Transmission, reflection and radiation at junction planes of different open waveguides

different open waveguides

Citation 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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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

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

Aan mijn ouders, aan oma

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.

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

4.2. 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

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

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

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.

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.

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.

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

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

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

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

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.

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

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

(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

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.

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)

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)

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

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).

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

(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.

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 z

1 < 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.,

'=

<{rT), 11-= Jl(!,T), where

'T

+

zi .

- -1l

Fig. 3.1. Uniform section of an open waveguide.

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 <₁ and IJr The domain outside

(}!iJ is denoted by !iJ•. In !i), the values of< and/or p differ from !heir values in !iJ'.

Dependent on the specification of < and p .. functions of !T E !i!, several types of waveguides are distinguished: ste!bindex, where ' and p are constanis in !iJ :

multi-step=index, where < and IJ are piecewise constant fanelions of !T in

!iJ=

U~=

2

fiJn; and graded-index, where 'and IJ are continuous functions of !Tin !iJ

(Table II). Some special waveguide shapes aften encountered in practice are: the rotationally symmetrie libre and the planar waveguide for integrated opties (Fig. 3.2).

Table II. Permittivity and permeability distribution in a straight waveguide section.

type of waveguide

step-index

multi-<tep-index

graded-index

permittivity and permeability !:T E !iJ <(;:T) =

'2

~Tl=

1'2

r E !iJ' -T <(!T) = '1 !'(!Tl = ~'1 <(!Tl =

'n'

!TE

!iJnt

<(;:Tl '1

l

n=2, ... ,N !'(rTl

=

~'n• !T E fiJn !'(rTl = ~'1

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).

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

.

.

~ ~

~(!_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,z

₂

)5(z-z

₂

). 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

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)1

1

2 (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 f i s taken from the branch point"= k1 towards infinity, and

-211'{~~.!!~}

denotes the 11_jump11 _{in {iE,:ÊI:E} across the branch cut}

_ff;

_{this jump is defined as the}

__ _,

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<) -~,..._ / 1 I 1 ... f".. I / I 1 ""''Xl (x• '··· · 1 • ~~~ ..,, - 1 I

rx'

rx'• r;;\ ·Re<kzl ' ,..,.., '--.l " ' ' f ' 1(31(21<1: \ I \ I \ I \ I \ _{\ ,} I

'

'

_{'... , /} ~

/

... f , / ... _____ L---''"" a+

Fig. 3.3. Complex kz -plane with branch points kz

=

:1:kl' branch cuts

.#

and

.:1ï

(Im(k~

-

k~)

1

/

2 = 0) and surface-wave poles

{~~;n}

and

{-~~:n},

which are either

{~~;~}

and

{-~~;~},

or

{~~;~}

and

{-,.;~}.

Also shown is the contour for the application of Cauchy's integral formula.

difference of the values of

{~E,H:E}

at the branch cut on the side where

Re(k~- k~)

1

_/

2

_>

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)

(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.

(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

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

/

2

1

~

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)

(3.28)

(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;

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(-jK0z) witb propagation coefficient K0

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)

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)

(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.

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

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 {e

1,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;2Up+l -Up •u' Upl

-®

p+2 p+1 p p~1 interface layer Up p Up1 -p-1 Up~2U q + 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 u

0 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 u

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(u_{1) 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 u

0, 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 u

0. 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:

- 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.

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 8

1

=

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 {e

1,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.

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').

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 e

1 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

e

1 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 h

1 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

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' ,~~:)

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

₁

x•=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 .

t 1

Ijl·

J

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( E

1p.1)1

1

2. 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