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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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https://doi.org/10.6100/IR315847https://doi.org/10.6100/IR315847https://research.tue.nl/en/publications/transmission-reflection-and-radiation-at-junction-planes-of-different-open-waveguides(fe5e9380-efe0-45a6-92c8-416ab47024c0).html
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TRANSMISSION, REFLECI'ION AND RADlATION AT JUNCI'ION PLANES
OF DIFFERENT OPEN WAVEGUlDES
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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
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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.
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Aan mijn ouders,
aan oma
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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.
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-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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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.
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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
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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.
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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)
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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.
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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)
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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)
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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
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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
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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