASI +ASz + A83, sum of top, side

and bottom surfaces of 5V, is reduced to

 

 

E = __(__z zjuie 3) 11m 1 - a/b ]
5V+o J(a/b)2 + 1

_ 2(2 - J)

- jwe '

(2.3.2)
It is observed from the above expression that, as long as

a/b<<1, the contribution to the electric field at center of

 

 

 

 

 

 

 

 

 

Figure 2.3 Cylindrical principal volume 6V with a<<b,

centered at field point (the origin), having

its axis parallel with the principal axis
of propagation z.

23
5V is insensitive to the shape of its cylindrical cross
section.

To further enhence the above observation, a rectangular
pillbox of sides a, b and c, centered at origin, is shown in
Figure 2.4. Assume that c/a<<1 and c/b<<l, then given the
same procedures as before, electric field at the center of

5V can be obtained as

 

 

E = 11m - fig- tan"l [ ab ]
5V+O cqu + b2 + c2
- 2(2 - 3)
- we °
3 (2.3.3)

Note that in (2.3.3), if a==b=c, i.e., a rectangular cube was

given instead, then a well known result is obtained:

33mg . (2.3.4)
Hence, the induced charges on the surface of 6V,
although artificially created, do produce a finite electric
field at the singular source point. However, the magnitude
of this electric field which is essentially the value of the

depolarizing integral involving the depolarizing dyad

17'"
HI
HI

l)=.__

(2.3.5)
will vary, depending upon the shape of 6V [48,49] which is

chosen to best suit the geometry of the source region.

24

 

 

  
 

 

2c
3... 1
++++++ ’Y
++ ++
+++ ++++

 

 

Figure 2.4 Rectangular pillbox as principal volume with
c<<a, c<<b, centered at field point (the
origin), having it axis parallel with the
principal axis of propagation z.

25

2.4 Scattering Coefficients and Mode Conversion

 

Consider a multi-mode waveguide which has a transverse
dimension large enough to support the propagation of more
than a single surface-wave mode. Scattering of an incident
surface—wave mode by discontinuities along such a guide
results in the excitation of additional discrete and radia-
tion modes in the scattered field. This mode conversion
phenomenon is usually undesirable, since energy is radiated
through coupling to the continuous spectrum or just simply
carried away by non-principal guiding modes. It is there-
fore of practical interest to calculate these scattering
coefficients.

Using the configuration as indicated in Figure 2.5, let
the region of discontinuity be bounded by two reference
planes, ime., 2221, the input terminal plane and z=22,
the output terminal plane. Assuming that a single surface-
wave mode of m'th order propagates down the open boundary
waveguide and is subsequently scattered by the discontinui-
ty. We define the reflection coefficient at z=zl as the
amplitude ratio of the back-scattered n'th surface-wave mode
to the incident m'th surface-wave mode

- +jBnzl
ane

 

mn = -j8mz1
Eoe
(2.4.1)
where a; is the surface-wave mode amplitude of the
normalized backward scattered nlth mode while E0 is the

normalized amplitude for the incident wave. .Similarly the

26

PRIMARY SOURCE DISCONTINUITY
_ REGION

Je | I

I l

 

 

 

 

Figure 2.5 Locations of input ans output terminal
planes at 21 and 22 for the definition of
reflection and transmission coefficients
appropriate for the incident m'th surface-
wave mode and the scattered n'th surface-
wave mode.

27
transmission coefficient Tmn at z=22 is defined as
amplitude ratio of total transmitted n'th surface-wave mode,
i.e., including both incident and scattered wave, to the
incident m'th surface-wave mode at z=zl,
T = (Eodmn + a;)e Janz
mn -ijzl

Eoe (2.4.2)

where a; is the surface-wave amplitude of the normalized
forward scattered nlth mode; 305mn represents the contribu-
tion of the incident field at the exit plane, it is nonzero
only if n=m, as would be the case in mono-mode waveguide.

Both a}: and a; can be obtained from E5 of (2.2.9)
which, in mono-mode waveguide, has the discrete component

jk - = .. _ _ ..
-—JQ 6n2(r')G (rlr') - E(r')dV'.
20 V’ d

d
(2.4.3)

The finite sum in the discrete Green's dyad is specialized
to extract the contribution by the n'th surface-wave mode,

Edn° When Edn is expressed in terms of its tranverse and

longitudinal components as

Edn(E|E') = - fi§(f)§;(5')

- [atn(6) 1 2e2n(5)][etn(5') ; gezn(5.)]

-jBn|z-2'|
Xe

be. for all EEVdI
(2.4.4)

28

it is clear that the transverse component field in the
reflection coefficient Rmn is produced by

_ - _ _ A _ +jBn(z-z')

etn‘°’(etn‘p" + zezn(p')]e .

(2.4.5)

Notice that the lower sign of superscript 'z'in equation
(2.4.4) is selected for the backward travelling wave, this
is because z is to the left of the input terminal plane,
such that zg 215 2'. By substituting (2.4.5) into (2.4.3),
the discrete portion of the scattered field, we then obtain

the backward scattered wave for the n'th mode as

_ _ _ +jsnz
an etn(p)e =
jk +jB z_ _ _ _ _ A _ _ _ -j8 z'
-(-z-:.-)e 1" etnmfv 6n2(r')[etn(p')+zezn(p')]'E(r')e " dv'.
d (2.4.6)

Following the same procedure, the total field for the n'th
mode to the right of exit plane z=zz, is expressed as the sum

of the transmitted and forward scattered waves as

-jan

(a a + a )étn(6)e

-jB z
(p)e “

= E05mnetn

jk ’38 Z_, _ _ _ _ A _ _ _ +j8 z'
—-2)e n etn(p)j;,6n2ir')[etn(p')-zezn(p')]'E(r')e n dv'.

-(
20 a
(2.4.7)

29
By substituting the transverse field amplitudes obtained
from equations (2.4.6), (2.4.7) and the transverse component
amplitude of incident m'th surface-wave mode from
-jB z

Etm(f) = Eoétm(5)e

into definitions of scattering coefficients given by (2.4.1)
and (2.4.2), we obtain Rmn and Tmn at the corresponding

input and output planes as

j(Bm+Bn)z1

 

'jkoe f 2 - -.- - -
R = 6n (r')E (r') ' E(r')dV'
mn E020 Vd n
(2.4.8)
and
'j‘anz'Bmzl) jko 2 -. -- -. - -. .
Tmn=e [Um-ET (Sn (r )En(r) ° E(r )dV].
0 0 Va '
(2.4.9)

Use was made of the normalization relation (2.2.5) for étn'
the transverse wave component, over the infnite cross sec-
tion of the waveguide.

For the case of a mono-mode dielectric waveguide, ime.,
a waveguide that supports only single dominant-mode propaga-
tion, there is no excitation of higher-order discrete
surface-wave modes in the scattering process. However, mode
energy is lost through backscattering, radiation coupling to
the continuous radiation mode spectrum and increased dielec-
tric loss due to the existence of standing wave, etc. Equa—

tions (2.4.8) and (2.4.9) can be specialized for the mono-

38

mode dielectric waveguide by letting m=n to obtain

. 2jBmzl
R = 6n (r')E (r') ' E(r')dV'
m E Z m
0 0 Vd

 

(2.4.10)

and

T ==e

-ij(z2-21)
m

jko 2 - -- - - -
1"E'—z (Sn (r')Em(r') °E(r')dV' .
o 0 Va
(2.4.11)

Finally, it follows from the conservation of energy, the

relative radiation loss as'a result of scattering is

Power radiated _ 1 _ IR

2
Incident power - mI

-l'r|2-

m
(2.4.12)

2.5 Solutions to Electric Field Integral Equation

 

This section describes various closed-form and numeri-
cal approaches to approximate solutions of EFIE (2.2.16).
Without loss of any generality, the dielectric waveguide
considered here is assumed to support only dominant mode
propagation and the incident field consists of a single
surface-wave mode.

First to be discussed is the case of a small discon-

tinuity such that contributions from radiation spectrum can

31
be neglected; solution for the total electric field which
has a longitudinal dependence of exp (iijz), is then
obtained by the Fourier Transform Method. This longitudinal
dependence with phase constant 86 is again assumed for
with unknown amplitude coefficients to implement an appoxi-
mate radiating solution. Subsequent exploitation of this
total field E(E) in (2.2.16), complete with the continuous
radiation component in the Greenis function, yields these
unknown coefficients after enforcing the EFIE at interior
points of the discontiniuty region. Hence, radiated power,
though negligible for small perturbations, can be quantified
to confirm the results obtained otherwise.

Numerical approaches, involving manageable matrix
sizes, are often utilized in solving integral equations to
obtain solutions of higher accuracy. Therefore, it is
appropriate to describe the Method of Moments in the case
where discontinuity is of resonant size or smaller.
Alternative solution based upon iterative process is then

pursued for discontinuities of larger dimensions.

2.5.1 Approximate Radiationless Solution to EFIE

For a small axially-invariant discontinuity (described
by 5n2(5), i4», not necessarily uniform) extending from z =
-2 tot , an approximate closed-form solution can be obtained
for the field in the discontinuity region if radiation is
neglected. This result provides limiting reflection and

transmission coefficients which can be used to confirm more

32
accurate MoM numerical solutions, as well as the zeroth—
order discontinuity field required to initiate an iterative
solution.

Neglecting radiation from the discontinuity 5n2(5) in
region |z|<.2 along a monomode (single O'th surface-wave mode
with phase constant 80) dielectric waveguide leads to the
approximate electric Green's dyadic

-j8 lz-z'l
_ -: - -; -. 0
e0(o)e0(o )e -

IR

E(Elf')
(2.5.1)
If reduced dyadic (2.5.1) is exploited in EFIE (2.2.16), an

approximate IE for unknown E is obtained as

_ _ jk l -jB Iz-z'l _ _ _ - -

E(r) + 2‘0 e 0 dz' an2(p')e§(p)e3(p') °E(r')dS'
- _ "3.802 ..

E Eoeo(p)e ... for Izl g 2, o e CSd ,

(2.5.2)
where CSd denotes the transverse cross section of the
discontinuity region. In the case of a principal mode well
above cutoff, the transverse components predominate over
longitudinal components [12]; consequently, only satisfica—
tion of the transverse components of IE (ZJLZ) is enforced

to obtain

33

_ _ jko _ _ 2 'jBOIZ‘Z'I 2 - ; L _ -
E (r)-+-——-e (p) e dz' 6n (p')e (p')-E(r')dS'
t 20 't0 CS 0
d
-l
- - -jBoz -
z Eoeto(p)e ... for |z| g 1, p e csd ,
(2.5.3)
The preceding expression leads to
E(r) Btu) eto(o)w(z)
(2.5.4)

where longitudinal wave function w(z) satisfies the 1-d IE

from (2.5.3)

-jBOIz-z'l -jBoz
W(z) + jkOC w(z')e dz' 2 Eoe

-2

... for [2] 5 l,

 

(2.5.5)
where C is defined as
“2(3); (Bi-é (Bids
CS t0 to
d
C = zro I
2 E (B)-é (ENS
jrzw(p) t0 to
(2.5.6)

with zw(3), the wave impedance of the surface-wave mode
éto(5). It is observed that C is independent of normaliza-

tion chosen for 5to°

34

A closed—form solution to approximate IE (ZJLS) can be

obtained by exploiting a Fourier-Exponential transform for

w(z) as

(2(2) = 5%] i(3(niejnzdn .

CD

(2.5.7)
Substitute (2.5.7) into (2.5.5) to obtain
m , l - I
. - -JB Iz-z |
_1_. ” 3'12 ° ’3“. 0 dz'] dn
2," W(n)[e +jk0Cfe e
-m - -Q
-38 z
= E e 0

0

By selecting the field point z , such that -5L,<_ zSIL,
the above integration over 2' can be carried out and terms
of common functional dependence are collected as follows:

co

2
2jB k c .
~ 0 0 an _
fw(n)[1—————2 2]e dn-o.

 

 

-m n - B
0 (2.5.8)
-j802 , Q~ e-jn£
— jkoce W(n) j(n + so) an = ZWEO ,
- (2.5.9)
m~ ejnl - 0
w(n) j(n _ Bo) dn — .
-m (2.5.10)

Since (Emmi 0, expression (2.5.8) leads (after invoking the

Fourier Transform theorem) to discrete values of allowable

spectral frequency

35

 

_ . _ 2
n - i 80 _ iNIBO + ZBOkOC

(2.5.11)
The transform solution therefore consist of the

discrete spectrum

~

¢(n) = 2fiA6(n + 85) + Zanin - 86)
(2.5.12)

with the corresponding wave function
-j862 jBéz
W(z) = Ae + B e
(2.5.13)

Substituting (2.5.13) into (2.5.9) and (2.5.18) results in

 

. -j(86—Bo)£
A = (80 - Bo)Eoe
koC(l - 92)
(2.5.14)
0 = E = 86 - BO e-ZjBéfi
A 80 + 80
(2.5.15)

where p is defined as the reflection coefficient for the
longitudinal wave function ¢(z).

Since E(P)= Eto(5) w(z), if input and output terminal
planes are defined at 28:2, scattering coefficients R00 and

T00 of (2.4.8) and (2.4.9) become

ej‘Bo'Bo’2 . -2j86£
R00=-j29 2 [$111 (86+Bo)l+e

1 - 9

sin (85-so)2].

36

I

2
_ 0
Too — e

 

_ jZe . ,_
l [Sln (8o BO)£-+p e

_- -j(B'-B )2 .
32802{ . o o 2 238
2

l - o

x sin ‘36 + 30"]] . (2.5.16)
ZJLZ Approximate Solution to EFIE for Radiating
Discontinuity
The preceding approximate radiationless solution was
based upon the assumption that the radiation contribution
to the Greenls dyadic is negligible. When it is desired to
consider such effects, the following Greenis function should

be employed

'jBOIZ'Z'l

CHI
'1]
HI
ll

- 53(6)50(6')e

.‘ _ _ _ _ _ _ -jB(€)lz-z'|
- PVJ[I. e3(o.€)eg(o'.€)e d2€

(2.5.17)
Again, in a simplified situation as described in Section
2.5.1, with only a principal mode supported (negligible
axial field components) the field in the discontinuity
region can be approximated as

_ - _ _ 'jB'z jB'z
E(r) s eto(p)[a1e 0 + aze 9

]

(2.5.18)

37

where 88 is that obtained in (ZJLll) and a1, a2, are the
associated unknown amplitude coefficients when radiation
effects are considered.

Recall the integral equation (2.2.16) as

am - 3Z3 5.1260355» -§(E')dv' = rim

0 V3
(2.5.19)

where E(?) is now approximated by (2JL18). Then, the IE
above has two unknown constants a1 and a2, therefore in
order to reduce it into two algebrac equations to solve for
these amplitude coefficients, an integral-operator with

weighting function 6p is used to pre-dot multiplying into

above IE and integrate over the discontinuity region Vd as

‘ G ..{ } av, p = 1,2 .
.lv' P

d (2.5.28)

For the purpose of simplicity, if delta-function is chosen

for a? such that

‘ = G 6(E - E )

P P P (2.5.21)
where BP is a unit vector, subsequent operation of (2JL28)
upon (2JL19) with the above delta-function weighting yields

a matrix equation with pal and 2 as

2
2: (2 a = b
l=1 9“ 2 9 (2.5.22)

38

where the matrix elements Cp and b are

2 P .
-j862
_ A .' ‘ P
Cp1 - up eto(rp)e
jko 2 _ A = _ _ _ _ 'ijZ'
- -——- 6n (p')u. °G(r Ir“) -e (r')e dV' ,
20 p p to
Va
+jB'z
.. ,. . - - 0 p
sz — up et0(rp)e
jkO 2 - A = — — _ _. +j862'
- -——- 6n (p')u °G(r |r') -e (r')e dV',
20 p p to
Va
b = " ~Ei .
9 up ”13’
(2.5.23)

ZJLB Moment-Method Numerical Solution for Discontinuity
Field

The integral-operator method described above for scat-
tering by a discontinuity along a dielectric waveguide is
particularly suitable for numerical solution, especially
when the region of discontinuity is heterogeneous in nature,
i.e. 5n2- 6n2(f), and arbitrary in shape. In this scetion,
EFIE (2.2.16) is reduced to a matrix equation by Method of
Moments (MoM) technique [50]. In this method, the unknown
field E(F) is first expanded in an appropriate basis set
(expansion functions); then, subsequent to taking the term-
by-term inner product with an appropriate set of weighting
functions (testing functions), the EFIE is discretized to a

matrix equation.

39
To begin, a volumetric pulse—function basis set is
selected because the resulting MoM matrix-element computa-
tion is simplified (numerical matrix fill-time minimized);
the pulse function expansion for E, the unknow field, is

therefore

3 N,
E(r) = él (12:1 xvquPqU r)

(2.5.24)
with

l ... for r 5 AV

8 ... for r t Avq

= volumetric pulse function spanning
q'th volume element AVq for q= 1,2, ..., N

where Vd is partitioned into N volume elements Avq. Use of

this expression in EFIE (2.2.16) leads to

jk - U
Elqu[ q(r) ' 20 AV 6n (r )G(r|r ) xvdV ] E (r)
q

... for all ? Eva.
(2.5.25)
Expression (2.5.25) is discretized, to obtain an MoM matrix
equation for the an' by operating term-by-term with the
d-function integral testing operator (?p at the center of

AVP)
3 N

uz=1 :41 L ME-EP) 52“ .{ }dv

d (2.5.26)

which provides

48

N .
(A) a =El(E)
v1; uquvq )1 P

. . for u 1,2,3 and

p = 1,2, ...; N,

 

 

 

 

 

 

(2.5.27)
a 3N by 3N matrix equation for unknown avg, or
r i ' 1
I i i -
| l axq Ex(rP)
I | i -
= E r
3N.l I(A11V)PQI am y( p)
I l i ..
a E (r )
z z
_ ' ' . L q . L p .
h—..——u
3N
(2.5.28)
where the MoM matrix elements are
(A ) =6 5 -ik—°f anZG'n? -G(1-:|1-:')-de' .
)1qu 11qu 20 AV )1 p v
. q
(2.5.29)

By applying standard matrix methods, the numerical solution
of MoM matrix equation (2.5.28) for avq leads to field E in

the discontinuity region through expansion (2.5.24).

41
2.5.4. Iterative Solution to EFIE

Since the polarization EFIE

.. _ _i _ jko 2 _ = - .. - _
E(r) = E (r) + 7?—- 6n (r')G(rIr')'-E(r')dV'
0 V6
(2.2.16)

is essentially a linear Fredholm integral equation of the
second kind, it leads naturally to iterative solutions [51]
for the total electric field E(F). This approach provides
an alternative to the MoM solution, which remains feasible
when the discontinuity region exceeds resonant size. If E
is the field of the 2th iteration, then EFIE (2.2.16)

provides

E - = 331(3) + 72f 6n2(f')(=;(fli") -E2(E')dv' .
O va
(2.5.38)

The iteration series provided by the above relation

converges if IIE£+l - EII+ 0 for large I. Convergent rate

depends strongly upon the initial selection for E the

0!
field of the O'th iteration, which might be estimated as

~Ei(?) .ufor small discontinuities, or
80(3): approximate ”radiationless" solution for
“Simple” discontinuities where (in2 s 6n2(5).
(2.5.31)
Choice of E = E1 leads to the classical Neumann series and

0
the associated resolvent kernel [51]. Due to the complexity

42
of Green's function G, a practical iterative field solution
will require adequate convergence after only serveral itera-
tions; an accurate choice for the (Nth-order field of the

first iteration E0 is consequently important.

CHAPTER III

APPLICATION TO SCATTERING BY SLICE GAP DISCONTINUITY IN A

DIELECTRIC SLAB WAVEGUIDE

3.1 Introduction

 

The one-dimensional slab waveguide considered in this
chapter for the applications is shown in Figure 3.1. The
slab extends infinitely in both directions of y and z with a
width of 2d. A discontinuity region Vdv being both y and z
invariant, occupies the longitudinal cross section of Va; it
is centered at origin and consists of a dielectric material
of refractive index of n3(x,z) which provides a contrast
with those of the slab and the surround which are n1 and n2
respectively. Single dominant TE surface-wave mode
incidence from z<<0 with amplitude E0 is assumed. The
infinite dimension along y insures the y-invariance for all
the field quantities, iue., 3 /BY - 0. Scattering by the
discontinuity is interpreted as arising from the equivalent
polarization current induced within the discontinuity and

prOportional to the index contrast 5n2

For a planar, step-index symmetric slab waveguide, the
well known characteristic equation [42] for eigenvalues of

natural TE surface-wave modes is

43

44

 

 

X
M
z=-z0 z=z0 CLADDING
= d / I I - - SLAB
I
‘-JRJP\/\r—#»-- j;;/§dl_515k_ _______

<— . =-d "3

DIELECTRIC n2
SLICE

 

 

 

Figure 3.1 Slice discontinuity region Vd of length 220,
width 2d along an one-dimensional slab wave-
guide having refractive index n3(x,z).

45

ZYK

2_Y2)

tan (Kt) =

 

(K

where t is the thickness of the slab. The characteristic
transverse phase constants K and Y are related to
longitudinal phase constant by the definitions K2=k%‘82 and
Y2=32-k%, where kl, k2 are wavenumbers of the slab and
surround respectively with k1=n1ko and k2=n2ko. A graphical
solution to that eigenvalue equation is shown in Figure 3.2
for a slab waveguide with n1=l.6, n2=l.0. As illustrated in
Figure 3.2a for TE even modes and Figure 3.2b for TE odd
modes, the later exhibits a lower cutoff frequency while the
principal TE even mode does not. It is evident that only a
finite number of discrete modes are allowed as contrast with
the‘infinite number of eigenmodes in conventional closed-
boundary metallic waveguides. This particular characteris-
tic allows us to vary the waveguide dimensions such that
either mono-mode or multimode fields are excited in the
scattering process. It is the purpose of this chapter to
demonstrate the use of various solutions discussed in the
last chapter. Relative merits of each technique are
evaluated and qualitative conclusions are drawn regarding

scattering phenomena associated with discontinuities along a

dielectric slab waveguide.

 

 

46

Kdtaan

 

 

 

 

xd
(a). TE even modes.
Yd -chotxd
2174
d/l=2.0
(yd)2 (xd)2=6k2d2
n-
d/A=.5 2d/A=1-'
‘ A ' f d
o «'12 ‘ 3n): "
(b). TE odd modes.

Figure 3.2

Graphical solutions to eigenvalue equation

for even and odd symmetric-slab surface-wave
modes for case of n1=1.6, n2=1.0, thickness
of slab t=2d.

47
3.1JL TE Propagation Mode for Slab Waveguide and Their
Normalization

DISCRETE SURFACE-WAVE MODES

Since the TE surface-wave mode fields are of the form

Fjan
n(x)e

Yey
(where subscript "n" in ey(x) and Bdenote the n'th discrete
mode) i.e., a travelling wave in the axial direction 12

with phase constant 80, its corresponding transverse

electric and magnetic fields are

étn(E) = yeyn(X)
.. .. 1 ,. .. _-,.
htn(r) = ET; (2 X etn) - xeyn(x)/ZTE
= xhxn‘x’ (3.1.1)

where ZTE=muo/Bn=zoko/Bo,zo= uo/eo, the wave impedance
with wavenumber Mam/”080' From (2.2.4) of Chapter 2, the

assumed normalization for eyn(x) is

” 1 w 2 1
-f eyn(X)hxn(x)dx = Z3]. eynhddx = 3

(3.1.2)

It is well known that the tranverse electric field eyn(x)

for the slab waveguide can be written as [28]

48

[rA cos (KX) ... |x| 3 d

e (x) =
Yn -
AeY(d le) cos (Kd) ... w 3 [XI 3 d

(3.1.3)
where it shows, in the transverse plane, there exists a.
standing wave inside the waveguide and a rapidly decaying
wave outside. The amplitude coefficient A is obtained by

exploiting (3.1.3) into (3.1.2),

1/2
A _ koz-o
" 2

sin 2Kd cos Kd
.ZBnd‘l + “—2731— + T)

(3.1.4)
CONTINUOUS RADIATION MODES

From (2.2.5) of Chapter 2, the normalization for con-

tinuous radiation modes in slab waveguide is given by

f 65'] 2' [éy(xt€) X Ex(X'I€')] dx
61: " 5.1 l _ 4
=. f . ..., -_
-.... zTEu; ) 2mm

with zTgszoko/Eflf). Similary, the continuous radiation

 

(3.1.5)

fields inside and outside the slab can be written as [28]

49

,/%-% cos (ox) ... [XI
(12[ [cos £(x -d) + a] |°°|

Amplitude coefficient c is obtained by exploiting (3.1.6)

|/\
D.)

ey(X.E;)
lxl 2 d

(3.1.6)

IV

into (3.1.5)

1/2

c = [1 + (E)2 sinzod]
(3.1.7)

where o,cxand u are defined by the following relations:

a = tan-1[-% tan (od)]

3JL2 Scalar EFIE for TE Mode Scattering along Slab
Waveguide
For the one-dimensional slab waveguide in Figure 3.1,

equation (2.2.16) of Chapter 2 can be expressed as

- jko 2 = -
E(x,z)-1r—» 6n (x')G(x,z|x',z') -E(x',z')dx'dz'
0 LCS '

= yEoeyn(x)exp(-j6nz)
(3.1.8)

where LCS denotes the longitudinal cross section of the
arbitrarily-shaped discontinuity region along the slab
waveguide. By carrying out the dyadic dot product of G-E

in the integral above, we obtain the following component

50
form scalar equations for (3.1.8).

x component:
2
E (x,z) - -——;/~ 6n (x')[G (x,z|x',z')E (x',z')
x z LCS xx x

+ ny(x,z|x',z')Ey(x',z')
+ ze(x,z|x',z')Ez(x',z')]dx'dz'
= 0

(3.1.9)
y component:

B (x z) -J—]:9-f 6n2(x')[G (x zlx' z')E (x' z')
y I zo LCS. yx I I x I

+ ny(x,z|x',z')Ey(x',z')

+ Gyz(x,z|x',z')Ez(x',z')]dx'dz'

= Eoeyn(x)exp(-j6n2)

(3.1.10)
2 compgnent:
E (x,z) - 33] dn2(x')[G (x,z|x',z')E (x' z')
z 20 LCS zx x '

+ Gzy(x,zlx',z')Ey(x',z')
+ Gzz(x,z|x',z')Ez(x',z')]dx'dz'
= 0

(3.1.11)

51

It is observed that due to the y—invariant property of
the slab waveguide, its Greenfs function, specifically, the
dyadic elements ny(x,z|x',z'), ny(x,z|x',z'), Gyz
(x,z|x',z') and Gzy(x,zlx',z') are equal to zero aided by
the fact that =G=(x,z|x',z') is constructed from electric
field eigenfunctions (both discrete and continuous)
consisting of TE modes with only y component or TM modes
with only x and z components, and the TE and TM modes are
orthogonal. Therefore, (3.1.9), (3.1.10) and (3.1.11) can

be reduced to

0 2
E (x,z) - ——— 6n (x') G (x,z|x',z')E (x',z')
x 20 LCS [ xx x
+ ze(x,z|x',z')Ez(x',z')]dx'dz'
= 0 (3.1.12)
jko 2 I I I I I I U
Ey(x,z) - 7%; LCSGn (x )ny(x,z|x ,z )Ey(x ,2 )dx dz
(-jan)
= Eoeyn(x)e
(3.1.13)
E (x z) - 159] 6n2(x')[G (x zlx' z')E (x' z')
2 z0 LCS zx x

+ Gzz(x,z|x',z')Ez(x',z'k]dx'dz'

= 0
(3.1.14)

52

A study of the above expressions reveal that the x and
z component equations (3.2.12) and (3.2.14) couple ii:x and E2
but are uncoupled from Ey; furthermore, zero forcing
function in both implies there exists only trivial solution
such that ExsEz=0. The y component-equation (3.1.13) is
independent in Ey with a non-zero forcing term in the right
hand side, therefore Eyfo. The TE mode only problem is then

reduced to that of solving the y-component integral

equation:
jko 20 d 2
Ey(x,z)-1;; 5n (x')ny(x,zIx',z')Ey(x',z')dx'dz'
-20 -d
-j an
= Eoeyn(x)e

... for all (x,z)e Vd
(3.1.15)
This is a specialized EFIE for unknown Ey(x,z) in the
discontinuity region due to scattering of incident TEn
surface-wave mode. In this IE, electric Greenhs function
G consists of both the discrete and continuous radiation

YY
component fields:

-jBnlz-.-z'|
ny(x,zlx ,z ) = Z - eyn(x)eyn(x )e
n
1 '” ' -je(a)lz-2'|
- I 0 ZTE(£)ey(X.€)ey(x .Eie d6
(3.1.16)

The absence of principal value evaluation in (34Ll6) above

is evident from the fact that TE mode propagation in this

53
slab waveguide does not support any currents in the z direc-
tion. Hence, there is no contribution to the correction
term I for the otherwise would be present source point
singularity in the continuous spectrum integral of

Gr(x,z |x',z') of ny.

3.2 Approximate Treatment for Scattering along Slab

 

Waveguide

 

3.2.1 Analytical Solution without Radiation Contribution

A closed-form solution was obtained for E(E), the field
inside the discontinuity region, from Section 2.5.1, by
neglecting the continuous radiation in a mono-mode

waveguide, as

Ill

E(E) 5t0(3)w(z) = ey0(x)w(z)§

_jB'z
O +

pejséz]

= §Aeyo(x)[e (3.2.1)
where A and p are given by (2.5.14) and (2.5.15). The

constant C given by (2JL6) is evaluated here for TEO mode
field under the assumption that the discontinuity region is

of homogeneous nature, i.e., n3(x,z)-n3, a constant, such

that

 

0
ll

d
3(“3 ' ni)ko 2
z e o(x')dx'
0 . _ Y

d

. 2 2 2
3“‘3 ' n1’ko

$ cos2 Kd + (Kd +

sin 2Kd)
2

sin 2Kd)
2

(Kd +

 

 

(3.2.2)

54
Then, 88' the phase constant of E(E), has the following

eigenvalue

. _ . 2

which depends upon the values of 6n2=n§-n§, the dielectric

 

contrast, through the definition of C above.

Shown in Figure 3.3 is BUt' the normalized propagation
constant, as a function of the value of 6n2=n§-n§. Figure
3.4 displays the relationship between.65t and the values of
refractive indicies of the discontinuity region, n3.

Amplitudes of the reflection and transmission coeffi-
cients [RI and |T| thus calculated from (2.5.16) are shown
separately in Figure 3.5 and 3.6. These coefficients are
plotted as a function of n3 with parameter zo/d(iJL, length
of the discontinuity region is normalized w.r.t. the width
of slab) having the values of 0.1, 0.5 and 1.0. It is noted
from these results that as the contrast of refractive index
increses, the discontinuity presents itself as a better and
better reflector. Furthermore, oscillatory nature in the
obtained values of IR] and |T| shown in these figures due to
larger value of zo/d certainly agrees with the intuition
that the ”strength” of the discontinuity is becoming
”stronger“, this can also be verified from the axial field
distribution in the MoM.solution to be presented in the

following section.

 

55

 

 

Figure 3.3

6.0

Normalized phase constant as a function of
index contrast for the axial propagation

in the slice-discontinuity region of a slab
waveguide.

56

 

 

 

6.0—
4.0-4
2.0—
0 o 2T0 410 610 8.10
“3

Figure 3.4 Normalized phase constant as a function of
refractive index n3 of the slice region for

the axial propagation in the slice discon-
tinuity region.

 

57

 

 

 

2 = .
d 0 310
.. A 2 /d: 0.1
1 0 o O 0.5
E! 1.0
nl=1.6 A
n =1.0
2 {A
0.8“ D
0.64
D
D D .
0.4“
In
0.27 I
D s are not connected due
to imcomplete data
0
0.0 2.'o 41.0 630 310
n3
Figure 3.5 Magnitude of reflection coefficient as a

function of its refractive index n3 for

three values of the normalized lengths of
the slice-discontinuity region.

0.] _.

58

   

 

 

2d = 0 3x0
.. A 2 /d: 0.1
1.0 (D 0 0.5
[3 1.0
n1:
n2=
0.8“
o
0.6"
|T|
D
o
0.4-
o's and [3's are not
connected due to incomplete (3
data
0.2“1
I
I I i T |
0 0 2 0 4 O 6.0 8.0 10 0
n3

Figure 3.6 Magnitude of transmission coefficient as
a function of its refractive index n3 for

three values of the normalized lengths of
the slice-discontinuity region.

59
3.2.2 Approximate Solution for Radiating Discontinuity
When radiation coupling in the scattering process is
considered, an of (2.5.18) assumes that the discontinuity
field maintains the eigenfield distribution and yet pos-
sesses axial propagation constant from the analytical radia-
tionless solution. IFrom the resulting matrix equation of

(2.5.22) and (2.5.23)
2
= = 1,2
géi Cp£a£ bp p

where the matrix elements, when specialized for a mono-mode

slab waveguide, becomes (p=l,2)

-jB(')z jk f =
- P _ __9, 2 I I |
cp1 — ey0(xp)e zo Lcsén (x )G(xp,zp|x ,z )
I I U
x ey0(x )e dx dz
+j80zp jk0 6n2(x')8(x ,z |x',z')
c 2 = e 0‘x ’9 ’ T P p
P Y 9 o LCS
+j862'
x (x')e ‘ dx'dz'

eyo
bp = Eoeyo(xp)e JBOZP
(3.2.3)
Two matching points for the delta-function at EP =(xp,zp) =
(0,-20) and (0,20) are selected for the calculation of two
unknown.coefficients a1 and a2. The results are shown in
Figure 3.7 for the subsequently obtained values of [RI and

|T|; in the Figure, zo/d = 0.5 is chosen in order to compare

with results of analytical solution of the last section.

60

 

 

1.0-,
.9-
z /
|T| n°___
.8J 1-
n2-
D : analytical solution
(w/o radiation)
.71 o : approximate solution
(w/ radiation)
.4“
.3‘
lRl
.2-
.1‘

 

 

 

1.0 2.0 3.0

Figure 3.7 Comparison of reflection and transmission
coefficients obtained for slice-discontin-
uity in a slab waveguide with 2d/l0= 0.3.

61
Although the comparison shows no significant change in
reflection coefficient between the two solutions, there is
substantial increase of radiation when n3 is greater than 2
(as can be seen from the sharp increase of transmission
coefficient). Similar phenomenon exists in Figure 3.8 for
discontinuity of smaller size (zo/d = 0.1), there appear to
have less amount of coupling to the radiation when n3
increases from 2 to 3; this should be expected since discon-
tinuity‘s size is now only'1.5% of the scattered wavelength

3.3 Moment Method Numerical Solution
3.3.1 Discretization of scalar 2-d EFIE to MoM Matrix
Equation
Refer to Figure 3.9, in which the region of discon-
tinuity Vd is uniformly partitioned in both the x and z
directions. 'The two-dimensional pulse-function expansion of

unknown field Ey(x,z) is defined as

N
P
Ey(x,z) = Z: Enpn(x,z) with
“‘1 (3.3.1)
1 ... for (x,z) e (AS)n
pn(xrz) =
0 ... for otherwise
(3.3.2)
where NP is the total number of partitioning with NP:

NxXN23 Nx is the number of partitions of length Ax = 2d/Nx

along x; N2 is the number of partitions of length Az=Zzo/Nz

62

 

1.0-F
.9 q
2 /d = 0.1
|T| 0
n1=l.6
.8 J, n2=l.0
: D : analytical solution
I (w/o radiation)
5 .' O : approximate solution
. (w/ radiation)
.4“
.3 “
IRI
.2 '1
.1‘

 

 

1.0 2.0 3.0

Figure 3.8 Comparison of reflection and transmission
coefficients obtained for slice—discontin-
uity in a slab waveguide with 2d/l0= 0.3.

63

 

 

 

 

 

Ax=2d/N

 

X | Vd
— I .--
3 fl.)
l--l
Az=220/Nz

Figure 3.9 Partitions of slice-discontinuity region
Vd for the application of Method of Moment

solution.

64
along 2. (AS)n is the surface element centered at 5: En
xxk+zz£ where xk = - d+(k-0.5)Ax and z£= -zo+(£-O.5)Az;
n is the running index covering all partitions, ime., n =
k+(£-1)Nx. From equation (ZJLZS) the three-dimensional
integral equation is then reduced to the following two-

dimensional form for the slab waveguide

N 2
P jcSn k0 d
_ _ ' I, I d I I
2 En Pn(x,z) 20 (As) ny(x zlx z ) x 2
n=1 n ,
-jBoz
= Eoeyo(x)e
(3.3.3)
. Multiply (3.3.3) by the delta-funtion operator
N
p .
Z: LAB) 5(x-xi,z-zj){ }dxdz ,
n=1 n
(3.3.4)

this essentially forces IE (3.4.3) to be satisfied at
discrete matching points (xi,zj) or at Sm=§xi+zzj where
m=i+(j-1)Nx, 15 i 511),, and 1 _<_ j 5N2; thus completing the
discretization of the EFIE to yield NP linear algebrac

equations in En as

N 2

p _ j6n k0 ' '
2 En pn(pm) - T— ny(xi,zj|x',z')dx dz

0 (As)
n=1 n
-'B 2.
= E e (x.)e 3 0 3 ... for 1 < i < N
0 yO 1 - - x
1 g j 3 NZ .

(3.3.5)

65

With pn(5m)=6mn from (3JL2), above expression can be

written in the more concise form

N
P
2 AmnEn= Bm for m= 1,2, ..., Np

“‘1 (3.3.6)
or in matrix representation
- W [- ‘4 r' _

Amn En = Bm

 

 

 

 

 

 

L I" b - - -I (3.307)
where the elements Amn of the Nx by Nz coefficient matrix

are defined as

. 2
Jk05n f ( I I l)d Id I
A = 6 - — G X-,Z- x ,Z X Z
mn mm 20 (As)n YY 1 3
(3.3.8)
and
- J
Bm - Eoeyo(xl)e .
(3.3.9)

The unknown expansion coefficients En can therefore be
obtained as the non-trivial solutions from matrix equation
(3.3.7).

To facilitate the numerical calculation, we can further
decompose (3JL8) into the sum of AS“, the contributidn from

r
discrete modes and Amn' that of continuous modes

.. d r
Amn - Amn + Amn .

(3.3.10)

66

By re-defining the integration limits to be the center of

its respective cells

X
wI+

on-

d
Amn can

area dx

H-

Ax/2

I
Az/2 ,

H-

(3.3.11)
be evaluated over the longitudinal cross section

'dz' as follows

. 2 z x...
d jkodn k d ' ' ' .
Amn — Smn - -—§—_— 2. x- ny(xi,zjlx ,2 )dx dz

where e

+ +
X

jkosn2 p.22 I ' -jBo|zj-z'| ' '
amn + ——§3__ ,z; .x; eyo(xi)eyo(x )e dx dz

kod 2 cos(dei)cos(dek)sin(KdAx/2)

€03) Kd 2
(73) os (Kd)4-[Kd4-

2 2
Gmn+2(n3-nl)(

 

sin(2Kd)]
2

. ‘jBodlzj‘ézl . ~ .
{3e Sin (BodAz/Z) ... for 2 ¢ 3

'jBodAz/Z
(l - e ) ... for l = j

(3.3.12)

igenfunction eY0(x) is expressed as (34L3) in the

above derivation. It is noted that the final form of Ag"

above is expressed in terms of normalized quantities, ine.,

§i=xi/d, zj=zj/d,Ax =Ax/d and Az=Az/d as well as Kd and Yd,

67
for the convenience of numerical calculation. Following the

same procedure, Ag“ can be reduced to

r
Amn

+ +
jk05n2./.m 21 xk ' 'jB(€)IZj'Z'I
= —-4—-— 0 ZTE(€)d€ - - ey(xi’g)ey(x Ig)e
22 xk

dx'dz'

 

 

2(n2 - n2)(k d)2 w ~ cos(5x.)cos(5x )sin(5A§/2)
= 3 1 o dégz 1 k
w 0 (E2 + 02s1n25)§25
~3§|5--5 | - -
je J i sin(BAz/2) ... for j # 2
x -j§A5/2 .
_(1 - e ) ... for j = 1

(3.3.13)
with normalized variables 5=od, §=€/d and B=Bd. Also
noted that the integrand in the above expression for Afin has
a singularity at Eakzd, i.e., when B=0 since §2=(k2d)2 -§2
by definition. In order to evaluate this improper integral,

let the integration be divided into three subintervals

w ~ kzd-E k2d+€
1H...) J
0 0 kZd-e

jsin(§Az/2) ... for j ¢ 1

+f .R(§) 5%! ,- ..
k d+€ B (1,_e‘JBAZ/2)

2 for j = 2

:1 +1 +I
3
1 2 (3.3.14)

68

where

~ Ezcos(5xi)cos(5xk)sin(55x/2)
Rm= 2 ~2 ..

 

l ... for j = 1
(3.3.15)
In the limit as (3+0, R(E) is regarded as a constant, taking

its value at Ea kzd. Furthermore,

lim j sin<éA§/2) = jéAE/z and
8+0

lbm (1 - e'jBAZ/Z) = jéAE/z .
8+0

12 of (3.3.14) then becomes

. ~ k d+e
jAzR(k d) 2 ~
lim :2 2 19$
k

'B+0 2 2d-s B

 

 

 

. ~ -k d ~
jAzR(k2d) [ 2 d5

2
kzd-e yfikZd)2 _ g2

 

 

(3.3.16)

69

Both integrations in (3JL16) above can be evaluated by

changing the variables: E= kzdsint in the
E =k2dcosht in the second to yield
jA£R(k2d)

lim I =
0 ‘2 2

1 E

-—)
kzd

 

- sin- (1 -

B”:

+ j COSh-1(1 4’ FEE) .
2

Finally Ag“ has the form

2(n§ — n§)(kod)2
Amn = 1T (11 + 12 + I3)

 

(3.3.14) and (3.3.15) as

kzd-e ~ jsin(éA£/2)
I1 9-] g—g R(g)[ .~ ~
0 B (1 _ e-jBAz/Z)

first and

(3.3.17)

(3.3.18)

and I3 can be written from

for j ¢ l

for j = 2

(3.3.19)

for j # l

for j = 1

(3.3.20)

70
where the infinite upper limit of I3 integral is replaced by
a finite value in the numerical process as long as suffi—

cient convergence of end result is obtained.

3.3.2 Numerical Results
MONO-MODE SLAB WAVEGUIDE

First to be studied is a slab waveguide with refractive
indices n1 =- 1.6 for the core region, n2 = 1.0 for the
cladding and n3 = 3.0 for the slice discontinuity. With
1 other guide parameters also indicated, Figure 3.10 shows the
distribution of Ey(x,z), the field inside the dielectric
slice discontinuity region, in the transverse cross section
at various axial locations. It is evident that the almost
uniform distribution of FY near the entrance plane at z/zo =
-0.875 as compared with those toward the exit plane which
possess a cosine type distribution leads to a prediction
that strong radiation is expected at the slab-discontinuity
interface. This is verified from Table 3.1 that indeed a
relative radiated power of 24% is obtained. Listed in Table
3.1 are values of reflection and transmission coefficients
and its relative powers including the radiated powers for
various value of parameters; Of all the cases calculated,
shown here are primarily the cases of n3 =- 1 and 3, each
with two axial discontinuity lengthes, 0.1 and 0.5. Results
obtained from approximate solutions of Section 3.2 (both
radiationless, G§y=0, and radiating, G§y¥0) are also

included for comparison.

nOrmalized amplitude of Ey(x,z)

71

E;(x.z) = Eoeyo(x)exp(-jeoz)

 
  

~ ... 30(x)=cos(xx)

 

 

 

 

 

 

 

 

' l 1 T
.125 .375 .625 .875
normalized location (x/d) along x :11 = 1.6

' n2 = 1.0
n3 = .

2d/Ao= .3

zo/d - 0.5

Figure 3.10 Distribution of field E (x,z) excited in
the slice-discontinuity region of slab
waveguide by TEO incident mode wave.

Table 3.1

CONFIGURATION
AND TYPE OF
SOLUTIONS
n3=l.0

z /d=0.l
analytical

(G =0)
approximate

(G 0)
:6: (G =0)

r
MoM (ero)

n =3.0

z /d=0.l
analytical
(G =0)
approximate
(Grfo)

MoM (Gr=0)
MOM (613m)

n3=l.0
z /d=0.5
analytical
(G =0)
approximate
(G #0)

M (G 80)
MoM (630)

n3:3.0

z /d=0.5
analytical
(G =0)
approximate
(Gr#0)

MOM (Gr=0)
MoM (GrfO)

72

Reflection and transmission coefficients
for TEO slab-waveguide mode incident upon

dielectric-slice discontinuity of various
configurations, as calculated by several
methods; resulting reflected, transmitted,
and radiated powers included. (2d/AO=O.3,

nl=1.6,n2=l.0)
2 2 2 2
R0 TO RO T0 l-Ro-T0
.08050 .9968 .00648 .9936 -.000090
.0782 .9956 .0061 .9912 .0027
.08072 .9967 .00651 .9934 .000073
.07871 .9956 .00620 .9912 .002585
.3063 .9519 .0936 .9061‘ .000067
.3459 .9029 .1196 .8152 .0651
.3071 .9518 .0943 .9059 -.000234
.3288 .9203 .1081 .8470 .044938
.3445 .9388 .1187 .8813 -.000026
.2972 .9319 .0883 .8684 .0432
.3465 .9385 .1201 .8808 -.000845
.3047 .9247 .0928 .8551 .052088
.4137 .9104 .1711 .8288 .000024
.3698 .7459 .1367 .5564 .3069
.4186 .9098 .1752 .8277 -.000296
.4482 .7482 .2009 .5598 .239313

73

Note that the results of the radiationless analytical
and MoM.solutions agree perfectly (as they should), thus
confirming the correctness of both; the small (positive and
negative) radiated powers are due here totally to numerical
truncation errors. Discrepancies between the accurate
radiating and approximate radiationless results increase
with the I'strength" of the discontinuity; radiation increase
for ”stronger“ discontinuities with progression downward in
Table 3.1. These results indicate those paramaters which
constitute a “strong“ discontinuity as well as conditions
for a ”small“ discontinuity where either the closed-form
radiationless solution or the approximate solution with

radiation might be adequate.

DUAL-MODE SLAB WAVEGUIDE

Here the MoM solutions is implemented to study the mode
conversion process discussed in Section 2.4, several
waveguide parameters are chosen to allow the excitation of
two principal modes, the T130 and TE2 modes. Assuming a T130
mode incidence, the results are shown in Figure 3.11 through
3.13. The plot of axial variations of Ey(x,z) in Figure
3.11 for the low contrast case of (6n = 0.15) shows the
increase of standing wave pattern as well as rapid phase
change in longitudinal direction when the length of the
discontinuity zo/d is extended from a value of 0.1 to 0.5.
As a comparison, the cases of higher contrast (5n = 1,4) is

shown in Figure 3.12 with zo/d = 0.2 and 0.4. While holding

(x/d=.063,z)

Y

normalized amplitude of E

 

 

 

D. \ \\
\ \
\
.7— 100° ‘0. \ )2
m \ \ \ \
.'> 0 fl\ \ \
vim \ \
6‘ :12 . d — \ \
‘Or: z /d = 0.5 ‘ \
H 0 \ ‘\B'\ \
Q \ ‘ \ \
.5-4 20° \ \ Ck \ \
\ \
\\ \ ‘EJ\\ E)
‘ \\ \ wt
04— \
\ Q
\ \
\
b \
.3‘ -20° \ \
\ \
l l I F' l f I I
-.875 -.625 -.375 -.125 .125 .375 .625 .875
normalized location z/zo n1=1.05
n2=l.0
n3=1.2
2d/A=4.0

Figure 3.11 Relative amplitudes and phases of fields
in the slice-discontinuity region as a
function of normalized axial locations.
The slab supports dual modes with a TEo
mode incident.

 

75

 

1.0.. n 0
amplitude :
phase : -------
9 180° D 3 Zo/d = ,
—_(
Q “ O: zo/d = 0.4 Q
‘5 \
‘ \
5 ‘ \
\ .
8 '8- \ ‘ \
8 a \ \
E “\95‘ ‘ \
V O b 8‘ k
m>.7_ 100 \ a“; 0.-
~ \
‘ ND
0 \ a #3\
3 go) \ ‘ El
3 06— "3:3 \ a \\
"4 (OJ: \ a a
H Hg,
04 0) \
5 u
\ a
'U \ fl
0 a
.5" °
:2 20 Q
m \
E \
O c \ u
a .4" \
\
a \
Gk
\
\
.3- - °
20 a \GK
\
1 l l j l l l I F
-.875 -.625 -.375 -.125 .125 .375 .525 .375
normalized location z/z0 “1‘1-5
n2=1.0
n3=3.0
2d/A=l.0

Figure 3.12 Relative amplitude and phases of fields

in the slice-discontinuity region as a
function of normalized axial locations.
The slab supports dual modes with a TEo
mode incident.

(X/d=.063,z)

Y

normalized amplitude of E

o
co

0
\l

 

76

 

 

O amplitude
phase
B‘ ‘
~-. Q ~
. O
_. 100
CL
\
\
G) c
> a)
_. wan
43c
or:
H 0..
w
u
— 200
— -20°
l I l l l l l 7
-0875 -0625 -0375 -0125 0125 0375 0625 .875 ..
normalized lacation z/z
0 n1=1.6
n2=1.0
- 2d/A=1.0
zO/d=0.2

Figure 3.13 Relative amplitudes and phases in the

slice-discontinuity region with various

indices for the slice region, as a func-
tion of normalized axial lications. The

slab supports dual-mode propagation.

77
zo/d constant and varying n3, Figure 3.13 shows the
existence of a stronger axial standing wave for n3 = 3.0 and
is also evident from Table 3.2, there it shows a 32¢4% of
radiated power. A study of Table 3.2 reveals that when zo/d
is changed from 0.2 to 0h4, there is a increase of mode
conversion from TEO to TEZ mode, as evident from the almost
equal percentage content of reflected and transmitted power
in TE2 mode; in fact, more power is converted to T82 mode
and transmitted than TEO mode, at the same time, radiated
power is decreased by nearly 79%. Different phenomenon is
seen in changing n3 from the value of 1.0 to 2.0, in that
the reflected power of T32 mode is greatly reduced when
compared to that of TEO mode as well as the transmitted
power is dominated by incident TEO mode; accompanied with
slight increase in radiated power.

As a final observation, despite the infinite upper
limit involved in the continuous spectrum integral I3 of
CL3.20), no difficulty was encountered in obtaining
adequate convergence in all the above MoM results.

Moreover, various partitioning for the discontinuity region
ranging from 4 by 8 to 8 by 16 (leads to a 128 by 128 matrix
elements) rectangular cells were used and less than 60
seconds of computing time on the MSU CDC Cyber 750 system
was consumed for each case.(source listing and sample out-

put attached as Appendix A and B).

78

Table 3.2 Mode conversion coefficients (reflection
and transmission) for TE0 slab-waveguide

mode incident upon dielectric-slice dis-
continuity of various configurations;
slab width is such that it supports the
propagation of TEO and TE2 modes.

 

 

 

 

% power
CONFIGURATIONS zO/d RO/R2 TO/T2 radiated

n1= 1.6 .025 .297/.Ol9 .953/.019 .2

n2= 1.0 .075 .556/.068 .815/.O74 1.65

n3= 3.0 .2 .259/.251 .608/.42 32.4
d/XO= 0.5 .4 .42 /.39 .51 /.58 6.96

n3

n1= 1.6 1.0 .197/.123 .899/.145 11.76
n2= 1.0 2.0 .136/.0226 .920/.092 12.6
d/Xoa 0.5 3.0 .259/.251 .608/.42 32.4

zo/d= 0.2

79
3.4 Iterative Solution

The 2+l'th iterative solution 814,1(2) to EFIE (2.2.16)

has the form

= E1 (r) + gig] 0n2(r' ')G(r|r' ) ~E2 (r)dV' .
Vd

(2.5.30)
when the longitudinal cross sectional discontinuity is again
partitioned as indicated in Figure 3.9, the above IE can be

discretized for the field at (xm,zn) as

j Nz Nx
_ -i 2
2.+'zl(xm n - E (xm'zn) +1202: 2 6n (xi’zj)
j=1 i=1
X E(xm,znlxi,zj) °E£(xi,zj)AxAz

for (xm,zn) eLCS.

(3.4.1)
By substituting Green's function (3.1.16) appropriate for
the mono-mode slab waveguide, (3.1.3) and (3.1.6) for

ey0(x), and ey(x,£) respectively, into (3.4.1), it yields

_ i
E£+1(xm’zn) - E (xm’zn)

jk Nz Nx
O 2 2
- .2? A cosn<xm Z 2: 6n (xi,zj)E£(xi,zj)
j=l i=1
-:180|zn-2|

x cosucxi e zj AxAz

80

Nz

R2
‘53?”

2
0n (xi,zj)E£(xi,zj)

" P6sz

-jB(€)|zn-z.l

 

k -s w 3
2 coscx.coscx
(H J 1 12 “‘e ’
. 0 7k _€J mm (6)
2
+ mg = k2) 121- sin-1(l - f) + j cosh-1(1 + xi)
2 2

- 2j|zn - szsJ }AxAz

for m = 1,2,

n = 1,2, ...
(3.4.2)

where

COSOX. COSOX
Ru: = k2) = 3 m
c (E) g=k2

With incident field Ei, chosen as the initial O'th

order solution as discussed in Section 2.5, such that

-jBoz.
2 o(x ,z.) = EOA costi e 3 ,

(3.4.3)

31(xm,zn) can then be obtained from (3.4.2) and (3 4.3)

Subsequent iteration will yield E2(xm,zn) and E3(xm,zn) etc.

until convergent results are obtained.

81

NUMERICAL RESULTS

 

Several cases which were included in Table 362 (n1:
1.6, n281.0, n3=3.0) are calculated. Listed below are
number of iterations required for the total unknown field
E£+1 to converge to within 1% of En, using the incident
field as the O'th iteration field.

no. of iterations

I. d/1080.15 zo/d=0.2 5
0.3 6

11. d/lo=0.5 zo/d=0.025 5
0.075 6

0.1 7

Worth noting from above results are that cases II
support dual mode propagation yet the number of iteration
required are not significantly increased due tet'small'
discontinuity. Since most of the time is consumed in the
calculation of Greenls functions for all the partitions
(average 30 seconds on Cyber 750), the time required for
iterative process is far less in comparison. However, for
larger discontinuities, a gobd initial estimate for the O'th
iteration field in addition to sufficient partitions are
necessary for the correct converged results (when compared
with that obtained by MoM). (source listing and sample out-

put attached as Appendix C and D).

CHAPTER IV

INTEGRAL-OPERATOR FORMULATION OF COUPLED

DIELECTRIC WAVEGUIDE SYSTEM

In open-boundary waveguide circuits, dispersion
characteristics of the guided-wave modes are required to
determine bandwidth and signal distortion, while the field
distributions of these modes must be quantified to predict
their interaction with other devices and excitation through
coupling to other waveguides. Particularly, in Optical
communication, it is desirable to bunch many optical fibers
into one cable; because of their proximity, the individual
waveguides can exchange power, so that some part of signal
that is being transmitted in one guide can enter a neigh-
boring guide and interfere with the signal that is being
transmitted there. In other applications, such as the direc-
tional coupler, waveguide coupling offers the possibility to
feed power from one guide into another in a controlled
manner.

In this chapter, a coupled multi-waveguide system is
studied. The description of a single isolated waveguide,
based upon an electric field integral-equation formulation
for its unknown core field is first presented based upon an

induced equivalent polarization current [14] and an electric

82

83

Green's dyadic as its kernel. This is followed by the
generalization of that formulation to a system of N coupled
waveguides. Polarization sources which radiate into the
unbounded region and maintain the total field in the coupled
system are identified. Homogeneous EFIE's for natural
surface-wave modes supported by the coupled system are
obtained. When the coupling is weak, assuming that field
distributions in each individual guide do not change signi-
ficantly from those of the isolate guide, a perturbation
formulation is developed to obtain an approximate solution
for the system-mode prOpagation constants and coupling

coefficients.

4.1 Equivalent Polarization Description of Heterogeneous
Waveguide Core
The Maxwellls equations appropriate for the description
of electromagnetic fields E(E) and H(;) maintained in the
heterogeneous waveguiding system by Pe, the impressed

polarization source density are

V°(€E) =-\7-;‘>e
V XE = jwuofi

- . _e _ -
V><H = ij + waE

(4.1.1)

where the nature of low-loss dielectric media (graded-index

84
core and surround) is represented through the complex
permittivity s(T:)=e'(E)-j 6"(E) with e"(E)=0(E)/we' where
0(2) is the conductivity of the material. An equivalent
polarization density, which is induced due to the permit-
tivity contrast between the graded-index core and its
background surround cladding, can be identified from equa-
Ition (4JL1).. If terms involving the permittivity:c of the
{surround are added and substracted in Maxwellls equations,

they can be placed in the form

v- [ecfi + (s- ec)§]= -v-f>e
VXE = - jquP—I

VXfi = jwf’e + j(0(s - Ec)§ + jwscfio
7 -fi = 0 .

The equivalent polarization is consequently identified as

Peq = 66(r)E = [E(r) - 8C]E
(4.1.2)
Thus when the system (44LJJ is rearranged to emphasize the

total effective source densities, the Maxwellfs equations

become
. ' _ - , -e '
V (ECE) - V (P + Peq)
VXE=- jwuoI-i
VXf-i= jm(Pe+Pe ) +jw€E
7 -fi = 0

(4.1.3)

85

Applying a curl Operation to Faraday‘s law in equations
(4.1.3) followed by the substitution of Ampere's Law, leads
to the well known-wave equation for the electric field E(r)
2

.. 2- - ..e .-
V x V X E - kCE - w u0(P + Pe ) .

q

(4.1.4)
where kc=w\/TIo—ec = ncko, is the wavenumber of the surround
cladding of refractive index nc. Expression (4.1.4)
indicates that total field E along the waveguiding system is
maintained by primary source 5e augumented by equivalent,
induced ieq both radiating into a uniform, unbounded region
with wavenumber kc- It is through this interpretation of
equation (4JLA) that the original problem involving a
bounded, graded-index dielectric waveguiding structure has
been replaced by an equivalent polarization density fieq
radiating into an unbounded surround medium. Equivalent
induced polarization seq =606n2(f)fi is prOportional to
total field E in the waveguide core region V; it is non-zero
only in that core region where index contrast 6n2(f) =

n2(1':)-n,23 is non-vanishing, i.e., for points E e V where

6n2¥0.

4.2 Electric Field Integral Equation Description for Guided
Waves Supported by Open-Boundary Dielectric Waveguide
System

The electric type Hertzian potential if, which is
maintained by the total effective polarization density

fitot'§e+§eq embedded in a uniform medium of permittivity

86

8c satisfies the vector Helmholtz equation

 

(4.2.1)
It is therefore appropriate to express the EM fields in the

unbound region of homogeneous surround as [52]

V(V -fi) +k§II

MI
ll

jwecv X H .

:m
ll

(4.2.2)
The solution for Hertzian potential in the above three-

dimensional Helmholtz equation is

P (5')
fi =f t°t G(E|E')dv'

s
V

 

C

(4.2.3)
with the well-known scalar, 3-d Greenfls function for an
unbounded medium

GIEIE') = e __ _

4nR(r,r')
(4.2.4)

where Ralf -Eq is the distance between a source point at E'

and the field point at E. Therefore (4.2.2) can be written

in the form of linear lntegral Operator as

87

p (E')
= 077- + k:)f t°t G(E|E')dv' .

 

E
C

V (4.2.5)

The fact that primary, impressed polarization 5e
maintains an incident field Ei(E) while secondary, induced
polarization Eeq excites the scattered field ES(E) due to the
non-uniform waveguiding region motivates the following field

decomposition [53]

E(E) = Fain?) + 83(2)

6453(5)} + e£{1'>eq(§)}
(4.2.6)

which leads.to the 37d, linear integral-operator equation

_ - - _ _ -i _
E(r) -dC{Peq(r)} — E (r)
(4.2.7)
Substitution of ti from definition (4.2.5) and 5eq 8 66E into

operator equation (4.2.7) leads to

E(E) - (vv- + tbfiié—rll E(E')G(E|E')dv' = 816:)
V C

(4.2.8)

where refractive indices satisfy's=n260, scangeo and

88
53=eo(n2-n§) =600n2. In terms of spatially dependent
wavenumber quantities defined as k2=n2kg, kg=ngkg and

5k2=k2-kg, integral equation (4.2.8) becomes

2 -' - = - - -. -
E(E) - (k: + vv-) I $5—%§—1 E(r')G(r|r')dV' = 81(r)
k ,
‘V c

... for Ee:V
. (4.2.9)

where V is that waveguiding region where 0n2#0 and 6k2¥0.
Expression (4.2.9) is a 3-d, volume, electric-field integral
equation (EFIE) for unknown field E(E) excited in that
waveguiding region by the impressed field E1 due to 5e.

EFIE (4.2.9) can be expressed in terms of electric
dyadic Green's function by carrying the differential
Operator (k?: + VV-) through the integral operator over V;
the resulting integral must be evaluated in an apprOpriate
principal-value sense. This leads to the conventional
relation between the electric type dyadic Greenfis function

69(2):!) and scalar Green's function E(Elf') [54] as
= _ _ _ 2 = _ _
Ge(rlr') — (kc I + VV)G(rIr')

where 1 is the unit dyadic 52x + yy + 22, and Ge(rlr') is the

solution to

(4.2.11)

89'

in the unbounded medium of wavenumber kc. We may rewrite

EFIE (4.2.9) compactly in terms of Ger taking into the
consideration that the volume integral be evaluated in a
principal value sense, such that a correction term from the

excluded principal volume is required [48] as

_ _ 2 -3 = _ _ _ _ 2 = _ _ _‘ _
E(r) -pv [El—(2L3- Ge(rlr') - E(r')dV' +-5—k-2— L'E(r) =El(r)
\I kc kc
(4.2.12)
where
= A.“
L = lim 11;] 9—213 dS
56+0 s R
6 (4.2.13)

represents a three-dimensional depolarizing dyadic; S6
encloses principal volume V6 with outward normal as
indicated in Figure 4.1. The contribution of this carrec-
tion term is discussed in detail in Section 2.3 of Chapter
2.

Equation (4.2.12) is then the basic 3-d volume
electric-field integral equation (EFIE) which describes the
open-boundary dielectric waveguide systems. *This vector
EFIE is an in homogeneous Fredholm integral equation of the

second kind [51] for unknown field E(E) excited in the

heterogeneous waveguide core region V (where 6n2#0) by
impressed field_Ei(E) due to excitatory polarization P3.
All following develoPments are based upon this fundamental

EFIE.

90

 

 

Y

Figure 4.1 Three-dimensional configuration of principal
volume which contributes to the depolarizing
dyad of the 3-d EFIE for a open-boundary
waveguide.

91

4.3 Homogeneous EFIE's for Natural Surface-Wave Modes along

 

Coupled Waveguide System

A natural, surface-wave-mode field is that field which
can exist on the open-boundary waveguide system in the
absence of excitation, ime., the non-trivial solution for

E120. EFIE (4.2.12) then become the homogeneous equation

(4.3.1)
In the case where the waveguide has a transversly graded
dielectric profile with longitudinal-invariant dielectric
properties, 6k(?) becomes 0k(5), where B=x§ + yy is a two-
dimensional position vector. Eigenfield solutions having an_

axially-travelling-wave nature with phase constant 3

(4.3.2)
are supported by such a system, as demonstrated below.
The principal-value integral in EFIE (4.3.1) can be
evaluated by exploiting E(E) and 58(EIE') based upon

expressions (4.3.2) and (4.2.10), reSpectively, as

92

(4.3.3)
Making the change of variables u=(z-z') for the component

z'-integra1 of result (4.3.3) above, leads to [55]

' m -jch
' '
[.5382 e—mr’ 62'

co. "jkcfl-‘6'I27'u2
__ 1 :sz ijBu e
-m JIE-p'l +11

-—oo

 

du

 

 

isz _ _
Ejfir— KO(YID"O'|)

(4.304)
where K0 is the modified Bessel Function of second kind and

Y is an eigenvalue parameter defined as

_ 2_2
H);

The asymptotic exponential decay of K0 for large real

 

(4.3.5)

arguments leads to the expected surface-wave-mode field

confinement. A condition for real Y is therefore identified

93
as 3> kc. This condition is consistent with the well known
result [9,11]; it permits the phenomenon of total internal
reflection at the core-cladding interface and, as a result,
the existence of confined, guided surface-wave modes. .Each
term in EFIE (4.3.1) is therefore proportional to exPfisz),
and expression (4JL2) is therefore indeed an eigenmode

solution.

Depolarizing dyad i is defined as [48]

5")

AI
9755'
R

SO '
=-4lTr-f dl‘fi' f 32-d2'
c 00R

0

I." II
II

_1_
4n

(4.3.6)

where C6 is the principal contour as shown in Figure 4.2.
The integration over 2' in (4.3.6) can be carried out by

changing of variable from 2' to u:(z-zF) such that

ii I .. (IFS-E.) +211
-ao .. [MD-”DU + u ]

 

I
k)
L'C)
|
D
.5

Let 235-5", then the depolarizing, or Green's-correction,

dyad becomes

94

 

 

Figure 4.2 Two-dimensional configuration of principal
volume shown as enclosed by principal
contour CG‘

95

IT'II

=lfmrd..=:
20 r

C6
(4.3.7)
for the two—dimensional, open-boundary wave guide geometry.
With results (4.3.2), (4.3.3), (4.3.4) and (4.3.7), the

three-dimensional volume EFIE (4JL1) then reduces to the

desired vector two-dimensional form

= 2 - = _ _ 2 -| = .. - _ _
kc CS kc

for all SECS,
(4.3.8)
51%)5la) represents the corresponding(B-dependent two--

dimensional Green's dyadic function and is defined as

EQE’IBIB') = [1.5+ (Vt$382)(Vt¥382)]K0W15-5'I)
(4.3.9)

where Pi is the unit dyad and the V operator has been
decomposed into the transverse Operator Vt and its longitu-
dinal component.

In a multi-waveguiding system as indicated in Figure
4.3, EFIE (4.3.7) can be generalized to describe surface-
wave modes supported by the coupled system of N waveguides.

Since the system-mode field propagates with common phase

CLADDING
(6 = (c, n = nc)

A

y‘ e+582

(:sl5L
ek

CLADDING
/N% 11 (e g 6c’ n 'I nc)
£863?” In each CSm
de'mfin lam

Figure 4.3 Configuration of N-coupled, open-boundary
dielectric waveguides.

 

 

97
constant B and scattered field as is maintained by polariza-
tion density fieq which exists wherever 6k2(5)#0, then it is
only necessary to extend the integration over each guide of

cross section CSn, n=1, 2, ..., N, to obtain

N
= 2 " = _ _ I 2 -l = ... - _ _
[1+5LL9). 2]“e(p) -Z PVf 6k (2 ) 9:2)(plp') - e(p')dS'
k
CS

=0 for all EEICSm

m8 1,2, ..., N.

(4.3.10)

Equations (4.3.10) are a system of N simultaneous EFIE's for
eigenmode fields 5m in each waveguide core. Non-trivial
solutions are obtained only for discrete phase-constant
eigenmodes 8 88m corresponding to the m'th surface-wave
mode. The coupled system of 2-d EFIE's (4.3.10) can be

expressed alternatively as

N .
[iii-(ii 7] -é(6) - Z pvj gigg- 338m?) -E(6')ds'
CSn nc
for?5 ECSm, m=1,2, ..., N,
(4.3.11)
where wavenumber k is replaced by refractive index n from
the relation k=nk0, '1' for fies is dropped but implied.

A similar result, yet provides better formats for

98
physical interpretation and the convenience in actual
computation (without the source point singularity of 5e) can
be obtained by first carrying out the divergence Operator in
(4.2.9) followed by the use of the symmetric property of the
scalar Greenfls-function VVG=-VV'G[59]. Subsequent applica-

tion of the same divergent identity leads to

N -
+ Z vflv' .[_613_<§;'_)_ E(E')G(E|E')]
n=1 V kc

.. [W

2 —. _ _
~§5—駗L E(E')]c(r|r')}dv'

5k2(E')E(E')G(E|E')dV' = fii(i)

1M2
£T‘-fia

for 'feVm, Ill-1,2, ..., N.
(4.3.12)
It is observed that the second term in the above expression
is the scattered field due to the scalar potential main-

tained by the surface charges and volume charges in the core

regions. Since V..[6_k_1(,£_)§(r )1 = w-u—i—Lnnu )1 =
k
-V'-§(r') due to V'-(e§$=o in a source freecregion,

together with V'-E=-V's-E/ec , and the invoking of

eigenfield (4.3.2), (4.3.12) above then yields [57]

5(6) +2 51-; E‘s—($119-5(6')(vtxjefimowIE-E'Imz'
C

99

 

M2

_ - 2 -
l . Ck I _ ' A _ .. '
2%] em 1:22. (p) WU 382%er p'“ds
cs 0)

1 n

:1
II

N
— Z 71;] 5k2(5')é(6')K0(yIB-B'|)ds' = o
“=1 05,1 for'p'e csm, m=1,2, N.
(4.3.13)

The effects of equivalent-induced charge and current are
readily seen in (4JL13); the contribution from the surface
charge due to the jump discontinuity in index contrast
between each core and its surround is evident from the
contour integral along I'Cn of the n'th guide; while the
volume polarization charge which is proportional to the
gradient of the continuous index profile (interior to each
core) is given by the second integral and the last integral
in (4JL13) represents the scattered field maintained by

equivalent-induced polarization current.

4.4 lntegral-Operator-Based Coupled-Mode Perturbation
Approximation
Guided-wave field §=é(§)exp(-jgz) supported by a
system of N dielectric waveguides (Figure 4.3) described by
refractive-index contrast 6n2(3) satisfies the coupled
EFIE's i

2 - N -
[I+—§§— £]°e(o)-Z PVf —-—2——geB(o|o')°e(p')dS'
CS c

100

for all Secsm,
m=1,2, ..., N.
(4.4.1)

The eigenfield En(6) for the n'th guide when isolated

satisfies EFIE

Hll

13

6n:(5) = _ _ 6n:(3') = _ _ _ _
+-———2—— 2 - enm) -PV T gen(plo')°en(o')d8' = 0
c

c S C
n

for a115€ csn
(4.4.2)

where 8:8“, the eigenvalue associate with isolated
eigenfield én(6), is implied in letting EeB = gen-
To obtain a system of scalar equations, Operate term by

term on eq. (4.4.1) with the integral operator

m ‘00. for 111:1,2, .00, No

(4.4.3)

where 5m is the isolated m'th guide eigenfield, to Obtain

101

... for m=l,2, ..., N.
(4.4.4)
Use was made of the well-known reciprocal property of the
electric Green's dyad [58] to Obtain (4.4.4); that prOperty

allows

(D
DI
thl

gem?) .am') =‘é(B') 388$ IE) -5 (5)

(4.4.5)
when it appears within the integrand of EFIE (4JL2)
subsequent to application of operator (4.4.3).

In a system with coupled—mode propagation, the system
mode phase constant B is embedded in Green's dyadic ZeB' TO
extract 8 in this approximate coupled-mode theory, the 2-d
Greenfs dyadic is expanded, retaining only the leading terms
Of a Taylor's expansion for 368 about its value at B=Bm for
the m'th isolated guide; this approximation will be adequate

given the condition that weak coupling prevails. Therefore,

(4.4.6)

 

102
Inserting (4.4.6) into the second term of (4.4.4) leads to

the following expansions

n=1 n
n m
N ' 6n (p') 6n£(p) - _
I o
_ 2: A81“ as n2 e(p) pv n2 agem em(p)dS
n=1 CSn C 'CSm C
(4.4.7)

Applying the defining EFIE for eigenmode field Emu-3') from

eq. (4.4.2)

= anim') = _ _ 4131(5) = - -
1+ 2 2 °em(o')=PV 2 gem'm'lo) °em(p)ds
n n
c CS c

in the first term of the R.H.S. of expression (4.4.7)

provides

N 5n:(5') _ _ . 6n (5) = - _ - -
- Z dS' -——-2—-f e(o') °PV ——2—— gem(p'lo)-em(o)ds
cs cs

C

Bro

:3

n C m

103

N an§(6') _ _ = Sniai') = - _
= - Z (15' ———2—— e(p') - I+———2—— 2. - em(p')
n
n=1 CSn
(4.4.8)
Since 6n:(5')ao for 5' eCSn, then the system (4.4.4)
leads, subsequent to use of results (4.4.7),(4.4.8), and the

use of f-Em(5')=em(5'), to

:1

N 6n§(5) _ _ _ _
Z -—-2—e(p) -em(o)dS
cs

C

N Gnih-D') - - - (311:1(0) = _' _ ._ _ d
+ ZABm ds —§—— e(o)-Pv ngemm Io) emm) 5
CS c CS

... for m=l,2, ..., N.
(4.4.9)
Equation (4.4.9) represents a system of N simultaneous .

equations, one associated with each of the N weakly coupled
dielectric waveguides; it involves N unknown fields, i.e.,
the é’(6)'s, one for each guide. Further approximation can
be made based upon the assumption that the field of each
guide in the coupled system will not differ significantly

from its isolated eigenmode field distribution as long as

104

the coupling is weak; i.e., the assumption can be made that

5(5) 3 an5n(5) for all 5 a CS“.

(4.4.10)
where an is an unknown amplitude coefficient that depends
upon the coupling. Moreover, the second term in (4.4.9) can
only have significant cOntribution when n=m since 6§em(5'|5)
is small when 5' {-ZCSi.1 while 5 E CSm when n¥m. With this

weak-coupling approximation, terms n#m in the second sum of

expression (4.4.9) are drOpped leading to
suits) _ _ _ _
2: an —72— en(p)-em(p)ds
CS

. 2 - - 2 -
+ AB (3.11—“fl; (-')'PV ' m 5- (‘IIE).§ (Skis
mam 2 m p a 2 gem 9, m
cs nc csIn 1“c

m

... for m=l,2, ..., N.
(4.4.11)
where subscript 'n' and 'm' are-used for index contrast 5n2
to indicate the summation index. Expression (4.4.11) can be

written in the form of matrix equation as

N
Z Cmn(8)an = 0
n=1

... for m=l,2, ..., N.
(4.4.12)

105

where matrix element Cmn is defined as

 

[(B"Bm)cmm ... for n=m
cmu = * 2 _
6nn(o) _ _ _ _
. 7— 611(0) 'em(p)dS ... for 11¢!!!
CSn c
(4.4.13)

and Cmmm a normalization constant depending upon eigenfield
am and aEem/as evaluated at isolated eigenvalueB-Bm is

defined as

~ an§(6)_ _ amid)» = __' _ _' '
Cmm= dS—T-em(p)-PV —-2-—Ggem(plp )-em(p )dS .
cs n cs -

(4.4.14)
The system mode eigenvalues are those 3's which lead to
a non-trivial solution to system (4.4.12) when det [Cmn(3)]
= 0. Relative modal amplitudes an are subsequently obtained
from the resulting homogeneous matrix equation of order (N-
1) obtained from system (4.4.12) after setting an=1 and

discarding one of the equations.

SPECIALIZATION TO COUPLED TWO GUIDE SYSTEM
When a coupled system consist of only two waveguides,

eq. (4.4.12) is reduced to

106

)- - ~ ‘1 " - )— 1
(B 81)C11 C12 a1 0

 

 

 

 

 

 

21 (8-82)c22..<1.a2. ..0 1

(4.4.15)
where the vanishing determinant for the coefficient matrix

leads to non-trivial solutions when

C C
12 21 _

~ ~

C11C22

82

By solving the above quadratic equation for system mode
phase constants B, we obtain

8 = E 1 68
(4.4.16)

where

ml
II
N
m
:3
O.)

as = (A2 + 52)5 with

B — B c c
A = 1 2 ' 52 = 12 21 ’

C11C22

The amplitude ratio of the coupled surface modes on each

guide can be found from eq. (4.4.15) as

107

C

 

mlm
A)

H

12
(4.4.17)

Since Cmms' are the self-coupling terms, it is convenient to
choose the normalization of Emfi)‘ in expression (4.4.14)

such that

(4.4.18)

subsequently, for degenerate or nearly degenerate coupling

81 g 82 3 Bo, and from (4.4.16),

(58 = VCIZCZI .

(4.4.19)

The ratio of coupled modal amplitudes then becomes

(4.4.20)
where superscripts '+' and '-' denote the coupled-mode
amplitudes associated with system modes having phase
constants 8’80 + 68 and Ba 30 - 68. The corresponding
longitudinal wave functions An(z), assuming the coupled-
surface-wave modes are well above cut off, can be written as

a linear combination of these fields as

108

aIe-j(8+68)z + a-e-j(B-68)z

A1”) = 1

A2(Z) age-j(8+68)z + age-j(B-68)z .

(4.4.21)
The initial values of An(z) at z=0 can be chosen
arbitrarily. Taking for simplicity|A1(0H2=0 and IA2(0)I2=1,
iuet, the initial power of the surface-wave mode in guide 2
being unity, susbsequent substitution of these initial

conditions into (4.4.21), and the use of relation (4.4.20)

yields
a+=—a-=l .C_1_2
1 l 2 . C21
(4.4.22)
C.12 -sz
A1(Z) = - j -E—-sin(682)e
21

A2(Z) = cos(<SBz)e-sz .

(4.4.23)

Expressions for 111(2) and 112(2) in (4.4.23) are the
solutions to the standard coupled mode equation with
constant coupling coefficients for a general (single mode)
coupled system [56]. It is noted from the expression for
A1(z) above that total transfer of power from.guide 2 to

guide 1 will occur at

109

This is the well known definition for 'coupling length';
furthermore, eq. (4.4.23) also indicates that power is
periodically exchanged between the two coupled parallel
-guides.

It is therefore concluded that the above results agree
with those from the more familiar, approximate differential-
operator coupled mode theory; ‘This serves to confirm the
validity of the integral-Operator analysis when applied to a
coupled waveguiding system. 'The later theory has a
conceptually-exact foundation prior to the coupled-mode
approximation, and leads to explicit expressions for the

coupling coefficients.

CHAPTER V

APPLICATION OF INTEGRAL-OPERATOR ANALYSIS TO

COUPLED SLAB WAVEGUIDE SYSTEM

5.1 Introduction

Applications using integral-operator based coupled mode
theory are sought for a uniformly-clad slab waveguide system
[15]. Physically, when modal fields become coupled, a phase
constant shift from.the B's of the isolated guides to that
of system modes occurs. In the degenerate coupling of the
uflth mode along two identical guides with isolated phase
constants er the system modes have phase constants
Bm'BOm 1 A8,

To begin, the general coupled system of EFIE is
specialized to a slab waveguide system. .A characteristic
equation for unknown system-mode phase constant 8 of a two-
guide system is Obtained using Fourier-exponential transform
method; parameters include slab refractive indices, dimen-
sions and spacings. Either exact or approximate solutions
can then be Obtained for phase-constant shift AB. Perturba-
tion solutions, which approximate the coupled fields in the
integral equations by the eigenfields of each individual
isolated guide as described in the last chapter, are

specialized for one-dimensional coupled TE modes. The

110

111
resulting coupling coefficients are obtained to implement
an approximate solution of the N coupled, simultaneous
EFIE's, resulting in a matrix equation for coupled-mode
amplitudes. Phase-constant shifts due to the degenerate TE
mode coupling of a weakly coupled two-guide system are
finally obtained.

Subsequently, numerical results are Obtained, using
both solution approaches, for several cases of degenerate'
and non-degenerate coupling in a two guide system. There,
the effect of coupling is demonstrated through the variation
of index contrasts in the system as well as the widths and

spacings of individual waveguides.

5.2 Specialization of EFIE for Coupled Slab-Waveguide System
The slab-waveguide system considered here has infinite
dimensions in both the y and z directions. It is assumed
that all waveguide parameters are both longitudinally and y
invariant. It follows that in (4JL13) using Vt 2 § a/ax,
5(6) = E(x) and assuming the transversely-graded index
profile n(5) - n(x), the integral of the y-dependent .
modified Bessel function K0 can be carried out to obtain a

one-dimensional Green's function g(x|x') as

m -y|x-x'l

1 - - , _ e ,
fif KO[Y|D’Q'I]dY " ZY =g(x|x) °

-ao

 

(5.2.1)

Also, the contour integral enclosing the transverse cross

section is reduced to contributions from surface charges at

112
both the upper and lower boundaries Of each guide as shown
in Figure 5.1. These specializations of (4.3.13) subse-

quently lead to the one-dimensional coupled EFIE's: .

 

 

 

' N 6k (x'= 2;) A d A
e(x) + Z 2 [ex(x2n) (x a; - sz)g(x|x2n)]
=1 kC
2 +
5k (x'=x )
n 1n A d , A
k2 [ex(x1n)(x a - sz)g(xlx1n)]
c
x2n dk:(x')/dx' A d A
- ex(x') 2 (x a; - sz)g(x|x')dx'
k (x')
x1n n

x
2n 2 _
-f 6kn(x')e(x')g(xlx')dx'} = 0
x

m=l,2, ..., N,
(5.2.2)

where the 5143, are evaluated at the boundaries x'stn and
x'arxin as the result of one-dimensional contour integration;
superscript '+' and '-' denote the interior side of the
slab/cladding boundaries. Decomposition of (5.2.2) into its

component equations leads to:

x-component:

113 )

 

 

 

 

 

c ‘,/
x2m" '-
I
" k Z
m

xlm- - k l’

” c
XZn' -

 

//
:3 kn PC 95
I

Xln-"’/k "A
j 3/ “
'3
a!
v .— - <1—~
2’
/ k

F
c

 

 

 

C

Figure 5.1 Contributions of surface charges which

arise from the index discontinuity between
each slab and its surround cladding.

114

 

 

N 2 -
5k (x'=x )
n=1 C
2 +
6k (x'=x ) ‘
- n k2 1n ex(xln)[‘Ysgn(x'xln)]9(xlx1n)
C

 

x2n dk§(x')/dx'
- ex(x')[-Ysgn(x-x')]g(x|x')dx'
x .

 

2
k (x')
1n n
x2n 2
- 6kn(x')ex(x')g(x|x')dx' .= 0 ,
x1“ (5.2.3)
y component:
N *2. 2
ey(x) - Z . 6kn(x )ey(x )g(xlx )dx = 0 ,
n=1 x1n ~
(5.2.4)

2 component:

N 2 I...
[ -jB 6kn(x —x2n)

 

 

 

ez (X) + Z 2 ex(x2n)g(x|x2n)
n=1 RC
2 +
6k (x'=x )
. n In
+38 . 2 ex(x1n)g(x|xln)
k
c
x2n dk§(x')/dx'
+18 2 ex(x')g(x|x')dx'
x kn‘x )

' 1n

115

-[xzn 6k2(x')e (x')g(x|x')dx' = o .

n 2
"1n (5.2.5)
The sgn function in eq. (5.2.3) has a value of l with
positive argument and -1 for negative argument.

A study of the above component equations reveals that
as a result of the one-dimensional slab system discussed
here, i.e., for natural coupled TE modes, ex(x)= ez(x)= 0,
and the only remaining y component equations are independent
in ey(x). In the case Of coupled TM eigenmodes, ey(x)=0,
therefore E(x) = xex(x)+ §ez(x) while ez(x) is coupled to
ex(x) in (5.2.5). Also noted that TM-mode coupling is
stronger due to the fact that there exist surface charges
from the normal component of the field, i.e., e(x). This
observation [57] parallels the discussion of scattering in
the one-dimensional slab waveguide in Section 3JL2. The
following discussions on various solution techniques will
concentrate on coupled TE mode systems; this is mainly due
to the simplicity of the component equation involved as
described above and the dominant nature of TE modes in
dielectric waveguides. These solutions will enable us to
understand the coupling phenomena in other more complex

systems.

116
5.3 Fourier-Expontial Transform Solution for Step-Index
slabs
5.3.1 Coupled TE Modes
Application of the coupled EFIE system (5.2.2) is
demonstrated by studying the coupled non-degenerate TE
surface-wave modes supported by the step-index slab--
waveguide system as shown in Figure 5.2. In this system,
the planar, slab waveguides have thickness (t1, t2),
constant refractive indices (n1, n2), wavenumbers (k1, k2)
and separation s. EFIE (5.2.4) appropriate for the TE modes,
having only y-component of electric field can be written for

guide fields ey1(x) and eY2(x) of slab 1 and 2 as

 

2 2 0
k -k
ey1(x) J{ ey1(x )e dx

 

 

 

2Y
t1
2 2 s '
k -k
_ 2 C v 'Yix-x'l ' =
2Y jr ey2(x )e dx 0
(5.3.1)
2 2 S
k -k
_ 2 c , -y|x-x'| ,
ey2(x) ZY ey2(x )e dx
s-t2
2 2 0
k -k .
_ 1 C I "le-X I I =
2Y I ey1(x )e dx 0
—t1 ... for s-tzg x5_s

(5.3.2)

117

 

 

-tl

 

 

 

 

 

-..)

Figure 5.2 Configuration appropriate for study of non-
degenerate TE surface-wave mode coupling
between two slab waveguides.

118
where kc is the wavenumber of the surrounding cladding with
refractive index nc.

Physically, it is noted that the third terms of both
the above expressions represent the coupling from the
neighboring guide. .As the separation between waveguides
increases, the effect of these coupling terms becomes
neglibible because Of the rapid decay of exp(-y|x-x%) (with
x' as source point in one slab and x as field point in the
other) since Y is positive real for the coupled surface-wave
modes. This coupled system then reduces to two independent
EFIE's for individual isolated slab waveguides.

For the purpose of simplicity, let k1 = k2, ime., both
slabs have the same refractive index; such a configuration
has common applications in practice such as symmetric direc-
tional coupler. With ki-kgskg-kga Akz, apply the following
inverse Fourier-exponential transform to represent unknown

fields in both (5.3.1) and (5.3.2)

- _L jnx
ei(x) - Zflfzime dn

-m i=l,2 for slab 1 and 2.

 

(5.3.3)
After integration over x', we have
slab l:
m . m jnx -(jn+y)t1-yx
21 jnx _ [e -e . ]
AszE1(n)e dn [E1(n)dn Jn+Y

m yx jnx
~ e -e
f E1(n)dn [ jn-Y ]

co

119

 

m (fin-y)s (jn-Y)(s-t2) '
_ e -e yx =
.l. E2(n)dn[ jn-Y ] e 0
'°° (5.3.4)

slab 2:

 

. - < ” jnx (jn+Y)(S-t2)-YX
2 JnX _ [e -e . ]
411.2 f 52mm dn f 32mm) an”

ac

 

m e(jn-Y)S+YX_ejnx]
- E2(n)dn [ jn-Y

m ' -(jn+Y)t1 .
-] E1(n)dn [l-e ]e-Yx = 0 .

 

jn+Y
(5.3.5)
By collecting ejnx terms of either (5.3.4) or (5.3.5)
above and exploiting the linear independence,of ejnx, er,

and e’Yx, it is concluded that

——Y—2 -—1 +-—.1 ] jnxd =0 .
f E1(n) [Akz jn+Y Jn-Y e n

This is essentially an inverse transform, and its vanishing
requires the bracketed quantity to vanish since E1(n)#0.
This then leads to the discrete allowable eigenvalues for

this nondegenerate coupled system

(5.3.6)

such that the transformed solution for Ei(n) associated with

120

ei(x) in (5.3.3) becomes

E1(n) = A6(n-K) + B5(n+K)
(5.3.7)

for slab l and

E2(n) = C6(n-K) + Dé(n+K)
(5.3.8)

for slab 2, where A, B, C and D are unknown amplitude

coefficients. The remaining terms from eq. (5.3.4) provide

°° ’(jn+Y)t1-Yx YX
E (n)dn[ e . - .e -
1 JU+Y 30+Y

 

 

m . (jn-Y)(s-t )
(3n-v)s 2
e -e YX =
.J{ E2(n)d" [ in-Y ] e 0

 

 

 

 

(5.3.9)
while from eq. (5.3.5)
m (jn+Y)(s-t )-Yx - -
d e 2 _ e(1n Y)S+YX]_
E2(n) n [ jn+Y jn-Y
m 1 ‘(jn+Y)t1
-e ‘YX = .
f E1(n)dn[ jfl+Y ]e. 0
(5.3.10)

If the transform solutions (5.3.7) and (5.3.8) are utilized
in eqs. (5.3.9) and (5.3.10) above, and linear independence
of the ein is exploited we then have a homogeneous system

of four simultaneous equations as

121

 

 

 

 

 

 

 

 

 

 

 

-(jK+Y)t1 (jK-Y)t1
e e _
A jK+Y B jK-Y ' 0
(5.3.11)
- _ (jK-Y)(s-t )
A + B + C e(JK Y)s-e 2
Y-jK Y+jK Y-jK
_(.Y+jK)s -(Y+JK) (s-tz)
+ D e '8 . = O
Y+JK
(5.3.12)
C _‘3:11: ., D e‘”"3"’: ... o
Y‘jK Y+jK
(5.3.13)
-(Y+jK)t (jK'Y)t (Y+jK)(8-t )
1 1 2
A e . ’1 + B e . ’1 + C e .
Y+jK Y-JK Y+JK
(Y-jK)(s-t2)
+ D e . = 0 .
Y'JK
(5.3.14)

To obtain a non-trivial solution for coefficients A, B,
C and D from (5.3.11) through (5.3.14), it is necessary that
their determinant be made to vanish. As a result, it yields

the characteristic equation for eigenvalue Y'Of the coupled

system as

122

jKt -jKt
(y+jK)2e 2‘(Y:jK)ze 2
jKt -jKt jKt -jKt

(e l_e 1) (e 2_e 2)

-2Y(s-t )
e 2 =

x (xth)2e -(Y-jK) e
(Y-jK)2(Y+jK)2 -
(5.3.15)
The same result can be confirmed by conventional differ-
ential-Operator and boundary condition technique; also, in
the limiting case where the separation between slabs is such
that coupling no longer exists, i.e., s+oo, (5.3.15) reduces

to the well known [42] characteristic equation of an

isolated slab,

" '1 (5.3.16)
Moreover, when tlstzat, a practical special case where both
slabs are identical (having same isolated eigenmodes),
characteristic eq. (SJLJS) can be shown to reduce for these

degenerate coupled TE modes, to

2 'Y(8-t)

= ZYKCOSKt4-(y2-K2) sinKt .
(5.3.17)

1 (y +K2) sinKte

The '1” signs should be properly chosen to correspond to
symmetric and asymmetric modal fields which exist on either

slab for this degenerate coupling.

123

AB FOR WEAK DEGENERATE COUPLING
In the case of weak coupling, an approximate solution,

utilizing the above characteristic equation, can be obtained
for A8, i.e., the shift of system mode phase constant
8= 30 +AB from the corresponding phase constant 30 of the
isolated guide. The approximation is that AB is small enough
such that (AB)2=0; this also enables us to express the
transverse wavenumbers K and Y in terms of (BO,AB) in
(5.3.17) as K=I<o 4» AK , and Y: Yo + AY. The latter follow
from definitions Kzski- 82, y2= 82-kg, where k1, kc are

wavenumbers for the slab and cladding regions, respectively.

Therefore,
K2 = (K0 + AK)2 = R; - (80 + AB)2
= K3 - 23018 - A82
which leads to
AK 2 - BOAB/Ko - (5.3.18)

Similarly from Y2=(YO+AY)2, we Obtain
AY 2 B AB/Y .
° 0 (5.3.19)
Substitution of above approximations to K and Y in
terms of K0, Yo, AK and AY into eq. (5.3.17), as well as
expanding both the Sine and Cosine function into their

corresponsing power series, yields

-Y (s-t)
i sinK0t(Y§+Kg)e 0

F(S.Bo)

 

AB 2
(5.3.20)

124

where

 

2(K2—y2) (KZ‘Y2)t
B - B COSK t 0 0 + -—9——9——
E(S’ 0) - 0 0 YOKO K0

2 2 “Yo‘s't)
+ Bo SinK0t(4+Zyot)1:BO(YO+KO)e

 

X

+

t COSKot' (s-t)sinK0t]
Ko Yo

(5.3.21)
5.3.2 Coupled TM Modes

Further demonstration of the Fourier transfrom method
is shown below for the case of degenerate‘TM mode coupling
between a pair of identical slab waveguides such that

2- 2 2- 2 , a - -
Akza k1 kc: k2 kc and t t1 t2. Since both symmetric and
asymmetric modes exist in a degenerate coupled system,

therefore

e2x("’ = * elx('x+s’t) ' (5.3.22)

From the independent x-component equation of (5.2.3)

Ak _ -Y(X+t) YX ]
e1x(x) + ——§-[e1x( t)e + e1x(0)e
2R2
Akz 0 -y|x-x'| .
_ __ I
2Y e1x(x )e dx
-t
2 _ -
+ AE— [e (s)eY(x S)-e (s-t)eY(x S+t)]
2 2x 2x
2k2
Ak2 s 7(x-x') .
- 7-y— e2x(x -)e dx - 0
s-t

-t < x 5 0.

(5.3.23)

125
Subsequent to application of the inverse Fourier-exponential
transform for elx(x) and e2x(x) as described in eq. (5.3.3)
followed by procedures similar to the TE-mode case above, we
obtain the allowable discrete eigenvalues for the transform

variable:

(5.3.24)

and the characteristic equation

2 2 2
. k k k
e 1 + Y(Y-jK) [1 + Y(Y-JK) t 1 Y(Y+JK)

[e-jKt-Ys_eY(t-s)'] }___

2 " 2 2
e-jKt [1 + —]:2-r-—] 1 + —-]:2.—— i [——k-2-.-—']
Y(Y-JK) Y(Y-JK) Y(YfJK)

x [ ejKt-Ys_eY(t-s)] I
(5.3.25)
Once more, when separation between slabs increases such that
s-vm, we recover the familiar characteristic equation of the

isolated TM mode slab waveguide as [42]

 

2 .
k1
, -2- (I) . . . symmetric mode
K
k1
tanKt/2 =3
L:E: ‘5 asymmetric mode
k2 Y "' ° (5.3.26)
1

126
It is therefore evident from above results that
Fourier-exponential transform method yields exact solution
to the coupled EFIEfls which describe a system of parallel
slab waveguides. These correct solutions offer evidence to
confirm the validity of the integral-operator formulation
and provide confidence for its application to more complex

problems.

5.4 Perturbation Approximation
5.4.1 Specialization for Coupled TE Modes
An appropriate perturbation solution was obtained in
Section.4.4 to the simultaneous EFIEfls describing a system
of N coupled dielectric waveguides having propagating modes
with exp (:sz) axial dependence. The result was a
homogeneous matrix equation for the coupled modal amplitudes
involving coupling coefficients Cmrr Recall system (5.2.4),
the EFIE's for the coupled TE modes supported by a slab
system,
N x2n 2
ey(x) - 21 6k (x')ey(x')g(xlx')dx' = 0
n=

x1n

m=l,2, ..., N.
(5.4.1)

The corresponding matrix equation for the one-dimensional

system can be written as

with

O
H

(B-B )C

02
I

1m

x2n 2
Cum = 6kn(x)emy
_ x1n

x2m
—-J{ 6k2(x)e
mm m my
x

127

... for m=l,2, ..., N.

(5.4.2)

... for an

x
2m
(x)dxj{ 6ki(x')emy(x') %%
x

B=B
1m .m
(5.4.3)
(x)eny(x)dx
... for n¥m
(5.4.4)

where emy(x) is the eigenfield of the m'th isolated guide

and is related to the coupled field eY(x) through the

assumption

Ill

ey(X) n my

a e (x)

no. for le SX SXZm'

m=l, 2, ..., N.

(5.4.5)

The eigenfields, supported by each individual isolated

guide, have the functional forms as discribed in Section

3.1:

emybc) =[

Am COSKmX

for x1 gxgxz

m m

-Ym(x-tm/2)

A.m COSKm(tm/2)e

coo >x x<x
for x._ 2m' __ 1m

(5.4.6)

128
where individual transverse wavenumbers Km and Ym are
defined by K%=K%-82 and Y£=B%-kg. Figure 5.3 shows the
configuration of two slabs, the nflth and nfth of a coupled
system of N slab waveguides, separated by a distance of sum
with widths of de and 2dn respectively. By exploiting

fields (5.4.6) in definition (5.4.4) we obtain

 

d
n -y (Ix-s I-d )
_ 2 m mn m
Cmn — AknAmAn Jr COSKnCOSKmdm e . dx
-d
n
= ZAmAnCOSKmdm(YmcosKndDSinhYmdn+KnSinKndncoshYmdn) x
2 2 2
1 + (em erg/2mm
e-Ym(smn-dm)

(5.4.7)
where Akiakifkgg similarly, for the diagonal matrix

elements, i.e., m=n

Cmm =

 

 

2 2 .
AkmBmdmAm [1 + 2 SinZKmdm
2

- cosZK d ] .
2Kmdm m m

Ym
(5.4.8)
In deriving (5.4.8), we utilized the following differential

operation

 

JL -Y|x-x'| _ _ _ l 3 -Y|x-x'|
BY e ‘ Ix x'l ysgn(x-x') ax' e
= - (x-x') %'§%T e—Y|X_x I.

(5.4.9)

129

 

 

 

 

 

 

 

 

‘x
m
5k2= o
x =d
2 m m
) 6k =Ak ¢ 0
m m
de T J, Z SLAB um"
‘ x =—d
m m
Smn
7x
n
X =
n dn
Zdn i ‘—-——-> Z SLAB "n"
2_ 2
dkn—Akn¢ 0
x =-d
n n
6k2= 0 x = x + S
n m mm

Figure 5.3 Configuration appropriate for study of non-
degenerate TE surface-wave mode coupling
between the m'th and the n'th guides in a
N-coupled slab waveguide system.

130
EL4.2 Degenerately Coupled TE Modes between Two Slab
Waveguides
For a pair of identical slab waveguide, the matrix
equation reduces to a 2x2 system (Section 4.4), such that
a2 stal, for even and odd surface-wave-mode coupling;

furthermore e1y(x) = e2y(x) a ey(x), and 81:82880. From eq.

(4.4.15)
C11 C12 a1T
= o
C C a
L 21 22._ L 2_

 

 

 

 

(5.4.10)
Since C12=C21, and Cll'CZZI the requirement of det [Cmn] a 0

for non-trivial solution of (5.4.10) leads to

O

(B-Bo)=AB=i-}'2'

11

0:

(5.4.11)

This is the shift of propagation constant from that of the
isolated slab in the presence of loose coupling to the
decaying field of the other slab waveguide. Substituting
(5.4.7) and (5.4.8) into (5.4.10) yields

AB =

-Yo(3'd)
2cosKod(YOCOSKodsinhYod+KOSinKodcoshYod)e

H-

 

2 2K d

Aszod 2in2K0d
1 + 2 - COSZKod
Y0 0

(5.4.12)

131
where 2d is the width of each slab waveguide. Relative
amplitude for the coupled-mode fields, a2/a1=:tl, can then
be deduced from (5.4.10) corresponding to the system-mode
prOpagation constant.8= 80: A8 for the symmetric and

asymmetric system modes.

5.5 Numerical Results

 

DEGENERATE TE MODE SOLUTIONS

 

Refer to Figure 5.4, which indicates two identical,
parallel slabs having normalized wavenumber k2t=32, and the
ratio of refractive index between slab and surrounding
cladding (index contrast) as nl/n2 a 1.01. These are the
parameters used in Marcuse‘s paper [31]. There are three
allowable prOpagating TE modes with Bot-432.06, 32.248 and
32.32. The normalized phase shifts (ABt), as calculated
from result of weak coupling approximation to the exact
eigenvalue equation (5.3.20), are shown in Figure 5.5 as a
- function of normalized separation s/t between guides. The
results agree very well with those solutions to the exact
eigenvalue equation.

Perturbation solutions, using both the delta-function
(in effect, a point matching technique) and the eigenfun-
ction field of (4.5.2) of the isolated slab (which weights
the solution across the width of the slab) testing operators
yield (ABt) as a function of s/t are shown in Figure 5.6.

It is as expected that more accurate results are obtained in

the simple point matching method from the solution

132

 

 

 

 

 

kzt = 32.0

nl/n2 = 1.01

Figure 5.4 Configuration appropriate for study of
degenerate TE surface-wave mode coupling

between two identical slabs.

(AB)t

133

  

=32.063

 

10-13_.

10-17_.

10'21“

 

 

10'25 r I l *T r l 1
1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0

s/t

Figure 5.5 Normalized phase constant shift for two
propagating modes of a degenerately coupled
two-slab system. (n1=1.6, n2=1.0)

 

Bot= 32.248

134

C) : perturbation solution
(point matching).

[a : perturbation solution
(IE formulation w/
eigenfield weighted)

   
 
 

A.: exact solution from
eigenvalue equation.

   

 

10'3 -
10‘5 -
4.1
1310'7 ‘
10‘9 "'
Figure

5.6

s/t

Comparison of resulting values for normalized
phase constant shifts from various solutions
in a degenerately coupled two-slab system.

(n1=1.6, n2=l.0

135
corresponding to a matching point at x/t=-0.5; this result
confirms the intuitive expectation that a matching point at
the guide center is most appropriate.

Perturbation solutions leading to A8 as derived in
expression (5.4.12) are shown in Figures 5.7 through 5.10
for another slab configurations. There, perturbation
results are compared with numerical solutions obtained from
the exact eigenvalue equation by Wilson and Reinhart [32].
In Figure 5.7, the small core/ surround index contrast
(ng/nc=1.05) results in a slowly decaying field outside of
each slab; consequently a stronger coupling is evident as
comparison are made with those corresponding results of
Figure 5.8 (ng/nc=l.6) and 5.9 (ng/ncsl.2). It is also
observed from Figures 5.8 and 5.9 that the perturbation
solutions converge to the exact solutions much faster in the
later case due to weaker coupling arising from larger value
of itfs decay coefficient‘y. The width of guide 2 is
increased by a factor of 5 in Figures 5.10 from that of
Figure 5.7 to study degenerate coupling between differing
guides. The refractive index of guide 2 is reduced to
ng/nc-l.02 as its width is increased to maintain an isolated
phase constant 802-601 equal to that of guide 1. Degenerate
coupling between dissimilar guides is consequently
implemented. .Although the decay constants of the isolated
surfce-wave modes are identical for such degenerate
coupling, the coupling coefficients are modified due to the

differing field distributions of the coupled modes.

136

 

O-
.5
n=n
parameters chosen by c
”2- Wilson and Reinhart
M

 

 

nc=3.2682 r—r‘i 2“ "”0 3
ng=3.uuso 3 n=nC

" d/AO=0.08673

23 L -
i ’20 n=ng E

TED-MODE COUPLING

 

 

 

 

 

integral-00eratar-based.coupled-mode
perturbation solution

numerical solution of
exact eigenvalue equation

normalized phase-constant shift (Amdxlo2
2 0

 

 

r j T 1 T

1.0 .L5 21) 2.5 2L0 355 1L0 1L5 SJ)
normalized slab seoaratlan S/d

OJ)

Figure 5.7 Comparison of results from integral-operator
based, coupled-mode perturbation solution
with numerical solutions to the exact eigen-
value equation for phase constant shift (A8)d
due to degenerate-mode coupling between iden-
tical slab waveguides with variable spacing s/d.

1.6

1.2

1.0

(L8

normalized phase-constant shift (AflldxlOl
0J4

0.2

 

3.0

1.0

Figure 5.8

   
   
   

137

 

 

 

 

c
I f g ‘2“ ”"9 E
nc=l 0 S
n #16
9 )
d/A0=0 15 ZS ‘ _ — n-n -—->-
- z
Bod=l 2710 l . C
i ,2d n=ng 2'

 

 

TED-MODE COUPLING

integral-00erat0r—based, coupled-mode
perturbation solution

numerical
solution of
exact eigenf
value equation

T

1.5 21) 2.5 110 3.5 (L0 4.5 5&0

T I

normalized slab separation S/d

Comparison of results from integral-operator-
based, coupled-mode perturbation solution with
numerical solutions to the exact eigenvalue
equation for phase-constant shift (AB)d due to
degenerate-mode coupling between identical
slab waveguides with variable spacing s/d.

normalized phase-conStant shift (Aflldxlol

Figure

1.6

1

1.0

l

1.2

l

1.0

1

0.8

l

0.6

00“

l

0.2

 

01)

1.

5.9

   
   
 

138

garameters chosen
y ones 5 I 2d ":09 3
nc=1 o 5 7
I S

n =1.8

g A
0/10=0. 175 25 L— - - - -—*-

fiod=l.8196 L— n=nc
{ {2d n-ng S

TED-MODE COUPLING

 

 

 

 

 

lntegral-operator-based. coupled-mode
perturbation solution

exact eigen-
value equation

I

0 1.5 :L0 2.5 31) 3J5 1L0 4.5 510
normalized slab spacing S/d

Comparison of results from integral-operator-
based, coupled-mode perturbation solution with
numerical solutions to the exact eigenvalue
equation for phase-constant shift (AB)d due to
degenerate-mode coupling between identical
slab waveguides with variable spacing s/d.

139

 

 

 

 

 

 
    
    

 

 

c
parameters chosen by
Wilson and Reinhart E r201 "="91 j
.9._ nc=3.2682
H n =3.llllSO " "' -n=n " ""
n92=3.3317 s s C 2
... a: dl/xo=0.08673 2
X
E (\3 _‘ 202 n=n92
3 ~ 1
t __
E 0. d
m -—1
g TED-MODE COUPLING
3; °?
c o "
O
U
&, integral-operator-based. coupled-mode
8 w perturbation solution
.2 .1
c1 0
8
j: .3. exact solution of
E O- .1 Wilson and Reinhart
é
N u
:5
O
O. I 1 1 I 1 1 I fl
14.0 16.0 18.0 20.0 22.0

normalized slab separation $21/d1

Figure 5.10 Comparison of results from integral-operator-
based, coupled—mode perturbation solution with
the exact solution of Wilson and Reinhart for
phase-constant shifts (AB)d due to degenerate
mode coupling between different slab waveguides

with variable spacing 521/d1°

140
Excellent agreement with numerical solutions to the exact

eigenvalue equation are again obtained.

NON-DEGENERATE TE MODE SOLUTIONS

 

Figure 5.11 through 5.15 demonstrate the coupling
between two non-degenerately coupled slab waveguides. Shown
in these figures are results for both system modes, i.e.,
3=§tt68. Once more, good agreement is obtained for phase
constant shift when compare the perturbation solution with
those from the exact eigenvalue equation (Figure 5.11).
Figure 5.13 shows the normalized coupled-modal amplitudes as
a function of normalized slab spacings, using the ratio of
the slab widthes as parameter. It is observed that when
both guides are in proximity of each other, strong coupling
results in almost equal modal amplitudes as expected
intuitively. Also, when dz/dl =- 1.0, i.e., the coupling
becomes degenerate, symmetric and asymmetric modes (having
equal amplitudes) are obtained on both guides (Figure 5.15)

as discussed in the last section.

141

 

 

 

 

 

 

 

 

”€33.2682 n=nc
ngl=n92:3.4450 ? 42(11 n.ngl
g- (ll/1080.1 .1— g
02/01-2.0 5 _- ___>.z
n-n
. c
__ “’1 L 1
O
— 20 n=n
x . 12 92
tg: a. n n
c C
n.
I a coupled
g} 5‘ IMMB
SE exact
z: -
m o-
G
g.
t; (fi-flzldz
: ‘L
8 a coupled
5 lmme ' p
g .. (3" 1)dl
a db
3
2 CXOCt
g é- _ TED-MODE COUPLING
O
C
2 f I 1 T I I I I
'0.0 1.0 2.0 3.0 4.0 5.0 0.0 7.0 0.0

 

normalized spacing S/dl

Figure 5.11 Results of integral-operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; phase-constant shifts for variable
spacing S/dl'

142

 

nc=3.2682
“91: ”92 = 3.1“450
a dl/AO=0ol

 

.4

 

 

 

0 2 O 3
1
MAMHVV PM?”
N
H—NQ-u
2
II
D
D
N

no!

 

 

 

., TED-MODE COUPLING

normalized phase-constant shift (fl-nflnldn x 101

 

 

0.0 110 2:0 3:0 410 510 0:0 710 0.0
normalized spacing S/dl

Figure 5.12 Results of integral-operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; phase-constant shifts for variable
spacing S/dl'

l

/O
. 7: 922

l

normalized modal amplitude a
3.0

143

 

 

 

 

 

 

 

n=nc 033.2682
‘ _ n =n =3.uuso
g ml n—ngl 0911 90 l
17' l (>’ '
S - - 44’-
1. n=nc z
i
i 02/01=2.0
n=n
c

   
 

TED-MODE COUPLING

 

 

 

I T
L0 2.0 8.0 7-0 0.0

3.0 4:0 5.0
normalized spacing S/dl

Figure 5.13 Results of integral-operator-based, coupled-

mode perturbation solution for non-degenerate

TED-mode coupling between differing slab wave-

guides; amplitude ratios for variable spacing
s/d .
1

144

 

 

 

 

 

 

     
 
  

 

 

 

nc=3.2682 n=nc
n =n = Am
91 92 3 50 E‘ZGI n=nl§
dl/lo:0.l g
If—
].5_ S/dl=3.0 S _ _ ‘)-
NO L n=nC
.; l 0 f
c ' ‘ coupled 202 n=n92
2 mode 1
C
n n=nc
3 0.5-1 eXOCt
1:; 0.0.. (p'pz)d2
E
g 415
O
? coupled (p fli)d1
3; mode
.8 -1.0.
Q
'8
j: TED-MODE COUPLING
;; -l.5-
E
a
C
’2.0 I I 1 1 I T j
01) 1.0 L2 IA) 1.6 1.8 24) 2.2

normalized thickness 02/01

Figure 5.14 Results of integral-operator-based, coupled-
mode perturbation solution for non-degenerate

TED-mode coupling between differing slab wave-

guides; phase-constant shifts (exact and
coupled-mode) for variable thickness dZ/dl'

145

 

 

 

 

 

 

 

   

nc=3.2682
3.0- - -
ngl—n92-3.4450
dl/AO=0.1
2.51 S/dl=3.0
2.0-
‘5 1.5-
:5”
8
.3 1.0-
"a
E
C
3 0.5-
'0
O
E
E’ 0.0- n=nc
‘6
E3
-0.5_
TED-MODE COUPLING
-1.a

 

 

0.0 1.0 1T2 130 11.6 173 2.0 272
normalized thickness d2/dl

Figure 5.15 Results of integral-operator-based, coupled-
mode perturbation solution for non-degenerate
TED-mode coupling between differing slab wave-

guides; modal amplitude ratio for variable
thickness dZ/dl'

CHAPTER VI

CONCLUSION

An integral-operator technique, representing an alter-
native to conventional boundary-value analysis, has been
applied to two classes of problems for EM wave prOpagation
along open-boundary dielectric waveguides. First, the
scattering of surface-wave modes by’a discontinuity along
the waveguide» Second, the coupling of surface-wave modes
in a multi-waveguide system.

In the construction of a volume electric field integral
equation, equivalent polarization current is essential to
its formulation. Identification of this current for the
device discontinuity region results from the index contrast
between the discontinuity and the unperturbed background
waveguide; it is this current which maintains the scattered
field. Using this scattered field in conjunction with the
incident field, the unknown total field is cast into an
integral equation and numerical solutions are subsequently
sought. For the coupling problem, a system of N coupled
waveguides is replaced by equivalent polarization sources
which arise from the index contrast between each core and
its surround cladding. This current then radiates into

unbounded space in the presence of other sources. Coupling

146

147
phenomena are the consequence of the total contributions of
all the fields maintained by these N coupled polarization
sources. -

Solutions to the resulting EFIEs' are demonstrated
through the application to slab waveguides. In the treat-
ment of scattering by dielectric-slice obstacles, neglecting
the contribution from radiation spectrum for small discon-
tinuies, a Fourier transform method yields the limiting
reflection and transmission coefficients which are further
confirmed by the Moment Method numerical calculation. Other
results, including the contributions from the radiation
spectrum, are also computed by MoM.solution and an approx-
imate solution assuming fields in the slice region have the
axial propagation constant obtained in the radiationless
case. In the coupling treatment, again, a Fourier transform
approach recovers the familiar eigenvalue equation for a
two-slab system. Together with the above treatment for
scattering, this demonstrates the versatility of the
transform.techniquec Subsequent coupled-mode perturbation
analysis yields the exact results which can be obtained from
the conventional differential-Operator approximation. This
new coupled-mode theory is applicable to relatively general
waveguide systems, since it requires approximation of only
the waveguide core fields.

In short, the contributions of this dissertation
research are that the applicability of Integral-Operator

analysis is clearly demonstrated and the correctness of the

148
obtained results are verified. This method, when
considering the abundance of unsolved problems associated
with the open-boundary waveguide structures in both the
Optical and millimeter regions, provides an invaluable tool
for future research in these areas. To further emphasize
this point, it was recently pointed out by Oliner, et al.
[30,37] that the approximate treatments [12,60] of surface-
wave modes supported by integrated dielectric waveguides
-neglect, due to an inadequate account for coupling between
TE and TM components of the hybrid modes, important physical
phenomena. Those new physical effects include both leakage
and sharp resonance phenomena not predicted by the conven-
tional approximate methods. It is clear that the integral-
operator analysis described in this research provides an
exact description of the hybrid propagation modes; it will
therefore expose the same new effects for a more general
class of graded-index dielectric waveguide systems having
cores of any cross-section shape.

Future research in the extention and application of
this powerful analysis should consider the complex configu-
rations of practical integrated waveguide systems i.e.,
isolated, or coupled systems of integrated guides as
indicated in Figure 6.1. In that Figure, waveguide cores
are deposited upon a uniform thin-film layer of index nf and
the waveguides are covered by a uniform cladding overlay of
index nc while the film layer is deposited upon a uniform

substrate of index ns. Recommended investigations of the

149
resulting rib or strip structures from such a configuration
include: i) surface-wave propagation modes supported by the
graded-index rib waveguide, by approximate perturbation, and
numerical methods, ii) study of system-mode surface waves
supported by coupled systems (both parallel and non-
parallel) of graded strip, channel and rib waveguides using
quasi-closed-form exact and approximate coupled-mode
approaches, iii) description of propagation modes supported
by electrOOptic integrated dielectric waveguides, iv)
analysis on the coupling of radiation to and from integrated
dielectric waveguide systems, including quantification of
continuous-spectrum radiation modes on rib-related waveguide
structures as forced solutions to the appropriate EFIEfls and
v) experimental confirmation of selected analytical predic-

tions.

150

CLADD I NG OVERLAY

n=nc

/
/

     

 

[\pea E35 i 4'5:
Es ,
// /

/////

/ , //
n==nIn (3) / / Amn‘é)
://T$m‘//' ~X/<>g%//://\
\f\m'th guide \ \‘n'th guide

i\\\n-Qf FILM LAYER \\\\\

_ \
/ / / /
WOVCQUHHI'IQ
/ axis
/ z SUBSTRATE
/ /n=ns /

Figure 6.1 Configuration of integrated, open-boundary,
dielectric-waveguide system consisting of
arbitrary-shaped, graded-index core regions
adjacent to the film/overlay interface depo-
sited upon a uniform substrate.

 

 

 

 

 

 

   
  

LI ST 0]? REFERENCES

LIST OF REFERENCES

[1] D. Hondros, and P. Debye, 'Elektromagnetische Wellen in
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[2] S.E. Miller, "Integrated Optics: An introduction." BSTJ,
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[3] lLS. Kapany, Fiber Optics, New York: Academic Press,
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[4] H. Kogelnik, “Limits in Integrated Optics,” Proc. IEEE,
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[8] C. Yeh, “Optical Waveguide Theory,“ Digest of North
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[10] N.S. Kapany and J.J. Burke, 0 tical Waveguides,
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[11] D. Marcuse, Theory of Dielectric Optical Waveguides,
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[12] EmliJ Marcatili, "Dielectric Rectangular Waveguide

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”Integral-Operator
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Hsu and D.P. Nyquist, “Integral-Operator Analysis
of Coupled Dielectric Waveguide System - Theory
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Johnson and D.P. Nyquist, "Numerical Solution of
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Katsenelenbaum, 'On the Propagation of
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Hsu and D.P. Nyquist, ”Integral-Operator
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Hsu and D. P. Nyquist, "Integral- Equation
Formulation for Mode Conversion and Radiation from
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1980.

D. Marcuse, “Radiation Losses of Tapered Dielectric

S.F.

B. Rulf,

Slab Waveguide," BSTJ, vol. 49, no. 2, p. 273,
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Mahmoud and J.C. Beal, “Scattering of Surface
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”Discontinuity Radiation in Surface
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1979.

L. Lewin, WA Method for the Calculation of the Radia-
tion Pattern and Mode-Conversion Properties of a
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1975.

R.W. Davis and J.N. Walpole, "Output Coupling for
closely confined pb1-xSn Te Double Heterostructure
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C.C. Ghizoni, J.M. Ballantyne and C.L. Tang, “Theory of
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Green's Function Approach,” IEEE J. Quant. E1ect.,

vol. QE-13, no. 10, pp. 843-848, Oct. 1977.

T.E. Rozzi and 6.3. In't Veld, “Field and Network
Analysis of Interacting Step Discontinuities in
Planar Dielectric waveguides,” IEEE MTT-s Trans.,
vol. MTT-27, no. 4, pp. 303-309, April 1979.

T.E. Rozzi, ”Rigorous Analysis of the Step
Discontinuity in a Planar Dielectric Waveguide,“
IEEE MTT-s Trans., vol. MTT-26, no. 10, pp. 738-
346, Oct. 1978.

C.Vassallo, 'On a rigorous Calculation of the
Efficiency for Coupling Light Power into Optical
Waveguides," IEEE J. Quant. Elect., vol. QE-13,
no. 4, pp. 165-173, April 1977.

S-T Peng and An A. Oliner, "Guidance and Leakage
Preperties of a Class of Open Dielectric
Waveguides: Part I--Mathematical Formulations,“
IEEE MTT-s Trans., vol. MTT-29, no. 9, pp. 843-
855, Sept.1981.

D. Marcuse, ”The Coupling of Degenerate Modes in Two
Parallel Dielectric Waveguides,“ BSTJ, vol. 50,
no. 6, pp. 1791-1816, July-August 1971.

L.O. Wilson and Fax. Reinhart, "Coupling of Nearly
Degenerate Modes in Parallel Asymmetrical
Dielectric Waveguides," BSTJ, vol. 53, no. 4,
pp. 717-739, April 1974.

154

[33] E.F. Kuester and D.C. Chang, 'Nondegenerate Surface
Wave Mode Coupling between Dielectric Waveguides,"
IEEE MTT-s Trans., vol. MTT-23, no. 11, pp. 877-
882, Nov. 1975.

[34] S.E. Miller, "Coupled Wave Theory and Waveguide
Applications,” BSTJ, vol. 33, no. 3, pp. 661-719,
May 1954.

[35] A.W. Snyder, “Coupled-mode theory for Optical Fibers,”
JOSA, vol. 62, pp. 1267-1277, 1972.

[36] J.A. Arnaud, “Transverse Coupling in Fiber Optics-Part
I: Coupling between Trapped Modes,“ BSTJ, vol.
53, 217-224, 1974.

[37] A.A. Oliner, S-T Peng, T-I Hsu, and A. Sanchez,-
'Guidance and Leakage PrOperties of a Class of
Open Dielectric Waveguides: Part II--New Physical
Effects,“ IEEE MTT-s Trans. , vol. MTT-29, no. 9,
pp. 855-869, Sept. 1981.

[38] R.F. Harrington, Time Harmonic Electromagnetic Fields,
New York: McGraw—Hill, p. 347,1961.

[39] A.L. Jones, “Wave Propagation in Optical Fibers," PhJL
dissertation, Ann Arbor, Michigan: University
Microfilms, 1964.

[40] Y. Rahmat-Sami, 'On the Question of Computation of the
Dyadic Green's Function at the Source Region in
Waveguides and Cavities,” IEEE MTT-s'Trans., vol.

MTT-23, no. 9, pp. 762-765, Sept. 1975.

[41] V.V. Shevchenko, Continuous Transition in Open
Waveguides, Boulder, Colorado: GoIem Press,

pp. 22-48 and 93-116, 1971.

 

[42] D. Marcuse, Li ht Transmission 0 tics, Princenton: Van
Nostran Reinhold, 1972, Chapter 8.

[43] D.P. Nyquist, D.R. Johnson and S.V. Hsu, 'Orthogonality
and Amplitude Spectrum of Radiation Modes along
Open-Boundary Waveguides,‘I JOSA, vol. 71, no. 1,
pp. 49-54, Jan. 1981.

[44] D. Marcuse, “Radiation Losses of Dominant Mode in Round
Dielectric Waveguides," BSTJ, vol. 49, no. 8,
pp. 1665-1693, Oct. 1970.

[45] R. F. Harrington, Time Harmonic Electroma agn netic Fields,
New York: McGraw—Hill, 1961, pp. 125- 128.

[46]

[47]
[48]
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[50]

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J. Van Bladel,

A.D.

R.M.

R.E.

155

"Polarization-Source
Formulation of Electromagnetism and Dielectric-
loaded Waveguides,“ Proc. Inst. Elec. Eng.,
vol. 119. PP. 1568-1574, Nov. 1972.

"Some Remarks On Green's Dyadic for
Infinite Space,” IEEE AP-S Trans., vol. AP-9,

n0. 6, pp. 563-566, NOV. 1961.

Yaghjian, “Electric Dyadic Greenfls Function in the
Source Region,“ Proc. IEEE, vol. 68, no. 2,
pp. 248-253, Feb. 1980.

Chen,
Function in Source Region,“ Proc. IEEE,

no. 8, pp. 1202-1204, Aug. 1977.

"A Simple Physical Picture of Tensor Greenfis
V01. 65'

Harrington, Field Computation by Moment Methods,
New York: MacMillan, Chapter 1 and 7, 1968.

R. Courant and D. Hilbert, Method of Methematical

R.E. Collin, Field Theor

R.E.

C.C.

Ph sics, vol. 1, New York: Interscience
Publ1shers, Chapter III, 1953.

of Guided Waves, New York:
McGraw-Hill, pp. 25-27, 1960.

 

 

Harrington, Time Harmonic Electromagnetic Fields,
New York: McGraw-Hill, pp. 113, 1961.

Collin and FhJ. Zucker, Antenna Theory-Part I, New
York: McGraw-Hill, pp. 41-43, 1969.

Gradshteyn and I.M. Ryzhik, Table of Integral
Series and Products, New York: Academic Press,
1965.

Arnaud, Beam and Fiber thigg, New York:
Academic Press, Chapter 3, 1976.

Johnson, Integral-Operator Analysis of Open-
Boundary Dielectric Waveguides, Ph.D. Disserta-
tion, Michigan State University, p. 23, 1980.

Collin, Field Theory of Guided Waves, New York:
McGraw-Hill, p. 62, 1960.

Johnson, Field and Wave Electrodynamics, New York:
McGraw—Hill, p. 8, 1965.

McLevige, T. Itoh, and R. Mittra, ”New Waveguide
Structures for Millimeter-Wave and Optical Inte-
grated Circuits," IEEE MTT-s Trans., vol. MTT-23,
no. 10, pp. 788-794, Oct. 1975.

APPENDICES

0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055

1556
APPENDIX A

SLABZ SOURCE LISTING

FTN4

0001') 0000 0000

GOOD

PROCRAH SLAB?

REAL N1,N2,N3,KOD,KPND,KPD

CONFLEx RN,TN,DUHHY ,

COHFLEx BOPD,C,BHB,BPB,RF,C4,AP,AH,R0,T0,HE,DIAG,DET,EY

DIMENSION XN(323,ZN(32),HE(32,33),EY(35)

DIHENSION GHND(2),KPND(2),BND(2),RN(2),TN<2)

COHHON/NATELIXN,ZN,KPD,BOD,N1,N2,N3,KOD,GHD,DXN,C2,DZN,E0,PI,NP,NX

1,N2,NRF1

COHHON/SCATO/zo,zoOD

COHHON/SHODE/RN,TN,GMND,KPND,BND,HHOD

COHHON HE

PI-3.141892653b

20-120.08PI

DPR-180.0/PI
311888!Itttttttttlitttlttt188831318881!tIttittttltttttttttttltttliltlttltxlt
READ FIRST DATA CARD FOR REFRACTIvE INDICES <N1,N2,N3> OF CORE, CLADDINC, AN
DISCONTINUITY RECIONS, RESPECTIVELY.
tittIIIIlttltltttttlltllttltt#183818883338tt¥¥¥ttt¥¥tttlttttttttt84833888#1!

URITE<1,61)

61 FORHAT<on,13HREAD N1,N2,N3)
READ(1,I> N1,N2,N3
1 FORHAT(3F10.3)
xttxtxxtxRx:xttx3:31:33xtxxxxxxxtxtxxxttxxxttxtxttRatttxxttxxtttttxxxxxxtxxx
READ SECOND DATA CARD FOR NORNALIZED SLAB THICKNESS DOLo AND NORHALIZED
EICENvALUE FARAHETERS (GHD,KPD,BOD) OF THE UNPERTURBED SLAB UAUEGUIDE.
txxtttxRxttxtxttttttrstttxxttttxtxttttxtxxxxxtxxxaxxxtttxxxttxtxxxttxttxxxxx

URITE(1,62)

52 FORHATt10X,21HREAD DOL0,GHD,KPD,BOD)

READ(1,x) DOL0,GHD,KPD,BOD

2 FORHAT<FIS.2,3EIS.O)

Ron-2.0xFIIDOLO
thttxxtIt:xxxxxttttuxtxxaxttxxntxxxxx:txxxtttttxtxxxttxtxxxxxxxtxxxxtxxxtx:
READ THIRD DATA CARD FOR NORHALIZED LENCTH 200D OF DISCONTINUITY RECION AND
AHFLITUDE Eo OF INCIDENT HAvE.
tttxtxxttxxxxtxxtxxttxtxxxRDSRSRRRRRRtxxtttttxtxxxxttxxxxxxxtxxtxxtxx:Sttxx:

HRITE(1,63)

53 FORHAT<IDx,17HREAD zoOD,Eo,HHOD)
READ<1,t> zoon,Eo,HHOD
3 FORHAT<F10.2,F10.1,II)
Rttxxxxxxxxxtxxxtxaxxtsxxxxxtttxttxxxxxxxxxxxxxtxxxtxtxxttxxxx:xsxxxxnxtxxtt
READ FOURTH DATA CARD FOR NUMBERS OF FARTITIONS (Nx,Nz) ALONG x AND 2
DIRECTIONS, RESPECTIVELY.
81881::tlttxtttttttttttltttttttttttxlttlttltllttltttxttxtttxlltxtittlttllttx

URITE(1,64)

64 FORHAT<on,IoHREAD NX,NZ)

READI1,I) Nx,Nz

4 FORHAT<I2,3x,Ia)

NF-NxxNz

NFFI-NF+:

DO 44 N-1,HHOD

URITE(1,6S)

65 FORHATton,27HREAD GHND(N),KPND(N),HND(N))

READII,S> GHND(N),KPND(N),HND(N)

41 FORHAT<3E:S.S)

1557

44 CONTINUE
C xx:tun::xtxtxxttxxtxtxxxxxtxxx99:11:11:xxxxxxtItRt:RIBS:tttxtxxxxxtxxxxxtxt:
C PRINT ESSENTIAL INPUT DATA.
C xxRxRRRRRBRRRIRRRRRRRBBB111::x111::xxxxxxxthxxxxx::xxxxxtxxxtxtxx:1:111:11:
URITE(6,S)
S FORHAT<1H1,//,10X,21HESSENTIAL INPUT DATA.,//)
URITE(6,6) N1,N2,N3
6 FORHAT<1H0,///,10X,4HN1= ,F4.1,2x,23H<SLAB REFRACTIUE INDEX),/,1ox
1,4HN2= ,F4.1,2x,27H<CLADDINc REFRACTIVE INDEX),/,10X,4HN3= ,F4.1,2
2X,32H(DISCONTINUITY REFRACTIUE INDEX))
URITE(6,7) DOLo,CHD,xPD,BoD
7 FORHAT<1Ho,/,1ox,bHD/Lo= ,FS.2,2X,32H<NORHALI2ED SLAB HALF THICKNE
188),//,10X,SHGHD- ,E14.8,2X,38H(NORHALIZED CLADDINC DECAY EICENvAL
2UE),/,1ox,SHKFD- ,E14.s,2x,ssH(NORHALIZED SLAB EIGENUALUE),/,10X,S
3HBoD- ,E14.8,2X,38H(NORHALIZED PHASE-CONSTANT EIGENVALUE))
URITE(6,8) 200D,E0,HHOD
s FORHAT<1H0,/,10X,6HZO/D- ,FS.2,2X,31H(RELATIUE DISCONTINUITY LENCT
1H),//,10X,4HEO= ,F4.1,2x,SH(v/H),2x,25H(INCIDENT HAvE AHFLITUDE)//
,,1ox,SHHHOD=,II,2x,23H<NUHBER OF HODES EXIST))
URITE(6,9) Nx,Nz
9 FORHAT<1H0,/,10X,4HNX= ,13,2x,2oH(FARTITIONs ALONC X),/,1ox,4HNz=
1,13,2x,20H<FARTITIONS ALONG 2))
DO 99 N-1,HHOD
URITE(6,91) N,CHND<N),RRND<N),BND<N)
91 FORHAT<1Ho,/,10x,2HN=,II,2x,SHCHND=,F1S.s,2x,SHKPND=,F15.s,2x,4HBN
DD-,FIS.B)
99 CONTINUE
C xxxan:9:993:19:txxxxtxtxxtxxxxxxxtxxxxt99:13:99:Rtxtnxxxxxxxxxxxxxxxxxxxxtxx
C CALCULATE AND PRINT THE AFFROXIHATE, RADIATIONLESS SLAB FIELD AND REFLECTION
C AND TRANSHISSION COEFFICIENTS.
c xxx11:1:a:RIB:xxxxxxxtxxtxtxxxtxxxxxtxxxtxxtxxxxtxxxtxxxxxxxxxxxxtxxx21::Bx:
CI-RPD+o.SBSIN(2.OBxFD)
C2=<KFD/CHD)R<COS(RFD))t:2+c1
C3=BDDBBOD+<N39N3-N12N1)xRoDBRoDxCI/Cz
IF<C3) 10,11,11
10 BOFD=CHFLx<o.o,-SQRT<-C3))
GO TO 12
11 BDFD-CHFLXISORT1C3),D.D)
12 C=o.St<N39N3-N1BN1)xCHFLx10.o,KoDxKoD/B0D)BC1/C2
BHBaBoFD-BDD
BPB-BDPD+BOD
RF-BHB/BFB
04-1.0-RFtRFtCEXP(CHPLX(0.0,-4.0)tBOPD*ZOOD)
AF-CHFLXTD.0,1.0)RBHBRonCEXF<CHPLx<o.0,-1.0)RBHBBZOOD)/<CRC4)
AH-AFSRFBCEXFTCHFLx10.o,-2.o)xBoPszoOD)
Ro-CHPLx10.o,-2.o)tRFtCEXP(CHFLX(o.o,-1.o)BBPszoOD)B(CSIN1BPszoO
1D)+CEXF<CHFLX(D.0,-2.o)xBoPszoOD)BCSIN<BHBtzoOD))/C4
To-<1.o-CHFLXTD.o,2.O)BCEx91CHFLx<o.o,-1.o)RBHBRZOOD)B(CSIN<BHszo
10D)+RFBRFBCEXF(CHFLX(D.0,-2.o)tBoPDRzoOD)xCSIN(BFBtzoOD))/CA)#CEXP
2<CHFLx<o.o,-2.o)tBoDRzoOD)
HRITE(6,13)
13 FORHAT<1H1,/////,10x,B9HRESULTS OF APFROXIHATE RADIATIONLESS SOLUT
1ION FOR SLAB FIELD AND SCATTERING COEFFICIENTS.)
URITE(6,14) Born
14 FORHAT(1H0,/////,20X,6HBOPD=(,E10.4,1H,,E10.4,1H))

 

0111
0112
0113
0114
0115
0116
[[117
0118
0119
0120
0121
0122
0123
0124
0125
0126
0127
0128
0129
0130
0131
0132
0133
0134
0135
0136
0137
0138
0139
0140
0141
0142
0143
0144
0145
0146
0147
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0165

DOG no 00 ODD 000

OD ant-Inn ('10

 

1558

APH=CABS<AP>
APP=DPRSATAN2<AIHAG(AP),REALIAP))
AHH=CABS<AH)
AHP=DPR¥ATAN2(AIMAG(AH),REAl(AH))
URITE(6,IS) APH,APP,AHH,AHP
IS FORHAT(1H0,///,20X,3HAP=,E10.4,6HEXP(J¥,E10.4,1H),//,20X,3HAH=,E10
1.4,6HEXP(JI,E10.4,1H))
RDH=CABS<R0)
ROP=DPR¥ATAN2(AIHAG(R0),REAL(R0))
T0H=CABS<T0)
TOP=DPR*ATAN?(AIHAG(T0),REAL(TO))
URITE(6,16) R0H,R0P,ToH,T0P
16 FORHAT<1Ho,///,20x,3HRo=,E1o.4,SHEXF(JR,E10.4,1H),//,20x,3HTo=,E1o
1.4,6HEXP(J¥,510.4,1H))
xxxxxxxtxxxrtttx991991991119:xxxxxxxxxxx:xxxxxxxxxxxxxxxxxxxxxxxxxtxxxxxxx
PARTITION THE DISCONTINUITY REGION USING NORHALIZED COORDINATE vARIABLES.
BBBBBRSBBBItxtxxxxtxttxxtttxxxxxxxxxxxxxtxxxtxxxxxxxtxxxxxxxxxxxtxxxxxxxtx
DXN-1.o/Nx
DZN-2.oxzoOD/Nz
DO 17 I=1,Nx,1
XNII)-(I-o.S)BDXN
17 CONTINUE
DO 18 J=1,NZ,1
ZN(J)--ZOOD+(J-0.S)¥DZN
18 CONTINUE
ttttttlttttttttltttttitttltttttltttlttltttltltltittltlttttttxtttttttttlxxx
GENERATE THE ELEHENTS OF HOH HATRIx HE(H,N).
xxxtxxxtxxxxxxxttxxttxtrxxtxxttxxxxtxtxxxxxxxrxxxxtxxxRxxtxxxxxxxtxtxxxx:x
CALL HATEL<HE>
HRITE (6,68) HE
68 FORHAT(1H1,1X,7HHE(H,N),/,1b3(13E10.3,/))
DO 19 H-1,NP,1
DIAGxHE<H,H)
URITE(6,71) H,H,DIAG
71 FORHAT<1X,3HHE(,Is,1H,,Iz,2H)=,2E1S.A)
DO 19 N-1,NPP1,1
URITE(6,77) H,N,HE<H,N),H,H,DIAG
77 FORHAT<1x,3HHE(,12,1H,,I2,2H)=,2E15.4,Sx,SHDIAC<,Is,1H,,Ie,2H)=,
-c2E1s.4,/)
DUHHY=HE(H,N)
HE(H,N>=DUHHY/DIAG
URITE(6,78) HE(H,N)
7s FORHAT<1x, BHHE/DIAG=,2EIS.4,//)
19 CONTINUE
HRITE(6,69) HE
69 FORMAT<1H1,1X,7HHE/DIAG,/,163(13E10.3,/))
xttxxxttttttxttxtxxtttxxxtttuxtxxxtxxxxxxxxtxtxxtxxxxxxxxxxtxxxxxtttxxtxx:
SOLVE THE HATRIx EQUATION FOR THE EY(N) AND PRINT THE RESULTS.
BBBBBBRR:Baxtnxxxttttxxxxthxxxtxxttttx99:19:ttxxxtxxxxxxxxxtxxxtxxxx:txx»
CALL CHATP(-1,HE,NP,1,DET,1.0E-35)
HRITE(6,70) HE
7o FORMAT(1H1,1X,7HHE(INV),/,163(13E10.3,/))
DO 20 N-1,NP,1
EY(N)-HE(N,NPP1)
20 CONTINUE '

0166
0167
0168
0169
0170
0171
0172
0173
0174
0175
0176
0177
0178
0179
0180
0181
0182
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0187
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0197
0198
0199
0200
0201
0202
0203
0204
0205
0206
0207
0208
0209
0210
0211
0212

1559

EYHAX=CABS(EY(1))
DO 202 N=1,NP,1
IF(CABS(EY(N))-EYHAX) 202,202,201

201 EYHAX=CABS<EY(N))
202 CONTINUE

URITE(6,21)

21 FORHAT(1H1,/////,10X,77HRESULTS OF NUMERICAL MOM SOLUTION FOR SLAB
1 FIELD AND SCATTERING COEFFICIENTS.)

HRITE(6,211)

211 FORMAT(1H0, 9X,46HPROGRAH UITH SYHHETRY AND HULTIHODE SCATTERING)

URITE(6,22)

22 FORHAT11H0,////,10x,SDHINDUCED FIELD EY(X,Z) IN THE DISCONTINUITY
1REGION.,/////,6X,1HI,6X,1HJ,6X,SHXN(I),6X,SHZN(J),6X,1HN,17X,SHEY(
1N),19X,6H£YA(N),7X,6HEYP(N),//)

DO 24 L-1,Nz,1

DO 24 K-1,NX,1

N=K+(L-1)¥NX

EYA-CABS(EY(N))/EYHAX

REY-REAL(EY(N))

AEY-AIHAG(EY(N))

EYP-DPRBATAN21AEY,RET)

URITE(6,23) K,L,XN(K),ZN(L),N,EY(N),EYA,EYP

23 FORHAT<Sx,I2,Sx,12,Sx,FS.3,Sx,FA.3,Sx,13,Sx,1H<,E11.S,1x,1H,,1x,E1
11.S,1H),SX,E11.5,SX,F6.1)

CONTINUE

C #818813fill8808883833388888littflltttt$3831!03018!IIIXIIUSIRINIIIXIRSIIIIIt!
C CALCULATE AND PRINT THE SCATTERING COEFFICIENTS RN AND TN.
C 8083*!IItIfittttllttltttiltIt*ltllltIllttltltttlHttttlttt#1840!!!4418004484

URITE(6,25)

25 FORHAT(1H1,/////,10X,86HSCATTERING (REFLECTION AND TRANSHISSION) C
1OEFFICIENTS DESCRIBING DISCONTINUITY REGION.)

CALL SCATC<EY,RN,TN)
Do 999 N=1,HHOD
RNHICABS(RN(N))
ARN-AIHAC(RN(N))
RRN-REAL(RN(N))
RNP-DPRRATAN21ARN,RRN)
TNH=CABS<TN(N))
RTN-REAL(TN(N))
ATN=AIHAG<TN(N))
TNP-DPRRATANztATN,RTN)
URITE(6,26) N,RNH,RNP,TNH,TNP

26 FORHAT11H0,////,2ox,2HN-,11,2x,3HRN=,E10.4,6HEXP(Ix,E10.4,1H),//,
XZSX,3HTN=,E10.4,6HEXP(J¥,EIO.4,1H))
999 CONTINUE

STOP
END

FTN4 COHPILER: HP92060-16092 REV. 1901 (781201)

It

NO UARNINGS #0 NO ERRORS #8 PROGRAM I 02711 COHHON = 04224

0213
0214
0215
0216
0217
0218
0219
0220
0221

0222

0223
0224
0225
0226
0227
0228
0229
0230
0231
0232
0233
0234
0235
0236
0237
0238
0239
0240
0241
0242
0243
0244
0245
0246
0247
0248
0249
0250
0251
0252
0253
0254
0255
0256
0257

1650

SUBROUTINE HATEL<HE)

REAL N1,N2,N3,KOD,KPND,K2D,K20HE,K20PE,KEDSQ,KPD

COHPLEx HE,Cb,C7,HED,HER,XCEXP,XHED,XHER

DIHENSION HE(32,33),XN(32),ZN(32)
COHHON/NATEL/XN,ZN,KPD,BoD,N1,N2,N3,K00,GHD,DXN,C2,DZN,E0,PI,NP,Nx

1,NZ,NPP1

COHHON/NER/K2D,vN,EPS,CB,K2DHE,K20PE,RHONH
COHHON/FHATL/HDXN,HDZN,K2050,UNSQ
K20=N2¥K00
UN=SORT1N1RN1-N24N2)4Koo
DZH=2N<NZ)-2N(1)

EPS=K20/1o.o

K2DHE-R2D-EPS

K2DPE=KED+EPS

RHONH-1o.oxK2D
CB-4.04(N3BN3—N13N1)RRDDSKOD/PI
HDXN=DXN/2.0

HDZNaDZN/2.0

KBDSOIKEDIKEI

vNSO-vavN

DO 1 H-1,NP,1

DO 1 N-1,NPP1,1
HE(H,N)-CHPLX(0.0,0.0)

1 CONTINUE

DO 4 J-1,NZ,1

DO 4 I-1,Nx,1
H-I+(J-1)¥NX

XNI-XN(I)

ZNJ-ZN(J)
XEYO=EYO(XNI,1)
XCEXPSCEXP(CHPLX(0.0,-BOD¥ZNJ))
HEIH,NPP1)~E0:XEYDBXCEXP
DO 4 L-1,Nz,1

DO 4 K-1,NX,1
N-R+(L-1)4Nx
XHED-HED(I,J,K,L)
XHER-HER(I,J,K,L)

GO TO 3

IF(H-N) 3,2,3
HE(H,N)-1.0+XHED+XHER

GO TO 4

3 HE<H,N)=XHED+XHER
4 CONTINUE

RETURN
END

FTN4 COHPILER: HP92060-16092 REV. 1901 (781201)

3! NO UARNINGS 03 NO ERRORS #0 PROGRAM I 00371 COMMON I 00000

0258
0259
0260
0261
0262
0263
0264
0265
0266
0267
0268
0269
0270
0271
0272
0273
0274
0275
0276
0277
0278
0279
0280
0281
0282
0283
0284
0285
0286

1

1(51

COHPLEx FUNCTION HED(I,J,K,L)

REAL N1,N2,N3,KPND,KPD,KOD

COHPLEx RN,TN,C6,C7 ,

DIHENSION GHND(2),KPND(2),BND(2),RN(2),TN(2)

DIHENSION XN(32),ZN(32)
COHHON/NATEL/XN,2N,KPD,BOD,N1,N2,N3,K0D,CHD,DXN,C2,D2N,E0,PI,NP,Nx

1,Nz,NPP1

COHHON/SHODE/RN,TN,GHND,KPND,BND,HHOD

HEDcCHPLx<o.0,0.o)

DO 99 N=1,HHOD

XX=KPND(N)

c1-xx+o.SRSIN(2.OBXX)

c2=<xX/GHND<N))R(COS(XX))492 +C1
C5=4.0*(N38N3-N18N1)I(KODIKOD/(BND(N)IBND(N)))lSIN(XXtDXN/2.0

C)/C2

C6ICHPLX(0.0,SIN(8ND(N)#DZN/2.0))
C731.0-CEXP(CHPLX(0.0,-8ND(N)#DZN/2.0))
XNI-XN(I)

XNK-XN(K)

ZNJ-ZN(J)

ZNLIZN(L)

IF(J-L) 2,1,2
HED'HED+C5¥COS(XXIXNI)ICOS(XX¥XNK)IC7
GO TO 99

2 HEDSHED+C5¥COS(XX¥XNI)ICOS(XXIXNK)0C6XCEXP(CHPLX(0.0
t,-BND(N)#ABS(ZNJ-ZNL)))
99 CONTINUE

RETURN
END

FTN4 COHPILER: HP92060-16092 REV. 1901 (781201)

#3 NO UARNINGS 80 NO ERRORS it PROGRAM I 00380 COHHON = 00000

0287
0288
0289
0290
0291
0292
0293
0294
0295
0296
0297
0298
0299
0300
0301
0302
0303
0304
0305
0306
0307
0308
0309
0310
0311
0312
0313
0314

1652

COHPLEx FUNCTION HER<I,J,K,L)

REAL N1,N2,N3,RDD,KPD,R2D,K20HE,K20PE,Keoso

COHPLEx C10,I1,12,I3

DIHENSION XN(32),ZN(32)
COHHON/NATEL/XN,2N,KPD,BOD,N1,N2,N3,K00,GHD,DXN,C2,DZN,Eu,PT,NP,Nx

1,NZ,NPP1

COMMON/FMATL/HDXN,HDZN,KBDSQ,VNSQ
COMMON/NER/K2D,VN,EPS,C8,K20HE,KRDPE,RHONM
COHHON/FHER/DZNJL,II,JJ,KK,LL
HER=CHPLx<0.0,0.0)

GO TO 10

12 II'I

JJ‘J

KK-K

LL=L

RHONIK2D

SIGN'SORT(VNtVN+RHONlRHON)
C9=COS<SIGN¥XN(I))3COS(SIGNIXN(K))ISIN(SIGN3HDXN)/((1.0+(VNSO/(RHO

1N¥RHON))USIN(SIGN)tl2)*SIGN)

C9XI<K20~EPS>I<SQRT<(K20)xta-(KZD-EPS)I¥2))
CIOBPI/Z.o-ATAN(C9X)+CMPLX(0.0,ACOSH(1.0+EPS/K2D))
I2-CHPLx10.o,HDZN)xC9xC10

DZNJLsABS<2N(J)-2N<L))

CALL CSIHC<1,o.0,K20HE,0.1o,20,I1,NOI1,R1)

CALL CSIHC(1,K20PE,RHDNM,0.10,20,13,N012,R2)
HER-CORII1+12+13)

10 RETURN

END

FTN4 COMPILER: MP92060-16092 REV. 1901 (781201)

It NO UARNINGS It NO ERRORS it PROGRAM I 00328 COMMON B 00000

 

163

0315 rOHPLFx FUNCTION r<1NDIx,RH0N)
0315 REAL KPD,K20,KPDSU
0317 COMPLEX BTN,C11
310 DIMENSION XN<32),7N<32)
0319 COHHON/NATEL/XN,2N,RPD,BOD
0390 COHHON/NrR/KPD,VN
0321 CUMMUN/FMATL/HDXN,HDZN,KPDSO,UNSO
0329 CUMMON/FMER/DZNJL,II,JJ,KK,LI
0323 IF(INDEX-1) 7,1,7 .
0324 1 RHUNSQ=RHON¥RHON
0325 SIGN=SORT<UNSO+RHONSO>
0390 IF(RHON-K2n) 2,3,3
0327 2 BTN=CHPLX<SORT(K2DSO-RHONSO),0.0)
0328 GO TO 4
0329 3 BTN=CHPLx<0.0,-SQRT<RHONSO—K2DSO))
0330 4 C11=RHONSOICOS(SIGN¥XN(II))lCOS(S]GN¥XN(KK))*SIN(SIGNRHDXN)/((RHUN
0331 1SQ+vNSOBSIN<SICN)::2)IBTNIBTNISICN)
0332 IF(JJ-LL) 6,5,6
0333 S F=011¥(1.0-CEXP(CMPLX(0.0,-1.0)#BTN#HDZN))
0334 GO TO 7
0335 6 F=C11¥CHPLX<0.0,1.0)4CEXP<CHPLx<o.o,-1.0)RBTN107NJL)4CSTN<BTN1HDZN
033A 1)
0337 7 RETURN
0338 END

FTN4 COMPILER: HP92060-16092 REV. 1901 (781201)

it NO WARNINGS *4 NO ERRORS #1 PROGRAM = 00322 COMMON = 00000

[1:13 (_;
0340
0541
054k
0343
0844
0 .445
0346
0347
034M
0349
0380
0351
035?
0353
0354
0355
0356
0357
0358
0359
0360
0361
0362
0363
0364
0365
0366
0367
0368
0369
0370
0371
0372
0373
0374
0375
0376
0377
0378
0379
0380
0381
0382
0383
0384
0385
0386
0387

300

100

200

u

164

SHUVHUIINF SLATL([T.RN,TN)

RTAT N1,N2,N3,000,RFND,TPD

LUMPIIX II,RN,1N,FTSB,LYSr,xCax0,YLIxP
LUHFLIx RR 11

runPLTx kx,Tx

01n1N010N R)(2),Tx<2)

DIHENSION F](3H),XN(3P),ZN(32)

DTHTNSJON CHND<2),KPND<2),0ND<P),RN<2),TNTP)
FOMMON/NATEL/'XN,ZN,kPD,B00,N1,NJ’,N3,P 00 (iral),v>.N.t .-’.D/N,[ 0,01 ,NP,N).
,N7,NPP1

LOHHON/SCATO/20,20O0
COHHON/SHODE/Rx,Tx,CHND,KPN0,BND,HHOD
DSN=DXN¥DZN

DO 99 J=1,MHOD

xx=KPND<J)

C01=xx+.S»STN<2.OBXX)
CC2=<xx/GHND<J))41COS<XX)):424CCI

BBwBND<J)

AJLSQRT(K004201XX/(2.0¥BH¥CC2))

llfJ.NE.1) GO TO 300

A1=AJ

CONTINUE

EYSB=CMPLX(0.0,0.0)

EYSF=CMPLX(0.0,0.0)

DO 1 L=1,NZ,1

DO 1 K=1,Nx,1

N=K+(L-1)INX

XEYO=FYO<XN(K),J)
XCEXP=CEXP<CHPLx10.0,—BND(J)47N(L)))
FYSB=FYSB+EY<N)RXEYORXCEXPRDSN
YCEXP=CEXP<CMPLX(0.0,BND(J)*7N(L)))
EYSF=EYSF+EY(N)XXEYOXYCEXPIDSN

CONTINUE

EYSB=EYSB*2.0

EYSF=EYSF#2.0

CR=<N3RN3-N1RN1)tKOD/(zoxFO)
RR=CHPLX(0.0,-1.0)tC34CEXP(CHPLx<0.0,-1.0:<BOD+BND<J))*2000))xFYSB
RN(J)=RR*AJ/Ai

IF(J.NE.1) GO TO 100
TT=CEXP<CMPLX(0.0,-1.0#(BOD4BND(J))#7000))*(1.0-CMPLX(0.0,1.0)¥C8#

#EYSF)

GO TO 200
T1=CEXP(CMPLX(0.0,-1.0#(BOD+BND(J))*ZOOD))4(0.0-CMPlX(0.0,1.0)NC8!

*EYSF)

CONTINUE
TN(J)=TT*AJ/A1

99 CONTINUE

REIURN
END

FTN4 COMPILER: HP92060-16092 RFV. 1901 (781201)

0388
0389
0390
0391
0392
0393
0394
0395
0396
0397
0398
0399
0400
0401
0402
0403

0404
0405
0406
0407

FTN4

It

1655

FUNCTION EYO(X,N)

REAL N1,N2,N3,KPND,KPD,KOD,KBD,KZDHE,KBDPE,K2DSO

COMPLEX RN,TN

DIHENSION GHND(2),KPND(2),BND(2),RN(2),TN(2),XN(32),ZN(32)
COHHON/SHODE/RN,TN,GHND,KPND,BND,HHOD
COHHON/NATEL/XN,ZN,KPD,BOD,N1,N2,N3,KOD,GHD,DXN,CC,DZN,E0,PI,NP,NX
1,NZ,NPP1

COHHON/SCATO/ZO,ZOOD

XX=KPND(N)

c1=xx+o.sxSIN<2.oxXXI

CB=(XX/GHND(N))¥(COS(XX))*#2+Ci

B=BND(N)

AssnRT(KonxzoxXX/(2.oxaxc2>)

EYO-AKCOS(XX¥X)

RETURN

END

FUNCTION ACOSH(X)
ACOSH-ALOG(X+SQRT(X¥X-1))
RETURN

END

COHPILER: HP92060-16092 REV. 1901 (781201)

NO UARNINGS 01 NO ERRORS It PROGRAM I 00030 COHHON I 00000

1656

0408 SUBROUTINE CHATPtIJOB,A,N,H,DFT,EP)
0409 COHPLFX A,R,DET,CONST,S,CN81,Z,U,x
0410 DINENSION 4(32,33>

0411 30 FORN4T<1x,42NTNE DETERMINAN1 OF THE SYSTFH EQUALS ZERO./
0412 11x,36NTHE PROGRAM CANNOT HANDLE THIS CASE.//)
0413 : 0ET=1.

0414 NP1=N+1

0415 NPH=N+H

0416 NN1sN-1

0417 IF(IJOB) 2,1,2

0418 1 00 3 I=1,N

0419 . NPI=N+I

0420 A<I,NPI>=1.

0421 f IP1=I+1

0422 no 3 J=IP1,N

0423 NRJ=N+J

0424 4(I,NPJ>-0.

0425 3 4(J,NPI>-0.

0426 2 DO 4 J=1,NN1

0427 C=CABS(A(J,J))

0428 JP1=J+1

0429 00 5 IaJR1,N

0430 D-CABS(A(I,J))

0431 IF(C-D) 6,5,5

0432 6 DET--0ET

0433 00 7 K-J,NPH

0434 8=A(I,x)

0435 A<I,K)s4(J,K>

0436 7 4<J,K>-a

0437 c=o

0438 5 CONTINUE

0439 IF(CABS(A(J,J))-EP) 14,15,15
0440 15 DO 4 I=JP1,N

0441 CONST-A(I,J>/4<J,J>

0442 no 4 K-JP1,NRN

0443 CNST=CONST$A(J,K)

0444 4 A(I,K)-4(I,x)-CNST

0445 IF<CABS(A(N,N))-EP) 14,18,18
0446 14 DETao.

0447 IF(IJOB) 16,16,17

0448 16 URITE<6,30)

0449 17 RETURN

0450 18 DO 11 I=1,N

0451 11 DET-DET#A(I,I)

0452 IF<IJOB) 10,10,17

0453 10 no 12 1-1,N

0454 x-N-I+1

0455 KP1-x41

0456 no 12 L-NP1,NPH

0457 s-o.

0458 IF(N-KP1) 22,19,19

0459 19 no 13 J-KP1,N

0460 2-8

0461 13 8-z+A(K,J)RA<J,L)

0462 c 22 U=A(K,L)-S

0463 C x-U/4(K,K)

0464 C A(K,L)=X

0465 22 U=4(K,L)

0466 A(K,L)=(U-S)/A(K,K)
0467 12 CONTINUE

0468 RETURN

0469 END

FTN4 COHPILER: HP92060-16092 REU. 1901 (78120.1)

U470
0471

0472
0473
0474
0475
0476
0477
0478
0479
0480
0481
0482
04133
0484
0485
0486
0487
0488
0489
0490
0491
0492
0493
0494
0495
0496
0497

0498
0499
0500
0501
0502
0503
0504
0505
0506
0507
0508
0509
0510

F0

31
32

167

SOHROUTINL CSIHC(1NDFX,X1,XFND,1FST,L]H,ARFA,NH1,R)
COMPLLx ODD rvLN,Aan1,ENDs,F,nREA
NOJ=0

ODD=CMPLX<U.U,0.0)

INT=1

U=1.o -
EUEN=CHPLX(0.0,0.0)
AREA1=CMPLX(0.0,0.0)
END5=F<INDEx,x1)+F<INDFX,XEND>
Hr<XENDwx1)/U

ODD=EVEN+ODD

x=x1+H/2.

EVEN=CHPLX(0.0,0.0)

D0 3 1:1,INT

EUEN=EVEN+F<INDEX,X)

x=x+H

CONTINUE
AREA=(ENDS+4.0REUEN+2.0xODD>#H/6.o
NOI=NOI+1
R=CA851(AREA1-AREA)/AREA)
IF(N01—LIH) 31,32,32

IF(R—TEST) 32,32,

RETURN

AREA1=AREA

INT=°¥INT

U=2.0¥V

GO TO 2

END

BLOCK DATA NATEL,SCATO,SHODE,NER,FHAT1,FMFR

REAL N1,N2,N3,KPND,RPD,K0D,K20,K2DNE,K2DPE,K2050
COMPLEX RN,TN

DIHENSION XN<32),2N<32)

DIHENSION GHND(2),KPND(2),BND(2),RN(P),TN(2)
COMHON/NATEL/XN,ZN,KPD,BOD,N1,N2,N3,KOD,GHD,DXN,C?,DZN,E0,PI,NP,NX
,N2,NPP1

COHHON/SCATO/20,700D
COMMON/SHODE/RN,TN,GHND,KPND,BND,HHOD
COHHON/NER/KBD,UN,EPS,C8,KBDHE,KZDPE,RHONH
COHHON/FHATL/HDXN,HDZN,KBDSQ,VNSQ
COHHON/FHER/DZNJL,II,JJ,KK,LL

END

6.-

F1N4 COHPILER: HP92060-16092 REV. 1901 (781901)

XX

NO WARNINGS it NO ERRORS **

BLOCK COHHON NATEL SIZE = 00156

BLOCK

BLOCK

BLOCK

BLOCK

BLOCK

COMMON SCATO SIZE = 00004
COMMON SHODE SIZE 8 00029
COHHON NER SIZE = 00014
COHHON FHATL SIZE = 00008

COHHON FHER SIZE = 00006

168

FTNU,L
SUHRUUTINE PLOT(DATx.~S,4uRVS.XSTARI.xSTEP.XMAY.ISYM.MAN,XOATA.Mv,
ANPTbolNCRmTI
CitttiitttttfiAAAAAIAAAAAOAAAIAOAAAOAAAflitti‘fifilitttiiiilifittittitfitiitifl
CtttttttttfitfiAttitifitifitttlIA.Altfitttitt.flit.it!tfifiifiififititfifiiitfitttfitit

CA A
CA x-Y PLOT SURRUUTINL A
C. A

CARA...AtttlttttttttitittttitfiAOAAAAA...A...tittttfitfifitiitttfiitttfiiii...
CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAAAAAIt...AAAAAI.Atlfilttttitttttttfit
CA A
CA A
CA A
C* A
CA A
CA SOURCE: SPL12 A
CA pELUCATARLE: RPLTZ A
CA A
CA THIS IS A STANDARD FORTRAN x-Y PLOTTING SUOROUTINE. IT CAN A
CA SIMULTANEOUSLY PLOT I? nTFFERENT CURVES (ONE IS A ULANK). THE A
CA x-AXIS CAN BE STARTED UR STOPPED AT ANY VALUE A INCREMENTED A
CA dY ANY AMOUNT. THE USER CAN ALSO SPECIFY WHAT INCREMENT THE A
CA x-AXIS VALuES ARE TO BE PRINTED. THE x-AxIs CAN BE SCALED BY A
CA ANY FUNCTION THE USER UESIRES. SUCH AS A LOG FUNCTION. x-AxIS A
CA SCALING IS DETERMINED BY THE MINIMUM AND MAXIMUM DATA VALUES OF A
CA THE FIRST CURVE TO BE PLOTTED. TUE Y-Axls CAN BE SCALEO LARGER A
CA THAN THE MAXIMUM VALUE RY FILLING THE FIRST ARRAY wTTH LARGE A
CA VALUES & PRINTING IT NITH BLANKS. IF THE x VALUE EOUALS THE Y A
CA VALUE THEN A s SYMSDL ls PRINTED. A

I

A

CA
CAA.AAAAAttttattttfitttttttttitttQtttttiiiAttttttfitttttttttitfitttttttttt
CA A A A
CA A PRINTnuT A INTEGER A
CA ARGUMENT DEFINITION A SYMOOL A DESIG. A
CA A A A
CtttttAttttttttttttttfittA...Atttttititttttttttttt‘ttttiitttttittttittil.
CA A A A
CA ARGI: ARRAY NAME UHERE CURVE DATA IS A 0 A I A
CA STORED A A A
CA ARGE: M DIMENSION UF ARQAY A A A 2 A
CA ARGS: NUMBER OF CURVES TO BE PLOTTED A X A 3 A
CA A984: STARTING POINT OF THE x-AXIS A A A a A
CA ARES: AMOUNT X SHOULD BE INCREMENTED A , A S A
CA ARGA: FINAL VALUE OF x A A A 6 A
CA ARGT: ARRAY NAME IN NHICH SYMBOL DATA A D A 7 A
CA 18 STORED IN A A A
CA ARGR: INTEGER INOTCATOR--0 TELLS PROGRAM A a A 8 A
CA TD GENERATE LINEAR x-AXIS. I TELLS A A A
CA PROGRAM THAT USER IS GENERATING A A A
CA THE AXIS A A A
CA ARGR: ARRAY NAME NHERE USERS X-AKIS A 0 A 9 A
CA IS STORED (USE 0 IF ARGB:0) A A A
CA ARGIO:VALUE TD RMICH ARGR IS DIMENSIONED A I A IO A
CA (USE I TF ARGO:0) A A A
C4 ARGIlzNO. DATA POINTS TO BE PLOTTEO IF A s A II A
c. AR58=T. 0THFRNISE U A ' ‘
CA AR512:1NCREMENT IN NHICH x-AXIS VALUES A BLANK A 12 A
CA ARE TO BE PRINTED A A A
CA A A A
CAOIAAOAOAAAQAOIAAAAAAAAAAAAAAAAAA!AA’AAAAQAIAAAAAOAOAAAAAQIOAAQ4.0.0...

169

CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
DIMENSION LINEA(TOI)oLTNE%(IOI)00ATX(eel)oISMoL(Ie).ISYM(IBIoIY(IZ
A).XOATA11)
DATA IRLNK/IH laIDT/IHI/.]SMRL/1HupIHA.IHx,1H+,1H,,1HA,1NU,1Hq,
ATHO.1NA,TH$.IN /.IA8AR/IN:/
OATA CENTRI/l.S/ottNTREISI.S/pLINEUIIOIAIH-I
KOUNszb
WRITE(6.9)
CALL EXEC(3.IIDEH.651

TAXIS=0
AMAX = 0.0
AMIN = 0.0
YVAL = xSTART
N0 = I
IN = INCRMT + I
00 b N=I.I2

6 TYTN) = 0
LINEB(T) = IAoAR
LINERIBOI = IABAR
LINER(51) = IARAR
LINER(76) = IABAR

LINEB(TOI) = IABAR
DO 10 T=I.IOI
Tu LINEA(II = IBLNK
NSTEPS = 1.0 A ((XMAX - XSTART) / XSTEP)
IFTMAN.E0.I) NSTEPS : NPTS
00 20 K:I.NSTEPS
IFTOATY(1.K).GT.AMAx) AMAx
IF(OATX(I.K).LT.AMTN) AMIN
20 CONTINUE
TF(AMIN.LT.0.0) GO TO 30
SCALE = IOO./AMAX
CENTR = CENTRI

DATX(10K)
DATX110K)

T1 : AMAX
Ta = 0.75 A AMAx
T3 = u.5 A AMAX
Ta = 0.25 A AMAX
T5 = 0.0
GO TO 60

30 AIMIN = -AMIN

IF(AMAX.GT.AIMIN) GO T0 40
IF(AMAX.E0.0.0) GO TO 35
SCALE : 50./AIMIN

TI = AIMIN
T2 = 0.5 A AIMTN
T3 = 0.0
T4 = 0.5 A AMIN
T5 = AMIN
GO T0 50

35 SCALE = 100./AIMIN
CENTR = IOI.5
TI = 0.0
T2 = 0.25 A AMIN
13 = 0.5 A AMIN
T4 = 0.75 A AMIN
T5 = AMIN
GO TO 60

do SCALE 8 50./AMAx
TI : AMA!

T? = 0.5 A AHAX

1170

T3 = 0.0
To = 0.5 A (-AMAX)
T5 = -AMAX

SO CENTR = CENTRE

60 VARA = ISTART

ICNTR : CENTR
DO 110 J=I.NSTEPS
KOUNT=KOUNTAI
IAX=0
IF(MAN.EO.I)VARx=xDATA(J)
LINEAtlcNTR) = IDT
Y = -XSTEP
IF(KOUNT.NE.66)GO TD 80
75 IAX=I
IF(KOUNT.EG.66)KOUNT=0
IAX=I
5 FORMAT(“A'oflonIO.2.A(ISXoFIO.2))
DO 70 N:I.IOI
70 LINEAtM) z LINEDtN)
80 00 BI L=IpKURVS
TY(L) = (DATxthJIASCALEIACENTR
LINEA(IY(L)) = ISNBL(ISYN(L))
BI CONTINUE
DO 83 H=IAKURVS
Do 82 N=I.KuRvs
82 IF(((IYtN).ED.IY(N)).AND.(M.NE.N)).AND.( .NOT.((ISYM(NI.EU.12)
A.QR.(ISYM(M).E0.12)))) LINEA(IY(N)) 8 ISNBL(II)
83 CONTINUE
IF(N0.E0.II so To 95
NRITEtboTI LINEA
IF(IAx.En.IINRITE(6.S)TS.TA.T3.T2.TI
7 FORMATtIOXoIOIAI)"
0 FORMATtlox.IOIAI.SX.F15.¢)
95 IF(HAN.E0.I)XVAL=XDATA(J)
IFTNAN.NE.I) VARx : VARx + XSTEP
IF(NO.NE.II Go To 105
NRITE(6.A) LINEAoXVAL
IF(IAX.ER.I)NRITE(6.S)TS.TA.T3.T2.TI
I05 NO 8 N0 A I
IF((VARx.GT.(xSTEPAo.25)).0R.(VARx.LT.(YA.25)))GO To 106
106 IFINO.EO.IN)ND=I
IFIMAN.NE.I) XVAL = XVAL A XSTEP
00 90 N:I.I01
90 LINEAIN) 8 IBLNK
IIo CONTINUE
00 IOT MsIoIOI
101 LINEA(N)=LINEB(N)
NRITE(6.4)LINEA
WRITE(605)TSAT‘0T3¢TEDTI
120 HRITE(609)
9 FORMATI'I'I
CALL EXEC(3.IIOSB.64)
130 RETURN
END
ENDS

1'7].

APPENDIX B
SLABZ OUTPUT SAMPLE

ESSENTIAL INPUT DATA.

N1 1.6 (SLAB REFRACTIUE INDEX)

N2: 1.0 (CLADDING REFRACTIVE INDEX)

N3: 3.0 (DISCONTINUITY REFRACTIUE INDEX)

D/LDA .50 (NORHALIZED SLAB HALF THICKNESS)

GHD: .37203240E+01 (NORHALIZED CLRDDING DECAY EIGENVRLUE)
XPD= .12473061EADI (NORMALIZED SLAB EIGENUALUE)

80D= .48693342£+91 (NORMALIZED PHASE-CONSTANT EIGENVALUE)
ZI/D: .08 (RELATIVE DISCONTINUITY LENGTH)

E0: 1.0 (U/H) (INCIDENT HAVE AMPLITUDE)

HHOD=2 (NUMBER OF HDDES EXIST)

NX= 4 (PARTITIONS ALONG X)
IZ= 8 (PARTITIONS ALONG 2)

N51 GHND= 3.72032400 KPND= - 1.24730610 BND= 4.86933420

N=u GHND= 1.62902900 XPND= 3.56971310 BND= 3.53883310

1'72?

RESULTS OF APPROXIHATE RADIATIONLESS SOLUTION FOR SLAB FIELD AND SCATTERING COEFFICIENTS.

BOPD=( .9269E+01, .0000E+00)

AP= .6313E+00EXP(It-.2070E+02)

AH= .1964E+00EXP(It-.1004E+03)

R0= .SbIIE+00EXP(JI-.1715£+03)

T0= .8277E100EXP(J¥-.8146E+02)

bwwuaumwaum»JumwawrowbuwwbuMHAwr-JM

1.713

RESULTS OF NUNERICAL HON SOLUTION FOR SLAB FIELD AND SCATTERING COEFFICIENTS.

PROCRAN HITH SYHNETRY AND HULTIHODE SCATTERING

INDUCED FIELD EY(X,Z) IN THE DISCONTINUITY REGION.

mWQQVVVVO‘O‘O‘O‘WWWM‘bLLwUML-INNNPOF‘WH»

XN(I)

.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875
.125
.375
.625
.875

ZN(J)

-.066
-.066
-.066
-.066
-.047
-.047
-.047
-.047
-.028
-.028
-.028
-.028
-.009
-.009
-.009
-.009
.009
.009
.009
.009
.028
.028
.028
.028
.047
.047
.047
.047
.066
.066
.066
.066

muomoum»

04040401) NNNNN mmwmumnmwwww

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

EY(N)

.55812E+0i
.30470E+01
.21479E+01
.299S7E+01
.60585E+01
.33614E+01
.23911E+01
.32602E+01
.63681E+01
35512801
.25461E+oi
.34382E+01
.6S7ISE+01
.36328E+01
.26078E+01
.36511E+01
.65197E+01
.35867E+01
.25728£+01
.36289E+01
.62106E+01
.34143E+01
.24422E+0i
.33644E+01
.SBUSSE+01
.31390E+01
.22213E+01
.3147SE+01
.52403E+01
.27464E+01
.19173E+01
.28481E+li

y...---------~9----vs.

.147S4E+01)
-.12984E+01)
-.11598E+01)

.18170E+01)

.30208E+00)
-.25116E+01)
-.21229E+01)

.13477E+01)
-.85719E+00)
-.36653E+01)
-.30431E+01)

.84815E+0|)
-.20382E+01)
-.47465E+01)
-.38931E+01)

.19691E+0')
-.30952E+01)
-.S7USIE+01)
-.46482E+ai)
-.30868E+00)
-.40789E+0i)
-.65113E+01)
-.52854E+01)
-.81147E+00)
-.48630E+01)
-.713$6E+01)
-.S7864E+01)
-.1204oE+01)
-.S4785E+01)
-.7S474E+01)
-.61364E+01)
-.15337E+01)

EYA(N)

.7187BE+00
.41239E+lo
.30392E+00
.43623E+Io
.75527E+00
.5224SE+II
.39812E+00
.43924E+l0
.80003E+00
.63543E+00
.49402E+00
.44092E+l0
.85667E+00
.74421E+|0
.58342E+00
.45526E+|0
.89860E+00
.83904E+|o
.66148E+00
.45346E+09
.92513E+00
.91541E+00
.72494E+00
.43091E+00
.9429ZE+00

.9706IE+00

.77172E+00
.41958E+00
.94393E+00
.10000E+0I
.80046E+00
.40277E+00

EYP(N)

1'74

SCATTERING (REFLECTION AND TRANSMISSION) COEFFICIENTS DESCRIBING DISCONTINUITY REGION.

n=1 RN: .sssze+oo£xp(J:¥.1723E+03)

TN= .8149E+00EXP(J0-.8109E+02)

N=2 RN= .6860E-01EXP(Jl-.ISS9E+03)

TN: .7356E-01EXP(10-.IS40E+03)

175

APPENDIX C

OSWDSC SOURCE LISTING

1=CMWWWW
2=Ci THIS PROGRAI CALCULATE DISCONTINUITY FIELD BY ITERATIVE SOLUTION 0
3=CHWWRWW

4: PROGRAI OSUDSCITAPEI,INPUT,0UTPUT,TAPE5=INPUT,TAPE2=OUTPUI)
5: REAL N1,N2,N3,KOD,KPB,K2D,K2DSO,K20IE,KZDPE

a: conRLEx A1,C,D,Il,IZ,IS,EL,£Ll,El,6£L

7: nxnsuszon C(4096),0!4096),ELI(8,0),ELIB,8),E118,8)

s: DIHENSION £RR18,81

9: nxnsnsxou XIIB),2N(8),XI(8),ZJ(8)

1o: connou1r1c1x2,vusn,xrn,xzn,xznsa,xn,zu,x1,21,u,u,1,1

11:

12:: i1111i11+i11111111111i1i111111111111R1i1111111111111111111111111111111
13:: READ FIRST para LINE FOR REFRACTIVE 1unxcss (NI,I2,N3) or cons, CLAIDI
14=c DISCONTINUITY REGIONS, RESPECTIVELY.

15=c 111111111111i11111i1R11i1i1111111i1111111111i1111111111111111111111111
16 READ 11,1) u1,u2,u3

17: 1 F0RRAIL:F10.31

1e=c 111111111111111111i111111111i1111i111111i11111111111111111111111111111
19:: READ sscoun 1111 LINE FOR IORIALIZED SLA! tuxcxnzss IOLO Run NORIALIZE
zo=c EISENVALUE PARAMETERS 1snn,xpn,nonn or THE OIPERTURIEI SLAB UAVESUIIE.
21=c I111111111111111i11111111111111111111i1i1i111a111111111111111111111111
22 READ 11,21 DOLO,6ID,KPD,BOD

23 2 FDRIAT1F15.2,3EIS.8)

24:: {iiiiifiiiiiiiiiffiiifiiiiiiiiiiiiiii0iiiiiiii{filiiiiiiiiiiiiiiiiiiii
25:c READ THIRD DATA LINE FOR IRRRALIZEI Lsustu 2001 or DISCONTINUITY REGIO
21=c AlPLITUDE so or INCIIEII HAVE.

27=c 111111111111111i111i111i111111111111111i11i111111111111111111111111111
28= READ 11.31 2001.50

29: 3 F0RRAI¢F10.2,F10.11

so=c 11111111i111111111i11111i111iii111i111i1111111111111111111111111111111
31=c READ rouRIR aura LINE FOR uuunsRs 0F RRRIIIIous th,IZ) RLoRs x All 2

32:: DIRECTIONS, RESPECTIVELY.

33=c 11111111R111111111111i111111111111i11111i1111i111111111111111111111111
34: READ 11,41 Rx,uz

35 4 FDRIATIIZ,3X,121

36=C 11111111ii11111111i1111i11i111iiii1i1i111111i1111111111111111111111111
37st ram ESSENTIAL mm mm.

33=c iiii§iiiiiiiiiiiiiifiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiifiii
39: URITET2,5)

‘40: s FORIATTlHl,II,10X,21HESSElTIAL INPUT DATA.,II)

41: 0R11512,e1 Il,l2,l3

42: a FORIATTIHO,III,IOX,4HII= ,F4.l,2X,23NISLAI REFRACTIVE 1Rm£x1,1,1ox
43: 1,4NI2= ,F4.l,2X,27H1CLAIlIIG REFRRcIIRE 1u9£x1,1,1ox,4uu3= ,r4.1,2
44: 2X,32HTDISCOITINUIIY REFRACIIVE 1u1£x11

45: URII£12,71 IOLO,6I0,KPD,IOD

41: 7 FORIA111N0,I,10X,6HDILO= ,r5.2,2x,32R1uoRRAL12£n s11) HALF qucxus
47. ossy,ll,1ox,susnn= ,E14.8,2X,38HTIORIALIZEI annnxus DECAY EISEIVAL

48=
49=
50:
51=
52=
53:
54:
55=

176

20E),I,10X,5NKPD= ,El4.8,2X,28H1NORIALIZED sLRI EIGENVALUEI,I,IOX,5
38300: ,EI4.8,ZX,38HINORNALIZED PHASE-CONSTANT EIBENVALUET)
URII£I2,81 zoon,£o

a FORNATIIHO,I,IOX,6HZOID= ,r5.2,2x,31R1R£LAIIVE DISCONTINUITY LEIBT
lH),II,lOX,4NEO= ,F4.I,ZX,SHIVINI,2X,25NTINCIIENT HAVE ANPLIIUDE)!
0R11512,91 IX,NZ

9 FDRIATIIHO,I,IOX,4HNX= .13,2X,20H(PARTITIONS RLnRs X),I,10X,4NN2=
1,13,21,20H1mnnous ALONG 1n

56=C ii000iiiiii0iiiiiiiiiiiil0fii00i0iiiiiiii0iiiiiiiiliiiiiiiiiiiiiiiiifi
57=C PARTITION THE DISCONTINUITY REGION USING NORNALIZEI COORDINATE VARIABL
58=C Efffifiiiii0i0iiiiiii0i0f0iiiiii00i00i0iiii0iiiiiiiiiiiifiiiiiiiiiif§i

59:16
60=
6I=
62=
63:

73:
74:
7s:

CONTINUE
PI=3.1415926536
20=12o.oIRI
R0n=2.onRIIn0Lo
NXH=KXIZ
nxn=2.01ux
nzu=2.onzoon/Rz
no 17 R=I,Rx,1
XNTN)=-l.0+(l-0.5)!DXI
Ian
x1111=xn1u1

17 courqus
no 18 N=1,NZ,1
zunnn=~zoon+1u-o.snnnzu
J=I
21111=zu1u1

1a CONTINUE

76=C liiiiiiiiiiiiiiiiiiiiiiiiiliiiiiiiiiffiiiiiiiiiii§0i0i00+§

77=C

DEFINE FREQUENTLY USED CONSTANTS 0

78=C 0iii0i0i0iiifiii0iiiiiifiiiiiiiiiiiiiiiiff}iiiiiiii00000§l

79:

DNSO=N1002-N2002
K20=N2§KOD
KZDSS=KZDfi2
VNSE=DNSO§K00§§2
SGNN=SDRTIVNSOAKZDSOI
RHONN=I0.0§K20
EPS=KZDII0.0
KZDNE=K20-EPS
KZDPE=K20+EPS
TER=0.0

[=0

90=C iFiIOFi0iii000iii0i00fii0iA00iiiiiiiiiifiiiiiiiiiiiii§iiii

918C

DEFINE CONSTANT COEFFICIENTS !

92=C 0*iiififiiiiifiifif0000{iii}!!!fiifi§§f§§*§!§i§i§§}i{90*}!

93:
94:
95:
96:
97:

A=COS€KPDTICOSTKPDIISNDASINIZEKPDIII2!KPDI*I
A=A!2!DOD '
A=SORTTNODEZOIAT
AI=CN°LXI0.0,'I.OTIIKODIZOF§A§!2§CNSRIBXN}IZN
CIK2=IAIVNSGIKZDSOIIISIN(SGRN:){42

177
98:: WW

99:1: CALCULATE mars mum m ALI. CELLS n
Ioo=c WM
101: no so 11:1,»:

1oz= no no H.112

103: no so I=1.Ix

164: no so mm

105: L=L+I

I06=C iii-WW

IW’C CICMTE IISCIETE CONTRIIUT I !

1088C WWW

I09= NLT=AIlCO$INPIIXITIIHCEXHC'LXI0.0,-I.01millll)-
IIO= -ZJIIIH

III=C WW

112:0 CALCULATE CONTINUOUS CONTRIIUTIONS 0

113=C mm

1143 CALL CSINCONTI,0.0,NM.0.10,20,II,NII,RII

IIS= CALL csmcouu.m.m,o.10.20.13.013331

1168 INOSIMXIIIII‘ICOSIMNNINHICINZ

II7=I IZ'IZNPIIZJ-ASINII'EPSIHI*C'LXIOJJJHACNHOEPSII-
III. *Z‘ICILXI0.0,I.0HAISIZNINI-ZJIJIHE'S)

119' C (“801“ I0.0,-I.0)NNO0I (20"!) HMIIIOIZAIIIMIMOI
120850 MINE

1218 EITNJIIEOIA'ICOSINPHNNIN)IiCEXPICmIO.0,-I.OI§NHZNIIH
122a El.IN,N)8CULX(.657,-.46MCEXHC'LXI.0,-2.423‘IZNINIH

1233 ELIN,N)*EUN.NT*CNPLXI-J,.MIKEXPIC'LXIJJJZIHNIN)H
1243C WWW
1258C! USE APPROXIMTEI FIELI ELIN,N) AS 0TH ITEIATIVE TOTAL FIELD
IMWWWWWW
1278C ELIN,N)*EIIN,NI

128860 comm

129: PRINT i.‘ '

130: PRINT i.’ '

131: PRINT I.‘ STTCCESSIVE ITERATIII moons: '
132: 11:0

18:70 n=m

134: TER=o.o

1358 TELMJ

136: BO

137: 00 110 III,“
138= I! 110 NII,NZ
139a EEL'CDLXIOJJ.”
1408 N I” 181,”
141* II 100 I'I,NZ

142= L'UI

143: GEL'GELHIILH'CILHIELII,”
I448

“#100 CWT“

1468 ELI 111.1018 II.NHEL

147s m1u.m-1ms1a11u.mIm-ImSIELInmImz
ms mmmanummmn

149- mmunssmnmmmn

150: mommomun

171=122
172:12‘

I78=128

178

TELNN=TELNN4IAIST£LTN,NTI
TER=T£R4£RRTN,N1
Pch=T£RIIooTTELNN
£LTN,N1=£LITN,N:

CONTINUE

IF TR.ST.21 GO TO 115

PRINT 4, ' '

PRINT n.' '.K.' TN IT£R4TIoN ERROR = -,T£R

PRINT i.’ PERCENTNsE ERROR Is: '.PCH6

coNTINUE

IF TPcus -1.C) 120,120.70

PRINT I. ' '

PRINT 2, ' CONVERSE To .01 ERRoR AFTER -,N.- ITERATTONS.‘
PRINT 1.' '

PRINT 4.' coNVERsEn nIscoNTINoITT TIELns: -

PRINT 1.- -

EYIAX=CABSTEL(1.1)T
no :24 N=1.NxN

no 124 N=1.Nz

IF TcNISTELTN,NTT-ETNNNT 124,124,122
£TNRN=CR25I£LTN,NTT

coNTINUE

no 128 N=1.NXN

no 128 N=1.Nz

EY=CABSTELTN.N))IEYNAX

NRIT£12.22T N.N.EY
FURNATTTHO.10X.3NEYT,11.1H,.11.3H)= .F7.4)
CONTINUE

179:CTiff}fifiiiiiii§§§§§0¥§111§0§0iiiiiIiii1ii0iiiiiiiiiiiifiiiiiiiiiiii

180=C§

INTERACTIVE ITERATION INPUTS 0

181=C111*!!!0f10{1iIii*iiiiiiiEEONEiiiiiiiiiifliiiiiiiiiiiiiiiiiiiiiiiiii

182=
183=
184=
105=
186=
187:
188=
189=
190=
191=
192=
193=
194=130
195=
196=
197:
198:
199=200

PRINT 1,' '

PRINT 4,-noLo=2.zoon:2.2oIT=o oR CONTINUE=1.... '
READ 15.11 noL.zoon.NN

IP1NN.£R.oT so To 200

IFTDOL.EO.DDLO) so To 130

noLo=noL
PRINT 4, -snn:2.NPn=?.Ion=?'
PRINT I.' -

READ 15.21 sun.NPI.non

PRINT 4.' enn2'.sun
PRINT I.' NPn=-.NPI
PRINT 1.' Ions-,non
CONTINUE

PRINT I.- -

PRINT 2.' IOLO='.DOL0
PRINT 2.' zoon='.zoon
so To 11

ENn

2018
2028
2038
2048
205-
2068
2078

2098
2108
2118
2128
2138
2148
21584
2168
2178
2108
2198
2208
2218

2248

2268

2288

2308
2318
2328
2338
2348
2358
2368
2378
2388
2398
2408
2418
2428
2438
2448
2458
246-
2478
2488
2498
2508
2518

2538

coma: mum PTInsx.RmN1
Twsnuxnguwum 179
TIPS CONPLEX oTN
nINENsIoN xNTsT.zNTaT.xITcn.ZIIoT
CDNNONIFICIKZ.VISI.KPI,KZD.KZDSO.XN.ZN.X1.ZJ,N.N.1,1
IFIINDEX-l) 7.1.7

1 RNoNsnsRNoNIRNoN
SIGN8SGRTTVNSOPRN0NSOT
c1=RnoNsaszoITsINTsIeNnm
C1=RHONSIICI
IPIRnoN-Nznn 2.3.3

2 ITN-cNPLszINTTN2252~RNoNsNT.o.RT
mTo4

3 ITNscNPLxTo.o.-sRRTTonoNsR-NzlsooT
P=cosTsIsN1NITITIncosTSIRNanTNTT
TsPIcEXPIcNPLxTo.o.-1.oTnnTNINIsTZNTNT-ZITITTT/TnTNT
P=P2c1 a

7 RETURN '
Ell

FUNCTION ACOSNIX)
ACOSNBALOGINOSIRTLNIN-IIT
RETURN

END

SUIPOUTIIE CSIICOIIIIIEX.XI.xEII,TEST.LIN.AIEA.IDI.RT
TYPE COIPLEX DII,EVEN.AREAI.EIIS,F,AIEA
lolso
ell-CIPLXI0.0.0.0T
IITSI
081.0
EVEN-CIPLXT0.0.0.0T
AREAI=CIPLXT0.0.0.0I
EllSsFTIllEX.XITOFTIIIEX.XEIIT

2 H=IXEll-XIIIV
OIIsEVElooDI
x=x11N12.
£VEN=CIPLX¢0.0.0.0T
no 3 I=I.IIT
EVEI!EVEIPPTIIIEX.XT
x=x4N

3 CONTINUE
AIEAPTEIISO4.OTEVEIPZ.Dialliifllb.o
I01=IOITI
PsCAISTIAREAI-AIEATINPENT
IFTNOI-LINT 31.32.32

31 IPTR-TssTT 32.32.4
32 RETURN

4 AREA18AREA
INT82§INT
V82.0!U
GO TO 2
END

180

ESSENTIAL INPUT DATA.
' APPENDIX D OSWDSC SAMPLE OUTPUT

N1 1.6 (SLAD REFRACTIUE mm
N2 1.0 1CLADDINS REFRACTIUE INDEX)
N38 3.0 IDISCONTIIIITY REFRACTIVE INDEX)

DIL08 .15 (MALIZED SLAD HALF THICNESSI

SNDI .853499M+00 (MALIZED CLADDII KCAY EISENVKLE)
NPD8 .8106980E000 (MALI- SLAD EISENUflIE)

80118 .12715050901 (WHEN PHASE-CONSTANT EISENUALUE)
10108 .10 IRELATIUE DISCIITINUITY LENGTH)
E08 1.0 WIN) (IKIDENT NAUE A'LITUKT

NN8 4 IPARTITIMS MM N1
N28 0 (PARTITIIIS ALM 21

SUCCESSIUE ITERATIN ERM:

1 TN ITERATIII ER“ 8 231.7218972309
PERCENTAGE ERRII IS: 130633504294

2 TN ITERATIM ERRN 8 20.75023329532
PERCENTAGE ERROR IS: 12.37396650678

CONVERSE TO .01 ER" AFTER 4 ITERATIMS.
CONVERSED DISCONTINUITY FIELDS:

£111.11: .2021
2111.21: .2077
2111.31: .2124
2411.41: .2124
2111.51: .2193
2411.11: .2212
2211.71: .2223
2111.21: .2222
2112.11: .9775
2112.21: .9224
2212.31: .9224
2112.41: .9921
2112.51: .9929
2112.41: .9921
2112.71: .9995
£212.21: 1.2022

00L0=?.20008?.IUIT80 oR coNTINu£-1....o.,o..o