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This is to certify that the

dissertation entitled

Non-Bound Contributions to the
Propagation-Mode Spectrum of Open-Boundary Waveguides

presented by

Jerry Michael Grimm

has been accepted towards fulfillment
of the requirements for

Electrical
Engineering

Ph ° D ° degree in

 

 

Major professor

 

Date _M_é/?2

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

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Michigan State

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TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE
n8 1) B 3009

 

 

 

 

 

 

 

 

9—1!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU In An Affirmative Action/Equal Opportunity Institution
chS-M

 

NON-BOUND CONTRIBUTIONS TO THE PROPAGATION-MODE
SPECTRUM OF OPEN-BOUNDARY WAVEGUIDES

By

Jerry Michael Grimm

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1992

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ABSTRACT

NON-BOUND CONTRIBUTIONS TO THE PROPAGATION -MODE
SPECTRUM OF OPEN-BOUNDARY WAVEGUIDES

This dissertation advances an integral—operator formulation capable of determining
the non-bounded contributions to the propagation-mode spectrum for a broad class of
practical open-boundary waveguides. These non-bounded contributions consist of the
continuous radiation spectrum and the discrete, non-spectral leaky-wave modes.

A transform-domain electric field integral equation (EFIE) is developed for open-
boundary waveguides in a planar, tri-layered substrate/film/cover environment typical of
integrated optics applications. Two complex (transform-variable) wavenumber planes are
implicated through Sommerfeld-integral representations of the associated Green’s
functions and the axial inverse transform to recover the space-domain field, whose
complete propagation-mode spectrum is recognized through a singularity expansion upon
the axial wavenumber plane singularities. The radiation spectrum is the continuous
superposition of solutions to the forced EFIE over a restricted spectral regime associated
with the axial wavenumber plane branch cuts.

Branch cuts in the axial wavenumber plane are hyperbolic and are chosen to
restrict migration of transverse wavenumber singularities, rendering forward transforms
convergent. A new component of the radiation spectrum is identified as being associated
with the surface-wave modes of the background structure.

The axial-wavenumber—plane branch cuts define a multi-sheeted Riemann surface

of which the top sheet is proper. Leaky-wave modes are the solutions of the

 

 

 

homogeneous EFIE on non-spectral Riemann sheets chosen by violating specific branch
cuts in the axial-wavenumber plane.

The integral-operator formulation is validated, and its usefulness demonstrated,
for a number of open-boundary waveguides. The continuous radiation spectrum is
quantified for the canonical symmetric planar waveguide to confirm the validity of the
integral-operator approach. Results are developed for asymmetric planar waveguides as
well. The radiation spectral surface current distributions for a simple microstrip
transmission line are determined. Finally, leaky modes of an integrated dielectric rib

waveguide are determined.

Copyright ° by Jerry Michael Grimm, 1992
All Rights Reserved

To my loving parents,
Bob and Sue Grimm

and

In memory of my grandmother Muriel Blauert,
who would have been so proud

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Acknowledgements

Great achievements are meaningless if no one shares in them. This pinnacle of
my efforts is not possible without the assistance and contributions of friends.

Foremost among those whom I thank is Dr. Dennis Nyquist, my advisor, whose
contributions are many. I will ever be grateful for his encouragement of my continuing
graduate education. His thought-provoking questions and valuable insights were of great
assistance during my research. He served as mentor to me, guiding my efforts as I
taught classes, conducted research, and published papers at conferences and in journals.
Finally, as a friend, he offered support throughout the entire process.

All who graciously served on my committee deserve thanks for their patience and
understanding during this process. Dr. Byron Drachman’s assistance on the mathematical
details within was invaluable. Drs. Ed Rothwell and Kun-Mu Chen provided advice and
assistance with both research and teaching efforts. Finally, I must thank Dr. Merv
Vincent of Boeing for his wonderful patience with me and for his questions and
comments.

I thank my family, parents Bob and Sue Grimm, and brother Brian, gave love and
continual encouragement. I thank Sean Caughman, whose friendship for the last ten
years has been one of the few constants in my life. I thank my extended family in Texas
as well, Teresa Patterson and Scott Merritt, for their support and friendship.

I recognize those who made my life in Michigan enjoyable. These include my
fellow grad students in the EM Lab, both old and new; my housemates Rob Butler and
Charles Scripter; my friends over the years at Tau Beta Pi Michigan Alpha chapter; and
my friends from science-fiction fandom. From these friends I take good memories of
my stay at Michigan State University.

Finally, a last special thank—you to my colleagues at CNA, whose emotional
support I enjoy as I finished publication of my dissertation.

vi

AClfllOu
list of l
List of

Chapte]

Chapter

Chapte

 

Table of Contents

Acknowledgements ...................................... vi
List of Tables ......................................... x
List of Figures ......................................... xi
Chapter 1
Introduction ...................................... 1
Chapter 2
Electric-Field Integral Equation Description of Open-Boundary
Waveguiding Systems ................................ 10
2.1 Development of an Electric-Field Integral Equation ........... 11
2.1.1 Equivalent currents ......................... 13
2.1.2 Fields within layered media .................... 14
2.1.3 Construction of the integral equation .............. 16
2.2 Development of the Hertzian Potential Dyadic Green’s Function . . . 19
2.2.1 Two-dimensional Fourier transform ............... 21
2.2.2 Principal Hertzian potentials ................... 22
2.2.3 Scattered Hertzian potentials ................... 24
2.2.4 Dyadic Green’s function for Hertzian potentials ........ 26
2.2.5 Electric-field dyadic Green’s function .............. 28
2.3 Axial-Transform Domain Electric-Field Integral Equation ....... 30
2.4 Development of a Transverse-Electric-Field Integral Equation for
Integrated Dielectric Waveguides ..................... 33
Chapter 3
Propagation-mode Spectrum for Open-Boundary Waveguides ........ 43
3.1 The Fourier Transform in the Complex Plane .............. 45
3.1.1 Fourier transform theory on the real-line ............ 45
3.1.2 Theory of analytic functions ................... 47
3.1.3 Regions of convergence for functions of exponential
order ................................. 48
3.2 Green’s Function Singularities ........................ 53
3.2.1 Transverse wavenumber plane (complex £~plane)
singularities ............................. 55
3.2.2 Considerations of forward transform convergence ...... 59
3.2.3 Axial transform-domain (complex {-plane) restrictions . . . 63

vii

 

 

 

Chaps

Apper

 

3.3 Propagation-Mode Spectrum for Open-Boundary Waveguides ..... 67

3.3.1 Bound modes ............................. 71
3.3.2 Continuous radiation spectrum .................. 73
3.4 Radiation Spectrum in the Low-loss Limit ................. 75
3.5 The Proper Role of Leaky-Wave Modes .................. 86
3.5.1 Identification of leaky-wave modes via the EFIE ....... 86
3.5.2 Usage of leaky-wave modes .................... 92
3.5.3 Physical interpretation of leaky—wave modes .......... 95
Chapter 4
Continuous Radiation Spectrum for Planar Waveguides ............ 101
4.1 General Considerations for Planar Waveguides .............. 102
4.1.1 Transverse uniformity considerations .............. 104
4.1.2 Uncoupled Transverse-Field EFIE ................ 109
4.2 TE Asymmetric Planar Waveguide Radiation Modes .......... 112
4.3 Results ...................................... 115
Chapter 5
Continuous Radiation Spectrum for Microstrip Transmission Line ..... 125
5.1 Application of the EFIE ............................ 125
5.2 Method-of-Moments Solution ......................... 129
5.2.1 MoM expansions .......................... 130
5.2.2 Excitation considerations ...................... 132
5.3 Results ...................................... 137
Chapter 6
Leaky-wave Modes for the Dielectric Rib Waveguide ............. 150
6.1 Application of the EFIE ............................ 150
6.1.1 Parity considerations ........................ 152
6.1.2 EFIE for integrated dielectric waveguide ............ 155
6.2 Method-of-Moments Solution ......................... 157
6.2.1 MoM expansions .......................... 157
6.2.2 Special considerations for the MOM expansion ........ 160
6.3 Spectral Analysis Considerations ...................... 163
6.4 Results ...................................... 166
Chapter 7
Conclusions and Recommendations ........................ 177
Appendix A
Electric Hertzian Potentials ............................. 185
A.l Electric Hertzian Potential .......................... 185
A.2 Hertzian Potential Boundary Conditions .................. 187
A.3 Interpretations and Considerations ..................... 189

viii

Apps:

Apper

Biblic-

 

 

Appendix B

Spectral Representation of the Dyadic Green’s Function ........... 193

B.l Principal Dyadic Green’s Function ..................... 193

B2 Reflected Dyadic Green’s Function ..................... 195
Appendix C

Proof of the Analytic Function Definition Theorem of Chapter 3 ...... 204

CI Theorem and Definitions .......................... 204

C.2 Proof of Theorem I ............................. 205
Bibliography .......................................... 207

ix

Table

Table .

Table

 

List of Tables

Table 3.1 Effect of the coalesced transform-domain branch cut upon
complex f-plane singularities ....................... 77

Table 3.2 Complex f-plane singularity arguments for j’ in the substrate
radiation regime ............................... 80

Table 6.1 Table of space-wave leaky wave poles (sheet 3) and full leaky
wave (sheet 4) for the dielectric rib waveguide where b/a=l. . . . 174

Figurq

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Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

List of Figures

(a) Typical microstrip transmission line. (b) Typical integrated
dielectric waveguide. ............................ 5

(a) Typical configuration for integrated dielectric waveguide.

(b) Typical configuration for microstrip transmission line. ..... l2
Hertzian potentials for a tri—layered environment. ........... 20
Configuration for axial-transform domain analysis of an

integrated dielectric waveguide ....................... 31
Waveguide geometry for development of TEFIE. ........... 39

Regions of convergence for functions of exponential order.
(a) Convergence in upper half-plane. (b) Convergence in lower

half-plane .................................... 50
Strip of convergence in transforrmdomain for physically

realizable functions of exponential order. ................ 51
Regions of convergence for exponentially increasing functions. . . 54

Singularity locations in the transverse-transform domain
(complex s-plane) ............................... 60

Strip of convergence in complex £-plane for forward transform
on x. Arrows denote singularity migration directions. ........ 62

(a) Migration of a E-plane singularity across real-axis through
contour of integration. (b) Migration of a £-p1ane singularity
across real-axis, treated correctly. .................... 64

Branch cuts in axial-transform plane (complex {-plane)
necessary to maintain convergence of forward Fourier transform
on x. ...................................... 66

Contour deformation in complex {-plane used to identify the
propagation-mode spectrum. Closure shown for z < z'. ...... 70

xi

Figure
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Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 4.1
Figure 4.2
Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Coalescing of complex {-plane branch cuts in low-loss limit.
Closure for case of z < z’. ........................ 76

Evaluation of complex-valued square roots. .............. 81

Complex f-plane singularity locations for the substrate radiation
regime. (a) Interior side of branch cut S. (b) Exterior side of
branch cut S. ................................. 83

Complex E-plane singularity locations for the full (cover)
radiation regime. (a) Interior side of branch out B. (b) Exterior
side of branch cut B. ............................ 85

Complex £-plane singularity locations for transverse-only
radiation regime. (a) Interior side of branch out I’. (b) Exterior
side of branch out P. ............................ 87

(a) Four-sheeted axial wavenumber (complex-f) plane.
(b) Nature of each Riemann sheet. .................... 89

Deformed inversion contour in complex f-plane used when upon

a non-spectral Riemann sheet of the axial-transform plane. ..... 91
Typical steepest-descent plane. ...................... 96
Physical interpretation of leaky-wave mode (plasma waveguide

example from Shevchenko). ........................ 98
Configuration of asymmetric planar dielectric waveguide. ...... 103
Contour used for evaluation of TEFIE. ................. 105

TE excitation of planar waveguide by line source at y=yo, z=0. . 113
Spectral radiation mode amplitudes of a symmetric planar

waveguide obtained by the integral-operator method compared to
Rozzi’s analytical closed-form results. ................. 118

Axial-wavenumber plane (complex if—plane) branch cuts for a
typical asymmetric planar waveguide. .................. 120

Spectral radiation modes in guiding region of asymmetric planar
waveguide for both substrate and full radiation regimes ........ 121

xii

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Figure 5.1

Figure 5.2
Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10
Figure 5.11

Figure 5.12

Figure 5.13

Figure 6.1

Figure 6.2

Effect of asymmetry (njn, = 1.2) on the radiation mode field

distributions, compared to the symmetric planar waveguide. . 122
Effect of small amount of asymmetry (tr/n, = 1.05) on the

radiation mode field distributions, compared to symmetric planar
waveguide. ................................. 123
Configuration of microstrip transmission line. ............ 126
Excitation of even radiation spectral modes for microstrip line. . 134
Excitation of odd radiation spectral modes for microstrip line. . . . 135
Axial wavenumber plane (complex {-plane) branch cuts for

low-loss microstrip transmission line. ................. 138
Normalized radiation regime axial surface current density, even

parity. Excitation is 5k above microstrip line. ........... 139
Normalized radiation regime transverse surface current density,

even parity. Excitation is 5% above microstrip line. ........ 140
Effect of source excitation distance on radiation regime surface

current amplitudes. ............................ 141
Radiation regime axial surface current amplitude, even parity. . . . 143
Radiation regime transverse surface current amplitude, even

parity. .................................... 144
Radiation regime axial surface current amplitude, odd parity. . . 145
Radiation regime axial surface current amplitude, odd parity. . 146
Typical surface current distribution within surface-wave

radiation regime, even parity. Excitation source 5). above

microstrip, §=kc. .............................. 148
Typical surface current distribution within surface-wave

radiation regime, odd parity. Excitation source 5). above

microstrip, (=19. .............................. 149
Configuration of a rib dielectric optical waveguide. ......... 151

Complex E-plane singularities for surface-wave leakage. ...... 164

xiii

Figur

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Figure 6.3 Singularities in the complex f-plane for: (a) space-wave
leakage (sheet 3). (b) full (space/surface wave) leakage
(sheet 4). ................................... 165

Figure 6.4 Dispersion curve for integrated dielectric rib waveguide. ...... 168

Figure 6.5 Field distribution for dominant EVu waveguide mode for guide
half-width of a=1.788x. .......................... 169

Figure 6.6 Field distribution for E3, waveguide mode for guide half-width
of a= 1.788% .................................. 170

Figure 6.7 Field distribution for £3, waveguide mode for guide half-width
of a=1.788>.. . . .' .............................. 171

Figure 6.8 Attenuation plot for E"ll leaky-wave mode in surface-wave-leaky
regime. .................................... 172

Figure 6.9 Field distribution for E"ll leaky-wave of rib waveguide
compared to a bound guiding mode. ................... 173

Figure 6.10 Comparison of field distributions of sheet 3 and sheet 4
leaky-wave modes. ............................. 175

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Chapter 1

Introduction

Open-boundary waveguides, such as microstrip transmission lines or dielectric rib
waveguides, are among the most fascinating type of waveguiding structures in common
use. Open-boundary waveguides (OBWG) physically differ from their more conventional
closed-pipe counterparts in one significant detail — electromagnetic waves are guided
along a preferred axis essentially by the mechanism of total internal reflection, instead
of being transversely confined by conducting walls. This characteristic makes open-
boundary waveguides indispensable for integrated optics and millimeter/microwave
integrated circuits (MMICs).

Of fundamental concern for waveguiding devices is a description of their
propagation-mode spectrum. If the complete propagation-mode spectrum is known, the
total electromagnetic field of an open-boundary waveguide can be expanded in terms of
its modes; this modal expansion in turn is used in analysis of excitation, coupling and
scattering problems [1,2]. Open-boundary waveguides have a significantly more
complicated mode spectrum than their closed-pipe counterparts; it is well-known that the
proper modal spectrum for open—boundary waveguides consists of a continuum of
orthogonal radiation modes [3,4,5 ,6] in addition to a finite number of discrete,

bound modes [7].

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is confined in near proximity to the guiding region; no power flows transverse to the
guiding axis. Bound modes are the hybrid guided-wave modes used for signal
transmission, and can be considered an analogue of conventional closed-pipe waveguide
modes.

A radiation mode, however, has no analogue in closed-pipe waveguide theory.
A radiation mode is not confined and bound to the guiding region; its electromagnetic
field possesses a standing wave pattern of finite, non-vanishing amplitude as approaching
infinity. This seems a non-physical solution, as no fields can exist at infinity. However,
Shevchenko [8] observed that radiation modes can never occur singly, but occur over
an entire continuum of propagation constants within a restricted spatial frequency regime.
While each radiation mode individually is non-vanishing at infinity, the superposition of
all radiation modes satisfies the radiation condition there. This superposition of radiation
modes (or continuous spectrum) forms the spatial radiation field. This spatial radiation
field, with non-vanishing transverse power flow, then models the loss due to radiation
effects from the waveguiding structure.

While the continuous spectrum is an important component of the complete
propagation-mode spectrum for open-boundary waveguides, very little work has been
done in conceptualizing it save for the simplest of examples. Snyder has determined the
radiation spectrum for the uniformly-clad circular fiber [9]. The radiation spectrum
for the symmetric planar waveguide has been quantified by Rozzi [10], while that of

the asymmetric planar waveguide has been presented in Marcuse [11]. The previous

examples are actually two-dimensional problems and possess closed-form solutions; as

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examples are actually two—dimensional problems and possess closed-form solutions; as
a consequence, the regime of the propagation-mode spectrum corresponding to radiation
spectral components is obvious.

Recent work [12,13,14] has focused on characterizing the radiation spec-
trum of practical open-boundary waveguiding devices. Uniformly-clad structures of
arbitrary cross-section or structures of finite cross-sectional width in a multi-layered
surround cannot be analyzed with conventional differential operator techniques due to
inseparability of boundary conditions. An integral equation, or integral-operator
formulation, is instead used to analyze practical open-boundary problems; the many
versions [15,16,17 ,18] of these integral-operator formulations share one com-
mon characteristic — for practical problems, solutions must be developed numerically.
Bound-mode determination is relatively straightforward using the integral-operator
technique; unfortunately, the proper regime for the radiation spectrum is not immediately
obvious with the integral-operator formulation. This is a major reason why the
characterization of the radiation spectrum for these devices is non-existent.

Another reason why research into the continuous spectrum has been ignored is
that leaky-wave modes provide a readily available approximation to the phenomena that
the continuous radiation modes model. A leaky-wave mode [19] is a discrete, non-
confined mode whose field distribution approaching infinity appears to increase
exponentially instead of vanishing. The leaky-wave solution is obviously not part of the
proper spectrum, and is denoted an improper or non-spectral mode. Leaky wave modes
become useful when constructing the total field by the method of steepest descents, which

is not a modal decomposition but an asymptotic technique to determine the scattered

field. After a suitable transformation of the propagation constant to the ”steepest-descent
plane' [20], the total field is constructed as the 'steepest-descents' contribution
evaluated at a "saddle-point", augmented where necessary by a number of other
contributions [21]. The most significant of these other contributions is the leaky-
wave mode. The leaky-wave mode contributes to the total field over a limited spatial
regime; within this regime, the leaky wave accounts for power loss from the waveguiding
structure. Consequently, much interest in the community is devoted to leaky-wave
modes [22,23,24,25,26,27,28]. This is where the majority of research
with open-boundary waveguides is being conducted.

The primary focus of this dissertation is on the characterization of the continuous
radiation spectrum for open-boundary waveguides. It investigates those open-boundary
waveguides operating in a planar, layered background environment, also known as a
layered surround. The basic guiding structure for MMIC’s (Millimeter/Microwave
Integrated Circuit) is the strip transmission line (also known as microstrip), where a
conducting strip serves as the waveguiding mechanism. This is depicted in Figure 1.1a.
As the operational frequency increases, conductor loss becomes prohibitive. Replacing
the strip with a dielectric guiding region forms an integrated dielectric waveguide
(IDWG), as depicted in Figure 1.1b. Doing so can reduce losses dramatically, allowing
usage at the intermediate microwave! optical frequencies. Most microstrip transmission
lines are fabricated in a conductor/film/cover environment, while a typical IDWG is
formed in a tri-layered substrate/film/cover environment.

Both waveguide types, integrated dielectric (IDWG) and microstrip (MS), are

assumed to be invariant and of infinite extent along the waveguiding axis. The

 

 

 

 

   

 

igigigggiPerfect electric conductor\

(a)

waveguide
n2 (1’ )e,J

       
 

cover (ec

substrate (e,)

(b)

Figure 1.1 (a) Typical microstrip transmission line. (b) Typical integrated dielectric
waveguide.

backg
direc:

tutti

 

 

background region planar layers are assumed to be infinite in extent in the planar
directions; practically, the transverse dimension of theiplanar layers is much greater than
that of the waveguiding structure. Each of the background layers is uniform with respect
to permittivity and permeability, but can possess dielectric and magnetic contrast from
layer to layer. All the interior layers are of finite thickness; the two outermost layers are
of semi-infinite vertical extent. It is apparent that the multi-layered background
environment in the absence of the guiding region is itself an open-boundary, planar
dielectric waveguide, consequently supporting its own guided wave modes and radiation
spectrum.

The second chapter develops an integral-operator formulation used to analyze a
general category of open-boundary waveguides. Based on the technique developed by
Bagby and Nyquist [16,29], an electric field integral equation (EFIE) is formulated in
terms of Hertzian potentials based upon equivalent sources that replace the open-
boundary waveguide of interest. A dyadic Green’s function for the Hertzian potentials
due to a current source radiating in a layered background environment is developed,
whose scalar components are two—dimensional spectral integrals on axial and transverse
spatial frequencies (also denoted as axial and transverse wavenumbers). These are
identified as belonging to the class of Sommerfeld integrals, highly oscillatory and
possessing singularities depending upon both spatial frequencies. No approximations are
made, thus rendering the dyadic Green’s function exact. The axial invariance of the
waveguiding system is exploited to reduce the dimensionality of the Green’s functions;
this allows an axial transform-domain EFIE to be developed. This transform-domain

EFIE will be used throughout the dissertation. Finally, another form of the axial

transform-domain EFIE is developed for the specific case of integrated dielectric
waveguides in terms of the transverse field components only.

In the third chapter, the complete propagation-mode spectrum of an open-
boundary waveguide is determined. First, a rationale to locate the branch cut
singularities within the complex axial-transform plane is advanced, based upon
observation of a subtle and usually ignored consequence of utilizing Fourier transforms
for analysis. This is new, and a major contribution of this dissertation. This rationale
is then applied to locate the desired singularities; a consequence of locating the
singularities in the axial transform domain is that all the singularities associated with the
dyadic Green’s function are located as well. The propagation-mode spectrum for the
open-boundary waveguide is then identified. The spectral components are found to be
associated with the axial-transform domain singularities; the nature of each component
of the propagation-mode spectrum is consequently discussed. A new component for the
continuous radiation spectrum, a surface-wave radiation regime, arises from the presence
of the integrated background environment. Lastly, the proper use of leaky-wave modes
is addressed. Their relationship to the propagation-mode spectrum, in particular the
radiation spectrum, will be discussed.

In the fourth chapter, the techniques developed in Chapter 3 will be applied to
canonical planar structures to verify their validity and gain insights into their application.
The transverse-field EFIE, developed at the end of Chapter 2, will be used to determine
the spectral field components. A numerical method-of-moments solution will be

implemented, and compared to the known canonical results.

 

1131151?
contir.‘
by G.
tune:
states

ruese

differ
“are
wflll
wfll

Resu
fions
dyad
denc

”133 7

phiJ

 

The fifth chapter determines the continuous spectrum of the microstrip
transmission line. This is a basic, practical structure, for which no results regarding the
continuous radiation spectrum have been published. The appropriate EFIE will be solved
by Galerkin’s method-of-moments, using basis functions with the well-known edge
current singularity built-in. Appropriate source excitation is chosen to exploit the parity
states for microstrip surface current distributions. Surface current distributions will be
presented and discussed for each different regime of the radiation spectrum.

As a final example, the sixth chapter sees the theory applied to determining the
different types of discrete modes, both bound and leaky-wave, for a rib dielectric
waveguide, a common structure in integrated optics. The axial-transform domain EFIE
will be solved using method-of-moments techniques. Parity states for the rib waveguide
will be taken into consideration to avoid the near-degeneracy of the bound modes.
Results from this work will be presented and discussed.

Chapter 7 presents the conclusions of this dissertation, and provides recommenda-
tions as to the future of this research.

Throughout the dissertation, the following notational forms will be observed. All
dyadics will be overstruck by a double-headed arrow, while vectors will be overstruck
by an arrow. Also, with respect to any complex quantity 2, the following holds: 2’, 2,
denotes the real part of z, Zr denotes the imaginary part of 2, while z” denotes the
magnitude of 2,. Consequently, 2, may have any sign, but 1” > 0. The following

physical assumptions also hold throughout this dissertation:

Undc

Cont

can i

 

 

(1) All media are linear and isotropic;

(2) Inhomogeneities in conductance o(i') and permittivity 2(7) are confined
to loealized regions, i.e., the guiding regions;

(3) Harmonic time dependence 81'” is assumed and suppressed.

Under these assumptions, Maxwell’s equations take the form

VxEG') = -jmul7I(i') Faraday’slaw

Vxfitn = jozmém +36) Ampere-Maxwell Law (1.1)
V-(Etném) = pm Gauss'staw

V°H(i’) = 0 Magnetic Source Law

Conduction current density is 7C6“) =- o(‘r’)E('r‘); consequently, a complex permittivity

can be defined in the usual manner, e(i') = 2(1’) + o(i')/jar.

Comi

 

the b
deve
Way
and

[30'

me

all

Chapter 2

Electric-Field Integral Equation Description of Open-Boundary
' Waveguiding Systems

As observed in Chapter 1, practical open-boundary waveguides defy analysis by
conventional differential operator techniques, primarily because of the inseparability of
the boundary conditions imposed by the waveguide structure. This chapter describes the
development of an integral-equation approach to the analysis of open-boundary
waveguides. The work contained within this chapter has been advanced by other workers
and is included for completeness. The notable contributions are by Johnson and Nyquist
[30], which advances the usage of the polarization integral equation; Bagby, Nyquist
and Drachman [16,29] which develops the approach for integrated background environ-
ments; and Viola and Nyquist [31], which clarifies source-point singularity consider-
ations and advances a reduced-component integral equation for optical waveguides [32].

The key to this development is replacing the guiding region with equivalent
sources. First, equivalent sources will be identified, and an electric field integral
equation (EFIE) for arbitrary open-boundary devices, dielectric or microstrip, will be
developed in terms of those sources. Secondly, the necessary Green’s dyadic for
electric-type Hertzian potentials supported by sources radiating in a planarly-layered
background environment will be derived. This dyadic contains all the physical

information about the background and makes no approximations; as a consequence, the

10

EFIE

 

lliir

in axi

axial

2.1

 

micr
field
the t

dielq

 

CUE
\Var

Ciel

EFIE is exact. The corresponding electric-field Green’s dyadic will also be developed.
Third, specific characteristics of waveguiding structures, in particular the infinite extent
in axial direction, will be exploited to develop a computationally-simpler two-dimensional
axial-transform-domain EFIE. Finally, a transverse-field EFIE for integrated dielectric

waveguides will be developed from the two-dimensional axially-transformed EFIE.

2.1 Development of an Electric-Field Integral Equation

Both the integrated dielectric waveguide (IDWG) system in Figure 2.1a and the
microstrip waveguide system in Figure 2.1b are open-boundary systems; that is, the
fields are not confined strictly to the guiding regions. For problems in this dissertation,
the open-boundary guiding structure is embedded within the cover layer of a tri-layered
dielectric background environment, typically at the cover-film interface. The planar
layers (cover, film or substrate) are homogeneous and uniform with permittivity of
e, = nfe, and free-space permeability n, = no, and of infinite extent in axial and
transverse directions. The film layer is of finite thickness in vertical extent, while both
the cover and substrate layers are of semi-infinite vertical extent. The substrate layer
becomes a perfect conductor for microstrip transmission-line problems. A coordinate
system is chosen such that the x and z axes are tangential to the planar interfaces, the y-
axis is normal to those interfaces, and the z-axis is specifically the guiding axis.

Consider the system depicted in Figure 2.1. Assume that a system of impressed
current densities I'Cr’) supports an electric field incident upon an open-boundary
waveguide of arbitrary cross-section, permittivity and conductance. This impressed

electric field E56) is the field that would exist in the layered-background environment

11

Ti?) waveguide
n2(?)eo - - a E“(?)

  
 

 

cover, (.g)‘

 

 

 

 

substrate (6.)

(a)

microstrip line >

 

 

 

(b)

Figure 2.1 (a) Typical configuration for integrated dielectric waveguide.

(b) Typical configuration for microstrip transmission line.

12

with Li
wavegr
The sur
system.

conditit

2.1.1 1'

Sources
(more g

be rep];
any P01;
Where ]
inhOmO

“mom

in (2.1

This is

 

with the waveguide absent. The impressed electric field induces currents within the
waveguide; those induced currents in turn support a scattered or re-radiated field E'G') .
The sum of the impressed and scattered fields yields the total field anywhere within the
system. This total field E6) - Eifi") +E'(i’) must satisfy the appropriate boundary

conditions for the open-boundary waveguide in question.

2.1. 1 Equivalent currents

Field equivalence principles state that an object can be replaced by its equivalent
sources. If equivalent currents can be determined for an open-boundary waveguide
(more generally, any open-boundary device, of which waveguides are a subset), it can
be replaced by those equivalent currents.

Equivalent currents can be identified by considering Ampere-Maxwell’s law at

any point within the layer in which the open-boundary device is embedded, namely,

Vxfim = i‘tmjwetnfitn (2°11
where 3° is the impressed electric current density and e is the complex permittivity. Any
inhomogeneities are localized and correspond to the open-boundary devices. The

uniform background layer with complex permittivity 6, can be explicitly accounted for

in (2.1), giving
VxH = I“ +jw[e(i’) -e,]E +jwelE.
This is easily recast into

VxH = 'r'°+ joeoan,’(r)fi+ 1'qu

13

where

definiti

from ti

equival

the equ

through
assume

Sources

0f Volur

(2.4) is

devi Ce ,

 

rddialir‘.

2.1.2

 

 

the He

 

where 6n,2 (1") - n26) -n,2 is the dielectric contrast (complex-valued). Based on its
definition, the contrast factor is obviously non-zero only inside the open-boundary device;

from this, equivalent polarization sources can be recognized as
1347) = eobn,2(i")E(i’). (2.2)

Inherent in the equivalent source definition (2.2) is the mechanism to define

equivalent sources for any of the open-boundary waveguides of interest. For IDWGs,

the equivalent sources are exactly as developed in equation (2.2), or
13.4?) = eoanfmétr) = corn’m-nhfitr) (23)
throughout the volume of the IDWG. For microstrip devices, the device conductivity is

assumed to be infinite, hence the internal field vanishes. The appropriate equivalent

sources are then surface currents, denoted as KG) , which are the surface specializations

of volume equivalent polarization currents, where

3,0,6) = jur‘rqm. (2.4)
The net effect of developing equivalent sources as given in relations (2.3) and

(2.4) is to remove the inseparable boundary conditions imposed by the open-boundary
device, leaving the much simpler problem of determining the fields supported by sources

radiating into a multi-layered planar environment.

2.1.2 Fields within layered media
The planar layered environment problem has much simpler boundary conditions;
differential operator techniques can be applied to obtain an exact solution. The electric-

type Hertzian vector potential (or Hertzian potential) will be introduced as an auxiliary

l4

potenti;

polarlz:

electric

where 1
current
bounda
in [emit

then be

0fthe1

Where

CffeCtS

be f0m

 

potential. This choice is prompted by historical considerations, as these potentials were
used by Sommerfeld [33] in his classic analysis of a Hertzian dipole over a lossy
earth. Additionally, the electric-type Hertzian vector potential is supported directly by
polarization currents which, as demonstrated in (2.3), have a simple relationship to the

electric field.

As developed in Appendix A, the Hertzian potential satisfies Helmholtz’s equation

'11:

v1fl+k=fi = ’ (2.5)

m |

where the polarization source current in (2.5) is the sum of the impressed polarization
current (5' ar/ju) and the equivalent polarization currents induced in the open-
boundary devices of interest. The sources in equation (2.5) can be alternatively written

in terms of electric current densities, as § = 3/j0 . The electric and magnetic fields can

then be represented in terms of the Hertzian potential as

E = k’fi +vv-1'i
it = jwerfi

(2.6)

The solution to Helmholtz’s equation in (2.5) must satisfy the boundary conditions

of the layered background environment. It can be written in the general form
fim = % [mm-me (2.7)
v

where G(i’ Ii”) is a Hertzian potential dyadic green’s function which accounts for all the

effects associated with the layered background environment. The total electric field can

be found as

15

2.1.3

SUppon

the tom
10 date

We)?“

Once l
kHOWE

 

 

 

deter”

f0r die

 

 

Em = (k2+vv->fc(r|r)-‘P (n+f’,,(r’)1 w (2.3)
Y E

for sources within the cover region. As defined previously, the impressed electric field
is supported by impressed polarization currents 13°, while the scattered electric field is

supported by the induced polarization currents is”. The total field is then constructed

as Em =- E‘m +317), where

E‘m = (k1+vv-)fé(r|r)-i5°—m dV’
y E
i3 (f) (2.9)
film a (k1 +vv-)fé(r|r’) 4:— dV’.
V

2.1.3 Construction of the integral equation

For the integrated dielectric waveguide, the total field as given in (2.9) is partially
supported by the equivalent sources in the guiding region. These sources depend upon
the total electric field within that region, which is unknown. Thus, an integral equation

to determine the unknown guiding region electric field is constructed by enforcing (2.8)

everywhere within the waveguiding region, or

dV’ = fi‘m; Vi'eV. (2.10)

 

bn2(i")fi('r")
1

n

it?) -(k3 + W-)f Gal?)-
V

c

Once the guiding region electric field is known, the equivalent polarization currents are
known as well, and the electric and magnetic fields elsewhere is space can then be
determined. Even though constructed for waveguide analysis, the EFIE in (2.10) is valid

for dielectric devices occupying arbitrary volume regions.

16

ofther

conditi

where
densitic
device,

the ele:

differer
The E1

integra]

Operatc

([24])

 

For microstrip transmission-line systems, the boundary condition upon the surface
of the microstrip device must be satisfied, viz. , t- (Bi +E') = 0. Enforcing this boundary

condition upon the total field as given in (2.9) results in

{11:3 + VV-)f CURB-Eds" = -t°fih°(i"); WES (2.11)

s Jwec
where electric current densities have been used, as opposed to polarization current
densities. Integral equation (2.11) is valid over the entire surface of the microstrip
device, and can be solved for the unknown surface currents 12(1) . Once RU) is known,
the electric and magnetic fields elsewhere in space can be determined.

The integral equations developed in (2.10) and (2.11) are technically integro-
differential equations, yet will be referred to as electric field integral equations (EFIE’s).
The EFIE for microstrip transmission line in (2.11) is an inhomogeneous Fredholm
integral equation of the first kind; the EFIE for integrated dielectric waveguides in (2.10)
is an inhomogeneous Fredholm integral equation of the second kind. Additionally, both
EFIE’s (2.10) and (2.11) are of the Wiener-Hopf type, in which the range is semi-infinite
while the domain is finite.

A true integral-equation representation can be developed by passing theUcc2 4» VV°)

operator through the integration over the source region. In doing so, care must be taken

to exclude the source-point, as the integrable singularity there will become non-integrable

([2.4]). As a consequence, integral equation (2.10) becomes

 

2(7) - f G‘am-“zazamdv’ = E‘a); vreV, (2.12)
V

"c

17

with si

functior

where r
exclud'n
sources

detail ir

den‘ce
mpreser
Conditic

(2.11”

H)

Ollly m

in this (

of Whit
I“ that

lOta] Sc

 

etiUIVa}

 

melts)

13m is rl

with similar results for integral equation (2.11). The electric-field dyadic Green’s
function is defined as
G‘mr’) .. P.v.{(k} + vv.)G(r|r')} +£a(r-r') (2-13)
where P. V. implies that integration of (2.12) is in the Cauchy principal value sense,
excluding the source point, while f. is a depolarizing dyad that corrects for any artificial
sources introduced by the exclusion of the source point [34]. This is given more
detail in Section 2.2.5.
If the microstrip device has finite conductivity, an electric field internal to the
device exists, and can be simply modeled as I?“ =Z‘K(‘r’), assuming a simple
representation for the surface impedance Z ‘ can be found. The appropriate boundary

condition is t-(E‘d?) = £13.“ on the microstrip surface, and the integral equation in

(2.11) becomes
*2 /R(7’)I“r “inc.
((1, + vv-)f man-Was - t-[Z Km) = +13. (f). vres. (2.14)
3’ °

Only microstrip transmission-lines with infinite conductivity (Z ‘ = 0) will be considered
in this dissertation, but (2.14) is included for completeness.

This technique can be extended to include many open-boundary devices, not all
of which need be waveguides, or even a mixture of dielectric and microstrip devices.
In that case, each open-boundary device is represented by an equivalent source. The
total scattered field is a superposition of the scattered fields supported by each individual
equivalent source; the resulting EFIE is enforced with each such individual device. This
extension, of value for coupled-device problems, is not of interest for this dissertation,

but is mentioned for completeness.

l8

242]

bang
SUPPC
unbor
metu
theso
inmrfi

pctent

Whme

Vflanz

aIEthe

 

2.2 Development of the Hertzian Potential Dyadic Green’s Function

Consider the tri-layered situation in Figure 2.2, with the source in the cover layer
being either equivalent or impressed currents, denoted generally as I . These currents
support a principal Hertzian potential radiating outward from the source in apparently
unbounded space. This potential propagates until it encounters a planar boundary, where
the wave is partially transmitted and partially reflected. The total Hertzian potential in
the source layer is the sum of that principal potential plus the scattered potential from the
interface. For other layers, the Hertzian potential is that portion of the principal

potential transmitted through the planar interface boundary.

As a consequence of the above analysis, Helmholtz’s equation can be written as
11' -3 . --
2 + l 8 ”we, 1-6 (2.15)
W "h{ n; 0 all z.

where fif is a principal Hertzian potential directly supported by current sources 3 (or
polarization current sources since i" = ilju) radiating in an unbounded medium, andfi:
are the scattered Hertzian potentials from the planar interfaces indirectly supported by
the sources in the 1'" (cover) layer. The total potential in any layer is the superposition
of principal and scattered Hertzian potentials in that layer; this total potential must satisfy

the boundary conditions on Hertzian potentials developed in Appendix A. A solution for

the total potential of the form (2.7) is sought, namely,
rim = —1—fé(r|r')-'J’(r')dv’.
1006 y

This section addresses the determination of the desired dyadic Green’s function.

19

 

 

F1311 n

 

 

 

 

Figure 2.2 Hertzian potentials for a tri-layered environment.

20

2.2.1

conditit
Recogr
directic

and z.

where i
be wn't-

Product

 

 

 

2.2.1 Two-dimensional Fourier transform

Solving the Helmholtz equations as given in (2.15) subject to the boundary
conditions developed in Appendix A is still difficult, but an exact solution does exist.
Recognition that the interface is infinite in extent in the transverse (x) and axial (z)
directions prompts a two-dimensional Fourier transform on those spatial coordinates x

and z. This transform pair is given as

 

11(1’) =- 1 . .fI(X;y)e""dzl
OWL!“ (2.16)

flag) a f;f;fi(i')e"x"d2r

where X - ii + C 2 is a two-dimensional spectral frequency. This transform pair can also
be written as an iterated set of one—dimensional transforms; this is apparent if the dot

product in (2.16) is explicitly carried out, and takes the form

fie) = fiqfif;fi(z.c;y)e"‘de]emdc
mam) = f_:[f;fi(r)e'/‘*dx]e-szz

The inner, bracketed quantities form a transform pair on x and transverse spatial

frequency £; while the outer transform pair is obviously on 2 and axial spatial frequency

(1 Application of this transform pair to the Helmholtz equations in (2.15) yields

3: _ p2 {fi'(i;y)} = {MED/106)} (2.17)
ay2 ‘ fi‘d’m 0 ’

the transform-domain Helmholtz equations. The quantity p, is a transformed wave-

number parameter, and given as
pi = [A2_ki2 = lg2+c2_ki2 . (2.18)

21

 

2.2.2

Where I

 

 

The square root in (2.18) will be chosen to enforce 81cm} > 0. Usage of transform
pair (2.16) converts partial differential equations in the space—domain to ordinary
differential equations in the spatial-frequency domain which possess relatively simple
solutions, while retaining the spatial y-dependence necessary for implementing boundary

conditions.

2.2.2 Principal Hertzian potentials

As defined, the principal Hertzian potential 11: is the potential directly supported
by polarization currents radiating in unbounded space, in this case the cover medium,
obeying the forced space-domain Helmholtz equation in (2.15). For unbounded space,
the solution of the forced Helmholtz equation is well-known; consequently, the solution

for 11: is simply
11%) = f G'(‘r’|‘r”):(Ti:2dV’, (2.19)
V c

where the principal Green’s function G’(i" | F’) is the usual free-space Green’s function

yr, |r-r’|
ortrlr’) = _‘___., (2.20)
4n If-f' I

Unfortunately, the spatial form given in (2.20) is not terribly useful in waveguiding
problems, nor for matching the boundary conditions for the total potential at a planar
interface.

A more tractable form of (2.20) can be developed by first solving for the

transform-domain principal Green’s function G” ( X; y - y’) , then constructingG’(i"| F’)

22

inthesp

inthetw

the solu

Where

inverse

function

BY unirl

fUncfiO

,

 

in the space-domain via the inverse transform as defined in (2.16). The Green’s function

in the two-dimensional Fourier transform domain solves the equation

31 ..
— -p.’ G'Utzyly’) = —a(y—y).
ay2

the solution of which is

é’(x;y—y’) zffl. (2.21)
2pc

where 8412c} >0 is required to satisfy the radiation condition. Application of the

inverse transform to (2.21) then recovers the desired space-domain principal Green’s

function in a spectral representation of

 

gran?) a ff__ 9b." e171: G-f’)d2;_ (2.22)

(2110’-..

By uniqueness, the spatial form (2.20) and spectral form (2.22) of the free-space Green’s

function are equivalent; this relationship is the Weyl identity [35], i.e.,

“-11 l'-"’l [I (r- 1") ml! 2 ’l
=ff ‘ ‘ d2). .
2(21r)2

41t|r- r’l _.

 

It is obvious from (2.20) that a singularity occurs at the source point, when
1' = i”. This singularity is contained in (2.22) as well, albeit not obviously. As f-oi",
the spectral form is slowly convergent. The obvious exponential decay in the integrand
due to exp( Iy-y’ I) vanishes, as do the destructive oscillations of the complex

exponentials as x-ox’, z~z’. At ‘1' =1”, the integrand takes the asymptotic form

d2).
2(21_n)2 [IT '

 

23

 

 

this 5e

nature

Open-)
w11111:
some:

which is obviously divergent. The singularity of order 0( If-i" I") in (2.20) thus

manifests itself as a divergent integral for (2.22).

2.2.3 Scattered Hertzian potentials
The scattered Hertzian potentials obey the homogeneous Helmholtz equation in

each planar layer. Solving the homogenous transform-domain Helmholtz equation (2.17)

is not difficult; a general solution in the i“ layer takes the form
fiId’J) = W,’(X)e""+w;(i)e"", (2.23)
and is easily recognized as a plane-wave type solution traveling in the :y direction. The

W: are vector constants that satisfy the Hertzian potential boundary conditions developed

in Appendix A. The space—domain scattered Hertzian potential follows naturally via the

inverse transform of (2.16),

trim = fnf[v's‘/,‘(X)e"’0+1)“v';(it’)e""’](‘71:):2 an. (2.24)

Determination of the vector constants W: appropriate for the layered-background
surround is a rather tedious process. The complete details can be found in Appendix B;
this section summarizes the major considerations, in order to provide insight as to the
nature of W: .

Designate the cover, film and substrate as regions 1, 2, and 3 respectively. For
open-boundary waveguides of interest in this dissertation, the guiding region is entirely
within the cover; consequently, only the total potential in the cover is of interest. Any

sources are assumed to be entirely in the cover region as well. This is the case depicted

in Figure 2.2. In the transform domain, the scattered Hertzian potentials are

24

travelins
by the 1

(assume

 

 

the sup

form 01
fit?)

Here,1
is be
brac

50m

YE:

5L

131(1)) - W16)?"
11:6.» = with” + WWW” (2.25)

fad.» = wane”
The cover (region 1) is unbounded as y» +00; consequently, only outward
traveling potentials W;(X)e"" in + y exist in region 1 (Rain) > 0 required), supported
by the reflection of the principal potential off the interface between regions 1 and 2

(assumed to be y =0). The total space-domain Hertzian potential in region 1 is then
fi,(i’) - fif(r)+fi’l(7), (2.26)

the superposition of the principal and scattered Hertzian potentials. By using the spectral

form of the principal Green’s function, the total space domain Hertzian potential becomes

f ] e-yI-r’evttm’
y iwel 2.010.)

I

(W + W{(X)e-p‘(m} d2}. 037)

 

 

 

l . x. ,(i),
fll""a§f.f" [

r.(l)|y-r'l

- i ’ . 1.
P“ )’e P|( )’ If

Here, the relation e y’ > y is exploited, since the interface

is below any source currents in the cover (if y’ < y, 3'00"",I = e" 1(1): [e 'P ((1)) ). The
bracketed quantity in (2.26) is denoted as 9(1”), and depends upon the location of the
source currents.

The total potential in other regions is just the scattered Hertzian potential in those
regions. The film layer (region 2) is bounded in y; the scattered potential therein is the
sum of a transmitted potential from region 1, W;(X)e°”’ , traveling in the -y direction,

and a reflected potential off the lower boundary, WKX)!” , traveling in the +y

direction. The substrate (region 3) is unbounded as y- -°°; it sees only a transmitted

25

potent

potent

applic
tediou
envirc
condit
Comp(
POtent

Obsen

 

2.2.4

 

 

“We:

 

potential, W;(i)e"” , from region 2 traveling in the -y direction. The space-domain

potentials in regions 2 and 3 are then

 

M?) = (2—1)7f]"“[ W;(X)e’="”+fi',’(i)e 1'1"” ] (12)., (2.28)
1! ..
fifi) = (2 f je1"'[ W3'(X)e”“” ] (121. (2.29)

The boundary conditions on Hertzian potentials are given by (A.l4); their
application to the potentials of (2.27) - (2.29) at each interface is straightforward though
tedious; details are given in Appendix B for a number of different background
environments. In brief, inspection of (A.l4) reveals that, at a planar interface, the
conditions for continuity of tangential Hertzian potentials (a =x,z) involve only tangential
components; while the conditions matching the normal (9) component of Hertzian

potential couples the normal and tangential Hertzian potential components. Based on this

observation, it is clear that the scatted potential in the cover takes on the form

Wei! 1“
=ffR,(X)V(r’)‘(2: d1; a=x,z
eh’e )l-Xm'
,sff 1241’) V (r)—— 232) 41 (2.30)
a + _. e-‘h’eflr 2
+(ax2 —2) [foam 5(21: d1.

2.2.4 Dyadic Green ’3 function for Hertzian potentials
With the recognition that V( 7/) involves the sources, the resulting potential in the

cover region can now be written as

26

where tl

The prir

while

These;

 

 

fix?) = . 1

just

fétrlr’)-J‘(r’)dV’ (2.31)
V

where the Green’s dyad can be decomposed as
6(2)?) -= é'(r|r’) + at; r'). (2.32)
The principal Green’s dyadic is
6P(FlF/) = iGp(Fli:/) (2.33)
while the reflected Green’s dyadic takes the form

r

6'01"”) = 3 6:! + i [:‘t + 6:5) +

(2.34)

 

 

30! . r
62 z + 2 G, 2.
The scalar components of the Green’s dyadic are two-dimensional inverse transforms on

spectral frequency 5: , and are given as

G’(?|i”) = ff 2m):

11 (14’) 1'. ly-r’l
‘ ‘ .411 (2.35)

 

30"” - R11) . - .
:(r‘lr’r = ff .(2) ‘flm‘zw ”.421. 0-30
:(rlr') "' CW 2‘2") ‘

 

 

These scalar components of the dyadic Green’s function are of the form of the notorious
Sommerfeld integrals, and possess the rapid oscillation for large spatial distances that
typify these integrals. The coefficients R‘().), R42.) and C0.) are reflection and
coupling coefficients specific to a given layered background surround. They are
functions of the magnitude of spectral frequency X (A = W) as implicated through
the wavenumber parameter p, given in (2. 18), and possess pole singularities correspond-

ing to the surface-wave behavior of that background structure, which itself is a planar,

27

waves
the 1)
respor
within
relates
maintz
influer
Finall;

nOrrna

2. 2.5

develc

deVeIc

 

where

 

dyad)
the el I

 

open-boundary waveguide. Appendix B lists the forms for a number of typical
background environments.

This dyadic Green’s function can be interpreted as the superposition of plane
waves propagating transversely with spectral frequency A (. W) and normally in
the :y directions with propagation constant pt. The dyadic Green’s function is the total
response in the cover region to a unit dyadic point source current 166-1") radiating
within the planar layered background surround. The reflected Green’s componentG,’
relates the influence of the background structure on the tangential Hertzian potential
maintained by a tangential point source. Likewise, G: relates the background structure’s
influence upon the normal Hertzian potential maintained by a normal point source.
Finally, G" relates the background-induced coupling of tangential point sources to

normal components of Hertzian potential.

2. 2.5 Electric-field dyadic Green '3 fimction
Even though the dyadic Green’s function for the Hertzian potential has been
developed, the electric field in the cover region is needed to satisfy the EFIE’s as

developed in (2.10)-(2.11). Passage of the operator (k: +VV°) through the source-point

integration is desirable, to obtain an electric field representation of the form

 

it?) - ,1 [G‘(r|r’)-i(r’)dv’ (2.37)
V

1‘“.

where G'G’IF’) is the electric-field dyadic Green’s function, also called the electric
dyadic Green’s function. This representation in (2.37) provides a compact notation for

the electric field amenable to algebraic manipulations. Passage of the spatial derivatives

28

through
A very (

key poir

quires t
reflecte
not. Tn
arising
can be

incorpc

The L)
inVole?

SOUFCe

Where

 

 

through the source-point integral requires special care, in particular the VV° operator.
A very comprehensive discussion of this procedure is carried out by Viola [31,36]; the
key points are reviewed here.

Passage of the operator VV- on field points through the source-point integral re-
quires that the integrand be uniformly convergent. The scalar components of the
reflected dyadic Green’s function possess this property; the principal component does
not. The presence of the absolute value function ly - y’ I gives rise to a singularity aty = y ’
arising from derivatives with respect to the normal coordinate variable y. This situation
ean be handled by defining the spatial integral in a principal value (P. V.) sense, and
incorporating an appropriate correction term [37].

The electric field in the cover region can be rewritten as
facet-K?) = (1:3 +W°)fG'°](F’)dV’
V
+ kffiorflrodv’ (2.38)
V
+ W-fiG'-](r’) dV’.
V

The third term in (2.38) demands careful attention. It can be properly evaluated by
invoking Leibnitz’s rule, and excluding a shape-dependent principal volume about the

source point, ie.
W-fG'-.7(F’)dV’ = RV. fW-é'io’mv' + EMF-f”)
V V

where P. V. designates evaluating the integration in a principal value sense, that is

29

whe:
dyad

113111)

2.3

fields
inclm
by a l
typica
uSnail
SUgge:
domal‘

Spatial

 

P.V.f{---]dV’ = 133301144“
V

v-v,
where V. is the shape-dependent excluded principal volume and l: is a depolarizing
dyad. For the planar layered background environment, a slice principal volume is

naturally assumed [2.4]; for this principal volume, 1. = 9?.

2.3 Axial-Transform Domain Electric-Field Integral Equation

The EFIE’s developed in (2.10)-(2.11) are in terms of three-dimensional spatial
fields, and are of use for any arbitrary-shaped obstacle. The dyadic Green’s function
includes the effect of the environment upon the obstacle; for spatial fields, it is defrned
by a two—dimensional inverse transform over the spectral frequency X = £5 +£C . The
typical waveguiding system of interest, microstrip or integrated dielectric waveguide, is
usually axially uniform and of infinite extent along the guiding axis. This symmetry
suggests that advantages might be obtained by solving the EFIE in the axial-transform
domain, and then recovering the spatial fields via an inverse transform on the axial
spatial frequency I.

Consider the situation depicted in Figure 2.3. Let CS be the cross-section of the
guiding region of an integrated dielectric waveguide. Then the volume integral for the
waveguide is an integral over the guide cross—section and the entire guiding axis
(-~ < z < 0a). Inspection of the dyadic Green’s function as given by (2.32)-(2.36) shows
that G(f|f’) = GUS I5’;z -z’). Taking explicit account of these observations in EFIE

(2. 10) gives

30

 

substrate (6‘)

x-O

Figure 2.3 Configuration for axial-transform domain analysis of an integrated
dielectric waveguide.

31

ml

The EFL

the guic

Applic."

Where

mar

 

thfborei'
t1’21!)st
(2.11 )

dOman

\Vhere

"3:1ng

 

 

- 2
2(7) -(kf + vv-)f f G(7-7’)-6"‘(?Zfimdz’ ds’ = E“(7); WEV.
a ..

"i
The EFIE possesses an easily recognized convolutional kernel over the infinite extent of
the guiding axis, suggesting an application of a Fourier transform on axial variable 7..
The axial transform pair is defined as

_ 1 ' -(
“TI-em “€de (2 39)

303.0 = f Renewal:
Application of the Fourier transform defined in (2.39) to EFIE (2.10) results in

6n, 2(1),) 3(5’ :C)

2
"r

(19’ = €"’°(5;C) (2-40)

 

awn) -(k.’ + W) I 2(b’lfi’x)
a!

where the lowercase quantities are the axially-transformed versions of their respective
spatial counterparts and V = V, +jCz‘. The Fourier convolution theorem (Faltung
theorem) allows the convolutional kernel on 2 to be replaced by the product of the
transformed Green’s dyad and the axial-transform-domain field. A similar analysis upon
(2.11) reveals that the EFIE for microstrip transmission lines in the axial-transform

domain is
A I- - -‘ dl‘ ‘
pa: + vv.)f gush-5’; )._k€_p’_c).d[’ = -t'é’m‘(i5;() (2.41)
c 1005,:

where C is the cross-sectional contour bounding the surface of the microstrip

transmission line.

The dyadic Green’s function in the axial-transform domain becomes

32

where

and 1114

The G
dimer]
backg

flmcti

2,4 ]

deScr

by (2

 

r5810

 

§(b‘|b".c) - §'(5|5’.C) + §'(b'lfi.() 0-42)

 

 

 

where
e'tb‘lm) = 78’(5li5 .C) (2.43)
§'(i5|5’.€) = 2 eff + y [8:2 + g"; + 1:3": + 2 g" ,3 (2-44)
and the scalar components are
' Ittx-x’) -r.lr-r’|
s'(b’|b‘:<) = f e ‘ a: (2.45)

‘1an

8:'(5|5';C)‘ .. R31)
“(x-fl) 1.0»)
gltb’lb";C)> = ff not) ‘ ‘ at: (2.46)

§.'(§I5’;C)J "' C”)

A_

 

 

 

The Green’s function scalar components are still Sommerfeld integrals, but are now one-
dimensional inverse transforms on transverse spatial frequency 5. Note that the
background reflection and coupling coefficients R,, R," and C are the same functions of
A as mentioned previously; consequently, the Green’s function scalar components are still

functions of the axial spatial frequency 1'.

2.4 Development of a Transverse-Electric-Field Integral Equation for Integrated
Dielectric Waveguides

In the previous section, an axial-transform domain EFIE was developed to
describe the fields associated with integrated dielectric waveguides. This EFIE as given
by (2.40) utilizes all three electric-field components, 2 = flex We), hiez . In a source-free

region, however, only two of the three electric-field components are independent, since

33

the elect
considei

only tw

domain
exampl

by det

aquatic

 

 

the electric field must satisfy Gauss’s law, V-(nzeo E) = 0. An appropriate question to
consider is whether the EFIE developed in (2.40) can be recast into an EFIE utilizing
only two independent electric-field components?

There is well-established precedence for this consideration. In the axial-transform
domain, only two of six field components (eye), 9,15,15,15) are independent. For
example, any waveguiding problem in a homogeneous, source-free region can be solved

by determining the axial fields (:1, ('2) independently, then satisfying the "Magic

3' 1 . e: . ”h:
1it - (kl-C2) JCV‘ hz quixV, -eez °

One can just as easily proceed to work with electric fields transverse to the guiding axis

equations" ,

 

 

(8, - to, +99) as the independent field components. It is well known from closed-pipe
waveguide theory that knowledge of the transverse fields is sufficient to characterize all
field behavior for a closed-pipe waveguide in a source-free region. Likewise, the same
observation about transverse electric fields can be made for planar dielectric waveguides
and uniformly-clad dielectric waveguides like the circular dielectric rod. It is desirable
to recast EFIE (2.40) into a form involving only the transverse electric fields. The
obvious advantage of doing so is a reduction in the number of unknowns to be solved
for, especially with MOM techniques. The other advantage, which is not as readily
obvious, is a reduction in the order of the source-point singularity that occurs in the
electric dyadic Green’s function kernel. This section presents that development,
originally performed by Viola [2.5]. Within this development, explicit dependence on ii

and r is suppressed, unless necessary for clarity.

34

electric

08
ll
08

Solvin g

 

where.

00mg

it'-

 

The fundamental step in deriving the Transverse Electric Field Integral Equation
(TEFIE) is invoking Gauss’s law in the axial transform domain to determine the axial
electric field in terms of the transverse electric fields. The electric field decomposes as
E = ‘e’,+£e,, and considering that the operator V =- V,+j€i, Gauss’s Law becomes

2.0116,) +an1ez = 0.
Solving to find the axial electric field ez gives

jCez = -v,oe, - a -e (2.47)

I t
where, to simplify notation, the vector quantity 3. is defined as

V 2
d = _,n = V,1nn2. (2°48)
I "2
The next step is to remove the axial component from the EFIE. First, the axial
component of (2.40) is recognized by premultiplying the EFIE in (2.40) with the dyadic

£2 ', resulting in

2 -°I
29(5) = safe) + ikff ——°" (,9 )[£°§(5lb”)-8(5’)]dS’
c" "‘ (2.49)

2 -°l
+ airy-f ——5" (,9 ’stela’)-a(6’)ds’.
cs

"4:

Subtracting (2.49) from the original EFIE (2.40) and exploiting the decompositions
V, = V-ij and 'e‘, = E - fez gives

6n’( b")
2
C

 

[ms lb”) 42-203 Ib”)]°é’(5’)dv’
n 2 (2.50)
../ ~
+ V. f ”LgL’V-s(rlfi')-6(a’)ds'
CS

"c

35

byt

Wher
notat

trans

The
this j

 

 

Next, the dyadic reductions that occur within (2.50) are expanded and grouped

by transverse and axial electric field components to reveal that
3-6 - z-E. +[91‘Cs.'+i(g'+s.')]¢z
522'? = i(s'+s.')e, = is”.
3.5-2513 g 2 .5t + y 1583; (2.51)

V's-a = (V.+1'Ci) '{fi’e} +[91'Cs.'+£(8’*8s'>]‘z
(2.52)

V

as.
=V°”'3+
'3 ‘ [6y

 

+e' +s.'](iCe,) = V,°§°€.+s,,jCe,

where the terms 3 ”=g' +3: and 821‘ -3' +3: +—— have been defined to simplify

notation. Substitution of (2.51) and (2.52) into (2.50) and separating integrals on

transverse and axial electric field components gives

atp)=61(p)+k3f —‘——"3)§(plp’)-.a(6’)ds’

"c

-/
+ V. f filingttlr’mtr’m’
a ’2’“, (2.53)
+ kff Jéfllyg.'(filfi’)[ne,(a’)]ds’
cs

"c

 

1 -°I
+ V. f 5" (,p)8u(5|5’)[1'(e,(b”)]ds’ .
C3 '3

C

The axial electric field component ez occurs explicitly in the last two integrals of (2.52);

this is removed by substituting (2.47) into (2.52), thus yielding

36

an El

under

Where
)(0n

the 5

 

 

 

 

e(b‘) - e.(p) +23] —”——°“”i(pl5 ’)e.rs( 'm’

n:

+ V. f fiflilst-stald’rata’m’
‘3 6": (2.54)
- 9k}! " ‘9 ,’s.'(plrs’)[v: ctis’) +3 .(r’) e(a’)]ds’
(3

C

 

 

2
-V.f °” (3)8utfilp’)[‘71e..(fi’)+3(5’) 6.p( ’)]ats’

C
an EFIE for transverse electric fields only in terms of transverse electric fields.
In this formulation, the derivatives of 5, inside the last two integrals are
undesired. Removal of these derivatives (the VI-E, term) is effected by integration by

parts. Using the vector identity v-(oii) = vortex shows that

memoir) View) =
V. -{6n’( 6') ma lb") 3.( (5’)} - 6n’(b”)[V.'s.(B lb") °8(i5’)] (2.55)
--g (5|5’)[V’n2(6') a .(7’)]

where 8. is either g" or 3a , and where the observation
stanzas) = v,(n1(5) 4.3) = v,n1(a) has been invoked to simplify results. Furthermore,

the on1( 'p") a_ /n3 product associated with the last two integrals of (2.54) reduces to

 

 

6n2(b")d. g flab-n} V.n’(b") _[ 1 1

n: 2:2 "2(5/) n: "2(5’)

)szw").

C

Taken together with (2.55) and the above result, the last two integrals of the EFIE in
(2.54) take the generic form of

37

one lei

(Greei‘

8“-
~q~

Wherd

andri

gathe

 

2 I
f—‘——"‘f’s.(5lp ’)[v:-,.55(5’)+a( ') 5,70 ’)]d:’

(3 I!
2
=fv’ {an—(Ls. (5Ip’)a (6’)}dr’ (2.56)

'16:" 2° ’Vdc.(5t5')-5.(5’)dr’-[5.(5I5’)3.-5.<5’>ds’
C3

cs n,

 

one term of which is amenable to application of the two-dimensional divergence theorem

(Green’s Theorem in the Plane). The explicit result for that integral is

2 --I
IV". 6n (:3 )8‘(§|§I)a‘(5l) dS’ = fr" on 2g" ———)'/8. (PIP’) [a] 3 (6’)]dl’ (2.57)
cs n: n:
where I‘ is the contour bounding the cross—sectional surface of the dielectric waveguide,
and n is the outward directed normal from contour I‘, as seen in Figure 2.4.

After all the manipulations mentioned in (2.55) through (2.57), terms can be
gathered to generate a nice form for a TEFIE of

2 -0/
61(5) = 51(5) + k3] thmflew’m’
c:

"c

+ VII __6n:(2ii’) [Vr'g +stni‘5.(fi’)ds’
f [51:33: + van] a.( 5’) -5.( 5') ds’

fr— 5_"___1(5')[yk’ 8: + rgu](fi"3r(5l))dl'
n

C

The bracketed quantities are new Green’s functions, but are not represented in their
simplest form, as certain derivatives will cancel out. It is well-known (and equally
obvious by inspection of (2.45) that the principal Green’s function behaves as

Vtg" = ~V,g’. The reflected Green’s functions are subtly different. While the

38

 
   
    

cover (so)

_ y-o

— y - .t
substrate (6,)

x-O

Figure 2.4 Waveguide geometry for development of TEFIE.

39

 

whe

Note

 

(War.

 

(2.6

6160'

"UI
I!

 

transverse behavior is the same as that of the principal Green’s function (ie,
331/31 s -ag:/ax’), inspection of (2.46) reveals that the normal derivatives will behave
as 63"]6)’ = agflay’. Under these considerations, the desired form of the transverse
electric field integral equation (TEFIE) is

5(5) = 51(5 )+k.’f°"‘——"——)§..(5Ip 5-,’)5(5 ’m'

C

 

+V.f 5" ‘9" 5.,(5I5’) 5 (5’)ds’

 

 

 

 

 

0’ "‘ (2 58)
+2.] (5| (5’)?! (5’) 5 (5')ds’ '
CS
-fl,"” "("2 55’)(fi’ 5 (5’))dz’
"C
where the following Green’s function quantities have been defined:
Eu 5' Ts" +533 + 9(81 + 68‘ )9 (2'59)
ay
, . as: (2.60)
5.. = V. 8mg. + 9
a)’
2 r r age, (2 61)
g. = ykegs +V‘ gp+gr + .

 

Note that both Green’s functions in and it, are dyadic, while 2, is simply a vector
quantity.

A physical interpretation of the newly developed Green’s functions in (2.59) to
(2.61) is possible by identifying the nature of the sources induced by the transverse
electric field. Transverse polarization current density is easily recognized as

i5t = 6112(5 )eoé,. It is readily apparent from considering (2.58) that both 5,, andfitv

40

 

relate
funct
wave

induc

kemc
introd
gradie
is rear
compt

well-b

integr

mem

 

SYSIEr

 

tTans."
the tr

tranyx

 

relate the polarization current density to the electric field. The vector 2. is a Green’s
function that relates the electric field to the polarization charge densities throughout the

waveguide cross-section and the waveguide boundary. From this consideration, the

induced polarization charge densities can be recognized as

pv = 101133.56,
(2.62)
p, = 6061321336,

The final form of the TEFIE as given in (2.58) is devoid of a highly singular
kernel, as desired. This is because the operator upon the principal Green’s function that
introduced the source-point singularity (V,V,-§") has been manipulated into only a
gradient operation (V,g'). This term is now on the order of If)’ - 5’I" as yey’, which
is readily integrable. As noted previously, there never was a problem with the scalar

components of the reflected Green’s dyadic. Consequently, all Green’s functions are

well-behaved and independent of the shape of the excluded source-region.

Summary

An electric-field integral equation for open-boundary waveguiding systems,
integrated dielectric waveguide or microstrip, in a planar-layered background environ-
ment has been developed. Exploitation of the axial uniformity of these waveguiding
systems allows the appropriate electric field integral equation to be solved in the axial-
transform domain. The necessary Green’s functions are one-dimensional integrals on 5,
the transverse spatial frequency, and given by (2.42)-(2.46). Finally, for the case of
integrated dielectric waveguide, an electric field integral equation based solely on

transverse field components can be developed. Regardless of which EFIE is used, the

41

lOlhi

 

spatial electric fields can then be constructed as the inverse transforms of the solutions

to the axial—transform domain EFIE’s.

42

dorr
of ti
Wav.

am

0f th(

IhCa

 

 

dwelt

 

(CO {I

Chapter 3
Propagation-mode Spectrum for Open-Boundary Waveguides

In Chapter 2, an inteng equation for the axial transform-domain fields of open-
boundary waveguides was developed. Subsequent to solution of that integral equation,
an inverse transform on axial spatial frequency 1’ is necessary to recover the space-
domain fields. One possible representation of the space-domain fields1 is a superposition
of the spectral modes of the open-boundary waveguide, where a spectral mode is any
waveguide mode that satisfies the Sommerfeld radiation condition. This chapter exposes
a method to recover the space-domain fields and recognize the propagation-mode
spectrum (axial eigenspectrum) of open-boundary waveguides.

The propagation-mode spectrum is found to be associated with the singularities
of the transform-domain field2 in the axial transform (complex-g) plane; hence, locating
the appropriate axial transform-domain singularities and determining their nature is of
vital importance. This task is non-trivial, as the Green’s function integrands are
dependent upon axial spectral frequency 1' and transverse spectral frequency 5 through

the relationship 12 = £2 + C2. It is not a priori obvious how singularities of the Green’s

 

‘The development in this chapter is performed for the electric field of the IDWG structure. This
development is equally valid for the surface current of microstrip transmission lines.

2Throughout Chapter 3, transfonn—domar’n without any qualification refers to the axial transform domain
(Complex f-plane).

43

 

 

transf

Comp

Esta!

 

is notl

Probl

field:
defo
Exp;

radi-

05<
re;

Q)

function in the transverse spectral frequency (complex-5) plane impact the location of
singularities in the axial transform plane, or vice versa.

Section 3.1 provides a necessary first step, developing conditions for a Fourier
transform pair to exist in the complex plane by consideration of analytic function theory.
In Section 3.2, these conditions are applied to the analysis of open-boundary waveguides.
The axial transform-domain Green’s function scalar components are inverse Fourier
transforms on transverse spectral frequency E. Requiring that the forward transform on
transverse position x converge serves to restrict the location of the singularities in the
complex f-plane; this restriction on the complex E-plane singularities locates and
restricts the singularities in the complex f—plane. The criterion developed in Section 3.1
is not new, just subtle; the application of this criterion to the open-boundary waveguiding
problem is new, and the major contribution of this dissertation.

Section 3.3 of this chapter develops a spectral representation for the space-domain
fields by evaluating the inverse Fourier transform on axial spectral frequency by contour
deformation into the complex f-plane. This propagation-mode spectrum is a singularity
expansion of the transform-domain fields from which the radiation field and continuous
radiation spectrum can be conceptualized. For open-boundary waveguides of finite
transverse extent in a layered background environment, a new regime of the radiation
spectrum will be identified. Section 3.4 is a discussion of the radiation spectrum for
Open-boundary waveguides with limitingly low-loss. The specific character of each
regime of the radiation spectrum, and the effect upon the complex E-plane, will be

discussed.

44

lflkyi

Hen“:
why t

311

3.1 l

for H
enght
Physi
ldosr
Hans

COmP

3NL}

 

 

Finally, Section 3.5 addresses a related topic — the usage of the non-spectral or
leaky-wave modes in the representation of the spacedomain fields. Leaky-waves are of
tremendous interest to the research community, yet almost no one adequately discusses
why they are of interest and importance; hence, this will be explicitly detailed in Section

3.5. Section 3.5 will also comment on their relationship to the radiation field.

3.1 The Fourier Transform in the Complex Plane

Little consideration outside the inclusion of generalized function theory is needed
for most typical applications of Fourier transform analysis that occur in electrical
engineering. Engineering problems analyzed via the Fourier transform usually possess
physical requirements that easily satisfy the existence conditions for the transform pair.
Most typical applications of the Fourier transform also deal with strictly real-valued
transform variables as well. As a consequence, when the transform variable becomes

complex, little, if any, extra consideration is given to the now complex-valued problem.

3.1.1 Fourier transform theory on the real-line

There is a great body of literature on the Fourier integral, from the basic
engineering-oriented considerations of Papoulis [38] and LePage [39] to in-depth
mathematical treatments [40,41]. The important conclusions, taken from Papoulis
[3.1], are summarized below. In this basic treatment, all functions f(x) are assumed to

be of bounded variation.

The familiar version of the Fourier transform pair, in this case on x and E, is

45

Cer
Cor
beh.
toni
absc

inter

ShOL

InVe

 

whh

 

f(x) - -— 21f F(E)e"‘d£ (3.1)

17(5) 3 fflx)e'j£‘dx (3.2)

O.

A sufficient condition for the Fourier integral (3.2) to converge is that the function f(x)

be absolume integrable, that is
f [f(x)|dx < a. (3.3)

Certain functions, such as sin(ax)lx, do not obey (3.3) yet possess a Fourier transform.
Consequently, a second condition for the convergence of (3.2) is that the function f(x)
behave as f(x) =g(x)sin(ax+¢) , where a and 4) are arbitrary constants, g(x) mono-
tonieally decreases as |x| ~00, and that f lf(x)x"|dx exists, that is, f(x)/x is
absolutely integrable. Under this conditionfthe Fourier integral in (3.2) should be

interpreted in the Cauchy Principal value sense, namely,

fr(x)dx= um I f(x)dx

r4-

One can refer to Titchmarsh [3.3] for more discussion on the above topics. It

should be noted that the starting point in Titchmarsh is actually Fourier’s Single

Inversion Integral (SII),

fix) 8 z met)_ sheer-0d,

1t (x-t)

which can be formally developed from the Fourier transform pair given in (3.2) and (3.1)

by substituting (3.2) into (3. 1) and exchanging the integration order.

46

3.1.2 Theory of analytic functions

With open-boundary waveguide problems, the spectral frequencies are often times
complex-valued. 0f necessity, the theory of complex variables is involved, in particular
the theory of analytic functions. Certain key observations and theorems are presented
below. In the following discussion, 7., w, and a are all complex variables unless
otherwise noted.

The definition of an analytic function is a complex fitnction f(z) of a complex
variable 2 is analytic at a point 20 if it is differentiable at every point within a
neighborhood of 20. A function is analytic in a region D if it is analytic at all points of
the region D (also denoted as regular in D); herein, the term analytic refers to functions
analytic in a region unless otherwise explicitly stated. Obviously, no singularities of
function f(z) exist in region D.

Application of analytic function theory to the Fourier transform pair requires a
theorem that allows for analytic functions to be defined by means of integration.

Theorem I: Let D be the region. Let f(z, w) be continuous in z and w
where z E D and w lies on a smooth contour C, possibly unbounded. Let
[(7, w) be an analytic firnction of z in D for each w on C. Let f f(z,w)dw
be uniformly convergent. Then c

15(2) = fcf(z.w)dw (3.4)

is an analytic firnction of z in D.
This elegant theorem is stated in slightly different form in [42]. A proof of Theorem

I is developed in Appendix C.

47

it

P:

wl

the
1m
IE;

thc

 

3.1.3 Regions of convergence for fitnctions of exponential order

The Fourier transform is used as a tool to solve the Helmholtz equation subject
to the boundary conditions of a planarly layered background environment. The solutions
to the Helmholtz equation are exponential functions. It is necessary, then, to determine
the region of convergence for the Fourier transform pair of a function of exponential
order. This follows the development from Mittra and Lee [43].

Let f(x) be a function of exponential order with a finite number of discontinu-
ities. These discontinuities are not of concern, and can be handled in the Cauchy

Principle Value sense. A function 1(1) of exponential order has the general behavior
Ae , x-ooo
|f(x)| < (3'5)

where A >0, B>O . To analyze the convergence properties of exponential order

functions, the forward transform in (3.2) will be decomposed as the sum of two parts,

0 .
HE) ff(x)e"'"dx + ff(x)e""dx
-- 0

(3.6)

1".(5) + Fifi)

Also, the following analysis assumes that E = o +j'c.

Consider where F,(£), the integral from O to on, is analytic. By Theorem 1,
the integral defining F,(E) is analytic wherever it converges uniformly. To show
uniform convergence, choose some to > r_. Consider now an x=T for T >R. It is

readily apparent that, if ]f(x) |< Ac” Vx>R where A and R are positive real numbers,

then the following

48

istr

Nov
inde
regit
that

of

no

the

C01

of

 

Ft

 

T 1'
ff(x)¢"°”"’dx < fAe“5""e’°‘dx = Il
R R

is true. Since c!" has a magnitude of l, and that 1' > to > i- , it is also obvious that

T T
[Ae“-""el°‘dx < [Ae'(‘°"5)’dx = I,
R R

Now, integral I, exists independently of E = a + j r in the region t>r_. Since this
independent upper limit exists, the entire integral for F .(5) converges uniformly in that
region; within that region then, F,(E) is analytic. This is shown in Figure 3.1a. Note
that if f(x) is non-zero for x>0, and zero for x<0 , then F,(E) is the Fourier transform
of f(x). The required inversion contour on 5 must then lie within the region r>t_.

A similar analysis can be conducted for F_(E) . It takes but little effort to show
that F_(£) is uniformly convergent for s in the region r < r, . Within this region,F.(£)
is analytic. This situation is shown in Figure 3.1b. It should be noted that if f(x) is
non-zero for x<0, and zero for x>0, then F,(E) is the Fourier transform of f(x) , and
the appropriate inversion contour lies within the region r <13.

Returning now to the decomposition in (3.6) reveals that the regions of
convergence for F,(E) and F_(£) must overlap if F(£) is to be the Fourier transform

of a function of exponential order, as demonstrated in Figure 3.2. This overlap region

is a strip in the complex f-plane, parallel to the real-axis, where
r_ < 1: < 1." (3-7)

F(£) is analytic in this strip and the Fourier inversion contour lies within it. The

Fourier transform of f(x) exists only if this common strip of convergence in if exists.

49

s. s
§ ' 'mrlE} " T-

W

//////// //

Figure 3.1

lm{€} - 15+

 

'mlE} ' 5.

figure 3.2 Strip of convergence in transform-domain for physically realizable
functions of exponential order.

51

CX

OI

of

tr

 

 

 

As mentioned previously, the Fourier transform is used in the process of solving

the Helmholtz equation. A typical solution to the Helmholtz equation takes the form
f(x) ., emu (3.8)
where a = a, +ja,. For these solutions to be physically realizable over all space, they
must be bounded as |x| 4 no. This will be true if f(x) is either: (1) a decaying
exponential function in x, requiring that
a, > 0 (3.9)
or (2), an purely oscillatory function in x, which requires

a, = o (3.10)

The effects of each of these cases on the region of convergence within the Fourier
transform plane will be investigated.

For case (1), f(x)-5e-” as x-ooo. Comparison with (3.5) reveals that
t_ -- -a,. As x-o -oo, f(x) ~e"’; it is apparent that r, = «3,. The Fourier transform
of (3.8) exists, and converges in a strip of finite width in E,

-a, < 3mm < a, (3.11)

This convergence strip in particular contains the real axis of 5, upon which the inverse
transform to recover f(x) would be taken.
With case (2), |f(x)| .. l as x .. 1:09. The asymptotic behavior in case (2) can

be viewed as the limiting case of a decaying exponential function, i.e.,

f(x) = lime""e"’"', v>0
v-O

52

The region of convergence for case (2) is the limit of case (1) as v (=a,) approaches
zero. In this limit, the strip of convergence of the Fourier transform contains only the
real axis, upon which the inversion contour lies.

Of course, it is possible for a, < 0 in (3.8), which leads to growing exponential
functions in 1:; these are obviously not physically realizable as they are unbounded as
|x| ~ co. Another consequence is that there is no common strip of convergence in the
Fourier transform plane, as t_ > 0 and t, < 0. This is demonstrated in Figure 3.3. By
previous considerations, no Fourier transform for this type of exponential function exists.

The conditions for the existence of the Fourier transform in the complex-plane for
functions of exponential order have been established. Assuming that these functions obey
the standard requirements for Fourier transformable functions, as stated in Section 3.1.1,
then the Fourier transform pair exists if the forward transform converges in a strip of
finite width in the transform plane. This strip is parallel to the real axis, and must
minimally include the real axis in the transform plane if the transformed function is to
be physically realizable, that is, bounded at infinity. Finally, this strip of convergence

is analytic and consequently, devoid of any singularity.

3.2 Green’s Function Singularities

As observed at the beginning of this chapter, the transform-domain (axially-
transformed) Green’s function singularities have a complicated and interrelated
dependence upon £ and I. As the Green’s functions are complex integrals, knowing the
location and nature of the singularities is vital to guarantee an answer exists. Application

of basic complex variables theory is sufficient to identify and determine the nature of the

53

 

 

%/

singularities.

The Green’s function is physically interpreted as the response of the background
environment to a point source excitation. It is desired that the Green’s function be a
spectral quantity; that is, it must satisfy the radiation condition at infinity. Imposing the
physical requirements upon the transform-domain Green’s function determines the
location of the singularities. Based on the development in Section 3.1, it is clear that
one of those physical requirements is that the forward transform on x, used to determine
the Green’s function, must converge. This last requirement is the key to resolving and
locating the singularities in both the axial and transverse transform domains unambigu-

ously, regardless of the relationship between £ and (5.

3. 2. 1 Transverse wavenumber plane (complex £—plane) singularities
Solving the requisite EFIE’s for the unknown axial transform-domain surface
current or electric field involves computing the axial transform-domain dyadic Green’s

function (2.42). Each scalar component of 2(5 | ii’;C) is a one—dimensional inverse

transform on transverse spectral frequency E, and can be represented generally as
83(5lb' ;C) = l f [F:(Ppy.y’)e"“']e"’dt (3.12)
21: __

where p, =¢ £2+C2-k,2 and F:(p,,y,y’) is the appropriate function involving
background reflection coefficients and the exponential behavior in y. Since 5 is
potentially a complex variable, pole and branch point singularities are possible; the

locations of the singularities in the transverse spectral frequency plane (complex E-plane)

55

must be known to guarantee that the integrands are single-valued and non-singular, i.e,
the integrals exist.
Specifically, the axially-transformed Green’s functions are given by (2.45) and

(2.46), namely

e'(b’|i>";C) = f

8-3, - cl ' Ii
em ) e P r r dfi (3.13)

 

4rtpc

l , l
8:(i5|5’;C) . l , - +

i8:(5|§’;()l If R:( ) amt-He no 1")
[8:75 lb”; ()1 5" C(r.)

d5 (3.14)

II
a
A
)5
v

 

4np‘

 

 

Reflection coefficients R,, R," and C relate the background environment effects, including
the surface—wave behavior, through wavenumber parameter p, = (l 62 + C2 -k,2 in each
layer, where l=c,f,s (denoting cover, film or substrate respectively). These coefficients

are detailed in Appendix B; for reference purposes, the coefficients for the conductor-

/film/cover background are reproduced here as

 

 

 

so) -
Z ”(1)
2
12.01) = Nfip‘ p’mw’t) (3.15)
2‘0)
2-
cm = 25w, 010‘
z (2025(1)
where
2‘00 = N; pxpfmhw) (3.16)

2"(1) = pc +pfcoth(p,t).
The following development assumes small losses in the background media, namely,
kt=ktr+jkw ku<o'

56

The wavenumber parameters p, are multi-valued functions of complex spectral
frequency 5, implying the existence of branch points and branch cuts in the complex
5-plane. The wavenumber parameters become multi-valued around the point where
Arg[p,] = 0, or where pf = E2 + (1 -k,2 = 0. A branch-point singularity in the complex
5-plane is thus located at

a“ a (k: _c2 (3.17)

It is obvious by inspection of equation (3.17) that the branch point locations in the
complex 5-plane directly depend upon the value of axial wavenumber 3' and migrate as
:- varies; furthermore, the branch point locations are related to 1’ through a square-root
in the complex {-plane. This is a very important observation, and will be dealt with
later.

A branch cut emanating from the branch point 5“ in the complex 5-plane is
necessary to define a single-valued function. This branch out indicates the joining of
Riemann sheets, upon which the wavenumber parameters are single-valued. Mathemati-
cally, any arbitrary curve in the complex 5-plane approaching a complex 6. will serve
to render the wavenumber parameters single-valued. However, the Green’s functions
were developed under the assumption that Stein} > 0; this must be enforced if the

Green’s functions are to have any physical meaning. Enforcing spectral behavior on the

Green’s function results in the more restrictive criteria of

M5,) = new 51-5; } >0 (3.1s)

being used to define branch cuts in the complex 5—p1ane and enforce spectral behavior

on the top Riemann sheet upon which the inversion contour must lie. This is the

57

equivalent of enforcing the Sommerfeld radiation condition on y, that is, the Green’s

functions vanish as Iyl -° 0. Criteria (3.18) implies that

-323 < limb/5241,} 5 3%. (3.19)

for the top Riemann sheet of the complex 5 -plane. The branch cut is the limiting case

of inequality (3.19), where fielpl} = 0; this results in the relations

(ti-rmdtiu-ti) < o
E,£‘-Ew£w = 0

sate-tit)
saute-ti.)

(3.20)

(the negative real axis in the complex 52-plane) which lead to a hyperbolic branch cut in
the complex 5-plane, initiating at the branch points of 5. 1:5”, and extending
asymptotically to infinity along the imaginary axis, such that

g smut.)
25,

 

5 we: It,| > ISmlfiull. (3.21)

Each wavenumber parameter p, (where l=c,f and s) obeys a branch out of this form.
But, since the branch cut emanates from the branch point 5“, they are also dependent
upon (5 in the same manner as the branch point. This is illustrated in Figure 3.4.
Inspection of the coefficients R,, R," and C, as given in (3.15)-(3.16), reveals that
they are even with respect to p,, the wavenumber parameter for the film layer.
Consequently, the branch out associated with p, is removable. This behavior also occurs
in the substrate/film/cover background as well; it generalizes for an N-layer structure,
where the branch cuts for all interior layers are removable [44]. It should also be
observed that the principal Green’s function, as given in (3.13), implicates only the

branch cut associated with the cover layer.

58

There are also pole singularities in the complex 5-plane where the denominators
in the reflection coefficients vanish. These denominators are functions of the
wavenumber parameter p, which itself must obey the branch cuts defined in (3.21). As
a result, location of any pole singularity in the complex 5-plane is implicitly dependent
upon the value of f. Observations in Appendix B reveal that the reflection coefficient
singularities are physically associated with the (possibly many) surface-wave modes 1:
of the background structure. Consequently, the pole locations are seen to be explicitly

dependent upon the value of f, since

5; a ’(1;)25C2 (3.22)

As observed with the branch points in (3.17), this pole location in the complex 5-plane
is dependent upon 1' through a square root. There can be any number of pole
singularities depending upon the background structure; for clarity, only one is shown in
Figure 3.4. This is the situation for an electrically-thin film layer in a substrate/film-

/cover environment.

3. 2. 2 Considerations of forward transform convergence

Figure 3.4 shows the location of the complex 5-plane singularities, save for the
removable branch out for p,. The singularities are fixed for the integration over spatial
frequency 5, but will migrate as axial wavenumber 3' varies, denoted by the arrows in

Figure 3.4. Furthermore, the sign convention for the square root on {is chosen to locate

poles and branch points in the lower half of the complex 5-plane, that is

59

5"“

 

.-’-'-O-O-e-e-o-e-O-O,OO

I
-5138 E. 6C .1! inversion contour

 

.f..-l-O-O-O-.-I-O-.-
.-

 

Singularity locations in the transverse-transform domain (complex 5-

Figure 3.4

plane).

Eu ' 'jxw X" ' vcz'ktz (3.23)

a; -= -rx;. x; ale-(11;)2

It is obvious from (3.23) that branch cuts in the complex {-plane are necessary if 1‘ is
complex valued; it is not obvious how the branch cuts are to be chosen.

As stated previously, a spectral representation for the Green’s functions (and the
subsequent waveguide fields) is desired. The Green’s functions are solutions to the
Helmholtz equation with a point source excitation and must vanish as the observation
point becomes distant; a spectral representation for the Green’s functions obeys that
boundary condition. From the analysis in Section 3.1, the forward Fourier transform on
x converges within a horizontal strip in the complex 5-plane. Also, the Fourier
transform is regular in this strip. Since the strip of convergence is regular, no
singularities can reside within it; the strip width is thus limited by the singularity nearest
the real axis in the complex 5-plane, as shown in Figure 3.5. It is obvious that as the
axial wavenumber r varies, the 5-plane singularities migrate, possibly narrowing or
widening the necessary strip of convergence. Yet, regardless of the 5-plane singularity
location, the analysis requires that the forward transform converge. This transform is
to represent a spectral Green’s function; the forward transform on x must converge in
a strip minimally containing the real axis in the complex 5—plane.

This requirement of convergence of the forward Fourier transform in x restricts
the migration of the complex 5-plane singularities. As long as any 5-p1ane singularity
does not migrate across the real axis, the forward transform on x converges and defines
a spectral mode. The transform converges even in the case where the 5-plane

singularities reside on the real-axis, since traditional real-line Fourier theory allows for

61

  

 

 

\
'U

 

be +€bs

""5'+

strip of convergence

 

 

.Ibo-I-o-Igo-oco-I-o..-.
‘0
\

Figure 3.5 Strip of convergence in complex 5-plane for forward transform on x.
Arrows denote singularity migration directions.

62

this possibility. Obviously, none of the 5 -plane singularities can migrate across the real
axis in the 5—plane; this is the criterion that defines branch cuts in the complex f-plane.

What can occur if a 5-plane singularity migrates across the real axis when I
varies continuously (no discontinuous steps). The first possibility is that a strip of
analyticity in the 5-plane, which includes the real-axis as an inversion contour, ean be
determined, as seen in Figure 3.6a. For this case, the singularity passes from below the
inversion contour to above the inversion contour; the inverse transform changes
discontinuously while I changes smoothly. This is undesirable; when considering the
Green’s functions, this is an equivalent to the physical problem changing discontinuously
[45].

A second possibility arises by not passing through the inversion contour on the
real axis in the complex 5-plane. Viewing the forward transform on x in the sense of
the decomposition (3.6) leads to Figure 3.6b; in which the Fourier integralF,(£)
converges for all 8mm > SIM +5“) , while F15) is convergent for all
8mm < SIM-Em} . There can be no common strip in which the Fourier transform
(sum of F,(£) and F_(£)) converges; in this case, a spectral representation cannot be

obtained.

3. 2. 3 Axial transform-domain (complex {-plane) restrictions
Considering the definition for pole and branch point singularities in (3.23), it is

observed that these singularities must remain in the lower-half of the complex 5-plane,

regardless of the value that 1' takes. Based on that definition, then, it is obvious that

aux“) > o (3.24)

63

 

 

i r
, r
s r
s l
2 s
. t
!
t t
!
0

Fifi) analytic

/ X
.5 E,

F_(E) analytic

 

 

 

(b)

(a) Migration of a 5-plane singularity across real-axis through contour of
integration. (b) Migration of a 5-plane singularity across real-axis,

treated correctly.

Figure 3.6

64

is required to restrict the branch points in the complex 5-plane to the lower half-plane;
thug“) > 0 (3.25)

is required to restrict the pole singularities in the 5 -plane to the lower half-plane. SinceXu
and x; are multi-valued functions of :, these requirements lead to branch cuts in the
axial transform plane. Inspection of (3.23) indicates that branch points in the complex
{-plane occur at C = :k, and at C =- al} Requirements (3.24) and (3.25) are similar to
the requirement of (3.18); consequently, little work is necessary to show that the

hyperbolic branch cut of

- butt?)
2:,

 

Ct +iC,; It,| > lIm{k,}| (3.26)

restricts the complex 5 -plane branch points (5,) appropriately, while the hyperbolic

branch out

___ but (192}

. 2“ +r'C,; |t,| > (mum (3.27)

restricts the complex 5-plane poles (6;). This is illustrated in Figure 3.7, where only
one branch cut arising from the complex 5—plane pole singularities is shown for clarity
(case depicted for a thin-film background environment).

It is curious to note that while the Green’s function possesses both pole and
branch point singularities in the transverse spectral frequency (5) plane, it possesses only
branch point singularities in the axial wavenumber (0 plane. It should be explicitly
observed that a pole in the transverse spectral frequency plane (complex 5-plane) leads

to a branch point and branch cut in the axial transform-domain (complex {-plane). This

65

 

 

 

 

 

Figure 3.7 Branch cuts in axial-transform plane (complex {-plane) necessary to
maintain convergence of forward Fourier transform on x.

66

is a new observation, and difficult to accept at face value; only when the propagation-
mode spectrum is recognized in the next section does this observation make sense.

By defining branch cuts in the axial transform plane (Figure 3.7) as per (3.26)
and (3.27), the locations of the singularities of the Green’s function in the transverse
spectral frequency plane are fixed, and their migration is restricted such that the forward
transform on x converges. The axial transform-plane branch cuts emanating from
C = tk‘ , the wavenumber in the cover, locates transverse spectral frequency branch point
Ck; similarly for the branch cuts from C = :k,. If the substrate becomes a perfect
conductor (nl - -j~), this branch point is unnecessary. The axial transform-plane branch
cuts emanating from C . :1; locate and restrict the transverse spectral frequency poles.
These branch cuts are associated with the surface-wave modes on the background
structure; a branch cut is needed for each individual surface-wave mode. Obviously, if
no surface-wave behavior in the background structure is possible, none of those branch
cuts are necessary. Finally, these branch cuts define a multi-sheeted Riemann surface
in the axial transform-domain, separating the spectral (top Riemann sheet) sheet from the

non-spectral (all other Riemann sheets) sheets.

3.3 Propagation-Mode Spectrum for Open-Boundary Waveguides
Solutions to the axial transform domain EFIE (2.40) can now be obtained, since

the axial transform-plane branch cuts guarantee that the Green’s function comprising the
kernel of (2.40) exists. For convenience sake, the EFIE is reproduced here
6n2 "’ é’ " ;

.( p ) (p C ) ds

2
c

 

5(5)-f §°(5‘|5”;C)- warm, ...vaecs (3.2s)
CS

n

67

using the transform-domain electric dyadic Green’s function
i“(5l5’:t) = (k3+W-)z(5l5';o+'2‘5(5~5’)
where B(§|§’;C) is defined by (2.42)-(2.46). The space-domain electric field is

recovered from the solutions of EFIE (2.40) via the application of the inverse Fourier

transform on axial wavenumber t, that is,
_ l . -o -o 1‘:
5(5) - — f e(p.t)e at (3-29)
21: --

A number of approaches are available to evaluate the inverse transform (3.29).
A pure numerical solution would utilize a fast Fourier transform (FFI’) on the real-line
inversion contour in the complex {-plane to obtain the total space-domain field at each
spatial point of interest. This approach offers no insight into the modal spectrum or the
waveguide physics. As spatial coordinate z grows large, the complex exponential
becomes very oscillatory, and amenable to asymptotic evaluation techniques like the
method of steepest descents. This approach, detailed in the next section, is not a modal
expansion but does offer insights into the waveguide physics.

The inverse transform can also be evaluated by contour deformation into the
complex {-plane; this is a singularity expansion of the waveguide field. This singularity
expansion determines the entire prOpagation-mode spectrum of the device; the bound
hybrid guided-wave modes are associated with {-plane poles, while the continuous
radiation spectrum field is associated with the {-plane branch cuts.

That the transform-domain field 865,0 can possess a finite number of isolated
singularities (poles of order m) is not difficult to accept. Yet, any branch cuts within the

axial transform-plane arise only from the Green’s functions. If these complex f—plane

68

branch cuts are to be used in a modal expansion, then 8(§,C) should share the branch
points of the Green’s functions. This is intuitively obvious when considering the physical
interpretation of the Green’s functions as a point source response within the layered
background environment. A waveguide field 'e‘(b‘,C) in this environment is thus
assembled as the superposition of equivalent point source responses; naturally, this field
shares the branch points of the Green’s functions. This can also be demonstrated by an
indirect proof [46].

The details of performing the inverse transform (3.29) by contour closure are now
presented. Without loss of generality, a spacedomain current source can be decomposed

as 7’ jo(5)6(z-z’), which becomes jo(p’)e"“’ in the transform—domain. It is

apparent that the Fourier kernel in (3.29) takes the form e’“‘“" . Closure in the f-plane
will be performed such that the integrand vanishes upon the contour at infinity. This
implies closure in the lower half—plane for z < z’ and the upper half-plane for z > z’ ; the
integrand and complex exponential vanish on the infinite semicircle by consideration of
Jordan’s lemma. This closure is shown in Figure 3.8 for the case of z < 1’.
Application of Cauchy’s residue theorem states that the closed contour integral

of an analytic function is proportional to the sum of the residues at the enclosed pole

singularities, namely,
fcé‘(5.C)e"‘dC = 21:12 Rw.{5(5. 05”} (3-30)

This closed contour is the sum of the real-line inversion contour, the contour at infinity,

and all the deformed contours around the branch cuts; consequently, Cauchy’s residue

theorem allows the inverse transform in (3.29) to be written as

69

(i
E.

  
 

-A°
-c c -'
p f a
x -ks -kc

Cr.

 

 

 

it)"

1111],) +C
6
+ 0
A'13
z < z '

Figure 3.8 Contour deformation in complex f-plane used to identify the propagation-
mode spectrum. Closure shown for z < z'.

70

5(5) =r£Rw.{5(5.C)e“5} - —‘— ] 3(6.C)e’°“dt (3-31)
21! C).
where the contribution from the contour at infinity vanishes by Jordan’s Lemma. For
the case of Figure 3.8, the space-domain field’s singularity expansion is

N , ,
2(5) = 5r: 5,.(5.2c;)e‘"‘""' - -1-f5(5.5)e5“'55z 'dt
n-O 21!

Cl

(3.32)
_ 1 ~ -J<lz-z’| - 1 ~ -rctz-z’t
2n £319.05 4?: 2n £319.05 0"!

3. 3.] Bound modes
The pole singularities of 8 comprise the discrete spectrum of the waveguide field.
The axial transform-domain field '6 takes on the behavior

a,é°.,(b‘.C,,)
(C - 6,)“

“5.0 " (3.33)

 

for {very near a pole of order m. At this time, assume m =1; under this assumption, 3:,
is the residue, denoted 'e'm. Substitution of this field into the EFIE results in
“g -. 61:20)") ~ -. -.
3.49:0 5 --—-— §’(p lp’;C)-5’,_(p’;0ds’ = 5‘(5,c) (3.34)
(c-c,) fa "2 p

c
The impressed field E’( if, C) supported by independent source currents is regular for
C = C,, the pole of the waveguide field. The left-hand-side of (3.34) must be
indeterminate if the EFIE is to be satisfied; the bracketed quantity in (3.34) must vanish
for C = (p. In this situation, 3,. satisfies the homogeneous form of EFIE (3.28),

2 -o
545:9) — [afl%§°(filfi’;C)-5m(§’;c,lds’ = o (3.35)

C

71

It is obvious that for C = C,, the residue field is a natural mode of the waveguide.

Any poles used in the modal expansion of the waveguide fields are located on the
top sheet of the axial transform plane. On the top sheet, all modes are spectral and
satisfy the radiation condition; this implies that the natural-mode fields exponentially
decay outside the waveguide. It has thus been established that poles in the transform-
domain waveguide field lead to the guided-wave modes of the open-boundary waveguide.

If the electric field pole is of a higher order, the previous analysis does not arrive
at the correct residue field for that pole. An iterative technique for poles of higher order
is presented in [3.46]; also presented is a methodology for determining the order of the
pole. This is not necessary for any waveguiding problem in this dissertation.

The field excited on the waveguide can be expanded in terms of the hybrid

guided-wave modes [47]. The excitation coefficient is

— 2 -.
, = F‘ f—"—"—‘P—5,(5.t)5‘(5.t)4: (3.36)
P C3 "c

an overlap integral of the impressed field with the guided-wave mode 8965;) weighted

by an appropriate normalization coefficient given by

c, = [450325)
mes ¢

n

 

 

e,(5.C,)]--§E§'(fi I6 .0

 

2 -o
[ 5" (9') Ep(§’,CP)]dsds’ (3.37)
-c'

n3
Details of this development can be found in [16] or [48].
Finally, it should be noted that the transform-domain field E(p’,C) has poles that

occur on the other non-spectral Riemann sheets. This possibility, and the leaky-wave

modes associated with these non-spectral poles, will be discussed in Section 3.5.

72

3. 3.2 Continuous radiation spectrum

The sum of the contour integrations along the branch cuts comprises the total
continuous spectrum of the waveguide radiation field. The transforrn—domain waveguide
field at any point along the branch cut is the solution of EFIE (3.28) for a given
excitation.

To recognize that the continuous superposition along the branch cut determines
a radiation spectrum, consider that these branch cuts represent the limiting values of
acceptable (“defining a spectral mode. Recall that a spectral mode satisfies the radiation
condition at infinity. In this limiting case, the behavior at infinity is bounded and
oscillatory but non-zero; this is obviously a radiation spectral component. As observed
by Shevchenko [49], the superposition of these modes vanishes at infinity; hence, this
superposition of solutions along the branch cut represents the radiation field of the open-
boundary waveguide.

There are a number of branch cuts defined in the axial transform plane. Up to
two of these cuts are associated with the wavenumbers of the semi-infinite in y
background layers (cover and substrate), and control the spectral behavior in those layers
via control of the complex 5-plane branch point singularity migration. For those two
branch cuts, the limiting acceptable spectral behavior is bounded oscillation in y.
Consequently, the branch cut from C = k. defines the cover radiation spectrum while the
branch cut from C = 1:, defines the substrate radiation spectrum.

The remaining branch cuts in the transform-domain control the spectral behavior
of the background surface-wave modes by controlling the location of the complex 5 -plane

pole singularities. A branch cut is needed for each surface wave mode the waveguide

73

can support. The radiation spectrum associated with these branch cuts is ealled a
transverse radiation spectrum; as it arises from the existence of surface wave modes, it
will possess the same field structure. These transverse radiation modes thus are
exponentially decaying in y like surface waves but possess the characteristic radiation-
mode standing-wave pattern in x. This transverse-radiation regime is believed to
constitute the observation of a new physical wave phenomena.

The continuous spectrum solution is directly dependent upon the impressed
electric field. It is desirable to not re-evaluate the forced EFIE in (3.28) for each
separate excitation. A conceptual dyadic Green’s function of the radiation modes can be
advanced to construct a formulation for the radiation field which is useful for arbitrary
excitation.

Let Two) - i. 6( i5 - 50) , a unit-amplitude point source excitation current ati)‘0
directed along 1. restricted to spectral frequency I along any of the branch cuts. This
excitation supports an impressed field upon the waveguide of ‘i(fl,/k¢)§°( ii I [50;C ) at. ,
which in turn induces a elementary spectral field component B. ( 5 I 50; C ) WhiCh satisfies

5.(5 I 50:0 - fmfigflreweo-Era’lam' = lf—‘r°(5t5,;c)-2, (3.33)
c c

If a radiation spectral dyad is defined as

fi(5’I5,:C) = )3 fi.(5’I5.)2, (3.39)

e -x,y.z

then the radiation field is the continuous superposition of all spectral field components

i51(5.z)= rife dCe’“ f4 )3 Ii-,,.(5I5’)]-3"(5')ats' (3.40)

I '10,:

74

where the contour CM is any or all of the appropriate branch cuts. Expressing the

excitation current in (3.40) as the forward transform of space-domain current 3‘ yields
Elm == fronmr’) -i(r’)dv’ (3.41)

in which the radiation Green’s dyad is identified as

3 L1. . ('1‘)
Gail?) 2“ [CA 2: E.(§|§’,C)£.e’“ dC (3.42)

I '10.!
This is clearly a continuous superposition of the elementary spectral field components as

defined by (3.38).

3.4 Radiation Spectrum in the Low-loss Limit

Specialization to a limitingly low-loss (8mik,} - 0; l = c,f,s) waveguide facilitates
identification of the physical characteristics of each regime of the radiation spectrum.
Furthermore, many practical structures are fabricated from low-loss materials. In the
low-loss limit, the hyperbolic branch cuts in the axial wavenumber plane coalesce and
become ”dog-leg” branch cuts, as depicted in Figure 3.9. Throughout this analysis, only
one surface wave mode will be assumed to propagate; more surface wave modes could
easily be included.

There are subsequently three regions of interest upon the coalesced branch cut.
The first region is denoted P; this branch cut only affects the location of 5;, the
background surface-wave pole singularities. The second region is denoted by S; this
branch cut affects the location of the substrate 5-plane branch point 5r.- Branch cut S
also affects the location of C; as well, as branch cut 5 is a continuance of branch out P.

The last region is denoted B and controls the location of the cover 5-plane branch point

75

 

 

 

 

+kc +ks +1; 45%

.- -------- I -------- {5%. ()9 (re

 

 

 

 

 

 

 

Figure 3.9 Coalescing of complex {-plane branch cuts in low-loss limit. Closure for
case of z < z’.

76

53:3 branch cut B also affects the locations of both it. and 5;. Table 3.1 summarizes
the effect of different portions of the coalesced transform-domain branch cut upon the

5-plane singularities.

 

Table 3.1 Effect of the coalesced transform-domain branch out upon complex
5-plane singularities

I Singularity Formula

surface-wave pole

Restricted By:

 

 

 

substrate branch point

 

cover branch point

 

 

 

 

 

 

Based upon Figure 3.9, the total field for this specific case is constructed in terms

of its propagation-mode spectrum as
' N
5(5) = 2 5.,(5) + 5:"(5) + 53(5) + flit?) 0-43)
k-O

The first term in the representation (3.43) is the sum of the discrete modes (pole

residues) of the waveguide structure, where each discrete mode is specifically

id?) = 515,454”) e‘l‘W'z’l , (3.44)

77

Closure in the lower half-plane for z< z’ encloses +C” clockwise, giving rise to -j
weighting the residue at +Cu; upper half-plane closure for z<z’ encloses +C"
counterclockwise, etc. The absolute value on 2 -z’ arises from the consideration that in
the lower half-plane, (z-z’) = -lz-z’| while in the upper half-plane (z-z’) = lz-z’l.

The remainder of the terms in (3.43) comprise the total radiation field arising
from the branch cut integration. The total radiation field is composed of three

components, each with a different modal characteristics. The first component is denoted

a surface-wave radiation field, given as

t,
3&3) 5 ”51:1 [5(5.t*) 55(5.C°)]e”"""'dc (3.45)
‘0

and consisting of the superposition of continuous radiation modes in the regime
k, < C,< 1;. The second component is the substrate radiation field,

1:,
EzAb(f) = -'2Ln'f [3153?)55(fi.C°)]e-lclz-z'|dc (3.46)
k. .

and is a superposition of radiation modes in the regime kc < C, < lg. The final component

is the cover radiation field,
1 .j. - I
52mm -—- -— f [8(5.C‘) - 6($.C°)]e"""' '5: 0-47)
21:
k,

and is the superposition of radiation modes in the regimes 0 < C, < k.- and -~< C, < 0.
In (3.45)-(3.47), the notation C‘ denotes a axial wavenumber I located on the
interior of the transform-domain branch cut; the interior being defined as the side nearest

the origin of the complex {—plane; likewise, C" denotes values of f on the branch out

78

exterior (outside), where the exterior side is not the interior side. The integration
contour is then parameterized along the interior side of the branch out from C = 1; to
C = -j~; the branch cut integrand then becomes 305,?) - 8033C"), as given in (3.45)-
(3.47), which then explicitly accounts for the exterior branch out contribution. This

difference will be denoted
83(53C) a 8(538) - 61-93(0) (3'48)
and is the spectral radiation mode for axial wavenumber (5.

The characteristics of each of these three radiation field components depend upon

the behavior of the Green’s functions in each of the three regimes. For ease of

reference, the general form (3.12) is reproduced here
33(5I5’;C) = 2i f R:(P,.y’.x’)e""e"‘d5 (3.49)
n --

slightly modified to show explicit exponential dependence upon y. Consequently, the
spectral mode characteristics are highly dependent upon the location of the singularities
in the complex 5-plane. Locating these singularities and determining their effects
involves evaluating a complex square root quantity of the generic form 11:32-23 1"”;

this evaluation is achieved by factoring the square root as

 

Q = 22-23 = Jz-zont-zo (3'50)

and considering the argument of each individual square root as. influenced by the
respective branch cut and branch point. The total argument for the singularity locations

will then be simply

79

--"-+Arlel = «3+ (WW)

1
2 2
where 0’ is the argument about the branch point at z - +20, 6' is the argument about
z = -zo, and «rt/2 arises from -j. This method is depicted in Figure 3.10.

Applieation of this technique is similar in each radiation regime; details will be
presented for the substrate radiation regime, while general conclusions will be discussed
for the remaining regimes. The substrate radiation regime is he < C, < k“ in which {lies
along the portion of the transform-domain branch cut denoted S. This portion of the
branch out restricts the migration of the substrate branch point 5a and the background
surface-wave pole singularity E; . The complex 5-plane singularity locations will be

computed as per (3.50); the appropriate branch-cut arguments are summarized below.

 

Table 3.2 Complex 5-plane singularity arguments for r in the substrate
radiation regime

IIII a... '
Ea -‘r 0 -1'

D

I-n;
e x 0 0

’5;

P

 

 

   

 

 

 

 

 

 

 

 

 

Arguments for C‘ Arguments for C"

80

\J
‘t
\.
N

 

.20 w 6'
Z

 

+ l'
0
I +
,- z o
I
U;
z .-
I
I.
'.
'.
’O
I.
’0
"
; D

Figure 3.10 Evaluation of complex-valued square roots.

81

Figure 3.11 shows the location of the complex 5-plane singularities for this regime,
where the region about the real axis has been exaggerated. Based on the table, it is
apparent that as I switches sides of branch out S, both C; and 5a. switch location in the
complex 5-plane. For a C‘ on S, both the pole C; and branch cut 51- are located on the
positive real axis; the inversion contour should remain above both these singularities, as
shown in Figure 3.11a, since choice of sign was made to locate the pole in the lower
half—plane. Enforcing the radiation condition on p, for values of 5 results in another
“dog-leg” branch cut, this time in the complex 5-plane. This branch cut associated with
p, runs parallel to the real axis towards the origin, then heads towards -j°°. For a I" on
S, the behavior is reversed; in particular, the p, branch cut still runs parallel to the real
axis towards the origin in 5 but then heads towards +jw, as shown in Figure 3.11b.
Meanwhile, for 53:: its argument does not change as { switches sides; consequently, it
stays in the same location. The branch out enforcing the radiation condition on p, is
easily determined, and coincides with the branch out for p, along the imaginary axis.
The Green’s functions can be evaluated as the sum of the real-line integral along
the branch cut from -E~< C,< 5w the real-line integral from 3:5,. to infinity, and the
small deformation around the poles at 3:5} For a C‘ (Figure 3.11a), the argument of
p, is +1/2 along the branch cut in 5 and 0 beyond the branch point, on the real-line
contour to j; on. For a C‘ (Figure 3.11b), the argument of p, is 45/2 along the branch
cut but still 0 beyond the branch point. Constructing the radiation spectral mode as per
(3.48) indicates that there will be an standing wave pattern in y within the substrate,
arising from that portion of the integration path along the branch cut. A similar analysis

indicates that p, has an argument of 0 upon the real-line contour.

82

Figure 3.11

 

 

+51); +5“ +EP
(51)
em c- Co
in! '51. -E°

 

(b)

Complex 5-plane singularity locations for the substrate radiation regime.
(a) Interior side of branch cut S. (b) Exterior side of branch cut S.

83

The pole deformation integral is proportional to the residue at that surface-wave
pole. Consider the deformation along the positive real axis in the 5-plane. For a f on
the interior side of S, that portion of the Green’s function is proportional to ‘41:}, a
phase front propagating in the -x direction. As I switches sides of the S branch cut, the
pole locations switch and the inversion contour deforms about -E; as in Figure 3.11b.
The contribution of this pole is now proportional to e '1 i" . The superposition of both
sides of branch cut S indieates a standing wave pattern in x. Naturally, the same applies
for the pole on the negative real axis.

For any value of ]' upon S within the substrate radiation regime, the Green’s
functions, and consequently the electric field, are oscillatory in x and in y for y < 0 but
exponentially decaying in y for y > 0. This corresponds to a radiation mode that can
carry energy away into the substrate and transversely away from the waveguide by an
excited surface-wave mode of the background structure. Also, as the value of r is real,
each of the radiation spectral components are propagating radiation modes.

The second regime of interest is the cover radiation regime defined by the portion
of the branch cut denoted B. Radiation modes in this regime are either propagating, with
axial wavenumbers 0 < C, < k, , or evanescent for -jo° < C l < 0 . The effect of branch cut
B upon the complex 5—plane is shown in Figure 3.12; notably, all the types of
singularities are affected. Based on the previous analysis, it is observed that a standing
wave pattern in y will occur for y in either the cover or substrate. The background
surface-wave pole is still intercepted, and consequently, a standing wave pattern in x
occurs as well. These cover, or full, radiation modes carry energy away from the

waveguide into the cover and substrate regions.

84

-E° -EI'I -£b¢

 

———@—n—t—' + , 5,,
+5.” +51, £5
(a)
Em C - CO

“Ebc “a“ -a; a
l $ “u "4....‘..... l as re

+5? +6“ +£bc

 

(b)

Figure 3.12 Complex 5-plane singularity locations for the full (cover) radiation regime.
(a) Interior side of branch out B. (b) Exterior side of branch cut B.

85

The last regime of interest is the surface-wave radiation regime. This is
characterized by k, < C, < 1;; the effect of the transform-domain branch cut 1’ upon the
complex 5-plane is shown in Figure 3.13. Radiation modes in this regime are
propagating modes, possessing evanescent behavior in y for y in both the cover and
substrate regions, but still possessing oscillatory behavior in x. This radiation mode will
then carry energy transversely away from the waveguide within an excited background
surface wave mode. Note that in this case, the energy is confined to the film layer of
the background structure. Whether this portion of the radiation spectrum is a significant
contribution depends upon the background structure. For thin-film structures, it is

expected that this radiation component will be small.

3.5 The Proper Role of Leaky-Wave Modes

A leaky-wave mode is a discrete mode of the waveguide that possesses non-
spectral behavior. A leaky-wave mode is a solution of (3.35) whose field distribution
exhibits exponential growth transverse to the waveguiding structure in either x or y.
Obviously, these modes cannot physically exist over all of space; consequently, these
modes certainly are not part of any proper eigenmode expansion of the waveguide field.
Yet, these leaky-wave modes are of tremendous interest to the research community, and

much effort is expended to determine the leaky-wave mode solutions.

3. 5.1 Identification of leaky—wave modes via the EFIE
For the discussion in Section 3.5, the background environment is assumed to be

a conductor/film/cover configuration. The film is assumed to be thin and hence only one

86

 

 

 

 

(b)

Figure 3.13 Complex 5-plane singularity locations for transverse-only radiation

regime. (a) Interior side of branch cut P. (b) Exterior side of branch out
P.

87

surface-wave background mode (TMO) is supported. This is the situation depicted in
Figure 3.14a, where the axial transform-plane branch cuts are defined to enforce spectral
behavior. This section addresses finding leaky-wave solutions to the EFIE (3.35).

The branch cuts in the axial transform plane serve to define a four-sheeted
Riemann surface, as depicted in Figure 3.14b. The branch cuts are the limiting case of
spectral behavior. The top sheet, denoted (1), is the spectral sheet. Upon this sheet lies
the inversion contour and the bound guiding modes (proper modes) of the waveguide.

The second sheet is reached by intentionally passing through the branch out P.
When this branch cut is violated, the background surface wave pole in 5 migrates above
the real axis and introduces non-spectral behavior (exponential growth) in x. The branch
out B is still obeyed and exponential decay in y is still maintained. Solutions on this
sheet are called surface—wave leaky modes, as the non-spectral behavior is confined to
the background planar interface.

The third sheet is reached by intentionally passing through branch cut B from the
top sheet. In this case, the 5-plane branch point migrates above the real axis and
introduces non-spectral behavior in y. The branch cut P is still obeyed, and the
background surface wave is not excited. Solutions on sheet (3) are called space—wave
leaky modes, as the leakage effect is directly into the cover region but not into the
surface wave. Sheet (3) cannot be directly reached from locations on sheet (2). This
implies that a surface-wave leaky mode cannot evolve into a space-wave leaky mode.
The last sheet, sheet (4), is reached by violating both the P and B branch cuts. On sheet
(4), energy leaks into both the cover and the background surface wave. Sheet (4) can

be reached from either sheet (3) or sheet (2).

88

 

 

 

(a)

 

Spectral (top) sheet

°B P
(2) f Surface-wave leakage
BT i

 

 

 

(3) Space-wave leakage
(4) Surface- & Space-wave
leakage
(b)

Figure 3.14 (a) Four-sheeted axial wavenumber (complex-0 plane.
(b) Nature of each Riemann sheet.

89

Unfortunately, by allowing the 5-plane singularities to migrate above the real-
axis, the original Fourier transform on x becomes non-convergent on the real axis. The
inversion contour must be deformed to stay within an analytic region in the 5-plane.
Complicating matters is that the analytic regions do not overlap (Figure 3.6b). To
maintain convergence of the forward transform, the integral needs to be considered in
the sense of (3.6), with a contour lying within each convergent half-plane. By using
analytic continuation, the analytic function defined where each Fourier integral in (3.6)
converges can be extended uniquely (Monodromy theorem, [50]) until a singularity
is encountered. An analytic continuation can therefore be defined for each region of
convergence from (3.6); within this analytic continuation lies the deformed inversion
contour, as shown in Figure 3.15. By the process of analytic continuation, the forward
transform on x remains convergent for leaky-wave modes.

The above discussion suggests the method in which to use the EFIE of (3.35) to
find leaky-wave modes. First, a choice of sheet is made, which chooses the nature of
mode leakage that is to be of interest. The appropriate branch cuts are violated
depending upon the nature of the leaky-wave mode of interest, and their associated
singularities are allowed to migrate across the real axis. The inversion contour must be
deformed and kept above these singularities. Once the 5-plane 'singularities and
inversion contour are known, the Green’s functions are evaluated, and a solution to the
homogeneous EFIE is then determined. More specific details on this topic will be dealt

with in Chapter 6.

9O

5i

i; Inversion contour

   

 

 

 

 

Figure 3.15 Deformed inversion contour in complex 5-plane used when upon a non-
spectral Riemann sheet of the axial-transform plane.

91

3. 5.2 Usage of leaky-wave modes

Up to this point, this dissertation has followed the traditional presentation of
leaky-wave modes within the literature, namely, how to find these solutions. It now
departs from tradition and explains how to use the leaky-wave mode solutions. It should ,
be stated that a number of good references are available that treat this topic [20,51,
52]; however, these references restrict analysis to a two-dimensional problem.

The total field of the waveguide is often times desired, especially for determining
the radiation patterns of open-boundary devices. The Green’s functions have the form
(3.49), from which it is clear that as x becomes large the Green’s functions are highly
oscillatory and very difficult to compute numerically. This nature, while rendering
numerical integration techniques useless, is readily amenable to an asymptotic expansion

evaluation via the method of Steepest Descents. Briefly, the method of Steepest Descents
(SD) is a saddle-point method applied to evaluate integrals of the type

[Korma (3.51)

o
where a is a large parameter. General details can be found in Matthews and Walker
[53]. In brief, Cauchy’s theorem allows the original inversion contour to be de-
formed to one upon which the exponential in the integrand has constant phase and rapidly
vanishes (the Steepest Descent Contour or SDC), which allows the infinite contour to be
approximated by a contour of finite length. This SDC is defined by the relationship

Mlfllfl = 8m(f(zo)l (3.52)

where the SDC passes through the saddle point 20, the point where f’(z0) = 0; this saddle

point region is the dominant contribution to the integral.

92

Application of the method of steepest descents to this problem allows the
determination of the scattered field from the open-boundary waveguide at large distances.

The space-domain seattered field is an inverse-transform on r of the transform-domain

scattered field. In the transform domain, the scattered field is

2 -o
°" (,p)§‘(5l5’.C)-?,(5’.C)ds’
"C

 

a5o=f
(3

where E, is the field in the guiding region that serves as the equivalent source supporting

that scattered field. Obtaining the spacevdomain field means that it is necessary to
evaluate integrals of the type

E.(r) = [ e.(p.C)e"‘dC 1353)

where e. is a scattered field component. In two-dimensional problems, this scattered
field integral is converted into polar coordinates in both space coordinates and in spatial
frequency, which allows for identification of a saddle-point with specific physical
interpretation. For a three-dimensional problem of the waveguide, a transformation into
a spherical coordinate system should be effected.

Observe first that the Green’s function is an inverse transform on 5. Also note

that the waveguide field is a function of i)" and r, but not of 5. An interchange of the

spectral integral on 5 with the spatial integral over the waveguide cross-section gives

6.0030 = f F.(E)H,(£.C)e"‘e""de (3.54)

where 11 is the source field integrated over the waveguide cross-section and F is a

Green’s function coefficient. Depending upon p, = [£2 - Calm” , and the appropriate F,

93

equation (3.54) can represent the field component anywhere in the layered background.
This formulation is explicitly a function of 5 and r, in which the spectral integral on 5

can now be evaluated asymptotically. A change of variables is made to polar

coordinates, namely,

E = Eusimt’ x = psint
p, =i£ucost’ y = 90054)
after which the square root upon 5 for p, is no longer implicated. Equation (3.54)
becomes

R,(5.o = f r,(£,,.4’)II,(t,,.¢’.t)e"‘""°°""”’d¢’

An asymptotic evaluation of the integral upon 4)’ shows that the saddle point occurs when

9 ‘9’; consequently, equation (3.54) takes the form

RAM) = i.(5.¢:£,,.t).-rt.p (3.55)

When (3.55) is used in the inverse transform on g5 of (3.53); equation (3.53) becomes
I.(r) = fR,(p.4;t,,.C)e"‘""e1“dc (3.56)

This form is again amenable to evaluation by the method of steepest descents, using

another set of polar variable transformations on complex angle 6’ = o +jn ,

c = k,sine’ (3.57)

EN = “(,2 _ c2 fleOSBI (3.58)

94

The transformation in (3.57) is similar to the traditional SDC mapping for two-
dimensional problems. Evaluating (3.56) using (3.57) and deforming into the steepest

descent contour gives insight into the usage of leaky-wave modes.

3. 5 .3 Physical interpretation of leaky-wave modes

The ”Steepest-descents” plane in Figure 3.16 is a mapping of 3' through (3.57)
and (3.58) which removes a branch cut in (5, in this case, there is no mechanism to
separate spectral from non-spectral sheets. There are 8 quadrants in the steepest-descent
plane; their relation to the original axial transform domain depends upon (3.57) and
(3.58). Finishing the spherical coordinate transformation by usingz - rsin 0, p - rcosO

results in
I.(r) = f 3.0.5.6):""'°°“°‘°"de’

The saddle point is at 0 =0’ ; when the inversion contour is deformed to the SDC as

defined by (3.52), the final asymptotic form is
= .11" ~ “‘12 .
r,(r,e,¢) e fsxR,(r,¢,z)e dz (3 59)

As the observation angle 0 changes, the saddle point moves along the real axis in the
steepesbdescents plane. The SDC naturally follows the saddle point movement; a portion
of the SDC now lies upon a non-spectral portion of the steepest-descents plane. A leaky
mode then contributes to the waveguide radiation field only if it is intercepted by the
steepest-descent contour. This is the only situation for which a leaky-wave mode is
useful. The contour only intercepts the leaky-wave pole over a restricted spatial regime;

in this restricted spatial regime, the leaky-wave mode now possesses propagating

95

 

 

 

 

 

Figure 3.16 Typieal steepest-descent plane.

96

exponential behavior away from the waveguide as r becomes large. In that regime, the
leaky wave mode augments the scattered field. Once out of the contributing regime of
the leaky waves, the scattered field is just the saddle-point contribution.

A nice physical picture of the leaky-wave effect is given in Shevchenko, and is
reproduced in Figure 3.17. Based on (3.57), the spatial observation angle 0 can be
written as

e = tan5‘(5)
y

for any p on the y-axis. As y increases, z/y decreases, and 0 decreases. The saddle
point moves towards the origin, and at some point then leaky-wave pole is not
intercepted anymore. In this sense, it can be seen that leaky modes contribute to the
waveguide radiation field in a restricted spatial regime, and shut off at an angle of 0.
This angle is often called the leakage angle. Based on the way the angle is defined, it
is rather obvious that the leaky-wave mode is useful only near the waveguide-background
interface.

As observed before, in this regime, the leaky-wave mode propagates away from
the waveguide, thus it is used to model the waveguide radiation and transverse power
flow away from the waveguide. It is through this interpretation of the leaky-wave pole,
in the method-of-steepest descents, that the leaky-wave relates to the radiation spectrum.

Because of its field structure, a leaky-wave mode is never part of the proper
eigenvalue spectrum and cannot be used in a modal expansion of the waveguide field.
Rather, the leaky-wave mode is useful in the excitation problem of determining scattered

fields.

97

\

 

 

 

l

 

l

a:
0‘ .—— --____
o g.
0‘ l—...._........_
N

   

 

 

Figure 3.17 Physical interpretation of leaky-wave mode (plasma waveguide example
from Shevchenko).

98

Summary

This chapter has presented a formulation for the complete propagation-mode
spectrum of a general open-boundary waveguide. The discrete spectral components
correspond to bound, hybrid guiding modes and are associated with first-order pole
singularities of the axial transform-domain fields. The continuous spectral components
correspond to the radiation modes of the guiding structure and are associated with
hyperbolic branch cuts in the axial transform domain. The branch cuts in the axial
transform domain are chosen to restrict the migration of singularities in the transverse
transform (complex 5) plane and guarantee that the forward transform on x converges.
Branch cuts in the complex 5-plane are chosen to satisfy the Sommerfeld radiation
condition, while poles in the complex 5 -plane incorporate the surface-wave behavior of
the layered background environment.

A new component of the continuous radiation spectrum is identified as being
associated with the surface-wave modes of the background structure. These radiation
spectral components have a standing wave pattern in x but remain bound to the surface
of the background; this will account for energy carried away by excited surface wave
modes in the background structure. In the limiting low-loss case, this surface-wave
radiation spectrum is confined to a finite range of axial wavenumbers.

Finally, the use of leaky-wave modes was addressed. Leaky-wave modes are
discrete modes with non-spectral behavior, and are associated with poles of the
transform-domain field located on all non-spectral, improper {-plane sheets. These poles

are not part of the proper eigenvalue spectrum, as they possess exponential growth

99

transverse to the guiding axis. Their importance is linked to the evaluation of the
scattered field in a waveguide excitation problem via asymptotic steepest-descent-contour
techniques. In this case, leaky-wave modes characterize the scattered radiation field of
the waveguide in a limited spatial regime near the waveguide surface, when they are

captured by the steepest-descents contour.

100

Chapter 4

Continuous Radiation Spectrum for Planar Waveguides

In Chapter 3, a method for using an integral-operator formulation to identify an
open-boundary waveguide’s continuous radiation spectrum was advanced. Central to this
method is the criterion for cutting the axial transform plane; the continuous radiation
spectrum is the superposition of all spectral modes along those branch cuts. Confirma-
tion of this theory is desirable, but few canonical examples exist to compare with. For
waveguides in a planarly-layered background, only the simple planar waveguide
possesses a closed-form, canonical solution for its radiation modes [10].

This chapter uses the integral—operator method to identify the continuous radiation
spectrum for a planar waveguide structure, and to determine the individual spectral
components of the radiation field. The planar waveguide supports either TE fields
(where 'e' = £5) or TM fields (where 'e' = y‘ey +£e,). This simplified set of electric-field
components is particularly amenable to analysis by the transverse-field EFIE as advanced
in (2.58); in each case, the TEFIE (2.58) reduces to a single, uncoupled integral
equation, easily solved using the Method of Moments and expanding the unknown field

in terms of subsectional-domain pulse basis functions.

101

4.1 General Considerations for Planar Waveguides

A typical planar waveguide considered in this dissertation is depicted in
Figure 4.1. It is comprised of three planar layers, of infinite extent in the transverse (x)
and axial (z) directions. The substrate and cover layers are semi-infinite in extent in the
normal y direction, while the film layer is of finite thickness t. The film layer is the
guiding region, with refiactive index of n(y) uniform in x; consequently, the
background environment is a simple two-layer interface. Canonical solutions exist for
a planar waveguide with a homogeneous film layer; for this case, the film layer
refractive index is denoted n,. The cover refractive index is n, and substrate refractive
index is n,. Guided waves are assumed to propagate in the :7. directions; hence, the

waveguide fields are invariant in the transverse x direction.

From chapter 2, the Transverse-field EFIE (TEFIE) is given as

2 -°I
5,(5) = 51(5) + k?f‘—"—‘;"—’2.,(5I5’)~5,(5')4w’
a e
2 -/
+ [ 151‘,—"13..(5I5’)-5,(5’)ats’
(3 "e

5 15(5lfi’)3.(5’)-5.(5')a./ (4.1)
(3

2 */
f,——°" (,9 ’ 2,(5I5’) (5’-5,(5’))dz’
"C

where the Green’s functions are defined by

102

9

cover, nc z

 

Figure 4.1 Configuration of asymmetric planar dielectric waveguide.

103

 

' A 7 age,
in = Ys'+is.£+y s.+ ay )3"

 

.. r r as! 4.2
3.. = V,[s.+s. + a), )9 ( )

'2', = ikstW, s’+s.'+ 6:;

 

and d, - Vzlnn2(y’). The scalar components for the above Green’s functions are
defined in (2.45) to (2.46) and take the general form of

ejzu‘x’)

. (4.3)
21: d5

 

s!(xlx’:yly’) = ff!(t.y.y’)

4.1.1 Transverse uniformity considerations

As observed before, the waveguide fields are x-invariant, making the planar
waveguide essentially a two-dimensional problem. An obvious specialization in this case
is any spatial derivatives on x vanish, i.e., a/ax =5 0. The Green’s functions in (4.2)
are specialized by observing that V, = 93:- . Invariance in x also corresponds to a spatial
frequency in the complex 5-plane of zero; hence, the 5 -plane behavior is simply 6(C).
The scalar components can be a priori specialized by taking the Dirac delta function
behavior into account when performing the integration over 5, which produces the
integrand f:(E,y,y’) evaluated at E =0. Since effectively a two-dimensional problem,
the cross-section surface integral reduces to an integral over the guide thickness in y, and
the contour integral reduces to point contributions at the edges of guiding region.

The intuitive conclusions will now be developed rigorously. The EFIE in (4.1)
is applied to the planar waveguide as shown in Figure 4.2. There are two types of
integrals in (4.1) to consider when the waveguide cross-section becomes infinite in

104

 

Figure 4.2 Contour used for evaluation of TEFIE.

105

9

N,

transverse extent - the two-dimensional surface integral over the now-infinite waveguide
cross-section and the contour integral around the waveguide cross-section as edges 3 and
4, as defined in Figure 4.2, approach infinity.

First consider a typical cross-section integral, of the form
1 g F I. I 5-5 r /
cs s.(xlx .ny )1 ,(y )4: (4.4)
cs

where f,(y’) is any of the equivalent current sources in (4.1). Substitution of the
Green’s function general form (4.3) into (4.4) and allowing the cross-section to approach
infinity along x results in

R t
8 . I eJKOI-l’) . I I /
[cs Eiflffiflw) 2n d5]1,(y)dxdy

 

and a simple exchange of integration order gives

_}_ f alter-1’ )dx/
2n

Ics 5 [12005197135335 d5 (4.5)
0 «-

 

 

It is obvious that the bracketed quantity in (4.5) is the Dirac delta function 6(5). The

integral 0"“ spatial frequency 5 becomes trivial, and (4.5) becomes
3
1c. = fr,(y’)g!(yly’)dy' (4.6)
o

where

g!(yly’) = 13(5 =o,y.y’) “-7)
by the sifting property of the delta function. In this chapter, a Green’s function which
is explicitly written as g: (y | y’) is independent of x or x’ and is evaluated as per (4.7);

that is, at C =0.

106

The contour integral presents more challenges in its handling. It takes the general

form of
1,. = f 83(xlx’;yIr’)[r’,(y’)-fi’,.]dl’ (4.8)
P

where a", is the outward normal to the contour, and in which the vector nature ofg:

(since only 2, involves the contour integral) is suppressed for clarity. This contour

integral decomposes into 4 components as suggested by Figure 4.2, namely,
Ir = Il + I2 + I3 4» I, (4.9)

where the various components take the form

R
It 5 2m f8:(x|x’;yl0)[1?.(0)5(-i)]dx’ (4.10)
“-a
R
I. = 13m feitxlx’alt)[f.(r)-9]dx’ (4.11)
“-n
I: 5 Limfsflxlkyly’)[i°,(y’)°£]dy’ (4.12)
“'0
[slim ’x- "5' ’--£ ’ (4.13)
. , [at I 15ny )[r,(y )( )]dy
..o ‘

Integrals I, and I, are similar and will be dealt with first. Substitution of the
Green’s function general form (4.3) results in

_L ferret-3’)de
2n

45

 

 

[1,2 = ‘9'}..(2’13) ff:(E’y’yl,2)

where yl’2 denotes the location of contour I or contour 2. From the previous evaluation

of the cross-section integral, it is easily recognized that I1 and 12 become

107

I 2 = 5? ~i”,(y,,,) stlym) . (4-14)
The remaining two contour integrals I, and I, are trickier to handle. Since the
field and sources are x-invariant, it is expected that the contribution from each of the two

sides at infinity should cancel each other out; as written in (4. 12) and (4.13), the sum of

I, and I. should vanish. The sum is then just

I: 5 I. 5 “mfb’.(y’)°£)[s!(x|&yly’) - 83(xl-kyly’)]dr’

,r_t_x_e-r

 

ml (16") i) f [fliers u-f'fiosy’) a?! ]d€dy

where, as before, (4.3) has been substituted for the Green’s function of interest. This

becomes, after interchanging the order of integration and algebraic manipulation,

1, + I. - [(1166-12) 13m [ 2f!(t.y.y’)sin(tx)sin(R5)dt
o ""..

 

 

If PIC) = 2f:(€.y.)")sin(Cx), then the bracketed spectral integral on 5 is simply

33f rammed:
which vanishes because of the Riemann—Lebesgue lemma. The conjecture is correct, and
13 + I, = 0 (4.15)
A final representation for In can be given as
IF = 951035.065) - 9-1’.(0)g!,t..(yI0) (446)

From this point onwards, any time a Green’s function is referred to, it will be assumed

to be of the form g (y ly’) unless explicitly stated otherwise.

108

4. 1.2 Uncoupled Transverse-Field EFIE

The transverse-field EFIE can now be specialized to explicitly account for the x-

invariance of the planar waveguide structure. In this case, (4.1) becomes

51(5) = 53(5) + 1: f 5n’ty’)5.,(yly’)~5,(y')dy’
0

‘ 2 r
+ f—°" (I ’2..(yty’)5.(y’)dy’

° "5 (4.17)
+ fe,(yly’)[3,(y’)-5,(y’)]dy’

0

- %[bnz(t)(,9'3,(t))§.0’|t) - 5"2(°)(Y'3:(°))§¢("°)]

C

where the Green’s functions are now

 

 

 

5., = 13' + 23:: + 9(3: + 65% ‘4'“)
5.. = 93%: + g: + 38"]9 “'1”
a)’ a?
e. = 9(538! + -§y-[s.' + s.’ + 65)] “'20)
and d. = 911%. Under close scrutiny, the dyadic Green’s functions in (4.18)-

(4.l9) are observed to be diagonal, and (4.17) can be written as

109

51(3) -- 5,6) - f5n’ty’)§,(yIy’)°5,(y’)dy’
0

5 f°"’(y’)§y(yIY’)°'¢'t()")d>"
0

t (4.21)
- {2,0IY’)[3,(y’)"c',(y’)]dy’
+ finkogyylorext) - 6n’(0)§,(y|0)9°3,(0)]
where
#532,, + g") = 'g'll + g, = 32g,“ +993” (4.22)

The contribution from the edges at y=0 and y=t involve as source terms only normal
electric-field components (e,). Also note that dn(y’) -8,(y’) = d”(y’) e,(y’) and thaw.
is normally directed. This behavior, taken together with the diagonal dyadic Green’s
function, indicates that (4.17) decouples into two independent scalar integral equations.

The first scalar inteng equation involves only the 2 component of the electric

field, and is given as

40') 5 9.0) 5 [MysAyly’kxfldy’ (4.23)
o n,
where
smwly’) 5 k3 [s'(yly’) + s.'(y|v’)] (4.24)

This is the integral equation for TB waves on an asymmetric planar waveguide.

The second scalar integral equation involves only the ey component of the electric

field, and is

110

t

2 I
5:0) = 5,0) - f a" (y )
0

2
"e

 

t
gm(y|y’)e,(y’)dy - {MyIy’)d,(y’)e,(y’)dy(’4m

. —15[8n2(t)g”(ylt)¢,(t) — 5n1(0)g,,(yI0)e,(0)]

C

 

 

 

 

 

where
as: as.’ a2
= k2 ' + + i + + —+k2 ' (4.26)
gm C (8 8:) ay a W (W2 C )8‘
= 38' + 68" + a: +k2] r] (4.27)
g” 5y [5) (ayz 5 ‘5

This is the integral equation for TM waves on an asymmetric planar waveguide.

The EFIE for TM modal behavior is much more complicated than that for the TE
modes. The extra terms are fields due to induced charge distributions within the
waveguide. As observed in Chapter 2, tine, is a volume polarization charge; this arises
from an inhomogeneous film layer. Also, 6n2(yo)e,(yo) is a surface charge arising
from the discontinuous jump in dielectric constant at the waveguide-cover and waveguide-
substrate interfaces.

Finally, the Green’s function scalar components appropriate to (4.24), (4.26) and
(4.27) take the form

 

 

r ‘ I ‘
i3: (Yiyl) R1- I
’ng’YII -y,(y+y )
s'(yly’) = e 27 ; (e.'(yly’) I = (RN I‘ 2y (4.28)
,e[(yly’), .C ,

 

 

 

 

Y1 = [(2 _ k} ; I = S, C (4.29)

where the following coefficients are specifically for the two-layer interface

111

Yc -7:

 

 

R, = (4.30)
Ye+Ys
N17 -7

Rn ... 55—5.: (4.31)
Nit/.57.
2(rtr2 -l)y

C = " 2 c (4.32)

(r, + 1,)(N..Y, + 5r.)

4.2 TE Asymmetric Planar Waveguide Radiation Modes

There are two major considerations in determining the radiation spectrum of the
planar waveguide. The first is choosing a method to solve for the unknown field
distribution within the film guiding region. The second is choosing an appropriately
located and directed source to maintain the impressed field upon the planar waveguide.
It should be noted at the outset that the TEFIE as presented in (4. 1) is not applicable to
determining the impressed field upon the planar waveguide; for this, the definition of the
electric field via the Hertzian potential (A. l l) is used in conjunction with the original
dyadic Green’s function (2.42) with scalar components as developed in (4.28)-(4.32).

Consider the configuration of Figure 4.3. The appropriate EFIE for analysis is

(4.23). The unknown field e, will be determined by a method-of-moments expansion.

The unknown field will be expanded in terms of pulse functions, namely

4)
" 1; Iy-y,| < —
9.0) 5 2 5,. ,(y) ; p,(y) = 2 (4.33)
M 0; elsewhere

where y, = (n -‘/2) Ay; Ay = t/ N . The film layer will be assumed to be homogeneous,
consequently, n’(y’) = n} and 6n2(y’) = n} - n3 = ANfi. Under these considerations,

112

 

Figure 4.3 TB excitation of planar waveguide by line source at y=yo, z=0.

113

the scalar EFIE for TB modes becomes

 

" Ak’
2 9+4» - 27 [him + R,h,'(y)]] = ef"(y) (4.34)
n-l e

2 2 2
where Ah ‘ ANkko and the expansion functions are

1.: emly-y’l
= f‘ p (y’)dy’ “'39
It: 0 e «.00’) '

Point match (4.34) at the center of each basis function by employing the testing operator

[My-y.) ------ dy; m=1,2,-«N (4.36)
0

The result can be written as

Me...) = [cw-)1 (m

where the matrix elements A... are defined by

Al:

2
2)! [h:(y_) + R,h,(y,_)] (4.38)

C

Au=a_-

 

and the expansion functions 11,, evaluated at x," are given as

r

- - , A
e ""' "'sxnh(v,—y) ; y..*y..

A
-7‘32

h:(y.) = :24
‘ 1 - e ; y~=yl (4.39)

 

L

2 - + . A
h.'(y.) = — e '4’- ”mun—2’1)

C

The selection of an impressed field has yet to be considered. From Appendix A,

the impressed field necessary in (4.37) can be calculated from

114

where, of course,

1
jwe‘

 

my) = [styly'>-i’°(y)dy’

All sources for the impressed field must be x-invariant; furthermore, the 3-9 operator

vanishes. Thus, the impressed field for TE excitation is
do) = 2.3150) (4-40)

The only source able to support the impressed TE field (4.40) is an i-directed current

of f'(y) afjfiy), as depicted in Figure 4.3. The specific form of the impressed TE
field supported by the line current is then

-jnck¢ ‘ ' I ' * ’ .0
a???) = ——27 [[e ”‘1" "+R1e "‘" ’ )1]; (Y’M'Y' (4’41)
c 0

4.3 Results

A method—of-moments code was implemented for a symmetric planar dielectric
waveguide of thickness I located within O<y< t. This is shown in Figure 4.3. The
guiding region (film layer) is uniform and homogeneous with a refractive index n , the
cover refractive index is nc , and the substrate refractive index n, = at.

It is desirable to compare to known results. From Rozzi [10], the radiation
eigenmodes of a symmetric planar waveguide of guiding refractive index n , cover

refractive index nc and thickness 2d are, for the TB even case,

9,0) = -2--l-cosoy. ly|<d (4'42)
nC

 

c= 1 + (1)2sin1ad (4-43)
\ p

where p = -j 1‘ is the wavenumber in the cover region, ranging from 0 < p < co. Also,

 

o2 =v1+p2 and v2 -k:(n} -n3). Note that the form given in (4.42) assumes the
waveguide is centered about the origin in y. An eigenmode expansion for the excitation

of a line-source located at ya is simple; the total radiation field is

E,'(y,z) -

 

 

kenolo ]‘ ”[00'1/2” “won't/2)] elCtMIz-z’l dp (4-44)
21: o C2 ((9)

where the origin has been shifted to the center of the previously mentioned waveguide.

To excite even eigenmodes in the planar waveguide, the impressed field must be

supported by an line source current in the transform domain of

j:(y229C) a 105(y-t/2)e'j(ll

located at the center of the waveguide (yo = t/ 2). This leads to the MoM implementation

of the impressed field
e;(y-) = Ice ’1‘1'———-J::k° [e ”CV-"’2' + Rre ”Jr-"liq (4.45)
C

Because of the symmetry of the background enviroment, only one branch out in

the axial transform plane is implicated. From Chapter 3, the total spatial radiation field

for the symmetric planar waveguide is constructed from the solutions to the EFIE as
1 "' ,
E.‘(y.z) = — f [e.(y.<‘)-e.(y.<°)]e1"” '4: (4-46)
21: t

For the case of limiting low-loss dielectrics, ye exhibits conjugate behavior on either side

of the branch cut. When this conjugate behavior is considered in equations (4.23),

116

(4.24), (4.28), (4.30), and (4.41); it becomes obvious that the radiation spectral mode

fields possess conjugate behavior as well, namely

abs?) = jkonolov,(y.l.')
e,(y.C°) = ik.nolov;(y.C)

Here, f without any superscripting is assumed to be the interior side of the branch cut.

The EFIE-determined radiation field then can be constructed as

£30 z) —onf1konol

[HxO- v;(y.C)]e"" “-2149 (4.47)

where the integration has been parameterized upon the wavenumber in the cover and
where 1'1: = p .

The integral-operator technique is now applied to a symmetric planar waveguide
where n,=1.5, nc=1.0, and t =0.25>\. Results are given in Figure 4.4, where the
integrands of equations (4.44) and (4.47) are compared for various spatial frequencies
in the cover p (normalized to lg). This corresponds to an axial wavenumber f on the

branch cut of

2 2
- ; 0< <
c: M” p " "° (4.4s)
-iko(/p’-n3 ; n,<p<°°

It is evident that agreement between the two techniques is very good. Both methods
show that the field periodicity within the waveguide increases as spectral frequency
increases. The spectral peaking associated with these radiation modes is also exhibited.
For the symmetric planar waveguide, the agreement between the integral-operator

analysis and classical differential operator theory is very good.

117

n,=1.o, n,=I.5, t/x=o.25

 

Yo=t/2- 20:0
40.0 -
‘ analytical spectrum
‘ Hut numerical MoM solution
30.0 -
. p=25
20.0 a .

10.0 -

radiation spectrum amplitude

 

 

 

0.0 0.4 ' .0
normalized location (xO/t)

Figure 4.4 Spectral radiation mode amplitudes of a symmetric planar waveguide
obtained by the integral-operator method compared to Rozzi’s analytical
closed-form results.

118

This same integral equation is now applied to an asymmetric planar waveguide.
This waveguide under consideration has the same physical characteristics as the
symmetric planar waveguide considered, save that now n, e n‘ . This introduces another
branch cut in the axial transform plane associated with k,. In the low-loss limit, the
branch cuts coalesce as shown in Figure 4.5, where an upper half-plane closure has been
chosen (case of z > 2’). It is obvious that there are two distinct regimes of the radiation
spectrum. The substrate radiation regime is associated with branch cut S, where
n‘: < (Jk, < n’. In this regime, 1, is imaginary and possesses conjugate behavior on S
while 7, is still real (or p is imaginary). The cover, or full, radiation regime is
associated with branch out B. In this regime, both 1, and ye are imaginary and possess
conjugate behavior.

Figure 4.6 shows the amplitudes of various radiation regime spectral modes as
defined in (3.47) for an asymmetric planar waveguide with nf= 1.5, tr, = 1.0, t =0.25k,
and n, = 1.2 (so njnc= 1.2). The excitation remains a line current at yo - r12 , the center
of the guiding region. The values of f correspond to z > z’ closure as in Figure 4.5.
It is obvious that the amplitudes associated with the substrate radiation regime
(-l.2 < C < -l.0) are small compared to those in the full radiation regime, indicating that
the full radiation regime dominates the non-evanescent portion of the radiation spectrum.
As 1’ moves deeper into the full radiation regime, the field periodicity increases as
expected.

Figure 4.7 and Figure 4.8 both show the effect of asymmetry in the background
environment (moderate asymmetry n,/nc= 1.2 for Figure 4.7 and small asymmetry

njn,=l.05 for Figure 4.8) on the radiation spectral modes by comparing to those

119

 

 

 

 

 

Figure 4.5 Axial-wavenumber plane (complex f—plane) branch cuts for a typical
asymmetric planar waveguide.

120

nc=1.o, “9:1.5, n.=1.2
YO=t/2. 20:0. t/A=O.25

 

110.0
100.0
90.0 C/ko
80.0 -1.19
...... -1.15
70.0 use: --1.10
e a e e e —O.95
60.0 4.29.4.9 —0.77
I I X I I —O 30

50.0
40.0
30.0
20.0
10.0

~ g '9‘." +1050
'___' r . ' +1302

 

 

radiation spectrum amplitude

 

.0
o

0.0 0.2 . 0'4 01'5 0.8 1.0
normalIzed locatIon (x/t)

Figure 4.6 Spectral radiation modes in guiding region of asymmetric planar wave-
guide for both substrate and full radiation regimes.

121

nc=1.0, n,=l.5, n,=1.2
yo=t/2. zo=0. t/>\=o.25

4:.
.0
O

Asymmetric waveguide
_- -- Symmetric waveguide

 

or
.0
o

 

radiation spectrum amplitude

 

 

 

F r I F I I I I

0.0 0'2 . 0.54 ole as 1.0
normalIzed locatIon (x/t)

Figure 4.7 Effect of asymmetry (n,/n,. = 1.2) on the radiation mode field distribu-
tions, compared to the symmetric planar waveguide.

122

n,=1.o. n,=1.5. n,=1.05
yo=t/2, zo=0, t/A=0.25

 

 

 

 

 

40’0 7 Asymmetric waveguide
q, 1 -_-.. Symmetric waveguide
.0
3 .

,4: .
330.0 -
E .
o .
E I
g 20.0 -
0 -' p=5.
o . , ’
a ., .
C 10 o - p=0 5
o .
‘5 ‘ Av -------------- ‘
.9 ' ~-
'0 . ..
o -
L.
0.0 Irrfirfir[rrrlfirr[rrrl
0.0 0.2 0.4 0.6 0.8 1.0

normalized location (x/t)

Figure 4.8 Effect of small amount of asymmetry (an, = 1.05) on the radiation mode
field distributions, compared to symmetric planar waveguide.

123

spectral modes of a symmetric environment. This comparison is only made over the full
radiation regime for normalized cover spectral frequency p (relation (4.48) still is valid),
as there is no substrate regime for the symmetric case. A noticeable shifi in the location
of the guiding region maximum towards the cover is observed over the propagating
portion of the full radiation spectrum (0 < (,lko < nc or nc > p > 0) for the asymmetric
case. The amplitudes increase as well. Both effects are more significant at low spectral
frequencies. Furthermore, these shifts are more pronounced when strongly asymmetric.

As p moves deeper into the full radiation regime and the modes become
evanescent, the maximum shift and amplitude difference disappear as seen in both
Figure 4.7 and in Figure 4.8. Deep into the radiation regime, p at C/ko, meaning that
1, -° 1, -jC . At high spectral frequencies, 1' dominates both kc, k,, and the background
environment is a slight perturbation effect on the impressed field. This indicates that
iterative techniques can be quite effective in determining the radiation spectral modes at

high spectral frequencies.

Summary

This chapter has demonstrated the implementation of the integral-operator
technique in determining the continuous radiation spectrum for planar waveguides. The
symmetric TE planar waveguide is a canonical problem for which closed-form analytical
solutions are readily available. The integral-operator technique was found to be accurate

and effective for determining the spectral radiation modes of these canonical problems.

124

Chapter 5

Continuous Radiation Spectrum for Microstrip Transmission Line

Having established that the transform-domain integral equation recovers the
correct radiation spectrum for a canonical planar waveguide structure, it can now be
applied to more common open-boundary waveguiding structures. This chapter applies
the theory to the case of an isolated microstrip transmission line. In this chapter,
numerical solution of the EFIE for a single microstrip line is implemented by Galerkin’s
method of moments with Chebyshev polynomial basis functions. Radiation-regime

current distributions are presented.

5.1 Application of the EFIE

A typical microstrip line is shown in Figure 5.1. In this case, the strip is
assumed to be a perfect conductor with infinitesimal thickness and a width of 2w, located
at the film/cover interface. The film layer is of thickness t, and may in general be a
lossy dielectric. This film layer is backed by a perfect conductor at y=-t and immersed
in an air cover region.

Radiation spectrum surface currents of the microstrip line are determined by

solving the EFIE for microstrip devices (2.43) under excitation; for the present case, the

appropriate EFIE becomes

125

'<>

X>

"e W

 
   

Perfect Electric Conductor

Figure 5.1 Configuration of microstrip transmission line.

126

lint“ f s°(xlx’;yIy’=0.C)-I'E(x’.odx’ = -i‘.e‘(x.y=0.o <5-1)
r0 -

in the limit where y ~ 0 on the surface of the strip and where the electric Green’s dyad
is used for notational compactness. Exchange of the limit y _. 0 with the source point
integration is permissible under the condition that the integral remains convergent. When
the source point and the observation point coincide, special consideration may be needed;
convergence properties will be examined later. The same observation applies to the
impressed field; however, it is of significance only if the source for the impressed field
is at y = 0.

The surface current on the infinitely thin strip has only tangential components in
the axial (z) and transverse (1) directions, namely

E(x.C) = :2 k,(x,c) + 2k,(x,c) (5.2)

One immediate consequence arising from this current distribution is the observation that
the depolarizing dyad for the electric Green’s dyadic is not necessary. Direct substitution

of (5.2) into (5 .1) and enforcing the tangential boundary conditions results in a pair of

coupled scalar integral equations

f{8£.(xlx’;y|0;C)k,(x’) + 3;(xlx’;y|0;C)k,(x’)}dx’ = -e,‘(x.y=0)
"" (5.3)

f{8£z(x|x/;YIO;C)k,(x') + g;(x|x’;y|0;C)kz(x’)}dx’ = —ez‘(x,y=o)

127

It should be noted that even though (5.3) does not involve normal (9) electric field

component this does not mean there are no normal electric fields. The scalar components

of the electric dyadic Green’s function are given as
8:.(xlx’.yI0) = lie (E,()e’“‘""e""d£ (5.4)
21: .. "

where a, B assume values of x and z. The coefficients C“ are functions of E and 1',
taking the specific forms of

2 2
(Np,- " 1) £2“; + (kc ‘ £2) (55)

6.45.0 = , , .
z (E)Z‘(£) z (a)

 

 

(Ni-um. _ (I: (5.6,
Home) 2‘05)’

 

(3:03.!) = (345.0 =

(N;- 1) :1)». + (k? -c’) (5.7,
More) Z"(E)

 

 

Ca(E.C) =

and where Z‘ and Zll are defined in Appendix B.
As the microstrip line is symmetric about 1 = 0 , invoking parity about this point
can simplify the problem. For the microstrip, the parity states depend upon the surface

current and surface charge density. The surface charge density in terms of k(x) is
- a 1: ° k
9.0:) - 312 ,(x) +1C ,(x)

which can be decomposed into even and odd behavior in x. Looking at the dependence
of the surface charge upon f(x), it is readily apparent that even axial surface currents
and odd transverse surface currents generate an even charge distribution; this will be
considered an even made of the microstrip transmission line. Consequently, the
appropriate parity states are then

128

even mode: kz( -x) = kz(x) ; k,( -x) = -kx(x)
odd mode: kz( -x) a -kz(x) ; kx( -x) = kx(x)

(5.8)

5.2 Method-of-Moments Solution

As with the planar waveguide, there are two issues to resolve at this point -
method of solution for the EFIE and choice of excitation. Solution of integral equation
(5 .3) is accomplished with Galerkin’s method of moments utilizing entire-domain basis
functions. These basis functions are chosen to explicitly accommodate the edge
singularity in axial current. Entire-domain basis functions are preferred in this situation
primarily because the current is represented in a compact form with a relatively small

number of unknown expansion coefficients.

The transverse and axial current components are expanded as

N N
’50:) = 2 k... n,.(x) k,(x) . 2: kn, uu(x) (5.9)
3'0 n-O

where the ie.-(x) ’s exist over the entire domain -w s x s w and have the axial current

edge singularity built in. Substitution of MoM expansion (5 .9) into (5 .3) results in

N N
2 knl;(x)+2 1:31;“) = -e,‘(x)

I;l a): (5.10)
2 ago»: raga) = -e,‘(x)
a-l a-l

to be enforced on the domain -w s x s w, and where

1.3.0) = lim f 8:3(xlx’;y|0;C)u,,(x’)dx’ (5-11)
r0 .,

129

In Galerkin’s method, the expansion basis functions are used as the testing

functions, resulting in a testing operator of

[hug-ml“) a a 1.7.

which leads to a set of 2N by 2N linear equations that can be written in matrix form as

At: An kn a: B: m," = 0,1,..." (5012)
A: A: k. B:
The matrix elements are
.43; = f u“(x)l:,,,(x)dx (5.13)
B: = f u._(x)e:(x)dx (5.14)

where a, 3 take the values of x or z and where 1:” is defined in (5.11). This MoM

matrix equation is inhomogeneous, and is easily solved for a given excitation.

5. 2.1 MOM expansions
Implementation of the method of moments solutions to develop the matrix
elements given by (5.13) and (5.14) is straightforward. In this implementation, the

spatial integrations over the basis functions will be exchanged with the spectral

integration defining the Green’s functions. Consequently, (5 .13) becomes

4:; = ling f e”"C.,(z)g.,.(of,,(ode a,p -.- x,z. (5.15)
Y" ..

where

130

f,.(£) -= f u,,(x’)e"“’dx’ (3 = x, z. (5.16)

3.,(15) . fuu(x)e"‘dx a = u, (5.17)

Matrix element (5.14) cannot be dealt with until a source excitation is chosen.

In this solution, Chebyshev polynomials with square-root edge factors are utilized

as basis functions where

u-(x) = T,.(1t/W)I/1-(x/w)2
u,,.(x) = T.(x/w)I\/r - (x/w 2

where T,(x/w) is a Chebyshev polynomial of order n of the first kind and k, and k, are

.. -w s x s w (5-13)

unknown expansion coefficients. One advantage of using Chebyshev polynomials as
basis functions is that the spatial integrals can be evaluated analytically in closed form.
Noticing the fact that Chebyshev polynomials of even order are even functions and odd
order are odd functions, the spatial integrals reduce to the following four generic types

and are evaluated as

' T2.(x/ w)

0 J1 - (xlw)2

w T2,.1(xlw) sin

0 l1 - (x/w)2

 

cos(Ex)dx = my 1‘5! Jh(£w) (5.19)

 

(Emu = (-1)"-’£2!Jm,(5w) (5.20)

f Thu/WW 1 - (x/w)2 cos(Ex)dx
o

(5.21)

= <-1>'-’;—" are») + isms») + g1......«w>

 

131

f T2M(x/w)y/l - (xlw 2sirr(EJc)dx
o

(5 .22)

= <—1>'1';‘3[I....<zw> + game» + p....«»»

where J,(x) is the Bessel function of first kind.

5. 2.2 Excitation considerations
Choice of an excitation will be made in the sense of Chapter 3; that is, point
sources will be used to identify basic behavior of the radiation spectral components. The

field supported by a point source within the layered background of the microstrip

transmission line is given by
err) = -jn.k. f ma I sen-ita’w
cs

For a point source current at p‘ = 50 flowing in the .9. direction,
f‘w) = 10505 ‘50”.
The appropriate fields incident upon the microstrip transmission line then become

4(a) = cox-mamas.

‘ (5.23)
31(5) 3 eo£'§‘(5lfio;C)'£¢

where :0 = -jr|ck‘.lo.

An arbitrary current density can be decomposed into its even and odd contribu-
tions. This fact can be exploited to reduce problem complexity. It is desirable to excite
the even and odd parity radiation-mode surface currents on the microstrip using a point

source excitation. For even parity on the microstrip line, axial surface current is even,

132

transverse current is odd, and the surface charge distribution is even about x=0. How
does the surface current parity affect electric field parity? Consider thatri 163(1)) = -p’(x)
at the surface of a perfect conductor, of which the microstrip transmission line is
assumed to consist. Since ti =9, obviously, e’(x) is even in x for even microstrip
modes. An analysis of Maxwell’s equations (6.3),(6.4) indicates that e,(x) will be even
while ex(x) will be odd in x. Obviously, the opposite situation prevails for odd
microstrip parity state.

Based on the above analysis, a point source excitation current within the system

must take on the form

even: 3.0000) = 250’ Wu) at "0 =0 (5.24)
0ch- j(xo,yo) = 35000) 0‘ "o '0

to excite even/odd modes of the microstrip. Figure 5 .2 and Figure 5 .3 show this set of

currents. Consequently, this results in an impressed field within the layered background

of

(5.25)

N)

away) = i'§°(x|(kylyo)-

N)

e.“""(x.y) = 2'§‘(xlo;ylyo)°
eI!"‘""(Jr.y) = £'§’(xl0.ylyo)°£

ef‘”(x.y) = i°§°(x|0;ylyo)°£
Once the source excitation has been chosen, the impressed field is then worked

into the Method-of-Moments expansion. Manipulating the forcing function matrix

element (5 .14) in the same fashion as (5 .13) results in

133

 

    

/
'9 act lectric Conductor

Figure 5.2 Excitation of even radiation spectral modes for microstrip line.

134

 

_y=0

  

2‘Y'4

   

’ erfect lectric Conductor

Figure 5.3 Excitation of odd radiation spectral modes for microstrip line.

135

B: = f e""°e”“°C.,(£)g,,(r)dz a,=x,z; (5.29)

where v is the excitation current orientation and 10 = 0 for parity state excitation.

One final point of interest involves the MoM form of the forcing function 8"," .
Whenever the excitation source is distant (y, large), or whenever the axial wavenumber
{is far into the evanescent radiation spectrum (C = j C" I, large), the e ""° term becomes
highly oscillatory as E -o E“. This situation is amenable to evaluation by asymptotic
methods about the saddle point.

Performing an asymptotic evaluation involves casting B: in the form
B: = IF:(£)e.f(0dE

where a = -y, and {(5) = pc - ‘/ 51-53,. The saddle point occurs when f’(5,,) -0; this
is at i =0. Deforming the integration into the steepestodescents contour in the complex

f—plane results in

3" = 2e""“-’°""’ f F:(jq5,,x)e”°"”‘:dn (5.30)

where a change of variables shows that

B: = 2 l Egg-“(530114)! F:( 1'45, P) e 41,» (5.31)
yo --

This could be indicative of problems as (4 becomes large.

136

5.3 Results

The method-of-moments analysis was implemented numerically. The results
presented here are typical. The physical parameters of the low-loss microstrip
transmission line under question are: film refractive index nf=3.13, cover index of
n,= 1, film thickness of t=0.0635>\, and a strip half-width w=2.85t. Even modes were
excited by a i directed unit current source at x=0 (Figure 5.2), while odd modes were
excited by a unit current at x=0 flowing in the 2 direction (Figure 5.3). All results
were determined assuming an upper half-plane closure (2 > z’). The radiation regimes
for this problem are seen in Figure 5.4.

Figure 5.5 and Figure 5.6 show the normalized current distributions for the axial
and transverse surface currents within the full radiation regime (branch cut 8). It is
obvious that as axial wavenumber r moves deeper into the full radiation regime that the
periodicity of the induced current on the microstrip increases. The amplitude appears
to have no discemable pattern at this time. Figure 5.7 compares the effects of
differences in excitation current location in y upon the radiation mode surface current
density. It is apparent that as the excitation source approaches the microstrip, the
transverse currents are strongly excited. This is because, near the microstrip, the x-
directed electric field dominates. This has significance for microstrip excitation problems
by near-proximity sources, as it shows that any quasi-TEM analysis, or analysis ignoring
the transverse current, will not be sufficient because of the dominant effect of that

electric field.

137

 

 

 

P +kc +Ae

 

 

 

EB """ E """"" " p C
— — I 0.0-0.0.0.0.0-000-0. re
A p k c r

 

Figure 5.4 Axial wavenumber plane (complex {-plane) branch cuts for low-loss
microstrip transmission line.

138

axial current, even modes

   
  

 

Y0 = 50
1.00 f ' ‘ a
{/ko ,’ \
o-o-o-o-o —08 / \
rag-ea —O.4 I \‘
He-H —0,1 ,’ \
0.80 H—t—H 0,0 I \
a, He... 10.1 I x
‘0 n---- 10.5 ’ \
3 e-oe-oe 11.0 I \
+4 I \
°; o-o-o-o—o 12.5 I
Cl0.60 "...... 15.0 ,
E I
O I
I
"C 9
Q)
.'_l‘ 0.40
C
E
$—
0
C

 

0.00 0.20 0.40 0.60 0.80 1.00
pOSItIon (x/w)

Figure 5.5 Normalized radiation regime axial surface current density, even parity.
Excitation is 5A above microstrip line.

139

transverse surface current density
Y0 = 50
1.00

w
\
g.

I
999

I:
liii
l

0.80

\\
/
I
i I
t l
6 +
mites .0

obeQodem

9
O?
0
re
0

0.40 ’ \

normalized amplitude

0.20 ’ .. ''''

....... ‘- ‘ \
[’1’ - ‘.
%.-.A-.‘~.-‘n - ....... .‘.‘§‘.

0. 40 0.60 0.80 1.00
position (x/w)

 

 

Figure 5.6 Normalized radiation regime transverse surface current density, even
parity. Excitation is 5A above microstrip line.

140

current density, {/k0=0.75

 

 

 

 

1.00 — k2. yo=10)\
o-o-o-o-o k2. yo: 1)\
"" k," yo=10)\
coo-04 km YO: 1)\
00.80
.O
3
.4:
0.
E060
O
8
No.40
6
E
00.20
C I
I
l
0.00 TwilllrllilrrllllllllilllIIlrlllllllrllllllllllrfia
0.00 0.20 0.40 0.60 0.80 1.00

position (x/w)

Figure 5.7 Effect of source excitation distance on radiation regime surface current
amplitudes.

141

Figure 5.8 through Figure 5.11 display the actual current amplitudes for the axial
and transverse radiation-regime surface currents in both even and odd parity states.
From these results, it is obvious that the radiation-regime surface currents acquire an
increasing periodicity across the strip width as I becomes more imaginary (deeper into
the radiation regime, farther out the branch cut).

The observed increasing amplitude behavior in Figure 5.8 through Figure 5.11
is troubling, as these surface currents do not seem be converging in spectral frequency.
This behavior is also completely contrary to the observed behavior in Chapter 4. This
observed behavior may not present a problem however.

This increasing amplitude behavior seems to be associated primarily with the
computation of the impressed field MoM elements from (5.29) as observed in the
asymptotic form of (5.31). An increasing amplitude trend as (“increases is obvious from
inspection of (5 .31). Yet, this impressed field exists in the space-domain. It is believed
then that the form of (5 .29) is unstable.

There are other possible explanations for the surface current amplitude trend.
Recall from Chapter 4 that only TE modes of the planar waveguide were considered;
those TE modes did not depend upon charges within the waveguiding structure. This
spectral amplitude behavior may be caused by the charges involved in this microstrip
transmission line.

Even so, this data trend is probably not serious. The amplitude behavior occurs
in the evanescent portion of the radiation regime. When 2 at z’ , the radiation spectrum
superposition over radiation spectral components, identified by contour-closure in the

complex {-plane, decays exponentially, which will annul any increasing amplitude trend.

142

50.00

4;
9
O
O

30.00

N
9
O
O

10.00

radiation spectrum amplitude

Figure 5.8 Radiation regime axial surface current amplitude, even parity.

 

$rrrrrrrlrrrrrrrrrlrrrrrrrrrl

 

Us:

— — —--——.—«—~~—a-~—---‘-

oxiol

current, even

C/ko

.

I
I I
bbbhh I
9V9*“9

oino'o'oU

0.20

 

0.40

060

 

0.80

position (x/w)

143

 

50.00

.3:
9
O
O

30.00

N
.0
o
O

10.00

radiation spectrum amplitude

tranverse current, even modes

  

. I I ..
. I .
“The“ .44”

0.00 ' ' '
0.00 0.20 0.40 0.60 0.80 1.00
posntIon (x/w)
Figure 5.9 Radiation regime transverse surface current amplitude, even parity.

144

50.00

as
9
O
O

30.00

N
.0
o
O

10.00

radiation spectrum amplitude

tranverse current, even modes
yo = 100 t/K

  

AVE». 7.. ‘4'“

0.00 ‘ ‘ '
0.00 0.20 0.40 0.60 0.80 1.00
posmon (x/w)
Figure 5.9 Radiation regime transverse surface current amplitude, even parity.

144

axial current, odd modes
yO == 1CK) L/A
50.00

‘_‘D

C/ko

i
l
I

.p.
9
C)
O

l

l
I
I
I
b .

 

8

e

8
989999
omooou

d

30.00

+
._—----——--—--——a

 

radiation spectrum amplitude

 

.rrrrrlrrrr111111441411rrrlrrrrrrrrrlrrrrrrrrrl

 

 

20.00
£- ‘
10.00 ,1 l,
U l
I I / I
I .-
”in. 4.4—”- "...aa. 4
0m‘ -—————~» 4 . .-

0.00 0.20 0.40 0.50 0.80 1.00
pOSItIon (x/w)

Figure 5.10 Radiation regime axial surface current amplitude, odd parity.

145

50.00

4:.
9
O
O

30.00

N
.0
o
O

10.00

radiation spectrum amplitude

0.00

0

Figure 5.11

tranverse current, odd modes
Yo = 100 t/X

 

blrrrrrrrrrlrrrrrrrrrjr‘

0 0.20 0.40 i 0.60 0.80 -
position (x/w)

Radiation regime axial surface current amplitude, odd parity.

146

 

tranverse current, odd modes

 

Yo = 100 t/k
50.00
{/ko
40.00 1:: {9%
---- 1 4.0
1 5.0
00-001 7.5
new 310.0

30.00

radiation spectrum amplitude

     

 

4 a
20.00 1 5‘
a h
1
10.00 3
I a
. \ d
1 .. .. / .
: l - -.. ' T «a
0.00 _: 3"ng - _
0.00 0.20 0.40 0.60 0.80 1.00

position (x/w)

Figure 5.11 Radiation regime axial surface current amplitude, odd parity.

146

Finally, the current distributions within the surface-wave only radiation regimes
were looked at. This is the limited portion of the radiation regime denoted P in
Figure 5.4. Figure 5.12 and Figure 5.13 give typical surface current distributions where
C = kc , the limiting edge of the transverse-only radiation regime. These currents are the
dominant magnitude currents in the transverse-only regime, and are considerably smaller
in amplitude than their counterparts in the full radiation regime. Other components
within the transverse—radiation regime have much smaller amplitudes than this limiting
case. As the current amplitudes within this regime are so small, it is conjectured that for
a thin-film layer, power propagated away from the guide carried only within the film

layer is small.

147

even made

 

 

 

 

0.08 1
3 nc = 1.0
Q) -I fly = 3.13
'0 j V} = 0.0635
3 . w t = 2.85
-"_'—_’ 0.06 -
Q. 2
E I
O -
E i
3 0.04 ..
4-1 -l
U .4
Q) -I
O- .1
(n ‘ .
j OXIal current
8 .
1:, 0.02 j
.9 ~
.0 .
o :
L -l
3 _______________________ transverse current
0.00 TI‘I'I.I.I'I.I I l I rrrr I I I I I I ITTI I I I I I II I I VT I I I I I I-I.I.I.I‘I.I-IS]
0.00 0.20 0.40 0.60 0.80 1.00

position (x/w)

Figure 5.12 Typical surface current distribution within surface-wave radiation regime,
even parity. Excitation source 5). above microstrip, §=k,.

148

odd mode

0.16
nc = 1.0
n. = 3.13
t/k = 0.0635
w/t = 2.85

.0
N

.0
O
on

 

""""" -. transverse current

 

radiation spectrum amplitude

rrrrrrrrrlrrrrrrrrrlrrrrrrrrrlrrrrrrrraJ

 

 

0.04 axial current
\\‘ I” \‘
s ,’ \
0.00 IIIIIIIIFTTIIIIITIIIIIIIIII\I’I]IIITIIITIIIIIIIIIIII
0.00 0.20 0.40 0.60 0.80 1.00

position (x/w)

Figure 5.13 Typical surface current distribution within surface-wave radiation regime,
odd parity. Excitation source 5). above microstrip, §=k,.

149

Chapter 6
Leaky-wave Modes for the Dielectric Rib Waveguide

The integrated rib dielectric waveguide is a common structure used in integrated
optics. While common and useful, the analysis of this waveguide is exceedingly
difficult. Simple approximate techniques, such as Marcatilli’s method [54], work
well for electrically-large rib waveguides, but fail as these waveguides become small and
approach cut-off.

Leaky-wave modes are used to model radiation loss via the method of steepest
descents. These modes describe the waveguide operating in cutoff. As approximate
techniques fail near cutoff, they cannot be used to determine leaky-wave modes. In
addition to modeling radiation loss, leaky-wave modes of the rib dielectric waveguide

may be important for coupling problems as well.

6.1 Application of the EFIE

Consider the optical dielectric waveguide as depicted in Figure 6.1. The
background environment is that of a conductor/film/cover, which is the same environ-
ment as the microstrip transmission line in Chapter 5. The waveguide cross-section
geometry is assumed to have symmetry about 1: = 0. Other than that, the EFIE as

developed in (2.40) is applicable in general until a solution is needed for a specific cross-

150

 

  

nc

El £8

_y=o

2_y'4

/
' erfect lectric Conductor

Figure 6.1 Configuration of a rib dielectric optical waveguide.

151

section. For the rib waveguide, the guiding region cross-section is rectangular.
The EFIE chosen to implement will be the original EFIE developed in (2.40),

namely,

 

- 2 -I .. /.
ism-(k? + W)! ammo-5" ‘p’f‘fi'c’dv' = 3’(b’:(). mvrecs
a

"c

This integral equation is notationally more cumbersome; yet, it avoids having to deal
with the source-point singularity problem and the necessary depolarizing dyad. Leaky-
wave modes and bound modes are discrete modes, thus requiring that the homogeneous

EFIE be solved. Then, the EFIE in (2.40) is
5(5)'(k,2+W‘)ii(p‘) = o (6.1)

where the Hertzian potential it is handled separately, taking the form

2
flat) = [Mflflxfiylyfifl °3(x’,y’,C)ds' (6.2)
cs

6

wherein an explicit dependence upon x and y has been shown.

6.1.1 Parity considerations

Parity considerations arise from previous analyses of the uniformly-clad
rectangular waveguide. Because of the symmetry in the geometry and the uniform
surround, the guiding modes are nearly degenerate. By pre-selecting a parity state, only
that parity mode of a degenerate set will be determined. Reasons to analytically consider
parity for the rib waveguide are not as overwhelmingly obvious as with the uniformly-
clad waveguide, as the background introduces a measure of asymmetry into the problem.

If the background is removed, however, the rib waveguide becomes the uniformly clad

152

rectangular fiber. Consequently, parity will be considered, so that this analysis can
recover solutions in the specialization to the uniformly-clad waveguide (in which the
reflected dyadic Green’s function vanishes).

Parity considerations exploit field symmetry about 1 = 0. For graded, non-
uniform guiding regions, this symmetry consideration assumes that the refractive index
is even in x, namely, n(x,y) a n(-x,y). The symmetry effects can be ascertained by

using the following equations obtained from the transform-domain Maxwell’s equations:

 

~ 1 - a -- i . -.i .. i

e' 1.30:.» NICE“ ’”"°ayh’) ii“ 6y“ 1”" axh‘H (6'3)
a. a.
_x-_z = -- (6.4)
6y ax NM‘

First, symmetry of the guiding region refractive index symmetry means that
k3(x,y) . t.)’|ltorI’(x,y)-C2 is even in x. From an inspection of (6.3), it is readily
observed that if e, is to be even in x, then e, must be add in x while It, is even in x.
But, if h, is to be even, then from (6.4) is obvious that e, must be add in x. This bears
out when considering e, as defined in (6.3). Since e, should be odd, ez must be odd,
which it is; while It, must be even, which it is. The symmetry states for this optical

waveguide can then be defined as

state ex C’ e:

1 evenoddodd
2 oddeveneven

from which the electric field can be viewed as é‘(-x) = a-erx) , where
3 = :[if-yy-ii] (6.5)
is defined as a symmetry dyad.

153

Application of parity to the EFIE necessitates decomposing the integral over the

waveguide cross-section into

fl...) = 1‘ (...)ds‘ + 1‘ (---)ds'
(:3 C5" ('3'

where CS‘ is defined for the cross-section of the waveguide with x>0, while CS is

defined for the x < 0 portion of the waveguide. Making the variable change ofx’ .. -x’

in the CS integral mirrors the limits of integration in x, that is

f(....x’)dg' . f(...,_x/)dg*
cs- cs'

Consequently, under parity considerations, the Hertzian potential in (6.2) can be

constructed as

in?) = fal— (x’y)[§(x|x’) 8(x’) + §(xl-x’) 3(- x’)]ds’ (6.6)

08' "c

Making use of the symmetry dyadic as defined in (6.5) allows (6.5) to be written as

2
11(5) = f —‘-——f”’§'(xlx°"’-;yly’;o e(x’.y coas' (6.7)

(3’ "e
where CS’ is the x>0 cross-section of the waveguide, and §'(x|x’;y|y’;() is the
symmetric Green’s dyad, explicitly expressed as
fi' = §(xlx’;yly’;0 + §(XI-x’;yly’;C)-3 W”
The symmetry dyadic 3 is diagonal; consequently, no directional change in the scalar

components of §'(x|x’;yly’;C) occurs.

154

6.1.2 EFIE for integrated dielectric waveguide

Upon recognition of the symmetry dyad, work can begin upon specializing the
EFIE to the integrated dielectric waveguide. In this development, explicit dependence
upon I and y will be suppressed unless necessary for clarity. Tensor notation will be
used for the dyadic (g =2 2 gwijp) and vector (1' = 2: f3.) components to
generate compact expressiairs ior the field components. Within this tensor notation,

summation indices 01 are assumed to take on the values of x, y, z.

In tensor notation then, the EFIE becomes
2 Me. - *3 n1} - We: = 0 (6.9)
c-nnz

The differential operator WV takes the form

.. .. 621:
we = 22.; 3:: a;

 

under tensor notation; consequently, (6.9) becomes

2 3
e. - 1.3.: - )3 a “t = o (6.10)
fl-sxz axaaxfl

 

for a = x,y,z. The scattered Hertzian potential is supported by equivalent sources

induced by the electric field; thus, the scattered Hertzian potential can be constructed in

tensor form as

a! = X 2,{ 2 1:1,} (6.11)

V'KYJ

where

2 I I
“i. = f ——b" (‘2’), )gl.e.49’ (6.12)
CS' "c

155

This renders the EFIE in (6.10) into

6%?
" k‘ E "" 2.3); ax.ax, °

 

for a = ,y,z.

The subsequent substitution of (6.12) into (6.13) will obtain an integral equation
for e,; before this is done, it is desirable to obtain the tensor form of the symmetric
dyadic Green’s function. The symmetric dyadic Green’s function is composed of a
principle and reflected part, namely

gt .. gaug- = hung-r (6.14)

In tensor notation, the idemfactor I = 2‘ 2’ 6"; and the symmetric Green’s dyad

takes the form
g’ = 2209"... + 3:.) (6.15)

Note that four of the components 3:, are zero.

There is one final trick left. The leading term in (6.13), e.( p’) , needs to be cast
in terms of the ”source" e,(p") . Exploiting the unit vector i, in tensor notation results
in e.. = E 0" e, , while judicious use of the Dirac delta function allows e.(p’) to

v-l

become
e.(1‘>) = fab..6(fi-fi’)e.(5’)dv’ (6.16)

Finally, substitution of (6.12), (6.15) and (6.16) into (6.13) results in the desired

EFIEof

156

v-l

3
El 5..5(5-5’) - kian’tp’Xs'b... +312)
CS. (6017)

2 ..,
_ 3 Sierra” + 8;)M}¢,(5’)d9' = o

 

&a 3 an n2

C

for a = x,y,z.

6.2 Method-of-Moments Solution

The EFIE developed in (6.17) is now in a form for which the method of moments
is readily applicable. The method of moments has been applied elsewhere in this
dissertation; this section will briefly touch on the significant differences in the application

to this waveguide.

6. 2.1 MOM expansions
Each component of the dielectric waveguide field (ex, ey, ‘2) will be expanded in
terms of subsectional basis functions,
1v
ev(x,y) = § e" p.(x,y) (6.18)
The basis functions used are two-dimensional pulse functions with the characteristic of

1; (x,y) e partition n
p,(x.y) = {

0; elsewhere

The pulse functions are centered at (x.,y~) and are non-zero for '3" < i x— 1.1 < A215

and %2 < ly-y‘ | < 32-2. There are N, elements along x and N, elements along y,

leading to the relations

157

Ax=

 

b
; Ay = .—
N:
The waveguide refractive index is assumed piecewise continuous, taking the form

N
n(p”) -- z n.p.(x’,y’). Recognition that n: = nfp: suggests that

n-l
6n’(b") = 0:3 - n3»: - an:p_(x',y'), Vfi’eA s_ (6.19)
Substitution of expansions (6.18) and (6.19) into the generic form for scattered potential

in (6.12) results in

s N 5": N
1!" = f3; 2: _—2p.(x/JI) z evapn(xl,yl)dx’dy’
CS’ and n: In!

This expression simplifies and becomes

. . .. ., " an:
n" = fm.g..(p|p)2 2
C

 

,(x’.y’)dr’dy’ (6.20)

ml n

because the pulse function product p.11. is zero where the pulses do not overlap (m en).
Passage of the summation on n through the integration on CS+ of (6.20) yields

the result

 

N an2
“iv = 2 - evn’cvn (6021)
ad )3:
where the MoM integral I“, is given as
I... = [3.1.05 lb”)P.(5’)49’ (6.22)
s

Paint-matching the MoM solution in two dimensions uses the testing operation

158

f atx-x,)60-y.)(-«)dxdy m = 1. 2. N (6.23)
a.

where location (x.,y.) is the center of the pulse function p...

The ultimate result of this operation is to turn the original EFIE in (6.17) into a

matrix equation. After some algebra, this matrix equation is

 

 

 

 

 

 

 

 

 

i m m1 P ‘ r 1
A: A” A: en 0
¢fi¢s=o “w
A: A: At: .‘U. L0‘
The matrix elements take the farm
A:=(A:'+(AS' (6'25)
with principle matrix elements given as
"" - 6 a 5": r’r' ‘92 1' (6 26)
(Ace ' up an - '7 c I'l(x.’y-) + Sax—$— Ben °
e P a p 3".)
and reflected matrix elements given as
(Amar=_bnn2k21r (x y.)+Z-—a—2—I' (627)
av 112 c an .9 p &.axp pvn '

 

 

e 3.).)

The reflected matrix elements vanish in the case of a uniform surround. The MoM

integrals within (6.26) and (6.27) are given as

1:" = buf[g'(x|x’;y|y’;C) + avg'(x|-x’;ny/;C)]P.(x’.yl)dx,d}” (6.28)
as"

II... = f [al.(xlx’alyfio + 0.8:.(xl-x’;yly’;6)]p.(x’.y’)dx’dy’ (6.29)

a.

159

Application of the method of moments has converted the EFIE to a system 3N by
3N homogeneous linear equations. This system has a unique, non-zero solution only if
det[A(C)] = 0. Any value of 1' that renders the determinant of (6.25) zero is the
propagation constant of the corresponding discrete mode. The field distribution of the

mode is then determined by finding the nullspace of the matrix in (6.25).

6. 2.2 Special considerations for the MOM expansion
In the numerical implementation of (6.24)-(6.29), certain tricks can be employed
to greatly improve computational efficiency. These tricks will be detailed in this section.
The first trick is to recognize that the MOM integrals of (6.28) and (6.29)

decompose into

IF! 3 IUI)’ + a, IUI)‘ (6.30)

3" IV. V IV.
because x-symmetry has been implemented. The principle MOM integral can then be

considered as 15:, - 6" 1?, where

If.” = fg’txlex’aly’Mr’dy’ (6.31)
A3

and AS. is the area of the pulse. The reflected MOM integrals take the form

I? = f sI.(xltx’.yly’)dx’dY’ (6.32)
AS

The Green’s functions will be cast into their spectral form in £ as in (2.44). Exchange

of the spectral and spatial integrals results in

e. t p
1:: = [fl (X,£)h. 0351‘) d5 (6.33)
, 2np,(€)

 

160

i:(x.£)h'0.t.a
R ' a (6.34)
I ..(t) 2np.(t) e

 

where the even behavior of the Green’s function integrands in 5 has been exploited. The

MOM function expansions are

m I
f.‘(x.£) = e!“ f em“ 4,)de
-m

4)“
hP(y’£,C) g fe'P¢|)"(y’-y.)ldy/
Am

An
h:()’95.0 3 fe.’¢(”0",.))dy, ,
Ayn
and take Specific forms of
r:(x.t) = -Sin(E—)cas£(xix) (6.35)
”chidmhiw] for ly-y I > AZ
2 .. 2
’ - 6.
hubrEst) ' -: E! A ( 36)
l'e 2 COShp C(y'y') ...for |y_y.| < _2.Z

 

h ,'.(v.t C) = —sintr(A , 31¢) "0 ’“~ (6.37)

Specific forms for the reflection coefficients Ru“) can be found in Appendix B.
The other trick is implemented to improve convergence of the principle MoM

integral If," on 5; the reflected MoM integrals all have exponential decay in y, as

evidenced by h: in (6.29), (6.37). While most of the Ii” terms possess exponential

decay in y and 5, the MoM expansion of h’ when 1?... -yn | < Ay/2 decays as onlyE"

161

in its asymptotic form. For some matrix elements, the convergence may only be as?1
due to the effects of spatial derivatives. This slow convergence can be analytically
handled by adding and subtracting the asymptotic form of If . This method is useful
only if the asymptotic form has a closed-form integral on i. If this is indeed the case,

then the asymptotic extraction yields

I.’ = [mo - v.(€)ld£ + f¢,(£)d£ (6.38)
0 o

The asymptotic form of If as £ -o on is easily recognized as

ME) = -isin(EAx/2)cosz(m,)
"i (6.39)

= --n1—£[sin£(xtx.+Ax/2)-sin£(x1-x'-Axl2)]

of which a closed form integral does exist, namely

0“:
NIH

sinaE dfi a gsgnm)
where sgn is the signum function. Application of this relationship to (6.39) gives
fl;- [sgn(x1x.+Ax/2) -sgn(x:tx_-Ax12)]

The only time the above sum is non-zero is when there is overlap; this only occurs when

lx —x. | < Ax/ 2. Consequently, the asymptotic integral evaluates to

° -1 ; (x,y) 6 partition n
= (6-40)
1; “‘0 d5 { 0 ; elsewhere

162

6.3 Spectral Analysis Considerations

If the film thickness is restricted such that only the TM.) background surface wave
mode can propagate, then the axial transform plane has the branch cuts as depicted in
Figure 3.14. There are consequently four Riemann sheets of interest, of which only the
top sheet corresponds to spectral, bound waveguiding modes.

Obeying both branch cuts B and P restricts the axial wavenumber f to the top
sheet; any solutions to the EFIE on this sheet are the bound guiding modes. No special
attention nwds to be paid to the spectral integrals on E. as all the singularities are located
below the real-line contour.

The second sheet determines surface-wave leaky waves. This sheet is reached by
violating the P branch cut. The corresponding situation in the complex E-plane is that
the background surface wave pole controlled by that branch cut is now located above the
real axis. The inversion contour to evaluate the spectral integral must still remain above
the pole. Deforming the contour to the real axis, as in Figure 6.2, thus captures the now
non-spectral pole. This contribution can easily be recognized as non-spectral, since
dbl" = e’ V e "f , which propagates in the -x direction but grows exponentially in -x
instead of decaying.

Either of sheets (3) or (4) is reached by violating the B branch cut in the complex
f-plane. In either of these cases, the branch point migrates across the real-axis. The
difference is whether or not the surface-wave pole, controlled by P, migrates above the
real axis. The necessary deformation of the integration contour to remain above the

original singularities for each of these cases is given in Figure 6.3.

163

. -.-O-O-o-a-o-0-0-O-O-:P.

-E bc ,
0°”
inversion contour
\

695.5;

\

Figure 6.2

 

Dome-0-0-0-0-.-.-
0-.

 

Complex £-plane singularities for surface-wave leakage.

164

.-.-.-.m....-.-.-.-.-.-.-.-..,.

 

  
  

   

C

(a)

Relpc} < 0

   

(b)

Figure 6.3 Singularities in the complex £-plane for: (a) space-wave leakage
(sheet 3). (b) full (space/surface wave) leakage (sheet 4).

165

The choice of the branch cut in Figure 6.3 merits discussion. As observed in
Chapter 3, the branch cut in the complex s-plane was specified to enforce the
Sommerfeld radiation condition as Iyl -° 0°. When the branch cut B is violated, this
requirement is meaningless, as these will be leaky-wave modes. When the migrates
across the real axis, an attempt to enforce spectral behavior (the Sommerfeld radiation
condition) on non-spectral modes results in a hyperbolic cut originating at Eu that
approaches infinity asymptotic to the imaginary axes but violates the deformed inversion
contour; consequently, this is not permissible. The criteria for the branch cut in this case
is less restrictive. Having disearded the idea of mapping the Sommerfeld-plane branch
cut, any cutting in the f-plane can be chosen, as long as it: 1) maintains the continuity
of the physical problem, and 2) does not violate the deformed inversion contour. These
considerations taken together dictate the branch out choice in Figure 6.3, starting at the
branch point and passing through the real axis, approaching infinity asymptotically along
the negative imaginary axis. Any branch cut obeying the above guidelines will not
separate spectral (where Steipc} >0) from non-spectral (where Stelpc} <0) sheets in the
E—plane. In Figure 6.3, the improper region on the complex f—plane is denoted by the

shaded area.

6.4 Results
A typical rib waveguide configuration was chosen for analysis. The waveguiding
cross-section has dimensions of width 2a (-a < x < a) by height a (O < y < a). The

guiding region is homogeneous, with a refractive index of 11,: 1.5, the same as the film

166

layer refractive index. The cover medium has refractive index n,= l, and the film layer
has a thickness of t=0.2 wavelengths.

A dispersion curve of the dielectric rib waveguide modes is given in Figure 6.4.
The bound modes are denoted as 3:. modes, where m and n designate the mode and
0: denotes the dominant component of electric field. Bound modes are found as solutions
on sheet (1) of Figure 3.14. It is observed that the dominant mode for the rib waveguide
is the principal if, mode. This 151’, has no lower frequency cut-off. At low frequencies
it merges into the m0 surface wave pole of the conductor/film/cover background.
Typical field distributions for the dominant and higher-order bound modes (1.23 < {/ku < 1.5
) are given in Figure 6.5 to Figure 6.7. More periodicity is naturally observed for the
higher-order modes.

Leaky-wave solutions have also been obtained by using the integral operator.
These show up on the dispersion curve in Figure 6.4 over the range C/ko < 1.23 . These
leaky-wave solutions can be of the three types mentioned previously. The ones shown
on the dispersion curve as extensions to the E“; plot are solutions upon Sheet 2 in the
complex {-plane, or surface-wave leaky waves. The attenuation constant for the modes
is shown in Figure 6.8, while a plot of leaky-wave field distribution within the
waveguide is displayed in Figure 6.9.

Two observations are apparent. First, the attenuation of the surface-wave only
leaky waves is very small. This implies that the loss mechanism of surface waves
propagating in the film layer does not carry much energy with respect to other loss
mechanisms. This is consistent with observations on the small amplitude of the

transverse-only radiation current amplitudes in Chapter 5. It is not correct to say that

167

1.50

 

 

 

 

O
x
\.
V1.40
E
O
4.2
(I)
€21.30
0
o
g TMO background pole
6 1'20 Ex“ mOde
O“
O
O.
O
L
Cl1.10
1.00 ITIIIIIUIIIIIITITIIIIIIIIIIUIIIIITIIFFIU]
0.00 0.50 1.00 1.50 2.00

half—width (Ci/h)

Figure 6.4 Dispersion curve for integrated dielectric rib waveguide.

168

ET, Field Distribution

o-1.788. b/o-I

l

.75 L90

Amplitude
"J

     

  

0.25 0.50 0

 

 

 

 

 

 

 

 

 

 

 

Figure 6.5 Field distribution for dominant E’u waveguide mode for guide half-width
of a= 1.788)»

169

Et‘, Field Distribution

o-1.788. b/o-l

 

Amplitude
0'1 0.;5 0);)0 0.75 Loo

 

    
 

0.6

Mb)

Figure 6.6 Field distribution for 1?."ll waveguide mode for guide half-width of
a=1.788>..

170

Eé, Field Distribution

0-1.788. b/o-l

Amplitude
0’l oas 0.150 0.75 Loo

 

 

 

Figure 6.7 Field distribution for E‘z, waveguide mode for guide half-width of
a=1.788>..

l7l

7.0E—003

6.0E-003 EX” mode

Ca/ko

5.0E-003

4.0E-003

3.0E-003

2.0E-003

attenuation constant

1.0E-003

0.0E+000
0.10 0.15 0.20 0.25 0.30 0.35 0.40

half—width (a/h)

Figure 6.8 Attenuation plot for F)” leaky-wave mode in surface-wave-leaky regime.

172

Ei‘, Field Distribution

b/o-l
8:} Leaky mode. a-0.255
‘” .n
.3 3‘
i .
E g
< .0
(~54
o

 

  

Figure 6.9 Field distribution for E“, leaky-wave of rib waveguide compared to a
bound guiding mode.

173

because a leaky-wave 5', is small that total loss is small; the :96 in Figure 6.8 are large
losses when operating at optical frequencies. More striking is the loss of the
confinement of the waveguide field for a leaky-wave mode.

Not many space-wave leaky modes (sheet 3 solutions) or full leaky modes (sheet
4 solutions) have been found. The following leaky wave poles have been determined and

are tabulated below.

 

Table 6.1 Table of space-wave leaky wave poles (sheet 3) and full leaky wave
(sheet 4) for the dielectric rib waveguide where b/a=1.

sheet 3 (space-wave) sheet 4 (full) leaky W
leaky IDOdO. {3 mo 1'. A: f: " {4 ;

0.86696 -j 0.17364 0.86969 -j 0.16913 0.00273 + j 0.00461

 

     
   
 

 

   

  

0.85391 -j 0.18829 0.85643 -j 0.18346 -0.00252 +j 0.00483

    

 

A plot of their guiding region field distributions is shown in Figure 6.10. These leaky-
wave poles are 52; leaky-waves based on their field distribution. Immediately obvious
from Table 6.1 and Figure 6.10 is that there is very little difference between propagation
constants and field distributions for a space-wave leaky mode or full-wave leaky mode
solution at a given guide half-width a. This means the effect of the surface-wave pole
on the field behavior is small. This is again consistent with observations about the thin-

film surface-wave contributions throughout this dissertation.

174

Eéz Leaky Wave Field Distribution

b/o n1.0.o - 0.5

 

8
.41
”to
U.
.33
a.
2?,-
(tn
”1
O
1..

 

Figure 6.10 Comparison of field distributions of sheet 3 and sheet 4 leaky-wave
modes.

175

Both space-wave leaky and full leaky waves can make significant contributions
to the far-zone scattered field. The contribution of a given leaky-wave mode depends
upon if it is intercepted by the steepest-descents contour. Given the similarity between
the space-wave and full leaky mode, is it possible to use them interchangeably if only
interested in the seattered-field above the rib waveguide. The usage of these leaky-wave
modes on sheet (3) or sheet (4) to determine the scattered field is a good topic for future

investigation.

Summary

The EFIE developed for dielectric waveguides in Chapter 2 has been applied to
a integrated dielectric rib waveguide. Bound and leaky wave solutions were obtained.
Of significant note is that the effect of the surface-wave pole of the background structure

is very small. This is consistent with other observations throughout this thesis.

176

Chapter 7

Conclusions and Recommendations

A rigorous, full-wave integral-operator—based formulation to characterize the
continuous radiation spectrum for a broad class of open-boundary waveguides has been
presented. This formulation is capable of characterizing the complete propagation-mode
spectrum for these open-boundary waveguides. Furthermore, this integral-operator
formulation provides a conceptual, unifying treatment of the relation of leaky-wave
modes to the propagation-mode spectrum and in particular the continuous radiation
spectrum.

This formulation is based upon the rigorous dyadic Green’s function describing
the Hertzian vector potential supported by an arbitrary current source immersed in a
planar layered background environment. This Green’s function is identified via Fourier
transform techniques; as a result, the Green’s function scalar components obtained are
inverse transforms of the Sommerfeld integral class. Knowledge of this Green’s function
allows determination of the associated electric and magnetic fields. The open-boundary
waveguide is then recognized as a set of equivalent sources within the layered
background; superposition over the set of equivalent sources then determines the

appropriate electric-field integral equation describing the problem.

177

The aforementioned integral equation is solved in the axial-transform domain;
recovery of the three-dimensional spatial waveguide fields from the determined
transform-domain fields is accomplished by an inverse transform on axial wavenumber
f. Evaluation of this transform by singularity expansion methods allows the identification
of the propagation-mode spectrum. This spectrum is comprised of two distinct types of
modes. There are a finite number of bound, hybrid waveguiding modes. These guiding
mode fields are confined to the vicinity of the open-boundary waveguide and do not carry
power transversely away from the waveguide. Such modes are associated with pole
singularities of the transform-domain fields; a guiding mode satisfies the homogeneous
transform-domain EFIE. Standard eigenfunction expansion theory can be applied to
determine the expansion of an impressed field in terms of the guiding modes.

The other component to the propagation-mode spectrum is a continuum of
radiation modes, which have field distributions that are not confined to the waveguide
vicinity. The radiation modes provide the mechanism to carry power transversely away
from the guiding structure, hence accounting for radiation losses. Radiation modes are
associated with branch cuts within the axial transform domain; at these points, a radiation
spectral mode satisfies the inhomogeneous transform-domain EFIE; consequently,
solutions for the radiation spectral modes are dependent upon the impressed field. The
total radiation field is the continuous superposition of EFIE solutions along the entire
branch cut; this superposition satisfies the radiation condition at infinity. The nature of
the Green’s functions used by the EFIE, and their complicated dependence upon the axial
wavenumber g- and transverse spectral frequency f, obscure the regimes in which

radiation modes would necessarily be continuous.

178

The regimes for the continuous radiation spectrum can be determined by enforcing
the requirement that the Green’s function scalar components must be spectral in nature;
that is, bounded or vanishing as approaching infinity transversely. Enforcement of this
behavior on the Green’s functions requires that the forward Fourier transforms converge,
in particular the transform an transverse coordinate x.

Mathematically, the forward transform must converge to an analytic function
within a finite strip in the transverse spectral frequency plane which is parallel to the
real-line axis; the inversion contour to evaluate the inverse transform lies parallel to the
real axis within this strip of convergence. As the Fourier transform is an analytic
function within its strip of convergence, no singularities of the Fourier transform can
exist within this strip of convergence.

For the forward transform to model spectral behavior, the strip of convergence
must minimally includes the real axis, upon which the inversion contour lies. Enforcing
this requirement restricts any singularities within the transverse spectral-frequency plane
to remain either within the lower- or upper— half-plane and not pass through the real axis
from lower to upper and vice-versa. This provides the criteria for choosing the branch
cuts within the axial wavenumber plane; consequently determining the regime for which
the continuous radiation spectrum of the open-boundary waveguide is defined. Enforcing
that the transform converge is the equivalent of enforcing the radiation condition.

For the limitingly low-loss case, the continuous radiation spectrum decomposes
into a number of identifiable radiation regimes. There are any number of surface-wave
radiation regimes, in which the radiation spectral components are propagating modes and

have the characteristic of bounded oscillatory behavior in x but exponential decay in y.

179

These surface-wave regimes thus model the power carried transversely away from the
open-boundary waveguide, within the interior (film) layers of the background structure,
by excited surface-wave modes of that layered background. Each of these surface-wave
regimes is associated with an excited surface-wave mode.

There are two other possible radiation regimes associated with the wavenumbers
of the semi-infinite cover/ substrate layers of the background environment. The substrate
regime is typically comprised of propagating radiation modes with bounded, oscillatory
fields in the transverse x coordinate and in the normal y coordinate within the substrate
layer, but with exponential decay in y within the cover region. The other regime is the
typical full radiation regime, in which the radiation mode has bounded oscillatory
behavior both transversely in x and in y within both the cover and substrate. The
radiation modes within the full-radiation regime can be either propagating or evanescent.

Substrate— and full-radiation regimes can be identified for open-boundary devices
of arbitrary geometry in multi-layered backgrounds; the surface-wave radiaton modes as
identified are new and specific to waveguiding applications.

The criterion defining the continuous radiation spectrum serves to define an n-
sheeted Riemann surface for the axial transform domain. Proper spectral behavior is
associated only with the top sheet; all other sheets are non-spectral. The transform-
domain homogeneous EFIE possesses solutions on the other sheets; these solutions are
the improper or leaky-wave modes of the open-boundary waveguide. This integral-
operator formulation is therefore capable of identifying the leaky-wave modes of these
open-boundary waveguides. Because of the branch cut choice, leaky-wave modes never

influence the proper spectrum of the waveguide. As leaky waves are non-spectral, they

180

cannot exist over all of space, but rather have meaning only in specific restricted spatial
regimes. Leaky waves are significant in the asymptotic evaluation (as in the method of
steepest descents) of the scattered field for an excitation problem. leaky waves augment
the saddle-point contribution to the waveguide scattered field in a restricted spatial
regime, defined wherever the SDC contour intercepts the leaky-wave pole. It is within
this regime for which a leaky-wave solution has physical significance.

This theory was then applied to determine the radiation spectrum for planar
dielectric waveguides. These planar waveguides possess canonical, closed-form solutions
determinable by differential operator techniques. The integral-operator formulation
agreed well with canonical results, giving confidence in its validity for more complicated
problems. It was observed that the deeper into the radiation spectrum, the more periodic
the waveguide field became. Asymmetric waveguides were also considered. The effect
of the asymmetry was rather significant upon the propagating radiation spectral modes
but insignificant upon the evanescent radiation modes.

The integral-operator formulation was then applied to determine the radiation
spectrum of a simple practical waveguiding structure in MMIC design, the microstrip
transmission line, for which no known results exist. Results for the spectral radiation
mode surface currents were numerically obtained at relatively small spatial frequencies
within the radiation regime. It was observed for a thin-film substrate that the surface
current amplitudes in the transverse-only radiation regime are very small compared to
the amplitudes within the full radiation regime. This indicates that the amount of
transverse-power loss, carried away from the microstrip by the excited background

surface wave, is very small. Again, increasing periodicity of the radiation modes was

181

 

observed deeper into the radiation regime. Deep into the evanescent portion of the full
radiation regime (that is, IC,| -° on), a disturbing trend of increasing radiation mode
surface current amplitudes was observed. This observed trend is most likely due to an
artifact from the calculation of the impressed field. Even though disturbing, the effect
of the increasing amplitudes is annulled when considering the total radiation field, as the
modes in question are evanescent and exponentially decaying.

Finally, the integral-operator formulation was applied to the determination of
leaky-wave modes of the integrated dielectric rib waveguide in a cover/film/conductor
background. There are no published results for any leaky-wave modes of these
structures. For a thin-film substrate, the axial wavenumber plane has four Riemann
sheets. The top sheet (Sheet 1) solutions are the hybrid guided-wave modes. Leaky-
wave solutions were found on each remaining sheet. Sheet 2 solutions are the surface-
wave—only leaky waves. These leaky waves have small attenuation constants relative to
the leaky-wave solutions on sheets 3 and 4. This indicates that power carried away only
in the film layer is small with respect to other radiation loss mechanisms. This is
consistent with results from the microstrip analysis. Finally, leaky-wave solutions on
sheet 3 or 4 were observed to have similar eigenvalues for either sheet. The attenuation
constants for these leaky-wave solutions indicate that radiation into the cover region
dominates radiation loss mechanisms.

Having confirmed the validity of the integral-operator technique, many extensions
to this research become obvious. First, more investigation as to the behavior of the
impressed field in the transform-domain is needed. Electric charges have been suggested

as a source for the increasing spectral amplitude deep into the radiation spectrum of the

182

microstrip transmission line. This can be investigated initially by applying the EFIE to
determine the TM radiation spectrum for a symmetric planar waveguide, as TM mode
behavior depends upon electric charge. As there are closed-form solutions for the
symmetric planar waveguide TM radiation modes to compare results to, this can yield
valuable insight into the behavior of the impressed field in the transform domain.

This integral-operator technique is easily applied to more complieated open-
boundary waveguiding structures. For example, thick-film substrates or non-conductor-
backed multi-layered environments are investigated by including all surface-wave poles
of the background structure. The simplest of these extensions is investigating thick-film
layers that support more than a single surface wave mode. Another simple extension is
to investigate the continuous radiation spectrum for a pair of coupled microstrip
transmission lines in a conductor/film/cover environment.

The preceding ideas can be accomplished without much effort. If one is willing
to develop new Green’s functions, many other possibilities open up. Dielectric channel
waveguides ean be easily analyzed by using a different dyadic Green’s function kernel.
The integral-operator technique can be applied to determine the leaky modes and
continuous radiation spectrum of microstrip transmission lines on an anisotropic
substrate. This necessitates developing a new set of Green’s functions, and could bring
to light new physical phenomena.

An intriguing application is an investigation of the effect of the propagating
continuous radiation spectrum upon microstrip discontinuity measurements. This has

potential benefits for applications such as microwave material characterization.

183

Finally, more investigation nwds to be carried out as to the contributions of each
type of leaky-wave mode to the total scattered field from a waveguide. This would
involve determining a solid methodology for applying the method of Steepest Descents
for three—dimensional problems. As mentioned before, the similarity of certain leaky-
wave solutions on different sheets suggests that there is much left to understand.

In summary, this integral-operator technique shows promise to conceptually
characterize the radiation spectrum for a wide variety of open-boundary waveguides.
The technique includes the excitation in a natural, straightforward manner. The integral-
operator technique also provides a methodology to identify leaky-wave modes of these
structures. Finally, this technique shows how the radiation spectrum and leaky waves

relate to each other, and consequently unifies both approaches.

184

Appendix A

Appendix A
Electric Hertzian Potentials

A.1 Electric Hertzian Potential
Maxwell’s equations govern the behavior of the electromagnetic fields of an open-

boundary waveguides, and take the form

VxE = -j(.) pH Faraday ’8 Law

that . jer + i Ampere-Maxwell Law (A.l)
V-B = p/e Gauss’sLaw

v-fi - 0 Magnetic Source Law

for linear, isotropic, homogeneous media and time-harmonic fields. Maxwell’s equations
as given in (A. l) are a set of overspecified, coupled partial differential equations, with
Gauss’s Law and the Magnetic Source Law embedded within the Ampere-Maxwell Law

and Faraday’s Law respectively. A direct solution from (A.1) is possible; after some

manipulation, the associated Helmholtz equations are found to be

v23 + 1:213. = 1'qu + WE)
G

V2171 +k2H = -V>J

(A.2)

with k2 = (021.16. The solutions obtained for equations (A.2) unfortunately possess a
fairly complicated dependence upon the source terms; development and recognition of
a Green’s function for (A.2) is complicated and difficult.

185

There is a more satisfactory approach. From the observation that no magnetic
sources exist, use of the standard vector identity V-(VxA) = 0 allows the magnetic field
to be formulated as the curl of an auxiliary vector potential. This is justified by
Helmholtz’s Theorem, which states that a vector field is uniquely specified to within a
constant if both its curl and divergence are specified everywhere. Based upon historical

precedence [55], the electric-type Hertzian vector potential 116‘) will be used.

Consequently, the Hertzian vector potential is defined as a vector whose curl satisfies

it = jwerfi. (IA-3)

Substitution of definition (A.3) into Faraday’s Law reveals that

we - 626.6) = 0 (AA)

and application of another standard vector identity, VxVV = 0, allows the introduction
of the electric-type Hertzian scalar potential 0‘; thus, the electric field is

E = k’fi - 170* (A5)

with k2 = «021.15. The sign of the potential is chosen for consistency with electrostatic

convention. Substitution of (A5) into Ampere-Maxwell’s Law yields the wave equation

VxVxfl -k2fi = —j—- - V¢‘ (A-G)

joe
which, by using the vector identity Vx Vxfi = VV-fi - V’fi, can be cast into the more

familiar form of

V211 +k211 = -—I— + V(V‘fi+¢‘) . (A.7)

jwe
No divergence has yet been specified for 11; by choosing to enforce the Lorentz gauge,

Qe = 'V'fi (A08)

186

equation (A.7) simplifies to

Vzfi + kzfi = J/jue . (A3)
Helmholtz’s theorem is satisfied and the Hertzian vector potential is completely specified
(though any choice for the divergence of 11 will satisfy Helmholtz’s theorem).

The Helmholtz equation for electric-type Hertzian potentials given in (A.9) has
as a source term electric current density 36'); while useful for some applications,
historically, the Hertzian vector potential has been supported by polarization currents
PG) . A relationship between 7 and P is easily derived. Recalling that polarization
charge density is p", = -V-P, and that charge densities are related to current densities
by V-IWl - -ja) p”, it is easily recognized that ij = 7". Thus, an equivalent equation

to (A.9), with polarization sources, is

v2fi +k2fi = - (A010)

“Ira;

Regardless of which version of the Helmholtz equation is used, the electric field is found

in terms of the Hertzian vector potential as

E 3 k2fi ... vv.fi (A.ll)

A.2 Hertzian Potential Boundary Conditions

An analysis of planarly-layered geometry, as depicted in Figure 2.2, could
proceed without developing specific boundary conditions on the Hertzian potential. The
electric and magnetic fields can be calculated from solutions to Helmholtz equation

(A.lO) by using equations (A.3) and (A.ll), then matching tangential-field boundary

conditions at a source-free interface,

187

0MBI -g) = 0
0x03, '13:) = 0

an indirect technique at best. If boundary conditions for the Hertzian vector potential can

(A.12)

be developed, then these can be applied directly to solutions of (A. 10), and intermediate

operations ean be avoided. This becomes significant for multi-layered geometries.
Essentially, the Hertzian potential boundary conditions are a disguised version of

the standard electric and magnetic field boundary conditions as given in (A. 12). Using

cartesian coordinates, the electric and magnetic fields in the 1', region, in terms of

Hertzian vector potentials, are

an an

El; = kfll,‘ + —a—V-IIi H“ = jmell {—5 - J]
a: By 67.
a 611

E" = k? 110 + a v.11‘ H _ __‘1] (A.l3)

£1: = #1211“ + % V'II‘ H

The interface is planar and source-free, while the regions on either side of the interface
are homogenous, have the same permeability in = 1‘0 , but possess dielectric contrast. The
boundary conditions in equation (A. 12) will be applied to the fields as given by equation
(A.l3), from which relations on the Hertzian potential will be deduced. This looks
difficult, as all components of 11 and their derivatives are involved. However, by
choosing orthogonally directed polarization currents, and using linear superposition, the
task can be accomplished. The results of this process are well-known in the literature

[16,29], having been stated as

188

n1. = N22111:. “ =5)"

 

 

 

 

anla 2 m2.
8 N on 8
6’ 21 a” a m (A.l4)
anl’_an27 = _(N221_1)[an21+an23]
3? 3? ax 5:

at a dielectric—dielectric interface, with N}. = 452 / el . The boundary conditions in (A. 14)

simplify at a dielectric-perfect electric conductor (PEC) interface, becoming

II“ 8 0
__ = 0
3y

for a = 1,2. For problems involving magnetic contrast, the boundary conditions of

(A.l4) need to be slightly modified; the results can be found in [56].

A.3 Interpretations and Considerations

Working through the process used to arrive at the boundary conditions in (A. 14)
and (A. 15), while seemingly redundant, reveals insights into the analytical method being
used, and the necessary form for solutions. For a vertical (normal to interface) excitation
of P = 91",, , only vertical components of Hertzian potential (fi = 915) are excited.
Application of the boundary conditions (A. 12) to the scalar field components in Cartesian

coordinates gives

 

 

E... gm) = a: 07-911..)
‘ " (A.ld)
003,111,) = men”)

m ax, 6x“

189

where a =x,z, B =x,z and 2.1. 12’ . Equality of tangential derivatives for any point (x,z)

on the planar interface implies

 

 

v-rr, =- v-rr2 ~

an” . an”
6y 8y (A.l7)

2
nly 3 N 21 [[21

where N3, = ezle, . The conclusion about tangential derivatives that leads to (A.l7) is

intuitive; it is also mathematically rigorous. If tangential derivatives of two functions are

equal at each point on the interface (consider the E. boundary condition in (A. 16)), then

the integrals of each tangential derivative over an arbitrary path on the interface will also

be equal, namely

 

 

A

A! 6x. I ax! .

from which it is evident that the conclusion reached in (A.l7) is valid. A similar
procedure can be applied to the II. relations.
Consider now a horizontal (tangential) excitation P a 11?, over a planar interface.

A naive assumption is that only horizontal components of Hertzian potential (fl = 211,)

exist. Application of the electric field boundary conditions in (A. 12) gives
E : 0 kfnn+—a—(V-mk) = a singly-mg) (11.18)
C I. ax. I. &‘

for a =x,z. Obviously, V-111=V'fiz - 11,5111, satisfies the boundary conditions for
a = z. Satisfying the boundary conditions for a =x also requires that k: 1113-2452 Hz:
— an obvious contradiction. It is apparent that a horizontal source cannot excite only
horizontal potentials over a planar interface, but should also excite vertical potentials as

well. This excitation is actually intuitive if careful consideration is given to the electric

190

field maintained by Hertzian vector potentials. An interface with dielectric contrast has
a net surface polarization charge density, implying a discontinuity of the electric fields
normal (vertical) to the interface. As evidenced in equation (A. 13), the electric field has
a strong component in the direction of Hertzian potential; as a consequence, there must
always be a vertical component of Hertzian potential at a planar interface. Another
consequence of this excitation is that any Green’s function satisfying the Helmholtz
equation (A. 10) in a layered background environment will be dyadic in nature.

With the horizontal (tangential) source exciting both horizontal and vertical

components of Hertzian vector potential (fl = in, +9I1,), application of (A. 12) reveals

Mild-t 2:11:14?
)2. kfrru+.§(v.fr,) = gaping icy-11,) = 362’1V'fi1) (11.19)
an”
fi= gen.) = £6.11.) 63-- i} = gens-en.)

where the first line is for electric fields and the second for magnetic fields. This implies

the following Hertzian potential boundary conditions

11,, = ~:,n,, v.11, = v-ri2
3111. 2 ant (“0)

2
n1, = Nzrnry 6y = 21-0;

Similar results arise for an excitation of P = 2P0. By superposition, the results in

 

equations (A. 17) and (A20) generalize to the desired boundary conditions on Hertzian
potentials presented in (A.l4).

Note that the condition V°fll = V-fi2 shows up for both vertical (A. 17) and
horizontal (A.20) excitation boundary conditions. As V-fl= -¢', this translates into
continuity of the scalar potential, supported by polarization charges, across the interface

191

 

(a comforting result). Obviously, the continuity of scalar potentials is the mechanism
that couples horizontal components of Hertzian vector potential to its vertical compo-

nents.
Determining the boundary conditions on Hertzian potential with a perfect electric
conductor in region 2 follows a similar process. Application of the appropriate boundary

condition, fixi’.l = 0, results in'the Hertzian potential boundary conditions of (A.lS).

192

 

Appendix B

Appendix B
Spectral Representation of the Dyadic Green’s Function

B.l Principal Dyadic Green’s Function
The principal Hertzian potential is the potential supported by a current radiating
in unbounded space. This potential satisfies the Helmholtz equation (A.9), written in

scalar form below (a =x,y,z),
V’If:I +1211: -= -J./j0)e 03-”

subject to the boundary condition that the potential vanish at infinity (the radiation
boundary condition). The Green’s function for this equation satisfies (B.1) with the
Dirac delta function as the excitation,

vzarmr') +k’G'(r'|f’) = -0(‘r’-i”) 03-2)
subject to the same boundary condition. Without loss of generality, a solution for

G'(f|'r”= 0) will be sought, and the final result shifted to an arbitrary i".

From Chapter 2, the two-dimensional Fourier transform pair is,

 

1 ' -. .
GP - 6'1 ”"d’l 03-3)
0') (21:): ff (.y)e
6'01.ka Gran-lira» (3.4)

193

where X = 125 +z‘C is the 2-D spatial frequency. Writing G'(F) as its inverse transform

and exploiting the relationship

 

 

(2110’ If e’" .121 = 6006(2). (1)-5)
equation (B.2) becomes
1 . 31 2 2 'r " {-7 2
—+k -}. G 1. +6 e’ d1=0, (3.6)
(2101 [I {lay J ( .y) 0)}

where the (V2 +k’) operator has been passed under the double spectral integral. Since

Y‘lml =0 - {...} =0, it is clear that (B.2) becomes
[1, -p’(i)]é'(i.y) = —60) 03-7)
By

in the transform domain, where p0.) =W. The solution of (B.7), G’ , solves the
homogenous problem for y a 0 , must vanish as I y I - 09, must be continuous across y=0,
and must have a step discontinuity in the first derivative at y=0. This last statement can
be confirmed by integrating (B.7) with respect to y about y=0, resulting in

~P 9.
lim—aG
«*0 a)’

 

= —1 + 11(2)] G'(X;y)dy

'0
where the RHS integral vanishes as e ~0 because it is continuous across y=0 (Midpoint

theorem). When these boundary conditions are met, the solution for G’ is

2p().)'

 

G" (1y) = (13.8)

194

where Rel 11(1)) > 0 to insure (B.8) vanishes as y-m. Inverse transforming equation

(B.8) then results in, after shifting to an arbitrary i”,

r- - - - I
G'(i’li") e H " (2'; :12? ”.11).. (3.9)
.. 1t )

 

the desired spectral representation of the principle Green’s function.

3.2 Reflected Dyadic Green’s Function

The reflected dyadic Green’s function arises from matching the Hertzian potential
boundary conditions in a planar layered background environment. This dissertation
considers two different background environments: a two-layer interface, serving as the
background for an asymmetric planar waveguide, and a tri-layered background
environment, which is appropriate for both the microstrip and integrated dielectric rib
waveguide analysis. The most general case of the background, a tri-layered configura-
tion, will be developed in this appendix; from this, the appropriate specializations to
simpler environments will be made.

The Hertzian potential boundary conditions are developed in Appendix A.

Enforcing these boundary conditions results in

 

 

 

 

 

11.. = sztnz. a we (3.10)
an, 2 5H2
" = N ' = (13.11)
By 21 6y (1 15.2
an” " 61122 = ”(N221 - ”[5112! + 6112‘] (3.12)
By 6) ax az

for the y=0 interface and

195

Hz. = ”322113. a :10); (3.13)

 

 

 

g;— = N3, 321;“ a =x,z (3.14)
£212-95! = _(N:2-1)[an1'+an3=] (3.15)
By 31' 6x 31

for the y=-t interface, where NU = 72,] n, and n, is the i‘I layer refractive index.
Assuming the sources are in the cover (region 1), the Hertzian potential in each

region, developed in Chapter 2, is written in scalar form as

 

 

 

 

_. 3 1 a I", J. e-jX-f’e-Pr(1)1Y'y’l I+ r ’hfl)! 2 $.16)
II,.(r) (2.621!“ i ,we‘ 291(1) dV W1.(}.)e d A

.. ___ 1 '11:.) W: 1 Mlb+Wr 1 H1)? (12).. (3.17)
112.0) (2.021!“ [ 2.1 )e ..( )e ]
113.0) (210111} [ M )c J

for a =x,y,z, and where Re (p, 1 >0 is enforced to satisfy the radiation condition.

For tangential components of Hertzian potential (0: =x,z in (B. 16)-(B. 18)), boundary
conditions (8.10), (B.ll), (B.l4) and (B.15) are appropriate; these will be enforced at
y=0 or y=-t for all x and 2. If (B. 16)-(B. 18) are to satisfy the boundary conditions for
an arbitrary x or z, then the bracketed inner quantities must satisfy the boundary

conditions. This is equivalent to solving the entire problem in the transform domain.

Regardless, matching boundary conditions leads to the linear system of equations

196

-W{. +~;,(w;, + We'.) = V.
2
"up:
1
W22: "=' + mic” -N,’,W;.e "" = 0

N322P3

2

 

W1: * (W2; " War.) 3 V.

 

W22; "1' - wage” - W,‘.e ‘5’ = 0

where, as observed previously in Chapter 2,

 

J 'r" 4” w’
V. "f '( )e e dV’ a=x,z

y [.061 2’71“.)

The system of equations (B.19) is solved to yield the following

W2: 3 —V

 

 

 

 

 

 

 

 

 

Ra'r 3 1’1 ‘pz’ R112 = P2 1’1
p1 +p2 p1 +P2
: 2sztp2 : 2pr
T21 = ’ T12 = 2
p ‘ +p 3 N21(P1+P2)
R3; _ Pz'pa T: 2p;

 

—_—+——, 23 = 2
p2 p3 N32(P2 ’73)

and

197

(B.19)

(13.20)

(B.21)

(3.22)

(3.23)

Where (B.23) is zero, pole singularities occur in the solutions given in (B.21). These
pole singularities correspond to the TE surface-wave modes of the background structure.
Enforcing boundary conditions (B.10), (B.12), (B.13) and (B.15) for normal
components of potential will lead to the linear system of equations
-W,', +N§,(W,; + W23) = V,
W.',+”—j<W.'.-W;.) = V,+F [nevus]

 

 

 

-:lt r )2: 2 t -p,t _ (3.24)
W2": 4»sz -N32W3,e — 0
W‘ye-‘h‘ W532" Eng? 73‘ = -G [jEVx +jCVz]
where
F : (N221 ‘ 1) T1‘2[1 +R3‘2e -2”!
P1 D'
T: e0) 7,):
G= (”32" l) T1T223 ’ 8 “P3! (13,25)
P2 D‘

J r’ -1I'r’ 'N’
V, . f f( )9 e
v 100e, 2pl(A)

 

and D‘ is defined previously. Note that the spatial derivative operators on x and z are
passed under the spectral integration in (B.16)-(B.18), resulting in the 1'2 and jg’ terms
respectively.

The system of equations 03.24) is solved to yield

198

 

 

 

 

 

 

 

 

 

 

 

W2; . inf—53V” [@2231 +C2] {1&3 [ji V, 4.1-{VI}
W2; ' girl, + Nz‘fc, '2’?” -2” [1'5 V, +1113] (3.26)
m; = R4: + ”if“ m V. + C. + 1.1mm? C”) . m [1'5 V. +1: VJ
W3; 153:2; V, + "QC: + 7'2';(AG-1261I’lf'zurcze -1”) [1.5 V: +12": ]} elh'h):
where

2
3 ”up: ’1’; R3. 3 ”322?; 'pg

 

a". , . 2 ,
"21p 1 ”’2 N37P2 ”'1’:
. 2p .. 2p
721 . __2__3_, T12 =3 2 l
N 12?: ’71 N21 (Pr ‘72)
Ta 2P2
23 8 ——
N323": +P3 (3.27)

. "221(N211‘1)T1‘2 1+R3‘29-2N

 

N221p1 +p2 D ‘
_ ”312(N322 -1) Tan;

 

"3221’: +1’3 D t
and
D" = 142111.362” 03-28)

Wherever (B.28) vanishes, pole singularities will occur in the solutions given in (B.26).
These singularities correspond to the TM surface-wave modes of the background
structure. Also, note that the coupling coefficients C, and C, also depend upon 0‘;

obviously, solutions in (3.26) possesses TE surface-wave mode singularities as well.

199

 

Rewriting coefficients in region (1) in terms of the scalar components of the

source excitation VG”) gives

W1: = Rrya
W,’, - RlV,+C[j£V‘+jCVz]

 

where
t _I__T2R’;2T21e-2p,¢
R. 321—0,
1' -
R _ R2: Trzksz 21‘ 21),:
D.
72.1 [ngNz-rzc "‘ C] -2 r
C = + 1 2 C ’2
DB

The total space-domain potential may be written as
fl1 ‘ fl: *fii'

Equation (B.29) is substituted into (B.16) to yield

01(7) = [6(r1r’)-’.—“'—’dV’
v 10):

I

where
Gmr’) = é’(r|r’) + G'(r|r’)

6'01?) = iGPmr’)

 

 

 

 

G' r r’ - 126')? 60:2 a' ' “
" 'I . (r-r’) 1'. LV -y’l
avmr’) = ff " ‘ d2}.

.. 2(21t)2 c

200

(B.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(13.35)

(B36)

 

r 1 l 1
Girlr’) . 1w) _ _ _ +6

16mm = If 112.00) "I‘M"? ’ .421. 03-37)
9.771%), "' (C (1)} 2‘2“) c

 

 

 

 

Obviously, by returning to space domain, the multiplications by j£ and j I return to spatial
derivatives on x and 2, as seen in (B.35).
If region 3 (the substrate) happens to be a perfectly-conducting material, then
a: - -j°°. Consequently, p3 ”jun; in this limit, the following interfacial coefficients
become
12;. ~ -1 R4; ~ +1
1;, ~ 0 73’; ~ 0

(3.38)

and the reflection and coupling coefficients become

p, -p, coth<p2 t)
Z ”(1)
N311). 192 who». 0
Z ‘ (l)
. 2 (N3. - up.
I ‘01-) Z ‘0)

 

R11) '

12.0.) = (B.39)

 

CO.)

 

where

Z ‘0) = N221 p1+p2tanh(P2t)
2"(1) = )21 +p2coth(p2t).

The coefficients given in (B.39) have been manipulated to provide a more stable form
when implementing numerically. Still, whenever Z ‘ (2.) or Z “(1) vanish, pole
singularities occur; these are associated with the TM even or TE odd surface-wave

background modes respectively.

201

For a two-layer background environment, there is obviously no separate interface

between region 2 and region 3. The reflection/transmission coefficients for the region

2-3 interface are
a; = o (”'4“)

and the coefficients (B.30) simplify dramatically, becoming

 

 

p1 ”p2
R: .
P1 +1’2
I N221p -p
R2. = —-,—‘——’ (3.41)
”211% *P2
C 3 2mm; -1)
1
(pr ’45) (”2211’1 +P2)

If dealing with lossy materials, the term N221P1*P2 can vanish; this corresponds to the
famous and contentious Zenneck wave [35].

It has been stated without proof that the zeros of the denominator terms D ‘ from
(B.23) and D" from (B.28) correspond to surface-wave modes of the background

structure. Noting that the tri-layer background structure is an asymmetric slab waveguide

motivates the following definitions

N

ngcg li-C
0=p,= A-k

P (I

N
N

(3.42)

. 2 2
x = -jp, = [cf-1.,

where y, (5, and I: possess positive real parts. After algebraic manipulation, D ‘ in (B.23)

is seen to become

202

mm = 5523981 (3.43)
-y

the well-known eigenvalue equation for TB surface-wave modes of the background
structure [11]. The solutions lie within the regime maxlk‘, k,} < A, < k,; these
solutions are indeed surface-wave modes of the background, for the fields are oscillatory
in the film (p,=jx) and exponentially decaying in the cover (pc =7) and substrate
(p: = 0 ). An analogous procedure can be carried out for D' in (B.28); it is apparent
that D " will become the eigenvalue equation for TM surface-wave modes of the

asymmetric planar waveguide.

203

 

 

Appendix C

 

Appendix C
Proof of the Analytic Function Definition Theorem of Chapter 3

Theorem I, the proof of which is presented in this appendix, is central to the
development of convergence criteria for the Fourier transform pairs used in the analysis
of open-boundary waveguides. While the proof of Theorem I does not warrant inclusion

in the main text, it is of interest nevertheless.

C.1 Theorem and Definitions
The following theorem, taken from Titchmarsh [42], defines an analytic function
by means of integration.

Theorem I: Let D be the region. Let [(2, )9) be continuous in z and w

where z e D and w lies on a smooth contour C, possibly unbounded. Let

f(z, 1v) be an analytic function of z in D for each w on C. Let f f(z,w)dw
c

be uniformly convergent. Then

F(z) = fc f(ewww (c.r)

is an analytic function of z in D.
Before stating the proof of Theorem 1, some preliminaries are needed. An analytic
function is defined as: A complex fitnction f(z) of a complex variable 2 is analytic at a
point 20 if it is dtfierentiable at every point within a neighborhood of 20 An analytic

204

function satisfies the Cauchy—Riemann conditions; namely, an analytic function
f(7.) = u +jv of z = x +jy satisfies u, = v, and u’ = "’3 at each point where analytic. A
function is analytic in a region D if analytic at all points of region D. Also important
is Cauchy’s Integral Formula:

Let f be analytic in a region D, and C be a closed contour within D.

Then at a point z, inside contour C

. .1. 1(1).
f(z.) 210‘ j zflock. (c.2)

C.2 Proof of Theorem I
Choose 1‘ as some contour in region D enclosing point a. As f(z,w) is

analytic,

 

2"“

, _1_ f(z.W)
f(a.w) 2n} ‘I dz (c.3)

by Cauchy’s integral formula. Also, f(a,w) is analytic, since a 6D.

First consider a finite (bounded) contour C. Thus

. = _1_ £2.21)
F(a) {renew j 210'! radzdw. (cu)

Interchange of integration order is valid, as the integrand is continuous over both

2 and w. Consequently,
F(a) = 5%]- ! ”Camel-1%; . (0.5)

But the bracketed quantity is just F(z), hence

F(a) = 1 f F(2)612. (C.6)

2njr z-a

 

205

Since a is enclosed by 1‘, and 1‘ lies within D, then (C.6) is Cauchy’s Integral
Formula. Since F(z) satisfies Cauchy’s Integral Formula as given in (C.6), F(z)
is an analytic function of z in D.

Consider contour C now becoming unbounded on the positive w axis. Choose

C. as that part ofC within the circle Izl =n, and let
F.(z) = [q f(z,w)dw (c.7)

From the prior result in (C.6), F,(z) is analytic. Furthermore, each F,(z) is
analytic, and F.(z) -° F(z) uniformly as n ~ co. Since F(z) is a uniform limit of

analytic functions, F(z) is analytic.

A consequence of Theorem I is that F(z) is continuously differentiable, or that all
higher order derivatives exist, at each point where F(z) is analytic. While the proof of
this statement, and of Cauchy’s Integral Formula (C.2), are important, they are not of

immediate interest and can be found elsewhere [57].

206

 

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211