Transmission, reflection and radiation at junction planes ofdifferent open waveguidesCitation for published version (APA):Ruiter, de, H. M. (1989). Transmission, reflection and radiation at junction planes of different open waveguides.Technische Universiteit Eindhoven. https://doi.org/10.6100/IR315847

DOI:10.6100/IR315847

Document status and date:Published: 01/01/1989

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TRANSMISSION, REFLECI'ION AND RADlATION AT JUNCI'ION PLANES

OF DIFFERENT OPEN WAVEGUlDES

TRANSMISSION, REFLECTION AND RADlATION AT JUNCTION PLANES

OF DIFFERENT OPEN WAVEGUlDES

PROEFSCHRIFf

1ER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNNERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. IR. M. TELS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN

OP DINSDAG 5 SEPTEMBER 19891E 16.00 UUR

DOOR

HELENAMARIA DE RUITER

GEBORENTERHOON

druk: wibro disserurtiedr'Ukkerij, helmend

Dit proefschrift is goedgekeurd door de promotoren:

Prof.dr. ir. A.T. de Hoop

en

Prof.dr. J. Boersma

CIP-GEGEVENS KONINKLUKE BIBLIOTIIEEK, DEN HAAG

Ruiter, HelenaMaria de

Transmission, reflection and radiation at junction planes of different open waveguides/Helena Maria de Ruiter. -[SJ. : s.n.]. Fig., tab. Proefschrift Eindhoven. Met lit.opg., reg. ISBN 90-9002864-1 SISO 539.1 UDC 537.874(043.3) NUGI 832 Trefw.: elektromagnetische golfvoortplanting I optische golfgeleiders.

-V-

Aan mijn ouders,

aan oma

-VI-

This study was performed as part of the research program of the professional group

Electromagnetism and Circuit Theory, Department of Electrical Engineering,

Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The

Nether lands.

-VII-

CONTENTS

ABSTRACT

1. INTRODUCTION 1

2. BASIC RELATIONS OF ELECTROMAGNETIC FIELD THEORY 7

2.1. Basic equations for the electromagnetic field quantities in an

inhomogeneons medium 7

2.2. The frequency--domain redprocity theorem 10

2.3. The electromagnetic Green's states 11

3. FIELD REPRESENTATIONS IN OPEN WAVEGUlDE SECTIONS 17

3.1. The straight open waveguide section 17

3.2. Modal expansion of the fields in an open waveguide section 19

3.3. Methods for the calculation of surface-wave modes in open waveguides 27

3.3.1. The integral-equation metbod 28

3.3.2. The transfer-matrix formalism 30

3.4. The computation of surface-wave modes in a planar open waveguide 36

3.4.1. The integral-equation metbod 37

3.4.2. The transfer-matrix formalism 57

4. INTEGRAL REPRESENTATIONS FOR THE FJELDS IN A STJ{.AIGHT

OPEN WAVEGUlDE SECTION IN TERMSOF THE TANGENTlAL

FJELDS IN THE BOUNDARY PLANES 67

4.1. Integral representations and the coupling problem 67

-VIU-

4.2. lntegral representa.tions conta.ining ;I# a.nd K # 67

4.3. Integral representa.tions conta.ining either ;I# or K # 70

4.3.1. Representations containing ;I# 70

4.3.2. Representa.tions conta.ining K # 71

4.4. The method of images 72

4.4.1. Representations containing ;I# 73

4.4.2. Representations containing K # 16

5. INTEGRAL EQUATIONS FOR THE FIELDSIN THE JUNCTION

PLANES OF SERIES-CONNECTED STRAIGHT OPEN WAVEGUlDE

SECTIONS 79

5.1. Configuration of series-connected waveguide sections 79

5.2. lntegral equations for the ta.ngential fields in the junction plane of

two series-connected straight open waveguide sections 81

5.3. Integral equations for the ta.ngential fields in the junction pla.ne of

three series-connected straight open waveguide sections 84

5.3.1. Integral equations conta.ining !!T a.nd !!T 85

5.3.2. Integral equa.tions conta.ining !!T 89

5.3.3. lntegral eqnations conta.ining !!T 91

6. REFLECTION, TRANSMISSION AND RADlATION AT THE JUNCTION

OF TWO PLANAR OPEN WAVEGUlDES 93

6.1. Description of the configuration 93

6.2. Integral equa.tions for the fields in the junction pla.ne of

two pla.nar open wa.veguide sections 94

6.2.1. Integral equations for TE--fields 96

6.2.2. Integral equations for TM-fields 97

6.3. Transverse Foutier Transformation of the integral equations 98

-IX-

6.3.1. Fourier-transformed integral equations for TE-fields

6.3.2. Fourier-transformed integral equations for TM-fields

6.4. Numerical methods employed

6.4.1. Numericalsolution of the integral equations and methods

of computation

6.4.2. Outline of the computational procedure

6.5. Numerical results

6.5.1. On-axis junction of two waveguides with different widths a.nd

100

102

102

105

112

117

equal permittivities er 5-10-3j 119

6.5.2. On-axis junction of two waveguides with different widths and

equal permittivities er = 2.25-10-3j 123

6.5.3. Offset junction of two identical two-moded waveguides, and

radiation from a terminating waveguide

6.5.4. Offset junction of two identical three-moded waveguides, and

radiation from a terminating waveguide

6.5.5. On-axis junction of two waveguides with different widths a.nd

different permittivities

6.5.6. Offset junction of two waveguides with different widths and

different permittivities

6.5. 7. Offset junction of two identical single-moded waveguides, and

radlation from a terminating waveguide ( dependenee on offset

and frequency of opera ti on)

6.5.8. Computation times and storage requirements

APPENDICES

A. On the branch cuts occurring in the spectral-domain field expressions

for open waveguides

A.l. The planar waveguide

125

135

142

148

152

161

165

165

167

-X-

A.2. The waveguide with bounded cross-section 168

B. Orthogonality properties of the modal field distributions 171

C. Symmetry properties of the Green's tensor elements of an infinite

open waveguide 189

D. Calculation of the axial and transverse Fourier transfarms of the

Green's tensorelementsof a multi-step-index planar waveguide 192

D.l. Calculation of the Fourier transfarms Ö~~(kx,k~,kz)

and Ö~~(kx,k~,kz) 193 D.2. Calculation of the free-space Green's tensors 208

D.3. Symmetry properties of the tensor elements 210

D.4. Behaviour in the complex kz -plane 211

D.5. Transverse Fourier transfarms of the modal fi.elds of a mul ti-step-

index planar waveguide 213

E. Expressions for the reflection a.nd transmission coefficients of the

junction of two open waveguides 215

F. Expressions for the directive gain of the terminating open waveguide 221

REFERENCES 231

ACKNOWLEDGEMENTS 239

SAMENVATTING 241

CURRICULUM VITAE 243

-XI-

ABSTRACT

The devices used for optical point-to-point communication typically consist of a

series-conneetion of sections of different types of cylindrical open waveguides. At a

junction of two different sections, one bas a discontinuity of the electromagnetic

properties, which results in reflection, transmission and radiation of electromagnetic

waves at the junction pla.ne. The main theme of the present thesis is the quantitative

analysis of these phenomena.

To start the analysis, both the propagation of electromagnetic waves a.long a uniform

(infinite) waveguide section and the interaction of waves at the junction planeneed to

bedescribed in mathematica! terms. This description is basedon Maxwell's equations

for the electromagnetic field, ihe frequency-domain reciprocity theorem, and the

electromagnetic Green's states. lt is shown that the fieldsin a uniform (infinite) open

waveguide section ca.n be represented by a modal expansion invalving surface-wave

modes and radia.tion modes. Two methods for the computation of surface-wave moda.l

fields are discussed and illustrated by numerical results for pla.nar open waveguides.

Next, integra.l representations are derived for the fields in a finite open waveguide

section in terms of the transverse fields in the boundary pla.nes, and for the fields in a

semi-infinite section in terms of the transverse field in the terminal plane and the

transverse incident field propagating towards the terminal plane. By means of these

representations, systems of integral equations are established for the fields in the

junction plane(s) of two (three) series-connected open waveguide sections.

-XII-

One of these systems of integral equa.tions bas been selected and solved numerica.lly,

for · va.rious combina.tions of two series-connected planar open wa.veguide sections and

fora semi-infinite waveguide terminating in free space, whereby the incident field is a

TE-surface-wave mode. More specifically, the system of integral equations is

subjected to a. spatial Fourier Tra.nsformation, whereupon the resulting Fourier

tra.nsformed system is numerically solved by the metbod of moments. The solution

obta.ined for the Fourier tra.nsform of the junction-pla.ne field, is used to calcula.te the

. transverse field in the junction pla.ne a.nd the reflection of the incident surface-wave

mode at the junction pla.ne. In a.ddition, the transmission at the junction plane is

computed for the series-conneetion of two waveguide sections, whereas for the

terminating waveguide the forward radiation from the terminal plane is determined.

-XIII-

-XIV-

-1-

1. INTRODUCTION

In communication engineering, optical systems for signal transmission are becoming of

ever increasing importance. As any communication system, they contain devices for

signal generation, signal transmission, signal detectión and signal processing. In the

present thesis we investigate in more detail the transmission of optical, i.e.,

electromagnetic, signals along waveguiding structures. In the early years, the

transverse dimensions of these structures were of the order of some tens of

wavelengtbs of the electromagnetic radiation employed, and, hence, they cou1d be

analysed with the aid of optical ray theory. However, the tendency is that the sizes of

the cross-sections will go down to the order of the wavelength; therefore, an analysis

based on the full electromagnetic equations becomes necessary. An introductory

overview of waveguide theory is provided insome standard textbooks on the subject;

we mention Kapany (1967), Marcuse (1974), Unger (1977), and Snyder and Love

(1983).

As far as the waveguiding structures are concerned, we concentrate on the cylindrical,

open, waveguides that are used in optical point-to-point communication systems.

Ideally, a single straight waveguide wou1d suffice, but in practice, a series-conneetion

of different types of waveguides is technically inevitable. As a consequence, both the

analysis of wave propagation along a straight section, and the interaction of waves at

junctions of two such sections are of importance. The junction of two different sec-

tions amounts to a discontinuity in waveguiding properties. At such a discontinuity,

reflection, transmission and radiation of electromagnetic waves take place. The

quantitative analysis of this kind of phenomena is the main theme of this thesis.

-2-

Now, for the calculation of electromagnetic fields, several methods are available. The

most direct one would be to solve, in practice numerically, Maxwell's equations,

taking into account the appropriate boundary conditions and causality conditions

(radiation conditions). In open-waveguide configurations, this metbod would require

a numerical solution of Maxwell's electromagnetic differential equations in the entire

IR3, since the fields in general extend considerably outside the directly wa.veguiding

region. Due to insurmounta.ble difficulties with regard to the stora.ge requirements in

the computer, this metbod is outside the range of practical a.pplica.tion. Hence, other

methods have to be called for.

First of all, we ca.n take adva.nta.ge, in an analytical manner, of the tra.nslationa.l

invaria.nce of the wa.veguide in the axial direction. For a straight open waveguide

section, the electroma.gnetic field can be decomposed into its axial-spectral

constituents by subjecting it to a.n axial Foutier Transforma.tion. This metbod leads

to the well-known modal description of the fields in a wa.veguide. For open

waveguides, two types of modes are distinguished, viz. the surface-wave modes {for

optical transmission the desired ones) a.nd the radiation modes ( usually of an

unwanted nature). In order to include the description of the excitation of the modal

field constituents by localised sources, we carry out the analysis by applying the axial

Fourier Transformation to the electromagnetic field equations in which souree terms

have been included. Then, upon a.nalytically continuing the axial Fourier tra.nsforms

into the complex kz -pla.ne (kz being the parameter of the axial Foutier

Transformation), the propagation coefficients of the surface-wave modes show up as

poles, a.nd the propagation coefficients of the radiation modes :6ll up on branch cuts in

the complex kz -plane, the Iatier being related to causal wave propagation in the

outermost medium. The former propagation coefficients are often referred to as the

discrete modal spectrum, the latter as the continuons modal spectrum. For the

computation of the propagation coefficients and the conesponding transverse field

-3-

distributions, several methods are ava.ila.ble. In the present thesis, the

integral-equation metbod and the transfer-matrix formalism are discussed.

For the computation of the field in the junction plane(s) of two (or more) open

waveguide sections, several methods have been presented in the literature. Firstly, we

mention the application of (semi-)analytical methods (Wiener-Hopf technique) to

the junction of two different semi-infinite structures ( Angulo and Chang, 1959;

lttipiboon and Hamid, 1981; Aoki et al., 1982; Uchida and Aoki, 1984).

A second method, which has been applied by many authors, is the full modal analysis,

which comprises the matching, in a junction plane, of both the surface-wave modal

fields and the radiation modal fields of the two waveguides a.t either side of the

junction plane. In an early paper by Angulo (1957), this metbod is used to derive an

integral equation · for the electtic field in the terminal plane of a terminating slab

waveguide. From it, Angulo derived variational expressions that yield upper and

lower bounds for the terminal admittance, and expressions for the forwardly radiated

power flow density. Ruif (1977) employed this metbod in the matching problem for

two semi-infinite slab waveguides. He reduced the problem to a system of singular

integral equations for the forward and backward scattering coefficients of the

surface-wave modes and the radiation modes. For small discontinuities in the

waveguides' properties or axial alignments, he obtained an approximate solution for

these equations by means of a perturbation analysis. Mostly, the continuons spectrum

is discretised by employing an expansion into a sequence of functions, the integrals of

products of which can readily be calculated ( Clarricoats and Sharpe, 1972; Mahmoud

and Beal, 1975; Brooke and Kharadly, 1976; Rozzi, 1978; Morishita et al., 1979; Rozzi

and In 't Veld, 1980). Then, systems of linear algebra.ic equations are obta.ined, which

can be solved by standard methods. A somewhat different metbod for solving the

equations obtained by mode matching was employed by Gelin et al. (1981) and by

Ta.kenaka et al. (1983); these authors determined the modal field coefficients by

means of an iterative procedure. For small step discontinuities, Marcuse {1970)

simplified the equa.tions for the scattering coefficients of the surface-wave modes: by

ignoring the backward scattered ra.diation modes, he obtained closed-form expressions

for the reflection and transmission coefficients of the surface-wave modes; next, by

ignoring the reflected surface-wave mode in the calcula.tion of the scattering

coefficients of the forward and backward scattered radiation modes, he obtained

closed-form expressions for the scattering coefficients of the radiation modes. The

samemetbod was applied by Ittipiboon and Hamid (1979).

The third metbod for the computation of the fields in the junction pla.nes of different

open waveguide sections employs surface-souree type integral representations for the

:fields in each of the joining waveguide sections. The latter fields are considered to be

excited by surface-souree distributions at the junction planes. These souree

distributions, which are simply related to the tangential electromagnetic fields in the

junction planes, enter into the integral representations mentioned, together with

appropriate Green's functions. By using, in each of the waveguide sections, these

integral representations for the fields right at the junction planes, and by imposing

the condition that the tangential fields should be continuons across the junction

planes, a system of integral equations for the fields in the junction pla.nes is obtained.

The kemel functions in these integral equations are the Green's tensorelementsof the

joining waveguide sections. This method was employed by Nobuyoshi et al. (1983)

and by Nishimura et al. (1983). These authors used approximate expressions for the

Green's tensor elements occurring in the integral equations, in the sense that they

either ignored the effect of the reflections at the transverse boundaries of the

waveguide (Nobuyoshi et al.), or partly ignored this effect and partly took it into

account by expressions based on geometrical opties or on image-metbod

approximations (Nishimura et al.). These procedures restriet the application of their

-5-

methods to wea.kly guiding structnres.

In the present thesis, the surface-souree type integral formalism that involves Green•s

tensors, is developed in a rigarous manner. Exact expressions are used for the Green's

tensor elements occurring in the integral equations. With it, a general metbod is

provided for the computa.tion of the reflection, transmission and radiation in a series-

conneetion of an arbitrary nnmber of waveguide sections (which can be nsed to model

other, more genera!, discontinuities in a waveguide). To calculate the as yet nnknown

field distributions in the junction planes, the integral eqnations are subjected to a

transverse Foutier Transformation. In this wa.y, the behaviour of the fields in the

junction plane, that may be both oscillatory and slowly decreasing away from the

guiding structnre due to the presence of continuons spectrum (radiation) field

components, can be accounted for. Another advantage of this Fourier-transform

computational metbod is, that the spatial singularities in the Green's tensors (cf. Lee

et al., 1980) are more easily handled in the transform doma.in. The Fourier-

transformed integral equa.tions thus obta.ined are solved numerically. From the

solutions, the scattering coefficients for the surface-wave modes are obtained, and the

forward radiation of a terminating planar open wa.veguide is determined. Subsequent

application of a Fast Fonrier Transformation yields the fields in the jnnction planes.

With this method, a number of confignrations has been analysed. A brief outline of

the contents of the subseqnent chapters coneindes this introduction.

In Chapter 2, the equations for the electromagnetic field, the frequency-domain

reciprocity theorem, and the electromagnetic Green's states fora general structure are

discussed.

Chapter 3 deals with the representation of the fields in straight open waveguide

sections in terms of surface-wave modes and radiation modes (discrete and

continuons spectrum). Two methods for the computation of surface-wave modal fields

(that will be taken as excitations for the discontinuities in the wa.veguide) are

discussed and results are presented for several types of pla.nar open wa.veguides.

In Cha.pter 4, integral representations for the fields in a straight open waveguide

section are derived. Depending on the conditions that are imposed on the Green •s

tensors, representations are obtained in terms of either the transverse electric field at

the boundary planes, or the transverse magnetie field at the boundary planes, or both.

In Chapter 5, the integral representations of Cha.pter 4 are used to derive integral

equations for the transverse fields {electrie, magnetic, or both) in. the junetion

plane( s) of two and three series-conneeted open waveguide sections.

In Chapter 6, the theory developed in Chapter 5 is a.pplied to the junction of two

planar (two-dimensional) open waveguide sections. The transverse Fourier

Transformation is a.pplied to the relevant integral equations. Numerical results are

presented for a number of configurations; a TE surface-wave modal field is taken as

the incident field. A compa.rison is made with the results obtained by Rozzi (1978).

Finally, the computing times involved are discussed.

Va.rious auxiliary caleula.tions and deriva.tions are given in Appendices A-F.

-7-

2. BASIC RELATIONS OF ELECTROMAGNETIC FIELD THEORY

2.1. BASIC EQUATIONS FOR THE ELECTROMAGNETIC FIELD QUANTITIES

IN AN INHOMOGENEOUS MEDIUM

In this section we briefly discuss the equations that govern the frequency-domain

electromagnetic field quantities in a medium with linear, time-invariant electro-

magnetic properties. The latter vary continuously with position, except at sufficiently

smooth surfaces, across which the electromagnetic properties may exhibit a finite

jump. Position in space is denoted by the position vector ! with respect to a fixed

reference frame. The frequency component with angular frequency w has a time

dependenee exp(jwt ), where j denotes the imaginary unit and t is the time coordinate;

the time factor exp(jwt) is suppressed throughout. In a domain in space where the

electromagnetic properties vary continuously with position, the electromagnetic field

quantities are continuously differentiable and satisfy Maxwell's equations

(2.1)

(2.2)

The quantities occurring in these equations are listed in Table I. SI-units are used

throughout the presentation. For a bounded domain, the electromagnetic field must

satisfy prescribed boundary conditions at the boundary of the domain; for an

unbounded domain, the field must satisfy the radiation condition at infinity (Felsen

and Marcuvitz, 1973, p.87). The medium under consideration is assumed to be locally

Table I. Quantities, symbols aild SI-units.

quantity time domain frequency domain

SI-units

electric field intensity V/m

magnetic field intensity A/m

electtic flux density C/m2

magnetic flux density T

volume density of electtic current A/m2

volume density of magnetic current V/m2

surface density of electtic current A/m

surface density of magnetic c~ent V/m

frequency-domain permittivity F/m

frequency-domain permeability H/m

* in vacuo E = Eo = 1/ J.toC~ wi~h c0 = 2.99792458 .. 108 m/s

**in vacuo p =Po= 4?r" 10-'7 H/m

symbols

E

H

D

B

!r Ky

!#' Kc#'

* f ** p

Fig. 2.1. Surface of discontinuity for the electromagnetic properties.

reacting, isotropic, and, as stated before, time-invariant. Under these circumstances

its constitutive equations are

(2.3)

B(!) = Jl{!)H(!). (2.4)

In general, E and p. are complex-valued, with Re( f} > 0 and Re(p.) > 0. Fora passive

medium, Im(E) ~ 0 and Im(p.) ~ 0. A medium is called lossy (dissipative) when

Im(e) < 0 andfor Im(p.) < 0; it is called lossless when Im(t) 0 and Im(p.) = 0.

Across a surface of discontinuity E for the electromagnetic properties the electro-

magnetic field quantities must satisfy the boundary conditions

(2.5)

(2.6)

that express the continuity of the tangential components of !! and _!!; !!. denotes the

unit vector normal to the surface of discontinuity E (Fig. 2.1). On the surface of

an electrically perfectly conducting object the condition

(2.7)

must hold, while on the surface of a magnetically impenetrable object

n" H = 0 - - - (2.8)

-10-

must be satisfied.

2.2. THE FREQUENCY-DOMAIN RECIPROCITY THEOREM

One of the most fundamental theorem.s in electroma.gnetic field theory is the Lorentz

redprocity theorem (Van Bladel, 1964). This theorem interrelates two different

electromagnetic states that can occur in one and the same bounded domain rand

have the same angular frequency w (Fig. 2.2). Each of the two states satisfies the

equations (2.1)-(2.6), applying totherelevant state.

Let us mark the quantities of state A by the superscript A and the quantities of state

B by the superscript B. Then, with the aid of (2.1)-(2.6), it can be shown that

l !!' (]!AxHB _ !B,.!!A]dA = J [-HB ·K\-- !A·:!B r- aA.KB"_+ !B ·:!A,l dV,(2.9) 8r r

where !! is the unit vector normal to 8 r, the bonndary surface of r, pointing away

from Y.Here it is understood that eA = eB and p.A = p.B for all! e r (Fig. 2.3).

V'

Fig. 2.2. Bounded domain rinspace with closed bonndary surface 8 Y;!!. is

the unit vector normal to 8 r, pointing away from r, and r' is the complement of ru 8 Yin 1R3.

-11-

Fig. 2.3. Identical bounded domains in spa.ce with the same permittivity and

permeability a.nd two different field distributions with the same angular

frequency.

Across surfaces of discontinuity for the electromagnetic properties the fields are

assumed to sa.tisfy the conditions (2.5) a.nd (2.6), while on the boundary surfaces of

impenetrable objects (2. 7) or (2.8) must hold.

2.3. THE ELECTROMAGNETIC GREEN'S STATES

From the reciprocity relation (2.9) we want to derive souree-type integral

representa.tions for the electroma.gnetic field qua.ntities. To that end, we consider the

fields genera.ted by (vectorial) unit point sourees with volume current densities

proportional to the three--dimensional unit pulse 6(!-!.'). The corresponding states are

denoted as the electric Green's state {~GE, !!GE, !!.G~, KG~} if

(2.10)

(2.11)

-12-

and as the magnetic Green's state {~GM, !!GM, !!_~,KG~} if

(2.12)

(2.13)

In an unbounded domain these Green's states a.re required to represent waves

travelling away from the souree point !.' towa.rds infinity, i.e., they must satisfy the

radiation condition. With the use of (2.1}-{2.4) we arrive at the following systems of

equations for the Green's states:

(2.14)

(2.15)

and

(2.16)

(2.17)

These equations a.re to be supplemented by the appropriate boundary conditions at

surfaces of discontinuity for the electromagnetic properties. In view of the linea.rity of

the governing equations, {~GE, HGE} and {~GM, HGM} may be written as

(2.18)

(2.19)

-13-

and

(2.20)

(2.21)

in which g are the so-called Green's tensors of rank two. The dep€ndence on the

position (of the point souree is explicitly indicated in the notation for g. Up to now,

the Green's stales are not unique. They can be made so by imposing appropriate

boundary conditions (in case of a bounded domain) or the radialion condition (for an

infinite domain). The equations for the elements of the Green's tensors follow upon

substitution of (2.18) and (2.19) into (2.14) and (2.15), substitution of (2.20) and

(2.21) into (2.16) and (2.17), and by taking for ~E and i!M the successive unit veetors

of !he coordinate system employed.

Let Y he a bounded domain with boundary surface 8 r, and let Y ' denote the

domain exterior to IJ 'Y. Consider an electromagnetic state {~, !!. ,! r• ! rl which

satisfies the equations (2.1)-{2.8). In !he Lorentz redprocity relation we take for state

A: {~A. !!A, !Ar• !Ar} = {~, !!, ~ r•! y}, and for state B the electric Green's B B B B GE GE GE GE .

state: {~ , !! , ,! 'Y'! y} = {~ , !! , ,! y,! y }. U pon usmg (2.10), (2.11), (2.18) and (2.19), wethen arrive at

I [gEM(~'.!l·!.,(!l + gEE(!',!)·,!&'(!)JdA(!) ar

+ J [gEM(!' ,IJ·!,._{!) + gEE(r' ,!) . ,! ,._{!)JdV(rl r

= {1,~, 0}~(!') when ~· E { r,a r, 'Y'}. (2.22)

-14-

Likewise, when we take for state B tbe magnetic Green•s state: {!!_B, _I!B, :!Br ~~} GM GM GM GM . = {]';_ , _!! , ! r , ~ r } and use (2.12), (2.13), (2.20) and (2.21), we arnve at

J [!;)MM(E'•!.H~!) + ~ME(!',!)·!&'(!)]dA(!) ar

+ J (~MM(t,!)· ~ "(!) + ~ME(f,!)·,! "(r)]dV(!) r

= {1, ~' 0}.1!(!') when !' E { 1( iJ 1( Y'}.

In (2.22) and (2.23) tbe surface current densities !&' and ~&'are given by

(2.23)

(2.24)

(2.25)

The factor 1/2 occurring in (2.22) and (2.23) applies to smooth boundaries, i.e., the

sw:face a Yis assumed to have a tangent plane.

In the preceding analysis we have assumed that r is a bounded domain with boundary surface a 'KWe can e.xtend the validity of the e.xpressions to cases in which ris an unbounded domain having (parts of) its bounda.ry at infinity, provided that

the fields involved satisfy the radialion condition. Then, the contribution of the pa.rts

at infinity to the surface integrals in (2.9) and (2.22), (2.23) vanishes. For the

unbounded doma.in exterior to a bounded closed surface only tbe contribution of the

latter surface remains (Fig. 2.4).

Ta prove redprocity relations for the Green's tensors, witb respect to their

-15-

Fig. 2.4. Domain r with boundaty a 'Y= a 'î, U '8 '2• with {j ':; ~ m. On the application of the Lorentz reciprodty theorem, the contribution of {j ":i vanishes, and only the contribution of IJ 'i remains.

dependenee on the two space arguments, we take 'Y= 1R3 in (2.22) and (2.23). We

then obtain

J [~EM(~',!J·~ "..(!) + ~EE(!,',!)·:!. ".(!JJdV{!) = ~(!'), r

J [~MM(~',!J·~ ".(D +~ME(!',!)·:!. ".(!JJdV(!) = !!(!'). r

(2.26)

(2.27)

By substituting for the field {~, !!}(f) in (2.26) and (2.27), the electric Green's field

due toa unit point souree at i.e., by setting:!."..(!) = !Eó(!-~"), ~ "..(!) = Q. and {~, !!J(t) ={!'!.GE, gGEH!.') and using (2.18)-{2.19), we arrive at

-16-

(2.28)

(2.29)

Simllarly, by substituting lor the field {E;, !!}(~') in (2.26) and (2.27), the magnetic

Green's field due toa unit point souree al r_", i.e., by setting! r(!l = Q, K r(!l = !Mt5(r_-r_") and {~. !!}(~') {!f:GM, ]!GM}(r_') and using (2.20)-{2.21), we obtain

(2.30)

(2.31)

From (2.28)-{2.31) we arrive at the reciprocity relations lor the Green's tensors

(Felsen and Ma:rcuvitz, 1973, p.92)

(2.32)

(2.33)

(2.34)

where the superscript T denotes tra.nsposition.

In subsequent chapters we shall use the inlegral representations (2.22) and (2.23) for

the electromagnetic field intensities at t E IJ 'Y, to descrihe the transmission and relleetien properties of secbons of straight open waveguides.

-17-

3. FIELD REPRESENTATIONS IN OPEN WAVEGIRDE SECTIONS

3.1. THE STRAIGHT OPEN WAVEGUlDE SECTION

In this chapter, the electrornagnetic Jields in a straight open waveguide section will be

investiga.ted. In Fig. 3.1, the pertaining configuration is shown. The axial coordinate

is z. The terminal planes of the waveguide section a.re the transverse planes z=z1 and

z=z2, with z1 < z2. The z-interval z1 < z < z2 is denoted by :i:'; the bounda.ry of >';

i.e., {z=z1} U {z=z2}, is denoted by i} :i:; {-w < z < z1} U {z2 < z < w} is denoted by

:%}. The configuration is translation invariant in the z-dîrection. This implies tha.t

the permittivity and the perrneability of the medium a.re functions of the transverse

position !.T only, i.e., '=

-18-

Outside the bounded cross~tional domain !iJ (see Fig. 3.1), whose boundary

contour is ê!i!, 'and IJ are constants, to be denoted by

-19-

3.2. MODAL EXP ANSION OF THE FIELDS IN AN OPEN WA VEGUIDE

SECTION

In this section the modal expansion of the fields in open waveguides is discussed. This

type of expansion is often used in descrihing the transmission properties of waveguide

sections. In the following, irrelevant dependences on coordinates will be suppressed in

the notation.

In order to investigate the transmission properties of the waveguide section, in which

the field distributions in the end planes serve as excitations, we subject the field

equations in a section to a finite Fourier Transformation with respect to the axial

coordinate. To this end we introduce

x-

~ 11 x-~ 111 x

Fig. 3.2. Planar waveguide (a), and rotationally symmetrie fibre (b), and

permittivity/permeability profiles: step-index (I), multi-step-index (11) and

graded-index (III).

-2Q-

z2

i:C!.T•kz) =I exp(jkzz) !(!.T·z) dz with kz e IR. zl

Inversely, we have

m

(3.2)

(2r)-l I exp(-jkzz) Ê(!.T,k2) dkz = {l,~,O}!(!T•z) when ze { ~8 ~ ~'}. (3.3)

-m

The electromagnetic field equa.tions (2.1} a.nd (2.2) then tra.nsform into

~ . ~ ~ Y x !!(!T,kz) - jwQ(!T,kz) = l ,{!T,kz) + l@"(!T'z2)exp(jkzz2)

(3.4)

... ... ... -Y I( ~(!.T,kz) + jw~.(!T,kz) = -K ,{!.T,kz) - K ~!T,z2)exp(jkzz2)

(3.5)

in which l# a.nd K # are given in (2.24) a.nd (2.25) with ~=~at z =z1 a.nd

n = i at z = z2, a.nd - -2:

(3.6)

Since é and p. in the waveguide are independent of z, (2.3) a.nd (2.4) tra.nsform into

. . Q(!T•kz) = é(!.T) ~(!.T,kz), (3.7)

-21-

~ ~

~(!_T,kz) = p(!_T) !!C!:.T,kz). (3.8)

The surface souree terms in (3.4) and (3.5) eau be regarded as the axial· Fourier

transforma over the interval -oo < z < ro of the transverse end-plane current sheets

with volume distributions of the electric type !&'(!.T•z1)5(z-z1), !&'(!.T•z2)5(z-z2),

and volume distributions of the magnetic type K &'(!_T,z1)5(z-z1),

!i~!.T,z2)5(z-z2). In the usual transmission case they serve as excitations, while the volume souree distributions ! 'Y and K 'Y in the interlor of the section vanish.

Consequently, our case is fully covered once the fields excited by a single transverse

electric current souree distribution !Tb"(z) and the fields excited by a single transverse

magnetic eurrent souree distribution KTb"(z) have been determined.

For a transverse electrie current souree !Tb"(z), the Fourier transforms of the fields

over the interval-ro < z < ro satisfy the equations

(3.9)

(3.10)

in whieh the superscript E indieates the type of excitation. By separating these

equations into transverse and axial parts, the symmetry properties of the field

components with respect to kz are readily established. Since !T is independent of kz,

the transverse component of the left-hand side of (3.9) must be even in kz, and we

arrive at

~E ~E

In order to reveal the modal strueture of the fields, the functions ~ and !! are

-22-

analytica.lly continued into the complex kz -plane. This analytic continuation is

assumed to have the property: I {~.E ,i,E}(kz) I -+ 0 as I kz I -+ m (by virtue of the Riemann-Lebesgue lemma, this assumption is met for real valnes of kz). From

experience with configurations for which the transformed quantities can be evaluated

analytica.lly, we expect iE and i_E to have the following singularities in the complex

kz -plane: a finite number of simple poles {n!h n = l, ... ,NE, (under certain circumstances, there may be no poles) and a branch point kz = k1 = w( f 1/l1)

112 (see Appendix A) in the fourth quadrant of the kz-plane; and, symmetrica.lly, a finite

number of simple poles {-"!} and a branch point kz = -k1 in the second quadrant of the kz -plane (Fig. 3.3). In genera!, k1 is complex-valued (lossy medium). The lossless

case is considered as a limiting case of the lossy one. The branch points kz = :k1 are

due to the occurrence of the square root (k~- k!)1/ 2 which is specified as that branch for which Im(k~-k~)1/2 5 0 (Appendix A). Aecordingly, we have the branch cuts ff a.nd §(on which Im(k~- k!)1/ 2 = 0) as shown in Fig. 3.3.

By use of Cauchy's integral formula for the functions iE and B:E and the contour shown in Fig. 3.3 (in the interlor of which iE and B:E are analytic functions of kz)

and by ta.king into account the symmetry properties (3.11) of the fields, we obtain

(3.12)

(3.13)

in which j{~!,H!} are the residues of {iE,B:E} at the polen!. The integration along

fis taken from the branch point"= k1 towards infinity, and -211'{~~.!!~} denotes the 11jump11 in {iE,:ÊI:E} across the branch cut ff; this jump is defined as the

-23-

__ _, s~-l!!l.i~t.. ;'.,.,.,. .... - f I - .......... ..., ,"" I I ,,

,. I I " / I f ,,

' I I '\ / } ' ' ' " ,, ' I I I: \

/ I I \ I I I 1

I I I \ I I I \ I / I

I -l( -1(2 -l 0 (indicated by a plus sign in Fig. 3.3), and the values of {~E,HE} on the side where Re(k~- k~)1/2 < 0 (indicated by a minus sign in Fig. 3.3).

By inverse Fourier Transformation of (3.12) and {3.13), evaluated by closing the path

of integration in the lower hal{ of the kz -plane, the electromagnetic field {~ .• !!} is

obtained as

(3.14)

When z < 0, ~E and HE can be obtained by using their symmetry properties

-24-

(3.15)

which follow from the symmetry properties (3.11) of {~E,:B:E}. In (3.14), the

summation over the poles can be interpreted as the contribution of the surface-wave

modes to the fieldsin the waveguide (discrete part of the spectrum); the surface-wave

poles x:! also appear as propagation coefficients of the surface-wave modes. The integration along ff represents the contribution of the radiation modes ( continuons part ofthe spectrum).

In the same way, we can analyse the excitation by a single transverse magnetic

current souree distribution with volume density KT6{z). The Fourier transforma of

the fields generated then satisfy the equations

(3.16)

(3.17)

in which the superscript M refers to magnetic current souree excitation. Since KT is

independent of kz, we now obtain the symmetry relations

As before, ~M and j_M are analytically continued into the complex kz-plane. We

expect ~M and :B:M to have a finite number of simple poles {:~:,..~}, n = 1, ... ,NM, ( that may be different from the poles { :~:,..!}) in the fourth and second quadrants, and

again the branch points kz = :k1.

Taking into account (3.18), the representa.tion a.na.logous to (3.12) a.nd (3.13) is now

-25-

(3.19)

(3.20)

From these expressions, the fields in z > 0 are obtained as

When z < 0, ~M and !!M can be found by using their symmetry properties

(3.22)

Again, the summation over the poles represents the contribution of the surface-wave

modes with propagation coefficients ~r.~, and the . integration along /b+ can be

interpreted as the contribution of the radiation modes.

From (3.14) and (3.21) it is apparent, that the fields due to an arbitrary excitation at

z = 0 can be represented by

and a similar representation for the fields when z < 0. In (3.23) the field contributions

due to transverse electric and transverse magnetic current souree distributions have

been taken together. In AppendixBit is shown that the modal field constituents

for z > 0, {~,!!n}exp(-j~r.nz), n = l, ... ,N, and {~~r.,H~r.}exp(-jK-Z), K E $+, form a

-26-

complete orthogonal set of functions. Next we introduce the normalised field

constituents fot z > 0, denoted by {~~!n} and {~x~!!x} 1 which satisfy the Lorentz

normalisation conditions

(3.24)

(3.25)

where 9J:r denotes the total transverse cross-sectionat domain of the waveguide and its surroundings, and ~ 1 = (k~-x

2)1 /21 ~ 1 = (k~-{x•)2)1/2 1 (note that ~ 1 and I I I

' kT 1 are real and positive). For z < 0, the normalised field constituents follow by '

applying the symmetry properties (3.15) and {3.22).

It can be shown that the Greenis tensors of the waveguide are ex:pressible in terms of

the Lorentz-normalised modal field constituents as

(3.26)

(3.27)

-27-

(3.28)

(3.29)

when z' > z (Blok and De Hoop, 1983; the difference in sign between their expressions

and (3.26)-(3.29) is due to a difference in normalisation). When z' < z, the

expressions ior the Green's tensors can be obtained by carrying out the appropriate

changes according to symmetry (ei. (3.15) and (3.22)).

In the next section we shall discuss some methods for caleulating the solutions of the

souree-free field equations that correspond to the surface-wave modes.

3.3. METHODS FOR THE CALCULATION OF SURFACE-WAVE MODES IN

OPEN WAVEGUlDES

Several methods exist ior the computation of the propagation coefficients and the field

distributions of the surface-wave modes in open waveguiding structures. We mention:

the direct numerical solution of the souree-free eleetromagnetic field equations (Mur,

1978); the numerical solution of the system of souree-type integral equations resulting

from the souree-free eleetromagnetic field equations (De Ruiter, 1980); the

transfer-matrix iormalism (for special geometries) (Suematsu and Furuya, 1972;

-28-

Clarricoats et al., 1966); and metbodsof an approximate nature, such as tbe weak-

guidance approximation {Snyder and Young, 1978). In this section, two metbods will

be treated in more detail, viz. tbe integral-equation metbod and tbe transfer-matrix

formalism.

3.3.1. The integral-equation metbod

The field of a surface-wave mode {!\1_0,!

0}exp(-jK

0z) witb propagation coefficient K

0

sa.tisfies tbe souree-free electromagnetic field equations

(3.30)

(3.31)

in whicb

(3.32)

and must be quadratically integrable over tbe total cross-sectional domain of the

waveguide and its surroundings. In fact, ~ is an eigenvalue of equaiions (3.30) and

(3.31). The deviations of the permittivity and permeability in the waveguide from

their values f 1 and p,1 i:p. the surrounding medium are now conceived as

z-independent disturbances. In accordance with this point of view, equations (3.30)

and (3.31) are rewritten as

(3.33)

(3.34)

-29-

where

(3.35)

(3.36)

Equations (3.33) and (3.34) have the a.ppea.rance of electromagnetic field equations in

a homogeneons medium with constitutive coeffi.cients t 1 and p.1, and with volume-

souree terms jn and !n· In terms of these volume sourees the solutions of these

equa.tions can be written as (De Hoop, 1977)

with

P.n(!T) = I g(!.T•!.±•""n) jn(!.±) dA(!±), !iJ

9.n(!.T) = I g(!.T•!±•""n) !n(!±) dA(!.±), !iJ

in which g is the two-dimensional free-space Green's function

with

(3.37)

(3.38)

(3.39)

(3.40)

(3.41)

-30-

(3.42)

For !:T E .!4 equations (3.37) and (3.38) constitute a. system of homogeneons integral

equa.tions. Upon solving these equa.t.ions (which, in genera!, ha.s • to be done

numerically), we obtain the propaga.tion coefficients {"n} a.s eigenvalues, and the

cortesponding modal field distributions a.s eigenfunctions.

3.3.2. The transfer-matrix formalism

For configura.tions in which the geometry, permittivity and permea.bility a.re functions

of a single coordinate only (e.g., the plana.r wa.veguide and the circula.rly cylindrical

waveguide), the problem of determining the surface-wave modescan be reduced toa

problem of solving ordina.ry differential equa.tions and a corresponding transfer-

matrix formalism can be developed. This formalism will be applied to a souree-free

configuration.

The configurations for which the transfer-matrix formalism can be used; are shown in

Fig. 3.4. The coordinate on which the waveguide properties depend, is denoted by u;

for the plana.r waveguide, u stands for the x-coordinate (--«~ < u < m), and for the

circularly eylindrical waveguide, u stands for the distanee p to the axis (0 ~ u < m).

The wa.veguide is divided into one or more layers, bounded by surfaces u= constant,

in which the permittivity and permeability a.re continuons funetions of u. Across the

interface of two suecessive layers, these quantities ma.y exhibit a finite jump. Now,

the four electromagnetic field components perpendicula.r to the direction of u a.re

continuons upon crossing these interfaces. They a.re combined into a column matrix,

the field matrix f.

Let u= up (p = 1,2, ... ,N-1) denote the location ofthe interfaces, then in the interlor

-31-

interfaces M.l "~~ of toyers ~~~

Fig. 3.4. Configurations to which the transfer-matrix formalism can be applied:

(a) planar waveguide with piecewise continuons permittivity !(x) and

permeability Jd.x); (b) circularly cylindrical waveguide with piecewise continuons

permittivity !(p) and permeability J'(p).

of the layer up-l < u < up, the field matrices at two positions u and u' are

interrelated by the transfer matrix l:p (Walter, 1976), viz.

(3.43)

in which x:n is the propagation coefficient of the surface-wave mode to be determined.

The columns of l:p are the special fund~ental solutions of the system of first-order

-32-

differential equations for the elements of! in the la.yer, that are uniquely defined by

(3.44)

in which J: denotes the unit matrix. Since f only contains field components that are continuous upon crossing the interfaces between successive layers, the field at a.n

arbitrary position u in the configuration can be expressed in terms of the field at

another arbitrary position u'. Let uq_1 ~ u~ uq and up-1 ~ u' ~ up, then we have

when q > p (Fig. 3.5)

A similar expression can be obtained in the case q < p.

The present relation between the field matrices at different positions is used in the

"interior" layers of the waveguide, i.e., u1 < u < uN_1. In each of the "outer"

domains, i.e., -m < u < u1 and uN_1 < u < oo for the planar waveguide, and

0 < u < u1 and uN_1 < u < oo for the circularly cylindrical waveguide, it is required

that the fields must remain bounded as u .... :t:m (pla.nar waveguide) or as u .... 0 and

u .... ro (circularly cylindrical waveguide). As an example consider the outer domain

- m < u < u1 of the planar waveguide. In this domain the goveruing differential

equations have four linearly independent solutions for the field matrix f consisting of

the transverse field components {e1

,ez,hy,hz}. Two of these solutions can be chosen

to be bounded as u -+ -oo, while the remaining two solutions grow exponentially as

u-+ -m. Obviously the latter two solutions must be excluded, which leads to two linear

relations to be imposed on the components of the field matrix f. By means of these relations two componentsof f(u1) can be eliminated. Similarly, by retaining only the

bounded solutions in the outer domain uN_1 < u < m of the planar wa.veguide, two

interface layer

q+1 Uq

OU q Uq-1

q-1 Uq~2------

up-;2------Up+l ------

Up •u'

Up-l ------

®

p+2

p+1

p

p~1

-33-

interface layer

Up

p

Up-1 ------p-1

Up~2------

Uq+2------q+2

Uq+1 ------

q+1

Uq

Uq-1 •U q

q-1

®

Fig. 3.5. Positions u and u' in the layers q and p, respectively, in a medium with

piecewise continuons f and p.: (a) when q > p, and (b) when p > q.

components of the field matrix !(uN_1) ca.n be eliminated. The same procedure also

applies to the solutions in the outer domains of the circularly cylindrical waveguide.

Thus we conclude that after elimination of two components as indicated, both !(u1)

and !(uN_1) contain two unknown field components only.

To determine the propagation coefficients and the field distributions of the

surface-wave modes we now proceed as follows. By means of the transfer matrices,

the field matrix at an arbitrarily chosen level u0 is expressed in terms of the field

matrix at u = u1 by

(3.46)

where ~ is a product of transfer matrices of the layers between the levels u1 and u0,

as in (3.45). At the sa.me level u0, the field matrix can also be expressed in terms of

-34-

the field matrix at u = uN_1 by

(3.47)

Since the field matrices at u = u0 must be identical, (3.46) and (3.47) lead to

(3.48)

This is a homogeneous system of four linear algebraic equations for the two unknown

field components of f(u1) and the two unknown field components of f(uN_1). This

system has a non-zero solution only for particular values of "n• which are called

eigenvalues. Having solved the resulting eigenvalue equation for "n' we can obtain the

unknown field distribution up to a complex multiplicative constant, which is

determined by imposing the normalisation condition. The field matrices at u = u1 and u = uN_1 are then known; the field matrix at an arbitrary position results by reusing the transfer-matrix formalism.

From (3.48) it is easily seen that the values of "n and of the field matrices do not

depend on the choice of u0, since

[I(uO,u1)]-1 = I(ul'uO), so [~(uO,u1)]-1·~(uo,uN-1) = ~(ul'uN-1), and the latter matrix, which is the transfer matrix from level uN_1 to level ul' is

independent of u0. In practice, the level u0 is chosen on computational grounds.

ldeally, this level should correspond to the maximum of the transverse field

distri bution of the mode under consideration.

The transfer-matrix formalism is particularly suitable for waveguides that consist of

layers for which closed-form expressions for the fundamental solutions are available.

Examples are:

-35-

- layers with a constant permittivity and permeability profile, for which the

fundamental solutions involve trigonometrie and exponential functions in the case of a

planar waveguide (Suematsu and Furuya, 1972), and Bessel functions in the case of a·

circularly cylindrical waveguide (Clarricoats et al., 1966);

- layers with a linear refractive index profile for which the fundamental solutions in

the case of a planar waveguide are expressible in terms of Airy functions

(Brekhovskikh, 1980, pp. 181 - 188);

- layers with an Epstein-type refractive index profile for which the fundamental

solutions fora planar waveguide are expressible in terms of hypergeometrie functions

or Reun's functions, depending on the type of polarisation (Blok, 1967; Brekhovskikh,

1980, pp. 164 -180; Van Duin, 1981).

Some authors have used a step-Cunetion approximation to an (arbitrary)

graded-index profile (Clarricoats and Chan, 1970; Suematsu and Furuya, 1972) and

have used the transfer-matrix formalism to perform computations of the propagation

coefficients and the field distributions of the surface-wave modes of a graded-index

waveguide. When the thickness of the layers used in the discretisation of the actual

profile is sufficiently small as compared to the transverse wavelength of the

surface-:wave mode under consideration and to the varlation of the profile, this

approach will yield good approximate results for the propagation coefficients of the

graded-index waveguide. Fora specific example, the influence of the number of layers

on the value of the propagation coefficient obtained for a particular surface-wave

mode in a circularly cylindrical waveguide has been invesUgated by Clarricoats and

Chan (1970).

In the next section, we shall apply the two methods discussed here to the wave

propagation in a planar open waveguide, and we shall present some numerical results

obtained by the two methods.

-36-

3.4. THE COMPUTATION OF SURFACE-WAVE MODES IN A PLANAR

OPEN WAVEGUlDE

In this section the methods of computation discussed in the previous section are

applied to the computation of the surface-wave modes in a planar open waveguide.

The configura.tion a.t hand is shown in Fig. 3.6. The geometry, permittivity and

permea.bility only depend on the x-eoordinate. The waveguide's thickness is d = 2a.. When -a ~ x ~ a., f and J.l are functions of x; outside the waveguide, E = fl and J.l = ~-'l are constants. In this configuration we investigate the fields that are y-independent;

then 81 = 0 and !.n = !xOx- jnn~· From {3.30) and (3.31) it is easily seen that the

field equa.tions separate into two independent systems of equations, viz. one system

for TE-fields with {e1

,hx,hz}-;, 0 and {h1

,ex,ez} : 0, and one system for TM-fields

with {hy,ex,ez} 'f. 0 and {ey,hx,hz} = 0. In view of the duality of the electtic and ma.gnetic field quantities, the equations for the TM-field quantities follow from the

TE-field equa.tions by ma.king the appropriate substitutions.

1- -·~:_ --I--~ Fig. 3.6. Straight planar waveguide and coordinate system. The slab thickness is

d=2a.

-37-

3.4.1. The integral-oouation metbod

In view of the y-independence of the configuration and the fields, the results (3.37)

and (3.38} for TE-fields simplify to

(3.49)

(3.50)

(3.51)

in which Pn and qn are now given by ,y ,x,z

(3.52)

(3.53)

Here jn and !.n are given by (3.35) and {3.36), respectively; d denotes the x-interval

occupiéd by the slab; and the one-dimensional freEHipace Green's function is now

(3.54)

with ~ given in (3.42). Alter inserting (3.52) and (3.53) into (3.49)-{3.51), the

orders of integration and differentiation can be interchanged. The operator öx acts on

the Green's function g only. This differentiation can be performed analytically. From

(3.54) it follows that ÖxS(x,x',,or,n) is discontinuons at x= x', and that ~g(x,x',~~:n) has a singularity -ó(x-x').

-38-

In most cases, the permeability of the waveguide is constant and equal to the

permeability of its surroundings, so that, according to (3.36) and (3.40), 9.n = Q.. Then (3.49), together with (3.52) and (3.35), provides a homogeneons integral equation for

e1

in the slab, while (3.50) a.nd (3.51) together with (3.52) and (3.35) are integral

representa.tions for hx and hz, respectively, in terros of e1

in the slab.

For TM-fields, on the other hand, the duals of (3.49)-(3.53) with 9.n :: Q.lead toa

system of homogeneons integral equations for ex and ez in the slab. The latter system

follows from the duals of (3.50) and (3.51), together with the dual of (3.53) and

(3.35). The dual of (3.49), togetber witb tbe duals of (3.53) and (3.35), tben provides

an integral representation for h1

in terros of ex and ez in the slab.

By diseretising the expressions (3.49H3.53) forTE-modes, (i.e., surfac;e-wave modes

ha.ving a TE field), or their duals for TM-modes, we arrive at a system of linear

algebraic equa.tions that is a.mena.ble to numerical solution. Tbe discretisation

procedure leads to a homogeneons system of the form

~·! = Q., {3.55)

in which f is a column matrix that is related to the field values used in the

discretisation scheme, and ~ is a square matrix, the elements of which are determined

by the diseretised versions of (3.49)-(3.53). The propagation coefficients "n are then

computed from det(~) = 0. Next, the field distribution of the corresponding surface-wave mode is obta.ined by substituting the value of "n into (3.55) and solving

this system, subject to a convenient norma.lisation.

The discretisation procedure to be used. bere is tbe metbod of moments

(Kantorowitsch and Krylow, 1956; Harrington, 1968). In this method, the field

-39-

quantities are expanded with respect. to the expansion functions { '1/Jj(x); j=l, ... ,J}.

Suppressing the subscript n referring tothemode number, we write

(3.56)

Upon inserting (3.56) into (3.49)-(3.53) and (3.35}-(3.36), the left- and right-hand

sides of the resulting expressions are multiplied by the weighting functions

{ IPk(x); k=l, ... ,J}, and integrated over the slab domain. Then by eliminating the

coefficients jj and !.j' a system of 3J equa.tions is obtained for the 3J unknown field

coefficients eJ. , h. , h .. In this system the left-hand sides contain the integrals of ,y J,X J,Z products of weighting and expansion functions 1 ~P:tc:(x)'I/J.(x)dx, while the right-hand

d J

sides contain the integrals

~~ ~P:tc:(x)g{x,x' .~)'1/Jj(x')dx'dx, ~~ ~P:tc:(x)Dxg(x,x' ,~)'1/lj(x')dx'dx,

11 ~P:tc:(x)~(x,x' ,lf.)'I/JJ.(x')dx'dx, and dd

~ ~P:tc:(x)( E-t1)'1/lj(x)dx, ~ IP:k(x)(~T-JLl)'I/Jj(x)dx,

whereby the latter two integrals are evaluated numerically. In case the weighting and

expansion functions are differentiable, the integrals involving Dxg and a;g can be

transformed by an integration by parts. We thus obtain

IJ ~P:tc:(x)DxS(x,x',lf.)'I/Jj(x')dx'dx =-IJ DxiP:k(x)g(x,x',lf.)'l/lj(x')dx'dx dd dd

+ [~P:tc:(x) J g(x,x',~>)'I/Jj(x')dx']~=~a' (3.57) d

in which x= -a and x= a are the boundary planes of the slab, and, since oxg(x,x' ,~~:)

= -Dx,g(x,x',~>),

-40-

IJ ~(x)a2~{x,x',~~:),Pj(x')dx'dx = -ll/Jx ~(x)g(x,x',~~:)/Jx,,Pix')dx'dx aa aa

[[ ( ) ( , )·1· ( ''J]x=a 1x•=a - ~ x g x,x ,~~: '~'j x x=--a x'=--a· (3.58)

For special choices of the expansion and weighting functions, the above integrals may

be evaluated analytically.

The simplest choice for the expansion functîons is

,p.(x) = Rect.(x) = { 1 when x e dj' J J 0 when x ~ dj

(3.59)

while for the weighting fundions we take

(3.60)

Here dj, j = 1,2, ... ,J, are the subintervals into which [-a.,a] is divided, and xk is an

interlor point of the subinterval dk (Fig. 3. 7). In our case, the subintervals have equal

lengths and xk is taken as the centre point of dk. Note that with this choice of

expansion and weightîng functions, (3.57) and (3.58) are not applicable. With this

choice, the metbod of solution for the integral equation is called the point-matching

metbod or the method of collocation. The integrals J g(x,x',~~:)dx', J /J~(x,x',~~:)dx' dj dj

and f a2 g(x,x',~)dx' occuning when using the point-matching method are calculated d.x

J .

analytically.

t 1 Ijl· J

-41-

DL-~----------------~----~----------~ XJ-1 Xj Q x-

' Fig. 3.7. The expansion function 1/Jj(x) = Rectj(x).

The zeros of det(~) are computed by using Muller's metbod (Muller, 1956; Frank,

1958) for the iterative determination of a complex zero. The number and location of

the zeros is frequency dependent. We have computed the zeros of det(~) that

correspond to some specific surface-wave modes. It appears that there is a tendency

for the diagonal elements of the matrix ~ in (3.55) for the case of TE-modes, to be

more dominant than the diagonal elementsof ~ for TM-modes, especially for higher

values of the contrast e{x)/ f.l - 1 and for values of /Çn relatively close to

k1 = w( E1p.1)112. In these cases, the TE-system of equations is better conditioned

than the TM-system, and hence the results for TE-modes will be more accurate than

those for TM-modes.

In the subsequent tables and figures numerical results are presented for various

waveguide configurations with symmetrie profiles of the relative permittivity

Er = E/ EO and the relativa permeability p,/ p,0. The resulting symmetry of the fields has been used in the computations in order to reduce the integration interval in (3.52) and

(3.53) to one half of the slab (De Ruiter, 1980).

First we have obtained results for the propagation coefficients /Çn of a step-index

planar waveguide. In this case there exists an analytica! expression for the eigenvalue

equation to be satisfied by the propagation coefficients and the field distributions

(Unger, 1977, pp. 93 - 100). To illustra.te the accura.cy of the present implementation

-42-

of the integral-eqna.tion method, we have listed in Table lil the valnes of the

norma.lised propaga.tion coef:ficients ".n/k0, with k0=w(E0p0)112, for a planar

waveguide with Er,2 = 1.01, Er,1 = 1, ~'r,2 = ~'r, 1 = 1 , for some modes. They have been obtained by the integral-equa.tion metbod with point-matching using 8 and 16

eqnally spaeed matching points in one half of the slab (a= d/2).

For a configuration with lossless media, the propagation coef:ficients i'i.n are real. The

lossless confignration can he considered as the limiting case of a corresponding lossy

confignration for vanishingly smalllosses. The modes are numbered in ascending order

of their cut-off freqnencies; the cut-off frequency of a specific mode is the frequency

below which the mode is non-existent.

In Fig. 3.8, the electtic field distribution of the TE5--mode at k0a=l.101xl02

(i'i.~/ko=l.0019914) is shown. The solid curve is obtained from the analytical

Table III. Some valnes of ".n/k0 for the step-index, Er=l.Ol symmetrical slab

wavegnide, obtained by the integral-equa.tion metbod and point-matching with 8

and 16 matching points; values from the analytical expression for comparison.

ko& mode /Çn/ko mode /Çn/ko

int.eq.J=8 !nt.eq.J=16 analy\ical int.eq.J=8 int.eq.J=16 analytical

3.142 TE0 1.0004370 1.0004369 1.0004369 TM0 1.0004298 1.0004297 1.0004297

1.101•102 TE0 1.0048987 1.0049014 1.0049025 ™o 1.0048984 1.0049012 1.0049023

1.855·101 TE1 1.0003129 1.0003146 1.0003152 ™1 1.0003088 1.0003106 1.0003112

1.101•102 TE1 1.0046327 1.0046432 1.0046477 TM1 1.0046314 1.0046425 1.0046472

8.168·101 TE5 1.0000512 1.0001317 1.0001614 ™s 1.0000551 1.0001297 1.0001602

1.101•102 TE5 1.0018485 1.0019487 1.0019914 ™s 1.0018388 1.0019440 1.0019884

-43-

eigenvalue equation and the pertaining analytica! expressions for the field

distribution; the field valnes at the matching points obtained by the integral-€quation

metbod with point-matching are indicated by " (8 matching points) and o (16 ·

matching points).

In Table IV, results are presented for a much larger contrast between the waveguide

and its surroundings, viz. e .2 = 2.25, f 1 = 1, # 2 = # 1 = 1. In general, the r, r, r, r, results obtained from the integral-€quation metbod using point-matching are in good

agreement with the results obtained from the analytica! eigenvalue equation.

Furthermore, the results pertaining to TE-modes turn out to be more accurate than

those pertaining to TM-modes. This better accuracy becomes more pronounced for

larger valnes of the contrast between the waveguide and its surroundings. This is

probably due to the TE-system of equations being better conditioned than the

TM-system, as observed previously.

Fig. 3.8. Electric field distribution ey(x) of the TE5-mode at k0a 1.101>~/ko = 1.0019914) obtained from the analytica! eigenvalue equation (solid curve) and with the integral-€quation metbod (x: 8 matching points, o: 16

matching points). The electric field distri bution is normalised such that its

maximum value is unity.

-44-

The same computational scheme has been applied to a plana.r waveguide with a

quadratic pennittivity profile, for which Er 2(x)=er max(1-2~x2/a2) when

' . -a < x < a. The value of e max is 1.01, while er 1 = 1, P.r 2 = P.r 1 = 1. Two values r, ' ' ' of ~ have been taken, viz. ~ = 2.475d0-3 and ~ = 4.950x10-3 (Fig. 3.9). Some

Table IV. Some valnes of Kn/ko for the step-index, er = 2.25 symmetrical slab

waveguide, obtained by the integral-eqnation method and point-matching with 8

and 16 matching points; valnes from the analytica! expression for comparison.

The question mark indicates tha.t the pertaining zero was not found numerically.

toa mode "nl"'o mode "niko

int.eq.J=8 int.eq.J=l6 analytica! int.eq.J=S int.eq.J=16 analytica!

2.811·10-1 TE0 1.01)323 1.05323 1.05323 ™o 1.01243 1.01243 1.01243

9.848 TE0 1.49255 1.49277 1.49286 TM0 1.4896'1 1.49148 1.49215

1.686 TE1 1.03836 1.03851 1.03864 ™1 1.01086 1.01118 1.01128

9.848 TE1 1.41000 1.47089 1.47121 ™1 1.45860 1.46575 1.46841

8.061 TE5 1.00693 1.01642 1.09492 ™s ? 1.00361 1.06482

9.848 TEs 1.20943 1.21975 1.22413 ™5 1.12894 1.17004 1.19876

Fig. 3.9. Permittivity profiles of:

---a step-index waveguide, er 2=1.01; '

--- a qnadra.t.ic-index wa.veguide, er niax=l.Ol, ~=2.475>

-45-

valnes of the normalised propagation coefficients, obtained with 8 and 16 matching

points in one half of the slab, are listed in Tables V and Vl. For the various planar

waveguides we consider the differences of the values of the propagation coefficients

obtained with 8 and 16 matching points. Then it appears that for the step-index

waveguide and for the quadratic-index waveguide these differences are of the same

order of magnitude. Hence, the differences between the computed and exact values of

the propagation coefficients of the quadratic-index waveguide are likely to be also of

the same order of magnitude as those for the step-index planar waveguide. In Fig.

3.10 the propagation coefficients of the TE0- and TEemodes for the three

permittivity profiles of Fig. 3.9 are plotted as functions of k0a. In Fig. 3.11, the

electric field component of the TEemode in the waveguide is shown for these profiles

at two different values of k0a.

Table V. Some values of "niko for a graded-index symmetrical slab waveguide

having a quadratric permittivity profile with t ax=l.Ol, A=2.475x10-3, r,m obtained by the integral-equation metbod using point-matching with J=8 and

J=16 matching points.

k0a mode "'niko mode "niko

J=8 J = 16 J=8 J = 16

6.288>c102 TE1 1.0039926 1.0040171 ™1 1.0039914 1.0040166

-46-

Ta.ble VI. Some va.lues of r;,n/ko for a graded-index symmetrica.l slab waveguide

having a quadratic permittivity profile with fr max = 1.01, t::. = 4.950>

I to >-

"' ®

-41-

,1.0

""' G>

Fig. 3.11. Lorentz-normalised electric field component ey(x) of the TEemode in

the slab waveguide (a) at k0a = 2.620xi01 (near cut-ff) and (b) at k0a

= 1.101x102 (far from cut-ff), for the step-index profile with Er 2 = 1.01 ( );

' fora quadratic permittivity profile, E ax = 1.01, D. = 2.475xl0-3 (-- ); r,m fora quadratic permittivity profile, fr max = 1.01, D. = 4.950x10-3 (-·-·-).

'

The integral-equatien metbod with point-matching bas also been applied to a

strongly lossy step-index planar waveguide, for which E 2 = 2.25-2.25j, E 1 = 1, r, r, #r,2 = #r,l = 1. Valnes for the propagation coefficients of the TE0-, TE1- and TE2-modes are listed in Table VII. In Fig. 3.12 the electric field distribution of the

TErmode at k0a = 9.848 as obtained with the integral-equatien metbod by

point-matching with 8 and 16 matching points, is compared with the electric field

distribution that has been obtained from the solution of the analytica} eigenvalue

equation. For lossy structures, too, a good agreement is observed between the valnes '

obtained from the analytica} expressions and those computed by the metbod of

moments, both for the propagation coefficient and for the field distribution.

We now brie:fly discuss the behaviour of the propagation coefficient and of the field

distribution of a surface-wave mode as a function of frequency in general. At a very

low frequency, only one TE-surface-wave mode (the TE0-mode) and one TM-

surface-wave mode (the TM0-mode) are present in a planar waveguide; the

frequency is below the cut-ff frequency of the other surface-wave modes. At the

k0a

-48-

Table VII. Values of the propagation coefficients "'n/k0 of the TE0-, TEe and

TE2-modes in a lossy step-index planar waveguide with fr 2=2.25-2.25j, '

~'r,2=ttr,l'=l, obtained by the integral-equation metbod using point-matching with J=16 matching points. Some values resulting from the

analytica! eigenvalue equation are given for comparison. Dashes indicate that the

· pertaining root of the eigenvalue equation is absent, i.e., the conesponding mode

is below cut-off.

,.niko TEo K.n/ko TEl ,.n/ko TE2

J=16 analytica! J=16 analytica! J=l6 analy1ical

1.048x10-1 9.82465•10-1 9.82465-IO-l

-3.14536·10-2j -3.14541•10-2j

2.096x10-1 9.50043·10-1 9.50057xl0-1

-1.28201•10-1j -1.28291d0-1j

2.515•10-1 9.43807•10-1

-1.82669•10-1j

4.192d0-1 1.00152 1.00153

--4.09812•10-1j --4.09782•10-lj

8.383-Io-1 1.28163

-6.48624•10-1j

1.061 6.24206•10-3 6.59612•10-3

-7.59690·10-1j -7.59000•10-lj

1.258 5.37179·10-1 5.37957•10-1

-7.31769•10-lj -7.31607•10-lj

1.670 1.550100 9.80910•10-l

-6.98459•10-1j -7.63830•10-lj

1.677 1.39102•10-2 1.42046•10-2

-1.39212 j -1.39097 j

2.096 1.54890 1.54887 1.22147 1.22174 5.41264•10-1 5.41954•10-l

--6.98394•10-lj -6.98459d0-lj -7.59070·10-lj -7.593llxl0-1j -1.02675xl0-1j -1.02668 j

3.144 1.59809 1.59810 1.44182 1.15639 1.15699

-6.93999d0-1J -6.94106•10-1J -7.32973oi10-1J -8.23075•10-1J -8.23970•10-1j

9.851 1.64182 1.64196 1.62312 1.59168 1.59306

-8.84585x10-1j -8.84730•10-lj -6.90547•10-1j -7.00866•10-1j -7.02160•10-lj

-49-

Fig. 3.12. Electric field distribution e/x) of the TEemode at k0a=9.848 in a

lossy step--index planar waveguide with er,2=2.25-2.25j, er,l=l, f'r,2=t.tr,l=l;

real part (-) a.nd ima.ginary part (-- -) of the exact field distribution deter-

mined from the analytica! eigenvalue equa.tion ("~/ko=l.49295-7.140llx10-1j);

x indicates the valnes from the integral-equa.tion method with 8 matching points

("~/k0=L47577-7.35179xl0-1j); o indicates the valnes from the integral-equation method with 16 matching points

(K~/ko=l.48875-7.37490xl0-1j). Normalisation is such that ey(O)=l.

cut-off frequency of a. surface-wave mode, the corresponding surface-wave pole is

located on the branch cut Im(kT,l) = Im(k~-k~)1/2 = 0, with k1= w(e1t.t1)112. With

increasing frequency the pole subsequently enters the Riemann sheet on which

Im(kT 1) < 0. Consequently, the modal field ha.s an exponential decay at infinity in , the transverse plane. At very high frequencies the modal fields have the tendency to

concentrate in those parts of the wa.veguide where the refractive index ( ert.tr)1/ 2 is

maximaL Then the value of "n approaches ko("rf'r)!~· The root loci of the TE0-, TEe and TE2-surfa.ce-wave poles for a lossy step-index planar waveguide, as shown

-50-

in Fig. 3.13, exhibit the outlined behaviour. It can be proved that these root loci lie in

a restricted part of the complex kz -plane only {De Ruiter, 1981).

The average computing times for finding the propagation coefficient and the field

distribution of a surface-wave mode with the integral-equation method combined

with the point-matching technique were 10 s for 8 matching points and 35 s for 16

matching points in one half of the slab. The computer programme was written in

PL-I and run on an IBM 370/158 computer.

In solving the integral equations by the metbod of moments, we thus !ar used the

simplest types of weighting and expansion functions, viz. delta functions and rectangle

functions, respectively. In order to investigate the effect of a different choice for the

:p

1.5 f 1.0

branch cuts 0.5

-1.5 0

-0.5

-1.0

-1.5

Fig. 3.13. Root loci of the surface-wave poles corresponding to the TE0-, TEe,

and TE2-modes in a Iossy step--index planar wa.veguide with \ 2=2.25-2.25j,

embedded in vacuum. The arrows along the curves indica.te the direction of

change at increasing frequency.

-51-

weighting and expansion functions, we now take '1/J}x)=cpix)=Trj{x), in which the

triangle function Trj(x) is defined by (Fig. 3.14)

j

(x-xj-l)/.1

Trj(x) = {xj+Cx)/.1

0

when x E dj-1

when x E dj

when x t dj-1 U dj

when x E d1

when x t d1,

' J=2, ... ,J-1,

(3.61)

here, the interval [-a,a] has been divided into an even number of subintervals

dj=[xj,xj+l], j=1,2, ... ,J-1, of equallengths .1. With this choice for the expansion and

~~~~ ~ lflj!'ij'l D j11.J

Ok---------------~--~-----------------

Fig. 3.14. The triangle functions Trj(x), used as expansion and weighting

functions.

-52-

weighting functions, the integra.tions by pa.rts in (3.57) a.nd (3.58) can be ca.rried out,

a.nd the resulting integrals ca.n be eva.lua.ted a.nalytically.

In the implementation of the metbod we have not used a possible symmetry of the

permittivity profile to reduce the integrations to one half of the cross-section, a.s ha.s

been done in the computations ca.rried out with the point-matching technique.

We have computed the propaga.tion coefficients a.nd the field distributions of some

TE-surface-wa.ve modes. It is understood tha.t p. = P.p hence !ln Q. in {3.49)-{3.51). Then (3.49), together with (3.52) and (3.35), provides a. homogeneons integral

equation for the field component e1

in the slab. Next, by discretisa.tion of (3.49) the

field coefficients e. a.re fonnd by solving a. system of J homogeneons algebraic J,Y

equations. Subseqnently, the valnes of hJ. and h. are determined by mea.ns of the ,X J1Z

relations that are obta.ined from the discretised versions of (3.35) and (3.50)-{3.52).

This procedure has been used. in the computations with tria.ngle expansion a.nd

weighting functions. Some results obtained with this procedure are listed in Tables

VIII - XII. For comparison, the results obta.ined with the point-matching technique

(PM) a.re listed as well. For the point-matching results the integer J indicates the

number of exparision fnnctions employed if the integration had been ca.rried out over

the entire cross-section without using the symmetry of the permittivity profile. Thus

J 2M-1, where M is the number of expansion functions a.nd matching points in one

half of the slab.

In Table VIII, some results a.re listed for the propagation coefficients of the TE0-,

TEe and TE5-modes in a step-index planar waveguide with tr 2 = 1.01, tr 1 = 1, I I P.r 2 = P.r 1 = 1. In Table IX, some results are listed for the propagation coefficients of

' ' the TE0-, TEe, and TE5-modes in a step-index planar waveguide with a higher

contrast, viz. tr,2 = 2.25, tr,l 1, P.r,2 = P.r,1 = 1. In Tables X a.nd XI, some results

-53-

are listed for the two quadratic-index planar waveguides with permittivity profiles as

shown in Fig. 3.9. Finally, in Table XII a comparison is made of the results obtained

with triangle expansion and weighting functions and with the point-matching

technique, for the propagation coefficient of the TErmode in a lossy step-index

planar waveguide in which f 2 = 2.25-2.25j. In Fig. 3.15, the analytically determîned r, exact field dîstrîbution e

1 of the TE4-mode is plotted, tagether witb the field valnes

e. at the posiiions x. obtained from the integral--equation metbod with trîangle 1Y J

·expansion and weightîng functions ( cf. Fig. 3.12).

From the results in Tables VIII - XII it appears tbat in general the moment metbod

using triangle expansion and weighting functions is superior to the point-matching

technique, in the sense that fewer expansion functions are needed to achieve a certain

Table VIII. Valnes of the propagation coefficients "'n/k0 in a step-index planar

waveguide with fr,2 = 1.01, fr,1 = 1, J.l.r,2 = J.l.r,l = 1, obtained wîth trîangle expansion and weighting functions, as compared with results from tbe

point-matching technique (PM) and with results from the analytica! expression.

The number of expansion functions used is J. Question marks indicate that the

pertaining zero was not found numerîcally.

k0a mode "niko

J=3 J=5 J=9 PM J=15 PM J=31 analytica!

3.142 TE0 1.0004383 . 1.0004370 1.0004370 1.0004370 1.0004369 1.0004369

1.010-102 TE0 1.0049014 1.0049024 1.0048987 1.0049014 1.0049025

J=4 J=7 J=13 PMJ=15 PMJ=31 analytical

1.885•101 TE1 1.0003083 1.0003148 1.0003152 1.0003129 1.0003146 1.0003152

1.101•102 TE1 1.0045998 1.0046429 1.0046474 1.0046327 1.0046432 1.0046477

J=9 J=17 J=33 PM J=lS PM J=31 analytical

8.168•101 TE5 1.0001565 l.OOOHI14 1.0000512 1.0001317 1.0001614

1.101xl02 TE5 1.0017825 1.0019817 1.0019911 1.0018485 1.0019487 1.0019914

-54-

Table IX. Valnes of the propagation coeffi.cients x,nfk0 in a step-index planar

waveguide with er,2 = 2.25, er,1 = 1, Pr,2 = Pr,l = 1, obtained with triangle expansion and weighting functions, as compared with results from the

point-matching technique (PM) and with results from the analytica! expression.

The number of expansion functions used is J. Question marks indicate that the

pertaining zero was not found numerically.

koa. mode t>n/ko

J=3 J=5 J=9 PMJ=15 PM J=31 analytica!

2.811•10-l TE0 1.05340 1.05324 1.05325 1.05323 1.05323 1.05323

9.848 TE0 1.49276 1.49285 1.49255 1.49277 1.49286

J=4 J=7 J=l3 PM J=l5 PM J=31 &nalytical

1.686 TE1 1.03783 1.03866 1.03862 1.03836 1.03857 1.03864

9.848 TE1 1.46717 1.47086 1.47124 1.47000 1.47089 1.47127

J=8 J=15 J=29 PM J=15 PM J=31 &nalytical

8.061 TE5 1.05882 1.09336 1.09487 1.00693 1.01642 1.09492

9.848 TE5 1.18608 1.22231 1.22402 1.20943 1.21975 1.22413

Table X. Valnes of the propagation coeffi.cients x,nfk0 of the TEemode in a

quadratic-index planar waveguide with e max = 1.01, !:::. = 2.475,.10-3, e 1 = 1, ~ ~ !Lr 2 = 1Lr 1 1, obtained with triangle expansion and weighting functions, as

' ' compared with results from the point-matching technique (PM). The number of

expansion functions used is J.

koa. Kn/ko TE1

J=5 J=9 J=17 J;33 PM J=15 PM J=31

2.620•101 1.0006930 1.0007656 1.0007809 1.0007846 1.0007834 1.0007850

1.101•102 1.0039028 1.0040196 1.0040260 1.0040264 1.0039926 1.0040171

-55-

Table XI. Values of the propagation coefficients K-n/k0 of the TEemode in a

quadratic-index planar waveguide with E ax = 1.01,!::.. = 4.950xl0-3, E 1 = 1, r,m r,

/.1, 2 = /.1, 1 = 1, obtained with triangle expansion and weighting functions, as r, r,

compared with results from the point-matching technique (PM). The number of

expansion functions used is J.

k0a K-n/ko TE1

J=9 J=17 PM J=15 PM J=31

2.620x10 1 1.0001274 1.0001418 1.0001557 1.0001553

l.lOlxlO 2 1.0036154 1.0036302 1.0035841 1.0036180

Table XII. Values of the propagation coefficient K.n/ko of the TErmode at

k0a = 9.848 in a lossy step-index planar waveguide with \ 2 = 2.25-2.25j,

Er 1 = 1, /.l,r 2 = /.l,r 1 = 1, obtained with triangle expansion and weighting , , , functions, as compared with results from the point-matching technique (PM) and

with results from the analytica! expression. The number of expansion functions

used is J.

J=7 J=13 J=25 PM J=15 PM J=31 analytica!

9.848 1.45171 1.49044 1.49281 1.47577 1.48875 1.49295

--û.74263j --û.73936j --û.74095j --û.73518j --û.73749j --û.74101j

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Fig. 3.15. Electtic field distribution e1

(x) of the TErmode at k0a=9.848 in a

lossy step-index planar waveguide with er,2=2.25-2.25j, er,1=1, ~'r,2=t-tr, 1 =1; real part and imaginary part (- - -) of the exact field distribution

determined from the analytica! eigenvalue equation; the valnes from the

integral-equation metbod with triangle expansion and weighting functions are

indica.ted by o, x, +, corresponding to the use of 7, 13, 25 expansjon functions,

respectively. Normal~sa.tion is such tha.t ey

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3.4.2. The transfer-matrix formalism

In this snbsection we apply the transfer-matrix formalism discnssed in Snbsection

3 .. 3.2 to the compntation of surface-wave modes in a multi-step-index planar wave-

guide. The configuration is shown in Fig. 3.16. The waveguide consistsof N-2 homo-

geneons layers {dp; p=2, ... ,N-1} in between N-1 planes {x=xp; p=l, ... ,N-1},

embedded in two homogeneons media present in the semi-infinite domains d1 :

-oo

-58-

matrices: !E and ~~ for TE-fields, and !M and ~~ for TM-fields. We shall restriet ourselves to the TE-case; the TM-case follows by malring the appropriate changes.

For TE-fields, the field components ey and hz are continuous upon crossing the

interfaces x= xp. Hence we have the field matrix

(3.62)

where the subscriptprefers to the layer in which x is located. From (3.30) and (3.31)

we derive the system of equations that has to be satisfied by ey and hz inside the

homogeneous, souree-free layer dp as

(3.63)

(3.64)

in which

(3.65)

and in which the subscript n referring to the n-th surface-wave mode has been

suppressed (~~: now denotes the propagation coefficient of some surface-wave mode).

The transfer matrix ~~(x,x') is easily constructed as

T (x x')= • x, ,p E [cos(kTp(x-x')) -(j/YEP)sin(kT (x-x'))l

-p , E , - -jY x, P sin(kT,p(x-x')) cos(kT,p(x-x'))

(3.66)

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with

(3.67)

where (3.44) has been taken into account.

The solutions of (3.63) and (3.64) in the lower and upper half-spaces can be written

as

(3.68)

and

(3.69)

respectively. By means of (3.68) and the transfer matrices of the intermediate layers,

the fields at