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‘ :THE ELECTRIC GREENS DYAD FOR THE CIRCULAR
~ ' ‘ _ DIELECImC FIBER it
A DIRECT COMPLEX ANALYSIS APPROACH

stsertatlon fer the Degree of M. S ‘ '7 '
MlCHIGAN STATE UNIVERSITY
PAUL FRANK HAVALA
1989

 

 

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293 00784 5971

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thesis entitled

THE ELECTRIC GREEN'S DYAD FOR THE CIRCULAR
DIELECTRIC FIBER: A DIRECT COMPLEX ANALYSIS APPROACH

presented by

Paul Frank Havala

has been accepted towards fulfillment
of the requirements for

 

 

Master of Science degree in Electrical Engineering

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THE ELECTRIC GREEN’S DYAD FOR THE CIRCULAR DIELECTRIC FIBER:

A DIRECT COMPLEX ANALYSIS APPROACH

By

Paul Frank Havala

A DISSERTATION

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

MASTER OF SCIENCE

Department of Electrical Engineering
1989

ABSTRACT
THE ELECTRIC GREEN’S DYAD FOR THE CIRCULAR DIELECTRIC FIBER:
A DIRECT COMPLEX ANALYSIS APPROACH
By

Paul Frank Havala

The electric Green’s dyad for the circular dielectric fiber, a
useful tool for the characterization of excitation and scattering
phenomena associated with these structures, is developed. A step-index
guiding region immersed in a uniform surround is investigated. The
longitudinal invariance of the electrical parameters allows both
impressed fields, maintained by an impressed current in either region,
and scattered fields, excited by interaction of the impressed fields
with the guiding structure, to be expressed as circular harmonic
expansions in the axial Fourier transform domain. Total tangential
fields are matched at the guiding/surround region interface to determine
the spectral expansion coefficients.

The spatial fields are recovered via the inverse Fourier
transformation. Evaluation of the inversion integral by an appropriate
complex plane analysis gives the complete propagation mode spectrum,
which permits construction of the electric Green’s dyad.

The electric Green's dyad is specialized for TB and TM modes. For
the TE case, the Green’s dyad reduces to a scalar Green’s function. The
TE Green’s function is employed to formulate an electric field integral
equation (EFIE), which, in turn, is used to classify scattering of the

TEO1 mode from a guiding region dielectric step discontinuity.

TO MY WIFE, JOAN

ACKNOWLEDGEMENTS

The author would like to express his sincere gratitude to Dr.
Dennis P. Nyquist for his patience, encouragement, and guidance

throughout this undertaking.

TABLE OF CONTENTS

Chapter Page
LIST OF TABLES ..................................................... vii
LIST OF FIGURES .................................................... viii
1 INTRODUCTION ............................................. 1
2 AXIAL FOURIER TRANSFORM DOMAIN FIELD REPRESENTATION ...... 4
2.1 Introduction ........................................ 4
2.2 Geometrical Configuration ........................... 4
2.3 The Axial Fourier Transform ......................... 6
2.4 Circular Harmonic Expansions for Scattered Fields... 8
2.5 Circular Harmonic Expansions for Impressed Fields
in Unbounded Guiding or Surround Region ............. 11
3 DEVELOPMENT OF THE ELECTRIC GREEN’S DYAD FOR THE
CIRCULAR FIBER WAVEGUIDE ................................. 16
3.1 Introduction ........................................ 16
3.2 Construction of Scattered Electric Green’s Dyad ..... 16
3.3 Construction of Impressed Electric Green’s Dyad ..... 24
4 SPECTRAL ANALYSIS FOR EVALUATION OF THE INVERSE AXIAL
FOURIER TRANSFORMATION ................................... 27
4.1 Introduction ........................................ 27
4.2 Deformation of the Real-line Inversion Integral ..... 27
4.3 Singularities in the Complex C-Plane ................ 32
4.3.1 Pole Singularities ........................... 33
4.3.2 Branch Point Singularities ................... 38
4.4 Summary ............................................. 46
5 APPLICATION TO AXIALLY-SYMMETRIC MODES ................... 50
5.1 Introduction ........................................ 50
5.2 Electric Green’s Function for TE Modes .............. 52
5.3 Electric Green’s Dyad for TM Modes .................. 58
6 APPLICATION TO SCATTERING FROM A GUIDING REGION
DIELECTRIC DISCONTINUITY ................................. 64
6.1 Introduction ........................................ 64
6.2 Formulation of EFIE for Scattering from a Guiding
Region Dielectric Discontinuity ..................... 65
6.3 Scattering Coefficients ............................. 69
6.4 Numerical Results ................................... 70

V

TABLE OF CONTENTS cont’d.

Chapter Page

7 CONCLUSION ............................................... 86
APPENDIX A ......................................................... 89
APPENDIX B ......................................................... 91
APPENDIX C ......................................................... 94
LIST OF REFERENCES ................................................. 98

vi

Table

2.4.1

2.4.2

3.2.1

4.3.1

4.3.2

5.3.1

5.3.2

LIST OF TABLES

List of K:B(plp’) terms. The upper expression in each

term is valid for p > p’, the lower for p < p’ ...........

List of R:B(plp’) terms. The upper expression in each

term is valid for p > p’, the lower for p < p’ ...........

Elements of E

“B I P _ _
List of Rn (plp ) terms, where 08 - Qg(C - CP) ...........

List of G:B(plp’) terms ..................................

List of R:B(plp’) terms for the TM guiding region
electric Green’ s dyad, where 0: =08 (C= Co ). Q0

-J Qs
QS(C= Co ), and A0 A .............................

TM= we H
0

List of G:B(plp’) terms for TM guiding region electric

. o_-J
Green s dyad, where ATM - we AE' The upper expression

in each term is valid for p 9 p’, the lower for p < p’...

vii

2m ..........................................

Page

14

15

26

48

49

62

63

Figure

2.2.1

4.2.1

4.2.2

4.3.1

4.3.2

4.3.3

LIST OF FIGURES

Page

Geometrical configuration of an axially-infinite,
step-index dielectric fiber of circular cross-
section. Cartesian coordinates (x,y,z) and
cylindrical polar coordinates (p,9,z) are oriented
so that the z-axis lies along the guiding axis.

Refractive indices ng and n5 are invariant along 2 ....... 5

Integration contour C when integrand has both pole
(Cp) and branch point (Cb) singularities. The
shaded region denotes the branch out emanating

from Cb .................................................. 3O

Fourier inversion contour when integrand has both pole

and branch point singularities. The infinite closure
+

path is denoted by c;, while the pole and branch out

respectively ........... 31

+
detours are denoted by C; and Ci,

Expanded view of the upper half-plane pole exclusion
contour c;, showing local polar coordinate represen-

tation of pole singularity ............................... 37

The complex C-plane. Shaded areas represent the
branch cuts emanating from the branch points at
1kg and tks. The surface wave pole detours Ct are

not pictured ............................................. 42

More detailed pictures of the complex C—plane,
showing (a) the sign convention for either Q on
both sides of its corresponding low-loss branch
cut, and (b) the upper half-plane integration
contour components as the branch cuts coalesce in

the low-loss limit ....................................... 43

viii

Figure

6.3.1

6.4.1

6.4.2

6.4.3

6.4.4

6.4.5

LIST OF FIGURES cont'd.

Geometrical configuration of uniform dielectric slice

discontinuity. Coordinate axes are aligned as in

Figure 2.2.1 ............................................

Radial distribution at z/d = 0.0, for various numbers
of radial expansion functions, of the electric field
induced within a uniform slice discontinuity along

a fiber waveguide with n8 = 1.5, ns = 1.0, n = 3.0,

d

d/a = 0.25, and a/Ao = 0.5 ..............................

Axial distribution at p/a = 0.5, for various numbers
of pulse expansion functions, of the electric field
induced within a uniform slice discontinuity along

a fiber waveguide with IIS = 1.5, nS = 1.0, n = 3.0,

d

d/a = 0.25, and a/AO = 0.5 ..............................

Dependence of TE01

normalized length (d/a) of a uniform slice discon-

scattering coefficients on the

tinuity along a fiber waveguide with n8 = 1.6,

ns = 1.49, 11d

Dependence of TE scattering coefficients on the

01
normalized length (d/a) of a uniform slice discon-

tinuity along a fiber waveguide with ng = 1.5,

ns = 1.0, nd

Dependence of TE scattering coefficients on the

01

refractive index nd of a uniform slice discontinuity

along a fiber waveguide with ng = 1.6, nS = 1.49,

d/a = 0.1, and a/Ao = 1.0 ...............................

ix

= 1.0, and a/Ao = 1.0 .....................

= 3.0, and a/Ao = 0.5 ......................

Page

66

76

77

78

79

80

Figure

6.4.6

6.4.7

6.4.8

6.4.9

6.4.10

LIST OF FIGURES cont'd.

Dependence of TE01 scattering coefficients on the

refractive index nd of a uniform slice discontinuity

along a fiber waveguide with n8 = 1.5, ns = 1.0,

d/a = 0.1, and a/AO = 0.5 ..............................

Radial distribution, at various axial locations, of
the electric field induced within a uniform slice
discontinuity along a fiber waveguide with ng = 1.5,

ns = 1.0, nd

Radial distribution, at various axial locations, of
the electric field induced within a uniform slice
discontinuity along a fiber waveguide with ng = 1.5,

ns = 1.0, n = 3.0, d/a = 0.25, and a/Ao = 0.5 .........

d

Axial distribution, at various radial locations, of
the electric field induced within a uniform slice
discontinuity along a fiber waveguide with ng = 1.5,

ns = 1.0, nd

Axial distribution, at various radial locations, of
the electric field induced within a uniform slice
discontinuity along a fiber waveguide with IIS = 1.5,

ns = 1.0, nd

= 3.0, d/a = 0.05, and a/Ao = 0.5 .........

= 3.0, d/a = 0.05, and a/Ao = 0.5 .........

= 3.0, d/a = 0.25, and a/Ao = 0.5 .........

Page

81

82

83

84

85

 

CHAPTER ONE

INTRODUCTION

As the high-frequency transmission of data along dielectric fiber
waveguides finds an increasing number of practical applications, the
electromagnetic analysis of the fundamental propagation phenomena
associated with these structures remains a pertinent research area. Of
particular importance is the study of radiation due to fiber excitation,
physical imperfections, and various waveguide interruptions. The
electric Green’s dyad for the circular dielectric fiber, a useful tool
for the characterization of fiber radiation, as well as other excitation
and scattering phenomena, is presented in this thesis.

Construction of the electric Green’s dyad for the circular
dielectric fiber requires understanding of its complete propagation-mode
spectrum, which consists of both discrete surface wave fields and
continuous radiation mode spectral components. The bound surface wave
fields, which transport energy indefinitely along a lossless waveguide,
are well-documented [1]. However, the radiation modes of the circular
dielectric fiber waveguide, characterized by oscillatory behavior far
from the waveguide and a continuum of axial propagation constants [2],
have received minimal attention. In general, many of the important
properties, including the orthogonality, normalization, and amplitude
spectrum, of radiation modes along open-boundary waveguides have been
described in some detail [3-6]. However, while Green’s function

techniques are available for the study of planar dielectric waveguides,

the current analysis method for circular structures remains the
conventional eigenfunction expansion approach developed by Snyder [2]
and pursued by Snyder and Love [7].

The electric Green’s dyad presented here provides an alternative to
conventional eigenfunction expansion techniques. While development of
the eigenfunction method is simple in comparison with that for the
Green’s dyad, its application often proves cumbersome. In the
eigenfunction approach, radiation modes are constructed from even and
odd symmetric combinations of incident transverse electric (ITE) and
incident transverse magnetic (ITM) components [2] to satisfy field
orthogonality properties. Awkward infinite integrals must be evaluated
to determine the orthogonality, normalization, and amplitude spectrum
associated with these mode combinations. Consequently, this method has
apparently been implemented only for axially-invariant applications.
Implementation of the electric Green’s dyad presented here circumvents
these difficulties.

These complications are avoided because the electric Green’s dyad
is constructed via a complex spectral analysis. In many ways, this
approach is analogous to that presented by Collin [1] for a lossless
dielectric sheet on a perfectly conducting plane. In this case,
however, the longitudinal invariance of the waveguide parameters prompts
an axial Fourier transformation, instead of the bilateral Laplace
transform invoked by Collin.

Fields are constructed as circular harmonic expansions in the axial
Fourier transform domain. The Green’s dyad is obtained by performing
the inverse axial Fourier transformation on these fields. A subsequent

complex spectral analysis of the Fourier inversion integral allows

identification of the complete propagation-mode spectrum. The Green’s
dyad is expressed as a sum of discrete surface wave contributions along
with an integral over the continuum of radiation mode components. The
fact that the entire propagation-mode spectrum may be extracted in a
unified manner is an inherent advantage to this approach. And, while
buried in its derivation are a number of analytical subtleties, once the
Green’s function has been identified it may be implemented rather easily
for a variety of applications.

The electric Green’s dyad presented here is well-suited for the
analysis of excitation and scattering phenomena associated with fiber
waveguides. In this thesis, an example of its application to the

characterization of scattering of the TE surface wave mode from a

01
dielectric discontinuity is offered. An electric field integral
equation (EFIE) technique is employed to provide conceptually-exact
descriptions of the discontinuity fields [8.9]. Field distributions are
determined from the method-of-moments (MOM) solution of the EFIE,
subject to numerical approximation error only. Convergence of the MoM
solutions is addressed, as well. The Green’s dyad may also be

implemented, along with many of these techniques, for the study of TM

and hybrid mode scattering and fiber excitation.

 

CHAPTER THO

AXIAL FOURIER TRANSFORM DOMAIN FIELD REPRESENTATION

2.1 Introduction

The aim of this chapter is to construct electric and magnetic field
representations which will ultimately aid construction of the electric
Green’s dyad. To meet this end, these field expressions should
accommodate integro-differential operations in cylindrical coordinates,
possess important orthogonality properties, and lend physical insight
into the components of the complete propagation-mode spectrum. Field
representations such as these may be developed by exploiting the axial
invariance of the salient electrical parameters of the waveguide, as
well as its circular transverse geometry. These properties invite
circular-harmonic field expansions in the axial Fourier transform
domain. First, however, the geometrical configuration of the dielectric

fiber waveguide is presented.

2.2 Geometrical Configuration

The configuration of the circular fiber waveguide is depicted in
Figure 2.2.1. The coordinate axes are aligned so that the direction of
wave propagation is parallel to the z-axis. All fields and sources are
assumed time-harmonic at radian frequency u, where the exp(Jwt)
dependence is suppressed throughout the duration of this thesis. Media
are assumed nonmagnetic, so that the guiding and surround regions may be

described by complex permittivities and wavenumbers (cg’kg) and (cs,ks),

 

Figure 2.2.1.

  
  

35:55:33: ==== SURROUND
" ,1, REGION

   

Geometrical configuration of an axially-infinite
step-index dielectric fiber of circular cross-
section. Cartesian coordinates (x,y,z) and
cylindrical polar coordinates (p,9,z) are
oriented so that the z-axis lies along the

guiding axis. Refractive indices n8 and 11s are

invariant along 2.

where k = cgu and k = wV c u . The excitatory volume current

3 o s s 0
density 3, which may reside in either the guiding or surround region, is
localized near 2 = 0, and represents either a real impressed current or
an equivalent current describing a heterogeneous discontinuity region.
The current 3 maintains impressed field E1 which is scattered by the
guiding/surround region transverse inhomogeneity, producing scattered

field Es. The total electric field in either region is given by the sum

of impressed and scattered components
26’) = EH?) + Est?) (2.1)

where E13 0 only in the region containing 3.

2.3 The Axial Fourier Transform

The invariance of the electrical parameters along the axial (9)
direction prompts spectral analysis of field quantities in the axial
Fourier transform domain. The spectral method, though requiring an
extensive complex—plane analysis, circumvents the cumbersome
normalization integrals necessary for the eigenmode expansion study of
dielectric fiber waveguides presented by Snyder [2] and pursued by
Snyder and Love [7]. Until the Fourier inversion integral is
investigated more rigorously in Chapter Four, all analysis is performed
in the transform domain. Here, the forward and inverse axial Fourier
transforms are defined, and several transform pairs are listed.

If the total electric field E(?) in either region is written in
generalized cylindrical coordinates as E(?) = E(3,z), where 3 is the
transverse position vector, then the transform domain electric field

3(3,C) is defined by the axial Fourier transformation

a)
2(3,c) = 92{E(3,z)} = I EIB.2I e-ch dz. (2.2)

Conversely, E(3,z) may be recovered from 3(3.C) via the inverse axial

Fourier transformation
a
EIB.zI = s;1{3(3,c)I = -l I 3(3.c) ech dC. (2.3)
m
The electric axial Fourier transform pair is denoted
E(3,z) e——————+ 3(3.cI.

Transformed fields are denoted with the same variables as their
space-domain counterparts, only in lowercase script. Axial Fourier
transform pairs for magnetic field, excitatory volume current density,

and the differential operator V may likewise be written as
R(3,z) e——————e H(B.C)

j(3,2) (—'———) 3(39C)

._ A6 ~-
V - Vt + 25; e——————e V - Vt + sz

where Vt is the transverse portion of V. From (2.1) the total transform
domain electric field in either the guiding or surround region may be
expressed as the sum of impressed and scattered components

ziI 35(*

zen = 3.0 + MI.

91

The impressed field e (3.0 is non-vanishing only in the region

containing the excitatory current.
The cylindrical transverse symmetry of the waveguide invites a more

detailed description of the scattered and impressed fields in terms of

circular harmonic expansions. The remainder of Chapter Two is devoted

to constructing these expansions.

2.4 Circular Harmonic Expansions for Scattered Fields

In this section, the scattered fields in both the guiding and
surround regions are expressed as circular harmonic expansions. These
expansions represent solutions of the homogeneous Helmholtz equation in
both the guiding and surround regions. The axial Fourier transform
domain Helmholtz equation in either region is solved for the
longitudinal scattered fields ez(p,6,z) and h2(p,9,z); the longitudinal
scattered fields are, in turn, employed to construct the transverse
scattered fields.

As discussed above, the expansions for the scattered fields arise
from the solutions of the homogeneous Helmholtz equation. In the axial
transform domain, partial differentiation with respect to z is replaced
by multiplication by JC; hence, in either region, the homogeneous

Helmholtz equation has the two dimensional form
vfwrp,e,c) + (k2- czIIIp,e,cI = o

where W(p,9,C) represents either ez(p,6,C) or hz(p,9,C). The quantity

k2 — C2 may be replaced by 02, leading to
2 2
Vt9(p,9,C) + Q W(p,9,C) = 0. (2.4)

The transform domain wavenumber parameter Q is defined by

In the guiding and surround regions, 0 may be specialized to

8 8
os= NICE—<2

where the subscripts g and 5 denote the guiding and surround regions,
respectively. Apparent ambiguities in the sign convention for Q are
resolved in the spectral analysis exposed in Chapter Five.

The appropriate separation of variables solution of (2.3) in

cylindrical coordinates yields

J n(QP)
\P(p.9.C) =2 V’n (c) eJne {14: (2) }.
n=-m (Qp)

Bessel functions are chosen for the radial dependence of the guiding
region (p < a) wave function since they represent standing waves finite
at the waveguide center (p = 0). In the surround region (p > a), the
Hankel functions are selected to ensure outward-propagating cylindrical
waves as p increases without bound (see Appendix C for some important
properties of Bessel and Hankel functions). This choice of radial
expansion functions provides the following expressions for the
longitudinal scattered fields in the guiding region
a
{ 82(P.9.C) } = Z { An“) } Jn(Qgp) ejn" (2.5a)

hz(p,9,C) n=_m Bn(C)

and the surround region
{ ezip.e,<) } = i{ Cn(C) }H1(12)(Qsp) ejne. (2.5b)
hz(p,e,C) n=_m Dn(C)
The: transverse scattered fields may be determined entirely from the

lorlggitudinal fields by

[CVtez(3.C) + wuo’z‘ x Vthz(3.C)1

1'"
A
P
J»
V
n
AI

:11.
1"
‘P
JR
u

A '9 ‘9
If? [U82 X VtGZIP.C) - Cvthz(p’g)].

In the guiding region, the transverse scattered fields are given by

 

es(* I - J An ( )J’(Q I + wu° n B ( )J (0 I Jne (2 6 I

P PIC " C C 8p n C D 8P 9 . a
=-oe

again = Z— Q:[5—:—p AnthJn (08p) + Juno B n(C)J’(Q gp) ] e5“ (2.6b)
n=-m

IPI" c) = Zoo—1— [3 A (CH (0 I + JCB (ClJ’(Q I ] eJne (2 50)

p p, 08 08p n n gp n n gp '
n=-m

O

s e _ 1 _ , _ _2S Jne

h9(P.C) - :Tg[ jwchn(C)Jn(Qgp) Qg" Bn(C)Jn(Qgp) ] e . (2.5a)
113-”

The surround region transverse fields may be determined in an analogous
manner.

The scattered fields in both regions are now expressed as circular
harmonic expansions. These expansions are specified in terms of
well-defined radial and angular functions, as well as the unknown
spectral amplitude coefficients An, Bn’ Cn’ and Dn' The expansion
coefficients are investigated in Chapter Three, where appropriate
boundary conditions at p = a are applied. The boundary condition
invocation, however, requires continuity of total tangential
fields--impressed and scattered--at the guiding/surround region
interface. Determination of the impressed fields, and expansions of
those fields in terms of circular harmonics, are the topics of the

following section.

10

2.5 Circular Harmonic Expansions for Impressed Fields in Unbounded
Guiding Region or Surround

The electric and magnetic fields impressed by an excitatory current
source located in either the guiding or surround region are expressed as
solutions of the forced axial Fourier transform domain Helmholtz
equation. Since no impressed fields may be maintained in a source-free
region, and the source is assumed to reside in only one region, the
impressed fields may be constructed from an equivalent current source in
a homogeneous medium with electrical parameters (c,k) characteristic of
the region containing the source.

The impressed fields are described in terms of the electric Hertz
potential. The electric Hertz potential, in turn, is written as a
circular harmonic expansion of the well-known scalar Green’s function
solution of the two-dimensional inhomogeneous Helmholtz equation. As
functions of the impressed Hertz potential in the axial transform domain

the impressed fields are [1]

3.C) + 66-21(3.CI (2.7a)

(N,
H.

) = k221(

pt
JV

31( (2.7b)

Jwefi x 3,C).

5:»

St

M
II

To retain generality, the electrical parameters (s,k) are not specified.
This Hertz potential description of the impressed fields entails the
two-dimensional inhomogeneous Helmholtz equation

2+1
n

9
t (3.CI + (k2 - c2)?1(3.C) = 1:19:51 (2.8)

V Jwe

where 3 is the transform domain equivalent current source localized near

(p’,9’,z’= 0). The solution to (2.8) is given by

11

ch

->.
21(3.<) = [319—51 3(3I3’) dS’. (2.9)
cs!

Here, CS’ represents the source cross-section in the (p’,6’) transverse

plane, and g(3l3’) is the well-known scalar Green’s function

2(3I3') = % 11:2)(le-3I). (2.10)

At this point, the impressed fields may be fully specified from (2.7).
with the impressed electric Hertz potential given by (2.9) and (2.10).
However, for reasons that become evident in the ensuing analysis, an
alternative expression for (2.10) is desired. The impressed fields must
be written in forms that accommodate operations in cylindrical
coordinates; more precisely, expressions for the impressed fields in
terms of circular harmonics are sought. This is accomplished by
expanding the two-dimensional scalar Green’s function in (2.10) (see

Appendix A) as

m
, J ?(QP’) H? )(Qp)
8(pI3’) = Lil—2f Hume . (2.11)
n.____ 2)(Qp’) Jn(Qp)

The upper expression is valid for p > p’ and the lower for p < p’. This
circular harmonic expansion for g(3|3’) yields the impressed electric

Hertz potential

.. H(2)
_ (Op) J n(Qp)’ _ ,
31(3’0 = 3%; Ze eJne [III-M H32) ’ e .1119 (15..
J:(Qp) (OP)
CSI

n=-o

The impressed Hertz potential, in turn, may be substituted into (2.7) to
give the axial Fourier transform domain impressed electric and magnetic
fields. After extensive algebraic manipulation, the impressed fields

may be written in the compact form

12

 

a
3103.0 = 2° 2 ejne Z A IX K:B(P|P')JB(3’,C)e-Jne’dS’ (2.12a)
n=-oo a CS'
m
3163.0 = % ZeJ“ 92a [Z xwmlp )JB(p, ,qu Jne'ds' (2.12s)
n=-oo a CS' B

where a and B may be p, 9, or z, and the K28 and Egg terms are presented
in Table 2.4.1 and Table 2.4.2. The K28 and R2? terms depend on the
radial distances p and p’, as well as the transform domain wavenumber
parameters C and Q(C). Note also that the radial dependence of each of
these terms has different forms for p > p’ and p < p’.

Since the transform domain impressed fields are now specified, the
total transform domain field in either region may be expressed as the
sum of the scattered and impressed field components developed in this
chapter, with the electrical parameters indicative of the region of
consideration. Determination of the total fields in both regions allows
the application of the boundary conditions at the guiding/surround
region interface, which provides a system of equations that may be
solved for the spectral expansion coefficients embedded in the scattered
field expansions. The solutions for the expansion coefficients lay the
foundation for subsequent construction of the electric Green’s dyad for

the dielectric fiber waveguide, presented in Chapter Three.

13

Table 2.4.1.

Kfipmlp’)
ngwlp’)
nguolp’)
K:p(plp’)
ng(plp’)
K:z(plp’)
K:p(plp’)

ze ,
Kn (plp )

zz ,
Kn (plp )

 

 

n2 {lemp’l’ézi‘l )+(Qp)}
szp’ (Op ) Jn(Qp)
. , (2)
:42 J n(Op I Hn (Op) _
Q” H‘2"(Op') I n(Op)
I (2),
49$ J n(Op I Hn (Op)
k2 H(2)(Qp ) J’(Qp)
2 J’(Qp’) 3‘2 )(Qp)
Inc
szp H;2)’(Qp’) I JOp)
n2C2 I n(Op I R? (Op)
kZQpr’ H(2)(Qp’ ) J n(0p)
-nc J n(Op ) Hf (Op)
k2 p H(2)(Qp’)n I JOp)
. , (2)
-I0c J n(Op I an (Op)
k2 H(2)’(Qp I I n(Op)
-n: I n(Op') Rf ’(OpI
kzp H(2)(Qp I J n(Op)
Q 2{Hn J n(Qp') H‘2 (0p) }
‘2' (2 ) ,
k an (Op ) Jn(Qp)

List of K:B(plp’) terms.

each term is valid for p > p’,

14

+2.7
Q” H(2)(Qp’)n J’(Qp)

szp’

+ n
H(2)
n

The upper expression in

the lower for p < p’.

}

2): (OP)
(Qp’) J’(Qp)

J! (Opt) an

H(2)’
n

)’(Qp)
(QP'J-V(QP)
(

5.
k2

{
Zl:

J n(Qp') HD
(2)

JDC

}

J n(Qp’ ) H: )’(Qp)

}

}

J’(Qp’) Him ’(Qp)

’(Qp’) J$(Qp)

Table 2.4.2.

i:p(p|p’)
fiftplp’)
ޤz(plp')
K:p(plp’)
i:9(plp’)
E:z(plp'I
E:p(plp'I
fi:°(plp'I

~zz ,
Kn (plp )

List of E:B(plp’) terns. The upper expression in
each term is valid for p > p’, the lower for p < p’.

I I (2) )I
22S {n I n(Op I “n (Op) } _ n: {k I n(Op I HQ (Op)
0” H(2)’(Qp I I n(Op) 0” an z’(Op') J’(Qp)

2 J ?(Qp') H? )(Qp) J'(Qp') HE )’(QP)
49-9— + J:
szp’ 2’(Op') I n(Op) HEZ)’(Qp’) J£(QP)

 

:42 {; J n(Qp’) H? )(Qp) }
p (2 ) ,
Hn (Qp ) Jn(Qp)

_~PB ,
Kn (plp )

~pp .
Kn (plp )

I n(Op') H(2)’(Qp)
O
H(2)(Qp’ ) J’ n(Qp)

= .42 I n(Op I HQ (Op)
9 M2’(Op'In I n(Op)

J’(Qp’) HQ )(Qp)
Q ?2) n
Hn ’(Qp’) Jn(Qp)

15

 

CHAPTER THREE

DEVELOPHENT OF THE ELECTRIC GREEN’S DYAD FOR THE CIRCULAR FIBER
HAVEGUIDE
3.1 Introduction

In the preceding chapter, the transform domain representations for
pertinent field quantities were introduced. In this chapter, these
field expansions, along with the boundary conditions at p = a, are
utilized to construct the electric Green’s dyad for the circular fiber
waveguide. The electric Green’s dyad 8e(?|?’) relates the components of
space domain electric field E at observation point ? to the components

of space domain current 3, residing throughout the source volume V’,via

the integral [8]

FIG?) = [teem-3(a) dV’. (3.1)
VI
Since the electric field may be expressed as the sum of impressed and

scattered terms, the Green’s dyad may also be written as the

superposition of impressed and scattered components

Elem?) (Tum?) + cesma). (3.2)

The scattered term is investigated in the following section.

3.2 Construction of Scattered Electric Green’s Dyad
The scattered term, or scattered electric Green’s dyad, is
constructed by relating the components of scattered electric field to

components of excitatory volume current. This relation is most readily

16

seen in the transform domain. Although the motivation is not obvious at
this point, this development hinges on determination of the spectral
expansion coefficients An, Bn’ Cn’ and DD in terms of components of the
transform domain current j(3’.§).

To determine the expansion coefficients, application of the
boundary conditions at the guiding/surround region interface is first

required. Continuity of total tangential fields at p = a demands

9-[3:(a.e.c) + 3:(a.e.c)l 9-[3:(a,e,c) + 3:(a,e,c)1 (3.3a)

t-[K;(a.e.C) + K:(a,e,c)1 t-IR;(a.e,c) + B:(a,e,c)1 (3.3b)

where t is the unit tangent vector, and the g and s subscripts denote
the guiding and surround region fields, respectively. For the purposes
of this exposition, the entire source current volume is assumed to
reside in either the guiding or the surround region, but not both.
Since the scattering applications in Chapter Six examine an equivalent
current distributed throughout a thin slice of the guiding region, the
following analysis assumes a source-free surround. An analogous
development may be performed for a source-free guiding region, although
a slight modification in the electric Green’s dyad will result. Under
91 91

the previous assumption, eS = h5 = 0, since no impressed fields may be

maintained in the source-free surround. Thus (3.3) reduces to

e:z(a,9,C) - e:z(a,e,C) = -e;z(a,9,§) (3.4a)
e:9(a,9,§) - e:9(a,6,C) = -e;9(a,9,C) (3.4b)
h:z(a,e,C) - h:z(a,e,§) = -h;z(a,9,C) (3.4c)
h:e(a,9,§) - h:9(a,e,§) = -h;e(a,e,C) (3.4a)

17

where the e and z subscripts denote the 9- and z-components of the
electric and magnetic fields. After substitution of (2.5) and (2.12a).

the Joining condition for ez (3.4a) becomes

Q m
jne _ (2) Jne
Z An(C) Jn(an) e 2 Cum) Hn (Qsa) e
n=-m n=-m
W m
= T° 2 J“ I Z K:B(alp’) JB(3',c) e’Jn" dS’. (3.5)
n=-w CS’ 8

Since the infinite summation over exp(Jn9) is common to all three terms
of (3.5), the orthogonality properties of exp(Jn9) [1] may be exploited

to yield a set of independent equations of the form

2)

(
An(C) Jn(an) - Cn(C) Hn (Qsa)

:20- IZ K:B(alp’) JB(3’,c) e-Jne’dS’
CS’ 8

to be solved independently for each n. From (2.5). (2.6). (2.12), and
(3.4), it is evident that the remaining Joining equations for e9, hz,
and he all include infinite summations over exp(Jn6). Hence the same
orthogonality argument may also be applied to these equations, producing
three more sets of independent equations to be solved for each n. If
only lnl s N harmonics are considered, where N is determined by
convergence criteria, what remains is a system of 2N + 1 forced 4x4

matrix equations to be solved independently for the spectral expansion

coefficients A , B , C , and D . Each has the form
n n n n

18

 

 

h d

where the elements of [E

(

 

L

An(C)
Bn(§)
Cn(C)

Dn(§)

 

d

 

5

lm] are listed in Table 3.2.1.

Furthermore, because the cross-sectional integration over

-Jn9’d

CS' B
”U _ I
-4—0 I Z K:B(alp' )JB(3'.C)8 jne dS’
7%— I Z i:3(a|p')38(3' C)e Jn" as

:— I Z EgBtaIp'uBtb” .c)e "Jn" as

 

exp(Jn6’), a linear operator, is common to all the forcing vector terms,

it may be temporarily suppressed, leading to the modified 4x4 forced

matrix equation

 

L

'The spectral expansion coefficients are related to the solutions of

(3.6) by

 

 

h

(3%)
B (3%)
~ g,
Cn(p .C)

(33¢)

q

 

_ -Jn9’ ,
vn(c) — I Vn(p’ .c) e as
CS,

where V may be A, B, C, or D.

 

L

wuo

T xff’talp' )jB(3’.C)
B

wu

79 Z K:B(alp')JB(3’.C)
B

7% Z E:B(alp')JB(3'.C)

-J ~98 . 9.

 

d

It is precisely this intermediate

(3.6)

(3.7)

reprresentation that invites decomposition of the spectral coefficients

19

into superpositions of contributions from each component of current,

JB(3’.C). For example,

>
{2
JV

II

N ~B I _)I
n X An(p ,C) JB(p .C) (3.8a)
B

u:

'6‘

J»
u

§B(p’.C) J (3'.c) (3.8b)
n B
B

where B may be p, 9, or z. The guiding region expansion coefficients
An(C) and Bn(§) may be retrieved via (3.7). The elemental spectral
amplitude coefficients Kg(p’,C), §fi(p’.C). Eg(p’.C). and fi§(p’.C) are

determined from the solutions of the 4x4 matrix equation

 

 

 

 

 

. . . . . )
Zfi(p',c) 3:2 K:B(alp’)
§fi(p'.c) 3:9 K:B(alp’)
E = . (3.9)
gm Efi(p'.c) 2%— K:B(a|p')
[ fifitp'.c) 2%— E:B(a|p')

 

The explicit expressions for the Kg and fig terms, obtained by straight-

forward manipulation of the Cramer's Rule [10] solution of (3.9), are

"'8 I
KB( I C) = an(p ’C) (3 103)
n P , 7T5)"— .
§B bfi(p’.§)
n(p .C) = 713—— (3.10b)

The denominator MC), common to both terms, is the determinant of [Elm]

given by

20

2 J’ (Q a)
_ (2) 2 nc 1 _ 1 _ 2 g H(2)
MC) - [an (Qsa)] {[71] {—2 —2] k O[ngn 08a (05 a)

n ZJ’ (Q a)

n2
1 H(2), g n g (2) ns H(2),
-— mm)” Hn (Qa)- — (Qsa)]
QS Qan an 5 0S

The numerators ag(p’.C) and bg(p’,C) are expressed as

 

 

. ‘ww
afi(p’.() = 4° {[ 0'2“); H(2)(QS a) KZB(alp ) + “(2 ’(os a) x°B(a|p )] Hg)
8
5m
H(2), 28
+ 088 Hn (QS a) Kn (alp’ ) A H(C)}

* -J— [H H(2)(Qs a) KZB(alp’ ) MH(§) + H(2 2)

~93 .
4 (Qsa) Kn (alp ) AH(C)]

6%: ,c) - ——9 [nfiz’msm K:B(alp’) MEm + Hi2)

GB .
4 (Qsa) Kn (alp ) AE(C)]

— 731— {[19 nr‘f’msa) fiff’mp') + Hff’msa) 12:3(alp' )] Hz)

 

02a
Jun
+ .522 HéZ)’(QSa) K:B(a|p’) AE(C)}
where
_ n 1 _ 1 (2)
F(C) - —a— [$2 ?] Jn(an) Hn (Qsa)
5 8
J“"“o n 1 (2) 1 (2)
MH(C) Q Q —a- [T Jn(an) HT) (053)" ‘0— Jn(an) Hn (Qsa) ]
8 S 8 S
ME(§) = cho .2. [ .2: J (Q a) H(2 )’(QS a)- n: J’(Q8 a) 8(2) (Qs a) ]
0808 a 8 n 8 08 n
_ H(2), _ _1_ (2)
AH(() quo[ -6; Jn(an)Hn (QS a) 08Jr’18(Q a) HD (Qsa) ]
n2 n2
_ s (2), _ g , (2)
AE(C) - cho[ —5; Jn(an) Hn (Qsa) _Q; Jn(an) Hn (Qsa) ].

21

A similar procedure may be used to determine Cfi(p’,§) and fifi(p’.C).
Since the guiding region expansion coefficients An(C) and Bn(C) are now
fully specified in terms of components of the transform domain current
through expressions (3.7). (3.8) and (3.10), these relations may be
substituted into (2.5) and (2.6) to yield the transform domain guiding

region scattered field components

 

S = Jne ”B I I
ep(p,6,§) Ze [EB—— g[ JCAn(P .C)Jn(Qgp)
n="m
upon I
+ BB(p’ .C)Jn (Q )] J (p’ .C) e Jne dS’ (3.11a)
03" g” B
e:(p.9.c) = Z“ eJDGI’ZB% 80” “C ABUJ’ .ncn (Q g)0)
n=-m
. jwuo§§(p’,C)J£(Qgp)] 33(3',c) e'Jne'as' (3.11b)
e:(p,9,§) = Z eJDGIZA AB(p’ .0) n(Qgp)JB(p)’ 3;)e’Jne dS’.(3.11c)

n=-m

These, too, are fully expressed in terms of the components of the
transform domain current. To obtain an equation analogous to (3.1) for
the scattered electric field, the space-domain scattered electric field
gs(?) is recovered via the inverse Fourier transformation (2.3). Next,
the transform domain current 3(3’,C) is written as the Fourier

transformation of 3(?’)
Q
J(3'.() = [362% e-ch dz’ (3.12)
-m

and substituted into (3.11). The cross-sectional integral in (3.11) and

the axial integration in (3.12) combine to yield a concise expression

22

for the guiding region scattered electric field as the volume integral

€S(?) = I Ces(?|?')-3(?') av' (3.13)
VI
where Ces(?l?’) is the guiding region scattered electric Green’s dyad.

The elements of Ces(rlr’ ) are given by

(new)?! )1”

eJn(e-G’) ,
Z—lnze I 08 [JcAfi (p .0.) n((28):)

 

n=-m
wuo n
+ Qp BB(p' CUn (Q 8p)]e eJUz 2 )d d: (3.14a)
8p
m m
es 9 a, 98 _ ejn(6- 9’ ) 1 DC "8 I
(c (rlr )) - ‘27: Xe I (2.—8 [6—8 p An(p .C)Jn(Qgp)
=-co
+ jwpo§g(p',C)Jg(Qgp)] eJC‘z'z')ag (3.14b)
m a:
[Ges(-)|->,)]zB = 2_n ize eJn(9- 6’ ) J‘ '71ng B,(p 0% (Q 8p)eJUz-z )dC
n=-ao
(3.140)
where fies (rlr’ ) is written
case)?» = Z Z .1 :3 [Ges(?l?’)]°‘8 (3.15)

and a and B may be p, 9, or z. This expression for the scattered
electric Green’s dyad is valid under the assumption of a source-free
surround. The next step in obtaining the total Green’s dyad for the
dielectric fiber waveguide is determination of the contribution from the
impressed fields. The guiding region impressed electric Green’s dyad is

evaluated in Section 3.3.

23

3.3 Construction of Impressed Electric Green’s Dyad

In this development, the guiding region impressed electric Green’s
dyad is derived. Fortunately, the impressed electric field in Section
2.4 was presented in an extremely useful form for these purposes. The
impressed electric Green’s dyad, which may be written in a form
analogous to that for the scattered electric Green’s dyad (3.15), is
superposed with the scattered dyad to yield the total guiding region
electric Green’s dyad.

Because the transform domain impressed electric field was written
in such a compact form in Section 2.4, determining the impressed
electric Green’s dyad reduces to merely substituting (3.12) and (2.12)
into (2.3). This results in an expression for the guiding region

impressed electric field in terms of the volume integral

EN?) = Jae’('r’|‘r")-3(?') dV’. (3.16)
VI
In analogy with (3.15), the guiding region impressed electric Green’s

dyad Ge1(?l?’) is written
fieifi’l‘r”) = Z Z a B (Ge’(?|?'))°"3 (3.17)
a B
where
“‘0” on no
[861(?|?')1°‘B = Tan—0 2: eJ“(9'° ) IKSBUJIP’) 835(2-2 )dc. (3.18)
n=-oo -oo
It is important to recall that the K23 terms, listed in Table 2.4.1,
have different forms for p > p’ and p < p’.
The total guiding region electric Green’s dyad may be written as

the sum of impressed and scattered components given in (3.2).

Alternatively, in the compact notation presented in Sections 3.2 and

24

3.3, (3°(‘r’l‘r”) is written as
(PEN?) -- ZZ 313 (8"(?I‘r"))"“3 (3.19)
a B
where
16mm)“ = (13“31(‘r’fir"))"“B + ((3‘3'5('r’|1~"))"“E3

and the [Geilafl and [ceslaB terms are given in (3.14) and (3.18)

In this chapter, the guiding region electric Green’s dyad was
specified under the assumption of a source-free surround. Through the
integral (3.1), this Green’s dyad relates the guiding region electric
field a to the excitatory current 3, which resides wholly within the
guiding region. This Green’s dyad may be readily modified to yield the
surround region electric Green’s dyad. Furthermore, if a source-free
guiding region is considered, a similar procedure may be followed to
produce the guiding and surround region Green’s dyads.

From (3.14) and (3.18), it is evident that the elements of the
Green’s dyad all contain the Fourier inversion integral. This inversion
integral requires special attention. In its present form, the Fourier
inversion reveals little insight into the physical components that
comprise the electric field; however, if the inversion integral is
properly evaluated, the complete propagation-mode spectrum may be
extracted. Appropriate evaluation of the Fourier inversion integral is

the focus of Chapter Four.

25

Table 3.2.1. Elements of [Ebml’

 

E11 = Jn(an) E12 = 0
_ _ (2) _
E13 - Hn (Qsa) E14 - 0
Jun
_. -n_C _ o .
ann g
_ nc H(2) _ - H(2),
E23 - 02a Hn (QS a) E24 - __5;—Hn (Qs a)
5
E31 = 0 E32 = Jn(an)
_ _ _H(2)
E33 - 0 E34 - (QS a)
-ch
_ 8 . _ DC
E41 - Q Jn (Qg a) £42 - 2 JD (Qg a)
8 an(
J“"33 (2) n____c_H
E43 = Q H (Qsa) E44 = 2 “(Q a)
s Qsan

26

 

CHAPTER FOUR

SPECTRAL ANALYSIS FOR EVALUATION OF THE INVERSE AXIAL FOURIER
TRANSFORMATION

4.1 Introduction

In the previous chapters, the important field quantities--namely,
the electric and magnetic fields, the excitatory current, and the
impressed electric Hertz potential--were represented in the axial
Fourier transform domain. These transform domain representations
facilitated construction of the electric Green’s dyad for the dielectric
fiber waveguide. However, because the Green’s dyad §(?|?') is a
three-dimensional quantity that relates §(?) to j(?’), the inverse axial
Fourier transform must be embedded in the development of C(?l?’). This
inverse transformation appears as the real-line Fourier inversion
integral common to all the terms of the electric Green’s dyad. This
real-line integration, although compact in form, offers little intuition
about the nature of the electric and magnetic fields. The aim of
Chapter Four is to evaluate the Fourier inversion integral in a manner
which identifies the physical components of the electric and magnetic

fields.

4.2 Deformation of the Real-line Inversion Integral

The complete spectrum of the waveguide electric or magnetic field
is comprised of both discrete surface wave and continuous radiation mode
components. Both the surface wave and radiation field contributions may

be identified by deforming the Fourier inversion integral off the real

27

C-axis and subsequently performing a complex C-plane analysis. The
inversion path deformation is outlined here.

If the integration path is deformed into the complex C-plane, then
the Cauchy-Goursat Theorem [11] may be invoked. This powerful theorem
states that the integral of a complex function f(C) around any closed
contour in the complex C-plane within and on which f(C) is analytic is

identically zero. More succinctly,

§ f(C) dC = o (4.1)
C

where C is any path in the complex C-plane within and on which f(C) is
analytic. Thus the contour C must not enclose any singularities of the
inversion integrand in (4.1). If the integrand is assumed to possess
both pole and branch point singularities, then C must exclude all poles,
branch points, and branch cuts emanating from the respective branch
points. An appropriate contour is pictured in Figure 4.2.1. Note that
the contour C excludes the pole at C = Cp, the branch point at C = Cb,
and the branch cut emanating from Cb.

This analysis is readily adapted when f(C) is taken to be the

Fourier inversion integrand in (2.3). From the Cauchy-Goursat Theorem

§ 3(3.<) eJCZ dC = o. (4.2)
C

If the contour C includes the real C-axis, then (4.2) may be employed to
express the real-line Fourier inversion integral in terms of integration
along some path (or paths) in the complex C-plane. For the entire real
C-axis to be included, the integration path must be closed with an arc of
infinite radius. That infinite semicircle must detour around the pole

and branch point singularities, as well as the branch cuts. A general

28

Fourier inversion contour is depicted in Figure 4.2.2. Because the

+ + +
inversion contour C’ is the sum of the paths Cr’ c;, C , and C', (4.2)

1
P
may be restated as

I 3(3.C) ngz dC = - I 3(3.C) equ dC — I e(3.§) ech dC
+ +
c C" C’
r a p
- I 3(3.<) eJCZ dC. (4.3)
t
Cb

The path for infinite closure may be determined by recalling the
electric Green’s dyad terms presented in Chapter Three. All the
scattered (3.14) and impressed (3.18) Green's dyad elements contain
exleC(z-z’)] as part of the inversion integrand. Thus, the integrals

present in the total electric Green’s dyad are all of the form

a)
I(p,z|p’,z’) = I Wc(p|p’) eJC(Z-z’) dC
~00

where Wc(plp’) has both pole and branch point singularities. To

determine how the integrand H (plp’)exp[JC(z-z’)] behaves along the

C

contours Czand c;, C is written as the sum of its real and imaginary

4.
parts and examined on the contours c;

w (plp’) ejC(z-z ) dC = I W (plp’) exp[J(C +JC )(z-z’)] dC
C C r i
t t
C C
w co
= I w (plp’) expl-C1(z-z’)] exleC (z-z’)] dC.
+§ r
C-
0
If the path c; is chosen when 2 < z’, the integrand is unbounded as C1
tends to m. Likewise, the integrand in unbounded as C1 tends to -m if
c; is selected when 2 > 2’. Thus, infinite closure is made in the upper

half-plane when 2 > z’, and in the lower half-plane when 2 < z’. This

29

Ci

\(::\\~\‘$*~—---~_1 {b Cr

Figure 4.2.1. Integration contour C when integrand has both
pole (Cp) and branch point (Cb) singularities.
The shaded region denotes the branch cut

 

 

emanating from Ch.

30

 

Figure 4.2.2.

 

Cr

 

Fourier inversion contour when integrand has
both pole and branch point singularities.
The infinite closure path is denoted by Ci,
while the pole and branch cut detours are

+ +
denoted by CB and C' respectively.

b,

31

convention requires the integrand to decay to zero along the paths Ci,

so that

I w (plp') e3“?Z ) dC = o.
+ C

C‘

on

+

Thus, integration along c; does not contribute to the evaluation of the
Fourier inversion integral if the inversion contour is closed by 0; when
2 > z’ and by c; when 2 < 2’. With this convention, the expression in

(4.3) is given by

c ct

I 3(3.C) ech dC = - 3(3.C) ech dC - I e(p.C) dch dC (4.4)
3'.
Cr b p

i
b

point singularities, and the choice of branch cuts associated with the

+

The paths CE, and C depend on the locations of the pole and branch
branch points. Once these are determined, the Fourier inversion may be
evaluated. These are the topics investigated in the remainder of

Chapter Four.

4.3 Singularities in the Complex C-Plane

In this section, it is shown that the Fourier inversion integrand
has both pole and branch point singularities. The locations of these
singularities in the complex C-plane are determined to permit
1 in (4.4).

b
Specification of the inversion path allows identification of the surface

4.
specification of the Fourier inversion contours CB and C

wave and radiation field components arising from the pole and branch

point singularities, respectively.

32

4.3.1 Pole Singularities

The transcendental eigenvalue equation for the discrete surface
wave poles, denoted Cp, is developed in this section, as well as a
method for evaluating the spectral expansion coefficients when C = Cp.
From Chapters Two and Three, it is evident that the fields impressed by
a current source do not possess pole singularities (the apparent pole at
Q = O is really associated with the branch point singularities
investigated in the following section). Therefore, pole singularities
should appear only in the scattered field terms. This is Justified by
examining the expansions for the scattered fields.

In Chapter Three, the guiding region scattered fields are expanded
B

n

in terms of X (p’.C) and §g(p’,C), which are related to the spectral
amplitude coefficients An(C) and Bn(C) through (3.7) and (3.8). From
(3.10), it is evident that the expansion coefficients share the common
denominator A(C). Thus, the surface wave poles occur when the

denominator common to all the expansion coefficients becomes zero.

Simply put,

A(C = (p) = 0 (4.5)

is the condition which implies complex surface wave poles at C = C

 

 

 

 

p'
This leads to the relation
p 2_ p 2
[DCP (083) (083) ] =
k p 2 p 2
0 (08a) (Qsa)
. p (2). p 2. p 2(2). p
[ Jn(an) _ Hn (Qsa) ][ nan(an) - nan (Qsa)
p p p (2) p p p p (2) p °
(an)Jn(an) (Qsa)Hn (Qsa) (an)Jn(an) (Qsa)Hn (Qsa)
(4.6)

33

P = P = = - J’ 2 - 2
where 08 08(Cp) and Q8 Qs(Cp) J Cp kS . Upon substitution of
(C.12) and (C.13) (Appendix C), the eigenvalue equation (4.6) for the
discrete surface wave modes of the step-index circular fiber reduces to

the well-known form [1]

 

p 2 p 2
[ 352 Egga’ * (“s“) ]
ko (oga)z(q:a)2

 

 

 

I p I p 2 I p I
[ Jn(an) + Kn(qsa) ][ nan(an) + nixn(q:a) ]
(02a)Jn(an) (qga)xn(q:a) (02a)Jn(o:a) (qga)xn(q§a)
(4.7)
where q: = jQ:= i C: - k: and Kn(x) is the modified Bessel function of

the second kind (Appendix C). The locations of the surface wave poles
in the complex C-plane may thus be determined from the solutions of
(4.7).

When the surface wave poles are specified--the number of poles will
depend, in general, on the electrical dimensions of the waveguide--the
contribution to the Green’s dyad may be evaluated as a sum of integrals
along the contours Ci. Since the impressed fields do not possess pole
singularities, the impressed electric Green’s dyad is not implicated.
From (3.14), the elements of the scattered electric Green’s dyad may be

rewritten as

m m
es 9 a, a8 _ 1 Jn(9-6’) a "B :
[c (rlr )1 - 5&- Z e [[smcp.<) Anna .0

Il='m —m

+ s;n(p,c) fifitp’.c) ] eJC‘z’z ’ dC (4.8)

where, for example

34

21mg” giJutop)

Qg"

qu
9 _ o
an(p.C) — Qg

     

n(Qgp).

From (4.5), Ag and fig, and thus the spectral integrands, have pole
singularities when A(C = Cp) = 0. Since the integrand is analytic along
the straight segments of each pole exclusion contour Ci, these segments
are allowed to coalesce so that integration in opposite directions
produces cancellations. Therefore, only the small loop surrounding each
Cp contributes to integral along its corresponding Ci. This allows A(C)

near each Cp to be expanded in a first-order Taylor series as

A(C) = A(C= Cp ) + (C - Cp ) (4.9)

32 c=cp
to facilitate evaluation of the integral along each Ci. The first term
in (4.9) is identically zero from the condition (4.5) for the existence
of surface wave poles. The resulting expression for A(C), along with
(4.4), are substituted into (4.8) to yield the surface wave pole
contribution to the scattered Green’s dyad elements as the sum

es (6 |?, a8_ ejn(e- 9’ ) a "B I
[ad )1 =Zp’12n 1:6 I: [smtmm an(P .C)

:JCp (z-z’ )d
a ~B , dC
+SBn(p.C) bn(p .C) ]e 6A

(C - cp) 6C

(4.10)

C= Cp

The subscript d denotes that this portion of the scattered Green’s dyad
elements is the discrete sum of contributions from pole singularities.
Pole singularities do not contribute to the impressed Green’s dyad.

The task of analytically evaluating the integral along each contour

13+

C' remains. Consider first determination of the integral in (4.10) at

35

axial locations 2 > 2’. When 2 > z’, infinite closure is made in the
upper half-plane, corresponding to the pole-excluding contour 0;. An
expanded view of this contour is depicted in Figure 4.3.1. Because the
pole singularity has been written explicitly in (4.10), and the rest of
the integrand in (4.10) is analytic along c; for all a and B. the
surface wave pole contribution to the scattered Green’s dyad from each

Cp may be expressed as

on
es 9 a, «B _ -1 Jn(6-O’) _ , a ~B ,

[de(r|r )] - E? E: e exleCp(z 2 )l[ SAn(P.Cp) an(p .Cp)

n=-m
a "B I 1 dC
+ an(p’€O) bn(P .Co) ] —§§T— I+TE—:-E;T (4.11)
a: _ c
C-Cp P

When the integrand in (4.11) is expanded in local polar coordinates about

the point C = Cp, as in Figure 4.3.1, and c is allowed to shrink to 0,

it is easily verified that

dC = _
LI: -' cpi 2'”
C
P
resulting in

es 9 a, a8 _ Jn(e-e’) _ , a ~B ,
[de(r|r )] - X e eprJCp(z 2 )l[ Sm(p.Cp) an(p .Cp)

a "B , J
+ SBn(vap) bn(P .Cp) ] —§§T— . (4.12)
3: =
C Cp

A similar analysis may be employed for integration along each c; when
2 < 2’. Because the integrands in (4.12) posses either even or odd
symmetry about 2 = 2’, contributions from every c; and C- are indicated

in the concise expression

36

 

 

 

Figure 4.3.1. Expanded view of the upper half-plane pole-
excluding contour C;, showing local polar

coordinate representation of pole singularity.

37

m

[c§s(?l?'))°‘3 = Z Z ejn‘e'e" 12:3(plp') eprCplz-z’l) (4.13)

p n=-m
where the st terms are listed in Table 4.3.1. Note that the terms
which are odd about 2 = z’ are multiplied by sgn(z-z’) to preserve this
property.

In this section, the pole singularities were identified.
Understanding these singularities allowed evaluation of the Fourier
inversion integral along the paths Ci. The result was an explicit
specification of the surface wave contribution to the scattered electric
Green’s dyad. The radiation field components of the total Green’s

dyad-~both scattered and impressed elements--are determined by

+
evaluating the Fourier inversion integral along the contours CS.
4.
Specification of the paths C’ and identification of the radiation field

b
fie a a,
contribution to (rlr ) are the topics of the following section.

4.3.2 Branch Point Singularities
In this development, the branch points of the inversion integrand
are identified. The corresponding branch cuts, which define the

+
contours C’, are specified as well. The behavior of the branch points

b
and branch cuts is investigated as the physical media exhibit near-zero
conduction loss. The radiation field component of the electric Green's
dyad is then expressed in terms of the Fourier inversion integral
evaluated along the low-loss branch out detours. Following a
qualitative complex C-plane analysis, the Fourier inversion along c: is
transformed to integration over the parameter Qs’ which is purely real

along the low-loss branch cuts.

Mathematically, the branch point singularities arise from the

38

transform domain wavenumber parameter Q implicated in both the impressed

and scattered field representations. In Section 2.3, Q was defined by

Q = 2V k2 - C2 = ink + Cfi'V/k - C‘.

 

This indicates the existence of square-root branch point singularities
at C = 2k in the complex C-plane. For the purposes of this discussion,
subscripts have been dropped to preserve generality. More specifically,
however, Q may be Qg or QS, leading to branch points at R8 and ks,
respectively. In physical media with non-zero dissipation, the
wavenumber k may be written in the complex form k = kr + Jki’ where kr >
0 and k1 < O for waves which decay along their direction of propagation.
Thus k may be represented in the complex C-plane, although in close
proximity to the real C-axis when the dissipation is small.

Ensuring the convergence of the inversion integral at locations far
from the origin requires a Judicious choice for the branch cuts
emanating from tk. Because the radial dependence of all fields is
expanded in terms of Bessel or Hankel functions or both, the
large-argument expansions of the Bessel and Hankel functions must be
considered. From Appendix C, the large-argument expansion of the

second-kind Hankel function is given by

H

1/2
] expl-J(Qp - nn/Z — n/4)].

_ 2
n (Qp) - [1703
This represents an outgoing wave for Re(Q) > 0, and an incoming wave for
Re(Q) < O. Physically, this wave must decay along the direction of
propagation. Adherence to this physical constraint demands that Im(Q) <
0. Examination of the large argument-form of the Bessel function

(Appendix C) also yields the condition that Im(Q) < 0. This condition,

39

imposed to ensure decaying waves, and thus, a convergent inversion
integral, implies that the branch cuts have the conventional hyperbolic

form [1]

k k
r
C

I‘

 

c1 1.
This describes the branch cut emanating from C = k subject to the
restrictions lCrl < lkrl and 'Cil > 'kil' Furthermore, the condition
Im(Q) < 0 avoids the branch cuts of the Bessel and Hankel functions, so
that only the square-root branch point singularities are implicated.

The complex C-plane, including both pole and branch point singularities,
is shown in Figure 4.3.2.

In analogy with the developments of the previous section, the
radiation field contribution to the total electric Green’s dyad may be
determined by evaluating the Fourier inversion integral along the branch
out detours Ci. Since the branch point singularities, unlike the pole
singularities, are present in both the impressed and scattered field
expansions, integration along the branch out detours c: will contribute
to the total electric Green’s dyad. From (3.14), (4.4) and (4.8), the

radiation field component of the total electric Green’s dyad is given by

e a 9, «B _ -1 Jn(e-6’)
(chm:- )1 ”52:9 H
n=-oo C;

u

-w
o
4

 

«B .
Kn (plp )

n

+ s:n(p.C) x3(p',c) + sgn(p,c) §§(p'.c)] eJc‘z‘z ’ dC.

(4.14)

The spectral integration path in (4.14) may be simplified greatly by
considering the behavior of the branch cuts when the physical media

exhibit near-zero conduction loss.

40

As the dissipation in both the guiding and surround regions
approaches zero, Im(k) = R1 tends to zero as well. Thus, the branch
points move limitingly near the real C-axis. It may also be shown that
the two branch cuts move infinitesimally close together and coalesce at
the real C-axis for Q < k and the imaginary axis for Q > k. The
behavior of the branch points and branch cuts in the low-loss limit is
pictured in Figure 4.3.3(a). Note that the parameter Q = iv k2 - C2
becomes purely real along the low-loss branch cuts, with the sign
convention shown in Figure 4.3.3(a). Also, the upper half-plane

inversion contour C+ is decomposed into the four contours pictured in

b
Figure 4.3.3(b). Several important observations about the inversion
integrand may now be stated by examining the branch cut detours in the
low-loss limit.

From the previous development, the low-loss branch cuts emanating
from k8 and kS clearly coalesce so that Q8 and QS are purely real.
However, it is essential to observe that these branch cuts remain
analytically distinct. Therefore, for C1 9 C2 9 C3 in the low-loss
limit (see Figure 4.3.3b), Qg(C1) = 'Qg(C2), while QS(C1) = QS(C2);
conversely, 08(C2) = Qg(C3), and QS(C2) = -QS(C3). With this in mind,
the inversion integrand at axial locations 2 > 2’ may be examined along
the path c; = F; + F; + F; + F; shown in Figure 4.3.3b.

Consider an excitatory current residing wholly within the guiding
region. The transform domain guiding region total field--the sum of
impressed and scattered components--may be shown to be identical for
equivalent C on opposing sides of the Q8 branch cut. This is done by
comparing the transform domain guiding region total field, or

alternatively, the integrand in (4.13), evaluated at (08,05) and at

41

 

 

 

Figure 4.3.2. The complex C-plane. Shaded areas represent the
branch cuts emanating from the branch points at
1kg and iks. The surface wave pole detours C1

are not pictured.

42

(+): Re(Q) > 0
Ci (-):Re(Q) < 0

 

Q

o—-&

 

 

df‘o

|
l
I
/

 

 

 

C:

(a) (b)

Figure 4.3.3. More detailed pictures of the complex C-plane,
showing (a) the sign convention for either Q on
both sides of its corresponding low-loss branch
cut, and (b) the upper half-plane integration

contour components as the branch cuts coalesce

in the low-loss limit.

43

(-Q8,Qs). The surround region total field--in this case, scattered
component only--may likewise be shown to possess even symmetry about the
Q8 branch cut. Because the integration paths F; and F; are in opposite
directions and the integrand is identical along these paths, integration
along the entire Qg branch out yields no net contribution to the
inversion integral. The value of the inversion integral is determined
solely by integration around the QS branch cut, or the paths F; and F;.
Furthermore, since all fields are even about the Q8 branch cut,
integration along F; and th (F; truncated so that C2 2 k2) is
equivalent in the low-loss limit; the identical value of the radiation
field contribution to E(?) is obtained by integrating along the contour
F = F; + th.

An identical argument may be made for a source-free guiding
region, so that, in either case, the radiation field contribution is
determined by evaluating the Fourier inversion integral along the
contour F. In the low-loss limit, Q8 and Q8 are purely real along F;
Qg’Qs < 0 along th, while Qg’Qs > 0 along Pg. In general, the total
fields in both regions will differ at equivalent points on F; and th.
However, realizing the analytic continuation relation for the Hankel
function (Appendix C) , the impressed and scattered fields are seen to
have simple conjugate relations across the two branch cuts. An easy way
of determining these relations in outlined in Appendix B. Contributions

from F; and F; may be combined into an integral along F; as

t

44

m
e a a, a8 3 -1 Jn(9—e’) -w“o a8 ,
[Gb(r|r )1 E; X e L{ [—4 Kn (plp)
=—m [‘5

+ s§n(p.c) Kficp'.c1 + same) fifitp'.c1]|0 >0

8.8
-wu ~
- [—33 Kgswlp’) + sfinmx) Afi(p’.C)
+ sgn(p,c) §fi(p’,C)]| } e3§(z'z') dC. (4.16)
Q8 <0

An analogous development may be derived for integration along c; when

2 < z’. The integrands in (4.16), as was the case for the surface wave

pole components, possess either even or odd symmetry about 2 = 2’ so

+ -

 

that contributions from both Cb and Ch are implied in the expression
G
e a a, a8 _ -1 jn(9-9’) a8 ,
[Gb(r|r )1 - Z X e I [Gn+(plp 1
+
=-m r
s
- c:§(plp')] eJC'Z’Z ' dC (4.17)
where
G“B(p1p') = walp’)
n+ n
Q >0
8.5
Ga8(plp’) = G“B(plp’)
n- n
Q <0
8.5

and the G28 terms are listed in Table 4.3.2. Note that, in analogy with
the R23 terms, odd symmetric terms are multiplied by sgn(z-z’) to
preserve this symmetry. The conjugate relationships between the 6:5 and
6:? terms are tabulated at the end of Appendix B. Finally, after the
integration variable is changed from C to Q5, the expression for the
radiation field contribution to the total electric Green’s dyad is given

by

45

m m
(c§(?|?')1°"3 = g; X eJ“‘°'°" I [cfi’fwm

11:-” O
Q
_ a8 , JClz-z’l s

where

(AN

-V k - Q ...0 5 Q8 5 k8
C(Qs) = .
JV Q - k ...Q8 2 kS

(0N
VIN IBM

The elements of the total guiding region electric Green’s dyad are now

specified as superpositions of the surface wave and radiation field

components
(ce(?1?'11°‘3 = (c§s(?1?')1°‘8+ [c:(?|i-")1°‘B.

These elements are described explicitly by combining the expressions

(4.13) and (4.18) to yield

[Ge(?l?’)]°‘8 = Z ejn‘e’e" {Z 11:5(plp') eXpUCplz-z’ I)
P

n=-m

°° o
_ 1 “B I _ “B I JCIZ-Z’ I S
55 I [ Gn+(plp ) Gn_(plp ? ] e C(Q;) dQs }°
0

(4.19)

4.4 Summary

In this chapter, the physical components of the electric Green’s
dyad were extracted by deforming the Fourier inversion contour into the
complex C-plane. Evaluation of the inversion integral along the
contours c: and Ci entailed identification of the surface wave and

radiation field contributions, respectively. These contributions are

46

evident in the expression (4.19) for the electric Green’s dyad.

Through integral equation techniques, the electric Green’s dyad in
(4.19) may be employed to quantify electromagnetic scattering in
dielectric fiber waveguides. Chapter Six is devoted entirely to the
scattering of TE01 modes incident on dielectric discontinuities in fiber
waveguides; Chapter Five serves as a transition by specializing the

Green’s dyad in (4.19) for both the TE and TM axially-symmetric modes.

47

Table 4.3.1.

PP I
an (plp )
PG I
Rn (plp )
P2 I
Rn (plp )
GP I
Rn (plp )
96 ,
Rn (plp )

Oz ,
Rn (plp )

2P I
Rn (plp )
26 ,
Rn (plp )

22 ,
Rn (plp )

“B I P _ g
List of BD (plp ) terms, where Q8 - Qg(C Cp).

 

 

       

 

       

             

 

 

J 1 I p ”P I W n "P I
an p [ jCJn(Qgp)an(p .cp) + p 8p)bn(p .cp1]
-— Q 08p
33 c=cp 3
J p1 I p "9 I Won
29] up [ JCJn(Qgp)an(p .cp) + 03 p ]
ac _ g g”
c-cp
sgn(z-z’) 6g 1p[ JCJ£(Q:p);:(p’.Cp)
63 c=cp 3
an (0p )6: (p c p)]
Q:P 3p

 

a: 1 [ g2: Jn<0291a§(P WC ’ + 3““ J"°ZP’3§‘P C ,1]

 

_. Op 13

33 c=cp 3

J p1 'DC p "9 , ’ p ~e I

.63 Qp [ap— Jn(Qgp)an(P .Cp) + quan(Qgp)bn(p ,cp)]
6C g_ 4p g

 

6A Qp
6: §_ Cp 08 33

sgn(z-z’ ) J? 1W[ nC J n(Qgp)a: (p’ .Cp )

I jwqu£(QZp)bn(p .cp1]

sgn(z-z’) a2 Jn(o§p)§§(p'.cp)
5? =
c cp
sgn(z-z’) a: Jn(Q:p);:(p’ICp)
53 =
c cp
J p ~2 I
29 Jn(Qgp)an(p .cp)
3‘ <=c

48

Table 4. 3. 2.

PP I
Gn (plp )

P9 I
Gn (plp )

sz

(plp’ )

9P I
Gn (plp )

69 ,
Gn (plp )

ez ,
Gn (plp )

29 I
an (plp )

ze ,
Gn (plp )

26 ,
Gn (plp )

       

 

 

List of G:B(plp’) terms.

__12 K:p(alp') + -é; [ JCJgtogp1Kfi(p’.C>

I Q;3H(omfimzo]
'”“o 1 ~e
n (alp’) + ——— [ 1:);(ogp1An(p'.c)

wuo n
03”“

W N
sgn(z-z’) { "‘42 K:z(alp’) + —%— [ JCJ£(Qgp)A:(P'IC)
8

J n(08pm: (p’ .C)]

 

wuo n ~z
Q g3 Jn(Qgp)Bn(p .c)]}

-wu p
0 9p , 1 'nc "P I

.
+ quoJ3(ogp)B§(p c1J

O 99 , 1 -nC "9 I

+ qung(Qgp)B: (p’ C)

 

 

I -wuo 92 I 1-n—c ~z I
sgn(z-z ) K (alp ) + ——— J (Q )A (p ,C)
4 n Qg Q gp n gp n

+ quoJ£(Qgp)§:(p’,C)]}

-wp ~

sgn(z-z’) [ __42 K:p(alp’) + Jn(Qgp)A:(p’IC)]
-wp ~

sgn(z-z’) [ -—32 x:°(a1p') + Jn(Qgp)A:(P’IC)]

W0 KZZ

4 n(alp’ ) + J n(08pm: (p’ .C)

 

49

 

CHAPTER FIVE

APPLICATION TO AXIALLY-SYHHETRIC MODES

5.1 Introduction

The previous three chapters share a common goal. Ultimately, each
of the developments in Chapters Two, Three, and Four has somehow
facilitated construction of the accessible form of the guiding region
electric Green’s dyad (4.19) for the dielectric fiber waveguide. This
chapter serves as a bridge between the strictly analytical developments
leading to (4.19), and the numerical scattering applications
investigated in the following chapter. Since the scattering

applications examine the axially-symmetric TE case, this transition is

01
made by specializing the electric Green’s dyad when all fields are
axially-symmetric.

In general, the form of the Green’s dyad allows the electric field
E(?) to be spatially oriented in any manner, depending on the
orientation of the source volume current 3(?’). Because no field
orientation is preferred for an arbitrary volume current, the electric
and magnetic fields will both, in general, contain an axial component.
When neither Q-E(?) nor Q'fi(?) is zero, these modes are termed hybrid,
or HE and EH modes. However, when n = 0, it is possible to have either
Q-E(?) = 0 or 9-3(?) = 0. because the Green’s dyad elements, and
therefore the electric (and magnetic) field components, no longer vary

with 9. This allows separability of the boundary conditions (3.4) at

the p = a interface. The decoupling effects of this axial symmetry

50

may be seen by investigating the forced 4x4 matrix equation (3.9). When

n 8 O, (3.9) may be decomposed into the two forced 2x2 equations

 

 

 

 

 

 

 

P Q Q r Q
m -qu ( w
o . o (2). ~ . o 98 .
-Q—— JO(an) T HO (Qsa) B§(P ,C) T KO (alp )
8 = (5. 13)
Jomga) -H(()2)(Qsa) fifiwum % fi:B(aIp')
. . L . . .
r I I Q r w“ T
Jo(an) 41:2) (05a) A§(p’ ,c) 112 K:B(alp’)
= (5.1b)
-Jwe ch , - ~
Q 8 Jam a) 73—5 HéZ)’ (Qsa) cfimao % KgBmlp’)
L S 8 S u n d b u

 

 

 

 

 

 

Since the guiding region electric Green’s dyad was developed
previously, attention is primarily focused on the guiding region
coefficients Ag and fig, which may be determined by solving (5.1). It
may be seen from the ensuing development that the solutions of (5.1a)
represent transverse electric (TE or H) modes, while the solutions of
(5.1b) are transverse magnetic (TM or E) modes. The solutions of (5.1)
are employed to construct the scattered electric Green’s dyads for both
TE and TM modes. Of course, the TB and TM Green’s dyads may be
extracted from the axially-symmetric specialization of the guiding
region Green’s dyad (4.19). However, the exposition presented in this
chapter affords more physical insight into the nature of the axially
symmetric modes, and is ultimately consistent with (4.19) when n = 0.
The remainder of Chapter Five is devoted to constructing the TB and TM

axially-symmetric guiding region electric Green’s dyads.

Sl

5.2 Electric Green’s Function for TB Modes

When n = Q, the fields have no angular dependence, so TE and TM
modes exist. In this section, the guiding region electric Green’s dyad
for TB modes. used in the scattering study in the proceeding chapter. is
developed. It is shown that the TB electric Green’s dyad contains only

the [Ge]96 term. The impressed field contribution is constructed by

specializing [Geilee

when n = 0, while the scattered field portion is
classified in terms of solutions of (5.1a). Alternatively, the
equivalent result for the scattered field contribution may be reproduced
by specializing [Gesl99 when n = O.

A qualitative physical argument indicates why, for TB modes, (5.1b)
is not implicated, and only [Ge]99 is nonzero. Since E , E2, and H9 are
not supported by TE modes, these field components certainly cannot be
present in the impressed fields. This prompts inspection of the
axially-symmetric guiding region impressed electric Green’s dyad.

This may be constructed by specializing (3.17) when n = 0, resulting in
861(p,zlp’,z’) = ZZaBlGe1(p,zlp’,z’)]aB (5.2)
a B

where

[Ge‘(p.zlp'.z')1°‘8 =

 

w
my I
o «B , JC(z-z )
8" I KO (plp ) e dC.
'0
Since the impressed electric field is recovered via (3.16), the condition
that guarantees Ep = E2 = O is 3-3 = 9-3 = 0. Since K29 = K:9= 0, when
only 9-3 is nonzero the guiding region impressed electric Green’s dyad
reduces to the scalar Green's function
Gei(P.ZIP’IZ’) = 8—1:) I ng(plpl) eJC(Z-Z ) dC

52

which relates the lone component of electric field, E9, to the component
of volume current J9 which excites TE modes.

Moreover, when 3°? = 903 = O, the solution of (5.1b) is Ag = CE =
0. Because A0 and Co are the expansion coefficients for £2, £2 = O and
the scattered fields are TE as well. Only the solution of (5.1a) is
therefore implicated. Since 303 = 9-3 = 0, the only nonzero element of

B5 is the B = a term, yielding

~6 , _ -1”oK96 (2 )
Bo(P .c) - —o—H [:4- (plp ) no (Qsa)
. _4 iZo (plp ) ——0: no (0 5a)] (5.3)
where
_ 1 (2), 1 (2 )
AH - Jwflo[ —6-S- JO(an) H0 (053)“ T J08(Q 3) H0 (083)].

An equivalent result may be obtained by specializing (3.10) when n = 0.
From (5.4), it is evident that B: has a pole singularity when AH = 0.

The condition

AH(C = co) = 0

along with (5.3) and the relations (C.12) and (C.13) (Appendix C) lead
to the relation
I P _ I P
Jo(an) Ko(qsa)

p —p = p p (5.4)
(an)Jo(an) (qsa)Ko(qsa)

 

 

where Q: and q: are defined in Section 4.3.1. Equation (5.4), which may
be obtained from (4.7) when n = 0, is precisely the transcendental
eigenvalue equation for the TB surface wave poles of the step-index

circular fiber waveguide [11.

53

The scattered electric Green’s dyad for TB modes may, in analogy
with the developments of Section 3.2, be constructed by substituting
(5.3) into the expressions for the guiding region scattered field
components (3.11). With the specializations n = Q and B = a, this

results in

e:(p.C) = o (5.5a)
S .. ”9 I I I I
ee(p.C) - Juno I Bo(P .C) Jo(Qgp) J9(p .C) as (5.5b)
cs’
e:(p.C) = o. (5.50)
Since e: = e: = Q, as expected for TB modes, only the [Gesl99 term of

the scattered electric Green’s dyad will be present. Substitution of
(3.12) and (5.5b) into the inverse Fourier transformation relation (2.3)
provides an expression for the guiding region scattered electric field

as the volume integral

§S(p.z) = 9 I Ges(p.2|p’.2’) J9(p’.2’) dV’
VI
where Ges(p,z|p’,z’) = [Ges(p.zlp’,z’)]ee is the guiding region

scattered electric Green's function for TB modes, given by

 

qu " ~ _ I
Ges(p.2|p’.2’) = Zno I Bg(p’.C) Jo(Qgp) '3‘,“2 z ) dC-
-ou

The guiding region scattered electric Green’s dyad element may be
combined with the expression for the guiding region impressed Green's

dyad term to yield the total scalar Green’s function for TB modes

on
-wu
Ge(p.2|p’.2’) = 5% [ -—33 ng(plp’)
+ quofigtp'.c) Jgto p)1 eJC‘z‘z" dc. (5.6)

54

When the explicit expressions for K:9(plp’) and §g(p’,C) are inserted,

(5.6) becomes

a)
up
Ge(p,zlp’,z’) = —§% I [ -%— Jo(Qgp) Jo(Qgp’)°
A
-oo TE

1 (2) (2), _ 1 (2), (2)
{—5; no (Qsa) no (08a) ‘6; n (05a) no (an)]
J'(o p') n‘2"(o )
+ { ?2)g ° 8p } ] eJC(z-ZI) dc (5.7)
no ’(Qgp’) Jo(QgP)

where the upper expression in braces is valid for p > p’, the lower for

p < p’, and

0 _ “J
ATE(C) — an; AH(C).

The scalar Green’s function (5.7) may be given a more useful
interpretation by deforming the inversion integral into the complex
C-plane.

In analogy with the developments in Chapter Four, the surface wave

and radiation field components of the scalar Green’s function may be

t
b

By expanding AH in Taylor series about each Cp, the discrete surface

+
identified by integrating along the paths co and C in the C-plane.

wave pole contribution to the scattered TE guiding region electric

Green’s dyad is given by

wu

 

es I I = O I. I I 1 (2) (2)!
Gd (p,zlp ,z ) Z —8_1? t Jo(Qgp) Jo(Qgp ) [73; Ho (Qsa) Ho (an)
p Cp JU )
z-z’
- —%— no2)'(osa) noz’tooa)] e dC (5.8)
s

a o
(C ' (p) EE[ATE] -
C-Cp

Since the TE surface wave pole singularities are only present in the

55

scattered fields, the impressed field component of Ge(p,zlp’,z’) is not
implicated. Evaluation of the integral in (5.8) along the upper
half-plane paths c; and inclusion of contributions from the paths c;
results in the compact expression for TE surface wave pole contribution

to the guiding region electric Green’s dyad

G:s(p,zlp’,z’) = Z Rge(plp’)exp(JCplz - z’ I)

p
where
Jun
99 I - o I p I p I 1 (2)I p (2) p
no (plp ) - ——4— Jo(Qgp) Jo(Qgp ) [—Q—o no (Qsa) no (an)
S
_ _1_ (2) p (2), p 1
0" no (08a) no mom] —a o
8 52(ATE] -
C-Cp

This expression may also be derived by specializing (4.13) and Rge(plp’)
when n = 0.
Likewise, the radiation field contribution to the guiding region

electric Green’s dyad is obtained by evaluating the integral in (5.8)
i
b'
in the upper half-plane, only the paths F; and th (see Figure 4.3.3b)

along the branch cut detours C In Section 4.3.2, it was shown that,
contribute to the net value of the inversion integral. Furthermore, the
analytic continuation relation for the Hankel functions (4.15) allows
contributions from F+ and F-
s gt
path F;. In this manner, the radiation field contribution to the

to be combined into an integral along the

guiding region electric Green’s function is given by

-l

e ’ ’ = _—
Gb(p.2|p .2 ) 2"

ea , _ 99 I JClz-z'l
+[ Go+(P|P ) Go_(plp )] e dC (5.9)

s
where

56

up
99 , _ o 1 , I
Go (plp ) - 4 -—— Jo (Q 8p) J o(Qgp’ )[i

ATE

J’(Q ) H? )’(Q
-'7%—H H(2)’(Qsa)Ho (2 )(Q ga)]+ 2)gp 8p)
m’(Qgp’ ) J’(Qgp)

The upper expression in braces is valid for p > p’ and the lower for

(2)
(Qsa) Ho

 

H(2) I
Ho (08a)

p < p’. Both scattered and impressed field contributions are present in
(5.9) since both possess branch point singularities. The identical
result may be derived by modifying (4.17) and 6:9(plp’) for axially-
symmetric modes. The relation (8.4) in Appendix B may be specialized

when n = 0 to yield
69 , _ _ 99 I .
Go_(plp ) - Go+((P|P )1 . (5.10)

When (5.10) is substituted into (5.9) and the integration variable is
changed from C to 05, the expression for the radiation field
contribution to the guiding region TE electric Green’s function is given
by

Q

m
e , , _ -1 96 , jClz-z’l s
Gb(p,zlp ,2 ) - T I RelGo+(P|P )] e a? dQS. (5. 11)

0

It is evident that the real part of the bracketed term in 6:9 is the
product Jo(Qgp )Jo(Qgp) for all values of p and p . Thus, (5.11) may be
written as

wuo

e I I .. ’ I o
Gb(p,zlp ,z ) - Tn Jo (Q 8p’ ) J o(08p) [1 + Re(Cc)]
O
Q
JClz-Z’l s
e 80—) d0.-

S

where

57

_ 1 1 (2) (2). _ 1 (2), (2)
Cc - F [—Q_ no (Qsa) no (08a) —Q— no (age) no (08a)].
TE 3 8

Finally, in analogy with (4.19). the guiding region TE electric Green’s
function is constructed by superposing the TB surface wave and radiation

field components to yield

Ge(p,z|p’.z’) = Z Rge(plp’)exp(J€plz - z’l)

P
Q
up 0 d0
_ O I I I JC'Z’Z’| S S
O
(5.12)

This is precisely the form of the guiding region Green’s function
employed in the TE01 scattering study in the Chapter Six, where only

the lowest-order TE surface wave pole is considered.

5.3 Electric Green’s Dyad for TM Modes

In this section, the guiding region electric Green’s dyad for TM
modes is constructed. In analogy with the previous section. the
impressed field contribution is expressed as the axially-symmetric
specialization of (3.17), while the scattered field component is
characterized by the solutions of (5.1b). A physical argument is also
presented to demonstrate that only the four elements [GelaB, where a and
B may be p or z, are nonzero in the TM electric Green’s dyad.

Transverse magnetic, or TM, modes consist of the field components
E2, Ep, and He. Therefore, only these three components may be present

in the impressed fields. Examination of the axially-symmetric impressed

electric Green’s dyad (5.2) provides the physical constraint 9-3 = 0.

9

oz = 0, so that when 9-3 = 0 the guiding region TM

Furthermore, Kgp = K

impressed electric Green’s dyad is given by (5.2) where a and B may be p

58

or 2 only.
Moreover, under the previous constraints, the solution of (5.1a) is
fig 8 fig 8 O. This indicates the presence of TM scattered fields, which

are classified in terms of the solution of (5.1b), given by

n 0
"p , _ C S 8 _ (2) (2), I I
Ao(p ,g) - TT— [ ———— 1 ] no (Qoa) n (Qsa) Jo(Qgp ) (5.13a)

2 o
nng
JQ n20

”2 I - 8 s 8 (2) (2).
Ao(P .C) - ——— [ ;§6— Ho (08a) Ho (Qsa)

g S

(2) (2),
- no (Qsa) no (an)] Jo(Qop) (5.13b)

where

h)

(2)

O

(

n5 n2 2)
AE(C) = ijO[ —6; JO(an) H (Qsa) ’ —6; J°(an) HO (Qsa) ].

Alternatively, (5.13) may be derived by specializing (3.10) when n = 0.

The condition

AE(C = (p) = 0

results in the transcendental eigenvalue equation

2 I _ I P
ngJo(Q§a) - niKo(qsa)

 

 

(Qpa)J (Qpa) - (qpa)K (qpa) (5'14)
8 0 8 s o s
for the TM surface wave poles of the step-index circular fiber
waveguide [1]. This, too, may be derived from (4.7) when n = 0.
As in the previous section, the TM scattered Green’s dyad may be
constructed by inserting (5.13) and (3.12) into the expressions for the

guiding region field components (3.11), then substituting (3.11) into

59

the inverse Fourier transformation relation (2.3) This sequence of
steps entails expressions for the guiding region scattered electric

field components as the pair of volume integrals

s ___ es I I 98 I I I

Eo(p.z) I Z [G (p.2lp .z )] JB(p .z ) dV
V’ B=p.z

E:(p,z) =I Z [Ges(p.zlp’,z’)]zB JB(p’.z’) dV’
V’ B=p.z

where the elements of the scattered guiding region Green’s dyad are

[Ges(p.ZIP’.Z’)]pB

«FIR

~B I I JC(Z-Z’)
Ao(p ,C) Jo(Qgp) 8 dc

N|
a»-

[Ges(p,z|p’,z’)]zB

"B I JC(Z’2’)
Ao(p .C) Jo(Qgp) e dc.

Nl
:H

8%889-fi8

This result may be combined with the expression for the TM impressed
guiding region electric Green’s dyad elements to yield the four
components of the TM guiding region electric Green’s dyad.

In adherence to the progression of Sections 4.3 and 5.2, the
surface wave and radiation field components of the TM guiding region
electric Green’s dyad may be identified by deforming the Fourier
inversion path into the complex C-plane. Evaluation of the inversion
integral along the pole exclusion contours c: provides the TM surface

wave pole contribution to the guiding region electric Green’s dyad

G§s(p,zlp’ , z’) = Z RzBWIP’ )exp(JCplz-Z’ I) (5. 15)
P
where a and B may be p or z, and the R28 terms are presented in Table

5.3.1.
The radiation field component, on the other hand, is determined by

+
integration along the branch cut detours CB. After an appropriate

60

spectral analysis and change of integration variable, the radiation
field contribution to the TM guiding region electric Green’s dyad is

expressed, in analogy with (5.9) for the TB case, as

Q
Q
8 I I “B _. '1 “B I “B I JCIZ‘Z'I S
[co(p.z|p ,z )1 - TI [Go+(plp ) - co_(p|p )1 e W dog
0

(5.16)
where, again, a and B may be p or 2 only. The 6:3 terms are given in
Table 5.3.2. The guiding region TM electric Green’s dyad may now be
written as the sum of its surface wave (5.15) and radiation field (5.16)
components.

In this chapter, the axially-symmetric TE and TM guiding region
electric Green’s dyads were specified, in relatively compact form, as
superpositions of surface wave and radiation field contributions.

The dominant mode of the step-index circular fiber waveguide is the

hybrid HE11 mode; unfortunately, the coupling nature of the 9-dependence
of the HE11 fields precludes any major simplifications to the general
guiding region electric Green’s dyad (4.19). Although the TED1 mode can

not propagate without the HE mode present as well, for the sake of

11
simplicity the scattering application in Chapter Six utilizes the TE
Green’s function developed in Section 5.2; the author hopes that the TM

Green’s dyad presented in this section will gain usage for TM scattering

and excitation problems, as well.

61

Table 5.3.1. List of R:B(plp' ) terms for TM guiding region electric
Green' s dyad, where Qp= Q8 (C= Cp), Qp= Qs (C = Cp ).

..IIO =;—i;AH
qu

an I - o I p I p I (2 ) p (2)I p 1
no (plp ) - "Z’ Jo(Qop) Jo(08p ) Ho (08 a) no (08a) 9. A°

3: TM C=<

p
J:u

npz(plp ) = sgn(z-z’)— °Jo(ogp) J o(Qgp) [k2 5‘ H(2)’(Qpa) H(2)(Q:a)

 

 

 

JCQ:
2 n‘z’topa ) no ”'(opa ) ]o
kao H° ——IA::]p
c= cp
Juno
R2p(plp’) = sgn(z-z’) Jo (Qpp) Jo(op -:§ n‘z’topa)
0 4 8p Qp::
kS
H(2)'(Qpa) o
—— A
eel m] .
c CP
22 quo p p Qp H(2) p (2) p
Ro (plp ) = Jo(Qgp) Jo(Qop ) [g k2 Ho (Qg a) Ho (Qsa)
S
(op)2
-o82n no2’(opa ) n‘2 "(o:a ) ]§
08 k A0 ]

62

Table 5.3.2. List of G:B(plp’ ) terms for TM guiding region electric
Green’ s dyad, where ATM = :—— AE' The upper expression
in each term is valid for p “9 p’ , the lower for p < p’

 

               

 

 

 

wuo
cgp(plp') = %[ Jo(Qgp) Jo(Qgp ) n‘2 ’(08 a) H(2)’(Qsa)
I (2),
+ c n{ Jo (0 8p ) no (08p)
k: H(2)’(Qgp ) Jo(Qgp)
pz I _ , JC B(2), (2)
Go (plp ) - sgn(z TMo(Qop) Jo(Qgp J)[ 38° (08a) Ho (Qsa)
S
Jco8 Q: J (O 8p ) n‘2"(o )
2 Ho 38 R2 (2)8
08 kg k8 no (Qgp’) Jo(Qgp)
-0“
G:P(plp') = sgn(z-z’) 4° H[ 1 Jo (0 8p) J H(Qgp [E -J§
ATM 08 k: ks
I (2),
n‘2’(o a)n n‘2 )’(Qs a) - 3355 J °(Qgp ) H° (03p)
° 3 k: nQngp')ongm
—wu Q
22 , _ o 1 , g (2), (2)
Go (plp ) - 4 [ o Jo(Qgp) Jo(Qgp ) [7 Ho (an) Ho (05 a)
ATM 1‘s
02 02 Jo (Q 8p' ) n‘2’(o 8p)
3 (2) (2), 8
2 (083.) HO (Qsa) +-2' H?2)8
05kg k8 (Qgp ) J 0(Qgp)

63

 

CHAPTER SIX

APPLICATION TO SCATTERING FROM A GUIDING REGION DIELECTRIC DISCONTINUITY

6.1 Introduction

In this chapter, an application is provided for the guiding region
electric Green’s dyad developed in the previous chapters. More
precisely, the axially-symmetric TE Green’s function (5.12) is employed

to characterize scattering of the TE mode from a guiding region step

01
discontinuity. This analysis provides an ideal vehicle to demonstrate
the utility of the electric Green’s dyad, since the Green’s dyad is
well-suited to polarization electric field integral equation (EFIE)
descriptions of discontinuity fields [8,9]. Moreover, the scattering of
surface waves from discontinuities in circular dielectric waveguides
remains a pertinent research area because dielectric discontinuities may
be used to model many practical waveguide interruptions, such as [8]:
waveguide imperfections resulting in radiation attenuation, splices or
Junctions between waveguide segments, and devices such as modulators or
detectors.

This chapter exploits EFIE techniques to quantify surface wave
scattering of the axially-symmetric TED1 mode. A general EFIE is
developed for the unknown discontinuity field, then specialized for the
TED1 mode. Although this mode cannot propagate without the principal
HEll hybrid mode, the simplicity of the TB Green’s dyad--it reduces to a

scalar Green’s function with no angular dependence--prompts its primary

study. The scattering coefficients--reflection, transmission, and

64

radiation--are formulated in terms of the unknown discontinuity field.
Finally, method-of-moments (MoM) numerical solutions are implemented to
determine the discontinuity fields and scattering parameters for various

discontinuities in both strongly and weakly guiding fiber waveguides.

6.2 Formulation of EFIE for Scattering from 3 Dielectric Discontinuity

In this section, the general EFIE for surface wave scattering from
a guiding region step dielectric discontinuity is developed. This result
is simplified by expanding the unknown discontinuity field in an angular
Fourier series and exploiting the orthogonality properties of exp(Jn9)
[1]. The EFIE is easily specialized for the lowest-order axially-
symmetric, or T801, mode. The geometry of the discontinuity is pictured
in Figure 6.2.1. The discontinuity occupies the region V’, defined by
p s a, 0 s 9 5 2n, lzl S d.

From the deve10pments of Livernois and Nyquist [8], the electric
field excited along the dielectric fiber waveguide and within the

discontinuity region is

§(?) = I Ee(?|?')-lje(?') + 3eq(?')1 dV’ = Ei(r) + 25(r). (6.1)
VI

The impressed electric field Ei(?) is maintained by a primary excitatory
current 3e(?’). It is assumed that 3e is a remote source so that E1 is

in a spatial steady-state [7] and consists only of one or more discrete

surface wave modes. Likewise, the induced polarization current jeq(?’),
given by [8]

jeq(?') = jwco(n§ - n:)E(?') = Jaeoanzfi(?')

maintains scattered field ES(?). which, from (6.1), may be expressed as

65

 

 

 

 

 

 

Figure 6.2.1. Geometrical configuration of uniform di-

electric slice discontinuity. Coordinate

axes are aligned as in Figure 2.2.1.

66

ES(?) = jwcodnz I Ge(?l?’)°E(?’) dV’. (6.2)
VI

For arbitrary discontinuities or waveguide configurations, 6n2 = 6n2(?)
and may not be passed through the integral in (6.2); however, no graded
profiles are considered here.

The volume EFIE for unknown 3 excited by interaction of 31 with the

guiding region discontinuity, given by
1%?) - choanz I Ee(?|?')-€(?') dV’ = BR?) (6.3)
VI

where ? e V’, is obtained by rearranging (6.1) and expressing as in
terms of 3 within V’ via (6.2). This vector equation represents a set

of three scalar equations to be solved for Ep, E , and 82. Each has the

9

form

Baa?) - choanz J‘ E [ce(?|'r"n°“3 E (“r") dV’ = £16?) (6.4)
VI 3

where a may be p, e, or 2. Since the Green’s dyad elements all contain

8

an infinite summation over exp(-Jn9’), the volume integral in (6.4) may
be reduced to integration over the discontinuity longitudinal cross-

section by expanding each Ea in an angular Fourier series as

Ed?) = ZEam(p’Z) e‘jme. (6.5)

When (6.2) and (4.19) are substituted into (6.4), the orthogonality
properties of exp(Jn6) [1] may be exploited to yield the set of EFIE’s

of the form

67

Z e‘jne { Ean(p’2) - choanz I z [ Z 21:12:8(plp’ )exp(J§plz-z’ I)
LCS’ B p

II='@
on

Q
_ “B I _ “B I JCIz-Z’ I S I I I I I
I [ Gn+(PIP ) Gn_(P|P )]e a? dQS] EBn(p ,2 ) p dp d2 }

O

= E;(p,e,z) (6.6)

where LCS’ is the longitudinal cross-section (in the p’-z’ plane) of the
step discontinuity. Equation (6.6) may be solved for the unknown field
components Ea induced within the dielectric step discontinuity by
interaction of the impressed field component B: with the discontinuity.

This result may be readily specialized when scattering from only
the axially-invariant TED1 mode is considered. The incident TEOI

surface wave field consists only of the e-component [12]
E1(p z) = J’(Qo ) exp(JC z) (6 7)
9 ’ o gp o '

where :0 is the TEO1 surface wave pole, and E; is normalized to unit
excitation amplitude. The TE guiding region electric Green’s function
(5.13) is, along with (6.7), substituted into (6.6) to produce the

scalar EFIE

Eeo(p,z) - choanz I [ ZnRge(plp’)exp(JColz-z’I)

LCS'
@
- 2J’(Q )J’(Q ') [1 + Re(C )leJC'z'z I 05 d0 E ( ' 2') 'd 'dz’
I o g” o g” c €16;T s so 9 ' P P
O
= J;(Q;p) exp(Jcoz) (6.8)

that describes the axially-symmetric discontinuity field Eeo(p,z)

excited by interaction of the incident TE surface wave mode with the

O1

68

discontinuity structure.

6.3 Scattering Coefficients
Solution of the EFIE (6.8) for the unknown field within the
discontinuity volume V’ entails determination of the scattered field
exterior to V’ via (6.2). The transverse terminal planes 2 = -d and
= d are defined as the input and output terminal planes, respectively,
TE01 surface wave scattering may be characterized by evaluating the

ratio of the scattered surface wave field and the incident TE field at

 

 

01
the terminal planes. Thus the TEOI reflection coefficient
Jkionz
R01 = exp(J2§od) 2 I Ro(Co)Jo(Qgp’ )eprCo z’ ) p ’dp’ dz’
LCS’
(6.9)
and TED1 transmission coefficient
ianz
To1 = exp(JZCod)[ 1 + 2 I RO(Co )J’ o(Qgp’ )eprCo z’ ) p ’dp’ dz’ ]
LCS’
(6.10)
where

_ 2 1 6 H(2). _1_ , (2) 6
no(co) — 9. A0 [F Ko(qsa)Ho (0° a)- ——o Ko(qsa) no (Qoa)]
C TE C’ 3 Q9»

may be determined from (6.2) and the solution of (6.8). Only the pole
contribution to the TE guiding region Green’s function is considered for
surface wave scattering applications. The fractional power radiated

from the passive discontinuity is calculated from (6.9) and (6.10) as

69

2 2

P = 1 - IR - lTOll

rad 01I

6.4 Numerical Results

In this section, numerical MoM solutions to the EFIE (6.8) are
presented. The scattering coefficients and discontinuity electric field
distributions are examined for various electrical lengths of uniform
slice discontinuities in both strongly- and weakly-guiding fibers. As
depicted in Figure 6.2.1, the slice discontinuity occupies the region
p s a, 0 s 9 5 2n, lzl s d. The contrast parameter 5n2 is equal to the

2

constant n: - 118 within the discontinuity region V’ and zero elsewhere.

The fiber radius p = a is chosen so that only the principal HE nd

11"1
the axially-symmetric TEO1 modes are supported, since the TE01 mode
cannot propagate without the principal mode present as well. However,
only the TE01 mode is considered in this axially-symmetric
specialization.

To implement the MoM numerical solution, the unknown discontinuity

field in (6.8) is expanded as

N
p
Eeo(p.z) = Z
n=1 J

“NJ J1(tnp) pJ(z) (6.11)

.Er\’1efz

where the pj(z) are pulse functions of length 2d/Nz centered at z the

J!
In satisfy

(Ina) Ji(tna) + J1(tna) = O

and Np and "z are determined by convergence criteria. The entire-domain

radial expansion is a specialization of the Dini series, defined by [13]

7O

f(R) = ZbJ (A R) (6.12)

n v n
n=1

where R = p/a, the An satisfy
RJv(R) + HJv(R) = O (6.13)

and v and H are real. The Dini series, unlike the Fourier-Bessel series
[13] commonly used for expansion of fields in circular conducting
structures, allows a combination of Neumann and Dirichlet boundary
conditions at p = a (6.13). This proves desirable because the
tangential fields are continuous, not vanishing, at the boundary of a
dielectric waveguide. For the purposes of this thesis, the combination
v = 1, H = 1 was selected to exploit the closed form integration

relation [12]

 

a
2
_ a
I J1(tnp) J1(Qgp) p dp - 2 2 [ (08a) J1(An) Jo(an) ]
A - (Q a)
o n 8
(6.14)

and orthogonality property [14]

a %[J1(An)]....m=n

I J1(Tmp) J1(Tnp) p dp = (6.15)

o O ............. m t n

where tn = An/a. This choice is also Judicious since, via (6.13) and
(C.4), An are the well-documented zeros of Jo(€) [14].

To determine the expansion coefficients anJ
testing operator must be applied to both sides of (6.8). Since the

in (6.11), a

(p,z) and (p’,z’) dependence of the integrand in (6.8) are identical,
the testing operator must posses the same radial and axial functional
dependence as the unknown field expansion (6.11). From the previous

argument, the testing operator

71

¢m1(p,z) J1(T p) p1(z) p dp dz

Q‘—1D-
(DH—59’

where 1 s m s Np and 1 s i s N2, is applied to both sides of (6.8).
This is known as Galerkin’s method. The EFIE (6.8) may now be expressed

as a set of Np-Nz equations of the form

szan2

o
anj { anTiJ - ——§——— [ Dn(Co) Dm(Co) "13(co) R°(Co)

(_I-

(v1.2
:1an

:3

I
DEB 1"

Q
s -
D (C) Dm(C) W1j(C) [1 + Re(Cc)] 876;) dQs — Dm(CO) 21(Co)

+
II

 

where
d AZ....i=J
T = p (z) p (2’) dz’ =
u j, 1 , o ..... H,
d d
"13‘“ — IIp1(z) pJ(z’) dz dz’
-d-d
ZJA
——z+—2—[1-exp(JCA )1 ........ i=3
C C2 2
{ -5— sin2(A {/2) e (JClz -z I) i s j }
g2 z xp 1 J ....
d
21(C) = I p1(z) ojCz dz = % sin(Az§/2) exp(JCzi)
-d
__ 2d
Az - N—
z
21 - 1
2i = -d + [ 2 ]A2

and, via (6.14) and (6.15)

72

a % [J1(An)]2 ....m = n
an = I J1(Tmp) J1(Tnp) p dp =

O

a

n (c) I Jo(ogp) J1(tnp) p dp
O

2
a

2 2
(08a) - An

 

[ (ooa) J1(An) Jo(ooa) ]

The set of Np-Nz equations represents a single matrix equation to be
solved for the anJ' These expansion coefficients, in turn, allow
construction of the discontinuity field via (6.11)

The spectral integrand in (6.8), corresponding to the radiation
field contribution to Ge(p,zlp’,z’), is singular at C = 0 when QS = ks.
Consequently, the spectral integration is computed as follows: The
integrand is approximated analytically for IQS - ksl < 0.00001ks, which
isolates the singularity. In the intervals 0 < Qs < 0.99ks, 0.99kS < QS
< 0.99999ks, and 1'00001ks < Qs < 1.01ks, the integral is evaluated via
adaptive Simpson’s rule integration. This technique of applying
adaptive Simpson’s rule integrations to small partitions on both sides
of the singularity improves both the accuracy of the analytical
approximation and the numerical computation time. Finally, the
remaining contribution from QS > 1.01ks is accumulated in partitions of
length ks until the desired accuracy is obtained. For the numerical
applications that follow, the spectral integral is evaluated to 0.1%
integration accuracy.

Convergence of the MoM solution for each case was determined by

numerical trial-and-error. Figures 6.4.1 and 6.4.2 indicate the

dependence of the radial and axial field distributions on the parameters

73

Np and "2' Convergence was checked for each data point in Figures 6.4.3
through 6.4.10. In general, three to six radial expansion functions
were required for radial convergence; for axial convergence,
approximately 15 to 25 pulse functions per discontinuity wavelength were
necessary.

The MOM solutions to (6.8) provide some interesting descriptions of
scattering of the TE01 mode by various uniform slice discontinuities in
both high- and low-contrast fibers. The following applications
examined either a high-contrast fiber with n8 = 1.5, n = 1.0, a/Ao =
0.5, and Coa = -3.73, or a low-contrast fiber with n8 = 1.6, n = 1.49,
a/Ao = 1.0, and Coa = -9.62. The weakly-guiding configuration is
modeled after a polysterene core in a methyl methacrylate cladding, a
common configuration for short, inexpensive links; the strongly-guiding
case is representative of a glass fiber immersed in a free-space
surround [15]. Figures 6.4.3 through 6.4.6 illustrate the dependence of
the TEDlscattering coefficients on the electrical length 2kd of the
slice discontinuity. In Figure 6.4.3 (low-contrast) and Figure 6.4.4
(high-contrast), the discontinuity refractive index nd is held constant,
while the discontinuity half-length d is varied so that at least one
discontinuity wavelength is considered. Conversely, in Figure 6.4.5
(low-contrast) and Figure 6.4.6 (high-contrast), d = 0.1, while nd is
varied from 1.0 , the refractive index of free-space, to 4.0, which is
well into the semiconductor range. Figures 6.4.7 through 6.4.10 display
some representative radial and axial field distributions for long and
short discontinuities in the high-contrast fiber.

Unfortunately, it appears that no previous scattering study for

the dielectric fiber is available for comparison of these results.

74

However, a qualitative check of the scattering coefficients may be made
against the results obtained by Livernois and Nyquist [8] for TEo
scattering in a dielectric slab, since the lowest-order TE mode is
fundamentally similar in both structures. Comparison shows that the
scattering coefficients are in excellent qualitative agreement. From
Figures 6.4.3 through 6.4.6, it is evident that reflection coefficients
and fractional radiated power approach zero as the discontinuity length
shrinks to zero; transmission coefficients tend to one in this limit.
Fractional radiated power generally increases as the discontinuity
electrical length increases, although some resonance effects are
observed.

Moreover, the discontinuity field distributions displayed in
Figures 6.4.7 through 6.4.10 offer further assurance of the validity of
these solutions. Radial and axial field distributions appear much like

the unperturbed TE incident field for short discontinuities; as the

01
discontinuity length increases, radial distributions exhibit more
variation, while axial distributions assume a standing wave character.

Physically, the radiation of incident power is associated with this

increase in axial standing wave content for longer discontinuities [8].

75

1.0

 

 

aaaaaNp :1
00000 Np = 2
AAAAANP = 3
_+Np = 4
_JNP = 5
N2 = 20

 

[1111111111111111

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (,o/a)

0.8 —
6 0.6 a
E _
O
“_J 2
> -
G) _
Lil 0.4 _
0.2 —
0-0 j j T
Figure 6.4.1.

Radial distribution at z/d = 0.0,
for various numbers of radial ex—
pansion functions, of the electric

'field induced within a uniform

slice discontinuity along a fiber

waveguide with ng = 1.5, n, =

1.0, nd = 3.0, d/a = 0.25, and
O/>\o = 0.5.

76

1.0

 

 

 

 

1
4
1
.4
0.8- I!-
J _:
1 '.
_ 1 I
g 0.6 -l
o 1 L
L.‘_-’ '
b 4
o _
LE 0.4 _
1
1 mason N2 = 4
0.2-j ooooo N2 = 8
1 A229: N2 = 12
- N,D = 5 t_+_++_: N2 = 16
1
-l
0.0 erjlllllllilllTll']
—1.0 -O.5 0.0 0.5 1.0

Normalized Axial Coordinate (z/d)

Figure 6.4.2. Axial distribution at p/a = 0.5,
for various numbers of pulse ex-
pansion functions, of the electric
field induced within a uniform
slice discontinuity along a fiber
waveguide with (1,, = 1.5, n, =
1.0, no = 3.0, d/a = 0.25, and
O/Ao = 0.5

77

—b
O

 

 

 

— oaooa |R01|
: ooooo |To1|
_1 AAAAA Prod
..l
0.8 - u
A ‘
E, 0.6 4
CL. 1
S I
5; 6.. -
8 I
M —l
G .
0.2 -‘
1D
Doo-lill'lllllllllllTlllllll]
0.0 0.1 0.2 0.3 0.4 0.5
NORMALIZED HALF-LENGTH OF
SLICE DISCONTINUITY
Figure 6.4.3. Dependence of TEm scattering

coefficients on the normalized
length .(d/a) of a uniform slice
discontinuity along a fiber wave-
guide with n9 =1.,6 n, = 1.,49
nd =1.,0 and a/)\ =1..0

78

'o

 

 

 

- aaaoaIle
‘ ooooolel
‘ AAAAA Prod
0.8q
A 0 '
"53 0.6- -‘
a. j '
.53 :
"1 0.4-
—P -l
O _ -.
01 ..
C - . -‘
0.24
I
0.0‘fiTrrlfiIIIIiIIIIIIII1
0.0 0.1 0.2 0.3 0.4

NORMALIZED HALF—LENGTH OF
SUCH DISCONTINUITY

Figure 6.4.4. Dependence of TE), scattering
coefficients on the normalized
length (d/a) of a uniform slice
discontinuity along a fiber wave—
guide with no = 1.5, n. = 1.0,
nd = 3.0, and a/Ao = 0.5.

79

1.0

0.8

0.6

lLJllllLllJ

0.4

(IR01IIIT01IIPrOd)

0.2

_ILJIJ;

l l

 

0.0 '-
1.0

Figure 6.4.5.

aaoao |Ro1|
ooooo ”01'
AAAAA Prod

 

 

IFTIITIIIIIIIIITIIIIII

1.5 2.0 2.5 3.0 3.5 4.0

Refractive Index of
Slice Discontinuity

Dependence of TE“ scattering
coefficients on the refractive
index nd of a uniform slice
discontinuity along a fiber wave—
guide with n9 = 1.6, n. = 1.49,
d/a = 0.1, and a/Ao = 1.0.

80

 

1.0
1
0.8 -
A j
.0
E -I
O. 0.6 -l
'2; 1
l:. 1
_2-3 04 1
0: J
C '1
1
0.2 4
1
0.0 ‘4
1.0
Figure 6.4.6.

sauna |Ro1|
00000 ”01'
AAAAA Prod

1
\ _
I
A _‘
\
\
Q

 

 

IIIIIIIIrITIIIIIIIIITIrIj

1.5 2.0 2.5 3.0 3.5 4.0

Refractive Index of
Slice Discontinuity

Dependence of TE” scattering
coefficients on the refractive
index In of a uniform slice
discontinuity along a fiber wave—
guide with n, = 1.5, n. = 1.0,
d/a = 0.1, and a/Io = 0.5.

81

1.0

 

 

 

1
1
..l
0.8 '1
1
25‘: 0.6 1
o I
LLJ
_ 'l
E 4
(D d
9;] 0.4 -
0.2 -
_l
0-0 IIIINTIIIIUTTIIIIIII
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (,o/a)
Figure 6.4.7. Radial distribution, at various

axial locations, of the electric
field induced within a uniform
slice discontinuity along a fiber
waveguide with ng = 1.5, n, =
1.0, nd = 3.0, d/a = 0.05, and
O/xo = 0.5.

82

'o

 

 

 

‘ = 0.5
: = 0.0
o = -0.5
0.8 ‘1
..l
-;< '1
a 0.6 4
E ..
03
LE I
E 4
1.1—3) 0 4 1
I
0.2 l \ V
1 7
0.0+ TIIIIIITTIIIIIIWIII
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)
Figure 6.4.8. Radial distribution, at various

axial locations, of the electric
field induced within a uniform
slice discontinuity along a fiber.
waveguide with ng = 1.5, n, =
1.0, nd = 3.0, d/a = 0.25, and
O/Ao = 0.5

83

 

1.0

 

 

: W ; ;\
I mama/a = 0.1
0.8 1 2.0.22.0 ,0/0 = 0.5
.1 AAAAAp/O = 0.9
—X ‘1
o 0.6 -l
g 1
LIJ _
—' ‘1
> -I
o ..
L—l—J 0.4 " MW
'1
0.2 4
7 1R01| '-""- 0.45
: ”01' = 0.83
_l
0.0 IITIIITIIIIIIIIIIIII
-1.0 -0.5 0.0 0.5 1.0

Normalized Axial Coordinate (z/d)

Figure 6.4.9. Axial distribution, at various
radial locations, of the electric
field induced within a uniform
slice discontinuity along a fiber
waveguide with n9 = 1.5, n, =
1.0, nd = 3.0, d/a = 0.05, and
a/)\0 = 0.5.

84

aaaaap/Q

:—:::: 273

II II II
9.0.0
(DUI-f

0.8

0.6

0.4

 

L] I J L I I l [“1 LJ 1 1 LJ l_l LJ LJ [4]

 

 

0.2
1R0“ = 0.49
lel = 0.32
0.0 ITIIrTIIIIIIITIifiIII
-1.0 -0.5 0.0 0.5 1.0

Normalized Axial Coordinate (z/d)

Figure 6.4.10. Axial distribution, at various
radial locations, of the electric
field induced within a uniform
slice discontinuity along a fiber
waveguide with n9 = 1.5, n, =
1.0, nd = 3.0, d/a = 0.25, and
a/Ao = 0.5.

85

 

CHAPTER SEVEN

CONCLUSION

While dielectric fiber waveguides are finding a broad spectrum of
applications as the field of optical data transmission blossoms, many
facets of the excitation, transmission, and scattering phenomena
associated with these structures require further understanding. The
electric Green’s dyad introduced here is well-equipped to contend with
problems unique to circular dielectric structures, and offers an
alternative to the conventional eigenfunction expansion developed
previously.

The key to construction of the electric Green’s dyad was
characterization of the radiation mode spectrum of the circular fiber
waveguide, which, in general, is poorly understood. All analysis was
performed in the axial Fourier transform domain, Justified by the
longitudinal invariance of the electrical parameters and the geometry of
the guiding structure. This axial Fourier transform field
representation allowed construction of the electric Green’s dyad via
deformation of the Fourier inversion path and a subsequent complex
C-plane analysis. This complex analysis, in turn, extracted the
complete propagation-mode spectrum, including the well-documented
surface wave modes, and the radiation field spectral components.
Surface wave contributions to the Green’s dyad were identified as
residues from pole singularities in the C-plane; the radiation field

component was evaluated by integration along the branch out detour

86

 

depicted in Section 4.3.2.

The electric Green’s dyad, expressed in compact form in Section
4.3, is easily specialized for specific applications. Axially-symmetric
TE and TM modes were considered here. It was shown that for TM modes,
only four elements of the electric Green’s dyad are nonzero, while for
TE modes, the Green’s dyad reduces to a scalar Green’s function. The TE
scalar Green’s function was employed to construct an electric field
integral equation (EFIE) describing the electric field induced by
interaction of the TE01 surface wave mode with a guiding region
dielectric slice discontinuity. The solutions of the EFIE, obtained via
method-of-moments (MoM) techniques, were incorporated to characterize
scattering from such discontinuities. Scattering from dielectric
discontinuities constitutes an important class of problems, since
various types of waveguide interruptions may be modeled in this manner.

The utility of the electric Green’s dyad was demonstrated by its
application to scattering phenomena. The Green’s dyad is ideally suited
to EFIE techniques, and possible future applications are numerous.
Scattering characterizations of TM modes (involving the four nonzero
elements of the Green’s dyad and two coupled EFIE’s) and the principal
HE11 mode (involving all nine nonzero elements of the Green’s dyad and
three coupled EFIE’s) represent two higher plateaus of complexity in
this area. Moreover, Green’s function methods may also be applied to
the study of fiber excitation. The author encourages future use, for
these and other creative applications, of the electric Green’s dyad for
the dielectric fiber waveguide presented here, and hopes that it will

aid the development of this exciting area of study.

87

APPEND I CES

 

APPENDIX A

ALTERNATIVE REPRESENTATION FOR TRANSFORM DOMAIN SCALAR GREEN’S FUNCTION

An expression for the transform domain scalar Green’s function in
terms of circular harmonics is obtained with the assistance of the
“summation theorem" for Bessel functions. From (2.10), the transform

domain scalar Green’s function is given by

”(2)(

I» _ a,
o

“J
T QP Pl)

An alternative representation for H;2)(Q|3 ' 3'I) may be found by

invoking the "summation theorem" for Bessel functions [16]
on
M = Jn¢
e 2v(mR) Z Jn(mp) Zv+n(mr) e
n=-m

where

r > O; p > 0; ¢ > O

 

V

R = V/rz + p2 - 2rpcos¢

n
0 < w < —2—

With the specializations

= H(2)
V v

v = 0

m = Q

 

R = I; - B’l = V/pz + p’2 - 2pp’cos(6 - 9’)

the zero-order Hankel function is given by

89

 

m , (2)
H(2) ->, __ Jn(6-—6’) J n(Qp ) Hn (Qp)
(QIP I) - e “(2)
(Qp’) J n(0p)

n=-m
where the upper expression is valid for p > p’ and the lower for p < p’.

(2)

This relation for Ho (QIB - B’l) is substituted into the expression for

the scalar Green’s function (2.10) to yield equation (2.11)

m , (2)
(Qp’)n J n(0p)

which relates the well-known scalar Green’s function in terms of

circular harmonics.

90

 

APPENDIX B

DETERMINATION OF CONJUGATE RELATIONS OF SPECTRAL COEFFICIENTS ABOUT THE
LOU-LOSS BRANCH CUTS

The analytic continuation relation for Hankel functions (Appendix
C) suggests that spectral parameters may be simply related on opposing
sides of the inversion contour F+ = F; + th. This development
exemplifies the conjugate relationships of the elemental spectral
amplitude coefficients Ag and fig evaluated at equivalent C on the paths
F; and th. These relationships, in turn, are employed to show that
contributions from the paths F; and th may indeed be represented in an
integral along F; only, as in (4.17) and (4.18).

In this example, the behavior of the coefficients A: and E: is
investigated. If all sources are assumed to reside wholly within the
guiding region, then (4.17) may be verified for the [Ges(?l?’)]ae terms.
Specifically, the [ceS(?I?')]°° term is analyzed, since this term only

is implied in the TE scattering study developed in Chapter Six.

01
The behavior of A: and E: along the paths F; and th may be
determined by solving the modified matrix equation

I- 1 p I p 1

 

~9 ”no 29
I
wu
Be a 99 ,
E21 E22 E23 E24 Bu‘” C) ‘3' Kn (alp ’
= (8.1)
~9 -J ~29 ,
E31 E32 E33 E34 Cn(p' .C) -4— Kn (alp )
De -J ~00 ,
E41 E42 E43 E44 anp’ .C) L _4_ Kn (alp )
L a b I d

 

 

 

 

 

at equivalent C along the paths F; and th, respectively. Since QS'QS>

91

 

0 on F;, then for any C along F;, (8.1) may be written as

q

q

F

q

 

 

 

 

 

 

[ wu
Ne ’ O 29 ’
E11 E12 E13 E14 “n+(P 'C) '3' Kn (alp ’
0“
“'6 I O 99 I
E21 E22 E23 E24 Bn+(P 'C) ”h Kn (a'p )
= (8.2)
"9 I -.J "29 I
E31 E32 E33 E34 Cn+(P 'C) ”4 Kn (a'p ’
~9 , -J ~69 ,
L 1541 E42 E43 E44 Dn+(P 'C) ‘3‘ Kn (a'p ’
d . d n u
~6 ~0 ~6 ~6
where the + subscript denotes values of An, Bn’ Cn’ and DD on the

contour F;. However, because Qg’Qs < 0 along F- the Bessel and Hankel

8t’
function analytic continuation relations (C.6) and (C.7) may be invoked

to express (8.2) as

 

 

 

 

 

 

r ! P~e I F ”no 29 Q
a a _ III - e I _ - I I
(211) 1(812) (£13) +(814) An_(P .C) 4 (+xo (alp )1
up
a I - a _ III "9 I _° ._ 99 I a
$(E21) (£22) +(E223) (£24) Bn_(p .C) = 4 [Kn (alp )l
+(E )* (E )r :(E )* -(n )' 6° (p’ c) 22— I-Ez°(aIp')l'
' 31 32 33 34 n- ’ 4 n
_ ~0 -J -~ee
III III - a (I) I _ I a
(£41) 1(842) (£43) +(544) 0o_(p .C) 4 (+no (alp )1
(8.3)
~6 ~6 ~9 ~9
where the — subscript denotes values of An, Bn’ Cn, and Dn on the path
rot. The upper sign is valid when C is real, and the lower when C is
imaginary. The sign convention for the zero elements 812, E14, 831, and

E is arbitrary; however, the above convention was chosen so that

33
comparison of (8.2) and (8.3) yields

92

 

“'9 I -~ I
An_(p ,c) = +[A:+(p .c)1*

”9 I _ _ "9 I
Bo_(p .c) - (an,(p .C)l'
"0 , _ "9 ,
co_(p .c) — ilCn+(p .c)1*
"9 , _ "9 ,

Therefore, conjugate relationships of the elemental spectral amplitude

coefficients at equivalent C on F; and th

determined without explicitly solving (8.1). The relations for 32+ and

may, via this technique, be

83+, in particular, may be substituted into the expression for G29 in

Table 4.3.2 to give
99 , ___ _ ea , ,
Gn_(p|p ) [Gn+(plp )] (8.4)

for all values of C. From (8.4) and (4.17), the radiation field

contribution to [Ge(r)l?’)]66 may be expressed as

a
Q
e a a, 99 _ -1 Jn(6-6’) 00 , JClz—z’l s
[G (FIT )lb - T Z 9 I+Re[Gn+(PIP )] 8 8—0—7 dQS
n=-m FS 8

Expressions such as (8.4) for the remaining Green’s dyad elements may be

determined analogously, resulting in

[Ga8(plp’)]‘ ...... «B = p9, pz, 0p, 2p
G:§(plp’) = n28
-[Gn+(plp’)]' ..... a8 = pp, 99, Oz, 29, 22

for 0 s Q s k , and
s s

[GaB(plp’)]’ ...... a8 = p9, 9p, 92, 26
c:§(pIp') = “:3
-[Gn+(P|P')]' '''' «B = pp! P2. 999 2p: 22

for Q8 2 kg.

93

 

 

APPENDIX C

COHHONLY USED PROPERTIES OF BESSEL FUNCTIONS OF INTEGER ORDER

0.1 Background
The separation of variables solution in Section 3.2 yields, for the
radial dependence of the wave function u, a second-order differential

equation of the form

56

2
62 )
where E = Qp and n = 0, t1, 12.....Solutions to (C.1) are given by

= 0 (C.1)

Bessel functions of the first kind, Jn(€), of the second kind, Yn(€).

(1)

and of the third kind, HD (6) and Hi2)(€) (also called Hankel

functions), where [14]

(1) _
HD (6) - Jn(€) + JYn(€) (C.2a)
(2) _ _
Hn (E) - Jn(€) JYn(€) (C.2b)
Yn(€). Hél)(€), and Hi2)(€) are analytic throughout the g-plane cut

along the negative real axis, while Jn(€) is entire. Jn(€) is also
bounded as I€l —+ 0 [14]. The remainder of Appendix C serves to list
several important properties of Bessel functions of integer order which
are used throughout this thesis. All of these properties have been

extracted or derived directly from Chapter Nine of [14].

C.2 Bessel Functions of Negative Integer Order

When n is an integer,

94

 

6 (g) = (-1)“6o(§) (c.3)

-n

(1) (2)

where 8 may be J , Y , H , or H
n n n n

n

C.3 Recurrence Relations
Derivatives of Bessel and Hankel functions may be determined from

the recurrence relations

. _ 2
6o(g) €n_1(€) e 6o(g) (C.4a)

65(6)

n
8n+1(€) + E €n(€) (C.4b)

0.4 Analytic Continuation Relations
The analytic continuation relations for Bessel and Hankel functions

of integer order

Jn(€eJmn) = 63m“"1o(g) (C.5a)
Hé1)(€ejn) = -(e’3““)n;2’(§) (C.5b)
H;Z)(€e-Jn) = -(63“")no1’(g) (c.56)

are essential to the spectral analysis exposed primarily in Sections 4.3
and 5.3, and in Appendix B. If E is written as E = 5r + 361, 6 tends to
the real number Er as £1 approaches zero through negative values. As 6
approaches real values in this way, the relations (C.5a) and (C.5a) may

be specialized to

_ _ n = - n
JoI-g) - ( 1) Jn(€) ( 1) [Jn(€)]' (C.6a)
no2’(-e) = (—1)“’1no1’(§) = (-1)“’1(n;2’(§)1* (C.6b)

95

 

where the ' denotes complex conjugation and the second equalities hold

for |€1| —+ 0. Derivatives of (C.6) are given by

(-11)”bwa =(-11P”uwav my“

J£(-€)

“(2),(_ -€) = (_1)nH;1)I(€) = (-1) n[H(2)’(€)]' (C.7b)

The parameter E is allowed to tend to real values in this way to comply
with the developments of Section 4.3.2, where E = Qp and Im(Q) tends to
zero through negative values in the low-dissipation limit. This

(2)

convention also avoids the branch cut of HD (6) described in Section

C.1.

C.5 Large Argument Expansions

When n is fixed and |§l -+ m (Iarg(€)l < n),

2 1/2 nn H
Jn(€) 8 )[ RE ] cos(€ - ~§ - 4) (C.8a)
H(1)(€ 1/2 nu n
«[2 -E ] exp[J(€ - —§ - 3)] (C.8b)
1/2
nuke [—§]emnne-%-§n mam

C.6 Modified Bessel Functions of Integer Order
The modified Bessel function In(€) and Kn(€) are solutions to the

differential equation

282R an 2 2
z ——— - n )

+€—-(€
a€2 dz

In(€) is entire; Kn(€) is analytic throughout the 2 plane out along the

ll
0

(C.9)

negative real axis, and decays to zero as l€l —+ m for larg(€)| < E.

The derivatives with respect to argument of modified Bessel functions may

be determined by the recurrence relations

96

, - _ g

zn(e) — zn_1(g) E £n(€) (C.10a)
I _ n

£n(€) - 2n+1(€) + E £n(g) (C.10b)

Where £n represents In. (-1)nKn, or any linear combination. The

relation between Kn(€) and H(

n2)(€) is given by

Kn(€) = 2%! e’Jnn’Z H;Z)(€e-Jn/2) (c.11)

When g is allowed to approach real values as in Section C.4. (C.11)

specializes to

H(2)
n

_ - 21 '
(J6) - T: (J)“Kn(€) (C.12)

with derivatives

(2), _ _ —2 .

Hn ( J€) - -; (3)"xn(g) (C.13a)
(2)” _ _ -2.) II -

Hn ( 3e) - -E— (1)“xntg) (C.13b)

Relations (C.12) and (C.13) are particularly useful in determining the

surface wave eigenvalue equations (4.7), (5.4) and (5.14).

97

 

[1]

[2]

[3]

[4]

[5]

[6]

(7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

LIST OF REFERENCES

R.E. Collin, Field Theory of Guided Waves, New York: McGraw Hill,
1960.

 

A.W. Snyder, “Continuous mode spectrum of a circular dielectric
rod,“ IEEE MTT-S Trans., vol. 19, pp. 720-727, August, 1971.

D.P. Nyquist, D.R. Johnson, and S.V. Hsu, “Orthogonality and
amplitude spectrum of radiation modes along open-boundary
waveguides," JOSA, vol. 71, pp. 49-54, Jan. 1981.

C. Vasallo, "Orthogonality and amplitude spectrum of radiation
modes along open boundary waveguides: comment," JOSA, vol. 71,
p. 1282, 1981.

R.A. Sammut, “Orthogonality and normalization of radiation modes in
dielectric waveguides,“ JOSA, vol. 72, p. 1335-1337, 1982.

C. Vasallo, “Orthogonality and normalization of radiation modes in
dielectric waveguides: an alternative derivation,“ JOSA, vol. 73,
pp. 680-683, 1983.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, New York:
Chapman and Hall, 1983.

T.G. Livernois and D.P. Nyquist, "Integral equation formulation for
scattering by dielectric waveguides," JOSA, vol. 4, pp. 1289-1295,
1987.

J.S. Bagby, D.P. Nyquist, and B.C. Drachman, “Integral equation
formulation for analysis of integrated dielectric waveguides,"
IEEE MTT-S Trans, vol. 33, pp. 906-915, 1985.

G. Strang, Linear Algebra and its Applications, New York: Academic
Press, 1980.

 

R.V. Churchill, Complex Variables and Applications, New York:
McGraw Hill, 1984.

 

S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in
Communication Electronics, New York: John Wiley and Sons, 1984.

 

 

G.N. Watson, A Treatise on the Theory of Bessel Functions, New
York: The MacMillan Company, 1945.

 

M. Abramowitz and I.M. Stegun, Handbook of Mathematical Functions,
New York: Dover Publications, Inc., 1972

 

98

 

[15] G. Keiser, Optical Fiber Communications, New York: McGraw Hill,
1983.

[16] 1.5. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and
Products, Orlando: Academic Press, 1980.

99

 

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