Propagation, scattering, and dissipation of electromagnetic waves

ElEctromagnEtic wavEs sEriEs 36

Propagation,scattering and dissipationof electromagnetic waves

A. S. Ilyinsky, G. Ya. Slepyan and A. Ya. Slepyan

Peter Peregrinus Ltd. on behalf of the Institution of Electrical Engineers

IEE ELECTROMAGNETIC WAVES SERIES 36

Series Editors: Professor P. J. B. ClarricoatsProfessor Y. Rahmat-SamiiProfessor J. R. Wait

Propagation, scatteringand dissipation ofelectromagnetic waves

Other volumes in this series:

Volume 1 Geometrical theory of diffraction for electromagnetic wavesG. L. James

Volume 2 Electromagnetic waves and curved structures L. Lewin,D. C. Chang and E. F. Kuester

Volume 3 Microwave homodyne systems R. J. KingVolume 4 Radio direction-finding P. J. D. GethingVolume 5 ELF communications antennas M. L. BurrowsVolume 6 Waveguide tapers, transitions and couplers F. Sporleder and

H. G. UngerVolume 7 Reflector antenna analysis and design P. J. WoodVolume 8 Effects of the troposphere on radio communications

M. P. M. HallVolume 9 Schumann resonances in the earth-ionosphere cavity

P. V. Bliokh, A. P. Nikolaenko and Y. F. FlippovVolume 10 Aperture antennas and diffraction theory E. V JullVolume 11 Adaptive array principles J. E. HudsonVolume 12 Microstrip antenna theory and design J. R. James, P. S. Hall

and C. WoodVolume 13 Energy in electromagnetism H. G. BookerVolume 14 Leaky feeders and subsurface radio communications

P. DelogneVolume 15 The handbook of antenna design, Volume 1 A. W. Rudge,

K. Milne, A. D. Olver, P. Knight (Editors)Volume 16 The handbook of antenna design, Volume 2 A. W. Rudge,

K. Milne, A. D. Olver, P. Knight (Editors)Volume 17 Surveillance radar performance prediction P. RohanVolume 18 Corrugated horns for microwave antennas P. J. B. Clarricoats

and A. D. OlverVolume 19 Microwave antenna theory and design S. Silver (Editor)Volume 20 Advances in radar techniques J. Clarke (Editor)Volume 21 Waveguide handbook N. MarcuvitzVolume 22 Target adaptive matched illumination radar D. T. GjessingVolume 23 Ferrites at microwave frequencies A. J. Baden FullerVolume 24 Propagation of short radio waves D. E. Kerr (Editor)Volume 25 Principles of microwave circuits C. G. Montgomery,

R. H. Dicke, E. M. Purcell (Editors)Volume 26 Spherical near-field antenna measurements J. E. Hansen

(Editor)Volume 27 Electromagnetic radiation from cylindrical structures

J. R. WaitVolume 28 Handbook of microstrip antennas J. R. James and P. S. Hall

(Editors)Volume 29 Satellite-to-ground radiowave propagation J. E. AllnuttVolume 30 Radiowave propagation M. P M. Hall and L. W Barclay

(Editors)Volume 31 Ionospheric radio K. DaviesVolume 32 Electromagnetic waveguides: theory and application S. F. MahmoudVolume 33 Radio direction finding and superresolution P. J. D. GethingVolume 34 Electrodynamic theory of superconductors S.-A. ZhouVolume 35 VHF and UHF antennas R. A. Burberry

Propagation, scatteringand dissipation ofelectromagnetic waves

A. S. Ilyinsky,G. Ya. Slepyanand A. Ya. Slepyan

Peter Peregrinus Ltd. on behalf of the Institution of Electrical Engineers

Published by: Peter Peregrinus Ltd., on behalf of the Institution ofElectrical Engineers, London, United Kingdom

© 1993: Peter Peregrinus Ltd.

Apart from any fair dealing for the purposes of research or private study,or criticism or review, as permitted under the Copyright, Designs andPatents Act, 1988, this publication may be reproduced, stored ortransmitted, in any forms or by any means, only with the prior permissionin writing of the publishers, or in the case of reprographic reproduction inaccordance with the terms of licences issued by the Copyright LicensingAgency. Inquiries concerning reproduction outside those terms should besent to the publishers at the undermentioned address:

Peter Peregrinus Ltd.,The Institution of Electrical Engineers,Michael Faraday House,Six Hills Way, Stevenage,Herts. SG1 2AY, United Kingdom

While the authors and the publishers believe that the information andguidance given in this work is correct, all parties must rely upon their ownskill and judgment when making use of it. Neither the authors nor thepublishers assume any liability to anyone for any loss or damage causedby any error or omission in the work, whether such error or omission isthe result of negligence or any other cause. Any and all such liability isdisclaimed.

The moral right of the authors to be identified as authors of this work hasbeen asserted by them in accordance with the Copyright, Designs andPatents Act 1988.

British Library Cataloguing in Publication Data

A CIP catalogue record for this bookis available from the British Library

ISBN 0 86341 283 1

Printed in England by Antony Rowe Ltd., Wiltshire

Contents

Page

Preface viii

List of notations x

1 Introduction 11.1 Loss reduction in microwave waveguides and resonators 11.2 Maxwell's equations; constitutive equations; boundary conditions 41.3 Solution techniques for mathematical problems of electromagnetics 71.4 Accuracy control and computational instabilities 14

2 Surface-impedance technique for the study of dissipation processesin bodies with finite conductivity 182.1 The Leontovich impedance boundary condition 182.2 The surface impedance of normal metals for the anomalous skin effect 202.3 Surface impedance of superconductors 242.4 Surface impedance modification for structures with edges 262.5 The edge condition for an impedance halfplane located at media interface 32

3 Normal modes in waveguides with losses 373.1 Excitation of waveguides without losses 373.2 Excitation of waveguides with losses in the walls 413.3 Eigenmodes in waveguides; dispersion characteristics 433.4 Associated waves 503.5 Types of dispersion characteristics; a concept of anomalous dispersion;

complex waves in lossless waveguides 533.6 Excitation of TM modes in a parallel-plate impedance waveguide 553.7 Attenuation coefficients of eigenmodes 593.8 Attenuation in a generalised microstrip line; model of the infinitely thin

strip 653.9 Attenuation in a microstrip line; model of a strip of finite thickness 703.10 Attenuation in a microstrip line; numerical results 75

vi Contents

4 Normal oscillations in resonators with losses 834.1 Expansion of eigenoscillations of a resonator with losses in the walls in

terms of resonant modes of an identical lossless resonator 834.2 Resonance frequencies and (^-factors of eigenoscillations 864.3 Eigenoscillations and free oscillations in a resonator with a magneto-

dielectric absorbing body 904.4 Q;factor of a cylindrical cavity 974.5 (^-factor of spherical and conical cavities 994.6 Galerkin's method for calculation of a complex-shaped cavity resonator

in the form of a body of revolution 1044.7 Cylindrical resonator with dielectric slabs 1114.8 Q;factor of a cylindrical resonator with a coaxial insert 114

5 Electromagnetic-wave diffraction by finitely conducting comb-shapedstructures 1235.1 Diffraction of a plane wave by an array of impedance halfplanes:

H-polarisation 1235.2 Diffraction of a plane wave by an array of impedance halfplanes:

E-polarisation 1295.3 Diffraction by a finitely conducting comb-shaped structure 1355.4 Perturbation technique 1405.5 Effect of abnormally small absorption in periodic structures 1415.6 Absorption in inclined comb-shaped structures and echelettes 1495.7 Diffraction by a complex-shaped periodic structure: integral equation

method 1535.8 Diffraction by a complex-shaped periodic structure: Galerkin's incomplete

method with semi-inversion 166

6 Dissipation in comb-shaped structures in inhomogeneous andanisotropic media 1796.1 Diffraction by finitely conducting comb-shaped structure with a layered

dielectric filling; resonance absorption 1806.2 Wave diffraction by comb-shaped structures in gyrotropic media 1856.3 Nonreciprocal resonance effects 190

7 Eigenmodes in corrugated waveguides and resonators with finitelyconducting walls 1967.1 Eigenmodes in periodic structures f 1967.2 Equivalent boundary conditions for finitely conducting comb-shaped

structures 2017.3 Surface waves in finitely conducting comb-shaped structures 2067.4 TM modes in plane comb-shaped waveguides 2077.5 Attenuation in waveguides with azimuthal corrugation 2127.6 Projection method for calculation of propagation and attenuation

coefficients of corrugated waveguides with arbitrary shapes of cross-sectionand corrugation 216

7.7 Propagation characteristics of circular corrugated waveguides 2237.8 Attenuation characteristics of circular corrugated waveguides 2277.9 Millimetre-waveband high-quality corrugated resonators 2317.10 Radiation from a corrugated horn 240

References 248

Appendix 1 Shooting method and its modifications 257

Contents vii

Appendix 2 Expressions for current-density distributions in a microstrip line

with a strip of finite thickness 268

Appendix 3 General formulae for the coefficients a^ , P^, y^, <5j£J, 272

Index 275

Preface

Textbooks and monographs on microwave theory contain very little materialon methods of calculating microwave absorption. However, problems do existin this field which is confirmed by the vast amount of journal publications onthe issue, especially during the last decade. There are some mutually contradic-tory views concerning a number of aspects and arguments about resolvedquestions regularly arise.

This is largely due to the absence of a monograph dealing with the problemsof electromagnetic wave absorption in waveguides, resonators and periodicstructures. This book attempts to fill the gap.

The introduction reviews possible ways of reducing losses in waveguides andresonators, it formulates boundary-value problems of electromagnetics for caseswhen losses are considered, and gives analytical and numerical methods ofanalysis. Chapter 2 deals with the formulation of impedance boundary conditionsfor a high skin effect in various conducting media.

Chapter 3 presents the theory of regular waveguides with finitely conductingwalls, and Chapter 4 the theory of cavities when the surfaces and the mediuminside are absorbing. These chapters present the derivation of general formulaefor the attenuation coefficient and (^-factor and consider concrete practicalapplications. The principal errors associated with the solution of these problemsare also discussed.

Chapters 5 and 6 deal with wave diffraction by imperfectly conductinggratings in both free space and inhomogeneous anisotropic media. The mainemphasis is on the description of dissipative resonance effects, in particular theeffect of the abnormally low dissipation.

Chapter 7 considers the attenuation of eigenmodes in periodic waveguides. Itpresents results obtained for low-loss flexible corrugated waveguides and high-quality corrugated resonators of the millimetre waveband etc.

A number of theoretical results confirmed by experiment are included.

ix Preface

The book is intended for physicists and engineers performing theoreticalresearch and designing microwave and millimetre-wave devices, and for studentsand postgraduates involved in these issues. It can be used by a design engineerto select a suitable type of a waveguide (resonator) and methods of computingits parameters.

As similar problems arise in acoustics, much of the book's material can bedirectly used for sound waves; this makes it useful for specialists in acoustics.

The authors are grateful to Prof. E.A. Alkhovsky, Prof. A.A. Kuraev, Prof.V.A. Cherepenin, Dr. T.N. Galishnikova, Dr. V.V. Zarubanov andMrs. V.N. Rodionova who made valuable contributions to the research.

The authors greatly appreciate the support and encouragement given by Prof.PJ.B. Clarricoats which has enabled the book to be written.

List of notations

SI units are usedE, H electric and magnetic intensitiesexp (—jcot) time dependenceco cyclic frequency

j = V ( - 1 )k = 2n'll wave numberX wavelength£, \i permittivity and permeability of substancese = e' + ja"tan 3 = £"le'e0, ^0 permittivity and permeability of vacuum<70 static metal conductivity8 = e/e0 relative permittivityWo= (HQISQ)1^2 impedance of free spaceZs surface impedancew0 interior unit normalx,jy, z Gartesian co-ordinates2x? zy? h unit vectors in the x,y and z directions${x,y) longitudinal component of the electric (or magnetic) field in

2-dimensional problemsJm{x)> Nm(x) Bessel's and Neumann's functionsymn nth root of Jm(x)Umn nth root of J'm(x)a* complex conjugate of aI identity operatorA ~l operator inverse to A (A~x A=== I)5mn Kronecker deltaV2 Laplacian

Chapter 1

Introduction

1.1 Loss reduction in microwave waveguides andresonators

An important feature of microwave waveguides and resonators is heat losses.The latter determine, in particular, the transient time, knowledge of which isindispensable for designing digital microwave devices (Kiang, 1991). Waveguideswith low attenuation are required to create highly effective radio-relay, spaceand tropospheric-scatter communication facilities and to design measuringequipment (including that operating at the millimetre waveband and infrared).High-quality resonators are widely used in radio engineering (tunable oscillatorswith high frequency stability), in applied physics (wavemeters, equipment formeasuring dielectric coefficients and permeabilities of substances and for micro-wave heating etc.), in electronics (gyrotrons, orotrons, free electron lasers etc.)and in unique physical experiments (for instance, experiments on detectinggravitational waves).

Therefore, heat-loss control and loss reduction in waveguides and resonators,in particular, are of great importance. Certainly, the most radical way is to usethe superconductivity phenomenon. New possibilities in that direction havearisen through the discovery of high- Tc superconductors (Winters and Rose,1991). Superconducting waveguides and resonators have unique attenuationparameters (so that, for instance, the unloaded (^-factor of a superconductingcavity resonator may reach 108—109). However, the use of the cryogenic tech-nique is sometimes undesirable or even unacceptable. Moreover, in uniquephysical experiments, when an extremely high sensitivity of the measuringapparatus is required, the values of Q actually obtained are often insufficient(Braginsky et aL, 1981). This points to the need for further research in this area.

The use of low-temperature techniques in microwave devices is not confinedto the superconductivity phenomenon. For example, experiments with cooledhigh-Qjfactor ring-dielectric sapphire resonators without superconducting films(Braginsky and Vyatchanin, 1980) are of considerable interest.

Measures of a technological character, such as polishing of inner surfaces anddeposition of special coatings, are also important for loss reduction. We shall nottouch upon these questions here; we merely mention that the possibilities avail-

2 Introduction

able in this direction have been largely exhausted. Therefore, optimal selectionof the waveguide (resonator) type, configuration and working mode is of para-mount importance. It is necessary to emphasise that this formulation of theproblem is by no means an alternative to the low-temperature technique. Onthe contrary, the results obtained in this direction may widen the opportunitiesprovided by cryogenic-engineering facilities.

To solve this problem, adequate methods of loss calculation for waveguidesand resonators are needed. These methods form the main subject of this book,along with physical models, the most important numerical results and compari-son of the latter with experimental data. In these introductory remarks we shallnot try to summarise the contents of the book but merely dwell on the historicalaspects of the development of the subject [see also the review material inpublications by Ilyinsky and Slepyan (1983, 1990)].

Reports about allegedly abnormal low losses in some electrodynamic systemsof a complex configuration have occurred from time to time; these have notbeen confirmed (see below). If we take into account difficulties related both tothe calculation and to accurate measurement of low losses, such a situation isnot surprising. Two directions of research are possible when trying to reducelosses through changing the electrodynamic-system configuration. The first('direct') is to use waveguides and resonators with fundamental modes. Modeselection causes no problems and shape optimisation is used directly to reducethe losses. Rectangular waveguides with rounded corners, double-ridge wave-guides (Kotayama et aL, 1979, Alkhovsky et aL, 1986) can be mentioned asexamples. It is difficult to expect a significant gain in this way, but if the shapeis not too complex even a minor gain (by 25-50%) is valuable. Besides, thequestion is of principal interest.

The use of multimode systems, where effective rarefaction of the eigenmodesspectrum and suppression of lower (fundamental) modes are provided by shapeoptimisation, is another method (let us call it 'indirect'). In this case the workingmode has a high (^-factor and is higher-order with respect to its field structure,being, in fact, fundamental (or one of the fundamental modes).

The use of attenuation abnormality for the TE0/ modes in circular waveguidesis a classical example of this. By taking a large enough waveguide radius it ispossible to obtain the TE01-mode attenuation as low as desired (Okress, 1968).However, though attractive, this result can not really be used: it occurs inessentially multimode regimes. The situation is made worse by the fact that theTEOl- and TMj, modes are degenerate at any value of working frequency. Theoperational TE01 mode will be transformed into parasitic modes at the wave-guide discontinuities (bends, junctions, technological deformations of the cross-section etc.). The attenuation of these parasitic modes is much more significantthan that of the TE01 mode. The transformation effects are essentially enhancedby modes degeneration. As a result, the total losses increase and attenuationreduction may turn out to be illusive.

A similar situation is observed in cylindrical cavity resonators. The TE0£n

modes have the highest Q but they are degenerate with the TMllM modes forany relationship between the cavity dimensions, and can only be used in anessentially multimode regime. It leads to difficulties in excitation and tuning ofthe resonator.

A problem of parasitic-mode suppression therefore arises. In other words, it

Introduction 3

is a problem of rarefaction of the eigenmodes spectrum of a circular waveguide.One of the ways to solve it is to use spaced-disc or helix waveguides. Spaced-disc waveguides (circular waveguides with narrow ring slits) are open systemswhere a spectrum rarefaction takes place because of a strong parasitic waveradiation. If the parameters are chosen properly, the losses for the TE01 modeincrease somewhat, compared with a smooth circular waveguide, but consider-able suppression of parasitic modes is thus achieved. However, such waveguidesin low-loss regime are also multimode ones, and are therefore highly sensitiveto the waveguide irregularities.

To remove the degeneration of the TE0£ and T M U modes, a corrugation ofthe internal surface of a circular waveguide can be used. Considering cases whenthe period and depth of corrugations are small compared with the wavelength,Gent (1959) predicted an essential reduction (by an order) of attenuation of theTEOn mode. However, a more careful theoretical analysis (Katsenelenbaum,1959, Nefedov and Sivov, 1977) has not confirmed this highly promising result.According to the rigorous solution obtained by Katsenelenbaum (1959), theTE0n-mode attenuation in a corrugated waveguide grows but not significantly.Nefedov and Sivov (1977) have analysed the mistaken conclusions drawn byGent (1959) in detail. The idea had been forgotten for some time.

Fundamental investigations by Clarricoats and his coworkers (Clarricoats andSaha, 1970, Parini et aL, 1977, Clarricoats et aL, 1975a, 1975b) have revived theidea of using corrugated (comb-shaped) surfaces for loss reduction. But there itwas related to the H E n mode and the period and depth of corrugations weregreat enough—about a quarter of the wavelength. It was shown theoreticallyand experimentally that in such waveguides attenuation of the H E n mode isapproximately equal to that of the TE01 mode in conventional circular wave-guides (~0.01 dB/m at / ~ 3-20 GHz). However, unlike the latter, in corru-gated waveguides the weakly attenuated mode can be fundamental andnondegenerate.

In such a way the effect of abnormally small dissipation of electromagneticwaves in periodical structures was discovered. Further work in this directionwas continued by the authors of this book. The results are presented in Chapters 5and 7. The emphasis was laid on the physical nature of the phenomenon, namelyon the specific mechanism of the decrease in the induced surface currents andalso the loss-calculation methods for electrodynamic systems with corrugated(comb-shaped) surfaces.

This effect has already been implemented in practice. Flexible low-loss corru-gated waveguides based on this effect with various (rectangular, circular, ellipti-cal, double-ridge) cross-sections are manufactured in the UK, USA, France,Russia, Germany, Canada and Japan. These waveguides are convenient for usein both stationary and mobile communication facilities because it is possible tobend them and to reel them on a drum many times. The theory, design,manufacture technology and examples of application of flexible corrugatedwaveguides were described by Alkhovsky et aL (1986). A broadband multi-modecorrugated low-loss waveguide (y"<0.01 dB/m at f~ 75-535 GHz) was usedto measure the parameters of electron cyclotron radiation in the TFTR Tokamak(Carallo et aL, 1990).

High-quality tunable resonators for the millimetre waveband (Luk et aL, 1988,Rodionova and G. Slepyan, 1989, Rodionova et aL, 1990) are another example

4 Introduction

of application of an abnormally small dissipation effect. To reduce the losses inthe walls, the internal surface is partially corrugated. The TMOmn modes of theresonator are operational. To rarefy the eigenmodes spectrum, special filteringslits are provided. The unloaded Qjfactor of 105 is achieved for / ~ 3 2 -53.57 GHz without any cryogenic-engineering means. The results of theoreticaland experimental investigations of such resonators are presented in Section 7.9.

The corrugated surfaces have another important application in antenna engin-eering. Using them as a basis, highly effective feeds for reflector antennas havebeen developed (Clarricoats and Olver, 1984).

Another possible way of reducing the losses in metallic walls is to apply metal-dielectric structures (Kazantsev et al., 1974, Kazantsev and Kraftmakher, 1979).The idea is to introduce a dielectric insert of a special shape which leads to areduction in the currents induced on the metallic surfaces. Although dielectricinserts cause some attenuation, the total losses in a metal—dielectric system maybe lower than those in a similar system having no dielectric filling. A character-istic example is a hollow circular waveguide with a coaxial dielectric bushing(Kazantsev et al., 1974). Another example is a metal-dielectric resonatordescribed in Section 4.7. The physical nature of the loss reduction in this caseis similar to the abnormally small dissipation effect in corrugated surfaces.However, in periodic structures there is also a highly effective mechanism ofparasitic-mode filtration due to transmission and stop bands.

Of other types of highly effective transmission lines, we mention single groove-guides, coupled groove-guides and dielectric ribbon waveguides. According toMeissner (1990), attenuation in a single groove-guide at the band f~40—120 GHz is about 0.1-0.4 dB/m, whereas in a conventional single-mode rec-tangular waveguide it is more than 4 dB/m. A coupled groove-guide has similarcharacteristics: 0.1 dB/m for the even mode and 0.2 dB/m for the odd mode.Optimisation of the configuration (Yeh et aL, 1990) has permitted an attenuationof 0.02 dB/m to be obtained in a dielectric ribbon waveguide at millimetre andsubmillimetre wavebands using a material with tan 3 ~ 10 ~4.

1.2 Maxwell's equations; constitutive equations; boundaryconditions

Problems of calculation of electromagnetic fields in microwave systems are solvedon the basis of macroscopic electrodynamics. The behaviour of macroscopicelectromagnetic phenomena is governed by Maxwell's equations

curl H=d-

curl E = - —dt

(1.1)

where E and H are vectors of electric and magnetic intensities, D and B arevectors of electric and magnetic flux densities, J is the macroscopic-conductioncurrent density in the medium andjext is the current density induced by externalsources.

In the electromagnetic theory, time-harmonic fields like

-E(r,tY E{r,co)l

H(r,(0)j

Introduction 5

(1.2)

are of fundamental importance. Similar relationships can be written for the restof the quantities in eqn. 1.1 (co is a real number called cyclic frequency). Thequantities E(r, co) and H(r, co) are called complex amplitudes of the correspond-ing vectors.

From now on, we shall study electromagnetic fields described by eqn. 1.2 andshall operate with complex amplitudes only, with no special notations for thelatter; the only exception will be the problem of free oscillations in a resonatorwith a magnetodielectric absorbing body (see Section 4.3). Harmonic fieldsexpressed by eqn. 1.2 are good models for the description of steady-state nar-rowband processes and, in fact, they have a much more general meaning. Thefact is that an electromagnetic field with an arbitrary time dependence in alinear system can be expanded in the Fourier series or integral in terms of time-harmonic fields.

Maxwell's equations for complex amplitudes can be written as

curl H = -jcoD + J + Jext )J J \ (1.3)curl E = jcoB J

Within the scope of macroscopic electrodynamics, eqns. 1.3 are completed byconstitutive equations which, for example, can be in the form

D(r)=e(r,co)E(r)

J(r) = tr{r, co) E(r)

(1.4)

The quantities s(r,co), fi(r, co) and <r(r, co) are called the permittivity, per-meability and conductivity of the medium. They are scaiars for isotro-pic mediaand tensors for anisotropie media.

Constitutive equations given by eqns. 1.4 arise in electromagnetic theory asa result of macroscopic averaging, i.e. field averaging over a physically infini-tesimal volume.1 This averaging leads to smoothing of microscopic details of thefield structure: sharp field oscillations are averaged at an atom-molecular scale.As experience shows, these details are completely insignificant for the determi-nation of macroscopic parameters of electromagnetic processes such as powerabsorbed, diffraction patterns, eigenmode spectra etc. But, in this case theproblem is essentially simplified. The parameters of substances c, fi and a areassumed to be specified within the scope of macroscopic electrodynamics. Theproblem of their determination for various media is solved irrespective of thefield-calculation problem. These parameters can be calculated from some struc-tural models of the media or measured in experiments. The physical fuhda-

1 By physically infinitesimal volume we mean a volume with dimensions much larger than atom-molecular dimensions but small compared with the wavelength, body size and other macroscopicquantities (Tamm, 1976).

6 Introduction

mentals of macroscopic electrodynamics and the macroscopic averagingtechnique are described in Tamm's book (1976).

Still more general forms of constitutive equations are possible; for example,nonlocal connections may occur between D and E, B and H, J and E:

D(r) = 8(r,r',co) E{r) dVJ

B(r)= U(r,r',(o)H(r')d3r'

J(r)= \a(r,r',(0)E(r')d3r'

The nonlocality mentioned reflects a phenomenon of spatial wave dispersion(Agranovich and Ginzburg, 1979). Further, we shall not take into accountspatial dispersion in magnetodielectrics. In Section 2.3 we shall consider in detailone of the spatial dispersion effects in metals, i.e. the anomalous skin effect.

The quantities e and fi are real numbers for lossless media. For lossy magneto-dielectrics they are complex: s = s' + je,", [i — \i' + jfi". The form of the constitut-ive equations in this case is the same as for lossless media. Because of this, analysisof electrodynamic systems with lossy magnetodielectrics can be reduced toreplacement of real e and \i by complex ones in the final expressions obtainedfor nonabsorbing structures.

A perfect conductor is characterised by the boundary condition

( M O X £ ) | S = 0 (1.5)

where S is the surface of the conductor and n0 is a unit vector normal to S. Weshall take into account finite conductivity of the metal on the basis of theimpedance boundary conditions. It means that instead of eqn. 1.5 the boundarycondition

noxE= -Zsfro x (»o x H)} ( l - 6 )should be used. Here n0 is the unit inward normal to the metal surface and ^sis the given surface impedance. Chapter 2 deals with the derivation and substan-tiation of the impedance boundary conditions for various cases.

Certainly, a conducting medium could be formally considered as a 'conven-tional' dielectric characterised by the permittivity e = G/JCO. Unfortunately, theuse of such an approach is complicated by the need to describe the field insidethe metal volume, although it has a significant magnitude in the thin near-boundary layer only (skin effect). Use of the impedance boundary conditionsin the form of eqn. 1.6 exempts us from this necessity. The field within theconducting medium is not considered, but the full information about this fieldis contained in the magnitude of the surface impedance and its frequencydependence.

However, variations in the boundary-condition form compared with the caseof perfect conductivity fundamentally change the mathematical statement of theproblem: some methods giving excellent results for perfectly conducting struc-tures become unacceptable for impedance structures; others require a consider-able modification. Therefore, a major part of this book will be dedicated to

Introduction 7

solution techniques for electrodynamic problems with impedance boundaryconditions.

It should be noted that £s is a small parameter which is in many casesinteresting from a practical point of view. This leads to the idea of applying theperturbation method to calculate the dissipation characteristics. However, thesmall size of £s means that this technique cannot yet be applied. In many casesits direct application leads to nonsense, although sometimes the fact is masked.We analyse the causes of failures in detail and suggest correct modifications ofthis approach. Finally, within the framework of the perturbation techniquevarious schemes are possible. They are equivalent in principle but differ incomputational efficiency. We therefore pay special attention to the comparativeanalysis of various modifications of the perturbation method and the use of thelatter for solving specific applied problems.

1.3 Solution techniques for mathematical problems ofelectromagnetics

The purpose of the following introductory remarks is to characterise the problemsof electromagnetics studied in this book and the solution techniques used. Fromthe mathematical point of view, these problems are boundary-value ones forpartial differential equations. W7e shall consider both diffraction problems (exci-tation of electrodynamic systems by fields or currents given) and spectral prob-lems (eigenmodes of smooth-wall and periodic waveguides, free oscillations ofresonators). Figures 1.1 and 1.2 show typical examples of such structures.

The diffraction problem can be presented in an operator form as

Au{r)=f(r) (1.7)

where r e V, A is the differential operator, u is the function sought and f is agiven free term. Let us assume that the boundary condition

Bu(r)=Q (1.8)

is imposed on the surface S bounding the volume V (r e S, B is some differentialoperator).

In the general case A is Maxwell's operator

= Tcurl -jcofil

\_jco£ curl JA =

and

is an unknown vector of the electromagnetic field depending on three spacevariables.

For finitely conducting structures, the impedance boundary condition givenby eqn. 1.6 plays the role of eqn. 1.8.

For a wide class of problems the electromagnetic field does not depend onone of the Cartesian co-ordinates and can be expressed in terms of a scalar

//////v//////

(ii)

(vi)

Figure 1.1 Examples oj finitely conducting waveguides and resonators considered in thebook

(i) Rectangular waveguide(ii) Circular waveguide(iii) Microstrip line(iv) Cavity in the form of a body of revolution with an arbitrary

-generatrix(v) Cylindrical resonator with coaxial insert(vi) Corrugated waveguide

function.2 In this case V is a plane area, S is the contour bounding this area,A = V2 + k2 is the Helmholtz 2-dimensional operator and u = ift is the unknown

2 This also takes place when the field depends on this co-ordinate but this dependence is knownfrom the symmetry properties.

Introduction 9

I I I Ii i i |

a

E

I

~-H

7/ / / / / / / / / s /

6 f

Figure 1.2 Examples of diffraction structures considered in the booka Array of impedance halfplanesb Impedance comb-shaped structurec Finitely conducting echelette gratingd Periodic structure with a complex configuration of the periode Absorbing body in parallel-plate waveguidef Open-ended irregular plane waveguide

longitudinal-field component. Eqn. 1.6 is then transformed into a boundarycondition of the third kind:

= 0

i.e. B — (d/dn) + jrj, where rj is a given complex number.

10 Introduction

The area V can be either bounded or unbounded (external in relation to S).In the latter case the solution of the boundary-value problem under discussionshould satisfy the radiation condition at infinity.

The formulation of a spectral problem is different. Instead of eqn. 1.7, thefollowing homogeneous operator equation is considered:

Au(r) =Aqu(r) (1.9)

Here q is a given weight operator and k is a constant. It is necessary to findnontrivial solutions of eqn. 1.9 (they are called eigenfunctions). The values of/ , when such solutions do exist, are called eigenvalues. These nontrivial solutionshave to satisfy the required boundary condition on S.

A specific feature of systems with dissipation (i.e. when 8, [i and Zs a r e

complex numbers) is that the problems in question are non-self-adjoint due tothe structure of the operators A and B. The theory of non-self-adjoint operators(Keldysh, 1951, Neimark, 1954) is much more complicated than that of self-adjoint ones. Very unusual properties may be typical for non-self-adjoint oper-ators. It is enough to say that they may have no eigenfunctions at all, and thatwhen eigenfunctions do exist their systems may not form the basis in L2(V).However, as it will be further shown in this book, many electrodynamic problemsfor systems with dissipation which are interesting from a practical point ofview lead to non-self-adjoint operators which are small perturbations of self-adjoint ones. Their spectral properties are very close to those of self-adjointoperators: the specific character of non-self-adjoint problems is shown eitherpartially or not at all. This makes it possible to use the conventional solutiontechniques.

In this book we apply mathematically rigorous solution methods, i.e. methodswith no assumptions based on physical intuition. The approximate analyticalexpressions obtained in the book are a result of some parameters beingsmall and the use of the perturbation method. The applicability limits forthese expressions are indicated and their accuracy is evaluated. In this bookwe apply exact analytical, direct numerical and analytical-numerical methods.Of a variety of exact analytical methods, the method of separation of variables(Jones, 1986) and the Wiener-Hopf technique (Noble, 1958, Mittra andLee, 1971) are used. Examples of structures to which the two methods areapplied are shown in Figures 1.1 (i) and (ii) and Figure 1.2a, respectively.The capabilities of these approaches are limited by very specific configurationsof the objects.

The use of direct numerical methods provides far more opportunities. In theframework of these methods the original boundary-value problem can be reducedto a standard mathematical procedure performed by a computer. We can listfour such procedures:

(i) finite-order matrix inversion;(ii) determination of eigenvalues and eigenvectors of a finite-order matrix;(iii) solution of a transcendental equation;(iv) solution of a 2-point boundary-value problem for a linear system of

ordinary differential equations by the shooting technique.

Let us dwell upon the statement of problem (iv). It can be written in theform

dy

dt

By{O) = b

Dy(T)=d

Introduction 11

(1.10)

where 0 < t< T, y(t) is an unknown vector of m components,/, b and d aregiven column vectors of m, m — r and r components, respectively, and A, B andD are matrices of the orders m x m, (m — r) x m and r x m. The ranks of thematrices B and D are m — r and r. There are a few modifications of the shootingmethod which differ in their computational characteristics. Their descriptionand comparative analysis are given in Appendix 1 (see also Roberts andShipman, 1972).

Of direct methods, we use the moment method (Harrington, 1968), Galerkin'sincomplete method (Sveshnikov, 1969) and the integral-equation technique(Poggio and Miller, 1973).

Let us consider an elementary scheme of the moment method for the spectralproblem given by eqns. 1.9 and 1.8. Let the set of basis functions {us} (s =1, 2, . . . , oo) complete in L2 {V) be specified. In addition we assume that eachof the functions us satisfies the required boundary condition. The solution of thespectral problem is sought in the form of a series

u(r)= £ cpup(r) (reV) (1.11)P=i

where cp are unknown coefficients. Introducing another set of functions {vs}complete in L2(V), we replace eqn. 1.9 by a system of orthogonality relations

(Au-Aqu,vs) = 0 (1.12)

(s= 1, 2, . . . , oo). Eqn. 1.12 results from the fact that a function orthogonal toall functions of a complete set is identical to zero (Kantorovich and Akilov,1977). The parentheses in eqn. 1.12 mean the scalar product in L2{V) defined

JJSubstituting eqn. 1.11 into eqn. 1.12, we come to an eigenvalue problem for

infinite-dimensional matrices like

00 00

£ cp{Aup,vs)=A. X Cp{qup9vs) (1.13)

The particular case vs = us corresponds to Galerkin's method. At the final stage,eqn. 1.13 is reduced to a set of a finite order: jV first terms in the series and jVequations with s = 1, 2, . . . , JV are taken into account. Then, eqn. 1.11 is trans-formed into

u(r)= t cûp(r) • (1.14)

Eqn. 1.14 is approximate due to the truncation of the highest terms with/? > JVin the series, and since the first jV coefficients are calculated approximately froma finite set of equations (this fact is emphasised by the superscript JV in thecoefficients in eqn. 1.14). An important question arising here is that of conver-gence of the approximate eigenvalues 1{N) to the exact ones and the convergence

12 Introduction

of the representations, given by eqn. 1.14, to eigenfunctions, when jV->oo. Anumber of works has been devoted to mathematical proofs of this problem (e.g.Mikhlin, 1970).

This elementary scheme is used in Section 4.6, in particular, for the structureshown in Figure 1.1 (iv). The case where the basis functions do not satisfy therequired boundary conditions is more typical for systems with dissipation. Inthis case the orthogonality relations given by eqn. 1.12 have to be modified sothat the expansion in eqn. 1.11 would provide both the fulfilment of eqn. 1.9and the boundary condition given by eqn. 1.8. Examples of such modificationsare presented in Sections 3.2 and 7.6.

The idea of Galerkin's incomplete method is to expand the function soughtu(qx, q2, q$) in terms of basis functions {us(q1, q2, ^3)} forming a complete setin each section of V by the surface q3 = constant:3

Z p Jp=l

Here, in contrast with eqn. 1.11, the coefficients cp are functions of one of theco-ordinates. The orthogonality relations now are written on a set of sections ofthe domain V by the surface q3 = constant. A system of differential equationsfor cp(q3), which can be reduced to eqns. 1.10, results from these relationships.In Sections 5.8, 7.6 and 7.10 Galerkin's incomplete method is applied to struc-tures shown in Figures l.l(vi), l.2d and 1.2/.

The integral-equation method is applied to a 2-dimensional problem of wavediffraction by a finitely conducting complex-shaped grating (Section 5.7). Usingthe methods of potential theory, the following Fredholm integral equation ofthe second kind was obtained:

(1.15)

where fi(P) is an unknown function specified at the contour C of one period ofthe structure. The essential property of this integral equation is that the kernelK(P, P') has a logarithmic singularity when the arguments P and P' coincide.To solve it numerically the Krylov-Bogolyubov method (Kantorovich andKrylov, 1958) is used. The unknown function is expanded in a series in termsof pulse functions, and eqn. 1.15 is reduced to a set of linear algebraic equations.An important advantage of this method is a high degree of universality asregards the shape of the contour C.

The methods mentioned above place practically no limitations on the con-figuration of the areas. However, there are classes of problems related to areasof special forms, which are of great importance for applications. To solve theseproblems it is possible to employ the techniques leading to specialised compu-tational algorithms which, however, have a higher computational efficiency.These are so-called 'co-ordinate' structures where the boundaries of areas con-sidered consist of parts of co-ordinate lines (surfaces) of any classical system oforthogonal co-ordinates (Cartesian, cylindrical, spherical etc.). In this case thearea under consideration may be divided into subareas for which general solu-

3 In particular cases us may be completely independent of the variable g3.

Introduction 13

tions of the equations being solved are written in a convenient form.4 The fieldsmatching at the interfaces between the subareas results in sets of linear algebraicequations or dual integral (series) equations. A microstrip line [Figure 1.1 (iii)]is an example of such a structure. Its cross-section is decomposed into twosubareas in the form of layers over and under the strip. Solutions for each layerare presented as Fourier integrals. The imposition of the boundary conditionsresults in dual integral equations, which, unlike eqn. 1.9, are nonlinear in thespectral parameter A. A numerical solution of these equations is carried out bythe moment method (see Sections 3.8 and 3.9).

For regions of special form the numerical-analytical methods (Mittra andLee, 1971, Shestopalov et al., 1984) are of great importance. Their essence isthat the analytical techniques are not applied to find the solution in a closedform but to transform the equation solved to a form allowing a more effectiveapplication of numerical algorithms.

Thus, for wave diffraction by a comb-shaped structure (Figure \.2b) it isconvenient to separate the area under consideration into a halfspace over thestructure and internal cell areas. In the first subarea the field is expanded in aseries in terms of Floquet's harmonics; in others it is expanded in terms offorward and backward eigenmodes of the corresponding plane waveguide. Thefield matching leads to the infinite set of linear algebraic equations

Ax=f (1.16)

where x is an unknown infinite-dimensional vector, f is a known infinite-dimensional vector and A is the matrix operator (see Section 5.3).

It is often ineffective to solve such sets numerically by the truncation methodsince eqn. 1.16 belongs to a class of ill-posed problems: operator equations ofthe first kind (Tikhonov and Arsenin, 1977). In this case the limit, to which thetruncation method converges, depends on the method of truncation and maynot coincide with the solution required [a phenomenon of 'relative convergence'studied by Mittra and Lee (1971), Shestopalov et al. (1984)]. The correct methodof truncation is unknown a priori but even if it is found the convergence will beslow. In addition, the sets obtained are ill-conditioned.

Therefore, an analytical regularisation of eqn. 1.16 is carried out. The operatorA is presented in the form A = Ao + Au where Ao is an analytically invertible'main part' of the operator A causing difficulties in numerical solution ofeqn. 1.16. For wave diffraction by a comb-shaped structure, Ao is a convolutionmatrix operator5 of the form

A0,mn = p —

where m = 0, ± 1, ±2, . . . , ± oo, n = 0, 1, 2, . . . , oo, r m =0( |m | ) and yn =O(n). HAQ1 operates on the left-hand side and the right-hand side of eqn. 1.16,the result will be

î)* = ô7 (1-17)

4 These solutions are in the form of Fourier series, Fourier-Bessel series with unknown coefficients,or Fourier, Hankel, Mellin, Mehler-Fock and other integrals with unknown spectral amplitudes.5 The procedure of its analytical inversion is described by Mittra and Lee (1971) and Shestopalovet al. (1984).

14 Introduction

Eqn. 1.17 reflects the essence of regularisation from the left or semi-inversionbeing developed by Shestopalov et al. (1984).

Another possible approach is a method of regularising substitution or regularis-ation from the right. The idea of this method is to introduce the linear substi-tution of the form x = ô^1^, where g is a new vector of the unknowns. Then,the following equation is obtained for g:

(I+A1Ao1)g=f (1.18)

In some cases a linear-fractional substitution of the unknown quantities isexpedient to simplify the operator structure in eqn. 1.18 (see, for example,Sections 5.1, 5.3).

The efficiency of a regularisation scheme depends on the particular featuresof the problem considered. However, if the 'main part' is reasonably defined,both of these provide the convergence of the approximate solution, obtained bythe truncation method, to the accurate solution. The rate of convergence is veryhigh.

These methods of analytical regularisation were applied to various classes ofoperator equations: matrix, integral, dual integral and dual series equations.Certainly, the possibility of extracting an analytically invertible 'main part' iscreated by their special form imposing restrictions on the configuration ofstructures under consideration. Normally the 'main part' has a clear physicalmeaning corresponding to the extreme value of frequency or one of the geometri-cal parameters of the problem. To invert it, various methods, such as the Wiener—Hopf technique (Noble, 1958), the method of the Riemann—Hilbert problem(Shestopalov, 1970) or the residue-calculus technique (Mittra and Lee, 1971)are applied.

In this book the numerical-analytical methods are developed in two direc-tions, one for structures with impedance-boundary conditions, and the other forstructures with a complex non-co-ordinate shape of the surface (Figures l.2dand 1.2tf). The latter case is notable because the diffraction problem is reducedto a 2-point boundary-value problem for an infinite set of ordinary differentialequations like eqns. 1.10, and the semi-inversion procedure is carried out at oneof its boundary conditions (see Sections 5.8 and 7.10).

1.4 Accuracy control and computational instabilities

In the previous Section we characterised briefly the mathematical methods usedin the book. These methods are based on different ideas and differ in generality,computational efficiency and labour content of analytical transformations. Thesurvey gives rise to the following questions. What criteria should be used whenchoosing a method to solve a particular problem? How should we monitor thecomputational accuracy obtained?

One of the most fundamental aspects in all methods under consideration isthe truncation, i.e. an approximation of an infinite set of algebraic or differentialequations by a finite set. The elements of matrices and free-term vectors involveintegrals, series and infinite products which often cannot be calculated in aclosed form. Errors due to rounding and other factors invariably occur whencalculating such elements and solving these sets numerically. Despite the errors

Introduction 15

being small in each particular step of the algorithm, they may accumulate and,as a consequence, considerably distort the final result.

In the light of this we formulate the following key criteria for method efficiency:

(i) high rate of convergence;(ii) simplicity of calculation of matrix and free-term-vector elements; and(iii) stability against computational errors.

Let us suppose that as a result of application of any numerical method aboundary-value problem has been reduced to a matrix equation of the JVthorder:

Ax=f (1.19)

Assuming that we have the following equation due to computational errors

{A + SA)x=f+Sf (1.20)

instead of eqn. 1.19. The solution of eqn. 1.20 can be presented as x = x + Sx,where x is the solution of the unperturbed linear system (eqn. 1.19). We canevaluate the error Sx (Fadeev and Fadeeva, 1963) as

| |5*| | <C, \\SA || + C2Iwhere C1>2

a r e some constants.6 The question is: how do C\ and C2 behavewhen JV grows? If they are independent of JV, then the algorithm will be stable.If C12 still grow when JV increases, then there will be no stability againstcomputational errors. This question should be cleared up before the computerprogram is worked out. If an algorithm is unstable, it is impossible to providea desirable accuracy, even when the approximate solution converges to an exactone at jV-> GO. In fact, any improvement in accuracy caused by an increase inJV may be neutralised by a greater influence of computational errors. Startingfrom some JV0 the latter factor may prevail, in which case the accuracy of thesolution may deteriorate when JV> JV0.

Two kinds of computational instability should be distinguished. The first iscaused by the fact that the original mathematical problem is ill-posed (e.g. aFredholm integral equation of the first kind). In this case the inversion operator,as known, is unbounded (Tikhonov and Arsenin, 1977), and an attempt toapproximate the latter by means of a sequence of finite-dimensional matrixoperators A"1 causes C1 2 to increase. To overcome instability here, it is necessaryto use a special technique, e.g. Tikhonov's regularisation method.

Another case is when the problem is well-posed, and the instability arisesbecause of poor algebraisation, for example, inappropriate choice of basis func-tions in Galerkin's method.7 To provide computational stability, the basis-function sets should have special properties, e.g. 'strong minimality' (Mikhlin,1966).

/ JV \l/26 Here the following norms are used: 11*11 = 1 £ \xj\2 ) ôr vectors, and

/ * N y/2 ^ j = 1 /M H = I Z \Aij\2) for matrices.

On the other hand, the contrary can be illustrated, i.e. when an ill-posed problem is solved byGalerkin's method. A special choice of basis functions may play the role of analytical regularisation.

16 Introduction

To control the accuracy of numerical results obtained the following meansare applied:

(i) rigorous mathematical substantiation of the method;(ii) 'intrinsic-convergence' check (stabilisation of the results when the trun-

cation order N is increased);(iii) check that fundamental physical laws are satisfied (energy conservation,

reciprocity theorem etc.); and(iv) comparison of particular calculational results with those obtained by other

methods.

Mathematical substantiation of a method (proof of convergence to a real solutionof the problem) is very important. In any case, it can not be substituted byreference to 'common sense'. A good illustration is the phenomenon of'relativeconvergence' when a certain class of infinite algebraic systems is solved by thetruncation method (Mittra and Lee, 1971): when the method of truncation isarbitrary there is a convergence, but the limit is not a real problem solution.

Mathematical proofs are rather voluminous and we therefore do not presentthem in full in this book. We only demonstrate a general idea of the proof andrefer the interested reader to works where this proof is described in detail.

Normally, a mathematical proof states that there is a truncation order JVallowing one to achieve a computational accuracy as high as desired. This,however, does not produce a specific technique to select JV providing anyrequired accuracy. The dependence of jV on frequency and geometrical param-eters of the electrodynamic system makes this selection still more difficult. Thecommon practice here is to perform a computational experiment solving theproblem at various JV and to compare the results obtained. Verification ofthe intrinsic convergence is an indispensable element of accuracy control but byitself it does not guarantee anything.8

The approximate solutions often identically satisfy the energy balance andreciprocity theorem, though it is not always obvious. In particular, the poweris totally conserved for fictitious solutions obtained as a result of relative con-vergence (Shestopalov et al., 1984). The energy conservation and reciprocitytheorem can therefore be used only for detecting casual errors.

Thus, each of the enumerated means of accuracy control alone does notprovide any reliability of the data obtained. They can only be relied upon whentaken in combination. When presenting results of mathematical simulation weshall pay special attention to the problem of computational-accuracy controland reliability of calculation results.

In the following discussion infinite sets of linear algebraic equations of specialform will also be of importance:

xn + A £ Amnxm=fn (1.21)m = l

where n = 1, 2, . . . , oo and X is a preset number. Let us assume that the matrix

8 We emphasise that it is impossible to replace the mathematical proof of convergence by an 'intrinsicconvergence' check. This was also confirmed in the paper by Heitkamper and Hienrich (1991),where the diverging algorithm was described though its divergence was very slow [as O(lnJV)]. Inthat case JV varying within definite limits does not lead to any considerable change in computationresults which, as Heitkamper and Hienrich pointed out, may be mistaken for convergence.

Introduction 17

elements and free terms satisfy the conditions

t 1 \Amn\2<co £ \fn\

2<cc (1.22)m = l B = 1 n—1

Then the Fredholm alternative is valid for the set given by eqn. 1.21: either /is the eigenvalue for a corresponding homogeneous set (when/,, = 0), or eqn. 1.21has a unique solution. This solution satisfies the condition

and may be obtained by the truncation method. Stability against errors incomputation of Amn and fn is provided (Kantorovich and Akilov, 1977). Suchsets are called 'sets of the second kind' or Fredholm sets. This reflects a similarityof their theory to that of Fredholm integral equations of the second kind. Theconditions given by eqns. 1.22 do not cover a class of Fredholm sets as a whole.For example, infinite sets with a spectrum within the unit circle are also relatedto the latter.

Fredholm sets often arise as a result of application of numerical—analyticalmethods to boundary-value problems of electromagnetics. Typically, inequalitiesgiven by eqns. 1.22 are comparatively easy to check. It is important to notethat they contain a full mathematical substantiation of the method. When thesets are obtained by numerical-analytical methods, there are very reliable waysof choosing the truncation order N according to the initial data of the problem(Shestopalov et al., 1984). We have already pointed out that these methodsprovide very fast convergence and that therefore the machine time required isvery short. This is an important factor but the development of computersgradually reduces its role. A much more important factor is that numerical-analytical methods have a considerably higher reliability than do numericalprocedures because of the above-mentioned characteristic features. Therefore,these results, in particular, give nontrivial tests for software debuggingand accuracy control when working with direct numerical methods. All thesepoints justify the complex analytical transformations involved when numerical-analytical methods are used.

Chapter 2

Surface-impedance technique for thestudy of dissipation processes inbodies with finite conductivity

Dissipation of electromagnetic waves in metal structures, due to their finiteconductivity, is usually investigated employing impedance boundary conditionsof the form of eqn. 1.6. The use of these boundary conditions allows the problemto be simplified, as the field inside the metal is not considered, which is theessence of the surface-impedance method. Penetration of microwave fields intothe metals is accompanied by a marked skin effect and this stipulates thefollowing special features of eqn. 1.6, which are important for the conditions tobe used in the future:

(i) surface impedances of metals in ordinary situations are small: l-(ii) the impedance Zs calculated for a plane wave does not in practice depend

on the angle of incidence [this means that the conditions given by eqn. 1.6can be used for the field of arbitrary (unspecified in advance) spatialstructure].

It is important to note that the surface-impedance method is a fairly universaltechnique which covers practically all interesting cases when dissipation charac-teristics of microwave and millimetre-wave components are to be determined.In this Chapter, expressions of the surface impedance for such cases will beconsidered.

2.1 The Leontovich impedance boundary condition

When deriving the Leontovich boundary condition the simplest constitutiveequation of metal is used:

J(r,co)=o0E(r,co) ' (2 .1)

where a0 is the conductivity of metal for static electric fields. The key task tobe examined is the incidence of a plane wave on the infinite flat interface'vacuum-metal'. In fact, it is sufficient to characterise the metal by a conditional

Surface-impedance technique for the study of dissipation processes 19

complex permittivity £=jo"0/co and use Fresnel's formulae (Weinstein, 1988).At <70 > co£0 we find from the latter that eqn. 1.6 is fulfilled at

Zs= (cofil2ao)ll2(l -j) = WokAo(\ - » / 2 (2.2)

where Ao = (2/co^cr0)1/2 is the field penetration depth into the metal (we would

like to underline once again that the angle of incidence is not involved ineqn. 2.2).

Though impedance boundary conditions are usually derived for a planeinfinite surface, they are also applicable to curvilinear surfaces, if only the radiiof curvature are large compared with Ao (in this case the boundary can beconsidered as locally flat).

Physically the validity of this assertion is rather obvious though it can beproved quite rigorously if we solve the problem for a body with a high conduc-tivity and curvilinear boundary by the integral-equation technique, as has beendone by Mitzner (1967). In this case, more general boundary conditions can beobtained which contain corrections corresponding to the curvature of the surface.In the particular case of Ao < R these conditions are reduced to the impedancecondition expressed by eqns. 1.6 and 2.2.

Tisher (1974, 1976, 1978, 1979) in his works conducted experimental researchabout the Leontovich boundary conditions' correctness for the calculation ofattenuation in waveguides. An attenuation coefficient in rectangular waveguideswas measured over a rather broad frequency range {f— 25-200 GHz) and theresults of these measurements were compared with those of the calculations(Tisher, 1979). The value <J0 was estimated from measurements at direct current(for copper at T=20°C the value <r0 = 5.73 x 107 S/m has been received).Secondary factors, such as admixtures, temperature influence on the attenuationduring hardening of the waveguide, surface roughness etc., were also taken intoconsideration. A satisfactory coincidence of theoretical and experimental datahas been obtained. Thus, for example, the highly precise measurement carriedout by Tisher (1978) has showed that for copper at the frequency/= 35 GHzthe measured surface resistance is 1.129 ±0.02 times that calculated.

The measurement of the influence of metal surface roughness carried out byTisher (1978) is of some interest. The roughness was controlled by a profilometergauged using the photomicrographic method. It was discovered that the influ-ence of roughness is equivalent to a certain increase in the surface impedance:Re ^ s =/(o)/x/2cr0)

1/2, where / is the surface impedance increase factor.Table 2.1 represents the results of measurements of x from Tisher5s work for/ = 3 0 G H z .

Theoretical research in roughness influence is somewhat difficult. The surfacemicrorelief is determined, in the first place, by technological factors and should

Table 2.1 Results of measurements of % (Tisher, 1978)

Mean-square dimensions of inhomogeneity Surface-impedance increase factor

({Am) x

0.2 1.1

0.5 1.2

1.0 1.3

20 Surface-impedance technique for the study of dissipation processes

be described by a random function. The problem of wave diffraction by arandom rough surface was considered, for example, by Bass and Fuks (1979)and De Santo and Brown (1986). A simpler model to take the roughness intoaccount is suggested in works by Mende and Spitsyn (1985) and Baryshnikovet al. (1988). It presents the surface microrelief as a periodic structure with aperiod and depth equal to a mean-square length and height of inhomogeneity,respectively. This is the very model which leads to impedance Zs being indepen-dent of the angle of incidence with the numerical factor % (Mende and Spitsyncalled it a roughness factor).

Methods of calculation of % differ significantly for two cases: when geometricalparameters of microrelief greatly exceed Ao, and when they are comparablewith Ao. In the first case we have a problem of wave reflection from the finitelyconducting periodic structure, on the surface of which the Leontovich conditionis imposed. As microrelief parameters are significantly less than / , this problemcan be solved in a quasistatic approximation which considerably simplifies it.For a rectangular microrelief profile, Mende and Spitsyn (1985) obtainedapproximate formulae for % which give results of the same order as those obtainedby Tisher (1978). In the second case the problem is significantly more compli-cated: it is necessary also to solve Maxwell's equations for the electromagneticfield inside the metal. For this case Mende and Spitsyn (1985) presented numeri-cal results obtained by the finite-difference method. Note that the microreliefcan be anisotropic, for example consisting of long thin rulings. In this case,instead of the numerical factor x, a tensor of roughness should be introduced.

The local constitutive equation given by eqn. 2.1 and, consequently, theLeontovich impedance obtained on its basis, are correct at the condition / <l Ao

(/ being the mean free path of electrons in the metal), which corresponds to thefrequencies/<^/°r, where

fcr^(aolil2ny' (2.3)

(Lifshitz and Pitaevskii, 1979, Abrikosov, 1987). Note that/° r depends indirectly(through a0 and /) on the temperature (/°r corresponds to the condition' = A0).

If/ were comparable to, or exceeded Ao, then the constitutive equation givenby eqn. 2.1 would be replaced by a more general nonlocal relationship

/(r,o>) = \<j(r,r'\a))E(r',Q))d3r' (2.4)

in which the metal conductivity has the form of an integral operator. In thiscase the skin effect has a number of specific features and is called anomalousskin effect. The surface impedance in eqn. 1.6 is also expressed differently. Thenext Section deals with this problem.

2.2 The surface impedance of normal metals for theanomalous skin effect

The nonlocal connection of the current density and the electric-field intensityin eqn. 2.4 indicates the presence of a spatial dispersion of the medium. For ahomogeneous infinite metal <r(r, r'\co) = o^ (r — r ' | co), i.e. the kernel in eqn. 2.4


depends only on the arguments difference. Then

J(r,co) = L^-r 'M^^dV (2.5)

Presenting J and E as plane-wave spectra

r, co))> exp( jar) da

£(r,eo)J J [E{q,(o)\and using the convolution theorem for eqn. 2.5, we obtain for the spectralamplitudes the relationship

where o^ (q, co) is the Fourier transform of the kernel a^ in eqn. 2.1.9

The specific feature of eqn. 2.6 (unlike, for example, eqn. 2.1) is dependenceof a^ on the wave vector q (spatial dispersion). The nonlocal connection betweenJ and E can be physically explained, for example, by the fact that electrons ofconductivity, acquiring a considerable velocity component in the direction nor-mal to the metal surface as a result of electron collisions, rapidly leave the skinand thereby do not contribute to the microwave current (the concept of'ineffectness' by Pippard).

The partial conductivity aao{q,co) in eqn. 2.6 can be found from the self-consistent solution of Maxwell's equations and the kinetic equation for freeelectrons. Next, well developed methods of electrodynamics for media withspatial dispersion can be used to calculate the impedance Zs (Agranovich andGinzburg, 1979).

Suppose that the E-polarised plane wave E = iy exp{jk(z sin 3 - x cos 9)} isincident from x > 0 on an infinite plane boundary vacuum-metal (the axes arechosen so that the interface is the x — 0 plane). Considering the spatial dispersionwhile solving boundary-value problems, we need to impose the so-calledadditional boundary conditions (Agranovich and Ginzburg, 1979). This resultsfrom the boundary being responsible for the difference between the kernels a ineqn. 2.4 and o^ (corresponding to an infinite medium). Moreover, the kernel6 remains indefinite even after a^ has been calculated. Additional boundaryconditions are physical assumptions a priori which allow us to specify the relation-ship between 6 and 0"^. To obtain these conditions, the assumption of a mirrorcharacter of the electrons reflection from the boundary can be made. Then, forEy inside the metal we receive the integral-differential equation

2 dJk

(2.7)

where aQ0(R — Rf) corresponds to a homogeneous infinite metal and thusdepends only on the difference between the arguments, and R and R' are radii-vectors in the plane Oxz.

Solution of eqn. 2.7, using general methods for problems of electrodynamics

9 We consider only isotropic metals, where the tensor o^ (q, co) becomes a scalar; the surfaceimpedance £s in eqn. 1.6 is therefore also a scalar.


of media with spatial dispersion, allows us to obtain the following expression forthe reflection coefficient:

l+j*cos^(£sin3)l-7*cosS^(AsinS) { '

where Z(k sin 9) is the partial impedance corresponding to the angle of incidence9 and is expressed by

From eqn. 2.9 it follows that for good metals the partial impedance £(v),with a high degree of accuracy, does not depend on v (and hence the angle ofincidence S). This means that the impedance £ in eqn. 2.8 can be identifiedwith the surface impedance Zs to be found for the impedance boundary conditiongiven by eqn. 1.6. Assuming v = 0 in eqn. 2.9 and substituting the specificfunction cr^dgrl), we can obtain the required expression for £s.

In case of an extreme anomalous skin effect when /^>A0, according toAbrikosov (1987), we can write

Then from eqn. 2.9 we obtain

j (2.10)

where £ = (Ofi(70l~1 and Aeff = {ljcofiG0)

1/3 is the effective depth of field penetra-tion into the metal.10

The qualitative character of frequency dependence of Re Zs on f is shown inFigure 2.1. The region / < ^ / ° r (/°r is evaluated by eqn. 2.3) is the region ofnormal skin effect described by the Leontovich impedance expressed by eqn. 2.2.The region / ^ > / ° r corresponds to an extreme anomalous skin effect (theimpedance is defined by the eqn. 2.10). Most of the difficulties arise in theintermediate region, where / ~ Ao and where it is rather difficult to obtain simpleanalytical relationships.11 However, it is necessary to take into consideration thefact that the transition from normal to anomalous skin effect takes place fairlysmoothly [this is also confirmed by experiments (see Abrikosov, (1987)]. Thisalready allows us to consider the surface impedance in the intermediate regionas also independent of the angle of incidence. The value of Zs c a n De estimatedby means of a simple approximation using two extreme cases given by eqns. 2.2and 2.10 (broken line in Figure 2.1).

It is important to note that the surface impedance for the anomalous skineffect turns out to be independent of the angle of incidence in spite of thenonlocality of the constitutive equation. Because of this, the frequency depen-dence of the surface impedance differs from that in eqn. 2.2.

10 For the anomalous skin effect the microwave field when penetrating further into the metaldiminishes nonexponentially. This is why the value Aeff ceases to have a direct physical meaning,unlike the value Ao in the normal skin effect.11 The analysis of this case has been carried out by Kittel (1963), but the expression obtained for£s is rather complicated.


Figure 2.1 Sketch of surface-impedance active component against frequency

1 Normal-skin-effect region2 Extremely anomalous-skin-effect region

approximation in intermediate region

The theory of anomalous skin effect is much more complicated if the characterof the electron reflection from the boundary is a diffuse one.12 However, thefinal expression for the impedance does not depend greatly on the characterof the electron reflection. Thus instead of eqn. 2.10, at /^>A0 we obtainan expression that differs from eqn. 2.10 by the constant numerical factor §(Abrikosov, 1987).13

Numerical evaluations show that the above-mentioned Pippard mechanismof anomalous skin effect takes place in the microwave and millimetre wavebandsonly at very low temperatures; for room temperatures it is essential only at veryhigh frequencies (infra-red and optical band).

In the work by Wang (1978) an attempt has been made to explain why theexperimental data (Tisher, 1978) differ from the results of calculations accordingto eqn. 2.2. Wang interpreted this as a result of anomalous character of the skineffect. He also suggested that the mechanism of anomalous skin effect involvesa Coulomb's screening of the donor potentials by the electrons of conductivity.The attempt to calculate the surface impedance for this case was also made bySlepyan (1984). In this respect it is necessary, however, to note the following:in the presence of the spatial dispersion even an isotropic metal becomes aniso-tropic due to the wave propagation; the anisotropic properties are defined by

12 Under diffuse reflection we understand such a reflection when electrons do not preserve a'memory' about their original states.13 Physically the assumption about a mirror character of the electron reflection appears to be morerealistic. This is because for the electrons moving inside the skin for a long time the reflection lawis close to a mirror one and these are the very electrons which contribute most to the microwavecurrent (Lifshitz and Pitaevskii, 1979).


the orientation of the wave vector q. That is why one should distinguish betweenlongitudinal and transverse conductivities of the metal (Landau et at., 1984). Ineqn. 2.6 and, hence, in eqn. 2.9 just the transverse conductivity appears, whereasCoulomb's screening affects the longitudinal conductivity of the metal.14 Theexpressions for Zs offered by Wang (1978) and Slepyan (1984) are thereforeerroneous, as is the suggestion of the influence of Coulomb's screening on thecharacter of the skin effect.

2.3 Surface impedance of superconductors

As is well known, microwave engineering and low-temperature physics areclosely related. On the one hand, the physics of superconductors finds newmaterials with unique physical characteristics for microwave applications; onthe other hand, microwave methods are highly effective for measurements ofsuperconductor parameters. Undoubtedly, the discovery of high-temperaturesuperconductivity in barium ceramics (see, for example, Ekholm and McKnight,1990) will contribute even further to their mutual influence.

Superconducting waveguides, resonators and microstrip lines are widely usedin microwave and millimetre-wave devices (powerful highly stable tunableoscillators, wideband delay lines, charged-particle accelerators etc.). Thus theproblem of electrodynamic modelling of these components is important. Cer-tainly, our task here is not to give a systematic description of the high-frequencyproperties of superconductors: that has been done, for example, by Lifshitz andPitaevskii (1979), Abrikosov et al. (1958, 1959), Ginsberg (1966), Mattis andBardeen (1958) and Mende and Spitsyn (1985). We wish only to demonstratethat these properties are described by means of the surface-impedance concept.Hence, all the information in this book applies equally to electrodynamic systemswith normal metals and to superconducting systems.

The constitutive equation for a homogeneous infinite superconductor is usuallywritten as a linear relationship between Fourier transforms (with respect to timeand all space variables) of the current density and the vector potential A

J(g,O})=-d(q,CO)A(q,O)) (2.11)

where QJ,q, co) is the so-called generalised susceptibility, calculated on the basisof a superconductor physical model. Eqn. 2.11, strictly speaking, is written forthe transverse components J and A, and Q{q, co) is transverse susceptibility.

In the theory of superconductivity the displacement current inside the super-conductor can be neglected, i.e. divj= 0. This means that q*J(q, co) = 0 and,hence q.A(q, co) = 0. Then the vector potential A is related to the electric-fieldintensity by the formula A — E/jco. Then eqn. 2.11 is reduced to eqn. 2.6, wherethe partial conductivity and the generalised susceptibility relationship is(JQO (q, co) = jQ,{q, co)/co. Therefore if Q(q, co) is determined from the physicalmodel, the scheme described in Section 2.2 can be applied to find the surfaceimpedance. For this, as for normal metals, the impedance does not depend onthe angle of incidence of the plane wave.

To determine o^ (q, co) one can use simple phenomenological theories of

14 The authors' attention was drawn to this fact by Prof. F.G. Bass.


superconductivity (London's model, Pippard's model) and the microscopic BCStheory as well. For example, in London's model a superconductor is presentedas a 2-component electron liquid (one component corresponds to 'normal'electrons, another to 'superconducting' ones). The 'superconducting' currentdensity Js (current caused by the motion of 'superconducting' electrons) isdescribed by the formula

where JVS is the density of'superconducting' electrons and e and m are the chargeand mass of the electron. Within the framework of this model

0"™ = ' V^s + ^ { 1 + (COT)2} a)TJVs + JVn|_ JVS {l + (C0T)2}

(2.12)

where <70 is the normal static conductivity, JVn is the density of'normal' electrons,and I = //yF where vF is the velocity of electrons on the Fermi surface. Thespecific feature of eqn. 2.12 is independence of o^ from the wave vector q (thismeans that within the framework of London's model there is no spatial disper-sion). From eqn. 2.9 there immediately follows an expression for the surfaceimpedance:

'y /1 ' \ ' / o \ l / 2 / i • \ — 1 / 2 / o i o \

£s = ( 1 —j ) {(Dfi0 JZGQ) ' \ g \ \ j g 2 ) (^-1^)where

gl

_ i K r , K (cot)2

The impedance expressed by eqn. 2.13 depends on the temperature through a0

and the ratio NsjNn. There is an approximate formula

JV. + JV- \Tc,where Tc is the critical temperature of the superconductor measured in kelvins.

It is far more complicated to determine the surface impedance on the basisof a rigorous microscopic theory. We shall present here only some final results(Ginsberg, 1966, Mattis and Bardeen, 1958). For the surface impedance ^s m

the extreme local limit, eqn. 2.13 is valid, but the coefficients gl2 are expressed,according to BCS weak-coupling theory, as

Si = I 2 2—1/2 2 2 1/2

-£ 2 -A 2 )d£, /*slhco,A)

-Lf - A2yi2{(hco - Z)2 - A2}112

ha)} (Z2 + A2 + hco!;) d^


where

S(hco,A) =hco - A (hco < 2A)

(hco > 2A)

A is the energy gap, f{£) is the Fermi distribution and h is Planck's constant(its presence in the formulae points to a quantum nature of superconductivity).Through the energy gap A and the function / (£ ) , coefficients g12 depend onthe temperature.

Qualitative dependencies g12 (co) at various temperatures are shown inFigure 2.2 for YBCO ceramics (Ekholm and McKnight, 1990).

As in Section 2.2, we can introduce the effective depth of penetration of themicrowave field Aejj , determining it from the condition Re Zs = êff Woj2.Then

e// 0 \ 6 l 62/ I \W I ) IT7^ ) )

where </> = tan"1 (g2jgi) and Ao = (2jcojiG0)112 is the normal depth of

penetration.A more detailed description of the high-frequency properties of superconduc-

tors is presented in the monograph by Mende and Spitsyn (1985). Apart fromthe physical models and formulae for £ s , it also contains extensive numericaland experimental data for various superconductors used (lead, niobium etc.).

2.4 Surface impedance modification for structures withedgesMany theoretical models of electrodynamic systems contain infinitely thinunclosed metal surfaces. Typical examples of these systems are microstrip and

2 • • • • . *

Frequency (Hz)

Figure 2.2 gU2 factors for YBCO (Ekholm and McKnight, 1990)

T = 4.2KT = 60KT = 85 K

Reproduced by permission of the IEEE © IEEE 1990


slot lines and their modifications, some types of open resonator, diffractiongratings etc. In general, it is impossible to take the metal losses into considerationby imposing the conventional impedance boundary conditions (when Zs *s

determined by any of eqns. 2.2, 2.10 and 2.13) on both sides of the unclosedsurface. The reason for this is that the local field and current-density distributionsnear the edge are described inaccurately by these impedance boundary con-ditions (Zhurav, 1987, Weinstein et at., 1986, Ilyinsky and Slepyan, 1986). Thiscan be seen from the fact that, in reality, field and current-density distributionsnear the screen edge are determined by its thickness and geometry and suchparameters are not involved at all in the model of an infinitely thin screen withthe impedance conditions examined above.

If the field is electrically polarised along the edge (H is parallel to the edge),the use of impedance boundary conditions considered in Sections 2.1-2.3, tocalculate such an integral characteristic as dissipated power, is relevant. This isbecause the current density near the edge in this case is rather small and itscontribution to the full dissipated power is insignificant. Therefore, the error incalculation of the current density is not important.

The situation becomes more complicated when there is a substantial compo-nent of magnetic polarisation {E is parallel to the edge). In this case the usualimpedance-boundary conditions near the edge are not applicable, so it is neces-sary to take into account directly the screen thickness and the actual edgegeometry; this significantly complicates the construction of effective numericalalgorithms. For small thicknesses some methods (for example, the integral-equations technique) result in algebraic systems with poorly conditioned matricesand, as a consequence, large computational errors (Galishnikova and Ilyinsky,1987).

This Section presents an approach, based on the replacement of the actualsurface of finite thickness by an 'effective' infinitely thin surface, on both sidesof which the boundary condition expressed by eqn. 1.6 with a modifiedimpedance Zs 1S imposed. The modified impedance is introduced in such amanner that it should describe satisfactorily both the dissipative properties ofthe metal and the induced currents in the edge vicinity. The modified impedancethen involves parameters describing the edge geometry (screen thickness, radiusof curvature near the edge).

We consider two cases: the rounded-edge geometry (Figure 2.3a) and therectangular-edge geometry (Figure 2.3b). Halfplanes with such edge configur-ations are the key structures to determine the modified impedance. The materialbelow is based on works by Ilyinsky and Slepyan (1986, 1990) and Slepyan(1988). To describe correctly the dissipative properties of the metal we have toconsider Re Zs having the value which follows from the skin-effect theory(Re Zs= kAeffWol2). This theory gives the value of Im Zs DUt) a s

Im Zs I Wo<^ 1, the influence of Zs o n waveguide mode dispersion, resonancefrequencies and diffraction characteristics of electrodynamic systems is negligiblysmall and is not usually considered. It will not therefore introduce a seriouserror in the calculation of these characteristics if we consider Im Zs a s arbitrarywhile the condition Im Zs I Wo<^ I is satisfied. The 'degree of freedom' thusobtained can be used to calculate correctly the losses within the framework ofthe infinitely thin impedance model. For this purpose, Im Zs should be chosenso that the current-density distribution near the edge would be the same as in


/ / / / \S

Is / y / S / / / / / y\/

a

7/ 1 tA

D

Figure 2.3 Geometry of thin unclosed screen

a Rounded edgeb Rectangular edge

an actual structure of finite thickness (we shall describe the methods of choosingthis later).

Most important for this procedure is the fact that the current density on theedge of an infinitely thin impedance halfplane has no Meixner's singularity.

Let us consider the solution of the problem of E-polarised plane-wave diffrac-tion by an infinitely thin impedance halfplane (Noble, 1958). The current-density distribution is given by the expression

iy {(l + cos,3)A;}1/2

2nK. {k cos 9) J ^ . ^ (w - k cos 9)K+ (w)(2.14)

where K+ are split functions regular in Im w> — k" and Im w < k"', respectively,and K{w) = K+ (w)K. (w) = 1 +jZs(">2 ~ k2)112/Wok.

Let us evaluate the leading asymptotic term (Zs ~^ ^) °f t n e integral

'jx+co {w -k)112 exp ( - jwx) dw

.„ (w - k cos 9)K+{w)(2.15)

at x—• +0. We assume that k — k' — jk" (k" > 0) and choose the brunch cuts inthe plane of complex variable w, as shown in Figure 2.4.


Figure 2.4 Brunch cuts and contour of integration in the complex w-plane

Deforming the integration path in the lower halfplane we can show that theevaluation sought is the same as that for the integral of the same function alongthe contour embracing the brunch cut. In this case eqn. 2.15 can be rewrittenat k" -* 0 as

f°° (Ju)exexp(j7r/4) — -

Jo (*~

(u) du

k cos 9 — ju)(2.16)

where

S(u) =2K.(k-ju)

l-(ZslW0k)2(u2l

Assuming that Zs = Z's ~JZs, Z's,Zs> 0, Z'i < Z!s and x = 0 in eqn. 2.16 andintroducing the variable v according to the formula v = JZsu!Wok, we obtainthe relationship

where

(1 + v2 + - cos 9)IWo)

To obtain the leading asymptotic term we have to carry out the limitingtransition Zs ~* 0 m t n e expression for Gx. Then

f{v) (2.18)


where

f(v) = lim K+ (jv) (e = e' + jE'\ E\ s" > 0)e->0

K+ (a) is a split function regular in

I m a > - e " and if (a) = T+ (a)Jf_ (a) = 1 + ( a 2 - s 2 ) 1 / 2

Finally, we have

I 1

The integral in eqn. 2.18 converges: G is a finite number independent of 9. Theintegrals in eqns. 2.18 and 2.19 cannot be expressed analytically. As a result ofnumerical integration, we obtain G= 2.3392.

Thus, it follows from eqns. 2.14 and 2.17 that

«= SH±l!!*>gV ,2,0,c o f i l 2 \ Z \ )

to an accuracy of 0 (| Zs 13/2) •We now deal with the current-density distribution in the quasistatic vicinity

of the edge determined by the condition kx<^\. Using eqn. 2.14, we obtain

% „ (2.21)In

Unfortunately the integral representation for F(x) resulting from eqn. 2.14 israther complicated for further use. We shall therefore construct a simple approxi-mation of F(x) on the basis of the following a priori information: first F(0), asfollows from eqn. 2.20, is finite; secondly, from the physical point of view it isevident that F(x) diminishes monotonously, and that F{x) ->2{njx)112 at x-> oo.The simplest approximation of the function satisfying all conditions mentionedabove has the form

F{x)^2nll2{x + 2\Zs\I^G2kW0)-x/2 (2.22)

This approximation is quite sufficient for calculation of the dissipated power.Let us now analyse the current-density distribution for a structure with a

rounded edge. We assume that

(i) the effective penetration depth Aeff15 is small compared with all geometri-

cal dimensions of the structure; and(ii) kR<4l,kD<\.

From assumption (i) it follows that the structure material can be considered asa perfect conductor. The scheme of analysis is as follows: beyond the bounds ofa certain small vicinity of the edge (x < S, kd <| 1) the induced current-densitydistribution does not differ from the corresponding distribution for an infinitelythin halfplane. Inside this vicinity the current-density distribution has a quasi-

15 For the Leontovich impedance, the value Ao is used as Ae


static character and, after simple curve approximation of the edge shape, canbe determined by the conformal-mapping method up to a constant factor. Thisfactor is determined by means of matching of the solutions at x ^ S.

According to the conformal-mapping technique, the current density in theedge vicinity can be expressed as

J=Ciy (2.23)

where /(£) is a function which describes the conformal mapping of the presetregion on the exterior of the unit circle and C is a constant.

The current density near the edge of an infinitely thin halfplane excited bythe E-polarised plane wave is determined by the formula

jco^ { nx )

The constant C can be found through matching of the representations given byeqns. 2.23 and 2.24 at large values of x.

We assume that the edge of a 'thick' halfplane has an elliptical shape(Figure 2.3a). The field structure near the edge will then be the same as thatnear the edge of the elliptical cylinder. The function /(£) is determined, accord-ing to Lavrentyev and Shabat (1987), by the expression

where £ = x — a + jz, c= (a2 — b2)112 and a and b are the large and small semi-axes of the ellipse. From eqn. 2.23 it follows that near the edge

R\~112

(2.25)

where R=p and p — b2\a is the ellipse focal parameter. Comparing eqn. 2.25with eqn. 2.24, it is easy to find C and obtain the expression for the currentdensity near the edge

exp(./7t/4) y * ( l + c o s 3 ) ] 1 / 2

A ' = o - l > con0 \ n ( X + R I 2 ) ] ( Z 2 6 )

If the edge has a parabolic shape described by the equation z2 = 2px, where pis a positive number, then £ = x ~p\2 + jz and /(£) = (y/Q ~ j{\f(Pl%)} which,taking into account eqn. 2.23, leads to eqns. 2.25 and 2.26.

Now we can obtain the expression for the impedance to be found. A relevantcriterion of equivalence of an infinitely thin impedance screen and the actualscreen of finite thickness is the equality of mean-square-current densities (andhence energy losses) in the edge vicinity of the order of the curvature radius R.For a parabolic cylinder, the dissipated power for both sides in this vicinity isgiven by the relationship

v l / 2

Determining the dissipated power in the edge vicinity x < R for an infinitely


thin impedance structure in a similar manner on the basis of eqns. 2.21 and2.22 and comparing the results, we obtain

Zs^k(Aeffl2-j£R)W0 (2.27)

where £=1.12. For the elliptic edge we obtain the same expression, but(£ = 1.04. Calculations for the edge of a rectangular shape are similar; in thiscase

Zs^k(Aeffl2-j0.025D)lY0 (2.28)

It is noteworthy that the impedances expressed by eqns. 2.27 and 2.28 do notdepend on the incident angle of the primary plane wave.

It is necessary to emphasise that the numerical coefficient in the imaginarypart of Zs ls involved in the expression for the dissipated power in the argumentof the 'large' logarithm (see Section 5.2), which makes its numerical specification(through considering fine details of the edge shape etc.) superfluous. The dissi-pated power is therefore only slightly sensitive to the dimension of the edgevicinity in which the comparison is drawn (the vicinity being determined in justthe order of its dimension). For example, Ilyinsky and Slepyan (1986) treatedstructures with a rounded edge on the basis of a much rougher criterion ofequivalence than that mentioned above: the equality of induced current densitieson the edge of the actual structure with finite thickness and its infinitely thinimpedance analogue. In this case (E = 4.0, but even such a difference in (E doesnot change an order of dissipated power and the qualitative character of itsdependencies on the parameters of the structures under study.

The proposed variant of impedance boundary conditions allows us to avoidsolving mathematically complicated diffraction problems for screens of finitethickness when calculating the dissipated power. This method can be used incombination with the Wiener-Hopf technique and its modifications and alsowith direct numerical methods, e.g. integral-equation or dual-integral-equationtechniques. To conclude, we emphasise that it cannot be used to take intoaccount the influence of the screen thickness on the diffraction or dispersioncharacteristics of electrodynamic systems. The thickness is only considered whencalculating the metal losses in the cases where diffraction or dispersion character-istics are described satisfactorily through the model of an infinitely thin surface.

Special attention has been paid to the choice of the sign of the imaginary partof the impedances expressed by eqns. 2.27 and 2.28. It may appear that thischoice is of no importance as long as the lost power P is determined only by/^5 and |/£s|. However, with the opposite sign for the imaginary part of theimpedance, the plane may support the fictitious surface wave with dispersionproperties defined by the edge geometry. In this case, when solving spectralproblems it is necessary to detect and remove such fictitious modes. In diffractionproblems the excitation of the latter may distort the distributions of the electro-magnetic fields calculated.

2.5 The edge condition for an impedance halfplane locatedat media interface

In Section 2.4 we obtained expressions for an impedance halfplane, allowing usto consider the influence of edge effects on the absorbed power. However, the


analysis was made for a halfplane in free space. A question arises: is it possibleto generalise eqns. 2.27 and 2.28 for the case of a halfplane located at an interfaceof media with different permittivities? To answer this question we shall analysethe edge condition for this case, using an approach different from that inSection 2.4—the modified Meixner method (Braver et ai, 1988). This will allowus to illustrate and compare various approaches to this class of problems.

Figure 2.5 shows the geometry of the problem. We introduce the cylindrical-co-ordinate system r, </>, the origin of which coincides with the edge of thehalfplane. Mathematical formulation of the problem is as follows: let i/ x = Ez inregion si (0 < (j> < 7i), \jj2

= Ez in region $ (n < (j) < 2n). We wish to find thesolution of the Helmholtz equations

Vr20î + A:2i//1 = O (2.29)

V?0^2 + £ 2 # 2 = 0 (2.30)

satisfying the boundary conditions

j$ #1 r #

# ! # 2

= 0

= 0

= 0

(2.31)

(2.32)

(2.33)

(2.34)<f> = n

where d = ,The Meixner method (Meixner, 1972) is applicable for the case of perfectly

conducting surfaces. The solutions sought are represented in the form of powerseries

*,*- (2.35)

Figure 2.5 Impedance halfplane located at media interface


where am((j)) and bm(<fi) are unknown functions and T an unknown positivenumber. Substitution of eqn. 2.35 into eqns. 2.29 and 2.30 leads to a recurrentchain of equations for am((j)) and bm((j)). Their solution and imposition ofboundary conditions results in a characteristic equation for T, the smallestpositive root of which defines the field structure near the edge.

But, as Braver et al. (1988) have shown, for impedance structures such anapproach enables us to obtain only a trivial solution: am((j)) = bm((j)) = 0. Itmeans that at d # 0 the expansions for ij/^ 2 given by eqn. 2.35 are not applicable.Braver et al. (1988) proposed a modification of Meixner's method which usesthe expansions

= L L0 0

{4>)ln"r (2.36)

instead of eqn. 2.35. Recurrent chains of equations and boundary conditions

= 0 (2.37)

T)Hmn(4>) + 2(m + x)(n+

+ (n+2)(n+ \)bm,n-2{,n-2{4>) + k £bm-2,n(4>) = 0 (2.38)

(2.39)

(2.40)

follow from eqns. 2.29-2.34, where the prime denotes differentiation with respectto (j). The coefficients with at least one negative index in eqns. 2.37-2.40 areassumed to be zero.

Let us write out the first few equations of the chain given by eqns. 2.37-2.40.These equations define the behaviour of the current density near the edge andhave the form

a'oo(<t>)

a"l0 ( = 0

The current density J is expressed in terms of ij/i 2 by

(2.41)

(2.42)

(2.43)

(2.44)

The equalities flOo(0) = ôol^) a n d ^ I O W = ^lol^) result from eqns. 2.33 and2.34. Solving eqns. 2.41 and 2.42 and taking into account eqn. 2.43 at (j) = 0we have

sin


flioM = Ao cos{(l + T)0} + Q10 sin{(l + T)(/)}

where Q oo? Clio a n d ^10 a r e the constants of integration. It follows from eqn. 2.43that T may be only integer or half-integer (for our purposes it is enough toconsider T = 0, •£, 1). Using eqns. 2.36, 2.39 and 2.40, we obtain the first termsof asymptotic expansions for the field in the vicinity of the edge:

T = 0, \j/ = Qwr sin (j) +

(2.45)

The constants dOo> Clio remain indefinite in this analysis. This situation occursbecause we did not specify field sources and consequently we did not solve anydefinite diffraction problem (the analysis carried out reveals the general structureof asymptotic expansions for the field near the edge of impedance halfplane).

Let us analyse eqn. 2.45. When T = 0 and x = 1 it follows from eqn. 2.44 thatthe longitudinal component of the current density J z is finite at the edge ofimpedance halfplane, which is in agreement with results of Section 2.4. WhenT = \ the current density is zero (this case is of no interest to us). We can applyMeixner's approximation Jz{r) ^ Cr~1^2 (C is a constant defined by the incidentfield) at some distance from the edge within the quasistatic domain kr <^ 1. Thuswe come to the approximation of the current density in the edge vicinity

cJ.(0)

which is similar to eqn. 2.22. A new element following from the analysis per-formed is that the approximation is quite applicable when there is an interfacebetween media with different permittivities. The ratio C/Jz(0) does not dependon the relative permittivity £. Consequently, eqns. 2.27 and 2.28 may be alsoused for an unclosed screen placed at the media interface. That allows us, inparticular, to apply them to microstrip lines and other planar structures ondielectric substrata.

Completing our consideration of impedance boundary conditions, we wouldlike to mention the papers by Grinberg (1981), Schimert et al. (1991) and Dasand Pozar (1991), where generalisation of the Leontovich boundary conditionhas been implemented. In these papers an infinite plane-metal layer of thicknessd, characterised by the permittivity s = e0 — jaoj(D, is taken as the key structure.Then the boundary conditions which are imposed on the field at an infinitesi-mally thin surface are obtained and both the thickness d and the conductivitycr0 enter into the expressions for the coefficients in the boundary conditions. Ingeneral, these conditions may not, however, be compared with those obtainedin Section 2.4. They are intended for taking into account the field penetrationthrough the metal layer and do not consider the edge effects (the key structureis unbounded). The use of such conditions is justified for rather thin metal layerswhen the thickness d becomes comparable with penetration depth Ao (in thiscase the fields on both sides of the layer are interrelated). Attempts to apply


these conditions for calculation of the dissipation characteristics in structureswith unclosed screens, e.g. in microstrip lines (Das and Pozar, 1991), proved tobe problematic, as did the use of the ordinary Leontovich condition for thispurpose. Sufficient accuracy may be obtained, when the contribution of edgeeffects to the lost power is negligible (e.g. in a microstrip line with a very widestrip).

Chapter 3

Normal modes in waveguides withlosses

This Chapter deals with the theory of regular waveguides with impedance walls.Spectral problems for normal modes have been formulated. General character-istics of different types of normal modes (eigenmodes, associated modes) havebeen considered.

Calculation of attenuation coefficients of the eigenmodes for the case of smalllosses is discussed. General expressions for attenuation coefficients are obtainedwith the aid of the perturbation technique. Applicability cases for differentvariants of the perturbation method are analysed in detail (energy-perturbationmethod, perturbation technique for solving the dispersion equation). Theirmodifications are described for cases when classic schemes of the method are notusable (degenerated modes of D-multiplicity, models of waveguides withunclosed infinitely thin metal surfaces). Typical principal errors in attenuation-coefficient calculation are analysed.

Computational peculiarities of different approaches are pointed out, as wellas the expediency of their application for different cases. General principles areillustrated by specific examples (rectangular and circular waveguides, coaxialand microstrip lines).

3.1 Excitation of waveguides without losses

Here we consider the theory of excitation of regular lossless waveguides by givenarbitrarily distributed currents. This theory appears to be important from themethodological point of view: similar ideas and methods will be used later forthe analysis of waveguides with losses.

Suppose that we have a longitudinally regular waveguide in the form of aninfinite metal tube filled with a homogeneous medium. The axis of the tube isparallel to the z axis. The side surface of the waveguide will be denoted by Z,the cross-section in z = constant by SL, and its perimeter by C. Let the electric-

38 Normal modes in waveguides with losses

current density

J{x,j>, z, t) = Re{Je{x,y, z) exp{-jcot)}

be described by a finite function (Je 7 0 only in the interval Z\ <Z<Z2)have to find the solution of Maxwell's equations

curl E = jcofiH J

satisfying the boundary condition

( i t o xE) | I = 0 (3.2)

where n0 is the unit normal to X. In addition, E and //"should meet the radiationcondition at £—•ioo: that means that the fields at z> z2

a n d Z < Z\ arepresented as superpositions of waves moving away from the source. As there areno losses, z" = \JL' = 0.

The theory of excitation of waveguides is one of the applications of the spectralmethod, which is a rather general approach to solving a wide class of problemsof mathematical physics. Its essence is as follows: solutions sought are expandedinto series in terms of basis functions determined from specially chosen spectralproblems (eigenvalue problems). In our case, eigenmodes of the waveguideunder consideration are taken as basis functions. The expansion coefficients aredetermined from Maxwell's equations using such properties of basis functions ascompleteness and orthogonality.

There are different enunciations of the theory of waveguide excitation basedon the spectral method. Weinstein (1988) solved eqn. 3.1 directly; Kisun'ko(1948) proceeded from the fields intensities E and H to scalar and vectorpotentials (D and A and solved equations for them; Tikhonov and Samarsky(1977) constructed a solution introducing the electric and magnetic Hertzvectors. All these versions of the excitation theory are equivalent and can betransformed from one to another. We shall dwell hereupon the most convenientvariant offered by Weinstein (1988) and use it in our further analysis.

Let us introduce the Dirichlet's and Neumann's spectral problems in thedomain SL:

= 0c

where (#f'w)2 are the eigenvalues.These problems are classical ones; their properties are studied and described

in detail in many works on mathematical physics.Let us form the vector functions

Ees(x,y,z)-

\Tie,.. .. . , _ J« £

' (3.3)

x u)

Normal modes in waveguides with losses 39

, z)

" " » (3 .4 ,7/7'" \

where Af'w = {A:2 - ( / se 'm)2}1 / 2 , ^ = 1, 2, . . . and A: = o)(6/i)1/2. Alongside eqns. 3.3

and 3.4 let us introduce the functions Eei™, Hei™ obtained from eqns. 3.3 and3.4 through the replacement of he

s'm by ~he^. Let us combine the subsystems

Ees,H

es and E™,H™ in the united system Es, Hs (s= ± 1 , ± 2 , . . .):

It is easy to show that the functions Es and Hs are the eigenmodes of a givenwaveguide: they satisfy homogeneous (without sources) Maxwell's equations andthe boundary conditions required. The functions expressed by eqns. 3.3 are TMmodes of a given waveguide (//fz = 0); the functions expressed by eqn. 3.4 areTE modes (Ee

sz = 0). If the cross-section S± is not single connected, we shouldalso add TEM modes in {Es}, {Hs} (for N-connected domain S± there are N— 1TEM modes). As is shown by Weinstein (1988), the orthogonality relation

{(Es xHs.)- (Es, x Hs)}iz dSj. = <SS, -S>NS, (3.5)s±

is valid where Ns is the sth eigenmode norm.At z > Zi t n e excited field can be presented as

(3.6)

and at z < Z\

[E

where Ts and Rs are complex coefficients which at this stage are unknown.Eqns. 3.6 and 3.7 are an analytical form of the radiation condition.

Validity of representations given by eqns. 3.6 and 3.7 follows from the com-pleteness property of the systems {Es}, {Hs} (s = + 1, +2, . . .) in a class offunctions meeting homogeneous Maxwell's equations for an arbitrary cylindricalarea (including the ^-unbounded one) with the cross-section SL. E and H aredetermined by the same systems of coefficients Ts and Rs. This can be explainedby the fact that there are no sources in the area under consideration. As hasbeen shown by Tikhonov and Samarsky (1977), the series given by eqns. 3.6and 3.7 are uniformly and absolutely convergent.

The transverse components of the vectors Es and Hs can be used as basisfunctions to expand the transverse components of the vectors sought inside theregion containing the sources (z\ <Z<Z2)- Thus let us present

E = El + El H = H+Hl Je=Jet+Jel

superscripts t and / denoting the transverse and longitudinal components of a


corresponding vector. It can be shown that the following formulae are valid:

= j(OllHl

dz

curl Hl + [I x = -jcozEt+J"(3.8)

c u r l £ r - | i,x

curl Hx - I L xdz

= -jcosE1

(3.9)

Eqns. 3.8 are the transverse part of Maxwell's equations, and eqns. 3.9 thelongitudinal part.

Let us expand the transverse components of the electromagnetic field intoseries in terms of the transverse vector functions El and Hi:

(3.10)

Here C±s(z) are unknown functions of the longitudinal co-ordinate z to bedetermined. Substituting eqn. 3.10 into eqns. 3.9, we obtain the expansions forthe longitudinal components of the electromagnetic field:

fel

(3.11)

Hl=s>0

(3.12)

Combining eqn. 3.10 with eqns. 3.11 and 3.12 we finally obtain the expansions

E=

s > 0

(3.13)

(3.14)

Our task will now be to determine the coefficients C±s(z). We shall use theLorentz lemma for this purpose. Let the electromagnetic fields E1, Hx andE2, H2 be excited by the currents J\ a n d y | , respectively. Then the followingidentity is correct (Lorentz lemma):

{(E1xH2)-(E2xH1)}n0dS=S JV

(3.15)

where V is an arbitrary closed volume and S is the boundary surface of thisvolume.

Let us apply eqn. 3.15 to the infinitely small volume inside the waveguidebounded by two sections z = Z, Z = Z + d^ and part of the side surface Z enclosedbetween them. Taking into account that Ex and E2 satisfy eqn. 3.2, we obtain


from eqn. 3.15

-fd) {(E1xH2)-(E2xHl)}izdS=

(3.16)

This equation can be used for determining C±s(z). Assuming that El,H1 ineqn. 3.15 is the field to be found, presented by eqns. 3.13 and 3.14 (J\ =Je)and E2 = E_S, H2 = H_S, {Je2 = Q) and using the orthogonality relationexpressed by eqn. 3.5, we have

A (^ i r- p = - JeE_sdSL (3.17)dz Ms Js±

where s = ± 1, ±2, . . . , ± oo.Boundary conditions for eqn. 3.17 result from the radiation condition; in

accordance with eqns. 3.6 and 3.7 they have the form

s = 1, 2, . . . , oo. The fields matching in the z — £1,2 planes results in

Thus the problem of excitation of a waveguide by given currents is solvedcompletely.

3.2 Excitation of waveguides with losses in the walls

The classical formulation for the excitation equations of a waveguide by givensources, examined in Section 3.1, is based on the expansion of the field soughtinto a series in terms of the eigenmodes of the waveguide considered. Tosubstantiate this approach we have to prove the completeness (basisness) of thesystem of eigenmodes. For lossless waveguides it can be done with rather generalsuppositions concerning the form of the waveguide cross-section (see Section 3.1).In the presence of losses the situation is more complicated. Eigenmodes systemsof waveguides with losses may be incomplete and not forming a basis in therequired functional space. However, where we have a basisness, for waveguideswith losses we can repeat the scheme described in Section 3.1.

It would therefore be quite natural to construct a theory of excitation forsystems with losses using the expansions of the fields sought in terms of someother complete systems of functions, for example the eigenmodes system of alossless waveguide of the same shape. In this case, the field sought and the basisfunctions used for its expansion meet different boundary conditions, so generalis-ation of the method is needed. Later we shall describe this generalisation basedmainly on the paper by Arkadaksky and Tsykin (1975a).

We need to find the solution of nonhomogeneous Maxwell's equationssatisfying the boundary condition

( » o x £ ) + ^ | H 0 x ( n o x H ) } = 0 (3.18)

on the side surface S. Here, the impedance Zs ' s independent of the z co-


ordinate. The field sought should satisfy the condition 'at infinity', that forwaveguides with losses (Re Zs > 0) has the form

This condition is responsible for the uniqueness of the solution of the problem.Before solving the boundary-value problem given by eqns. 3.1, 3.18 and 3.19,

let us derive an auxiliary identity. We introduce two systems of fields E±, H1

and E2,H2 (excited by sources J\ andyf ) satisfying the boundary conditiongiven by eqn. 3.18 with the impedances Zsi a n d Zs2> respectively. Applyingeqn. 3.15 to the infinitely small volume d F = 5 ' i d ^ w e obtain for the given classof fields the formula

- I {(E1xH2)-(E2xH1)}izdS

-(Zsi-Zs2)i> {(noxH^xH^ttodC

(J\E2-Je2El)dV (3.20)

Eqn. 3.20 is a generalised version of eqn. 3.16 for the case where impedanceboundary conditions are imposed.

As a basis for our analysis we shall use Galerkin's incomplete method (seeSection 1.3). Let El9 Hx be the fields sought (J\ =Je, Zsi = Zs) and E2, H2

be the fields of the /?th eigenmode£_p,/f_p (Je2 = 0, Zsi = 0,P = ± 1 , ±2, . . .)•

Then eqn. 3.20 at various/? forms a chain of orthogonality relations equivalentto the original boundary-value problem given by eqns. 3.1, 3.18 and 3.19. Letus present the fields sought as expansions in terms of the eigenmodes {Es}, {Hs}of a waveguide of the same shape but without losses (the expressions for Es, Hs

are presented in Section 3.1). Corresponding expansions have the form ofeqns. 3.13 and 3.14. It should be mentioned that in the case considered theexpansions given by eqns. 3.13 and 3.14 (unlike those in Section 3.1) cannot bedirectly substituted into Maxwell's equations to determine the coefficients C±s(z)because it is impossible to apply the term-by-term curl operation to theseexpansions. However, this is not necessary; we avoid it through using eqn. 3.20.

Using eqns. 3.13 and 3.14 and considering the energy orthogonality of theeigenmodes (see Section 3.1), we obtain the following system of ordinarydifferential equations for the coefficients C±s(z)'.

l °° C

•/Vp s= - oo JC

The last term in eqn. 3.21 describes a mutual coupling of different eigenmodesdue to the surface impedance of the walls. Eqn. 3.21 is a formulation of excitationtheory in terms of coupling modes.


To make further analysis easier we introduce new variables according to theformula

Then for Xp we obtain the system of ordinary differential equationsAY °o

p - /? 4- V Y A

yP JSJeE°.pdSL

where

Asp=jL jy

PJc

E°p=Epexp(-jhpz) H°p=Hpcxp(-jhpZ)Proceeding to vector-matrix designations, we have

i^ti {H°_px(noxH°)}.nodC

(3.22)

(3.23)

where X and R are vectors of the coefficients {Xp} and {Rp}, respectively, A isa matrix of the elements Asp and the matrix A is independent of z.

The system given by eqn. 3.23 does not represent the full formulation of thetheory of excitation of waveguides, as it has no unique solution. This has to dowith the fact that eqn. 3.23 does not take into account the condition at infinitygiven by eqn. 3.19. This system should be solved under some auxiliary conditionson the vector X sought. However, the analytical form of these conditions interms of coupling modes is rather complicated. Therefore we postpone theirformulation until the next Section where the general solution of this system isexamined.

3.3 Eigenmodes in waveguides; dispersion characteristics

Suppose that we have a nonsingular matrix B which reduces the matrix A tothe diagonal form, i.e. implementing the transformation

r = BAB~1 (3.24)

where T is the diagonal matrix

o


and yt are some complex numbers. From the reciprocity theorem it follows thatthey can be numbered in such a way to fulfil the equation y_f = — yt.

Then, introducing new variables

we obtain from eqn. 3.23 the equation for g

^ = Tg + K (3.25)&Z

where K = BR.The field sought can be presented in terms of the new variables in the form

(Arkadaksky and Tsykin, \91bb):

Z {gs{z)Es{x,y)s>0 (3.26)

s > 0

where gs(z) is the sth component of the vector g(z) and

. O P ( ^ J ) } 1 (3.27)

bps are elements of the matrix B~x, and E°(x,y) and H°(x,y) are expressedby eqn. 3.22.

It can be easily seen that eqns. 3.26 are an expansion of the excited field interms of the eigenmodes of the impedance waveguide. Eqn. 3.25 can be writtenin the form

ns Js±(3.28)

Eqn. 3.28 coincides with that obtained according to a classical scheme (seeSection 3.1) when we introduce new variables: gs{z) = Cs(z) exp(jysz). By ns

and ys we mean the norm and the wave number of the ith mode of the impedancewaveguide.

Thus if there is a transformation given by eqn. 3.24 for matrix A, then theexcited field can be presented as a series in terms of eigenmodes expressed byeqn. 3.26 with coefficients determined from eqn. 3.28. This equation is easilyintegrated in quadratures. The functions Es, Hs defined by eqn. 3.27 are trans-verse field distributions of the sth eigenmode in the waveguide with losses (tobe more precise, their representation in the basis {E°}, {H®}).

As was pointed out in Section 3.2, to determine the unique solution ofthe problem some auxiliary physical conditions are required. Suppose that theexternal sources are localised in some finite area Z\ <z<Zj- Then, ifthe transformation expressed by eqn. 3.24 exists, to meet the condition at infinityin the areas z > Z2 a n d z < Z\ we should reject the increasing waves, i.e. add thefollowing boundary conditions to eqn. 3.25

g-,(z2) = 0, g.(zi) = O (3.29)

In lossless systems the solution sought can be selected by different methods: with


the aid of Sommerfeld's radiation condition, Mandelstam radiation energyprinciple, principles of limiting absorption and limiting amplitude. Analysis andcomparison of such principles applied to the problems of the dynamic theory ofelasticity, are contained in the book by Vorovich and Babeshko (1979). Wewould like to stress here an a priori and heuristic character of these principleswhich limits the area of their application. Only for the simplest problems arethese principles equivalent. The main difficulties in their use are met when thereare associated modes or waves with anomalous dispersion (see eqn. 3.45).

Thus, using eqn. 3.29, the electromagnetic field at z > li and z < Z\ can bepresented as

The eigenmodes of a waveguide with losses in the walls satisfy the orthogonalityrelations given by eqn. 3.5, and hence each of them in the area which is free ofsources propagates independently from the left. Thus eigenmodes have a clearphysical sense: these are the fields that can be excited in a waveguide outsidethe area occupied by sources.

Let us consider now the conditions of existence of the diagonalised transform-ation expressed by eqn. 3.24 for the matrix A. The theorems formulated hereare proved in the book by Fadeev and Fadeeva (1963) for a matrix of finiteorder. Transition to infinite matrices is valid under certain conditions whichjustify the operations carried out. Comprehensive substantiation can be givenonly for specific cases [see, for example, Arkadaksky and Tsykin (1915a)] whereit was made for a parallel-plate impedance waveguide. Below we shall considerthe corresponding conditions fulfilled without mentioning specifically the differ-ence between the finite and infinite matrices. The following theorem is correct:

Theorem 3.1: If all eigenvalues of the matrix A are different in pairs, then thetransformation given by eqn. 3.24 exists and diagonal elements of the matrix Tcoincide with eigenvalues of the matrix A.

The characteristic equation of the matrix A

0 (3.30)

defines the dependencies ys(co) at given waveguide parameters.Thus, eqn. 3.30 is the dispersion equation of the waveguide with losses.Eigenmodes propagating in waveguides can be divided into four groups:

(i) fast modes characterised by the condition Reys(co)<£ (vPh > c) (for agiven shape of a waveguide their spectrum is defined by the area of thewaveguide cross-section);

(ii) slow (surface) modes characterised by the condition Re ys(co) > k (vPh < c)\(iii) TEM modes (modes which do not have longitudinal field components)

characterised by the condition ys (co) = k = co/c; and


(iv) near-boundary modes; fast waves, whose spectrum at a given shape of awaveguide is determined not by the area but by the perimeter of thewaveguide cross-section (Nefedov and Rossiisky, 1978).

The modes of the first type are specifically conventional TM and TE modesof hollow rectangular and circular waveguides with a smooth metal surface.

The modes of the second type are typical for their field being concentratednear the guiding surface and rapidly transversely decreasing (without losses;exponentially). Such modes can exist in waveguides with impedance walls or inthe presence of dielectric insertions.

The TEM modes do not undergo any dispersion and their field structure isstrictly transverse. Such modes can exist in multiply connected lossless guidingsystems (e.g. coaxial lines). In the presence of heat losses in the walls there arelongitudinal field components and dispersion effects, but with small losses theyare notable only slightly.

Near-boundary modes can exist both in impedance waveguides and in idealwaveguides of a sufficiently complex shape of the cross-section (in the latter casethey should necessarily be TE modes). The theory of near-boundary modes isdescribed by Nefedov and Rossiisky (1978). Thus, for example, the TEml modesof a coaxial waveguide and biconical horn and some TE modes in a cross-shapedwaveguide can be considered as near-boundary. The important features of suchmodes are a rather rare spectrum and the ability to reflect totally from wideningsegments of a waveguide. Based on the same principle, effective selection ofparasitic types of eigenoscillations in open resonators is possible.

One of the most important characteristics of eigenmodes is the dispersioncharacteristic: the dependence ys(co). As well as calculation of dispersion charac-teristics of a specific waveguide system, qualitative investigation of dispersionlaws, in the most general sense, is very significant. It permits discovery of generalphysical properties typical of different classes of eigenmodes and not connectedwith the special features of the waveguide shape.

An effective method of carrying out such an investigation, based on the theoryof analytical functions of many complex variables, was suggested by Krasnushkinand Fedorov (1972). Below we shall touch upon the problem's formulation,methods and the main results of these investigations.

Let us assume that the waveguide under study is characterised by the param-eters Wx, W2, . . . , Wn (these parameters describe the waveguide geometry,characteristics of its surface and filling media etc.). Then eqn. 3.30 can besymbolically written as

A(7, W1;\V2, . . . , Wn)=A(y, {Wt}n) = A(y,W) = 0

Let Wt be the co-ordinates of a point in the ^-dimensional complex space C"and (y, {Wi}n) the co-ordinates of the point A e C " + 1 . If the coefficients Asp areanalytical functions of {Wt}, then A is an analytical function of A. In this casethe dependencies ys(W) would be holomorphic functions of n variables {Wi)n

excluding the degeneration points in which

^ = ^ = . - . = ^ = 0 („<„) (3.31)dy dy2 dym K J K J


These points will be branch points of the mth order. When among eigenmodesare degenerated ones (with the same propagation coefficients), i.e. complyingwith eqn. 3.31, then the question on existence of diagonalised transformationexpressed by eqn. 3.24 is still open. The answer is given by theorem 3.2.

Theorem 3.2: If some of the eigenvalues of the matrix A are multiple, thetransformation given by eqn. 3.24 exists only when the multiplicity of eacheigenvalue equals the number of linearly independent eigenvectors correspond-ing to this eigenvalue (Fadeev and Fadeeva, 1963).

If the conditions of theorem 3.2 are fulfilled, the degenerated modes havedifferent field structures and their orthogonal linear combinations can be formed.Such degenerated modes are called modes of diagonalised multiplicity (D-multi-plicity modes). If all the degenerated modes are of D-multiplicity, the excitationtheory in terms of eigenmodes is also correct.

When affirmation of theorem 3.2 does not take place then the transformationexpressed by eqn. 3.24 does not exist and transition from coupled waves toeigenmodes is impossible.

In the general case, this transformation can be used to reduce the matrix Ato the Jordan canonical matrix form (Fadeev and Fadeeva, 1963):

X, 1

Thus, if My-multiple degenerated j th eigenvalue corresponds to m-j < Mj linearlyindependent vectors, j = 1, 2, . . . , JV, then the matrix F contains N Jordanblocks of the order of Mj — my + 1 belonging to the eigenvalues ky Degeneratedeigenmodes responsible for the Jordan blocks of the matrix F are called modesof Jordan multiplicity or, in short, J-multiplicity. It is worth pointing out thatJ-multiple eigenvalues can be found only with non-self-adjoint operators.

The dependencies ys(W) are presented in Cn + 1 by a countable set of analyticsurfaces. These dependencies are studied by Krasnushkin and Fedorov (1972)on sets of routes <£f (J, H), where s is a real parameter determining the locationof a point on a curve presenting the route and \i is the real index selecting thegiven route out of the set. To study the behaviour of branches ys(W) in theneighbourhood of branch points determined by eqn. 3.31 it is necessary totransfer from C" to some Riemann manifold £ on which the functions underexamination are single-valued. To construct such a Riemann manifold we should


introduce m copies of C", make cuts along their branch surfaces and glue differentedges of branch cuts to each other in pairs.

If at the wave 'movement' in the space Cn a branch cut is crossed, then atransfer to another 'sheet' of Riemann manifold S takes place [to another branchof ys(W)]. A secondary crossing in the opposite direction restores the initialsituation. Thus the global structure of branches of ys(W) is determined by thelocation of the branches' intersection points and their behaviour in the neigh-bourhood of these points. If we take a frequency CO as one of parameters {W^}n,then we can also study a structure of dispersion characteristics of waveguides.

As an example, let us analyse eqn. 3.30 assuming that only two modes areclosely coupled (these will be the modes with numbers s and/?). Then eqn. 3.30transfers into a 2nd-order equation and we can obtain the following expressionsfor the roots:

where

/$ = A — A l = ( ) ( A A \1/2

Fields' distributions of these modes are presented as

up=CppVp+CpsVs

where

and Cmn (m, n = s, p) are coefficients satisfying the formulae

(3.33)

Eqns. 3.32 and 3.33 are relatively similar to the corresponding formulae forplane-layered media studied through this method by Krasnushkin and Fedorov(1972). So, without going much into detail, let us formulate the main results.

In this approximation, it is evident that not more than a 2-fold degenerationof modes is possible. The points W2 (points of 2-fold degeneration) form in C2

a set determined by equation

Assuming that Wx = 5 and W2 = /, we find from eqn. 3.24 that the points ofJ-multiplicity Wj lie on straight lines d = ±1 and the point of D-multiplicity WD

is located at their intersection and thus is not isolated. Taking into account theabove considerations let us introduce two copies of the space C2 and cut themalong the surface £c spanned by the straight lines 5 = + /. Then we glue thecopies of C2 along both sides of Sc (Figure 3.1). In such space the function


s\\\\ \ P

. fLs'

a

Reh6

Figure 3.1 Schematic representation of points of 2-fold degeneration in C2 (Krasnushkinand Fedorov, 1972)

a The set £e{s, /i)b The dependencies y{W)

y(W) would be single-valued. Five types of the set J£?(J, /*) have been examinedby Krasnushkin and Fedorov (1972):

(i) S# does not cross WD and Sc (there are no wave transformations).(ii) Scg does not pass through WD but crosses Sc. In this case at \i = /Jo, when

S# crosses a point of Sc, the so-called exchange of partial systems bynormal waves takes place. The essence of this phenomenon is as follows:at a transition through a point of J-multiplicity the character of dispersiondependencies (see Figure 3Ab) and the field structure of the eigenmodeschange sharply because of the change of the sign before the square rootin eqns. 3.32 and 3.33.

(iii) S# passes through WD but does not cross Sc [for example, S# coincideswith the plane (5\ 6") and s = S\ /j = <5"]. In this case there is no field-structure alteration when passing through the degeneration pointS' = 5" = 0, as at this point there is no coupling between the partial waves.However, there is an alternative method of construction of branches ofdispersion characteristics, as in the previous case.

(iv) S# passes through WD and along one of the straight lines d = ±1 (degener-ated case). Then all routes passing through Wj (or WD) will cause branchesof dispersion characteristics that are separatrixes (solid lines inFigure 3.1£). In this case for each such route there is an alternative, as incase (iii) for \i = 0.

(v) S& passes through WD, at once reaching the area Sc. In this case, as incase (ii), for routes at any small fi there is an exchange of partial systemsby normal waves. It is caused by the fact that WD is not isolated from the


connected set of points W3. This case, however, differs from case (ii) inthat the wave shapes at W — WD remain different.

To conclude we wish once more to underline the fact that for the existenceof wave-transformation effects not only an existence of degenerated points isnecessary but also the presence of coupling between degenerated partial waves.When there is no coupling (Asp = 0), the phenomenon of wave transformationis not observed and points of multiplicity do not differ from the ordinary ones.We shall later call such a case a trivial multiplicity. The term 'D-multiplicity'will be reserved for cases when there is a coupling between degenerated partialwaves.

For example, let us consider modes of rectangular and circular impedancewaveguides and analyse dependencies yp(Zs)> where Zs 1S t n e surface impedanceof the walls. At £s = 0 for any operating frequency the TEwn and TMmw modes(m,n>\) in a rectangular waveguide and the TEOn and TM l n modes in acircular waveguide are degenerate. In a rectangular waveguide, when Zs ^ 0,the partial TEmM and TMmn modes are coupled (with the exception of a squarewaveguide a = b) because they are D-multiple modes. In a circular waveguidedegenerated modes have different azimuthal dependency and are not coupled;therefore, the point ^ s = 0 in this case is that of trivial multiplicity.

3.4 Associated waves

Let us examine in more detail the specific case when there is only a 2-folddegenerated eigenvalue that is of J-multiplicity (let us assume yl to be such aneigenvalue). Here we shall demonstrate the peculiar features of excitation equa-tions caused by the presence of eigenmodes of J-multiplicity.

Jordan canonical form for this case can be written as

r =77-2

r0772

(3.34)

where the matrix Fo is presented as

77i 1

77i

"77i

1 "77i

The corresponding equation for the variables gs has the form of eqn. 3.25, wherethe matrix F is given by eqn. 3.34. For all gs except J = 1, equations have an


ordinary form, so we shall in future limit our discussion to examination of thedegenerated mode. It is characterised by a four-component vector

gX

satisfying the equation

(3.35)

Eqn. 3.35, as is evident, cannot be reduced to excitation equations in the formof normal modes given by eqns. 3.28 and 3.29 but differs greatly from the latter.To understand the essence of these differences let us consider eqn. 3.35 in anarea, free of sources (Ko = 0). Writing eqn. 3.35 in a scalar form, we obtain

dz

The solution to eqn. 3.36 has the form

(3.36)

(3.37)

where C is an arbitrary constant. The amplitude coefficients g* (z) describe theeigenmode of the given type, and G * (z) is the so-called associated wave. Thus,in the excitation theory, the mode of J-multiplicity is presented as the sum ofthe eigenmode and the associated mode. The latter, though it disappears atinfinity, has a rather unusual dependency of the field on the longitudinal co-ordinate. When the associated wave occurs, the condition at infinity given byeqn. 3.19 cannot be defined in the form

H= 0{exp(-Ck| )}

at z~+ ±oo, where C is a constant. This condition is too rigid and, as followsfrom the analysis presented, the problem stated in this manner may have nosolution.

Determining the transformation matrix B in eqn. 3.24 (to be more precise,the bltP coefficients), we can obtain a representation of the associated wave inthe form of an expansion in terms of eigenmodes of a lossless waveguide of thesame configuration.

Let us now examine equations for the electromagnetic field in the presenceof modes of J-multiplicity. Assume that we have the eigenvalue problem

Lu-XQu = 0 (3.38)

where QJ1 L is a non-self-adjoint operator. As is known, if k is an eigenvalue ofmultiplicity m, then the set of eigenfunctions and associated functions of the


problem given by eqn. 3.38 satisfies the operator equation (Keldysh, 1951)

Taking m = 2 for simplicity, we obtain the chain of equations

(3.39)

where w(0) and w(1) are eigenfunctions and associated functions, respectively.Applying the general formulae given by eqns. 3.39 to waveguide problems,

we can write for transverse field distributions of eigenmodes the equations

± E{0)+jy{iz x £(0)) = > / ( 0 ) j

curl± Hi0)+jy(iz x H{0)) = -j(osE{0) j

where the /?(0) and H{0) satisfy the impedance-boundary condition of the formof eqn. 3.18, the symbol _L indicates that the derivatives djdz are replaced byzeros and X = — jy.

Transverse-field distributions of the associated modes satisfy the equations

curl± Ea) +jy(iz x E{1)) = jcotrit{1) + E(0) |

_ iz x H(1)) = -ja>eE{1) + H{0)

and the same boundary condition. It is noteworthy that similar equations canalso be written at m > 2.

In the area without sources, the forward and backward associated modes areexpressed by

„ + r = £ i W \^1] f ' N [- Gl (z) \&0), ' , (3-42)

(3.43)

whereg* (4;) and G * (z) are expressed by eqns. 3.37, J£(0>1)(x, j>) and ^0,1) (x? j )are determined from eqns. 3.40 and 3.41.

Using eqns. 3.40 and 3.41, we can show that the associated modes expressedby eqns. 3.42 and 3.43 satisfy Maxwell's homogeneous equations.

Note that the associated wave is not such a rare and exotic phenomenon asit may seem at first. Associated waves occur in conventional hollow waveguideswith perfectly conducting walls excited at a cutoff frequency. As y(coc) = 0, thewaves travelling in opposite directions become indistinguishable (linearly depen-dent) and the points co = (Oc become of J-multiplicity. Therefore the classicalexcitation equations for waveguides at the cutoff frequencies contain the in-determinate forms. The modification of excitation theory has been made byArkadaksky and Tsykin (1976) by means of the limiting transition co—>ooc] thusthe associated wave is extracted in an explicit form.


3.5 Types of dispersion characteristics; a concept ofanomalous dispersion; complex waves in losslesswaveguides

Let us consider typical dispersion characteristics of waveguides. As mentionedabove, their trend will be determined mainly by the specific points of themultiple-valued function y(co) that can be obtained from eqn. 3.30.

The most simple type of dispersion characteristics is observed in losslesswaveguides. In this case forward and backward modes of the same type may bedegenerated. It corresponds to the cutoff frequency coc [y(coc) = h(coc) = 0].This case is illustrated by Figure 3.2a. Branches BC and BC correspond to thepropagation area (co > coc); in this area Im h = 0, Re h > 0 for the forward wave,and Re h < 0 for the backward one. Branches BA and BA' correspond to thebelow-cutoff area (co < (oc)\ in this area Re h = 0, Im h > 0 for the forward wave,and Im h < 0 for the backward one. In the presence of losses both branches CBAand C'BA' become complex-valued at any 0) and may have no contact points.However, if the losses are small, their trend remains approximately the same asin lossless systems (broken lines in Figure 3.2a).16

The trend of dispersion dependencies in the propagation area depends essen-tially on the type of waveguide. For waveguides with smooth perfectly con-

Qc

Figure 3.2 Dispersion diagrams

a Cutoff vicinityb Propagation region

Qt

16 This is why the concept of cutoff frequency for lossy waveguides is not uniquely determined andhas no such direct meaning as for lossless systems. However, when the losses are small, it wouldbe advisable to use this concept, as it permits differentiation between two essentially differentoperational regimes of a waveguide.


ducting walls the trend of the dependency h(co) in the propagation area is shownby the solid line in Figure 3.2b. In this case the dispersion curve always liesunder the straight line Re h = co/c. In waveguides with impedance walls, anothertype of dispersion characteristic is possible (broken line in Figure 3.2b). At theintersection point of this curve with the straight line Re h = cojc transformationof the fast wave into a slow (surface) one occurs. At the point co = cot the surfacewave undergoes a cutoff: in this case the function h(co) has a pole owing to thefact that the function A (A, co) for the impedance waveguide may be nonanalyt-ical. It distinguishes this system from, for example, a layered dielectric medium.Note that for the modes shown in Figure 3.2b the so-called normal dispersion istypical, i.e. the following condition is fulfilled:

d /Reh(co)\— — >0 (3.44)dco\ co J

In complicated waveguide systems, for some types of modes an anomalousdispersion can be observed when the condition given by eqn. 3.44 is replacedby the opposite one

d {Reh(co)\— ^ <0 (3.45)dco \ co )

In some types of waveguide even when there are no losses the branch points ofdispersion characteristics can be located, unlike the case shown in Figure 3.2<2,not at the origin of co-ordinates but on the co-ordinate axes of the plane of thecomplex variable h. In this case more complicated branch systems are formedwhich can cause couples of complex waves in a certain range of frequencies. Forthese waves the -dependence is one of the forms

exp (±jh'z) exp (h"z)

exp{±jh'z)exp(-h"z)

where A'= |Re A| > 0 and h" = |Im h\ > 0.It may seem at first that such waves cannot be found in lossless waveguides.

However, we have to take into account the fact that complex waves are excitedin pairs with equal amplitudes17 and form two reactive (i.e. not carrying theenergy) waves, similarly to ordinary evanescent modes but having a sine spatial-amplitude modulation. For such waves, usual orthogonality conditions expressedby eqn. 3.5 are inapplicable (see Illarionov et at., 1980).

Complex waves in lossless systems were found for the first time by Miller(1952) in a waveguide with anisotropic impedance walls and more recently ina number of electrodynamic systems: gyro tropic plasma (Gershman, 1955),circular waveguide with a dielectric rod (Glarricoats and Taylor, 1964, Veselovand Lyubimov, 1963) and screened microstrip line (Kovalenko, 1980). Theproperties of these waves were studied by Beliantsev and Gaponov (1964) andKrasnushkin (1974).

17 This conclusion, generally speaking, cannot be drawn on the basis of an analysis of dispersioncharacteristics. For that we have to solve the problem of excitation of the waveguide by an externalsource. The solution of this problem (obtained by Beliantsev and Gaponov, 1964) proved thisassumption.


The existence of complex waves usually results from the interaction of partialguided waves with differently directed energy flows. Complex waves are essentialfor correct formulation of excitation equations in complicated waveguide systems,rigorous analysis of waves scattering by various discontinuities etc.

In the presence of losses, the branch points slip from the co-ordinate axes inthe plane of the complex variable y; in this case the degeneration is possibleonly for modes of different types (backward waves corresponding to the replace-ment y-• - y are shown in Figure 3.2a by broken lines). The degeneration pointscan be points of trivial, D- and J-multiplicity. In the last two cases the electro-magnetic field in these points is of a complicated character: in the first it is asuperposition of coupled degenerated eigenmodes, in the second a sum of theeigenmode and the associated wave (see Section 3.4).

3.6 Excitation of TM modes in a parallel-plate impedancewaveguide

Excitation theory of waveguides with ohmic losses in the walls, described inSection 3.2, is quite general, though analysis of the relationships derived is arather difficult task. The reason is that to arrive at an expansion in terms ofnormal modes of the waveguide under consideration it is necessary to analysethe dispersion equation given by eqn. 3.30 in the form of an infinite determinant.

There is, however, another approach to this problem which does not need ana priori modal expansion of the field sought: the integral-transform technique.The range of applicability of this method is considerably narrower than that ofthe approach examined in Section 3.2. However, the integral-transform methodallows one to construct and analyse the solution in a much simpler way. Weshall later examine it using as an example the problem of excitation of a planewaveguide with an impedance wall (Figure 3.3) by a 'magnetic' line source.

Mathematical formulation of the problem is as follows: it is necessary to findthe solution of the Helmholtz equation for the j-component of the magneticfield ij/{x,z)

VLiA + k2il/ = -jwed{x - x')8(z ~ z) (3.46)

t Jx'.z'ja 1*

J I zFigure 3.3 Parallel-plate waveguide with impedance wall excited by 'magnetic' line

source


satisfying the boundary conditions

f =0 (3.47)ux x = 0

= 0 (3.48)-a

and the conditions for field reduction at z~* ± oo. The complex parameter rj ineqn. 3.48 is expressed by the formula rj = k£sIW0.

This problem (set somewhat more generally) has been solved by means of theintegral-transform method by Markov and Vasilyev (1970). We shall later brieflydescribe the procedure of solution and analyse the formulae obtained. A solutionfor eqn. 3.46 can be sought in the form

where \j/1 is the field excited by a 'magnetic' line source in the free space and\j/2 is the so-called induced field satisfying the homogeneous 2-dimensionalHelmholtz equation and chosen in such a way that the boundary conditions forthe total field given by eqns. 3.47 and 3.48 should be fulfilled.

The field \//1 (x, z) is expressed by

where H^(u) is a Hankel function of the first kind and zero order, R ={(x — x')2 + (z~ Zr)2}112, x' and z' are the co-ordinates of the line source and

The field \j/2 {x, z) can be sought as Fourier's integral:

f00ij/2 =jcoe {A{y) cos Px + B(y) sin fe) cxp{jy{z - z')} Ay (3.50)

J - o o

where A(y) and B(y) are unknown functions which should be determined fromeqns. 3.47 and 3.48. Substituting eqns. 3.49 and 3.50 into the boundary con-ditions given by eqns. 3.47 and 3.48, we obtain a system of two equations forthe unknown functions A(y) and B(y). Finding A(y) and B(y) from this system,we can write down the solution of our boundary-value problem as

•"•"-C p{psm(Pa)+jt,cos(lla)}

xexp{jy(z-z')}dy (3.51)

Since the line source is located right on the impedance wall (x' — a) and z — 0,eqn. 3.51 can be simplified and will have the form

We shall use this formula as the basis for our further analysis. First note thatthe function under the integral is single-valued at every point of the complexplane y (it has no branch points). Therefore \j/(x, z) expressed by eqn. 3.52 can


be completely determined by residues in the poles of the function under theintegral. The poles, as can be seen from eqn. 3.52, are the roots of the transcen-dental equation

f(P) = P s'm(pa) +jr] cos(Pa) = 0 (3.53)

i.e. a dispersion equation for TM modes.First we assume that eqn. 3.53 has only simple roots. Then, calculating the

integral in eqn. 3.52 through the Jordan lemma, we obtain

>l>(x,z)= £ Cmexp(r/mk|)cos(j?mx) (3.54)m = l

where

cosfa

and Pm are the transverse-wave numbers of TM modes of the plane impedancewaveguide. Thus, eqn. 3.54 is an expansion of the field sought in terms of theTM modes of the waveguide under examination (TE modes are not excited).So the transition from eqn. 3.52 to eqn. 3.54 is only possible if there are nodegenerated modes (multiple roots of eqn. 3.53).

Now suppose that eqn. 3.53 has multiple roots. This means that apart fromeqn. 3.52 the following system of equities should be fulfilled:

= 0 (3.55)

p= 1, 2, . . . , JV— 1, where JV is the root multiplicity. Analysis shows that atN> 2 the chain of relations expressed by eqn. 3.55 for different/? is incompatible.This explains why no more than doubly degenerated modes can exist in a planeimpedance waveguide. It is evident that these waves will be of J-multiplicity.Equating the derivative d/(/?)/d/? to zero and using eqn. 3.53, we obtain

ja{P2-f]2)= -t] (3.56)

which, together with the dispersion equation given by eqn. 3.53 determinesconditions of J-multiplicity of the eigenmodes. It follows from eqn. 3.56 that nomore than one pair of modes can be degenerate.

Expressing jS from eqn. 3.56 and substituting the result obtained into eqn. 3.53,we have

/ :n \i/2 C/:n \l/2 -\

F(r})=(J- + ri2) tan^ - + >/2 a\+jri = 0 (3.57)

which determines the values of the surface impedance at which there is adegeneration of eigenmodes. With purely imaginary rj, solutions of eqn. 3.57 arenonexistent, i.e. points of J-multiplicity can only be observed in lossy waveguides.

At a sufficiently small \rj\ the associated wave can only be initiated by asymmetrical eigenmode with small \p\ ~ O(\rj\1/2). The function F(rj) is analyt-ical in a certain circle with the centre rj = 0, and at the same time F(0) = 0.This is why, according to the theorem of uniqueness for analytical functions(Evgrafov, 1965), F(fj)^0 in some neighbourhood of the point 7 = 0. Thedegeneration point rj = 0 is evidently fictitious and does not lead to the excitation


of an associated wave. Thus, for sufficiently small \t]\ associated waves are absent,whatever the sign of 77" and relations between r\' and r\".

The integral in eqn. 3.52, can be determined through the Jordan lemma asbefore, though it is necessary to take into consideration the fact that one of thepoles of the expression under the integral (for example, the rcth one) is a poleof the second order. After calculation, we obtain

^{x,z)= X Cmexp(jywU|)cos(j8mx)m = l

3B2acos(B a) c o s ^ w ^ GXPOyn\z\)

f n \ z \ )} V k | c o s ( ^ ) + ^*sin(iSnx)V (3.58)

The first two terms in eqn. 3.58 are a sum of eigenmodes; the third term, as caneasily be proved by a direct check, is the associated wave.

Note that direct observation of the associated wave in the experiment is arather difficult task. The reason is that the associated wave exists only at somediscrete values of the surface-impedance parameter rj determined from eqn. 3.57.For regular waveguides, owing to fluctuations of parameters, inaccuracies inmanufacturing etc. we shall practically always be in the situation where onlynondegenerated modes exist, though phase constants and field structures of twomodes can turn out to be close enough. In this case an associated wave is amathematical abstraction, convenient for description of wave-transformationprocesses when their phase constants and fields distributions draw nearer. Irregu-lar waveguide transitions, such as, for example, impedance waveguides with avariable surface impedance, are another matter. Ifri(z), when varying, passesthrough the point of J-multiplicity, new physical effects may occur due to theexcitation of an associated wave. For a parallel-plate waveguide this problemwas studied by Bichytskaya and Novikov (1979). The method of coupled waveswas used as a basis; the solution of the system of differential equations has beencarried out asymptotically in a zero-order approximation in the parameter ofsmallness e~\drjldz\- The main result is as follows: if the sth eigenmode isincident on the segment of a variable impedance and there is a point of J-multipl-icity for the sth and pth modes, then the transformation of the sth mode into apth mode takes place in a zero-order approximation in the parameter e.18 Notethat at the point of J-multiplicity no energy is carried by eigenmodes; energytransmission through this cross-section is implemented by the associated wave.

This effect has some similarity to that of reflection of a mode from the cutoffcross-section in a smoothly irregular lossless waveguide. This parallel is under-standable in the light of the fact that forward and backward modes of the sametype in the cutoff cross-section are of J-multiplicity and the reflection means atransformation of the forward mode into the backward one. Here, in the cutoffcross-section an associated wave is also excited (Arkadaksky and Tsykin, 1976).

18 Let us point out, for comparison, that when there are no points of degeneration the effects oftransformation and reflection of modes in a smoothly irregular waveguide transition have the order£<^1 (Katsenelenbaum, 1961).


It is important to note that the effects of associated wave excitations are ofgreat interest and are examined intensively, for example, in connection withradiowave propagation along the 'earth-ionosphere' waveguide (Bichytskaya,1990) and the theory of metallodielectric waveguides and wave processes incontinuous media (Gureev, 1990«, b).

3.7 Attenuation coefficients of eigenmodes

Let us proceed now to the dispersion equation of a lossy waveguide. Theconductivity of the walls is assumed to be sufficiently large (R.e ^g/Wo <^ 1).Neglecting intermodal coupling we can present eqn. 3.30 in the form

from which we obtainr

H_ nTHnT dCK7c"-PT PX

or

H.prHptdC

(3-59){(EpxH^p)-(E.pxHp)}izdS±

s.

H_ptHpzdC

fp~- - (3.60){(EpxH.p)-(E.pxHp)}tzdS1

Js±Eqns. 3.59 and 3.60 are general relations, applicable to any types of modeincluding below-cutoff and complex modes if \(O — coc\ is not too small. Forpropagating modes, using the formulae £_s = +£*, H_s = +Hf, we can trans-form eqns. 3.59 and 3.60 into

(3.61)

2 (EpxH*)hdS1}

(3-62)

2 {EpxH*)izdSLJSL

Eqn. 3.62 can be rewritten as y"p~P1j2P1, where P1 is the linear (per unit


length) power of losses in the walls, and P± is the average over the period T =2njco power carried through the waveguide cross-section S±. Eqn. 3.62 reflectsthe energy method of loss calculation based on the concept of perturbations ofthe electromagnetic field due to finite conductivity of the walls (Weinstein, 1988,Ilyinsky and Slepyan, 1983). The quantity y"p is called the attenuation coefficientof the jfrth eigenmode.

Attenuation coefficient is a complete characteristic of the absorption processonly for single-mode waveguides. For a multimode waveguide (i.e. a waveguidein which several eigenmodes can progagate) it is no longer so, since amplituderelations between different types of modes are of major importance.

It is seen from eqn. 3.61, that the propagation coefficient y'p due to finiteconductivity of the walls undergoes a correction which can be neglected owingto the small size of the losses and sufficient remoteness from the cutoff frequency,

For lossy waveguides the concept of a cutoff frequency does not have the sameclear meaning as in lossless waveguides. In fact, there is no real frequency co atwhich ys(o)) =y's(a>) + jy"p{(o) = 0 . Nevertheless, when the losses are small theconcept of a cutoff frequency can be introduced to divide two physically differentregimes of waveguide operation. However, the border between these regimes isnow not distinct and the cutoff frequency a>c is not uniquely defined.19 Theeasiest way to consider the cutoff frequency of a lossless waveguide is as that ofan identical waveguide with losses. Then at (O>a>c, yf

sf>yg (travelling waveundergoes a weak active attenuation), at co<a)c, y's^y's (the wave undergoesstrong reactive attenuation, i.e. below-cutoff mode). The losses lead to a weaksine modulation of the exponentially decreasing field along the £-axis. Later inthis book when discussing cutoff frequency of waveguides with losses we shallnot make any specific reservations.

To discover the applicability limits of eqn. 3.62 let us obtain a more precisesolution of the dispersion equation taking into account the coupling of twomodes with the nearest propagation coefficients. Assuming that these are the sihand pth modes, we can write the dispersion equation in the form

DetAss-jy ASP

Aps App-jy= 0

or

(Ass-jy) (App-jy) - AspAps = 0 (3.63)

Solution of eqn. 3.63 gives the propagation coefficients

.771.2= SS o P " ±2 "• { 4 ps

When there is no intermodal coupling (Asp = Aps = 0), we obtain, from eqn. 3.64,

JTi = Ass jy2 = App (3.65)

19 To be more precise we mean not the cutoff frequency but the narrow cutoff-frequency range.


which leads to eqn. 3.62. Now let us assume that the pth and sth modes arecoupled due to the losses in the walls but that inequality

\Ass-App\>2\ApsAsp\1l2 (3.66)

is correct. Ignoring the terms O[(Re£s)2] in eqn. 3.64 because of their

extremely small size, we once again obtain eqn. 3.65. When the condition givenby eqn. 3.66 is fulfilled, we can use eqn. 3.62 which results from the perturbationtheory.

In the neighbourhood of degeneration points of coupled modes (hs — hp->0),the condition given by eqn. 3.66 is not satisfied. In this case corrections toeqn. 3.65 have an order of Re Zs a n d should be taken into consideration.Introducing the attenuation mutual coefficient ysp by the formula

rRe 2

f ) 1 / 2

(EpxH*)i2dSL (EsxH?)izdSASi JSj. J

we can present the attenuation coefficients of D-degenerated eigenmodes as

: ± l (ys-y P ) + y ^ I 3 67^

where y'^p are 'partial' attenuation coefficients for the sth and jfrth modes definedby eqn. 3.62. Eqn. 3.67 is valid when these modes are not very close to cutoff.Thus, for degenerated modes, coupled due to losses in the walls (ypS7^0), theconventional perturbation method is inapplicable (Kato, 1966) and attenuationcoefficients should be determined according to eqn. 3.67.

When three or more modes in a lossless waveguide are degenerated, theanalysis can be carried out in a similar manner though it becomes morecumbersome.

Let us consider, for example, degenerated modes in types of waveguide widelyused in practice. In a circular waveguide the TMOn and TEl n modes aredegenerated but because of different azimuthal dependency of the fields of thesemodes the attenuation mutual coefficient is zero (trivial multiplicity). Thus theperturbation method can be applied to calculate the attenuation coefficients ofany modes in a circular waveguide at operational frequencies remote from thecutoff (see eqn. 3.62).

Expressions for fields in a circular lossless waveguide are well known and weshall not present them here. According to eqn. 3.62 we have

for TEfn modes and

for TMin modes (b is the waveguide radius).


In a rectangular waveguide the TEmM and TMmn modes (m> l,n> 1) aredegenerated at all operational frequencies and any values of a and b. Unlike thecase of a circular waveguide, these modes are coupled owing to losses in thewalls when a / b. Therefore the conventional perturbation method is inappli-cable.20 In fact, in a lossy rectangular waveguide at m> 1, n> 1 there are noseparate TE and TM modes. The modes excited have all six components of theelectromagnetic field and are linear combinations of the TE and TM modes.The amplitudes of the partial modes are comparable in the order of magnitude.In fact we have here a non-single-mode regime where, as already mentioned,attenuation coefficients of partial modes have only a conditional meaning anddo not characterise completely the total losses.

Separate TE and TM modes with m, n > 1 in the presence of losses can befound only in a square waveguide. This is why the formulae for attenuationcoefficients given by Chang (1989) are correct in this specific case.

The TE01 mode and TE10 mode in a rectangular waveguide are nondegener-ated (they are degenerated only at a = b but even in this case they are notcoupled owing to the losses in the walls). This is why for these modes we canuse the approach based on eqn. 3.62. In this case we obtain the following resultfor the TE10 mode:

For the TE01 mode a similar formula can be derived obtained from that giventhrough replacement of a with b and b with a.

Eqn. 3.62 is applicable also to calculation of the attenuation coefficient of theTEM mode in doubly connected systems,21 for example in a coaxial line. In thegeneral case of an jV-connected system eqn. 3.62 cannot be applied as the TEMmode in it is of the multiplicity JV— 1 and, hence, described by more complicatedlaws (see above).

Attenuation in a coaxial line has been examined by Daywitt (1990). In thiscase the following relation is obtained for the attenuation coefficient of the TEMmode:

,,_7 Wo 26 In (A/a)

where a and b are the radii of the inner and outer co-axial conductors,respectively.

Now let us consider the attenuation in a cutoff waveguide. Here the forwardand backward modes of the type under consideration are of J-multiplicity.Assuming that the operational frequency is not very close to the cutoff valuesof any other modes, let us use the formulae of a 'two-wave' approximation.Substituting p= —s in eqn. 3.64, we obtain an expression for the complex

2 0 For this reason the dependencies for the attenuation coefficients of the TEmM and T M m n modes(m, n> 1) in a rectangular waveguide, calculated using eqn. 3.62 and quoted sometimes in theliterature (see, for example, Chang, 1989), have no meaning except for the case a = b. This fact hasbeen pointed out repeatedly by Prof. L.A. Weinstein.2 1 In the presence of losses in the walls it is more correct to speak about a quasi -TEM mode as inthis case the longitudinal field components are not exactly equal to zero.


propagation coefficients to the accuracy O[(Re^s)2J :

yu2 ~ ± (h2s + 2jhs^ (j) |ff ° | 2 d c j (3.68)

where the sign + corresponds to forward and backward modes. Eqn. 3.68 iscorrect for all frequencies including CD — COC. If the inequality

is fulfilled, then on expanding the right-hand side of eqn. 3.68 into a powerseries in £s and retaining the terms of the first order, we reach eqn. 3.62.Another limit case resulting from eqn. 3.68 is attenuation right at the cutofffrequency (hs—>0). The norm can be expressed as JVS = coshsAls, where Ms issome frequency-independent coefficient. Then at hs—>0

(3.69)lk'* -\coeMs

At co-+coc the attenuation coefficients y'[j2 ~ (Re£s)1/2> i-e- losses in a cutoffwaveguide are considerably larger than those in a travelling waveguide of thesame cross-section.

Finally, we shall discuss another case where the conventional perturbationmethod for calculating losses cannot be used, i.e. structures with unclosedinfinitely thin metal screens, e.g. microstrip and slot lines. If the eigenmodeunder consideration has an electric-field component directed along the edge,then eqn. 3.62 gives an infinitely large value of attenuation coefficient at anyfrequencies, as the integral

\H°nI2 dCc

does not converge (Pregla, 1980, Heitkamper and Heinrich, 1991). This isbecause the magnetic field intensity H® has a quadratically unintegrable singu-larity at the edge if J?s ~ 0- For a finite impedance there is no edge singularity(see Section 2.4). Then the perturbation of the magnetic field due to an infini-tesimal impedance cannot be taken as weak in some neighbourhoods of theedge.

In fact, the problem under study goes beyond the limits of perturbationtheory. The problem is wider: to what extent are the impedance-boundaryconditions themselves and the model of an infinitely thin strip applicable? Realstructures, of course, have a finite thickness, exceeding sufficiently the skin depth,and an infinitely thin strip is simply a model giving an opportunity to applyhighly effective computational methods. If this model is not used and we considera strip of finite thickness with a real edge geometry, there will be no suchdifficulty, so that eqn. 3.62 will be valid. But in this case determination of Zsp, Hp

as defined in eqn. 3.62 and its integration over SL and C may become verycomplicated.


Within the framework of the model of an infinitely thin strip for a modehaving the current-density component directed along the edge, the Leontovichimpedance boundary conditions are not valid (see Section 2.4); it is necessaryto use the modified impedance expressed by eqns. 2.27 or 2.28. In this case7p ~ Re Zs m \Zs\> a n d the conventional perturbation method cannot be used.But the perturbation technique can be applied by considering Re Zs ( n o t Zs)as a small parameter; at the same time the terms 0 (Im Zs) should be attributedto a zero-order approximation.

There is one more opportunity that allows us to avoid moving beyond theborders of the model of an infinitely thin strip: modification of the energy-perturbation method, which can be interpreted as follows: at the integrationalong the contour C in eqn. 3.62 some neighbourhoods of edges are excluded.The size A of a neighbourhood excluded should be expressed in terms ofgeometrical parameters of the edge (thickness of plates, edge curvature radius).Nosich and Shestopalov (1980) were the first to suggest this approach; correctdetermination of A for different edge configurations was carried out by Lewin(1984). Similar results were later also presented by Weinstein et al. (1986).

Such a modification of the energy-perturbation method is closely connectedwith that of the surface impedance described in Section 2.4. It can be seen fromeqn. 2.22 that the quantity A is expressed in terms of Zs by means of theequation A = 2|<^|/7rG2A;J4/

0. Substituting the impedances expressed byeqns. 2.27 and 2.28 into this formula and taking account of the fact that\Zs\ — ~~ I m Zs> w e obtain

A = f (3.70a)

for the edge of an elliptical shape, and

for the edge of a rectangular shape.Eqns. 3.70a and 3.70b coincide with the corresponding formulae obtained by

Lewin (1984). This fact indicates a principal equivalency of the energy-methodmodification considered and the concept of a surface impedance in Section 2.4.However, from the computational point of view these methods are not identical:they are complementary to one another and each has its own field of effectiveapplicability.

Different cases of the perturbation-method application described above areshown in Table 3.1. Note that use of the energy method is not always expedienteven if it is thoroughly substantiated: everything depends on the waveguidegeometry and applied method of solving the boundary-value problem. Forwaveguides of simple configuration when separation of variables is possible, theenergy method will in all probability be the shortest way of obtaining analyticalformulae for attenuation coefficients. For waveguides of complicated shapesdetermination of eigenfunctions of corresponding lossless systems and subsequentintegration in eqn. 3.62 may be rather difficult. Then it would be expedient toobtain a dispersion equation for the waveguide with losses using the numerical


Table 3.1 Perturbation-method applications

Mode

Nondegenerated(o) is not near coc)

D-degenerated(co is not near coc)

Nondegenerated(co is close to coc)

Asymptotics of y"s

at Zs->0

O(Re Zs)

O(Re Zs)

O[(ReZs)1/2]

Method of calculating y"s

energy perturbation technique;eqn. 3.62

modified perturbation methodtaking into account the inter-modal coupling; eqn. 3.67

perturbation method is inappli-cable; eqn. 3.69

With the edge singularity O(Re Zs In \ZS\)of the current density

modified surface impedancetechnique; Zs is expressed byeqns. 2.27 and 2.28

or numeric-analytical methods for impedance-boundary-value problems.Attenuation coefficients can be determined, solving the dispersion equationthrough the perturbation method considering Zs o r ^ e Zs (f°r t n e case of modeswith edge singularity of the current density) being small. Further on we shallbe using both a direct solution of impedance problems and numerical integrationof fields for lossless systems. Both approaches supplement one another and weshall come across cases when one is preferable to the other. We shall try to drawreaders' attention to the reasons for this preference.

3.8 Attenuation in a generalised microstrip line; model ofthe infinitely thin strip

The cross-section of a generalised microstrip line is shown in Figure 3Aa. Letus consider the procedure for deriving the functional equations for determinationof normal modes (Zarubanov and Ilyinsky, 1985). To simplify our analysis weassume that the relative magnetic permeability of all layers is 1. Let the stripbe infinitely thin and the lower and the upper screens be perfectly conducting.Thus, we shall consider here the absorption in both the strip and the dielectriclayers. The first component of losses is the most important to evaluate and themost difficult to determine. It is not difficult to take into account the losses inthe screens in the framework of the technique described but, when comparingvarious methods of loss determination in a microstrip line, it would be moreconvenient to have separate components of the total losses.

In this section we shall describe an algorithm based on the impedance formu-lation of the problem. As we have accepted a model using an infinitely thinstrip, the Leontovich boundary condition is not correct. In each of the regionsSi where the permittivity is constant, we are looking for the solution of a


/ /

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^N

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y

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//

//

7Z / / / / / / / /

a.

V/ 4 //////// ////V

/////////////

Figure 3.4 Microstrip lines (sectional view)

a With infinitely thin stripb With strip of finite thickness

homogeneous system of Maxwell's equations in the form of normal modespropagating along the £-axis. Tangential components of fields should be continu-ous across the interfaces between the dielectric media and on the perfectlyconducting surfaces of the screens the tangential component of the electric fieldshould become zero. In addition, the edge condition should be satisfied.

Let us examine nonradiating modes of the line. This results in the additionalcondition that the field of the mode tends to zero when (x2 + j 2 ) 1 / 2 tends toinfinity in every cross-section z = constant. Introducing electric and magneticHertz vectors in the form of corresponding Fourier transforms with respect tox, expressing the field in every region in terms of the Hertz vectors, and matchingthe fields at the boundaries, we can obtain expressions for the components Ex

and Ez at the interface between the regions S-^ and S2 in the form of spectralexpansions

f°°2njo)soEx = P(a, y)Ix (a) exp (jctx) daJ - o o

f00

+ G(a,y)/Z(a) exp (jax) daJ - o o

2nja>£0E.f°°=

J -c

G (a, y)Ix (a) exp {jocx) da

Q(ot,y)Iz(<x)exp(jotx)d<x

(3.71)

Here a is a spectral variable, IXtZ(oc) are Fourier transforms from jumpsof z- and x-components of the magnetic field at the boundary,

— oo < x < oo,

P(<x,y) =

G(a,y) =

a2 + y2

aya2 + f


k2y2

.a , 7) =

^ j

,,2

+Ma)£2a2

e{g

j82 = a2 + y2 - A;2£j, ? = 1, 2, 3, 4, and et is the relative permittivity of the layers.Let us introduce now the main functional relations assuming that the

impedance boundary conditions are satisfied on the surface of the strip. Thenfrom eqn. 3.71 we obtain

P(a, 7)4 (a) exp(jax) da0

G(a, y)/z(a) exp(ja^J-oo

G{<x,y)Ix(<x) exp(jctx) da

J." (a) exp (jax) da = i

(3.72)

where ^ = 2njcoso^Si (i= 1, 2), ^ S 1 is the Leontovich surface impedance, ^ S 2

is the surface impedance expressed by eqn. 2.27 or eqn. 2.28, and— W12 < x < WJ2. Moreover, at |x| > W\2 the following functional relations arecorrect:

Ixz (a) exp (jccx) da = 0 (3.73)

We stress that eqn. 3.73 is a spectral formulation of the fact that the electriccurrent outside the strip is equal to zero and determines a permissible class ofsolutions together with the main functional relations given by eqn. 3.72.

To solve eqn. 3.72 let us use the moment method. We shall present the currentson the strip as expansions in terms of basis functions

/,(*)= J am4>m(x){\-(2xlW)2}2}'ll2=

Zm=0

m = 0

m=0

am<t>m(x)(3.74)


where am and bm are complex coefficients. For even (symmetrical in x) modes(j)m{x) = T2m{2xjW) and ij/n{x) = U2n+1 {2xjW)\ for odd modes (j)m{x) =T2m+1(2xlW) and \\jn{x) = £/2fI(2*/W), where Tt{x) and Ut{x) are Chebyshevpolynomials of the first and second kinds, respectively. Substituting eqns. 3.74into eqns. 3.72, we obtain

00 _ _

Z bnij/n((x) exp(jocx) da

_00 _ _ 00 _

= £i £ bn\ljn(x) (3.75)

G(a, y) X îfo (a) exp(jax) da

t = ^2Yu *»0nM (3.76)n = 0

We take the complete systems of functions {\\jn(x)} and {(j)m(x)} as weightingfunctions for eqns. 3.75 and 3.76, respectively. After some transformation weobtain an infinite system of linear algebraic equations for the am and bn

coefficients. To be numerically solved this system has to be truncated:*W/2 _ _

0'n{x)\jjp(x) dx )I -W/2

(3.77)

'a)0*(a) da

$m{x)<l>r{x)dx\=0 (3.78)' - T F / 2 J

where ^ = 0, 1, . . . , TV; r = 0, 1, . . . , M.The integrals with respect to x in eqns. 3.77 and 3.78 after presenting Un in

a trigonometric form are easily found. It can be proved, using the orthogonalityrelations for Chebyshev polynomials, that these integrals in eqn. 3.78 dXm^rbecome zero. The improper integrals are absolutely and uniformly convergent.Equating the determinant of the system given by eqns. 3.77 and 3.78 to zero,we obtain a dispersion equation for complex propagation coefficients:

Det A{y, {£,.}, {st}) = / (y , {£;}, {8t}) = 0 (3.79)

where 7 = 1,2; *"= 1,2,3,4.Numerical solution of eqn. 3.79 should be treated with caution. That is

because f{y,£2) is an analytical function only for a finite matrix; while itapproaches the limit M, jV-» oo the analyticity at the point £2

== 0 is broken;


hence the perturbation method considering Zs a s a small parameter cannot beapplied: there is no convergence in this case. However, we can use a modificationof the perturbation technique described in Section 3.7: ^Zsi = êZsi~— Im Zsi = Z's is a small parameter whereas the terms 0 (Im Zsi) a r e those ofthe zero-order approximation. When calculating the ohmic losses in the strip,the dielectric media are to be considered as being ideal (tan 3t = 0).

Assuming Z's t o De a small parameter, we expand the function f(y, Z's) m

eqn. 3.79 into a Taylor series in the neighbourhood of the point y — Jo,Z's ~ 0, where y0 satisfies the equation/(y0, 0) = 0. Neglecting the terms abovethe first order, we obtain

dy ' 8Z's= 0 (3.80)

The first summand is equal to zero by definition and as a result the followingformula for the attenuation coefficient can be derived from eqn. 3.80:

The subscript o is introduced to distinguish the attenuation in conductingsurfaces of the line from that in dielectric layers (we shall calculate it later); thetilde sign denotes that only the losses in the strip are taken into account.

It is essential that while using eqn. 3.81 it is not necessary to carry outcomputations requiring complex arithmetic: derivatives of f(y, Z's> {£;})? a r e

calculated with real arguments.When creating an algorithm from eqn. 3.81 we face the problem of differen-

tiation of the function given in the form of a determinant. Two methods arepossible: the first is numerical differentiation (it needs caution in calculations).The other is analytical differentiation carried out according to the formula(Fadeev and Fadeeva, 1963)

|£=fDet4 (3.82)

where K = JV+ M + 2. Here the matrix A\ follows from A through replacing thenh row by that of the derivatives dAipldx.

Which of these methods is better will depend on the labour required foranalytical calculations of the derivatives in eqn. 3.82. The surface impedancesZsi a n d Zsi enter into eqn. 3.79 linearly; therefore differentiation with respectto Z's c a n De carried out on the basis of eqn. 3.82. The parameter y is involvedin the matrix elements A in a complicated transcendental way and analyticaldifferentiation here implies rather cumbersome transformations. This is why thederivative with respect to y was determined numerically^by the finite-differencemethod.

As the basis and weighting functions chosen are different (for the integrals toconverge), so the matrix A will be asymmetrical. Only its top left-hand blockof the order JV+ 1 will be symmetrical.

Let us proceed to determination of attenuation in dielectric layers. Now weassume Zsi= Zsi~ 0 and the permittivities of layers to be complex:


et = £• + js". Then £•' are considered to be small parameters and the approachused is completely identical to calculation of y^. Omitting obvious intermediatecalculations let us write down the final result:

7e =dy ^ < (3-83)

Here y0 represents the solution of eqn. 3.79 at Zs\ = Zsi = si =0 (i= 1, 2, 3, 4).Differentiation of the function f with respect to st in eqn. 3.83 is also performedby means of the finite-difference method.

The total losses within the framework of the perturbation method can befound by adding the independently calculated components. This means that thetotal attenuation coefficient is expressed by

y" = y" + y" + y"f t o ] f £ l fas

where y"a and y"e are determined by eqns. 3.81 and 3.83, respectively, and y'^s isthe attenuation coefficient due to losses in the screen (its determination is notconsidered in this Section).

The algorithm presented allows us to calculate the attenuation in differenttypes of planar lines. It is characterised by fast convergence: three basis functionsare usually sufficient to approximate the longitudinal component of the currentdensity and two functions to approximate the transverse component. The analy-sis and comparison of the numerical results obtained by means of variousmethods (including that described above) will be presented in Section 3.10.

3.9 Attenuation in a microstrip line; model of a strip offinite thickness

The cross-section of the transmission line under study is shown in Figure 3Ab.Let us consider the problem of determining the even normal modes (analysis ofodd modes can be carried out in a similar way). For such a model we can usethe conventional energy-perturbation method (see Section 3.7). Here it is neces-sary to calculate the field distribution in the cross-section of a line (in particular,the current density distribution on conducting surfaces), corresponding to thecase of perfect conductivity. In this Section we shall describe the numericalalgorithm for solving this problem suggested by Zarubanov and Ilyinsky (1990).

The mathematical formulation of the problem is similar to that in Section 3.8.Let us divide the cross-section of the microstrip line into three partial regions,as shown in Figure 3Ab, introducing into each the electric TV and magnetic Ilm

Hertz vectors. The electromagnetic field is expressed in terms of Hertz vectorsby the formulae

E = grad div W + k2 W + jcou0 curl IIm 1, > (3.84)

H = grad div n m + k2 Um - j(oes0 curl IT JAs shown by Weinstein (1988), to represent an arbitrary electromagnetic fieldin a region without sources we can just use one component of each Hertz vector.One of the specific features of the approach described is as follows: because ofthe boundary conditions it is convenient to use different pairs of components


of the Hertz vectors in various partial regions. According to Section 3.8, welook for solutions whose ^-dependence is exp(jhz), where h is the pro-pagation coefficient to be found. Thus, for regions 1 and 3 let us assume in eqn.3.84 that IT 'm(*, j , z) = iyn

ey-

m{x,y) exp(jfe), and for region 2 thatITe'm(*,y, z) — ixll

ex'

m(x,y) exp(jhz) (in regions 2 and 3, £ = 1). For the functionsIle

x™(x,y) in various partial regions the following spectral representations takeplace:

In region 1 (— d1 <y < 0)

JO LUfj,0bnm[fJ1u1)

in region 3 (t <y < d2)

ceô_sh{P3(d2-y)}

3o to/io

in region 2 (0 <y < t)

) +Fcosh(02y)}o

where C ^ are unknown spectral densities. Expressing the field tangentialcomponents in each region in terms of Hertz vectors we obtain their represen-tations through the unknown spectral densities.

In this formulation the tangential components of the electric field are ident-ically equal to zero on the screens and end faces of the strip. Now we have toimpose the boundary conditions on the top and bottom faces and the edgeconditions.

Let us introduce the following four functions

at |*| < W/2:

M, (x) = Hx(x,t + 0) - Hx(x,t-O) =fz(x)

JV3 (x) = Hx(x, - 0 ) - Hx(x, +0) =j°z(x)

x9 -0)-Hz(x, 2


at |x |> Wj2

Nt(x) = 0 i = 1, 2, 3, 4

where7°'r(x) andj°'f(x) are unknown densities of the currents induced on thestrip. According to the edge conditions for a right-angle wedge (Meixner, 1972),the current densities on the strip can be presented as

j°/(x) = constant + {1 - {2xjW)2}2t?>^0^(x)

where (j)(x) and \j/(x) are smooth regular functions.Let us substitute the integral representations of the fields into the continuity

conditions for the tangential components of the electric fields at the interfacesbetween the partial regions, and into the expressions for the functions Nt.Multiplying the equations for the components Ex and Hz by cos{a'(x — Wj2)}and those for the components Ez and Hx by sin{a'(x — W/2)}, and then integrat-ing with respect to x along the positive semi-axis, we obtain, owing to theorthogonality of the corresponding trigonometric functions, a system of eightrelationships allowing us to express the spectral densities of the field componentsin terms of Fourier transforms of the functions Nt (from here on indicated by

From the system obtained after simple but cumbersome transformations, wecan obtain expressions for spectral densities in terms of the functions JV . Substi-tuting these expressions into integral representations for the tangential compo-nents of the electric field and using the boundary conditions on the upper andthe lower sides of the strip, we obtain a system of four homogeneous functionalequations of the first kind for the unknown functions Nt:

(G1N1 + PXN2 - GN3 - PN4) sin(ax) da = 0o

(Qj.^1 + Gi-#2 ~ Q A - GN^) cos (ax) da = 0

{G2Nl + P2N2 + G3N3 4- P4jV4) sin (ax) da = 0

1 + G2JV2 + Q^Y 3 - G3 JV4) cos (ax) da = 0

(3.85)

where |*| < W/2, and

1P

k2h2

G =ah

h2 k2a2

TU


P1

Pi

G2

di

1

7~tcosh(/?2/)'

—

tanh(j82r)

P

(I

coth(jMi)' . + .p

G

Q

—

G3

63

= cosh (j821) {tanh (jl21) coth (j92 d2) - 1}

1 (£P2a2

Q

G =ah ( sp2

£ n/2 _, / 2a2 + h2 \p, Tt fl2 T,

1

l - tanh2(j820

- tanh(j82/)

T =J?1

-» Pe cot

The functions 7"£ and 7^ have roots corresponding to the propagation coefficientsof the LM and LE modes of a 2-layer plane dielectric waveguide.

Taking into account the behaviour of current density near the edges of thestrip, we expand the current densities in the series in terms of the completesystems of Gegenbauer polynomials with the corresponding weight multipliers:

= I \an0\{l-(2xlW)2r»3Ctf(2XllV)

o la )

(3.86)

m = l ( ^ m

The coefficients tfj,'0, b^0 have to satisfy the conditions

n=0


The terms containing bft;

0 are added into eqn. 3.86 to satisfy the matchingconditions for the longitudinal component of the magnetic field at x = ± Wj2.

The Fourier transforms of the basis functions are calculated in the explicitform

(aW'12)1'6

w r(2m+l/3)- . w r(2m+l/3)2(2m+l)!r(7/6)27/6

Let us substitute the expansions for current densities given by eqn. 3.86 intoeqn. 3.85. Applying the moment method with the weighting functions <pr (r =0, 1, . . .), \j/s (s = 1, 2, . . .) and changing the order of the operations of inte-gration and summation [which is possible in this case (Ilyinsky, 1981)], weobtain an infinite system of homogeneous linear algebraic equations for thecoefficients a^0 and ij,'0. Note that at this stage the number of equations is twoless than the number of unknowns. Matching the expansions for the transversecurrents given by eqn. 3.86 with the integral representations for the tangentialmagnetic field Hz in region 2 aty = 0 andy = t, we obtain the required equations.

Using the Wiener-Paley theorem in a similar way to Ilyinsky (1981), we canshow that the infinite system of linear algebraic equations and the originalproblem of normal-mode determination are equivalent in the sense that the fieldof a normal mode gives rise to a nontrivial solution for the homogeneous systemof linear equations and, vice versa, a nontrivial solution of the latter enables usto calculate the field of the normal mode.

Equating the determinant of the system to zero, we obtain a dispersionequation to find the propagation coefficient. The a1^0 and b\;0 coefficients, andhence the functions jVt(a), are determined from the corresponding homogeneouslinear system. The attenuation coefficient due to losses in the metal surfaces wasdetermined by the formula

(3.87)

where

= f (ExH*-EyH$)dxdy

The integral in the numerator of eqn. 3.87 is taken along the contour aroundthe metal conductors in the cross-section of the line. The components of thecurrent density in the expression under the integral are presented by eqn. 3.86(see also Appendix 2). The integral in the denominator of eqn. 3.87 is takenover the cross-section of the line. Integration with respect toy was performedanalytically; later, using Parseval's theorem, we transfer from integration with

JSormal modes in waveguides with losses 75

respect to x to that with respect to the spectral variable a:

(EXH* - EyH*) dx dy = — \ PS(OL) da. (3.88)J X

The expression for the function Pz(<x) is presented in Appendix 2.In practice, for numerical computations the infinite system of linear equations

has been truncated. Inner convergence of the method is fairly fast: in practiceit always is enough to consider three terms in the expansions of the longitudinaland transverse current densities. Some matrix blocks are symmetrical. This factallows us to reduce the number of matrix elements to be calculated. It isimportant from the computational point of view, because every element isexpressed by an improper integral of a rapidly varying nonmonotonic function.The numerical integration was carried out to the upper limit Tbn\W by meansof the Gaussian quadrature formula. The relative error in calculating the inte-grals did not exceed 10~7. The search for a root of the dispersion equation wascarried out in the interval (hLM, ky/e), where hLM is the propagation coefficientof the lower LM mode and only the root closest to kyje was sought. Once theroot had been found a nontrivial solution for the homogeneous system of linearalgebraic equations was determined by means of the method of back iterations(Ilyinsky and Zarubanov, 1980). In practice, to achieve a high accuracy fiveiterations are sufficient.

3.10 Attenuation in a microstrip line; numerical results

In this section we analyse the numerical results obtained on the basis of algor-ithms described in Sections 3.8 and 3.9. Here we compare them with other datapresented so far in the literature. This clarifies which of the physical models andmethods are most effective for evaluating the attenuation in various planartransmission lines: slot, coplanar etc. To determine £s, the value a0 =5.9 x 107 S/m (copper) was used.

Tables 3.2 and 3.3 present the results of calculation of attenuation for a linewith H/ = 0.96mm, dx = 1 mm, d2= oo, 8 = 9.6 at / = 2 GHz a n d / = 12 GHzfor strip conductors of various thicknesses. The second, third and fourth columnscorrespond to the contributions (in percent) to the ohmic losses made by thecurrents on the upper, lower and side surfaces of the strip, respectively; the fifthcolumn gives the contribution of the current on the screen. The sixth and theseventh columns present the absolute value of losses (in dB/m) in the screen and

Table 3.2 Attenuation coefficients of microstrip lines: contribution of different componentsin total losses atf=2 GHz

t/w

10~3

10-2

10~1

1

7.347.386.220.31

y'oo (%)

80.8179.0878.8489.43

tfi (%)

0.392.645.252.75

v:.(%)11.4611.009.697.51

y"as (dB/m)

0.1560.1570.1630.191

yl (dB/m)

1.361.421.692.54


Table 3.3 Attenuation coefficients of micro strip lines: contribution ojdifferent componentsin total losses atf= 12 GHz

t/w

10~3

10~2

10"1

1

y"A%)

6.78

6.73

5.65

0.18

y;'o(%)

78.00

76.47

76.61

87.36

vi'i (%)

0.37

2.51

5.00

2.55

f,.(%)14.85

14.29

12.74

9.91

7L(dB/m)

0.487

0.489

0.506

0.544

y"ff (dB/m)

3.28

3.43

3.97

5.49

the total ohmic losses in the line (y"a = i'al + y^0 + i'al + fas = fa + fas). In spiteof the statement by Vymorokov et al. (1987) that losses on the end faces of thestrip for a 50 Q microstrip line make up 30% of the total losses, according toour calculations these losses are one fifth the size. Note that Vymorokov et al.calculated the losses through the static approximation.

Figure 3.5 shows the dependencies of total attenuation coefficient in themicrostrip line with W = 1.2446 mm, dl = 0.508 mm, d2 = oo, s = 9.35, tan 5 =2.5 x 10~4 and tjW = 0.001. Curve 1 is taken from the work by Gopinath et al.(1970). It was obtained for a model using an infinitely thin strip in TEM-approximation. Curve 3 (broken line) corresponds to the results of experimentsby the same authors. Curve 4 shows data by Nikolsky and Kozlov (1987b) andcurve 5 gives results according to the method in Section 3.9. Curve 2 is a resultof using the technique in Section 3.8, where £s indicates the Leontovichimpedance (the losses in the screen are taken into account). Such a model, aswas mentioned above, is not physically correct: by curve 2 we mean to showthe errors implicated in the use of the conventional model of an infinitely thinstrip. It is seen that curves 4 and 5 demonstrate best agreement with theexperiment. Curve 2 and the experimental data differ noticeably. Zarubanovand Ilyinsky (1990) studied transmission lines with wide strips (M^ = 9.15 mm,^ = 0.64 mm). In this case the model, corresponding to curve 2, leads tosufficiently accurate results. This can be easily explained physically: inapplica-

7 3 5frequency, GHz

Figure 3.5 Total losses in microstrip line calculated by various methods


bility of the Leontovich-impedance condition in combination with a model usingan infinitely thin strip results from incorrect description of edge effects. However,for such wide strips the contribution of edge effects is so small that errors intheir description are not significant. Let us underline that the model correspond-ing to curve 2 invariably leads to an overestimated value of the attenuationcoefficient. This can be explained by a more rapid increase of the longitudinalcurrent density in the neighbourhood of the edge than for a conductor of finitethickness.

In Figure 3.6 are shown the results of calculation of the attenuation causedby the losses in a strip (the parameters of the line are the same as for Figure 3.5).Curve 1 corresponds to the algorithm from Section 3.8 with the Leontovichimpedance (Zs2~~*Zsi)> curve 2 to the algorithm from Section 3.9, curve 3 tothe data by Nikolsky and Kozlov (1987b), and curve 4 to the energy-pertur-bation-method modification for an infinitely thin strip (Section 3.7) and to thealgorithm from Section 3.8 with the impedance Zs expressed by eqn. 2.28(calculation results for these methods coincide within the accuracy of presen-tation, and therefore in Figure 3.6 they are presented by one curve). Thisinformation allows us to conclude that the model of an infinitely thin stripmodified according to Sections 2.4 or 3.7 enables us to describe the edge effectscorrectly and to obtain accurate results.

When analysing the reliability of numerical data on losses, a qualitativeinvestigation of current distribution on the conducting surfaces is important.Figure 3.7 shows the current-density distributions for a line with W = 0.96 mm,d1 = 1 mm, d2

= oo, £ = 9.6 and different values of the strip thickness at f =12 GHz. Curves 1 correspond to t\W = 0.01, curves 2 correspond to tjW = 0.1,and broken lines correspond to an infinitely thin strip. In the currents desig-nations the superscript / corresponds to the upper side of the plate, 0 to thelower side, 1 to the lateral side and s to the currents on the lower screen. Whenthe strip thickness decreases, the total currents densities j® + j z and j ^ +J* tendto the corresponding current distributions for a line with an infinitely thin strip.

1 3 5frepuexcy, GHz

Figure 3.6 Partial losses in the strip calculated by various methods

Parameters of microstrip line as in Figure 3.5


Jz:t

f

1.1

C 0 J? 3 0 A -W/2T X

W/2

it 0X \

0.020.01

V

x

D £

-0.02

BA

yD

A

0.02

0.01

•<°<j*

- / -0.02

W/2

-D

0 1 2jz1 0 /y/ -0.1 0 jJ-0.1 0 jj

J I-ZW -W 0

-0.022W


Jz

0.8

OA

0

-OA

My0.2

0

-0.2

-2W ~W -0.02-

2W x

2W x

Figure 3.8 As for Figure 3.7 for microstrip line with tjW= 1

The strip thickness slightly influences the currents induced on the lower plate.The distributions of these currents are close to each other even at t\\V — 1(curves 3) and tjW = 0.01 (curves 2). In Figure 3.8 current distributions a t / =12 GHz of the same line with a square section strip are given. The broken linepresents a distribution of the longitudinal current on the lower plate for aninfinitely thin strip. Compared with the case of thin strips, the character oflongitudinal currents for the upper and lower sides of the plate has been changed.

It is known that for a line with a dielectric substratum a concept of character-istic impedance cannot uniquely be introduced. For microstrip lines the followingconcept for characteristic impedance is mainly used: %i = ^7^2> where / is thetotal longitudinal current of the central conductor and P is defined by eqn. 3.88.For an open microstrip line with an infinitely thin strip we can define the

Figure 3.7 Surface-current-density distributions for conductors of a microstrip line

i t/wôm, 2 t/w=o.\


impedance in the form (Brews, 1987):

where ZT ls t n e impedance of the line in the TEM approximation, and V canbe regarded as tension in the line:

"0

V =

Table 3.4 shows the values of impedances of the line with parameters correspond-ing to Figures 3.5, 3.6, determined using various formulae. The magnitude ofZT w a s calculated according to the method suggested by Lerer and Mikhalevsky(1983). A t / - > 0 , all the formulae produce very similar results. With increasesin frequency, Zi decreases and Z3 increases faster than Zi an<^ ZA-- For a stripof finite thickness at t\W = 0.01, the behaviour of Zi remains the same and itsnumerical value slightly diminishes (27 Q at / = 6 GHz). The small spread inthe impedances shows a good approximation of the field by the TEM mode inthis case.

Finally, let us dwell upon several general points about which models wouldbe acceptable for use in evaluation of losses in planar structures. First, let uspoint out that direct analysis of structures with strips of finite thickness (bothwith impedance and approximating to perfect conductivity) is rather compli-cated: to prove this we can compare algorithms described in Sections 3.8 and3.9. Moreover, in Section 3.9 a strip of rectangular geometry is examined (seealso Rawal and Jackson, 1991). In fact, the shape of the strip is determined bytechnological factors and may not be rectangular: rounded, trapezoidal etc. Inthis case opportunities for solving the problem become even more limited: useof the partial-regions method is out of the question (or at least, problematic).It would then seem natural to use the integral-equation technique, but for thinstrips (t^W, dli2) it may lead to ill-conditioned systems of linear algebraicequations (an example of this type as applied to a different class of problems isdiscussed in Section 5.7).

This is why it would seem very attractive to modify the model of an infinitelythin conductor, and attempts of this kind have been discussed in the literaturemore than once. In all probability, one of the first such attempts was made byMirshekar-Syahkai and Davies (1982). They suggested excluding the edge neigh-bourhood in the quadratic integration of the current density when using the

Table 3.4 Comparison of the values of Zi

'(GHz) £L(Q) £Ô) Z^Q) Z4(Q)

123456

30.1530.1030.04

29.9629.8829.80

30.0330.02

30.01

30.0030.0030.02

30.3930.58

30.81

31.0931.0931.75

30.2130.3030.41

30.54

30.7030.87


energy-perturbation method. The size of the neighbourhood A excluded wasdetermined by the number of basis functions 3 in the current-density expansion:A = WjN. It was asserted that the numerical results for different N confirm aconvergence of the method. It can be seen from the results of Sections 2.4 and3.7 (see also Pregla, 1980, Heitkamper and Heinrich, 1991) that this approachat N-* oo diverges according to 0(\nN). Obviously, this slow divergence gaverise to the illusion of convergence at .V—•oo. Another example is the work byRozzi et al. (1990), where attenuation in a slot line was calculated. The currentdensity on the surface of the plates was approximated with 'corrections' to thedistribution near the edge: instead of the singularity x~1/2 corresponding to theinfinitely thin screen, Rozzi et al. used x~1/3 corresponding to a perfectly con-ducting wedge with a right angle. In this case integration in accordance withthe conventional energy-perturbation method results in a finite value of y"a.However, this approach cannot be considered as physically justified, or theresults obtained reliable, because in the above current-density approximationthere are no parameters characterising the edge geometry (thickness, radius ofcurvature etc.).

There is one more class of model which does not imply use of the impedance-boundary conditions. Klimenko and Fialkovsky (1990), Vakanas et al. (1990),Nikolsky and Kozlov (1986, 1987a, 19876) and Kiang (1991) presented the stripas a dielectric body with permittivity s=jaolco. It can be assumed that thedevelopment of such models is partially stimulated by the difficulties arisingwhen using standard approaches (see Section 3.7). From the physical point ofview, these models are faultless, of course, but their practical application is verycomplicated. In the paper by Klimenko and Fialkovsky (1990) only a mostsimple key problem is considered-—plane-wave excitation of a finitely conductinghalfplane in free space. Analysis has been carried out with the aid of the modifiedWiener—Hopf method: through factorisation the problem has been reduced toa rapidly converging system of linear algebraic equations. However, the algor-ithm is so complicated that its generalisation for a more realistic model of theelements of microwave and millimetre-wave circuits is rather problematic. Inthe numerical solution we shall certainly face the problem of field presentationinside the conducting body, for example its expansion in terms of some basisfunctions. However, if the skin effect is strong this field is very inhomogeneous(it is large in a thin near-boundary layer and sharply damped beyond its limits).Therefore, when a conventional system of basis functions is applied, the conver-gence of the expansions obtained is very slow. The situation can be improvedto some extent, but only for very thin strips with thickness comparable with thefield-penetration depth (Vakanas et al., 1990, Kiang, 1991).

Owing to mathematical difficulties Nikolsky and Kozlov (19876) have modi-fied this model, considering a rectangular strip with face ends impermeable bythe field (they are covered with ideally conducting films). A heuristic hypothesisis used here which needs to be checked and proved. First the losses in the faceends are not taken into account. Though their dimensions include usually asmall part of the total strip perimeter, the current density near the face ends ishigh and their contribution to losses can be significant. Secondly, the changesin the edge condition caused by the arbitrary introduction of'impermeable' faceends can distort the currents on the side surfaces of the strip near the edge.

Analysis and comparison of various models show that the most promising are


the models of an infinitely thin strip modified in accordance with the real edgegeometry as shown in Sections 2.4 and 3.7. These models provide sufficientaccuracy in the loss calculation and, at the same time, do not lead to majormathematical difficulties (see also Barsotti et al., 1991). In addition, they arevery flexible as regards the edge geometry: its change results in that of thenumerical coefficient in the expressions for A or £s.

Chapter 4

Normal oscillations in resonators withlosses

This Chapter deals with the theory of normal oscillations in resonators withlosses in the walls, and also cavities filled with inhomogeneous dissipative disper-sive media. Concepts of eigenoscillations and free oscillations and corresponding(^-factors are introduced. General formulae for the (^-factors of eigenoscillationsand free oscillations are derived and the difference between them, resulting fromthe dispersion properties of the media, is considered.

The Qrfactors of various resonators (cylindrical, spherical, conical, biconicaland cylindrical with coaxial metal plug) are calculated. Different approachesare used to calculate (^-factor (energy-perturbation method, impedance-pertur-bation technique for solving the characteristic equation), and the advantagesand disadvantages of each method are discussed.

Comparisons are made of resonators of different shapes with respect to theirQ;factor. Ways of increasing the Q;factor through optimisation of the resonatorshape and filling are analysed.

4.1 Expansion of eigenoscillations of a resonator withlosses in the walls in terms of resonant modes of anidentical lossless resonator

Let S be a closed surface with the interior V (Figure 4.1). The solutions ofMaxwell's homogeneous equations

curl E = icouH )

curl H— —jcosE J

satisfying the impedance boundary condition

(»0 x E) + Zsin0 x (n0 x H)} = 0 (4.2)

on S are called eigenoscillations of the cavity V.As the eigenvalue the frequency co is used. The permittivity s and permeability

84 Normal oscillations in resonators with losses

Figure 4.1 Geometry of cavity resonator

[i are assumed to be frequency-independent, purely real and constant throughoutthe cavity (a more general case will be discussed in Section 4.3).

For complex £s the problem considered is not self-adjoint and the eigenvaluesare complex co(r) = a)[r) —jco'{r), r being the eigenoscillation number. The paren-thetical subscripts are used to denote the eigenfrequencies of the resonator withlosses in the walls, whereas cor are the eigenfrequencies of the resonator with aperfectly conducting surface.

Let us dwell briefly upon the physical meaning of the problem formulated.Eqns. 4.1 and 4.2 describe oscillations in the resonator after the external sourceshave been switched off. For the resonator with losses such oscillations areattenuated, and to analyse them we therefore need a nonstationary statementof the problem. In formulating the problem using eqns. 4.1 and 4.2 we in factpostulate the harmonic time dependence exp{— jco^t) or, in other words, theexponential character of the oscillation attenuation exp{ — co'^t}. Generallyspeaking, the existence of such solutions has to be proved as well as whether allsolutions are characterised by this form of time dependence. Such a proof canbe performed using the more rigorous approach described in Section 4.3. Theenergy losses complicate considerably the procedure of determining the eigen-frequencies and eigenfunctions even for regions with simple shapes. The relevanttechnique (compare with Section 3.2) for finding the eigenoscillations of aresonator with losses is to expand them in the series in terms of resonant modesof an identical lossless resonator (Mashkovtsev et aL, 1966, Weinstein, 1988).Then for the coefficients in such expansions we obtain a homogeneous infinitesystem of linear algebraic equations. The approximate solution of this system,obtained with the condition that the losses are small, results in simple formulaefor the electrodynamic parameters of eigenoscillations of the resonator withlosses.

The surface S will be considered as arbitrary and not containing infinitelythin edges. Such an analysis can also be carried out for a surface with edges of

Normal oscillations in resonators with losses 85

this shape but the modified impedance described in Section 2.4 should be usedin eqn. 4.2 instead of the Leontovich impedance.

Let us assume that the eigenoscillations of a resonator without losses satisfyingthe system of equations

curl Es=jcosfxHs

curl Hs = -ja)ssEs

and the boundary condition

(n0 x£s)\s = 0

are known. Here {cos} ( 5 = 1 , 2 , . . . ) is the sequence of eigenfrequencies of theresonator without losses in increasing order of magnitude

0 < col < co2 ' ' ' <cos< •••

Let us consider also the resonator-potential functions expressed as

K= 0

dn s

The potential functions can be treated conventionally as degenerated eigenoscil-lations of infinite multiplicity with the frequency cos = 0. According to Weinstein(1988) the systems of functions complete in Fare formed through the unification{Es} = {Es}®{es}, {Hs} = {Hs}®{hs}. The families {Es} and {Hs} are eachorthogonal:

e I EpEsdV=-ii I HpHsdV=3spmp (mp>0)Jv Jv

Thus, the sets {Es} and {Hs} are bases in V. Then the vectors E and H to befound can be expressed in the form of the expansions

s s

As and Bs being unknown constant coefficients. In accordance with a generalidea of the method of moments let us replace eqn. 4.1 by an equivalent systemof projection relations:

(curl E-j(OfiH)HpdV= 0

(4.4)

(curl H + jcosE)Ep d V = 0v

where p = 1, 2, . . . .To substitute eqn. 4.3 into eqn. 4.4 we exclude the differential operations on

E and H, as the expressions for curl E and curl H cannot be obtained byperforming term-by-term differentiation22 of the series for E and H. Using the22 This is because the series in eqn. 4.3 are not uniformly convergent on the boundary S, as thefields and the basis functions satisfy different boundary conditions. This kind of peculiarity of suchseries is also discussed in Section 3.3.


standard vector identities and eqn. 4.2, we rewrite eqn. 4.4 as

cuv\HpEdV-Zs<P Hp{nox (n0 x H)} dS

HHp dV=0

Hcur\EpdV EEpdV=0

(4.5)

Substituting eqn. 4.3 into eqn. 4.5 and using the orthogonality properties of thefunctions Es and Hs, we obtain the following system of equations for As, Bs:

(4.6)

(4.7)

a)pBp = coAp

jmp(copAp-coBp

where

Ps = 0

= (b Hp{n0 x(noxHs)}dS=- (b HpxHST dS

(the subscript t stands for tangential to S component of the vector H).From eqn. 4.6 it follows that coefficients Ap with potential functions ep are

zero {a>p = 0). However, the potential functions hp are preserved in the expansionfor the magnetic-field intensity.

Excluding Ap from eqn. 4.7 we obtain the homogeneous system of linearequations for Bp

jmp

c o 2p - c o 2

CO(4.8)

Nontrivial solutions of eqn. 4.8 determine eigenoscillations of the resonator withfinitely conducting walls. Complex eigenfrequencies are calculated as roots ofthe determinantal equation

DetCO *sp = 0 (4.9)

4.2 Resonance frequencies and Q-factors ofeigenoscillations

Let us suppose that among the modes of the lossless resonator there are nodegenerate modes (a)p>^(op) and the intervals between eigenfrequencies arerather large. Then, owing to the small size of Zs (\Zs\^ ô ) f°r t n e ^ t n

eigenoscillation it is sufficient to take into account only the pth equation in thesystem given by eqn. 4.8. Then the characteristic equation can be written as

(4.10)


Considering that the difference \cop — co\ is small, we can replace Zs(co) byZs{°*p) m ecln- 4.10. Then we obtain the expression for the eigenfrequency ofthe pth. eigenoscillation

^P)-(OP + JZS^ (4.11)

The real part of the eigenfrequency a>[p) determines the conditions of resonance;let us call it a resonance frequency of the resonator. According to eqn. 4.11

co'{p)ô)p-\mZs^ (4.12)

Later, when discussing resonators, we shall be considering the impedance £s a s

frequency-independent, using its value at the eigenfrequency cop. Usually,because of the small size of £S } the second term in eqn. 4.12 is negligibly small,and a>[p)^cop, i.e. the resonance frequencies can be determined assuming thewalls to be perfectly conducting. The rate of attenuation of the eigenoscillationdue to losses in the walls is characterised by the imaginary part of the eigen-frequency. The value

is called the ohmic (^-factor of a free oscillation (the superscript c indicates thatthe losses in the metal conducting walls are considered). Substituting theexpressions for cop and co"p) into eqn. 4.13, we obtain

H2pzAS

Eqn. 4.14 reflects the energy-perturbation method for resonators (Jones, 1986,Weinstein, 1988). The following interpretation of eqn. 4.14 is possible: let 0* bethe power dissipated in the walls, then 0* = — dM/d/, where W is the averageenergy stored in the resonator. As W(t) = W(0) exp( — 2a/7), we have

where His magnetic-field distribution in the resonator with losses. If the disturb-ance of the resonant-mode field due to losses is small, then assuming that H ~ Hp

in eqn. 4.15, we obtain eqn. 4.14. This assumption and, consequently, eqn. 4.14are not always valid, e.g. for the modes with close resonance frequencies. Neithercan this formula be used in the presence of infinitely thin diaphragms in theresonator, because of the singular behaviour of the field at the edges. The aboveconsiderations are similar to those in Section 3.7 which deals with the wave-guides, so we do not present any detailed discussion of these here. Eqn. 4.15,


generally speaking, is free from the limitations described but precise calculationof the magnetic field H is a rather complicated problem.

Within the limits of applicability eqn. 4.14 allows us to determine very easily(by calculating the integrals) the ohmic Q;factor if the field distribution of theresonant mode in the lossless resonator is known.

Let us consider now a case of two oscillations with close resonance frequencies((Op~cos). Then the two equations in the system given by eqn. 4.8

jmpmp~J° Bp + Zs(KPBP + APSBS) = 0

jms ^ - ^ - Bs + Zs (KsBs + AspBp) = 0

have to be retained.Assuming the determinant of this system to be equal to zero we obtain the

characteristic equation

(4.16)

To simplify the analysis consider (JL>P = (OS — CO0 assuming that Aco = coo — co issmall owing to the small size of the losses. Then eqn. 4.16 can be reduced to aquadratic equation for Aco:

(Aco)2 - AcoJ-^ F ^ + — - ^ 1 v pp ss psJ = 0 (4.17)2 \mp ms) 4msmp

When deriving eqn. 4.17, we take into consideration the relationship Asp = Aps.The roots of eqn. 4.17 are expressed by the formula

4 l\mp mAm A^V _ 4(AppAss-A2

s)|1 /2

P rns) msmp J

The real part of Aco determines a small shift of the resonant frequency underthe influence of finite conductivity of the walls. Such shifts are different for thetwo eigenoscillations, i.e. because there is finite conductivity the degeneration isremoved.

The ohmic (^-factor of coupled ^-oscillations is determined by the imaginarypart of Acol 2

^ ' z 2Acu'{f2

and can be expressed through the ohmic Q;factors of the/?th and ^th eigenoscilla-tions calculated according to eqn. 4.14 and the parameter

A _ cop(msmp)1/2


The latter can reasonably be called a mutual Q;factor of the pth. and stheigenoscillations. The final result is as follows:

i f i i \( i i V 4 I1'2!(4.18)

If Asp = 0 (Cisp = oo) eqn. 4.18 for each eigenoscillation transforms into eqn. 4.14.Thus, the approximate formula for the (^-factor given by eqn. 4.14 should becorrected only when the resonant frequencies of any two eigenoscillationscoincide (are close), and these eigenoscillations are coupled due to losses in thewalls (the mutual (^-factor is not infinitely large). A similar (but more cumber-some) consideration with identical conclusions can be carried out also for threeor more modes with close resonance frequencies. The analysis presented is closelysimilar to that for waveguides, given in Section 3.7.

Let us consider, as examples, two types of commonly used resonators: rectangu-lar (in the shape of a parallelepiped) and cylindrical (segments of a circularcylinder).

Let a parallelepiped have the dimensions a x b x /. The transverse electricand transverse magnetic types of modes will be determined relative to the faceax b. The TErnw and TMrnm modes with r ,K,m^0 are degenerate for anyrelationships between the dimensions. Let the multi-indices s and p correspondto the TM and TE modes, respectively; then for the (^-factors in eqn. 4.18 weobtain

c W02nabl

( '

W0{ab)2l (r2 «2+ u2

where Xp= 2{{mjl)2 + (rja)2 + {njb)2}~1^2 is the resonance wavelength.Thus the Q -factor of the mode in a rectangular resonator, when all indices

are nonzero, cannot be determined by means of eqn. 4.14, resulting from theenergy-perturbation method. Eqns. 4.18-4.21 are to be used instead, except forthe case a — b^ when, according to eqn. 4.21, Q^sp = oo.

In a cylindrical resonator, according to the calculations, the degenerateeigenoscillations are not coupled despite the losses in the walls and the modesfor which Asp / 0 cannot be degenerate at any relationships between the diam-eter and length. That is why the energy-perturbation method is valid for anytypes of eigenoscillations in a cylindrical resonator. The ohmic (^-factor canalways be determined according to eqn. 4.14.23 The corresponding formulaeare given in Section 4.4.

23 This is correct for resonators with ideally smooth walls. Even small deformations or roughness ofthe walls may result in the coupling of degenerate modes. In this case, the actual ohmic Q -factorfor degenerate modes can differ greatly from the Q;factor calculated according to eqn. 4.14 (seealso Section 4.4).


In a resonator with losses in the walls, apart from the eigenmodes, theassociated oscillations may occur. To analyse them we can use the results ofSection 3.4. Associated oscillations are mathematically described by the multipleroots of eqn. 4.9 (compare with Section 3.4).

4.3 Eigenoscillations and free oscillations in a resonatorwith a magnetodielectric absorbing body

In this Section we shall examine the oscillations in a resonator with perfectlyconducting walls but containing an isotropic absorbing magnetodielectric body.We shall take into consideration also the frequency dependence of the permit-tivity 8 and permeability ji of the body material.

Let us consider the problem strictly formulated as nonstationary, because theattenuation of oscillations in the resonator without sources is, in fact, a transientprocess. So we have to solve Maxwell's equations

curl/f = £—-dl (4.22)

dt

where s and jx are integral operators of the form

fe(<-T,r)•H, dr

describing the frequency dispersion of the body material (kernels e(t — z,r),fi(t~ T, r) are scalar functions).

We shall consider the moment of switching off the sources as the initial instant(t = 0) of our observation. Then the initial conditions for eqn. 4.22 have thefollowing form

e(-T, r )£(T, r )dT = C(r)— 00

where C(r) and §(**) are known functions.The boundary condition for eqn. 4.22 is expressed by

In addition, the tangential components of E and H must be continuous acrossthe surface of the magnetodielectric body.

In a similar way to that used by Weinstein and Vackman (1983), we represent


E(r, t) and H(r, t) in the form of Fourier integrals

1e x P ( "

1(4.23)

H(r, co) exp ( —jcot) dco0

These representations are valid, as it is physically obvious that

(4.24)

Substituting eqn. 4.23 into eqn. 4.22 we obtain the equations for E(r, w) andH(r, co)

cur\H= -jcos{co)E +

curl E = jcojj,(co)H— §

and the boundary condition

( n o x £ ) | s = 0 (4.25)

Thus we obtain for the Fourier amplitudes the problem of the resonator exci-tation by the currents given. The functions e(r, co) and //(r, co) are Fouriertransforms of kernels e(t — T, r) and fi(t — r, r), e(cw) and fi(co) being complex-valued functions:

8(co)=e'(co)+js"(co), fi(oo)=ii'(co)+jfi"(co), e"(co), fi"(co)>0

Assume that we know the eigenfunctions Es and Hs of the resonator underexamination, satisfying the equations

cur l i / s = -jcoss(co)Es

cur\Es=jcosfi(co)Hs

and eqn. 4.25. The eigenvalue is cos, and co is some free parameter. It is obviousthat Es = Es(r, co), Hs = Hs(r, co) and cos = cos(co). Let us present the solutionof eqn. 4.24 in the form

(4.26)

r,a>) = ftA,(<o)E,(r,a>)(4.27)

where As(co) and Bs(co) are the coefficients sought. Quasistatic terms in eqn. 4.27are rejected as

div (£ = div § = 0

Following Weinstein (1988), the coefficients As(co) and Bs(co) are expressed inthe form

AJco)= - Jns )v

{co%{r')Es{r\co)

(4.28)


(co -

The field sought is determined using eqns. 4.27-4.29 by means of the Fouriertransform according to eqn. 4.23. The result is

Ds{co)Es{r,co)

_«, (co2 - co2)ms(o)exp (—jot) dco (4.30)

where

{(0&(r')Es(r', co) - co,$(r')H,(r', to)} dV

A similar formula can be written for H(r, t).Let us analyse eqn. 4.30 in more detail. According to the principle of causality

(see, for example, Agranovich and Ginzburg, 1979), the functions s(o) andfi(co) are analytical in co in the lower halfplane;24 hence Es(r, co) and Ds(o)are also analytical. Taking into account the fact that the expression under theintegral has only simple poles [ms(o) # 0 in the lower halfplane], we obtain,according to the Jordan lemma, the expression for E(r, t) at t > 0

E(r, I) = Re r (r, &s

1 - •dcos

dco

where Qs satisfy the equation

Q =co.(QJ

(4.31)

(4.32)

As can be seen from eqns. 4.31 and 4.32 the electromagnetic field in the resonatorafter the sources are switched offcan be presented as a sum of the eigenoscillationsEs(r, o) and Hs(r, co) at co = Qs.

The functions Es(r, Qs), Hs(r, Qs) will be called free oscillations of the res-onator, the values Qs being their complex frequencies (Weinstein and Solntsev,1973).

The free oscillations satisfy Maxwell's homogeneous equations of the type

curl Hf= -jQs{Q)Ef)> (4 33)

curl Ef=jQfi{Q)Hf J

These equations present the eigenvalue problem nonlinear in the spectralparameter Q (the boundary condition for free oscillations in the case of perfectconductivity of the walls is the same as for eigenoscillations).25

2 4 Only nonconducting media are considered.2 5 This is correct also in the presence of losses in the walls w h e n the surface i m p e d a n c e is frequency-independent.


Free oscillations are harmonic and have the complex frequencies Qs

(Qs = Q's — jQ's, Q's > 0) and are therefore nonmonochromatic. They areamplitude-modulated oscillations with the amplitude decreasing exponentiallyas exp( — Cl'st). Thus the validity of the formal statement of the problem givenby eqns. 4.1 and 4.2 in Section 4.1 is obvious. Now we introduce the quantity

called the Qjfactor of the free oscillation in the resonator. This value characterisesthe rate of oscillation attenuation (after the sources are switched off) in theresonator due to dielectric and magnetic losses.

Let us calculate the Q;factor of the free oscillations. From eqn. 4.33 we canobtain the following 'energy' identity for the ith free oscillation:

(Q.*e*(Qs)\E{\2 - Qsn(as)\H{\2) dV= 0 (4.34)

Assuming that 0% < Q's, let us expand the functions fi(I2s) and ju(f2s) in Taylor'sseries in the neighbourhood of the point Q,'s (thanks to analyticity of e(co) andli(co) in the lower halfplane such expansions are valid). Neglecting terms abovethe first order in Q^, we have

A similar expansion can also be written for the function /x(fi). Substituting theexpansions for e(Qs) and jU(£2s) into eqn. 4.34, we obtain

Now we can easily derive the following formula for Qj:

OS = ^ ~c = L " (4-35)2

Vn

Eqn. 4.35 is sometimes used for active systems (in a linear operation) consideringthe losses as formally negative (e" < 0, pi" < 0). In this case Q< 0, which corre-sponds to the exponentially increasing amplitude of the oscillations [accordingto the factor exp(Qs//2|QJ)]. We would like to stress here that such an interpret-ation is purely formal and should be treated with caution, because the differencebetween passive and active systems even in linear operation is not merelyreflected by that in the sign of e" and //'. With active systems some additionaldifficulties arise. The first is purely mathematical: in active systems the fields


grow with time and Fourier's ordinary integrals to represent them do notconverge. This difficulty can be bypassed by shifting the contour of integrationto the upper halfplane (Weinstein and Vakman, 1983). However, this gives riseto another important problem: the integration path in eqn. 4.30 is in the upperhalfplane where the functions s(co) and fi(co) are not necessarily analytical. Theirsingularities when deriving eqn. 4.31 from eqn. 4.30 should be taken intoaccount. In this case the structure of the spectrum of free oscillations may changeconsiderably: new components which are not typical of passive systems may alsoemerge, in particular oscillations of a continuous spectrum.

Now we consider excitation of the resonator under consideration by the time-harmonic cur ren t ly , t) = Re{JE(r) exp(-jcot)}. Such a problem, even in thepresence of losses, can be stationary: losses in the resonator in this case arecompensated for by the exciting current. Thus, it is necessary to find the solutionof Maxwell's equations

curl H=-jcoe(co)E+JE~)

CUT\E=JCDIU(CO)H J [ ' }

satisfying the boundary condition given by eqn. 4.25.According to Weinstein (1988), when \co — cos\ < co, we can obtain the approxi-

mate solution of eqn. 4.36

E~ASES H~BSHS Bs{co) = —As{co)CO

AS(OJ)=- 2J0) I JEEsdV

{co2-cot)msjv

These formulae describe the phenomenon of resonance: a sharp increase in theintensity of the field excited when the resonant frequency of an eigenoscillationand the frequency of the source coincide. In this case the spatial structure ofthe field of forced oscillations is practically the same as that of the resonancemode. In the presence of losses the eigenfrequencies cos(co) are complex ones:cos(co) = CQ'S(CO) — jco's (co). The eigenfrequency cos can be presented in the form

COS(CO)=CO'S(CQ) ( l ~ j V ^ i

where QJ(co) = co'sj2co's is the eigen ^-factor of the resonator. This value directlydetermines the field amplitude at the resonance:

max \As(co) | ~1

msQJs(co)(o'sJEEsdV

Let us dwell on the derivation of the formula for unloaded (^-factor of the cavityconsidered. We introduce a system of eigenfunctions e®,h% of the identicalresonator without losses (c" = //' = 0). Having added corresponding potentialfunctions to these systems, we obtain the complete systems {eq}, {hq} in terms ofwhich the fields sought can be expanded. Let us present the eigenoscillations inthe form


The system of projection relations is

a; I e'{r)epE,dV+m,

co- ii'{r)hpHsAV+<os s(r)epEsdV=0J

(4.38)

where co® are eigenfrequencies of the corresponding lossless resonator.Substituting eqn. 4.37 into eqn. 4.38 we obtain a system of equations for a^

and bf\

a? ^ = 0mp

(4.39)

where

H"{r)hphqdV

\pq= e"(r)epeqdV

mp= z'{r)e2pdV=- v'{r)h2

pdVj J

The nontrivial solutions of the system represented by eqn. 4.39 determine theeigenoscillations in the resonator with a dissipative filling. The complex eigenfre-quencies can be calculated as roots of the corresponding determinantal equation.When the losses are small and the frequency cos is not degenerated and is notnear degenerate, we can obtain the following characteristic equation fromeqn. 4.39:

(4.40)

Assuming that Aesflms<^ 1 (the condition when the losses are small) we obtain

from eqn. 4.40 the approximate formula for cos

(4.41)— J

2ms

The real part of cos determines the resonant frequency:

R e co? = coK


The imaginary part of cos determines the unloaded Qj^ctov of the resonator;using eqn. 4.41 it can be presented as

c o ) = tl ( 4 < 4 2 )

((*)}2 + / / ( r , 0)){h°(co)}2] dV[£"(r, co){e°(Jv

Eqn. 4.42, like eqn. 4.15, is a result of an interpretation of the losses as pertur-bations (see Section 4.2). For degenerated modes eqns. 4.15 and 4.42 are notapplicable. The corresponding generalisation can be carried out in a similarway to that in Section 4.2.

Without dispersion eqns. 4.35 and 4.42 coincide; they determine the samevalue (QJ? = Qj) that can be presented as

stored energy

loss power

With a dispersive filling of the resonator QJ? ^ Qj. These quantities representthe different characteristics of the losses in the resonator: QJ specifies thetransmission band in the regime of forced-oscillation mode, and Qj theintensity of attenuation of oscillations in the resonator after the sources areswitched off. Neither QJ nor Qj can be presented by eqn. 4.43 if dispersion istaken into account. Moreover, for dispersive dissipative media the numeratorand denominator in eqn. 4.43, as pointed out by Agranovich and Ginzburg(1979), cannot be determined correctly at all within the framework of phenom-enological electrodynamics. When the losses are small, we can substitute E{0

and H{0, corresponding to a given resonator without losses [at e"(co) =fi"(co) = 0], for E{ and H{ in eqn. 4.35. In this case the numerator ineqn. 4.35 can be interpreted as the energy, stored by the sih free oscillation,in a similar resonator without losses.

When the losses occur in both the walls and the magnetodielectric filling, itis again possible to make an analysis similar to that described above. We do notpresent it here and only point out that within the framework of applicability ofthe perturbation method the complete losses can be found as a sum of partiallosses, so that the complete Q;factor is expressed as

1 _ 1 1

where QJ" is the 'magnetodielectric' Q -factor.Thus the partial Q^-factors—ohmic and 'magnetodielectric'—can be deter-

mined independently of each other, assuming that the losses of each type arethe only losses under consideration, and the complete Q^factor is determinedthrough the partial Q -factors according to eqn. 4.44.

When there are losses of a different physical nature—due to radiation intofree space, coupling to a load etc.—each of these types of losses can also becharacterised by a partial Q^factor determined independently from other partialQ^factors and included in the complete Q^factor according to eqn. 4.44. Note


that the condition of additivity of losses is not reduced to the smallness of allpartial losses. The additivity takes place only when there is no coupling ofdegenerate modes due to losses.

Eqn. 4.44 is not valid when losses of different physical nature are interdepen-dent. This is the case, for example, when we have a cavity coupled with regularsemi-infinite below-cutoff waveguides. When ohmic losses in waveguides areabsent then there are no radiation losses either but the finite conductivity of thewaveguide walls also accounts for radiation losses.

4.4 (?-factor of a cylindrical cavity

In this Section we begin a description of dissipative properties of simply shapedcavity resonators: cylindrical, spherical and conical. Let us underline that wedeal with calculation of the (^-factor characterising the losses in conducting walls(the cavities have no filling inside).

As in the determination of attenuation in waveguides (Section 3.7) it isnecessary to make a choice between two approaches: the energy-perturbationmethod based on eqn. 4.14 and the impedance-perturbation technique forsolving the characteristic equation of the resonator with finitely conductingwalls. In principle, both methods are equivalent, but we have to choose theone which implies the simpler calculations and more illustrative representationsof the final results. Note that for resonators of simple shapes the eigenfieldsare determined by the method of separation of variables and are expressed interms of well known special functions (in particular, the integrals in eqn. 4.14are calculated analytically). Eqn. 4.14 for the (^-factor can be presented inthe form Q= q/k0A0, where q is a dimensionless coefficient which does notdepend on the material of the resonator and is determined by its shape andthe type of oscillations. Later we shall analyse expressions for this coefficient.

In the case of perfect conductivity of the walls there are two classes ofeigenoscillations in a cylindrical resonator: TMmnp and TEmnp. For the TMmnp

oscillations

*o = {(vmnW2+ (pnll)2}1'2 (m,p = 0, 1, . . . ; » = 1, 2, . . .)

and

(4.45)rj _ jh 7 , hmn \ fcos(m0)l fpn

/ / =0

where b and / are the radius and the length of the cylinder, respectively. For

+ (pn/l)2}112 (m = 0, 1, . . . ; n,/, = 1, 2, . . .)

the TEww/7 oscillations


and

pnb

s m

We will not write out the electric components of eigenfields here as we do notneed them for the (^-factor determination.

The TEOnp and TM l n p oscillations are degenerate with any relationshipsbetween the radius and the length of the resonator. Moreover, for some ratiosbjl other modes can also be degenerate. However, it is possible to show that alldegeneration points in a cylindrical resonator are of trivial multiplicity (themutual (^-factor of degenerate oscillations is infinitely large). This means thatthe (^-factor of a cylindrical resonator for any mode can be determined by meansof the energy-perturbation method according to eqn. 4.14. Using eqns. 4.45,4.46, and 4.14 we obtain

for the TMmnp oscillations and

n2nn + (2bll) (pnbll)2 + {1 - (2bll)}(mpnblnmnl)

2nbll) + {1 - (2bll)}(mpnblnmnl) [ ' '

for the TEmwp oscillation.This does not mean, however, that the points of trivial multiplicity should

not be taken into consideration at all. Though degenerate oscillations in thesepoints are not coupled due to ohmic losses they can be coupled on account ofthe skewness and deformation of the walls, adjustment elements etc. Theseirregularities are usually rather small but they may result in a noticeableinteraction of degenerate oscillations. This is why the actual values of the Qjfactors for all degenerate oscillations can differ considerably from those calcu-lated by eqn. 4.14. A detailed study of the problem of mode interaction in cavityresonators was carried out by Shteinshleyger (1955).

In Figures 4.2-4.4 the dependencies of q on 2b 11 for different modes are shown.As expected, the TEOwp oscillations are characterised by the largest values of q.It is interesting that for these oscillations the Q,-factor is maximal when 2b = Ifor any n and p. The magnitude of this maximum grows slowly with n and pincreasing.

Eqns. 4.47 and 4.48 can also be obtained using the method of separation ofvariables for solving the interior-impedance problem for a cylindrical domain.In this case the original eigenvalue problem is reduced to the transcendentalcharacteristic equations for complex eigenfrequencies. The solution of the equa-tions obtained by means of the impedance-perturbation method results ineqns. 4.47 and 4.48. However, it is necessary to take into account here the fact


1.2

1.0

o.d

0.6

0 1.0 2.0 2b/lFigure 4.2 Ohmic Q^factor of cylindrical cavity for various modes

1 TE0 2 3 2 TE022 3 TE021 4 TEO U2 TE0226 TEQH

that in an impedance cylindrical resonator the oscillations with m # 0 are notpurely transverse magnetic and transverse electric (all components of the electricand magnetic fields are nonzero).

4.5 Q-factor of spherical and conical cavities

Apart from the cylindrical resonators examined in the previous Section, othertypes of cavity resonator can be treated by the method of separation of variables:rectangular resonators (see Section 4.2), resonators in the shape of a sphere,spherical cone, oblate and prolate spheroids (angular symmetric modes) etc.Below we present some of the results obtained using this method.

Spherical resonator: TM oscillations: Separation of variables is carried out in thespherical polar co-ordinates p, (/>, 3; the origin of the co-ordinate system is inthe centre of the sphere. The class of eigenoscillations under examination ischaracterised by the condition Hp = 0. If the Debye potential is introduced, the


0,8 -

5N>^ *

0.5-

// 3JZ- ^X

:==—£ _

9

N

6

7

—

1.0 2.0 2 b/IFigure 4.3 Ohmic Q^factor of cylindrical cavity for various modes

TE123TE112

TE122TE212

TE121TE211

5 TE,TE

problem is reduced to a scalar wave equation which can be easily solved by themethod mentioned above. The magnetic components of the TMm n p oscillationin the approximation of perfect conductivity of the walls are

sin(

(4.49)

where P%(x) are associated Legendre functions of the first kind.The resonant wave number £0 of the TMmnp oscillation is determined as the

pth root of the characteristic equation

JnJn- 1/2

(4.50)


to 2.0Figure 4.4 Ohmic QjJactor of cylindrical cavity for various modes

TMO2oTM1 1 2

TM0 2 3T M 1 U

3 TM0 2 2

8 TM0 1 3

TM021

TM0 1 1

510

TM1

where R is the radius of the sphere. Eqn. 4.50 follows from the boundarycondition E^ = 0 at r = R.

The characteristic equation given by eqn. 4.50 does not contain the azimuthalindex m. Thus every eigenvalue is of infinite multiplicity. As a result of somedisturbance the degeneration can be removed; in this case the eigenfrequenciessplit. These questions were studied using methods of group theory (Sapogovaand Kontorovich, 1971, Gaplevskii and Kontorovich, 1971). It is easy to provethat in this case the degeneration has a trivial character.

The (^-factor for this type of oscillation can be found by the energy-pertur-bation method according to eqn. 4.14. Substituting eqn. 4.49 into eqn. 4.14 weobtain after integration

n(n+ 1)

This formula coincides with that obtained by Fel et al. (1962). They solved theinterior problem for an impedance sphere by the method of separation ofvariables and applied the impedance-perturbation method for solving the


characteristic equation obtained. Fel et al. (1962) pointed out, in particular,that the corresponding formula in the well known book by Smythe (1950) wasincorrect.

Spherical resonator: TE oscillations: The solution procedure is completely similarto the previous case. The distributions of magnetic components of the TEmnp

eigenoscillation in the case of perfectly conducting walls are

n(n+l)(kp)

T KV(W>J.+1,2 (kP)l ~P7(cos 5) jsin(m*>dp v d# [cos (m(p)

— sin (m(f>)

'(4.51)

The resonance wave number k0 of the TEmnp oscillation is determined as thepth root of the characteristic equation

Jn+m(koR)=0 (4.52)

Using eqns. 4.14, 4.51 and 4.52, we obtain the expression for the coefficient q

q = k0R (4.53)

Thus, for example, for the TE011 oscillation k0R = 4.4934, i.e. the Q,-factor of aspherical resonator is the largest among those of resonators of simple configur-ations. For comparison, for a cylindrical resonator with the same type of oscil-lations qmax — 4.14 which is achieved when the length and diameter of thecylinder are equal.

In some works it is suggested that spherical resonators have the highest Qjfactor possible. This statement is supposed to be based on the approximateformula

d-T-7, (4-54)

where V and S are the volume of the resonator and its inner surface area (at agiven V the sphere has a minimal S). We would like to stress, however, thatsuch considerations are not very convincing. Eqn. 4.54 evaluates only the orderof magnitude of Q, and can be used to draw a rough comparison between cavityresonators and other classes of resonance systems (for example with quasistation-ary or quasioptical resonators) as regards their (^-factors. The nonuniformityand the specific features of the induced current-density distribution over S arenot reflected in eqn. 4.54. This is why it is impossible to compare Qj factors ofcavity resonators of different shapes using this formula.

Conical resonator: TEOnp oscillations: For this type of oscillation only the compo-nents EQ, Hp, H$ differ from zero. The separation of variables is carried out inthe spherical polar co-ordinates with the origin in the cone vertex. The


expressions for fields are

TTJ7P VVQ

-1/2 (cos 3^ > (4.55)

where Pv-i/2 (cos 3) = Pv-1/2 (cos $)> a n d v is determined from the boundarycondition £^ = 0 at 3 = 3 0 . The latter can be written in the form

The resonant wave numbers satisfy the following equation

For simplicity let us consider only small flare angles: 9Q/2 ^ 1- Then

(4.56)

and the expressions for the field components given by eqn. 4.55 can be rewritten

Using eqns. 4.14 and 4.57 and the equality

dp - Ko P

~ J2Ji(kR)}

2v + 1

Jv(kR)Jv + 1(kR) Jv(kR)Jv_1(kR)

2 v + l 2v— 1

kR{J2v.1(kR)-Jv(kR)Jv.2(kR)}\

2v- 1 j

R

we obtain the formula

(4.57)

(4.58)

for the coefficient q.


For negligible 3 0 (<90 < 1) we obtain, from eqn. 4.58,

q~k0R80

Physically the transition from eqn. 4.58 to this formula corresponds to neglectingthe losses on the cone end wall (p = R).

For the resonance wave number k0, from eqn. 4.56 we obtain the relation

k0R~v + 1.856v1/3+ 1.033v"1/3+ ••• (4.59)

the accuracy of which increases for decreasing 50 .The calculations according to eqns. 4.58 and 4.59 show that in the angle

range 5° < #0 < 15° q ~ 4 and depends weakly on 3 0 .It is necessary to make the following comment. Unlike the case of a cylinder

and sphere, for a cone with impedance boundary conditions the method ofseparation of variables cannot be applied. This is why the energy-perturbationmethod based on eqn. 4.14 is practically the only possible simple method fordetermining the (^-factor of a conical resonator.

Of other resonators to which the method of separation of variables can beapplied, we should also mention prolate and oblate spheroidal cavities. Theseparation of variables is carried out in prolate and oblate spheroidal co-ordinates, respectively, for angular symmetric modes only. The eigenfields areexpressed in terms of wave spheroidal functions. Unfortunately, the integrals forthese functions in eqn. 4.14 are not evaluated in analytical form. Numericalcalculation of these integrals is not simple to perform, making numerical methodsrelevant for resonators of more complicated shapes than those examined above.One such method will be described in Section 4.6.

Another kind of resonator of practical importance is the quasi-optical openresonator. The theory of such resonators is based on the parabolic-equationmethod (Weinstein, 1969<z). The ohmic Q,-factor can be determined by theenergy-perturbation method according to eqn. 4.14. The volume Fin eqn. 4.14is understood as the region of the field localisation with the caustic surfaces asboundaries, the surface S being the illuminated part of the mirrors.

Calculations of this nature were carried out by Hargreaves et al. (1991).According to them, for a resonator with spherical mirrors

q = ikod (4.60)

where d is the distance between the mirrors. It should be underlined thateqn. 4.60 does not allow us to calculate the complete (^-factor because it ignoresradiation losses. The question of the radiation Q determination has beendescribed at length (see, for example, Weinstein, 1969#) and we shall not dwellupon it here.

Generally speaking, resonators of this type are characterised by high unloaded(^-factors. According to the experimental data obtained by Kuraev et al. (1991)the complete unloaded Qât the b a n d / ~ 83-117 GHz is of the order of 105.

4.6 Galerkin's method for calculation of a complex-shapedcavity resonator in the form of a body of revolution

Let us consider a cavity resonator in the form of a body of revolution with anarbitrary continuous single-valued generatrix b(z) (Figure 4.5). Let us examine


the TEOrjp oscillations characterised by the field components E^, Hr, Hz\

H =

(4.61)

(4.62)

The analysis presented is based on a variant of Galerkin's method proposed bySlepyan (1977). The objective of the analysis, unlike that of Sections 4.4 and4.5, is to investigate resonators of essentially different shapes within the frame-work of a single approach. A special set of basis functions is used, each of whichsatisfies precisely the boundary condition for arbitrary b(z). Such functions caneasily be constructed for the boundary condition of perfect conductivity of thecavity material. However, such a construction is much more difficult to performfor impedance boundary conditions, so the (^-factor evaluation here will becarried out by means of the energy-perturbation method, on the basis ofeqn. 4.14.

Formulation of the eigenvalue problem is as follows: it is necessary to findnontrivial solutions of the equation

L0> = k2d> (4.63)

where L = ~Vfz + ?'~2- The boundary condition for O is

<S>(r,z)\s = 0 (4-64)

where S is the inner surface of the resonator.Let us introduce a system of functions

(4.65)

where X — {i, m], i, m = 1, 2, . . . , C{z) is a function, different from zero at theinterval 0 < <; < / with the exception of a finite number of isolated points, andSi are constant multipliers.

Figure 4.5 Cavity resonator in the form of a body of revolution with an arbitrarygeneratrix


It is easy to see that the functions CDA satisfy the boundary condition given byeqn. 4.64. Following the general idea of Galerkin's method, we replace eqn. 4.63by projection relationships

- LQ) + Jt2O)OA.r dr dz = 0o Jo

representing O as

(4.66)

(4.67)

where / ' = {/', m'}, JV={/, Af}, and Ck are unknown coefficients. Substitutingeqn. 4.67 into eqn. 4.66 we obtain the following matrix eigenvalue problem

<BC=(kN)2WlC (4.68)

where C is an jV-dimensional vector of the coefBcients Ck to be found, and 23and 9M are matrices of the order jV. The structure of C and 93 is as follows:

93]

C =

Cn

,93 =

»{.=

The structure of the matrix 9JJ is similar to that of the matrix 93. Matrix elementsare expressed by

z (4.69)o Jo

(4.70)o Jo

The inner integrals in eqns. 4.69 and 4.70 can be determined in a closed form.After some transformations we have

et


fdb d£\ , , .+ b£ I — £ + b —I {m' cos (m' ) sin (m ) + m cos (mz) sin (m'^)}

\d£ d£/

C,2b2mm cos (mz) cos (m'z) ) dz

"o.--ôi')e»fir Jo L tit' vlvd^ ,

x sin(m^) sin(m^) — —^C {mcos(m^) sin(m^)d^

— m'cos(m^) sin(m'^)}

j \£i JO

sin (mz) sm(m'z) &z

Here m = m7i// and m = m'njl.To derive the expressions for ©J^, and

dx =

we use the formulae

. - , ,2.

1=1'

The matrices S and 501 are symmetrical:

' JJl

The eigenvalues of the problem given by eqn. 4.68 determine approximately (asJV is finite) the resonator eigenfrequencies co^ ((0% ~ k^c, oc = {n, p} being the two-dimensional mode index), and the eigenvectors determine electric-field distri-butions of eigenmodes (in accordance with eqns. 4.61 and 4.67). The magnetic-field components of eigenoscillations are expressed in terms of O by eqn. 4.62.

The ohmic (^-factor of the resonator, as already mentioned, is calculated bymeans of the energy-perturbation method. The final result is

ITJo Jo

{a)(x,lg2(T)xdxdT


where

( I M a 7 / ,, \

H X I cim 7-7=—C(T) sinC » = 1 m = l

( . 1 = 1 HI = 1 £

g(T)=b(T)lbmax T = zll

go=g(0) gi=g{l) v«

The function £(£) and the coefficients et in eqn. 4.65 can be chosen specificallyto improve the efficiency of the computational algorithm, making it possible toenhance the convergence rate or to simplify the formulae for the matrix elements33^' and 5WAA' by presetting the function £(z) in an appropriate manner.

To ensure that the algorithm is correct it is necessary to carry out a mathemat-ical substantiation of all formal operations performed above. We shall not presentany detailed description here (see, if necessary, the work by Slepyan, 1977). Wemerely note that it contains the proof of convergence of approximate values ofkN and qN to the accurate values at /, M—• 00 and also the condition of algorithmstability against the errors in the matrix-element calculations (the concept ofstability is discussed in Section 1.4).

The proof of convergence is based on the properties of linear independenceand completeness of the function systems {OA} and {curl(zÔA)} in correspondingfunctional spaces. To prove the convergence of the Q-factor it is necessary alsoto substantiate the possibility of a term-by-term differentiation of the series ineqn. 4.67 at jV-» co (when the magnetic field is determined from eqn. 4.62). Itis essential that the matrices 93 and Wfl are positive definite. Finally, the necessaryand sufficient condition of the algorithm stability, according to Mikhlin (1966),is a strong minimality of the set of functions {O^} in the energy space of theoperator L. It takes place when the asymptotic evaluation e,- = 0 (//o(2) 1S fulfilled[it is convenient, in particular, to set ef = Jo(ôi)]-

As a test example, the eigenfrequency and the (^-factor of the TE011 mode inthe spherical resonator have been determined. When the matrix elements®AA?3ÂA' f°r a sphere and similar configurations (spheroid etc.) are beingcalculated, the following difficulty occurs: at the points z = 0,1 b(z)->0 anddb/dz—• 00, which leads to indeterminate forms (the expressions under theintegral are finite at these points). The method of evaluation of these indetermi-nate forms depends additionally upon the choice of the function C(z)- In ourcalculations £(z) — 1 and the function under the integral equals zero at z = 0and z —l> This is why the end sections with a length of AT = A/// are ignoredand the integration is carried out between the limits A/ and / — A/. The calcu-lation results for a sphere at AT = 0.01 and different values of/, M are presentedin Table 4.1. The values of q and the normalised resonant wave number obtainedfrom eqns. 4.52 and 4.53 are assumed to be accurate. Excellent agreement isobserved for even six basis functions ( /= 2, M — 3). Thus, the convergence rateof the algorithm is fairly high.

The eigenfrequencies of the TE0 1 1, TE0 1 2, TE013 modes of prolate and oblate


Table 4.1

Parameter

klQ

Convergencecavity

of normalised

M=3

9.03304.0091

resonant wave number

M=l

9.00814.3100

and Qjfactor

1=2M=3

8.98854.4883

for spherical

Accuratevalue

8.98684.4934

A/a

0.6 0.2 0.2 0.6Figure 4.6 Normalised resonant wavelength of prolate and oblate spheroidal resonators

as a function of eccentricity

spheroidal resonators have also been calculated. In Figure 4.6 the values of2njka {a being the minor semiaxis) are shown as functions of the eccentricity e.The calculations have been carried out at 1=2, M= 3, AT=0.01. The solidlines correspond to the data obtained by Semakov and Tereschenko (1975)through the method of separation of variables in spheroidal co-ordinates. Circles,triangles and squares represent the results of our calculations according to thealgorithm described in this Section. As in the previous example, there is againvery good agreement between the results obtained by the two approaches. It isnecessary to stress that we have taken only six basis functions for a wide rangeof eccentricities of both prolate and oblate spheroids. This allows us to concludethat our algorithm is effective for resonators of various configurations.

Using this algorithm we have determined the electrodynamic characteristics


of the TE011 mode in a symmetrical truncated biconical resonator. The compu-tational results and the sectional view of this resonator are shown in Figure 4.7(the solid line being the coefficient q, and the broken line the dimensionlessparameter a = 2/iOi (1 ~ So)lô^ t a n $o)- The calculations have been carried outfor g0 = 0.1, / = 1, M = 7. Numerical experiments have shown that the increasein / and M does not in practice influence the results described.

Of some interest is the comparison of cavity resonators of simple shapes(cylindrical, spherical, conical, biconical) examined in this Chapter with respectto the (^-factor of the TE011 mode. This is because the TE0np modes in cavitiesin the form of a body of revolution are characterised by the largest values of Qand the TE011 mode has the lowest resonant frequency among the TEOnp modes.For convenience, we shall compare the values of the coefficient q directly: thiscorresponds to resonators with the same resonant frequency and the samematerial of the walls. Spherical resonators have the highest Q of all the types ofresonator under study ( = 4.4934). For cylindrical cavities the coefficient qequals 4.14. The values of q for conical and biconical resonators are close tothat for a cylindrical resonator. The differences in Q, though, are not significantfor all types of configuration, i.e. spherical resonators cannot be recommendedfor use, taking into account the difficulties in their manufacture and frequencyretuning. For this reason we cannot yet gain much by way of optimising thegeneratrix shape of a body of revolution. The cylindrical cavity appears to bepromising for applications, but in this case higher modes in combination with

-3.0

$ 12 15 18 21 24 27 JO60, degrees

Figure 4.7 Normalised eigenfrequency and Qjfactor of TEoll mode in truncatedbiconical resonator


some means of spectrum rarefaction (for example, narrow azimuthal slits) shouldbe used.

4.7 Cylindrical resonator with dielectric slabs

In this Section we shall consider one specific case of a cylindrical resonator witha partial dielectric filling. We shall demonstrate that the insertion of dielectricbodies can increase the complete unloaded (^-factor for certain types of eigenoscil-lations (Ilyinsky and Slepyan, 1983).

Consider a cavity resonator of cylindrical shape with two identical dielectricdiscs made of a material with the permittivity s^s'+js" (&' > e") and per-meability }JL = jti0. The discs are positioned symmetrically as regards the z = l\2plane (/ being the length of the resonator). The geometry of the problem isshown in Figure 4.8 (i). Taking account of the fact that the z = 1/2 plane is aplane of symmetry for the resonator under consideration, the field distributionscan be either symmetrical or antisymmetrical with respect to this plane. Thisallows us to analyse the field in only half the resonator. Let us consider theTE0/ip oscillations which are described by the azimuthal component of theelectric field E^ according to eqns. 4.61 and 4.62. To determine the (^-factor weuse the energy-perturbation method, based on eqns. 4.14 and 4.42. For this wehave to calculate the eigenfields in a lossless resonator of the same shape. Usingthe method of partial regions [they are denoted by Roman numerals inFigure 4.8 (i)], we represent the fields in the form

r dr

dC(v)

where v (v = 1, 2, 3) is the number of a partial region. The functions C(v)(^) aresolutions of ordinary differential equations of the form

d2C(v) , , , , , , .2 l~ \k £(V) ~~ /ÔnfC 0 (4.7 1)

where £r(1) = ej3) = £0, £[2)

= £'-The boundary conditions for p even are

C ( 1 ) ( 0 ) = 0

C ( 1 ) ( t f ) -C ( 2 ) ( t f )=0

(4.72)

/ d C ( 1 ) _ d C ( 2 ) \I I — u\ dZ dz I z = fl


dC( 3 ) \

d^ / z=a + d

C(3)(//2)

= 0

= 0(4.73)

The last relationship in eqns. 4.73 should be replaced when p is odd by

dC(3)

= 0

Solution of the eigenvalue problem given by eqns. 4.71, 4.72 and 4.73 leads tothe transcendental characteristic equation for resonant-wave numbers whichallows us to determine the eigenfrequencies and eigenfields of the resonator. Weshall not present this analysis in a general form here. Our objective is to illustratethe typical physical effects in one specific case. Let d— a and

ha = TT/2

hd =71/2

(4.74)

(4.75)

where h and h are propagation coefficients of the TEOn mode in circular wave-guides of radius b, filled, respectively, with air and dielectric with the permittivity

where 8 is relative dielectric permittivity.Omitting simple intermediate calculations, let us write out the final expressions

for the electric field in partial regions:

(4.76)

where B is an arbitrary constant.One of eqns. 4.74 and 4.75 can be considered as a characteristic equation

[determining the resonant frequency for given a (or d)]; then from the other wefind d (or a).

The complete unloaded (^-factor can be obtained (see Section 4.3) fromeqn. 4.44

n = JL^L (4.77)

where Q? is expressed by eqn. 4.14, and Qj by eqn. 4.42 at \J!' = 0 (dispersive

2.1802.1362.520

0.6380.6360.645

557

.34

.11

.32

X

xX

104

104

104

787

.72

.14

.32

xxX

104

104

104

3.163.143.00

X

xX

104

104

104


Table 4.2 Calculational results for copper cylindrical resonator with poly core slabs

b(mm) a (mm) c/(mm) Q Qc Qz

12148

media are not considered). In this case eqn. 4.42 can be rewritten in the form:

e'EldV(4.78)

where Vd is the volume of the dielectric filling. In accordance with the energy-perturbation method we shall substitute the expressions of eigenfields in thecorresponding lossless resonator (they are given by eqn. 4.76) into eqns. 4.14and 4.78. Evaluating the integrals in eqn. 4.78, we obtain

Q E = ^ — A ' i U + ['i) } + £ \ (4-79)e tan 3 ( u ' x u ' ( ' v ;

where 3 is the angle of dielectric losses,

^=—{— .> / h P " (4-80)

Table 4.2 presents the results of numerical calculations by eqns. 4.77, 4.79 and4.80 for copper resonators with polycore slabs (s = 10, tan 3 = 10~4, Xo = 8 mm,TE0 1 3 operational mode). For comparison we point out that, for a cylindricalresonator made of copper without dielectric slabs, Qmax ^ 2 x 104 (Ao = 8 mm).Thus, the (^-factor of the resonator with dielectric discs is 1.5 times as high asthat of the optimal unfilled resonator. When the length of the resonator issignificantly larger than its radius, there is no gain in Q.

Physically, this result can be explained by the fact that the main part of thestored electromagnetic energy is concentrated in the central region of the res-onator; the electromagnetic field in the peripheral regions and inside the dielec-tric slabs is comparatively weak [see Figure 4.8(ii)]. That is why the surfacecurrents in the end walls are small, and so are the ohmic losses. So, at / < b Qîncreases, in this case essentially because the losses in the cylindrical wall aresmaller. Then the losses in the end walls are considerably reduced owing to theinfluence of dielectric slabs. Even though the losses in the dielectric filling leadto a decrease in the complete (^-factor, it remains fairly high—higher than themaximum (ôbtainable in unfilled resonators at the same operational frequency.

The increase in (Hs still higher when using dielectrics with tan 3 smaller than


d'

8&

a

m

(i) (ii)

Figure 4.8 Cylindrical resonator with dielectric slabs

(i) General view and main designations(ii) TM013-mode field distribution along the £-axis

with dielectric slabswithout dielectric slabs

those of polycore. Similarly, the ^-factor of the TE0Mp modes with p>3 can beincreased. Note that the greatest increase in Q,occurs when the conditions givenby eqns. 4.74 and 4.75 are fulfilled.

A similar physical mechanism is used by Kazantsev et al. (1974) to reduceattenuation of the TEOn mode in a circular waveguide. They studied a circularwaveguide with a hollow coaxial dielectric pipe. A reduction in losses is possiblecompared with those of ordinary circular waveguides.

4.8 (?-factor of a cylindrical resonator with a coaxial insert

The configuration of the resonator under examination and the main notationsare shown in Figure 4.9. Such resonators are widely used in microwave electronicdevices, measuring equipment etc. Many publications have been dedicated todetermination of their electrodynamic characteristics. Two types of coaxialinserts were studied, one in the shape of a hollow cylinder with an infinitelythin wall, and the other in the shape of a cylinder with a flat metallised endface. In most published work the walls of the resonator were supposed to beperfectly conducting; the Qjfactor was not determined.

A rough (as to the order) evaluation of Q,for the fundamental mode has beenmade by Kleev and Manenkov (1982). The problem of calculating the (^-factorof the TMOnp oscillations was considered in papers by Gubsky et al. (1982),Kleev and Manenkov (1984) and Rodionova and Slepyan (1986). In the firsttwo papers the energy-perturbation method was used, while the third dealt withsolving the impedance problem for a given resonator. Some results on theQ,-factor of cylindrical resonators with coaxial inserts, and explanatory notes tothem, can be found in the survey by Ilyinsky and Slepyan (1990).

When using the methods of integral equations and of partial regions, calcu-


r

m

/ / / / /

11

a

Figure 4.9 Cylindrical resonator with coaxial insert

lation of the field distributions of eigenmodes is rather complicated. This makesit difficult to use the energy-perturbation method to calculate Q (numericalintegration is needed). Therefore, Rodionova and Slepyan (1986) resorted toconsidering the impedance problem and developing an effective algorithm forits solution. As a result they obtained a characteristic equation for complexeigenfrequencies, the solution of which by means of the impedance-perturbationmethod allows determination of Q_. In this case an approximate analyticalformula for Q has been obtained that can not be derived easily in any otherway (for example, by the energy-perturbation method). The material of thissection is based on the paper by Rodionova and Slepyan (1986).

Let us examine the TMOnp oscillations which are of most interest. The fieldcomponents of such modes can be expressed in terms of the azimuthal componentof the magnetic field //^(r, z) that satisfies the equation

: ^ + + *2^--y = o (4-81)dr2 r dr dz

and the boundary conditions

ii -(r c

= 0•=a-0,0<z<L

= 0

(4.82)

(4.83)= a + 0,0<z<L


Z = L = O (4.84)z = - I

where rj = k£s / Wo.The solution of eqn. 4.81 should also meet the edge condition. We have to

find the complex eigenvalues k = k! + jk" of the eigenvalue problem given byeqns. 4.81-4.84.

At the first stage of solution, we use the partial-regions method. The partialregions are indicated by Roman numerals (Figure 4.9). In area I the field soughtis expressed as a superposition of the TMOn modes of the coaxial waveguide(with radii of pipes a and b) propagating in opposite directions. In regions II,III similar expansions are written in terms of the TMOn modes of circularwaveguides with radii a and b, correspondingly. The field matching is performedin the z~ 0 plane. As a result we obtain the infinite homogeneous system oflinear algebraic equations

= 0 (4.85)s=l \ l p P s

where

r2..,=7,

; -2 j f f s / )

p = 1, 2, . . . , xs being unknown coefficients.The values Xs, 9S and xs

a r e t n e roots of the transcendental equations

Jo(Z,The homogeneous algebraic system given by eqn. 4.85 presents the eigenvalueproblem nonlinear in the spectral parameter k. This system is characterised bya 'relative' and slow convergence, with low stability against roundoff errors.Because of this direct numerical solution by means of the truncation method isvery difficult.

Regularisation of systems of this type is usually carried out using the methodof semi-inversion (Shestopalov et aL, 1984) or the modified residue-calculustechnique (Mittra and Lee, 1971). The latter was applied to simplified spectralproblems of the form of eqn. 4.85, when as = 0, fip = 0,p > 1 (Al-Hakkak, 1978).


The comparison between these two methods was drawn by Shestopalov et al.(1984) for inhomogeneous boundary-value problems. It was established that themodified residue-calculus technique results in algebraic systems with the highestrate of truncation-method convergence. The approach presented in this Sectionis as effective as the modified-residue-calculus technique and, unlike the analysisgiven by Al-Hakkak (1978), is free from any of the limitations mentioned above.

The solution of eqn. 4.85 is sought in the form of a regularised substitution

m = l l m ~ Ps

where Fm are new unknown coefficients, and {ds} are nontrivial solutions of thesystem

Z " 0(4-87)

|w = fc (4.88)

where

with algebraic asymptotic behaviour at s-+cc.It can be shown (see, for example, Shestopalov et al., 1984), that

X~ [b — a) \n{bl(b — a)} 4- a In [bja) and C is an arbitrary constant.Substituting eqn. 4.86 into eqn. 4.85 and using eqns. 4.87 and 4.88, we obtain

the following homogeneous system of equations for Fm:

I Fm£>pm = 0 (4.89)m = l

where

°sdsm P m ^ w = r»

y s s

5=1 r m ~ J

(4.90)

The characteristic equation for complex eigenfrequencies is

</>(£) = Det \\Dpm\\=0 (4.91)

Approximate values of eigenfrequencies are obtained from eqn. 4.91 after beingreduced to the JVth order. As can be seen from eqns. 4.89 and 4.90 the matrixoperator D is a Fredholm operator in some Hilbert space which guarantees theexistence of the infinite-order determinant Det D. The proof of convergence ofthe approximate eigenvalues to accurate eigenvalues at JV—> oc is based on thetheory of finitely meromorphic operator-functions (Vainikko and Karma, 1974)and results obtained by Shestopalov (1980, 1983).

Let us apply the perturbation technique to eqn. 4.91 taking into considerationthe fact that \r}/k\<^l. The difficulty lies in the function (j>(k) being expressedby the infinite determinant and this function, according to the perturbationtechnique, is supposed to be differentiated, expanded in Taylor series etc.


However, all necessary operations, as in Section 3.8, can easily be performedwithin the framework of the theory of determinants. Let us present Dpn at small\f]\ as

where

The eigenvalue k to be found can be presented as k = k0 + Ak, where k0 satisfieseqn. 4.91 at r\ = 0 [Det \\DfJ\\ = (f)(ko)=0] where Ak is a small complex correc-tion due to finite conductivity of the walls. The (^-factor of the eigenoscillationunder examination is expressed as

a- *°2Im(AA;)

where

I Det||5jj>Ak = -jrj •

Z Det||Cg>||S = l

The elements of the matrices Z?(s) and C(5) can be determined according to therelationships

= s)dk

In practical evaluations, as mentioned above, the matrix D is truncated to theNth. order; in this case to find Ak we should calculate 2N determinants of thejVth order. When jV is large, computation of these may take a long time andthis technique can therefore be especially effective in combination with analyticalmethods of regularisation, resulting in rapidly converging systems of linearequations.

Let us consider the TM01p oscillations which are of particular practicalinterest. In this case it is possible, with some assumptions, to obtain approximateanalytical expressions for k0 and Q. Let l>> L, l>>b and ko<voilb. Then theoscillations under consideration are formed due to the reflection of the TEMmode from the coaxial-to-circular waveguide junction (z = 0) and the end wall(z = L) of the resonator (these modes are not actually affected by the other endwall z— —/). In the first approximation the characteristic equation is

— = 0 (4.92)dw w=ri 2 r

Applying the perturbation method (rj small) to eqn. 4.92 we obtain the following


formula for the dimensionless parameter q:

2jx{(2/>+l)+2yo(/:)}lnxg-[(l+x){(2p+l)ii+2e(k0)}+yx{\+y2P(k0)}\nx-1] [' '

where x = ajb, y = kob and p = 0, 1 , 2 , . . . . The values P{k0), y(k0) and s(k0)are expressed by

k°]^ £ ' " U ( v g , - ^ 1 / 2 ^ -v*rW

(4.94)

yx

(4.95)

V VY

^—(v2 — v 2 \ 1 l 2 - ^— (v2 - v 2 v 2 \ 1 / 2

s=l \VOs V O s

H - t a n " 1 ! vU — I - v z V I + t a n

(4.96)

The resonant wave number k0 is determined from eqn. 4.91 at r] = 0, JV= 1, or,which is the same, from eqn. 4.92 at f] = 0.

The results of calculation of q for the TM0 1 0 mode according to eqns. 4.93—4.96 are shown in Figure 4.10. Figure 4.10(i) shows the dependence of q onLjb for two values of the ratio ajb. When L/b->0 we have that q->y-+v01. AtLjb ^ 1 a sharp fall in this dependence is observed, which transfers at Ljb >> 1into a smooth decrease. The dependence of q on x = ajb at different Ljb is shownin Figure 4.10(ii). The q coefficient is maximal at , t~0.3 for any relationshipbetween L and b.

Also of interest is the case when / < L, I <^b. Then we can consider theresonator as quasistationary: area III in Figure 4.9 plays the role of capacitanceand area I that of inductance (in this case the electric and magnetic fields arelocalised in areas III and I, respectively). Evaluation of the resonant frequencyis simplest when the model of a coaxial insert with metallised end wrall is used.Then the capacitance, neglecting edge effects, is expressed as C ~ £0na2/2/, the


i/b(i)

0.5 0.7a/6

(ii)

Figure 4.10 Characteristics of fundamental mode in cylindrical resonator with coaxialinsert

(i) Normalised eigenfrequency and (^-factor against Ljbqy

(ii) Normalised Q,-factor against x

inductance being $£ — fi0L In x~l j2n, and the resonant-wave number is

k (wo\12 ( 4/ Y/2

°~\£ec) ~\La2\nx~l)The Q;factor of the resonator can be determined by means of the energy-perturbation method as the field structure of the eigenoscillation is relativelysimple. For this case, eqn. 4.14 can be written as

& = •

H%AV

"IJs

(4.97)

Hi AS

where by Fwe mean the volume of'inductive' partial area I, and by S the partof its surface which is metallised. Supposing that H^ — I\2itr (I being the totalcurrent) and integrating in eqn. 4.97, we obtain the following formula for Qj.

( 4 ' 9 8 )

which is the final result of the analysis.


Experimental investigations of the resonator described have also been carriedout. The structure of the experimental model and the block diagram of themeasuring equipment are described in the paper by Rodionova and Nekhay(1981). The resonator frequency retuning was carried out by means of a metalring moved in area III. The resonator surface was coated with silver andpolished to the 12th surface-finish class to remove the microroughness.

Three experimental models were examined with different values of Ljb andthe same ajb — 0.3. At the upper operating frequency^ = 2.14 GHz the retuningring was placed at a distance from the edge of the coaxial insert (the assumptionsmade when deriving eqn. 4.93 are fulfilled). At the lower operating frequency/ 0 = 1.07 GHz the retuning ring was located near the edge of the coaxial insertand played the role of the second plate of a capacitor. The eigenoscillation herebecame quasistationary and its ^-factor is expressed by eqn. 4.98. Thus, thevalidity of eqns. 4.93 and 4.99 was verified using one experimental model.

For two values of f0 the halfpower bandwidth 2A/ and the standing-waveratio SWR0 in resonance were measured. The experimental value of the Q -factorwas determined from the results of measurements according to the formula

Q=(l+SWR0)24/

Figure 4.11 Comparison of theoretical and experimental results for (^factor offundamental mode in cylindrical resonator with coaxial insert


Figure 4.11 shows a comparison between the results of measurements andcalculations (the solid line represents calculation according to eqns. 4.93-4.96,and the broken line estimation according to eqn. 4.98, circles and triangles:measurements a t / 0 = 2.14 GHz and fo = 1.07 GHz, respectively). As can beseen from Figure 4.11, there is good agreement between the theoretical andexperimental data.

Chapter 5

Electromagnetic-wave diffraction byfinitely conducting comb-shaped

structures

This Chapter is dedicated to the diffraction of plane waves by finitely conductingcomb-shaped structures. Finite conductivity is taken into consideration by meansof impedance boundary conditions. Periodic structures of comparatively simpleconfiguration (array of halfplanes, symmetrical and nonsymmetrical lamellargratings, echelettes) are examined as well as structures with complicated shapeprofiles. Highly effective numerical methods are described for the latter whichare universal in relation to the structure configuration.

Approximate formulae and numerical results are given for a linear (per-period) power of heat losses. Its dependence on the angle of incidence, wave-length and configuration of the unit cell is analysed.

For the case of H-polarisation considerable attention is paid to the effect ofabnormally small dissipation (the losses in a periodical structure may be smallerthan those in a smooth surface of the same material). For E-polarised waves thisdoes not occur. The physical mechanism of the absorption anomalies is analysed,and their practical use in reducing attenuation in microwave waveguides andresonators is discussed.

5.1 Diffraction of a plane wave by an array of impedancehalfplanes: H-polarisation

In this section we shall consider the problem of a planewave diffraction by anarray of finitely conducting halfplanes (Figure b.la). Let us assume halfplanesto be infinitely thin. It is permissible for the fields of electric polarisation (E ={Ex, 0, Ez}, H= {0, {//, 0}) on condition that the thickness of halfplanes is smallcompared with the structure period and wavelength, but exceeds considerablythe skin depth. Mathematical statement of the problem is as follows: we haveto find the solution of a 2-dimensional Helmholtz equation

vLiA + *2<A = o (5.1)

124 Electromagnetic-wave diffraction by finitely conducting comb-shaped structures

* d D

I I

F D

aFigure 5.1 Schematic representation of gratings

a Parallel-plate gratingb Comb-shaped periodic structure

satisfying the impedance boundary condition

= 0l/2)d,0<x< oo

(5.2)

(m = 0, + 1 , +2, . . . , and the signs ' + ' in eqn. 5.2 correspond to the right-hand and left-hand sides of the plates, q = k£sIW0) and the edge condition.From the latter it follows that

fJdV

dxdz< oo

where 3V is a bounded area in the plane Oxz containing the edge.At x-> oo we impose the 'extinction' condition

lim {//(x, z) = 0 (5.3)

and at x -> — oo \\i should satisfy the radiation condition [scattered fieldij/ — ij/0 being a superposition of outgoing plane waves, i/f° (x, z) =

exp{jk(x cos 9 + z sin 9)}].The problem of wave diffraction by the array of halfplanes may be considered

as a key problem for a whole class of periodic structures. We shall describe the

Electromagnetic-wave diffraction by finitely conducting comb-shaped structures 125

effective solution technique for this problem (Slepyan, 1980), which can easilybe modified for comb-shaped structures, echelette gratings etc. (see below).

Let us present the field over the structure as an expansion in terms of Floquet'sharmonics:

^{x,z)=^{x,z)+ E Rpexp(japZ + rpx) (5.4)P = — OD

where ap = 2{pn + u)Jd, u = ka sin 3, a = rf/2 and Tp = (a2, - £2)1/2.To satisfy the radiation condition a square-root branch in the expression for

Tp has to be chosen according to the condition Re Tp > 0 (if Re Yp = 0, thenImrp<0).

The field in the mth cell of the structure at x > 0 can be presented as a seriesin terms of eigenmodes of a plane-impedance waveguide of the width d:

+ T™$n(z-md)exp(-%x)} (5.5)

where_ $n(z)= cos(PnZ), $n(z) = s i n ( ^ ) , yn = (Pn - k2)1'2, % = (pn - A2)1'2,and Pn and j8n are the roots of the transcendental equations

^ i ( / T ) ( / f )

Applicability of the field representation given by eqn. 5.5, unlike the case ofperfectly conducting halfplanes, is not obvious and needs special substantiation.For a non-self-adjoint operator, corresponding to a homogeneous problem foran impedance-plane waveguide with a complex impedance of the walls, theeigenfunction set is generally speaking incomplete (see, for example, Keldysh,1951). Only the eigenfunctions and associated functions form a complete set.Whether or not there will be associated functions depends on the values of thecoefficient rj in eqn. 5.2. In Section 3.6 some limitations on r\ are formulated:with these limitations there are no associated waves. Later we shall considerthese limitations to be fulfilled, so the use of eqn. 5.5 will be quite relevant.

To fulfil the 'extinction5 condition at x—• oo a square-root branch in expressionsfor yn,yn should be chosen in accordance with the conditions Re yn > 0,

Owing to Floquet's conditions of pseudoperiodicity we have

f<M> = T<°> exP(j2mu), f «m) = f <0) exp(j2m«)

This makes it possible to examine only one unit cell of the structure.Using the continuity of i// and d\jj_[dx _across the x = 0 plane and the ortho-

gonality property of the functions {^n, \j/n} in the interval |^| <a , we come tothe following dual system of linear algebraic equations of the first kind (SLAE-1) for unknown coefficients Cp:

00 C 1

^ I V ^ T T V ^ ; (5-6)

& Cp(l + cop) _ 1 + co0

P=-oo r' -% TO + %


where

CO = —c o t u) s m 2w

C _ {-\)pRp(*p+jri cot «)p A: sin 3 + j?/ cot u

According to the edge condition a physically correct solution of the SLAE-1(eqns. 5.6 and 5.7) should belong to the class

{Cn)el2: £ |C | 2 (1 + | n | ) - 1 < o o (5.8)n = — oo

As shown by Slepyan (1980), the SLAE-1 has a unique solution in l2.Direct use of the truncation method for the system given by eqns. 5.6 and 5.7

is not particularly effective because of the 'relative convergence' phenomenon(Mittra and Lee, 1971). Its essence is the following: if we take JV\ equations ineqn. 5.6 and JV2 equations in eqn. 5.7 with the total number of unknown quantitiesJ\"=JV1 +^V2, then when JV tends to infinity, the result will depend greatly onthe ratio JV1 jM2. A correct limiting transition (when the condition given byeqn. 5.8 is fulfilled) can be obtained only at JV1 = JV2 (Ilyinsky and Slepyan,1983). But even in this case the SLAE-1 is characterised by slow convergence ofthe truncation method and low stability against computational errors.

The SLAE-1 therefore needs an analytical regularisation (see Section 1.3). Itis necessary to underline that the form of the SLAE-1 differs from the systemsexamined by Mittra and Lee (1971) and Shestopalov et al. (1984). This differenceis caused by the impedance character of the boundary condition and is thereason why the procedures of analytical regularisation suggested by these authorscannot be applied directly to eqns. 5.6 and 5.7.

The regularisation procedure described in this Section is also based chiefly onthe possibility of an exact solution for the matrix-convolution equations bymeans of the residue-calculus technique (Mittra and Lee, 1971, Shestopalovet al., 1984). We employ a regularising substitution of a special type (see eqn. 5.9)but, unlike that in Section 1.3, the relationship between the coefficients soughtand new unknowns in eqn. 5.9 is not linear but fraction-linear. Only this methodof analytical regularisation allows us to obtain the system of linear algebraicequations of the second kind (SLAE-2) with the most simple expressions formatrix elements and free terms.

Let us present the solution of eqns. 5.6 and 5.7 in the form

where

Fm being new unknown coefficients, and d= {dp} is the solution of eqns. 5.6 and5.7 at cop = 0 (del2).

The coefficients dp are expressed as


where the function f(w) satisfies the following conditions:

(i) f{w) is analytical everywhere in the complex w plane except at the pointsw = Tp, w = — Fo (p = 0, + 1 , +2, . . .) where it has simple poles;

(ii) f(yn) = 0, /(%) = 0 , / » # 0 at w = yn, %;(hi) f{w) decreases algebraically at w^-oo; and(iv) Res / (M;)L = _ r o = l .

The function f(w) can easily be expressed in terms of infinite products (Mittraand Lee, 1971, Shestopalov et ai, 1984).

It is worth noting that there is a relationship between the regularising-substitution method and the modified-residue-calculus technique. In fact, wecan present

The poles w = % of the function P(w) describe a displacement of the zeros of<p(w) relative to the zeros off(w). However, explicit formulation of a condition,modifying condition (ii) for the function f(w), and derivation of an expressionfor (j)(w) seem in this case to be rather difficult.

The coefficients Cp determined by eqn. 5.9 satisfy eqn. 5.6 and the conditionC= {Cp} 6 l2 as \P(w) | < constant. The unknown coefficients Fm in eqn. 5.10 canbe found from eqn. 5.7. Substituting eqn. 5.9 into eqn. 5.7 and changing(formally yet) the order of summation, we obtain the system of linear algebraicequations for Fm

f f dp(l+QJp) | l+co 0

Let us transform eqn. 5.11 to a more convenient form. Consider the integral

The integration path C^ is a circle of infinitely large radius.26 It can be easilyseen that Imn = 0. On the other hand, Imn can be evaluated according to theCauchy residue theorem. Equating the sum of residues with zero, we come tothe important identity

£ f r - f u r v w r +* wr -HM+*™A» = 0 ( 5 J 2 )

( i y ) { [ - - y ) { i + y ) [ i + y )where An = dfldw\w= =n. Using eqn. 5.12 and the properties off(w), it is easyto write eqn. 5.11 in a standard manner:

26 From the mathematical point of view it would be more rigorous to consider CVJ to be a so-called'regular system of contours' (Evgrafov, 1965).


t »m=P» (5.13)m = 0

where

+

r _

a0 = — exp I — In I I — [ [ — exp [

X fl (y r°wr"",rOx exP« = i ( 1 B - l o ) ( l _ „ - l 0 )

exp( r n -r p ) (r_ n -r p ) e x p l

12n

PL\ (Tp-%){T-,-%) ^[ pnoo (p)

Here y2m — ym, Jim+x ~ 7m5 t n e symbol Y[ denotes that the pth term of the

infinite product is omitted.Following Neimark (1954), we have that at n-> oo

(5.14)


Taking into account eqn. 5.14, according to Mittra and Lee (1971), we obtainthe asymptotic evaluation:

\An\ = O(n'1'2) (5.15)

KI = O(/T1/2) (5.16)

From eqns. 5.14-5.16 it follows that

l&J-Ofa-1*-1) (5.17)\Pn\~O(n-1) (5.18)

Using eqns. 5.17 and 5.18, it can be proved that the system given by eqn. 5.13is reduced Jo the Fredholm system by means of substitution of variables of theform Fn = Fnsn, where Fn are new unknowns and en= 0(na), a being an infinitelysmall positive number. This fact allows the following conclusions to be made:

(i) A solution of eqn. 5.13 does exist and is unique.(ii) For any r\, when there are no associated waves, the truncation method

for eqn. 5.13 is converging.(iii) The following asymptotic evaluation is valid:

If sin S — 0 or kd sin 3 = mn then cop = 0 (for any p). In this case it follows fromeqn. 5.13 that Fm = 0, and we have an exact analytical solution of the problemconsidered:

If these conditions are not fulfilled then the SLAE-2 after truncation should besolved numerically. At a small rj the iteration technique is also effective. Theproof of its applicability will be given in Section 5.4.

5.2 Diffraction of a plane wave by an array of impedancehalfplanes: E-polarisation

In Section 5.1 the diffraction by an impedance parallel-plate grating has beentreated for the case of H-polarisation. We shall now consider the plane-wavediffraction by the same geometry for the E-polarised case. This case has essentialdifferences: the Leontovich-impedance boundary condition is inapplicable forthe model of infinitely thin plates (see Section 2.4). Direct analysis of structureswith plates of finite thickness and taking into account the actual geometry ofedges leads to considerable complication of the solution techniques. We thereforeuse the model of infinitely thin plates, employing the modified impedancecondition described in Section 2.4 instead of the Leontovich condition. Thisallows us to apply highly effective analytical-numerical methods and take intoaccount indirectly the edge-configuration influence on the value of the powerabsorbed.

The mathematical statement of the problem is identical to that in Section 5.1,but now i// — Ey (x, z) • For convenience we rewrite the boundary condition in


the form

= 0 (5.19)l/2)d,0<x< oo

where 3 = ZslWok, Zs *s t n e modified impedance introduced in Section 2.4.Let us solve this problem using the Wiener-Hopf technique. According to

this method (Noble, 1958) we assume that k = k' +jk", k" > 0 (the solution forreal k is obtained by letting £"—>0 at the end of analysis). Let us define theFourier transform of the function ij/(x, z) with respect to x as

<D(a, z) = *, Z) exp (jocx) dx

Taking into account Floquet's condition of pseudoperiodicity, we obtain fromeqn. 5.1

(5.20)

where A(OL) and C(<x) are unknown functions, y = (k2 - ct2)1/2, u = ka sin 9 and

Let us introduce the Fourier integrals as

x, z) exp (jocx) dxf°=

Jo

and O+(a, +a) = lim Ê>+(a, ) (the symbol + denotes the limit approachedz-> ±a

from the right and from the left, respectively).It is obvious that the following relationships are true:

0+ (a, +0) + O_ (a, a) = 4(a)

0>_ (a, - a ) - <D_ (a, fl) = C(a)

Differentiating eqn. 5.20 with respect to ^ we obtain

(5.21)

(5.22)

<D'+(a, + a) + V-(a, a) = y-

O'+(a, - a ) + <DL (a, a) = y cot(y</)C(a) - ^ (

C(x)-A(a)y cot (yd) (5.23)

7 exp (— 2 iu)J^ (5.24)

where the prime indicates the derivative of a corresponding function with respectto Z-

Finally, presenting the boundary condition given by eqn. 5.19 in terms of


Fourier transforms, we obtain another pair of formulae:

<D+ (a, +a) -jW+ (a, +a) = .) ~ * " ° (5.25)j (k cos y + a)

0>+ (a, -a) +jd<t>'+ (a, -a) = ,' + **"* (5.26)j (k cos # + a)

It is possible to show (see Noble, 1958) that functions with the subscript 'plus'are regular in the halfplane Im a > —k" cos $, and functions with the subscript'minus' in the halfplane Im a < k". Then eqns. 5.21-5.26 can be considered asa system of functional equations for O'±(a, ±a), 4>± (a, ±a), A(<x), C(a) that iscorrect in the strip — k" cos 9 < Im a < k". Excluding A (a) and C(a) from thissystem and introducing new unknown quantities

we come to the following system of Wiener-Hopf functional equations( - r c o s S < I m a < r ) :

G(oc)S+ (a) +jSP(x)L+ (a) = 2O'_ (a, a) + 2*g ^ (5'27)k cos ^ + a

T ( a ) I + (a) + P(a)S+ (a) = 2O_ (a, a) + \-— (5.28)A: cos 9 + a

where

^ cos(2w)-cc-/-7N J

jsin(yrf)

cos(2w) — cos (7^)

The diffraction field is expressed in terms of the solutions of eqns. 5.27 and 5.28as

471 J _ +jb |_ [sin{7(af + z)}~ sin (yz) exp ( - 2ju)

fexp(2» cos (yz) -as |7 + [cos{7(rf + £ ) } - cos(y^) e x p ( - 2 »

exp (— jctx)

_ s+ [c

da (5.29)cos(2w) — cos(yd)

The upper line in the large curly brackets corresponds to the case z > 0, thelower line to ^ < 0, where b is an arbitrary real number satisfying the inequality-k" cos 9<b<k".

Thus, unlike the case of perfect conductivity investigated by Weinstein


(1969/?), the problem under examination is reduced to the Wiener-Hopf matrixequation.

The exact solution of the system of functional equations given by eqns.5.27 and 5.28 is possible only when P(a )=0 , i.e. for $ = sin"1 (nn/kd),n = 0, ± 1 , ±2, . . . . In this case the system is divided into two independentequations. Now we proceed to examine the normal incidence of a plane wave(S = 0).27

So, let us assume that sin 9 = 0 in eqns. 5.27 and 5.28. From eqn. 5.27 itfollows that

S+ (a) = 0>'_ (a, a) = 0

Then eqn. 5.28 can be written as

£(a)L + (a) -2O_(a , a ) = — ^ — (5.30)J{<X T K)

where g(a) — y~l cot(ya) — jS. The solution of eqn. 5.30 can be obtained in aclosed form by the factorisation method. The final result is (Noble, 1958)

where g± (a) are functions obtained as a result of factorisation of the function

The function g(a) is a ratio of two meromorphic functions and, in principle,can be factorised by the method of infinite products. However, the slow conver-gence of infinite products obtained in this case and the need to calculatenumerically a large number of zeros of the function g(oc) in the complex a planemake this method practically inapplicable. Much more effective is the integralrepresentation of g± (a) in the form suggested by Noble (1958).

From here on, let us confine our interest to the case kd < n. Substitutingeqn. 5.31 into eqn. 5.29 and deforming the integration path over the poles ofthe integrand in the upper halfplane, according to the Cauchy residue theoremwe obtain for x < 0:

{j/(x, z) — exp (jkx) + R(k) exp (—jkx) + • •

epresent an infinite sum of Floquet'stion coefficient is expressed as

= R0=~exp{-n(k)} (5.32)

where dots represent an infinite sum of Floquet's higher evanescent harmonics.The reflection coefficient is expressed as

where

m = —It is easy to see that if Re Zs ~ 0 then \R\= 1 (there is no absorption in thehalfplanes). If Re Zs > 0 then the magnitude of the reflection coefficient charac-terises losses in the structure. Taking into consideration the fact that

27 When the Wiener-Hopf matrix equation cannot be solved exactly it can be reduced to a SLAE-2 or to a system of Fredholm integral equations of the second kind.


s ) 2 > (Re^s)2 and neglecting the terms 0(|£s |

2), we have

where

(5.33)

2|l/2-11. du

u\ coth u- 2

fca tan w dw

Here it is relevant to recall the remark made at the end of Section 2.4 on theproblem of choosing the sign with the imaginary part of the impedance £s

determined by eqns. 2.27 and 2.28. When the choice of the sign is wrong,fictitious surface waves occur (as mentioned in Section 2.4). This may also leadto a physical nonsense for the problem considered here: even at kd<n andRe Zs ~ 0> l-^l^lj i-e- the magnitude of the reflection coefficient does notcharacterise the heat losses completely.

Let us proceed to the evaluation of the integrals in eqn. 5.33. The integral/ l 5 as can be easily seen, is reduced to that evaluated in a closed form (seeIlyinsky and Slepyan, 1983). To determine I2 let us examine the complex uplane with the cut between the points ±jka along the imaginary axis. We can

-KaFigure 5.2 Integration path for calculating I2


establish that

f f ~| tanh u du< 5 M i

G\ being the contour of Figure 5.2. As analytical continuation of the integrandin eqn. 5.34 tends to zero at \u\—> oo, the integral in eqn. 5.34 can be expressedthrough the sum of residues at the poles u = j (n + %)n, n = 0, 1, . . . , oo. Finally,in this case we obtain for v(k)

^ ] )In accordance with eqns. 2.27 and 2.28 for the rounded edge we shall consider\Zs\ — kpW0 (p being the curvature radius near the edge), and |£sl — 0.025kdW0

for the edge geometry near rectangular, where d is the thickness of plates, andC= 0.5772 . . . is Euler's constant.

It is necessary to underline the fact that the argument of the logarithmicfunction in eqn. 5.35 involves geometrical parameters characterising the edgeconfiguration. This fact points to the inapplicability of the Leontovich boundarycondition and the usual impedance-perturbation method. Eqn. 5.35 coincideswith that obtained by Weinstein et al. (1986) for an array of impedance halfplanesof finite thickness. Note that the analysis made by Weinstein et al. (1986) andZhurav (1987) is based on a rather complicated variant of the Wiener-Hopfapproximate technique. The use of modified impedance-boundary conditionsimposed on infinitely thin surfaces appears to be much simpler.

In Figure 5.3 are given the results of computation according to eqn. 5.35.Note that the absorption in such an array considerably exceeds that in a smoothmetal surface of the same material.

In the case of oblique incidence, as it was mentioned above, no exact solutionof the problem has so far been obtained. A simple approximate solution for anarbitrary angle of incidence 8 can be found when the structure period is smallcompared with the wavelength. In this case the equivalent surface impedanceof the structure should be independent of S.

The reflection coefficient R($), according to Nefedov and Sivov (1977), isexpressed through the equivalent impedance £h by the relationship

On the other hand, £* is expressed in terms of R(0) by the formula

tf(0)+_lZ R(0) - 1 W°

where R (0) is the reflection coefficient for the case of normal incidence deter-mined by eqn. 5.32.

These two formulae allow us to express R(S) through R(0); the final result


5= 6 -

OA 0.8 1.2Figure 5.3 Absorption in parallel-plate grating with rounded

E polarisation, Zs{kmin)IWQ = 10"4

/ta

ajp = 0.0075fl/p = 0.015ajp = 0.03

can be written as

R(9) =pR(0)~\

(5.36)

where p = tan2 (3/2).

5.3 Diffraction by a finitely conducting comb-shapedstructure

Figure 5Ab shows the configuration of the structure under study and defines thenotation. The formulation of the problem is completely similar to that inSection 5.1: it is necessary to find a solution of eqn. 5.1 satisfying the impedanceboundary conditions


= 0 (5.37)dz z = (m+l/2)d,D>x>0

=0 (5.38)

the edge condition and the radiation condition at #—• — oo.Such a mathematical statement of the problem describes also a similar acoustic

problem if by i , *7, ?/i we mean the corresponding physical values. This is why,with the reservation mentioned, all further considerations will be applicable alsofor acoustic waves.

Let us consider briefly the basic steps of the solution procedure. First we passfrom the original boundary-value problem to the infinite system of linear algebraicequations writing down the field expansions in partial areas and performing fieldmatching at the interface x = 0. This system is of the first kind (SLAE-1) and isineffective for a direct solution, i.e. it is an intermediate step. Then we transformthe SLAE-1 into the infinite system of linear algebraic equations of the secondkind (SLAE-2) using the regularising substitution technique described in Sec-tion 5.1. Finally, we derive approximate formulae for a physical analysis.

Let us represent the field above the structure, as in Section 5.1, in the formof an infinite series in terms of Floquet's harmonics. The field in the mth cellcan be expanded into a series in terms of eigenmodes of a plane impedancewaveguide of the width d, propagating in opposite directions:

ynx)+ pnexp(ynx)} (5.39)

where

h±J!!± {_2 D) (y2n = y n , y 2 n + i = yn)

the functions (j)n(z) being determined in Section 5.1.Let us assume that the rj coefficient satisfies the conditions defined in Sec-

tion 3.6. Then there will be no associated waves and the use of the expansiongiven by eqn. 5.39 is relevant.

The expressions for the field given by eqns. 5.4 and 5.39 satisfy the Helmholtzequation, radiation condition and impedance boundary conditions on thestructure surface. Because of the condition of pseudoperiodicity, 7^m) =Tj,0)exp (2jmu), and we can consider just one unit cell of the structure (forexample, with m = 0). The unknown coefficients Rp, T|,0) can be found from thecondition of continuity of \\J and di/z/dx across the x = 0 plane at the interval

fl. That leads to the following system of functional equations for Rp and

exp(jkzsin9)+ £ Rp^(japZ)= X T<° V« (z) (1 + Pn) (5.40)p = — oo n — 0

00

-jk cos $ exp (jkz sin 9) - £ TpRp exp {jotpz)

z){\-Pn) (5.41)

TpRp exp {jotpzp= — oo


If there are no associated waves, the function (j)n (z) form a complete orthogonalsystem in L2 {a, —a) (Keldysh, 1951), so that on multiplying the right-hand andleft-hand sides of eqns. 5.40 and 5.41 by (j)p(z) (/> = 0, 1, 2, . . .) and integratingover the structure period, we obtain a system of linear algebraic equationsequivalent to eqns. 5.40 and 5.41:

MOs+ £ RpMps=Ti°>(l+ps)nsp— — 00

-jAcosSA/o," I rpi?pMps=T<°>(l-ps)«sysP = — 00

Here

2(— l ) p cos(Psa) sin w^ , 2 s = 2 _ O-2 ( a p + N C O t W)

2j{- \)psin(^sa) cos M^ P , 2 S + I =

2 _ ^2 {ap-jri tan u)

(5.42)

Excluding Tj,0) in eqn. 5.42 and introducing new variables according to theformula

0Lp + jrj cot u

we obtain the system of linear algebraic equations for Cp

where

2;'w6,1 = . f0 ,, d2=jrj cot u

sin (2M)

To fulfil the edge condition, the vector C={Cp} has to belong to the class ofnumerical sequences meeting the condition given by eqn. 5.8.

The linear system obtained is a SLAE-1 of a more general form than thatexamined in Section 5.1. We can also apply for its regularisation the proceduredeveloped in Section 5.1. Through this procedure we reduce eqn. 5.43 to theSLAE-2 allowing an effective numerical and analytical investigation of theproblem.

The amplitudes of waveguide modes in the grooves Tj,0* can be expressedthrough the coefficients Cp according to the recalculation formula resulting fromeqn 5.40.

For the diffraction problem formulated in this Section a theorem of uniquenesshas been proved by Slepyan and Slepyan (1980<2). From this theorem there


follows a theorem of uniqueness for eqn. 5.43 in the class of numerical sequencesdefined by eqn. 5.8.

Then we easily obtain the relationship

f° , # f- lim Im i/^*—-d^ = Re rj(s) \y/(s) \ ds (5.44)

x ^ ° ° J -a CX JEFDC

where f](s) = rj on FE and CD and rj(s) = f/x on FZ) (J 6 EFDC).Substituting eqn. 5.4 into eqn. 5.44, we obtain the result

T | 1N (Sp0-\Rp\2)rpd\=P (5.45)

where JV± is the number of propagating Floquet's harmonics (positive andnegative, respectively) and

P=~^Re t]{s)\il/(s)\2 ds (5.46)2^ JEFDC

P being the linear per-period dissipated power.Eqn. 5.45 reflects the law of energy conservation. It shows that part of the

energy of the incident wave is distributed between the propagating Floquet'sharmonics and the remainder is dissipated in the comb-shaped structure, turninginto the Joule heat.

We continue now with the description of the regularisation procedure. Weshall present the solution of the SLAE-1 as a fraction-linear substitution givenby eqn. 5.9 with

P(w) = l+ li^~- (5-47)

The difference between eqns. 5.47 and 5.10 is in the fact that the summationin eqn. 5.47 is carried out in all m and not only in indices corresponding toantisymmetrical modes.

Substituting eqns. 5.8 and 5.47 into eqn. 5.43 and changing the order ofsummation,28 we obtain

P.

(5.48)

We shall need a few mathematical identities. In a way similar to that in

28 The correctness of this operation can be substantiated with the aid of the evaluations given byeqns. 5.14-5.18.


Section 5.1, let us consider the integral

= jm" 2

From the asymptotic behaviour of f(w) at |H;|->OO it follows that Imn = 0.Calculating Imn by the Cauchy residue theorem, we obtain the identity

00 d 1 f(-v )

pJi*>{Tp + yn){Tp-ym) {TQ-yn){TQ + yn) yn + ym

In addition, we shall use eqn. 5.12 and the well known formulae, proved byMittra and Lee (1971):

F^7" ^ fT7=/(~7") (5-50)l 7 l + y

F4 I TT ()Using eqns. 5.12, 5.47, 5.48 and 5.49, we proceed from eqn. 5.41 to the followingsystem of linear algebraic equations for F= {Fm} (Slepyan and Slepyan, 1980/2):

{I+B)F=t (5.52)

where

Vi l r o + 7m Vo + 7n To - yn,1 , Pn \ , pJ{~yn]

^ P — rD = — 00 A n /

(5.53)

The free terms tn are expressed as

* _ _

( 5 - 5 4 )

The infinite system of linear algebraic equations given by eqn. 5.52 is the resultrequired.29

Let us consider the properties of the matrix B as a linear operator in theHilbert space of numeric sequences. Using eqns. 5.15 and 5.16, we can show,

29 If the side walls of the cells are perfectly conducting (t} = 0), then the system of linear equationsgiven by eqn. 5.52 coincides with that obtained by the modified residue-calculus technique (Mittraand Lee, 1971).


as in Section 5.1, that the following asymptotic evaluations for Bnm, tn are correct:

IB^Ôim-^-112) (5.55)

UJ~0(«~1/2) (5.56)Then, it is easy to show that the system given by eqn. 5.52 is a Fredholm systemand its solution can be obtained by the truncation method. In this case

\FJ~o(m-112)

and this allows us to draw a conclusion that C expressed through F by eqns. 5.9,5.47 satisfy the condition given by eqn. 5.8. Thus, a full mathematical substan-tiation of the formal solution of the problem has been implemented.

5.4 Perturbation technique

First, we shall present some auxiliary information. Vector t(rj) will be calledanalytical in rj in the area 3) if all its components ti(rj) are analytical in the area3. Similarly, the matrix operator B(rj) will be called analytical if Bik(rj) areanalytical for any i, k. The following theorem (see Evgrafov, 1965) is correct:

Theorem 5.1: & sum of uniformly convergent series of analytical functions is ananalytical function at every interior point of that set where the series convergesuniformly.

Now, let us examine eqn. 5.52 in the light of perturbation theory. We extractthe operator Bo, which corresponds to the case of perfect conductivity of theridges, as a main part of B, i.e. Bo= lim B(rj). Assuming that k^nn/d, we

|f/| ~* 0

present eqn. 5.52 as

[/ + (B0 + I)~1{B(ri)-B0}]F= ( o + Z)"1*^) (5.57)

Let us construct Neumann's series for the vector F

F(r,)= £ (I+B0)-*{B(r,)-Boy(Bo + I)-1(-iyt(rl) (5.58)v = 0

The operator (I+B0)~l is bounded, (1+ B0)~

l (B — Bo) being completelycontinuous in la

2 (0 < cr < I)30 and its norm can be as small as required atsufficiently small rj.

Taking into account theorem 5.1, Weierstrass' criterion of uniform conver-gence (Korn and Korn, 1968) and evaluations given by eqns. 5.55 and 5.56,we find that the vector function F(rj) expressed by eqn. 5.58 is analytical in thevicinity of the point rj = 0, ifB(rj) and t(rj) are analytical in this vicinity. Letus prove now that B(rj) and t(rj) determined by eqns. 5.53 and 5.54 are reallyanalytical in the neighbourhood of zero.

The functions yn{Y\) at k / nnjd are analytical.31 Considering this fact we can

30 The symbol la2 means a Hilbert space of numerical sequences; the scalar product of two vectors

x andy in la2 is defined as (x,y) = £ xmy^(\ 4- m)~a.

m = O31 An exception can be made for the point r\0 corresponding to a two-fold root of the waveguidedispersion equation. Earlier, such points have been excluded from consideration. As is shown inSection 3.6, there is a circle of a finite radius where


easily [see Evgrafov (1965)] establish the analyticity of A(rj), s(rj) = / ( - ? „ ) anddp(rj) on the Riemann surface of the logarithmic function. The analyticity ofB(rj), t(rj) follows from theorem 5.1 and Weierstrass' criterion mentioned above.

Thus, F(r\) can be presented as a Taylor series converging in some neighbour-hood of the point rj = 0. Expanding t(rj) and B(rj) into a Taylor series with thepoint rj = 0 as a centre of the circle of convergence, and multiplying the seriesin eqn. 5.58, we obtain an expansion for F(rj) in the form of a series in integerpowers of rj. According to the theorem of uniqueness for analytical functions(Evgrafov, 1965) this series will be a Taylor series for F(rj).

It follows then that the coefficients Rn{f]), Tf){r]) and the function ij/(x, z, H)for any values of x and z are analytical in the vicinity of the point rj = 0. Thiswill not be true, however, for the derivatives of \jj with respect to x and z, asafter a term-by-term differentiation the series in eqn. 5.4 converges nonuniformlyabout the points x = 0, z = {m + \)d (m = 0, + 1, + 2, . . .). In fact, the analysisshows that at the points # = 0, z={m + j)d the analyticity of grad if/ty) isbroken.

For electrically polarised fields for metals in microwave and millimetre-wavebands the condition \rj\ < k is fulfilled. Therefore, it is reasonable to consider thelosses as a small perturbation. The analysis presented in this Section rigorouslysubstantiates the validity of such an approach when there are no associatedwaves and the frequencies are not too near to the cutoff frequencies for theeigenmodes in the grooves. Otherwise such an approach is not correct. In asingle-mode range (kd < n) the energy absorption can be characterised by boththe magnitude of the reflection coefficient |/?0| and the linear per-period dissi-pated power in the comb-shaped structure. These values are connected by theformula

P = ±d cos &(\-\RQ\2)W0 (5.59)

resulting from eqn. 5.44 if kd< n. In practice, it is enough to determine \R0\ upto the terms of order rj. It is possible, however, to calculate P directly and tofind \R0\ from eqn. 5.59. Here, when calculating P, we can use the fielddistribution corresponding to the case of perfect conductivity (zero-orderapproximation of the perturbation method). Otherwise, to calculate \R0\ directlyfrom eqn. 5.52, it is necessary to carry out complicated computations takinginto account small quantities of the first order in rj.

Unfortunately, for fields of magnetic polarisation a similar simplification isimpossible even for small losses. This can be explained by the breaking ofanalyticity of Rn(f]), T^^rj) in the neighbourhood of an infinite point. Thephysical sense of such a situation was discussed in Section 5.2. Here, we noteonce again that this greatly complicates loss calculations for cases of magneticand mixed polarisation.

5.5 Effect of abnormally small absorption in periodicstructures

Let us consider the incidence of a plane wave of electric polarisation on a finitelyconducting comb-shaped structure. We assume that the depth of the structure


is sufficiently large and its period is such that all EOn modes in the grooves areevanescent (the E01 mode is not very close to the cutoff). Mathematically theseassumptions are formulated as

T<n

(5.60)

(5.61)

(5.62)

where /I = D/d, T = kd and Zs *s t n e surface impedance of the material of theridges. Taking into account the inequality I^sl/H^ ^ 1? m accordance with theresults of Section 5.4, we shall use the perturbation theory. In other words, whencalculating the linear (per-period) lost power P, we substitute the distributionof the magnetic field corresponding to the case of perfect conductivity of thegrating material into eqn. 5.46. Then for P we obtain the expression (Slepyanand Slepyan, 1979)

Re Zs(5.63)

where

F± (0 = Tne*[exp(-hn[. - 2)}]

2-r2}112 ho=-jThn = {(nn)2-r

Tn is the complex amplitude of the nth eigenmode in the groove, calculated onthe assumption of perfect conductivity of the structure material.

In agreement with the conditions presented by eqns. 5.60-5.62 we canneglect the terms with n > 1 in the series for F± (£) and f{Q. This physicallycorresponds to the fact that only the contribution of the TEM mode to thepower absorbed is taken into account. The reason for this simplification is aconsiderable decrease in Tn when n increases. In addition, the fields of theevanescent EOn niodes are exponentially damped with respect to x. This causesthe terms ignored in eqn. 5.63 after integration to acquire an additional orderof smallness.

Integrating in eqn. 5.63 and performing some simplifications, we obtain thefollowing approximate formula:

(5.64)

Thus, what is left is to determine the magnitude of the coefficient To.


In the zeroth order of the perturbation theory we have for Tn

T"= \ "f id \~^d^nkoTod+h,

(5.65)

where

_ f sin ( r sin 5/2), n = 0, 2, . . .Vn~ \j cos (T sin 3/2), « = 1,3, . . .

The coefficients Fn are determined from eqn. 5.52 in which the terms of thehigher order of smallness in ^lk = sIW0 have been ignored. It is possible tofind an approximate analytical solution of eqn. 5.52 when the conditions givenby eqns. 5.60-5.62 are fulfilled (in this case the 'reflection' of the higher EOn

modes from the x = D plane will be negligibly small).32 Then ignoring theexponentially small terms in eqn. 5.52 and performing some transformations,we can easily obtain the following formula for To:

(5.66)

where p — tan2 (5/2). The function H(w) is expressed as

(5.67)

Now we can find the magnitude of To from eqns. 5.66 and 5.67. Consideringidentities (they can easily be checked)

1 J ' 2 sin2{(Tsin 9) 12} sin T

T sin2we obtain for | T0\2 the formula

1 Ul \+p2-2pcosQ{T)

where

sin 1

I = i ^ \2qn+ TsinS

r2qn - T sin Sj \qn

-s in" 1 - » (5.69)

32 The term 'reflection' in reference to evanescent modes is rather conditional; thus we use here thequotation marks.


Let us analyse the formulae obtained, taking for simplicity % = 1. The ohmiclosses in the comb-shaped structure can be characterised by the coefficient

( 5 - 7 0 )

where Ro = ^m &o *s t n e reflection coefficient for a smooth surface of the same

material, P = lim P(T, $, JJL) being the linear (per-interval equal to the structure

period) power of losses in such a surface.Let us consider first the case of normal incidence. Then | To\ = 1, and, taking

into consideration eqns. 5.64 and 5.70, we obtain

It is clear from eqn. 5.71 that at any T, \i a > 1, i.e. the absorption in the comb-shaped structure at normal incidence is greater than that in a halfspace of thesame material with a smooth surface. This result is rather obvious from thephysical point of view: at normal incidence the reflection from the ridges isextremely small.33 This is why the absorption in the grooves' bottoms is practi-cally the same as that in a halfspace without ridges. In addition, there areadditional conducting surfaces: ridges which significantly increase the total losses.

In Figures 5.4-5.6 are shown the dependencies a(5) (T, \i — constant), a(/i)(Y,$ = constant) and a(T) (//, 3 = constant). They are calculated by means ofeqns. 5.64, 5.68 and 5.69 at %= 1. It can be seen from Figure 5.4 that a(S) ismonotonically decreasing with increasing 9.34 Physically, such a trend in thedependence a (3) can be explained by the fact that the field is 'forced out' ofthe comb-shaped structure in the case of oblique incidence. This process getsmore intensive with increasing angle of incidence. As a result the currentsinduced on the metal surfaces decrease and this leads to diminishing of the losses.At #-»7i/2 the amplitudes of all surface waves \Rn\ (n>\) are strictly equal tozero, and the reflected wave accurately compensates the incident one. This isthe Rayleigh diffraction anomaly characterised by the fact that the longitudinalwave number of the fundamental space harmonic Fo is transformed into null.The Rayleigh anomaly can be considered as an extreme case of the field 'forcingout' from the periodic structure: 'forcing out to infinity'. A similar mechanismis typical for a plane wave scattered by a rough surface: if the wave incident isat a shallow angle, it in practice does not 'feel' any roughness while any otherwaves are scattered intensively.

Nevertheless, the use of the Rayleigh anomaly alone is not an effective methodof loss reduction in electrodynamic systems as it provides small ohmic losses onlyat grazing incidence. However, there is one more mechanism for 'forcing out'the field; to understand the physical nature of this we shall refer to eqn. 5.66.Let us present eqn. 5.66 as

33 In the case of perfect conductivity there is none.34 Such reduct ion takes place up to the angle # ~ c o s " 1 (\t]\jk).


0.2X' 40 do

6, degreesFigure 5,4 Absorption in comb-shaped structure against angle of incidence

H polarisation

1 r = 2.2, /i = 0.712 r = 1.57, ^ = 1 . 03 r = 1.26,/i =1.254 r = 0.94,/x= 1.665 T = 0.63, // = 2.5

where

= s o { \ + r o e x p ( 2 j k D ) + r\ e x p ( A j k D ) + • • • }

(5.72)

v0tan3//(-jTcos3)

2jkdln2\H(-jT)


a

2 -

a

10

0 1.0/

M 1

1

2 1 \ \

2.5 3.5 AFigure 5.5 Absorption in comb-shaped structure as a function of normalised groove depth

H polarisation, »9 = 60°

1 r = 0.942 r = 1 . 5 73 r = 1 . 8 8

Physically, the field formation in the grooves can be presented as an infinitechain of diffractions by the open ends of plane waveguides (here we do not takeinto account the transformation of the TEM mode into evanescent modes) andreflections from the x = D plane. It can easily be seen that eqn. 5.72 simplycorresponds to such interpretation: s0 is the transmission coefficient for the systemof halfplanes, r0 is the reflection coefficient of the TEM mode arriving from thewaveguide (phase shift of the incident wave per one period of the structure isA; sin 3), and the ft + 1st term on the right-hand side of eqn. 5.72 describes anrc-fold transition of the wave to the bottoms and back.

If the depth of the structures is chosen as

X d\n2- +4 n

(5.73)

then, according to eqn. 5.73, 'partial' waves corresponding to a single link of achain are suppressing each other in pairs while interfering (not completely, ofcourse, as their amplitudes are different). As a result, the field inside the groovesbecomes weaker. The first term in eqn. 5.73 corresponds to the phase variationon the way from the open ends to the bottoms and back, the second one to theinsertion phase at the diffraction by the open ends. Figure 5.4 shows that alreadyat 3 = 60° the losses in the periodic structure are approximately half those in asmooth surface of the same material. At 9 = 80° the decrease in losses is 10-15times. It happens because of the joint effect of the Rayleigh anomaly, reducing


0 OA 0.8Y/%

Figure 5.6 Absorption in comb-shaped structure against Tjn

H1234

polarisation, /B =10° 5# = 20° 6S = 30° 75 = 40° 8

i= 1

3 = 50°3 = 60°3 = 70°# = 80°

\so\, and the interference mechanism examined earlier, which reduces the absol-ute value of the second cofactor in eqn. 5.72.

It is necessary to pay attention to a sufficient dependence of a on the structureperiod. It can be seen from Figure 5.6 that cc(T) has a minimum at Topt ~n/2.The extremum becomes less distinct with increasing angle of incidence, thoughthe value of Topt is practically independent of 3. In the extreme case when theperiod of the structure is small compared with the wavelength (kd <^ 1), there isno gain in losses for all 9 except those very close to 7r/2, though the 'forcing out'of the field takes place at any kd. It can be explained by the fact that thoughthe field penetrating into the grooves is rather weak, it undergoes considerableabsorption on the surfaces of the ridges if kd <^ 1, D^> d. If T > TC/2, then theincrease in a is a result of both the increase in the TEM mode dissipation in thegrooves' bottoms and the influence of the higher EOn modes. The higher modesexcited in the grooves are not taken into account when deriving eqn. 5.64. Thisis why the increase in the dependencies OL(Y) plotted according to eqns. 5.64and 5.68 (solid lines in Figure 5.6) is for the first of the reasons mentioned. A


more rigorous analysis considering the higher modes (see below) leads to aneven faster increase in dependencies <x(Y) at Y > n/2 (broken lines in Figure 5.6).

Numerical evaluations show that when the conditions given by eqns. 5.60-5.62 are fulfilled, we can neglect the infinite series in eqn. 5.69. Then we obtainthe following simple formula for the dissipated power:

To check on the accuracy of the approximate analytical formulae obtained andalso lift the restrictions given by eqns. 5.60-5.62, the problem under consider-ation has been solved numerically. The coefficients Tn have been determined inaccordance with eqn. 5.65. The SLAE-2 given by eqn. 5.52 was solved in azeroth-order approximation of the perturbation method by means of truncationup to the JVth order. The dissipated power was calculated with the aid ofeqn. 5.60 but in the series for/(£) and F± (£) the first JV terms were retained.Specific calculations have been carried out for JV= 10 which, as shown bynumerical experiments consisting in the variation of JV, ensures, in this range ofparameters, an accuracy within 0.1%. The results of the numerical solution areshown in Figure 5.5 (insert) and in Figure 5.6 (broken lines). Comparison ofthese data with the results obtained by means of eqns. 5.64, 5.68, 5.69 oreqn. 5.74 shows that at ft > 1, Y < n\2 they coincide within the accuracy of thegraphical presentation. At Y > n\2 or \i < 1 a slight difference is observed whichcan be explained by the increasing effect of the higher modes in the grooves.Note that the qualitative character of the dependencies cc(Y), calculated byapproximate analytical formulae, also remains correct at Y> nj2.

A similar effect also takes place in acoustics: to reduce the energy losses atthe reflection of a sound wave from a rigid absorbing surface, the latter shouldbe corrugated. The theory and formulae for calculation in the acoustic case arethe same.

We return now to the case of E-polarised wave diffraction by the parallel-plate grating analysed in Section 5.2. At kd<n and, even more so, at kd<^ 1,the electrodynamic characteristics of such structures will not differ muchfrom those of lamellar gratings with sufficiently deep grooves (all modes in thegrooves are evanescent). Using eqn. 5.59 (which is correct also for the case ofE-polarisation) and eqn. 5.35 for \R\, we obtain

(1 -p2) cos ,9

" T2p2 - 2I> cos y + 1

where T and y are the magnitude and phase of the reflection coefficient.To evaluate P(9), the quantities T and y can be taken assuming that the

structure material is perfectly conducting. Then, according to Weinstein (1969£),for kd < 1

_ 1 l 2kd\n2 . . _ .F~l y~yo~n-\ (5.76)

The analysis of eqns. 5.75 and 5.76 shows that at $-»7i/2 (grazing incidence)sharply falls down to zero. Here, the Rayleigh diffraction anomaly demon-


strates itself as being independent of the wave polarisation. However, in thiscase there is no interference mechanism of the field 'forcing out', as is typicalfor electrically polarised waves. This is why the losses in the structure underconsideration are higher than those in the smooth surface of the same material.In fact, introducing the coefficient a(9) = P{S)jP0 (5) [P{9) being the linear(per-interval d) dissipated power in a smooth surface of the same materialilluminated by magnetically polarised plane wave with unit amplitude], weobtain from eqn. 5.76

1 + p - 2p cos y0a(0) (5.77)

Taking into consideration the fact that a(0) > 1 (see Section 5.2), from eqn. 5.77it follows that a($) > 1 for all #. In other words, under the conditions discussedabove, there is no abnormally small dissipation effect for E-polarised waves.

5.6 Absorption in inclined comb-shaped structures andechelettes

The configuration of the structures under consideration and main designationsare shown in Figure 5.7. The echelette grating is considered as a limiting caseof an inclined comb-shaped structure (D-+0). Let us limit the scope ofour interests to the analysis of the H-polarisation. We shall use the energy-perturbation method according to which the linear (per-period of the structure)dissipated power P is evaluated by the formula

R e ^ s | l / ' | 2 d / ( 5 ' 7 8 )

where if/ describes the current density distribution on the surface of a perfectlyconducting structure of the same geometry, and L is a contour AEFO corre-sponding to one unit cell of the structure (see Figure 5.7).

In this case for x > 0 we have00

ij/(x, z) = e*p{jk{z sin S - x cos 9)} + £ Rp exp(jocpz ~ Tpx)p= - c o

(5.79)

where ap = (2pn + u)/dsec 0, u = kd sin S sec/] and Tp= ((X2p~ k2)112.

For — D~md tan /? < £ < — md tan /? and md< £ < (m + \)d (m being the num-ber of the cell), we have

+ exp [-}>„{£ + W + (m + 2)d tan )8}]) (5.80)

where yn = {(nn/d)2 - k2}1/2 and T^ = T™ exp(jmu).Matching of the expansions given by eqns. 5.79 and 5.80 is performed with

the aid of a 2-dimensional version of the second Green's theorem for the trianglearea OAB (Mittra and Lee, 1971, Shestopalov et al., 1984). As a result we obtainan infinite system of linear algebraic equations for the coefficients Rp and the


At1

Figure 5.7 Inclined comb-shaped structure

recalculating formula for Tj,m\ Solution of this system and evaluation of thecoefficients Tj,m) are carried out through the modified residue-calculus tech-nique. For a conventional comb-shaped structure (corresponding to /? —> 0 inFigure 5.7) the representation of if/ by eqn. 5.80 is correct over all the contourL and calculation of P is reduced to substitution of eqn. 5.80 into eqn. 5.78 andnumerical integration, as in Section 5.5. In the case considered jS ^ 0, and theproblem therefore arises how to evaluate \// on the segment OB [representationsgiven by eqns. 5.79 and 5.80 in the triangle area OAB are not correct(Shestopalov et al., 1984)]. This difficulty can be avoided using the second Greenidentity for some area including the OAB triangle. This area can be the polygonONMAEF (see Figure 5.7) which is convenient from the computational pointof view. In this case we obtain, according to papers by Slepyan (1984) andSlepyan and Slepyan (1986)

(5.81)

where t£ represents the boundary of the area ONMAEF excluding the segmentOF,

G = {JV0(A/?1)+JV0(Afi2)} and RU2 = {(£ - f)2 + (£ + <T)2}1/2

The integral in eqn. 5.81 is evaluated numerically and \p and di/z/dn are


calculated from eqn. 5.79 on the segment AEF and from eqn. 5.80 on all otherparts of the contour J5f.

For an echelette grating a more effective method is possible (Slepyan andSlepyan, 1986). Apart from the structure under consideration an auxiliarystructure of this type with the same period and /?' = 7i/2 — fi is considered (infact it is the same structure but in performing the matching of modal expansionswe introduce a virtual-plane waveguide on its other facet). In this case ^(C)IOBJfor the structure under consideration, equals \JJ{^)\BA

m t n e auxiliary problemand hence is represented by an expansion in terms of eigenmodes of a virtual(D-+0) plane waveguide with the width d' = d tan /?.

Let us present some results of calculations obtained with the aid of thetechnique described. The parameters of the gratings are such that all spaceharmonics except those which are mirror-reflected are nonpropagating. In allfigures the a coefficient is defined by eqn. 5.70.

In Figure 5.8 the dependencies of a on the angle of incidence 5 are shown forechelette gratings with different X = kd sec /?. As for lamellar gratings consideredin Section 5.5, the effect of abnormally small dissipation (a < 1) is observed.However, in echelette gratings it occurs at larger angles of incidence. A similarregularity is typical for inclined comb-shaped structures.

Figures 5.9 and 5.10 show the coefficient a as a function of X at different $for inclined comb-shaped gratings. Unlike conventional comb-shaped structures,for some values of 3 the dependence a(X) at small S is increasing (curves 1—4in Figure 5.9). The area X>nsecP in Figure 5.10 is of considerable interest(in this area two waveguide modes are propagating). For this regime, a largeincrease in losses is observed and sharp oscillations of a take place. Physicallysuch behaviour of a can be explained by the interference phenomena for differentmodes propagating in the parallel-plate waveguides.

Figure 5.11 shows the dependence of a on the echelette blaze angle /? at twovalues of the angle of incidence. At 9 = 80°, [lopt = 45° corresponds to the regimeof minimal absorption. For 9 = 0° the situation is different: dmin= 1 at j?-»90°,and at ft = 45° the coefficient a is maximal.

To conclude, we would like to dwell upon one methodological question. Forall the structures examined above, in the case of H-polarisation the energy-perturbation method has been used. It is based on the integration of current-density distributions evaluated in approximation of perfect conductivity(eqns. 5.46 and 5.78). This approach led to rather convenient computationalalgorithms (most of the difficulties arose with inclined comb-shaped structures—see eqn. 5.81). Let us note here that constructive methods for solving theproblems directly in impedance formulation do exist: the semi-inversion tech-nique for echelette gratings (Shestopalov et al., 1984), and the method of reg-ularising substitution for inclined comb-shaped structures (Slepyan, 1984).

For E-polarisation the energy-perturbation method does not lead to a satisfac-tory solution. First, its direct application is impossible for all structures exceptechelette gratings—a modification described in Section 2.4 has to be carriedout. In addition, the algorithms obtained in such a manner need a rather largenumber of computations. Thus, for example, in Section 5.2 the induced currentdensity is presented as a Fourier integral. Its evaluation and the subsequentnumerical integration are very consuming of machine time. Of course, insteadof a Fourier integral we can turn to an expansion in terms of eigenmodes for


a

= 0.31

0 20 4-0 60 60d, degrees

Figure 5.8 Absorption in echelette grating against angle of incidence

H polarisation, j8 = 60°


a

o OA 0.6 X/SCFigure 5.9 Absorption in inclined comb-shaped structure against X

H polarisation, f$ = 60°, Djd=\

the current density but the convergence of this expansion would be rather slow.The reason is the singular behaviour of the current density at the edge vicinity(for H-polarisation there are no such singularities). Therefore for the E-polarisedfields it is preferable to solve the diffraction problem directly with the use of theimpedance formulation (see, for example, Section 5.2).

5.7 Diffraction by a complex-shaped periodic structure:integral equation method

So far we have examined absorption processes in periodic structures with specialconfigurations which allowed us to use special highly effective numerical-analyt-ical methods. The analysis of wave diffraction by periodic structures of anarbitrary configuration is of considerable interest. Development of analysismethods suitable for any grating profile would enable us, specifically, to solveproblems of optimisation of the shape of periodic structures to minimise thepower absorbed. Certainly, such methods can only be numerical, based on themodern achievements in computational mathematics.

154 Electromagnetic-wade diffraction by finitely conducting comb-shaped structures

a

12

8

0 OA 08 1.2X/51

16 2.0


(X

15

10

0.5

45 60 75 90J3, degrees

Figure 5.11 Absorption in echelette grating as a function of blaze angle

H polarisation, X = 1.79,9 = 0

The configuration of the structures under consideration and main designationsare shown in Figure 5.12. It is necessary to distinguish between transmitting[Figure 5.12(i)] and reflecting [Figures 5.12(ii) and (iii)] gratings. In this Sectionwe dwell mainly upon the transmission structures. The method of integralequations based on potential theory will be used. This method has already beenused extensively in mathematical physics and mathematical theory of diffraction,in particular Colton and Kress (1983), Parton and Perlin (1977), Poggio andMiller (1973) and Galishnikova and Ilyinsky (1987). For impedance-boundary

Figure 5.10 Absorption in inclined comb-shaped structure against X\n

H polarisation, $ = 0, /? = 45°D/d= 1.666D/d= 1.0


conditions, use of this technique is characterised by some specific features, whichwill be discussed in this Section.

Let us consider a problem of diffraction of electromagnetic field by an infinitelywide grating with period d obtained by translation of a cylinder with cross-section S, bounded by the contour C, along the z axis. Now let the incidentexcitation be simultaneously created by a line source at the point M(x, z) andthe plane-wave set of the type

where Tn = (oc2 - k2)112 (Im rn < 0; Re Tn > 0 if Im Tn = 0), and ocn and C°n aregiven constants. This form of the incident field is typical of diffraction electronics,in particular problems of radiation of charged particles moving across a periodicstructure [Smith-Purcell effect, Shestopalov (1976)].

The problem of wave scattering by such a periodic structure consists in solvingthe Helmholtz equation for the total field

V2i// + k2il/= -2nd{M,M) (5.82)

which satisfies the following boundary conditions on the structure surface:

— (P) -JY](P)\jj(P) = 0 (?eC) (5.83)

In addition, the radiation condition should be added; the analytical form forthe radiation condition will be formulated later.

One more peculiar feature of this statement of the problem is a nonperiodicalexcitation of the periodic structure owing to which Floquet's conditions ofpseudoperiodicity cannot be used. This difficulty can be overcome with the aidof the discrete Fourier transform.

Let us reduce the original diffraction problem to that in a strip:

d d- < £ < - -oo<*<oo;

Let us introduce the transform (Ilyinsky, 1973)

(5.84)

The function £/(T, X, Z) is defined for any quadratically integrable functioni//(x,z) in D1 and satisfies Floquet's condition of pseudoperiodicity with theparameter t:

00

U(T,x9z + d) =exp( j t ) £ \j/(x,z + nd) exp(-j/rr)

x,z) (5.85)

Because eqn. 5.85 is true, to determine {//(x, z) at any point of the space it isenough to determine U(z,x,z) in the strip Dx. The functions i//(x,z) andU(T, x, z) are related by the formula

^ M = ;r [2* U(T,X,Z)&X (5.86)2TT JO

The function U (T, X, Z) satisfies the equation


0)

(ii)

(iii)

Figure 5.12 Geometry of complex-shaped periodic structures

(i) Transmission grating(ii) and (iii) Reflecting gratings


= ~2n £ 5(x-x)5(z-z + nd)exp(-jnx) (5.87)

the boundary condition given by eqn. 5.83, and the radiation condition. Thelatter can be formulated as follows: let us consider the strip x1<x<x2 containingthe line source and the body S. The function U(r, x, z) at x < xx and x > x2 canbe presented as

U{x,x,z)= £ Tn(T)ij,n(z,z)exp(TnX) x<x, (5.88)

T,z) exp(-r^) x>x2n= — oo

(5.89)

where

<MT, Z) = cxp(jctnz)l{Jd) ctn=(z + 2nn)ld

Uo (T, x, z)being the transform defined by eqn. 5.84 for the incident field ij/0 (x, z)-The coefficients Rn{i) and Tn{z) are unknown and should be determined.Eqns. 5.88 and 5.89 can be regarded as an analytical form of the radiationcondition.

To obtain the integral equation for the function U (T, X, Z) let us introduceGreen's function g(M, P) satisfying Floquet's condition and the equation

V2g(M, P) + k2g(M, P)= - 2nS{M, P) (5.90)

at any point of the space. The solution of eqn. 5.90 can be expressed as

g{M,P)=J~ £ H^\k{(xM-xP)2

£ n— — oo

+ (zM + nd-zP)2}ll2]exp(-jnT) (5.91)

This function satisfies the following conditions of pseudoperiodicity:g{xM> ZM, XP> ZP) = exp{j(n - m)T}g{xM, zM + ™d, xP, zP + nd)

d d•Js—g{xM> ZM> XP> ZP) = exp (JT) - — g ( x M , zM ~ d, xP, zP)dzu vzM

d d^-g(*M, ZM, XP> ZP) = exp ( - 7 1 ) ~— g(xM, zM, XP> ZP ~ d)CZ cz

(5.92)

Applying the second Green identity to the functions U(z, x, z) and g(M, P) inthe area Dx [Figure 5.12 (i) j and taking into account the conditions of pseudo-periodicity (eqns. 5.85 and 5.92) and also the radiation condition (eqns. 5.88and 5.89), we obtain an integral representation for the total field

U0(T,M) MeD (5.93)


n being the unit normal vector directed out of S. Eqn. 5.93 enables us to findU(T, M) everywhere in the area D in terms of £/(T, P) and dU(T, P)jdnP on C.Assuming that rj(P) ^ 0, M e C and using eqn. 5.83 and the properties of simple-layer and double-layer potentials (Tikhonov and Samarsky, 1977), we obtainthe integral equation for dU(x, P)/dnP

2rj{P) dnP 2n Jc\r]{Pf) cnr

= -g(P,M)-U0(T,P) PeC ri(P)±0 (5.94)

^ oo, we similarly obtain the integral equation for U(T, P)

M) + U0(T,P) PeC i / (P)ôo (5.95)

Eqn. 5.94 at rj(P) = oo and eqn. 5.95 at r\(P) = 0 turn into integral equationsfor a perfectly conducting grating in cases of magnetic and electric polarisation,respectively. Eqns. 5.94 and 5.95 can be written in a general form

K(P, P')pi(Pf) &lF. = F(P) (5.96)C

where fi(P) is an unknown function.To construct a numerical algorithm for the solution of eqn. 5.96 accurate

enough for various shapes of the contour C, it is necessary to approximate theintegral operator in such a manner that changing of the contour will notinfluence the accuracy of approximation. The difficulty of natural integrationalong the contour C is in the fact that for an arbitrary contour the co-ordinatesof the points P and P\ on which the kernel in eqn. 5.96 depends, cannot beexpressed in terms of the length of the arc without any quadrature computation.For this reason let us introduce inside S an auxiliary contour C1 consisting ofstraight lines and segments of circles. Let r = r0 (cj)) be an equation of the contourC in the cylindrical polar co-ordinates r, <fi with the origin at the point 0 situatedinside the contour. Let us connect the system of co-ordinates 0££ with thecontour C\. The £ axis makes the angle p with the x axis of initial Cartesian-co-ordinate system. Let us assume, for simplicity, that the origins of the systemsOxz and 0££ coincide. Taking into account the character of the contour C, wecan choose the angle p and the dimensions for the contour Cx. If 5 is the anglein the polar system of co-ordinates with the £ axis, then (j) = p + d, and for thecontour C1 it is easy to express the angle S in terms of the length of the arc s,i.e. 6 = S(s). Establishing a one-to-one correspondence of the points P E C andP eC1 having the same polar azimuth </), we obtain a parametric description ofthe contour C as a function of the arc length of the contour Cl:

(5.97)= R(s) cos{p + d(s)}


The arc elements for the contours C and Ct are related as follows:

d/c = {R2(s) (S's)2 + (K)2}112 dsCl = I(s) dsCl (5.98)

Replacing the integration contour C in eqn. 5.96 by Cx and denoting the kernelby K(sP, sP') (sP and sP> are the values of s for the points P and Pf, respectively),we obtain

K { ) I { ) { ) d F ( ) (5.99)

To solve the integral equation given by eqn. 5.99 numerically, the Krylov-Bogolubov technique (Kantorovich and Krylov, 1958) is used. In the frameworkof this method the integral equation is reduced to an algebraic system of linearequations. For this purpose let us divide the contour Cx by points sOi sii . . . , sN

(s0 = sN) into JV subintervals approximating fi(sP) as a piecewise-constant func-tion. Computing the integrals by a quadrature formula, we can write eqn. 5.99in the form

Y [SPJ+1 K{sP,sP.)I(sP.) dsP, + fi(sp)=F(sp)

(5.100)

where

Assuming that sP = sPj+i/2 in eqn. 5.100, we obtain a system of linear algebraicequations for n{sPj+l/2):

l/2) (5.101)

where i = 0, 1, . . . , JV — 1, and

l CSpj+iA i j = 2 n ^ ( ^ + 1/2' SP)J(SP)

l CSpj+iij=2n ^ ( ^ + 1 / 2 '

The accuracy of solving the linear system obtained depends on the accuracy ofcomputation of the coefficients Atj defined by the behaviour of K(sP, sP>) on thecontour C1.

Green's function expressed by eqn. 5.91 has a logarithmic singularity whenthe arguments coincide; when calculating the matrix elements in eqn. 5.101 thissingularity should be extracted, i.e. we need to present Green's function as

g(p, n = gi (p, n+^ ®(p, n (5.102)

where g^ (P, Pf) is a regular function, and lim <£>(P, P') — 0.

With the aid of Poisson summation formula we present g(P, Pr) in the form

ZP -zP>)-Tn\xp- xP. I}

Let us consider the function g(P,Pf), having the singularity In(rpp*)'1 at


P=P' similar to g(P, P')\

I Zp Zp' \ \~* / Zp' P ) = exp 7 T ) COS 2/271

V d ) n=\ \x expI — 2nn

d J n

Summing the infinite series in this expansion, we have

. Zp ~ <

where ^>{P, P') are written in the form

+ sin2 | ^ (Zp ~ Z \<b(P, F) = sinh2 j -d (xP - xP.) 1

Using the formulae written previously, we can present Green's function in theform of eqn. 5.102.

Let us examine dg(P, P')jdnP^ the normal derivative of Green's functionexpressed by eqn. 5.102. It can be shown [see Tikhonov and Samarsky (1977)]that, if the curvature x(P) °f t n e contour C is continuous, then the normalderivative of Green's function has no singularity and

de(P, P') 1lim ^ V L = -

P^P' dnP> 2The series involved in the expression for dgx(P, P')jdnP> have an order ofconvergence of O(n~l). Therefore, in spite of the fact that dgl (P, P')jdnP> is aregular function, it is necessary to improve their convergence. One possible wayto do it is described by Galishnikova and Ilyinsky (1987). Let us note that thekernel singularity is completely determined by the logarithmic singularity ofGreen's function.

Thus while the elements of the matrix Atj (i ^j) in eqn. 5.101 are calculated,the integrals are evaluated by means of ordinary quadrature formulae with theaccuracy required. The integrals of the singular functions which appear wheni — j are calculated in the following manner:

Zp,)\ I(sP.) In *(/>,+1 / 2, F) ds

x\n{Q)(Pi+ll2,P')\sr-sPi + l/2\}dsP,

,/7 ~~ Zp1) \I{sp') In \sP> — sP. \dsP> (5.103)J

The first integral in eqn. 5.103 can be computed to the desired accuracy and• d


the second is expressed in a closed form:

In | sP' — sP. I dsP'

{spj s P i + l / 2 ) { l n \ s P j s P i + l/2\ 1 ) }

Solving the system given by eqn. 5.101, we find fi(s) in the points of the contourC\ and, as a consequence, the unknown functions in eqn. 5.94 or eqn. 5.95. Bymeans of eqn. 5.93 we can determine £/(T, M) at any interior point of the stripDx. Then, using eqns. 5.85 and 5.86, we can find the total field i//(x, z) at anypoint of the space.

It is possible to show that the diffraction field in the far zone is determinedby the coefficients Tn{x) and Rn(r), corresponding to propagating space harmon-ics. According to eqn. 5.88 the total field i//(x,z) in the area x<0 can bepresented in the form

*2n

Tm(T)4tm(T,z)exp{rm(T)x}dT

m+1)

= - oo J o

i oo (*2n(= — g271 m = -oo J2nm

x exp{Fm (T' — 2nm)x} dz'

where T' = T + 2nm.Let us introduce a variable — oo < X < oo by means of the formulae

2nn m = n, n> 0

r' m = 0

x' — 2nn m= —n,n>0

The expression for \j/(x, z) can be written as

(5.104)

where y{X) = {k2 - X2 ) 1 / 2 ,

frm(Xd-2nm) 2nm<Xd<2n(m+ l ) ,

T(X) = T0{Xd) 0<Xd<2n

-2nm<Xd< -2n(m+(5.105)


The representation given by eqns. 5.104 and 5.105 is the Fourier transform ofthe solution sought.

To determine the radiation characteristics in the far zone let us introduce thecylindrical polar co-ordinates

x = rcos</>, z = r sin (/), ~ < (j) < — , (x, z) e Dx

Using the saddle-point method, we obtain the following asymptotic represen-tation ofif/(x, z) when r tends to infinity:

Thus at r-> oo the field, transmitted through the grating, is a cylindrical inhomo-geneous wave diverging from the periodic structure. Using eqn. 5.105, we canexpress the radiation pattern FT((t>) in terms of amplitudes of the propagatingspace harmonics:

d1/2(-kcos(j)) Tn (kd sin <j) - 2nn) 2nn < kd sin (f) < kd

dll2(-kcos(j)) To (kd sin 0) 0 < kd sin 0 < 2n

d112 (- k cos (j))T_n(kd sin 0 + 2nn) - kd < kd sin (j) < ~2nn

(5.106)where

the symbol [a] standing for the greatest integer in a.A similar expression can be obtained for the reflected field. Designating the

radiation pattern for the reflected field as FR((j)), we obtain

d1/2 k cos (f)Rn (kd sin (j) - 2nn) 2nn < kd sin 0 < kd

dll2k cos (f)R0 (kd sin 0 < kd sin <f> < 2n

d1/2kcos (j)R-n (kd sin 0 + 2nn) —kd< kd sin (/> < —2nn(5.107)

where — 7r/2 < (f)< nj2.The total flux of the reflected and transmitted energy in the areas x <xx and

x > x2 is expressed in terms of the amplitudes of propagating space harmonics.The following energy identity is valid:

neN'(x)


c

X Imrn\B°n(T,x2)\2dT (5.108)

O neiV'(t)

where

-d/2

Summation in eqn. 5.108 is taken over the set JV'(T): neN'(x) if £2 — a2 > 0,i.e. JV'(T) is a set of integer indices corresponding to the numbers of propagatingspace harmonics. Using eqns. 5.93 and 5.94 we can show that under thecondition of the surface impedance being small for a sufficiently smooth contourC, the losses can be taken into account in the framework of the energy-pertur-bation method. In other words, we can solve the problem on the assumption ofperfect conductivity and then, having the currents distribution on the surfaceof the structure, determine the dissipated power in accordance with eqn. 5.46.

A modification of this method for the case of reflecting gratings [Figures5.12(ii) and (in)] is obvious. The approach described has been tested for a seriesof problems. For example, periodic structures made of rectangular and circularbars have been examined for cases of excitation by plane waves and line sources(Ilyinsky and Slepyan, 1983, Galishnikova and Ilyinsky, 1987). Here we shallpresent the absorption characteristics of periodic structures excited by H-pol-arised plane waves. In this case the algorithm described above can be simplifiedessentially: there is no need to use the T-transform defined by eqn. 5.84. Tocalculate the induced current density and then the dissipated power we simplysolve eqn. 5.94 assuming that g(P, M) — 0 and x = kd sin 3.

As in Section 5.5, it is possible to characterise the absorption in a periodicstructure by the dimensionless coefficient a defined by eqn. 5.70. In Figures 5.13and 5.14 the dependencies of a on the angle of incidence are shown for a comb-shaped structure with ridges of finite thickness (solid lines). Similar dependenciesfor structures with infinitely thin ridges, determined according to the method ofSection 5.5, are presented in Figure 5.13 by broken lines. When the ridges arerather thin {hjD < 0.04), the results are very close to the corresponding broken-line curves. However, for hjD = 0.1 the influence of the ridges thickness on theabsorbed power is already noticeable, especially for small angles of incidence(curve 1 in Figure 5.13).

Note that, using the integral-equation method, we cannot diminish infinitelythe thickness of the ridges h. Otherwise we may observe the computationalinstabilities caused by the fact that the matrix of the system given by eqn. 5.100is close to singular. Thus, for parameters corresponding to Figure 5.13 thenumerical algorithm described is unstable when hjD < 0.02. In this case thetrend of the dependence a (3) becomes nonmonotonic and the small-scale oscil-lations of current-density distributions occur. These oscillations do not have anyphysical sense and are caused entirely by the accumulation of computationalerrors. Thus, when using this method for structures with thin ridges, we canrecommend analysis using the current-density distributions (Galishnikova andIlyinsky, 1987).


OL

e0 45 90

Figure 5.13 Absorption in a comb-shaped structure with ridges of finite thickness againstangle of incidenceH polarisation, \i = 1

1 hlD = 0.\, T= 1.572 /z/D = 0.04, r = 1.573 A = 0, r=1.574 A/Z) = 0.04, r=2 .25 /z = 0, r = 2.2

Investigation of losses in the structure shown in inserts to Figures 5.13 and5.14 has been carried out by Gavrielides and Peterson (1979). They presentedcomputational formulae and some numerical results for both polarisationsthough they did not describe the effect of abnormally small dissipation.

In Figure 5.14, specifically, are shown the dependencies a(#) in the neighbour-hood of the points of origin of higher propagating harmonics (curves 4 and 5).A break in curve 5 can be clearly seen, similar to the well known Wood anomaliesfor diffraction characteristics.

Figure 5.15 shows the absorption coefficient a as a function of the incidentangle S for a structure with semicircular corrugations. For curves 1 and 2 afairly smooth trend is observed in the area of moderate 9 and a sharp fall withinthe narrow range of large values of #. When F_ x = 0 the characteristic breakalso occurs (curve 3) but the peak height is considerably smaller than that inFigure 5.14.

In this Chapter we have analysed dissipation characteristics for differentreflecting periodic structures: gratings with infinitely thin ridges and those of a


90Figure 5.14 As for Figure 5.13

1 h/D= 1, r = 7r2 h/D^ 0.5, T = n3 h\D = 0.25, T = n4 A/Z) = 0.25, r = 1.05TC5 h/D = 0.25, T= I .ITT

finite thickness, inclined comb-shaped structures, echelette gratings and gratingswith semicircular corrugations. The results obtained allow us to make generalphysical conclusions, to which we would like to draw the reader's attention:

(i) The phenomenon of abnormally small dissipation for H-polarised waves isa universal physical effect: it takes place for structures with any form ofthe period at sufficiently large angles of incidence. However, the strengthof this effect (specifically, the range of angles where it is observed) dependsgreatly on the form of the grating profile.

(ii) The peaks of absorbed power, as with Wood anomalies for diffractioncharacteristics, are in the neighbourhood of the points of origin of higherpropagating space harmonics. Their intensity is also determined by theshape of the grating.

5.8 Diffraction by a complex-shaped periodic structure:Galerkin's incomplete method with semi-inversion

In this Section we continue our discussion of problems of wave diffraction byperiodic structures with complicated configurations. We shall present anothermathematically rigorous method which is in many cases more effective than theapproach based on integral equations. This method has been developed in a


0 45 90Figure 5.15 Absorption in comb-shaped structure with semicircular ridges against angle

of incidence

H polarisation, 2rojd=0.b

r=1.57r=2.2T= 3.2987

number of works by the authors of this book [Ilyinsky et al. (1987), Kopenkinet al. (1987, 1988, 1989), Ilyinsky et al. (1989), Slepyan (1990), Kuraev et al.(1991)].

The geometry of the problem and main designations are shown inFigure 5.12(ii). Let the periodic structure be infinitely wide and homogeneousalong thej axis. Let us consider the case of excitation by an electrically polarisedplane wave. The magnetic field of the incident wave is

H° = iy\j/° = iy exp{jk(x cos 3 + z sin 3)}

The magnetic field scattered by the structure has a single component ij/(x, z) —Hy (x, z) satisfying the Helmholtz equation

V L ^ + k2{j/ = 0 (5.109)

To calculate the linear (per-period) dissipated power we shall use the energy-perturbation method:35

1 (5.110)

35 Note that the use of the energy-perturbation method is not the only possible way to calculate theabsorption characteristics of the structures considered: the approach in question can be easilytransferred to the case of impedance-boundary conditions.


where C is a contour of one period and \j/ corresponds to a model of perfectlyconducting structure. To determine \j/ we have to find the solution of eqn. 5.109satisfying Neumann's boundary condition

dn= 0 (5.111)

In addition, \j/(x, z) should satisfy the radiation condition at x-> + oo and theedge condition.

The main idea of this approach is a synthesis of Galerkin's incomplete methodand the method of semi-inversion. At the first stage the original diffractionproblem is reduced to a boundary-value problem for an infinite-dimensionalsystem of ordinary differential equations. This problem is ill-posed: it cannot besolved satisfactorily by means of truncation to a finite-dimensional problem anduse of the shooting technique. The reason is that the matrix operators in theboundary conditions are noninvertible, as for matrix operators in the SLAE-1to which diffraction problems in areas of simpler shapes are reduced [Mittraand Lee (1971), Shestopalov et al. (1984)]. As a result, as for such systems,computational instabilities arise and the relative-convergence phenomenon isobserved. Note that the convergence is rather slow even when the method oftruncation is correct. Compared with the approaches based on the direct solvingof the SLAE-1 the task is redoubled by the fact that the discretisation errors areadded and they may be essential.

To regularise the problem it is necessary to extract for the inverse in a closedform the matrix operators of convolution type in the boundary conditions aswas done earlier in relation to infinite algebraic systems of the first kind (Shesto-palov et aL, 1984). The boundary-value problem obtained can then be solvednumerically by the truncation method with the use of the shooting technique.

Let us consider auxiliary diffraction structure shown in Figure 5.12(iii). It iscomposed of sections of parallel-plate waveguides of width d— a1 (0) + a2 (0) andlength A linked to the original structure at the x = 0 plane. We shall pass to thelimit A -> 0 in the final algorithm.

As the first step let us use the method of partial regions; the division intosubareas is shown in Figure 5.12(iii) by broken lines. The field over the structurecan be presented as Floquet's expansion

il,(x,Z)=il,0(X,z)+ t Rpcxp{japZ-Tp(x-A)} (5.112)p= — oo

where Rp are the coefficients sought, ap = 2npjd + k sin $ and Tp = —j(k2-a^)1/2. The representation given by eqn. 5.112 satisfies the radiation conditionsat x—XX). The field at 0 < x < A can be expanded in a series in terms ofeigenmodes of a parallel-plate waveguide of the width d:

n (* - A)}

(5.113)

where A(nia) are unknown coefficients and yn = {(nnjd)2 — k2}1'2. In the area

— L < x < 0, due to its nonco-ordinate form, let us use Galerkin's incompletemethod. In accordance with this method, instead of eqns. 5.109 and 5.111, we


can write a system of projection relations in the area 3>

x,z)dz = Am(x) (5.114)

where

.___n , JL /.. ., [mn{z-a2{x)}x)}l—

Eqn. 5.114 results from the 2-dimensional version of the second Green identityapplied to the functions \\t and (j)m in the cross-section of the area 3) by the planex = constant.

Let us expand \jj into a series in terms of (j)m:

il/(x,z)= I Pn{x)4>n{x,z) (5.115)n = 0

Substituting eqn. 5.115 into eqn. 5.114 we arrive at a system of ordinarydifferential equations for Pn (x) which in the vector-matrix designations can bepresented as

P"(x) + H(x)P'{x) +B{x)P(x) = 0 (5.116)

the prime denoting the derivative with respect to x, and P(x) = {Pn(x)}. Theelements of the matrices H(x), B(x) are expressed as

i(x)

J -at(x) CX

Hmn(x)=hl(x)Smn+Nm\*>) J ~ai(x)

2

Note that Galerkin's incomplete method is used not as a means for solving theboundary-value problem as a whole but as a means of presentation of thefunction sought in one of the partial regions. Usually, Galerkin's incompletemethod is introduced as a finite-dimensional approximation of the boundary-value problem and in this connection a finite number of equities is retained ineqn. 5.114 and the same number of terms in eqn. 5.115 is taken into account.It is essential that our approach implies that eqn. 5.116 is of infinite order, andit is thereby actually equivalent to the original boundary-value problem given


by eqns. 5.109 and 5.111. The truncation of eqn. 5.116 will be carried out atthe next stage of the solution.

Boundary conditions for eqn. 5.116 at the x = 0 plane result from the con-tinuity of ij/ and dijj/dx across x = 0 and x = A. For x = 0, in accordancewith Surattean et al. (1985), we obtain the relations

P / (0 )±rP(0) + ^ + P(0)= ±2TE±AU2 (5.117)

where Al2 are vectors of unknown coefficients A^t2\ T and E± are infinite-dimensional diagonal matrices with the elements Smnyn and Smn exp( + 7«A),respectively, and £+ is a matrix with elements expressed as

(5.118)

where the symbol — 0 stands for the limit approached from the left.Matching of expansions for \\t and d\\i\dx defined by eqns. 5.112 and 5.113 in

the x ~ A plane can be carried out by a method similar to that used by Mittraand Lee (1971). In this case, excluding Rp, we arrive at an infinite set of linearequations for A^'2):

Z Y - + ? T +" = ~Jôsdtan9 (5.119)

where v2n = sin(w/2), v2n+1 = j cos(w/2), u = kd sin 9.Eqn. 5.119 can be written in a more compact matrix form

G~A1 + G + A2 = b (5.120)

where G± are matrix operators with the elements G*s = vn(Ys + yn) - 1 , and b

is a vector with the components bn = — jdnOd tan 9.Taking into account the fact that Ax and A2 are expressed by eqn. 5.117,

then eqn. 5.119 (or eqn. 5.120) is the boundary condition to be found.The boundary condition at x= —L follows from eqn. 5.111. In fact, from

eqn. 5.111 we can proceed to equivalent projection relations

<f)m{-L)dz = 0 (m = 0, 1,2, . . . , oo)c= -L

Substituting into these equities the expansion given by eqn. 5.115, we obtainthe boundary condition at x = — L in the form

(P' + r P ) l , = -L = 0 (5.121)

where the matrix £~ is expressed in a manner similar to eqn. 5.118 but for thecase when x= — L + 0 (the symbol +0 denoting the right-hand limit).

Note that eqns. 5.116, 5.119 and 5.121, generally speaking, do not form aproblem having a unique solution. They should therefore be supplemented bythe edge condition. If, according to this condition, we demand a finiteness ofelectromagnetic energy in any bounded area containing an edge, then, as inSection 5.1, we arrive at the condition

£ <co (5.122)n = 0

where îj,1'2* are defined by eqn. 5.117. Together with this condition, eqns. 5.116,


5.119 and 5.121 form a 2-point boundary-value problem for an infinite-dimen-sional system of ordinary differential equations. However, such a problem cannotbe solved numerically, e.g. by means of truncation to the finitely-dimensionalproblem and use of the shooting method. This method does not lead to successiveresults as the boundary-value problem is ill-posed because the boundary con-dition at x = 0 contains the matrix convolution operator G ~ and the solutionprocedure involves its numerical inversion. The operator G~ is responsible forthe unfavourable computational properties of the algorithm obtained in such amanner (slow and 'relative' convergence of the truncation method, compu-tational instabilities etc.). In fact the situation here is very similar to thatoccurring in Sections 5.1 and 5.3 for systems of linear algebraic equations of thefirst kind (see also Section 1.4). Thus, regularisation of the 2-point boundary-value problem given by eqns. 5.116, 5.119, 5.121 and 5.122 is needed. It canbe provided by means of analytical inversion of the operator G" in eqn. 5.119.

Let us dwell upon the procedure of semi-inversion in more detail. Let ustranspose the term with A^ to the right-hand side of eqn. 5.119 and considerthe relationship obtained as an equation for A^\ Taking into account the edgecondition expressed by eqn. 5.122, this equation can be solved exactly with theaid of the residue-calculus technique (Mittra and Lee, 1971, Shestopalov et al.,1984). In this case we obtain

t (5.123)m = 0

where {cn}, { J,m)} are the solutions of the systems

n=o i s — yn

oo Jim)V

rko Ts -yn To + ym

{s = 0, + 1, . . . , ±oo, m = 05 1, . . . , oo) having an algebraic asymptote atn-* oo. The coefficients cn, d^ are expressed in a closed form through the infiniteproducts; the corresponding formulae are well known and we shall not presentthem here (see Mittra and Lee, 1971, and Shestopalov et al., 1984).

The procedure described above can be written in a matrix form. Usingeqn. 5.120 we can write eqn. 5.123 as

^ i + {G~)-1G + A2= ( G " ) " 1 * (5.124)

Combining eqns. 5.117 and 5.124, we obtain a regularised boundary condition.Note that eqn. 5.124 is similar to systems of the second kind obtained withinthe framework of the semi-inversion method (see Section 1.3).

The 2-point boundary-value problem given by eqns. 5.116, 5.121 and 5.124can be solved by the standard numerical method (truncation to the finite order2JV and use of shooting method). At this stage of the solution we can putA = 0, and thereby return to the initial geometry of the structure. We can alsoexplain the purpose of introducing the auxiliary geometry with A # 0. Becauseof this the convolution matrix operator in the boundary condition given byeqn. 5.119 has been extracted in the explicit form. Such a formulation of theboundary condition makes possible the application of the regularising technique


based on analytical inversion of the convolution operator. This procedure is nota new one: introduction of auxiliary structures has been employed many times(Mittra and Lee, 1971, Shestopalov et al., 1984).36 However, in these paperspartial areas had sufficiently simple shapes and the problems have been reducedto linear systems of algebraic equations.

Complex amplitudes of space harmonics are expressed through the solutionof the 2-point boundary-value problem as

2drsRs=(-iyas t f^L + ^L) (5.i25)

To make calculations more convenient we can substitute eqn. 5.123 intoeqn. 5.125 and perform some identical transformations (Kopenkin et al., 1988).

Let us consider the series given by eqn. 5.115, truncated to the Nth term, tobe an approximate solution of the problem in the area Q) [the coefficientsP^ [x] are determined from the 2-point boundary-value problem given byeqns. 5.116, 5.121 and 5.124 truncated to the order 2JV]. Complex amplitudesof the diffraction harmonics Rp determined in terms of / ^ ( O ) , P(

nN)f (0), will be

designated as R^K A question arises about convergence of the approximatesolution to the accurate one at jV—•oo, i.e. about a rigorous mathematicalsubstantiation of the method described above. Such a substantiation is carriedout in papers by Kopenkin et al. (1987, 1988). Here we shall expound the mainideas of this substantiation without mathematical details. The most simple caseto be examined is A 0.

The difficulty of substantiation is that we arrive at a boundary-value problemfor an infinite system of ordinary differential equations. A general theory of suchproblems in mathematics has not so far been developed, and we cannot thereforeuse the ready-convergence criteria as, for example, for infinitely-dimensionalproblems of linear algebra [however, specific cases of such problems have beenexamined by Sveshnikov et al. (1965) for smoothly irregular waveguides]. It isnecessary to take into account, however, the fact that truncation of this problemis done in two directions: first truncation of matrices / / , B, £* describing thefield forming inside the grooves, and secondly truncation of matrices(G~) ~1, (G~) ~1G+ in eqn. 5.124 describing the radiation from the groovesinto free space. In fact, in the course of computations the truncation process isindivisible, but to mathematically substantiate the algorithm developed, trun-cation procedures, for convenience, should be considered separately. Toimplement this let us present

Pn(*)= t ^.W^J1)exp(-y,A) (5.126)i = 0

or in matrix form P=V(x)E + A1, where values {Vin} satisfy the 2-pointboundary-value problem given by eqns. 5.116, 5.121 and

V'(0) + (r + C) V(0) = 2T (5.127)

Then from eqn. 5.117 it follows that A2 = E+{V{0) - I}E + A1. Usingeqns. 5.116, 5.121, 5.124 and 5.126, we obtain the infinite system of linear

' In this book an example of its use is given in Section 5.6.


algebraic equations

Ax + {G-)'1G + E + {V{0)-I}E + A1 = ( G " ) " 1 * (5.128)

Thus the problem of substantiation of the method is split into two basic parts:

(i) proof of convergence of the truncation method for computation of thematrix F;

(ii) proof of applicability of the truncation method for solving of eqn. 5.128.

The first problem coincides with that examined by Sveshnikov et al. (1965)by means of the method of energy identities: it is considerably simpler than theoriginal one because of a special form of boundary conditions given by eqn. 5.127.The second problem is that of linear algebra; we can show that eqn. 5.128 is ofthe second kind. This allows us to prove applicability of the truncation method.

It is necessary to note the relationship between this approach and the general-ised scattering-matrix technique (Mittra and Lee, 1971). In fact, we can presentthe structure under consideration as a junction of two successively locateddiscontinuities: an array of halfplanes at x = A and sections of smoothly irregularwaveguides with short-circuiting end walls at x = — L. In fact, this connectionis expressed by eqns. 5.126-5.128. It was used to substantiate our approach.However, in practice, the method described here is far more effective: to findthe scattering matrix of the second discontinuity of the jVth order [it is expressedas F(0) — /] it is necessary to solve JV times the 2-point boundary-value problemgiven by eqns. 5.116, 5.121 and 5.127 with different boundary conditions atx = 0 and then carry out matrix operations in accordance with eqn. 5.128. Thewhole computational procedure of our algorithm consists of a single solution ofthe 2-point boundary-value problem but the boundary condition at x = 0 has amore complicated form (see eqns. 5.117 and 5.124).

Let us consider the results of mathematical simulation of wave diffraction bycomplex-shaped reflecting gratings obtained through the method describedabove.

First, the characteristics of trapezoidal-groove gratings have been calculated(the grating is shown in the insert to Figure 5.17). All calculations have beencarried out for j8 = 30°, S = 30°, L/d=0A2. The 2-point boundary-value prob-lem given by eqns. 5.116, 5.117, 5.121 and 5.124 is solved by the conventionalshooting method (see Appendix 1).

Figures 5.16 and 5.17 show the magnitudes and phases of complex amplitudesof space harmonics as functions of the truncation order JV for kd= 10. Anabsolute error of active power balance d comprises a value of an order of 10 ~7.As in our calculations a1( — L)+a2( — L)<4d,L, the geometry under study isrelatively close to the geometry of a conventional echelette grating, which iswhy, from the physical point of view, we can expect numerical results for thetwo structures to be close to each other. It is of interest to compare them. InFigures 5.16 and 5.17 the computational data obtained by Shestopalov et al.(1984) for echelette gratings with the aid of the semi-inversion method are shownby broken lines. Here, there is good agreement with the results of our calcu-lations. It is worth noting that the rate of convergence of the truncation methodis rather high; sufficiently accurate results can be obtained at jV= 5-6. Differ-ences between these data and the results for the echelette gratings can beexplained by the errors of the shooting method (they can be reduced, using


0.9

0.8

0.7

0.6

— 0.5

"0.4

0.3

0.2

0.1

L-2 /

— 1 \ /

\ / — " " • • ^ ^ / \

A

\ H \ l i i 1 1

2 3 4 5 6 7 8 9 N

Figure 5.16 Convergence of complex amplitudes of space harmonics (modulus) fortrapezoidal-groove grating

more accurate finite-difference approximations of differential operators andmodifications of the shooting method described in Appendix 1).

It will also be of interest to investigate the influence of the numerical inte-gration step on the accuracy of the results obtained. Corresponding data aregiven in Table 5.1 for kd = 6.28 (autocollimation scattering). Here M is a numberof steps on the integration interval and N= 9. Let us point out, for comparison,that the magnitude and phase of complex amplitude of the first backward spaceharmonic for the corresponding echelette grating are 1.0 and 179.9, respectively.It can be seen from Table 5.1 that the phase is far more sensitive to the choiceof integration step than the magnitude.

Periodic structures with the profile described by the equations

(5.129)

(5.130)

have also been examined. In these equations/? is a numerical coefficient charac-terising the degree of groove-profile irregularity. The upper sign in eqn. 5.130


160 r

-120-

-160\-Figure 5.17 Convergence of complex amplitudes of space harmonics (phase) for

trapezoidal-groove grating


T a b l e 5,1 Influence of integration step on R^1 — \R~i\ exp (j(j) _ 1)

M 1 (cleg)

48163264128

0.9990.9991.0001.0001.0001.000

176.1178.5178.9179.0179.0179.1

2x 10~3

1 x 10"3

5x 10~4

1 x 10~4

2x 10"5

5x 10~6

4.0

J.O

2.0

1.0

V,

4.0 4.25 4.5

Figure 5.18 Frequency characteristics of complex-shaped gratings

L/d= 0.3

2 /> = 0.3 p = 0.


10

0.5

0

n=0

450, degrees

Figure 5.19 Magnitudes of reflection coefficients against angle of incidence

T=5, L/d=03

90

= 0.05

corresponds to a 'symmetrical' profile, the qualitative character of which isshown on the insert to Figure 5.18; the lower sign in eqn. 5.130 corresponds tothe 'nonsymmetrical' profile (see the insert to Figure 5.19).

In Figure 5.18 are shown the amplitudes of two space harmonics as functionsof kd at kL = 0.3, 9 = 30° for three different values of the parameter p. Thecomputations have been carried out at JV = 10, M= 10. Solid lines correspondto the mirror-reflected harmonic, broken lines to the first backward harmonic.Curves 1 are calculated at p = 0, which corresponds to the symmetrical lamellargrating with infinitely thin ridges. These curves are completely identical withthose obtained by other rigorous methods (semi-inversion, modified residue-calculus technique). In Figure 5.18 we can see an angularity of the curves for\R0\=f(kd), corresponding to Wood's anomalies. The active power-balanceerror 8 did not exceed 10~4.


In Figure 5.19 are shown the dependencies \Rn\ (n = Q, — 1) on fl for twovalues of p (p = 0 as it was mentioned already, corresponds to the case of aconventional lamellar grating). The values of the parameters are given in thecaption for Figure 5.19.

The results given have an illustrative character: their purpose is to demonstrategeneral opportunities and peculiar features of this approach. As a result, it ispossible to make the following conclusions:

(i) Convergence of this approach is fairly fast; the diffraction characteristicsare determined to an accuracy within 1% when jV*= JV0 + 3 [No is thenumber of propagating modes in a parallel-plate waveguide of widthma.x{a1(x) +fl2 (*)}]•

(ii) This approach allows us to calculate complex diffraction effects in theresonance frequency range: Wood's anomalies, autocollimation scatteringetc. That makes it an effective means of investigation of periodic structureswith complicated shapes, e.g. phased antenna arrays from the hornradiators.

(iii) The method is very effective from the computational point of view: as waspointed out by Kopenkin et al. (1988), the run time required for thealgorithm based on the integral-equation technique for a number ofexamples is several times greater than that needed for the algorithm inquestion.

Chapter 6

Dissipation in comb-shaped structuresin inhomogeneous and anisotropic

media

In Chapter 5 we examined the problem of wave diffraction by a comb-shapedstructure with a finitely conducting surface in a homogeneous isotropic medium.The main physical result of this study was resonance dissipation effects, specifi-cally the effect of abnormally small absorption.

When a comb-shaped structure is placed into a layered or gyrotropic medium,one can observe a number of interesting features of the effects mentioned above.This Chapter deals with the following problems:

(a) H-polarised plane-wave diffraction by an imperfectly conducting rec-tangular-groove grating with a plane-layered dielectric halfspace over itor plane-layered dielectric filling of the grooves;

(b) H-polarised plane-wave diffraction by an imperfectly conducting comb-shaped structure in a gyrotropic medium.

For the first problem a rigorous numerical method to calculate the dissipationpower has been developed. Resonant absorption phenomena resulting fromdistortions caused by the grating in total internal reflection from the dielectricboundary are discussed. For the second problem an approximate analyticalsolution has been obtained. The latter is applicable when the period of thestructure is small compared with the wavelength. It is shown that the absorptionresonant effects for comb-shaped structures in gyrotropic media become non-reciprocal, and, owing to their significant (^-factors, a high nonreciprocity isobserved even when the medium gyrotropy is weak. These effects can be usedto design new types of nonreciprocal devices and to measure the gyrotropicmedia parameters.

The material is based on papers by Ilyinsky and Slepyan (1983), Slepyanand Slepyan (19806, 1981), and Slepyan (1983).

180 Dissipation in comb-shaped structures in inhomogeneous and anisotropic media

6.1 Diffraction by a finitely conducting comb-shapedstructure with a layered dielectric filling; resonanceabsorption

Let us consider the incidence of the electrically polarised plane wave

(A0 (*, z) = exp{jk(y/g) (x cos S + z sin 3)}

from the halfspace x < 0, filled by a homogeneous lossless medium (s = so8i, fi =fi0) on a metallic finitely conducting comb-shaped structure in an jV-layerdielectric medium (Figure 6.1).37 The ith layer is characterised by the thicknessAf — A,--! and permittivity eo£j (all layers, for simplicity, are considered to beisotropic, nonabsorbing and nonmagnetic), i = 1, 2, . . . , JV. It is assumed thatthe ridge thickness is significantly smaller than the wavelength, structure periodor thickness of each layer, but significantly exceeds the skin depth. The energylosses in the comb-shaped structure can then be taken into account within theframework of skin effect theory, in a method similar to that in Section 5.3. Thelinear (per-period) dissipated power in the periodic structure is expressed byeqn. 5.46.

To calculate P, as in Section 5.5, we use a perturbation technique (£s small),

Figure 6.1 Comb-shaped structure in layered dielectric medium

37 Similar problems for cases of perfect conductivity of the periodical structures were examined byAmitay et at. (1972), Shestopalov et at. (1986) and Andrenko and Shestopalov (1979), whereresonance effects of 'reflection-transmission' were studied. This class of problems is of interestbecause there are both traditional (polarisers, antenna radomes, dielectric parameters measuringequipment) and new applications (for example, logic elements based on such periodic structures).

Dissipation in comb-shaped structures in inhomogeneous and anisotropic media 181

i.e. as a first approximation we replace the actual \\jj\2 by the value correspondingto infinite conductivity. This can be done for frequencies sufficiently remotefrom the cutoff ones (coin = niic/d^/Si), which will be assumed throughout. Pre-senting the field in each layer as an expansion in terms of forward and backwardeigenmodes of the plane waveguide with a corresponding dielectric filling, weobtain the formula for dissipated power

where

Dp7 f

' = 2

Ft (x) =

d/2

-d/2(6.1)

3 ~ o )rexP(~A«.j»AJ»)

Ani = {(nnjd)2 — k2^}112 and Ao = 0. The coefficient Tni is the complex ampli-tude of the ?zth forward eigenmode in the zth dielectric layer, and pni is thereflection coefficient of the nth mode from the interface between the ith and thei+ Ith layers. The coefficients Tni and pni are expressed by recurrent relation-ships in terms of Tni and pnN (pntN being the reflection coefficient of the nthmode from the metal plane x = AN). These relationships are obtained using thecontinuity of field tangential components across the layer interfaces and havethe form:

Pn,i+ 1 lit e XP ( ~e x p ( ~

(6.2)

e x p ( - An,i+! A,) + pnA exp(AM>i

(6.3)

(6.4)

To calculate the coefficients Tnfl we use the modal-expansion technique. Thefield in halfspace x < 0 can be written as a series in terms of space (Floquet)harmonics (see eqn. 5.4). The field in the region 0 < x < Ax is presented as aseries in terms of eigenmodes of the plane waveguide with dielectric filling ofpermittivity s = sos1. The amplitudes of forward modes are equal to Tnl, thoseof the backward modes are pni Tnl (pnl being the reflection coefficient of thenth waveguide mode from the set of dielectric slabs lying on the metal planex = AN). The use of the continuity of the tangential field components across x =0 (as in Section 5.3) leads to a system of functional equations connecting theamplitudes of Floquet's harmonics Rp with the coefficients Tnl. Then, per-forming certain transformations, we obtain the system of linear algebraicequations


where

u = kdty/Si) sin 9

The solution sought is to belong to the numerical-sequence space

£ (I + \p\)\Rp\2 < co (6.6)

p= — oo

which provides a correct field behaviour near the edge. The coefficients Tn 1 areexpressed through Rp using the formula38

where v2n = sin u, v2n+1 — j cos u.

Now we solve eqn. 6.5 in the required space of the numerical sequencesdefined by eqn. 6.6 and determine Tnl using the modified residue-calculustechnique (Mittra and Lee, 1971). The latter allows us to reduce eqn. 6.5 to aninfinite system of the second kind with simple matrix elements. As the methodsof obtaining such systems are standard, we shall present only the final equationswithout going much into detail. The unknown coefficients Tn x are expressed by

- £ I-TF)} ' (6'7)

where sn = pn,J{~Vi)/A«> An = d//d«;|w = V l , and f(w) is expressed as inSection 5.1 after the replacements yn -> hntl and rM -• Tn. The coefficients Fn satisfythe infinite system of linear algebraic equations

(6.8)

Our computational algorithm thus has four basic stages:

38 A system of linear algebraic equations for the unknown coefficients Tnl can be obtained, whichallows us to avoid calculation of Rn. However, this system seems less appropriate for the modified-residue-calculus-technique application.


(i) solution of eqn. 6.8 using the truncation method;(ii) calculation of Tnl (n = 1, 2, . . . , JV) according to eqn. 6.7;(iii) calculation of the coefficients to be used in eqn. 6.1 through the recurrent

eqns. 6.2-6.4;(iv) calculation of the linear (per-period) dissipated power P using eqn. 6.1.

The system given by eqn. 6.8 at Ax ^ 0 is of the second kind, which followsfrom the evaluation sm= 0{exp( — mA1 )}.39

Let us consider in more detail the case of a 2-layer medium, when eqns. 6.2-6.4can be reduced to the simple explicit expressions

i e xP ~ 2 V i A i 6-9

[e x p ( — hn i A x ) + pnl exp(hntl A 1 ) ~|

2cosh{An,2(A2-A1)} J

n12 — (e2/6i )1 /2 being the relative refractive index.

The energy losses, as usual, are characterised by the factor

<x(8, rf, A l 5 A2, »12) = P(9, d, A 1 ; A2, B 1 2 ) / / *

where P = lim P.A2-+0

(a) n12 > 1: In Figure 6.2 are shown the dependencies of a on A2\d for theangle of incidence 3 = 60°, n12 — ^/l0 and various values of kd. The interfacebetween the dielectric layers is in the x = Ax plane (Ax = 0.05^). These curves,like those for homogeneous media, are wavy in shape; however, contrary to thecase of homogeneous media (see Section 5.5), a > 1 for all values of A2/d. Thuswhen a substance of larger refractive index is used to fill the grooves, we havea considerable increase in energy losses. This result can be physically explainedby the fact that the dielectric layer with e2 > £i serves as a matching transformerbetween the medium with sx and the corrugated metal surface.

The idea of filling the grooves with substances of larger refractive index lookedpromising for the development of compact low-loss corrugated waveguides andresonators. However, as the calculations showed, this idea proved irrelevant:reduced sizes result in the loss of the desirable effect (decrease in dissipation dueto the surface corrugation).

In Figure 6.3 the dependencies a(9) for incomplete (A1 = A2/2) grooves fillingwith the dielectric (n12 = y/\0, A1/d= 1) are shown for two values of kd. Thesedependencies differ from the corresponding curves for homogeneous media,having a more smooth trend at small and moderate angles (9 < 40° — 60°) anda shift of the zone of sharp decrease in a towards the region of large angles ofincidence.

(b) nl2 < 1: In Figure 6.4 are shown the dependencies of a on A2\d at 3 =60°, n12 = ^, A1 = 0.05d and different values of kd. Figure 6.5 shows the frequencydependencies of a at 9> = 60°, A2 = 4d for two values of n12. We note that these

39 At Al = 0, the truncation method is also applicable. This follows from the fact that the matrixoperator on the left-hand side of eqn. 6.8 belongs to the operator class with a spectrum inside theunit circle (Sirenko, 1983).


a

7

0 1 2 A2/dFigure 6.2 Absorption in a comb-shaped structure as a function of A2/d

H polarisation

1 r = 0.2982 r = 0.59613 r = 0.4967

sets of parameters correspond to the condition of total internal reflection. Thecomb-shaped structure, as it can be seen from the above data, distorts thisphenomenon at certain groove depths. This is indicated by the absorptionresonances: sharp peaks of the curves Qc(A2/d). Similar data for w12

= 1/ /3 areshown in Figure 6.6.

In Figure 6.7 are shown the dependencies a(S) at n12 = i, A2\d — 3, Ax = 0and different values of kd. As for homogeneous media, a(3) is diminishingmonotonically with 8 increasing up to 5 ~ cos"1 (\f\\jk) at any values of s2 andkd. It is necessary, however, to note that the specific trends of the curves a(fl)at different kd vary to a larger extent than for homogeneous media.

The most important result of the above consideration is evidently the effectof the resonance-wave absorption in comb-shaped structures (Figures 6.4 and6.5). It is necessary to stress that this phenomenon results from the joint effectof the periodic structure and media interface. It occurs only at n12 < 1; thoughat n12 > 1 the absorption variations (dependent on the structure depth) can beobserved, the absorption peaks are washed out and the range of variation of ais considerably smaller.

The effect of resonance absorption in the comb-shaped structures is applicablein many cases. It can be used, for example, for the suppression of parasitic modesin waveguides and resonators.

The case of negative resistance Re ^ 5 < 0 is of special interest, as it corresponds


a

0 JO 60d} degrees

Figure 6.3 Absorption in comb-shaped structure with dielectric layer against angle ofincidence

H polarisation

1 T = 0.2982 T = 0.5961

to amplification of the incident wave by an active (for example, semiconducting)film. When we have a corrugated active surface the effect of resonance amplifi-cation occurs. Then, unlike the case of a smooth surface, the amplification factorat the working frequency increases greatly and the out-of-band radiation issuppressed essentially.

6.2 Wave diffraction by comb-shaped structures ingyrotropic media

Let us consider the following problem. The extraordinary plane wave

H°y (*, z) = exp{ -jke {x cos 9 - z sin 3)} (H°x = H° = 0)

is incident on a comb-shaped structure placed in a gyrotropic medium with thetensor

0

7^2

(6.11)


a

7

6

0 4 5 A2/dFigure 6.4 Absorption in comb-shaped structure placed near dielectric halfspace as a

function of A2/d

H polarisation

12

1 = 2.5132i = 1.8849

Here ke = 2n{ (ef — z\) js1 }1/2 /A, X being the wavelength in free space (Figure 6.8).

Suppose that the dielectric medium is lossless (el5 e2, S3 real), and the thicknessof the ridges A is small compared with A, d, and D, though it exceeds considerablythe field-penetration depth into the metal (in this case ridges can be consideredas infinitely thin).

The linear (per-period) dissipated power P for a structure of a highly conduc-tive material can be calculated using the perturbation technique (see Sec-tion 5.4). In the case considered there is a nonreciprocal energetic effect:P(#) T^P( —9) (note that for reflection from a smooth surface this effect doesnot exist; there is only a nonreciprocal phase shift). This effect can be character-ised by a factor of nonreciprocity

p1(»)=P(&)IP(-9) (6.12)


a

7

5

J

1

-

-

-

2

0 0.2 0.6 1.0

Figure 6.5 Resonance absorption of H-polarised plane wave in comb-shaped structureplaced near dielectric halfspace

5 = 60°, 62= 1, A2/rf=41 «12 = 0.3872 «12 = 0.331

a

1.5

0.50.5 1.5 2.5

Figure 6.6 Absorption in comb-shaped structure placed near dielectric halfspace againstA2 Id

1 7\/ei = 1.88492 TyJ&i = 0.9424


20 60 909, degrees

Figure 6.7 Absorption in comb-shaped structure placed near dielectric half space againstangle of incidence

£i = 0.3167i = 0.6333

3 r7£i=0.95= 1.2667

, = 1.5833i = 2.21671 = 2.85

i = 3

Let us suppose that ked< n, i.e. only the fundamental mode can propagate inthe grooves. In this case

where A (9) is the amplitude of the forward fundamental mode in the grooves.The field-formation process inside the grooves can be presented as a chain of

successive diffractions on structure ridges and reflections from bottoms. There-fore, the higher evanescent modes in this chain can be ignored since for largeD (which we consider as such), coming from the point of their excitation, theydo not in practice reach the other discontinuity.40 In this case A(Q), according

40 However, the higher modes should be taken into consideration when determining the reflectionand transmission coefficients.


aFigure 6.8 Periodic structures in gyrotropic medium

a Reflecting gratingb Transmission grating

to the generalised scattering-matrix technique (Mittra and Lee, 1971), isexpressed by

+ s0Rexp(2jyQD) + s20R

2 exp(4jy0D) + •••}

- so(9)R exp(2jy0D)r1 (6.14)

where to(&) and so(S) are the transmission and reflection coefficients of thefundamental mode for the array of semi-infinite parallel-plate waveguides; /0

corresponds to the case of plane-wave excitation, s0 corresponds to the periodicexcitation by fundamental modes (phase change per unit cell equals ked sin 9),R is the reflection coefficient of the fundamental mode from the bottom of thegroove,41 and y0 = 2n(y/e1)/l is a fundmental wave-phase constant.

Taking into account eqn. 6.14 and also the fact that

(6.15)

where r= \R\ and (f) and (j)s are phases of the coefficients R and 0 respectively.The values \so\ and (j)s are determined exactly using the Wiener-Hopf technique

41 We reserve the traditionally used names for R and sQ(S) (see Wu, 1967), though physically theterm 'transformation coefficient' is more correct as, in the presence of gyrotropy, the transverse-field distributions of forward and backward modes are essentially different.

we can transform eqn. 6.13 as


(Wu, 1967); an approximate expression for R was obtained by Popov (1981):

r ~ e x p { - 2nde2

The nonreciprocal effect considered is strongest when for the angle of incidence9 the resonance condition cos{2y0^ + 0s($)} — 1 *s satisfied, and for the angleof incidence — 9 the accuracy of meeting the antiresonance condition

+ ^s (~~ $)} — ~ 1 is maximised. Then

where A(j)s{9) = {(j)s(9) + <£5(-3)}/2; according to Wu (1967) we haveP _ (P2 _ p 2 \ l / 2 __„ Qo^ \^1 " 2 / LUa i7

A(/>s (S) = - 2 tan~x [tan (*e</ sin S/2) coth{27T</e2/Jî)}] (6.18)

6.3 Nonreciprocal resonance effects

In Figure 6.9 the dependencies /3™ax (9) are shown for different values of ked,calculated by means of eqns. 6.16-6.18. Analysis of these curves shows that atsufficiently large angles of incidence (3 > 70-80°) a significant nonreciprocaleffect occurs. It is important to stress that nonreciprocity, because of a specificresonance character of the effect, can be very strong even at a rather lowmedium gyrotropy (in our calculations v = £2/£i = 0.1). The effect is reducedwhen the cutoff condition ked— n is approached.

As for the physical meaning of this effect, there is a nonreciprocal phaserotation when the wave is reflected from the periodic structure ridges. It istherefore possible to fulfil both the resonance condition at the angle of incidence9, and, at the same time, the antiresonance condition at the angle of incidence— 3. This is not enough, however. The phase conditions become essential onlywhen the large-amplitude partial waves interfere. This is why the effect describedtakes place only at large values of 9 when the reflection from ridges is substantial.

It would be of interest to study the trend of the dependencies /3™ax (9) in thedomain of large angles of incidence (9 > 80°), absent in Figure 6.9. Analysis ofeqns. 6.16-6.18 shows that at 3-»90° fi™ax increases monotonously tending toa certain finite value. At the condition 2nde2l^s1 < tan(A;e*//2), which in thiscase is fulfilled, it is not difficult to obtain the following:

lim Pâx(9) = 1 + 16 cot2{kedl2) (6.19)d ~*7r/2

The magnitude of the limit in eqn. 6.19 decreases when ked increases.Proceeding from a certain analogy between interference mechanisms of wave

formation in the grooves and wave interference in layered media we can alsoexpect similar nonreciprocal effects to occur in layered media. To analyse themconsider a plane-wave reflection from a metal plane placed under the interfacebetween two media (Figure 6.10), one of which for simplicity is assumed to beisotropic. This will result in two cases of the problem (Figure 6.10a and b).

There are exact solutions of the wave-propagation problems in plane-layered


40 606, degrees

60

Figure 6.9 Peak nonreciprocity coefficient for comb-shaped structure placed near dielectrichalf space against angle of incidence

6 ! = 1 0 , £ 2 = 1

1 ked=0.05n2 ked=0.2n3 ked=0.5n4 ked=0.Sn

media. The latter are normally obtained by means of fields matching at theinterfaces. We prefer, however, to use a method of successive reflections as thisform of solving the problem is more convenient to compare with the resultsobtained for the comb-shaped structures. This is also a rigorous solution, becausein layered media there are no higher modes. It is equivalent to the Jones calculuswidely used in optics (Yariv and Yeh, 1983).

To begin we consider the structure shown in Figure 6.10<2. The tensor ofpermittivity IT for a gyrotropic medium is defined by eqn. 6.11. As in the previousSection, we obtain

omaxPi — ( i - k l ) 2 (6.20)


77777777a 6

Figure 6.10 Layered structures with gyrotropic media

a With gyrotropic halfplaneb With gyrotropic slab

where s0 = \so\ exp(j(j)s) is the reflection coefficient of the plane wave, comingfrom an isotropic medium with permittivity e. Eqn. 6.20 corresponds to theisotropic-layer 'resonance' thickness; the resonance condition has the form

COS{4TLD(1 - sin2 % 2 ) 1 / 2 /A + &} = 1 (6.21)

where p = {elel(el ~ e|)}1 /2. For so(9), the following expression is obtained:

+jv/> sin fl

+7v/> sin 3 ( 6 > 2 2 )

-/> cos

cos , cos

where v — e2 /î and cos 0 = (1 - sin2 &/p2)1/2.For the structure shown in Figure 6A0b eqn. 6.20 is also valid, if so(9) is

expressed by the formula

p cos (j> + jvp sin (j) — cos i

p cos (j) ~ jvp sin (J) + cos i(6.23)

where cos (j) — (1 — p2 sin2 3)1/2, and thereby the resonance condition given byeqn. 6.21 will be slightly changed.

In Figure 6.11 the dependencies /?™a*(#) for the structure shown inFigure 6.10<2 are presented for different values of the parameter p. First notethat the nonreciprocal effect here is much lower than in periodic structures(with curve 1 as an exception, corresponding to a very high gyrotropy parameterp of 10). In addition, these dependencies have quite a different character fromthose for periodic structures: they have an extremum. When 9 tends to 90°,from eqns. 6.20 and 6.22 there follows

lim pmax( = l+4/>2v2


1.060 d, degrees

Figure 6,11 Peak nonreciprocity coefficient of layered structure against angle of incidence

Thus, no obvious analogy is observed between nonreciprocal effects in comb-shaped structures and layered media, unlike the resonances in the case ofisotropic media. This means that comb-shaped structures cannot be looked uponas layers of artificial dielectrics. This results from the difference between thedependencies of the nonreciprocal phase component on the angle of incidencefor periodic and plane-layered structures. Optimal phase and amplitude con-ditions in the latter (intensive partial-wave reflection from the interface betweenisotropic and gyrotropic media) cannot be achieved simultaneously—they corre-spond to different angles of incidence.

The dependencies of the nonreciprocity coefficient on the gyrotropy parameterv = £2/ei f°r corrugated (solid line) and layered (broken line) structures arepresented in Figure 6.12. The trends of these curves are different: for comb-shaped structures the curve j?(v) has an extremum, whereas for layered structuresit is monotonic. This underlines once again the difference between the nonreci-procity mechanisms in the two cases. For corrugated surfaces the parameter j?reaches values of the order of 10 with a much lower gyrotropy than for layeredstructures.


/J.O

9.0

5.0

10

0.1 0.3 0.5 0.7

Figure 6.12 Peak nonreciprocity coefficient as a function of £2/ei

comb-shaped structure (e1 = 10, kd= 0.21)layered structure (p =

For the structure shown in Figure 6.10b, nonreciprocal effects are very weak(as follows from eqns. 6.21 and 6.23); at v<0.1 they can hardly be observed.

Consider now the reflection of a plane wave from a smooth finitely conductingsurface in a homogeneous gyrotropic medium. The simplest way to obtain anexpression for the reflection coefficient lies in the replacements (/)—•$, 9—^(j),p =

l/Zsw m ecln- 6.23. Here Zs 1S t n e surface impedance of the metal,w = {(ei ~~ £2)/eiMo)1/2 a n d Mo is the free space permeability (all media arenonmagnetic). We therefore have

cos S + jv sin $ — Zsw

cos 5 — j v sin 3 + Zs w (6.24)

As Zsw ^ 1? w e c a n reduce eqn. 6.24 to a more simple form neglecting the termsabove the first order of infinitesimals, which corresponds to the ordinary pertur-bation technique. Therefore

\R0 (8) |2 - 1 - 4Zsw cos 9 (6.25)

Then, to an accuracy of the terms of the order of Zsw> t n e nonreciprocalenergetic effect is absent (only nonreciprocal phase rotation takes place).

Resonance nonreciprocity is also observed for 'reflection—transmission' reson-ances in transmission periodic structures. The simplest model for this case isshown in Figure 6.8£: the extraordinary plane wave falls at angle 3 on a periodicstructure of vertical metal strips. Under certain conditions the transmission


coefficient T may depend greatly on the path in which the incident wave arrives(at identical absolute value of 9). The nonreciprocity factor in this case can bedefined as

Calculation of j83 (9) is completely identical to that in Section 6.2; the resultswere presented by Slepyan (1983). The effect has its maximum when theresonance condition is satisfied, i.e. when yeD + (/>s(9) ~ n. For example, at ex =10, e2 = 1, 9 = 80°, ked= 0.2 the coefficient âx is of the order 5.

Finally, we note that the nonreciprocity of 'reflection-transmission' effects issomewhat lower than that of energy losses in comb-shaped structures at thesame parameters of the medium, angles of incidence and structure periods.

The effects described above are promising for the design of nonreciprocaldevices in microwave and millimetre-wave bands. They can also be used as abasis for developing equipment for measuring gyrotropic media parameters.

Chapter 7

Eigenmodes in corrugated waveguidesand resonators with finitely conducting

walls

This Chapter deals with the electromagnetic fields in systems with corrugatedsurfaces: waveguides, resonators and horns. Special attention is given to theinvestigation of dissipation characteristics: attenuation coefficients of differentmodes in corrugated waveguides and (^-factors of eigenoscillations in resonatorswith corrugated walls. Application of corrugated surfaces of both step shape andsmooth is considered. When a period of the structure is small compared withthe wavelength, a method of equivalent impedance-boundary conditions is used(in Section 7.2 a generalisation of this method for finite conductivity is pre-sented). Low-loss corrugated waveguides (Sections 7.6-7.8) and millimetre-waveband high-quality resonators with a rare spectrum of eigenoscillations(Section 7.9) based on the effect of abnormally small absorption in periodicstructures are described. In Section 7.10 a rigorous method of calculation ofelectrodynamic characteristics of corrugated horns, which are highly effectivefeeds for microwave antennae, is considered.

7.1 Eigenmodes in periodic structures

Let us first consider the properties of eigenmodes in periodic structures of arather general type. We define a periodic structure as an area V, internal orexternal in relation to a conducting surface 2, given by the equation F(x,y, z) =

0, where F satisfies the condition of periodicity F(x,y, z) — F(x,y, z + d). Anothervariant of a periodical structure is an inhomogeneous medium with periodicallyvarying permittivity and permeability

e(x,y, z) = e(x, y, Z + d) //(*, j , z) = v{x,y, z + d)

Finally, a combination of both cases is possible: a periodically inhomogeneousmedium inside the volume V.

Eigenmodes of periodic structures are nontrivial solutions of Maxwell's equa-

Eigenmodes in corrugated waveguides and resonators with finitely conducting walls 197

tions satisfying Floquet's condition of pseudoperiodicity:

[ E ( x , y , z + d ) ) ,. . f£ (* ,y , .

where y is the phase change coefficient.Eqn. 7.1 corresponds to an intuitive idea of waves in periodic structures: from

one unit cell to another fields differ only in phase factor. The phase change perunit cell for the periodic structure illuminated by a plane wave is defined bythe angle of incidence $, i.e. y = k sin #, (see Figures 5.1 and 5.12). For eigen-modes y can be regarded as a spectral parameter.

Using eqn. 7.1, we can represent the eigenmode sought as a space-harmonicexpansion

where the functions em(x,y) and hm(x,y) are determined by substituting eqn, 7.2into Maxwell's equations and boundary conditions. Thus we obtain a character-istic equation for y.

It is noteworthy that each of the space harmonics is not an actually existingwave. Only eigenmodes containing an infinite set of space harmonics are aphysical reality. The fact that under certain conditions one of the space harmon-ics can prevail over others (for example, at d < X only zero-order harmonics areessential) is not contradictory to this statement.

The conditions of pseudoperiodicity given by eqn. 7.1 are, to a certain extent,taken a priori and need to be rigorously substantiated. First, do such solutionsof Maxwell's equations exist at all? Secondly, do all possible solutions satisfyeqn. 7.1? In the 1-dimensional case the answers to these questions are givenby the theory of ordinary differential equations with periodical coefficients(Floquet-Bloch theorem). For distributed systems of general type there is norigorous proof of this theorem analogue.

As an answer to the first question we can consider, for example, pseudoperiod-ical solutions for specific periodic structures. As for the second question, whensolving a dispersion equation, imaginary values of y can be obtained (we areconsidering now systems without losses!). Finally, in the work by Beliantsev andGaponov (1964) complex modes in periodic structures were found. They arecharacterised by a complex y even for lossless systems. For such modes thecoefficient yd cannot be interpreted as a phase change per unit cell. It can beshown that waves with imaginary y do not carry energy along the z axis; in thisrespect they are similar to evanescent modes of ordinary smooth waveguides,42

Such solutions correspond to stop bands of periodic structures (there is an infinitenumber of alternating transmission and stop bands). Complex waves are alwaysexcited in pairs and form reactive fields (i.e. energy is not carried along the zaxis), the spatial distribution of which along the z axis is not periodical.

A formal solution of dispersion equations for systems with dissipation leads toa complex y (y = y' +jy"). In this case it follows from eqn. 7.1 that the field

42

axisUnlike the case of evanescent modes in smooth waveguides, in this case fields vary along the zis as / ( s ) e x p ( - \y\z), where f(z) is a periodic function with period d.

198 Eigenmodes in corrugated waveguides and resonators with finitely conducting walls

diminishes with exp( — y"z)> A question arises: to what extent is this law generaland do nonexponential solutions exist? To answer these we shall not impose theconditions of pseudoperiodicity a priori. In this case the periodical structure willbe considered as an irregular waveguide, and the theory of such waveguides willhave to be applied (Schelkunoff, 1955, Katsenelenbaum, 1961). Now we considerthis approach for lossless structures.

The fields sought are written as the expansions

E(x,y,z) =(7.3)

where {ep(x, y)}, {hp(x,y)} are systems of vector-functions, complete in an arbi-trary cross-section of the waveguide, and cp (z) and bp (<;) are the coefficientssought. Substituting eqn. 7.3 into Maxwell's equations and boundary conditions,we obtain a system of ordinary differential equations for the functions bp(z) andcp(z) which can be reduced to the normal Hamilton form (a specific exampleis given in Section 7.6):

where x is an unknown vector and H(z) is a periodical matrix Hamiltonian,i.e. H(z) = H(z + d). Eqns. 7.4 have a form similar to Hill's system and theapplication of general methods of Hill's theory (Yakubovich and Starzhinskii,1975) to eqns. 7.4 would be promising. However as the system given by eqns.7.4 is infinite-dimensional, the transition to a finite-dimensional analogue shouldbe substantiated in detail for every specific case. For example, in the paper bySveshnikov (1963) there is an appropriate proof for circular irregular wave-guides. In the papers by Koroza (1970) and Shankin (1974) the possibility ofthe transition to a finite-dimensional system is postulated with a subsequent useof Hill's theory. It is shown that the solution of eqns. 7.4 satisfies the relationship

where A is called the monodromy matrix.43 Eigenvalues pk of the matrix exp(A)are called multipliers. They can be expressed by

where yk is the propagation coefficient of the kth eigenmode.If matrix A satisfies the conditions of theorems 3.1 and 3.2 (Section 3.3), it

43 For this method to be used, the monodromy matrix should exist. For some complex periodicalstructures the monodromy matrix may not exist at all.


can be reduced to a diagonal form (all multipliers in this case are either simple,or D-multiple). Thus, we come to eqns. 7.1 and 7.2. If some of the multipliersare of J-multiplicity, then the monodromy matrix can be reduced to a normalJordan form (see Section 3.4). In this case, as shown by Shankin (1974), apartfrom the eigenmodes there are associated modes not satisfying the conditions ofpseudoperiodicity. Such modes can not be presented in the form of eqn. 7.2.This can be looked upon as an example of a nonexponential solution of Maxwell'sequations for periodic structures. If pk = exp(jykd) is a multiplier, thenpk = exp(— jykd) is also a multiplier. The numbers pk correspond to forwardmodes and pk to backward modes.

In periodic structures without losses the multipliers are usually located on thecircle of unit radius or on the real axis. The first case corresponds to a trans-mission band, and the second to a stop band; band-edge frequencies satisfy theequations

exp{jyk((o01 )d} = +1 (7.5)

The general theory of wave transformation developed by Krasnushkin andFedorov (1972) is applicable to periodic structures. As shown by Krasnushkin(1974) and Shankin (1974), the frequencies co0tl determined from eqn. 7.5 arepoints of J-multiplicity. At these points transformation of two propagating modesinto two evanescent ones (and vice versa) occurs and associated modes emerge.We note that the points of J-multiplicity do not necessarily correspond to thevalues pk — + 1. They can also be located at an arbitrary point on the unit circlein the plane of the complex variable p. This occurs when one of the degeneratedmodes is characterised by normal dispersion, and the other by anomalousdispersion, which gives rise to complex waves (Krasnushkin, 1974).

Considering the eigenmodes in a periodic structure with losses in the walls,we have to solve Maxwell's homogeneous equations with impedance-boundaryconditions. We assume that eigenmodes of the same structure but without lossesare known (let us denote them by En,Hn). The method, described below, isbased on the expansion of eigenmodes in the system with losses in terms ofeigenmodes of the same lossless system and is a generalisation of the techniqueused in Section 3.3 for periodic structures. Using the Lorentz lemma we easilyobtain the following relationship for En and Hn (Weinstein, 1988):

(EnxHn,-En.xHn)izdSx = O

where hn is a propogation coefficient of the nih mode in the system withoutlosses.

From the latter it follows that either hn + hn> = nmjd (m integer), or the integralover Sj_ equals zero. The first condition at m 0 corresponds to the band-edgefrequencies. In the case of m = 0 we come to a condition n = ~n. Thus, weobtain

(En x Hn - En, x Hn)iz dS± = dnt __.7Vn

This is known as the orthogonal property of eigenmodes. From the Lorentzlemma in differential form (see eqn. 3.20) it follows that in lossless structures Nn

does not depend on z-


Now we assume that systems {£„}, {Hn} are complete in an arbitrary cross-section î(^).4 4 Then eigenmodes of the system with losses can be presented (asin Section 3.3) by

where Y±n(z) are functions to be determined.To determine T±n(z) we apply eqn. 3.20, substituting the expansion given by

eqn. 7.6 as Ex, Jând E_p, Hp as E2, H2 (JlU2 = 0, Zsi = Zs, Zsi = 0) • After

some transformations we obtain the following system of differential equationsfor T±n(z):

dT Z fE M £ r()& {(noxHq)xH-p}nodc (7 .7 )

where C{z) is the perimeter of the cross-section S±(z).Replacing the unknown functions Tn(z) by bn(z) £xp{~jhnz) m eqn. 7.7, we

have

where Bqp (z) are periodical functions with period d, expressed by the formula

Bqp = (b exp0"(AP ~ hq)z}HqzH_px dc

y) dc (7.9)

C(z)

Let us consider an approximate solution of eqn. 7.8 on the following assumptions:

(i) there are no degenerated modes among the eigenmodes En,Hn (excepttrivial multiplicity cases);

(ii) frequencies considered are distanced from the band-edge ones;(iii) surface impedance of the periodic structure material is small (\Zs\l

In agreement with the assumptions mentioned above we can reduce thesystem described by eqn. 7.8, as in Section 3.3, to the equation of the first order;the latter has the form

Taking into account the periodicity ofBpp(z) expressed by eqn. 7.9, the solution

44 Our case is confined to areas internal as related to the surface S. In other cases, the theory ofopen waveguides should be used (Weinstein, 1969a), which takes into account the functions ofcontinuous spectrum.


of eqn. 7.10, in accordance with the Floquet-Bloch theorem, can be written as

) twhere cpr are constant factors.

By direct substitution of eqn. 7.11 into eqn. 7.10 we can show that, withinthe framework of the perturbation theory (£s small), the eigenvalue yp isexpressed by

7 Cd

yP = hp+j—\ Bpp(Q d£, (7.12)

J^P Jo

For modes in a transmission band45 we can transform eqn. 7.12 (as inSection 3.7) into the formulae

y'p = hp- — ^ (7.13)

Id Re {EpxH*)iz&SLJs±

Re^s f \Hpx\2dS

y;= — ^ (7.14)

U Re (EpxH*)iz&SL

where Zd is the metallised surface of a unit cell of the structure.Eqn. 7.14 coincides with that obtained by application of the energy-pertur-

bation technique for periodic structures. It is similar to eqn. 3.62 for regularwaveguides. Eqn. 7.14 is also valid when areas external to the surface Z areconsidered, though its derivation becomes more complicated. For E with edges,when the induced current density has singularities, eqn. 7.14 should be modifiedin a manner identical to that for regular waveguides. The use of eqns. 7.13 and7.14 is relevant when the integrals in the numerator and denominator have aform convenient for calculation. Otherwise, a direct solution of the originalproblem with impedance-boundary conditions would be recommended (seeSection 3.7).

7.2 Equivalent boundary conditions for finitely conductingcomb-shaped structures

The equivalent-boundary-condition technique is widely used for the analysis ofelectromagnetic fields in periodic structures when the period of the latter ismuch less than the wavelength. This method consists of the imaginary replace-ment of the periodical structure by a smooth (plane) surface where special

45 Note that in the presence of absorption the boundaries of transmission and stop bands becomeslightly washed out in a similar way to cutoff frequencies of waveguides (see Section 3.7).


boundary conditions are imposed. The conditions are introduced in such amanner that the fields at large distances from the original periodic structureand the corresponding smooth surface would coincide. In this Section generalis-ation of this method for structures with finite conductivity will be presented.The material is based upon the work by Slepyan (1981).

Let a comb-shaped structure (Figure b.\b) be illuminated by an electricallypolarised plane wave of unit amplitude under the incidence $. It is easy to seethat the field in the far zone will be identical to that reflected from the smoothplane surface x = 0, where the tangential-field components Ez and Hy satisfy theboundary condition

(E:-ZeHy)\x = o = 0 (7 .15)

The impedance £e is expressed by

where F and y are the magnitude and phase of the reflection coefficient Re(S),respectively.

In actual cases losses are rather small (F is close to 1) and y differs slightlyfrom y0, the reflection-coefficient phase corresponding to the comb-shaped struc-ture with perfect conductivity. Then eqn. 7.16 can be simplified in the followingmanner:46

JW0 cos 9 tan ^ (7.17)1 + cos y0 2

where Zs *s t n e surface impedance of the structure material, and a (9) = PjP[P being the power dissipated per unit cell of the structure, P = lim P(9)].

The formula for a(9) is obtained in Section 5.6; here we will write it in aform more convenient for further calculations:

cos2,9{M8>

where a = Q(kd)/2, the function Q(a) being defined by eqn. 5.69.The reflection coefficient Re(S)at ^ s = 0 is expressed by

p-so

where

(7.19)

so = exp(2jkD)

p = cot2 (912)

46 This simplification is correct when the condition tan{y(l - F2)/2} < 1 is fulfilled. This conditionis not satisfied in a certain narrow neighbourhood of the point y0 = n, where more accurate analysisis needed.


and f(w) is introduced in Section 5.1. After some algebraic manipulations wecan obtain, from eqn. 7.19,

y ~ 2 tan" l [tan{Q(fo/)/2}/cos 9] - £l{kd cos 9) + 2kD cos 9 (7.20)

When the structure period is small compared with the wavelength (kd <^ 1)

Q(kd)^2kD (7.21)

Then, we rewrite eqn. 7.17 in the form

An important special feature of eqn. 7.22 is that the impedance £e does notdepend on the angle of incidence 9. This allows us to consider the boundarycondition given by eqn. 7.15, originally introduced for a plane wave, indepen-dent of the field spatial structure, and to apply it for an arbitrary electricallypolarised field. This is only correct when the replacement according to eqn. 7.21is valid. As numerical calculations show (see also Kalhor, 1977), eqn. 7.22 canbe used when the value of kd does not exceed nj2.

Eqns. 7.18 and 7.20 are obtained for plane waves if/0 (x, z) = exp{j(kxx + kzz)},where kx and kz are real (kx = k cos 9, kz = k sin 9); meanwhile an arbitrary fieldcan also contain partial surface waves formally corresponding to imaginary valuesof cos 9. For such waves, generally speaking, there is no reason to assert that ata moderate kd the impedance ^ does not depend on 9. Nevertheless, in the theoryof spatial normal modes in corrugated waveguides (Sections 7.4 and 7.5) we willuse the boundary condition given by eqns. 7.15 and 7.22 with the proviso thatthe contribution of partial surface harmonics into spatial normal waves is negligiblysmall. When analysing surface normal waves we will assume that kd <^\.

For structures without losses (Re^ s = 0), eqn. 7.22 is reduced to the known(see, for example, Weinstein, 1969£) formula £e ~ — jW0 tan (kD) which iscentral to the elementary theory of perfectly conducting comb-shaped structures.

For high values of kd there is always a dependency of /^ on 9 which does notallow us to introduce impedance boundary conditions similar to eqn. 7.15.

For magnetically polarised fields the equivalent boundary conditions can alsobe obtained if the solution of a corresponding key problem for a plane wave hasbeen found. The simple formulae are obtained only for the case of normalincidence (9 = 0) and on condition that the structure is sufficiently deep (seeSection 5.2). However, if kd <^ 1, the impedance should not depend on the angleof incidence and can be determined from the solution of the key problem at9 = 0. Then

_ l + * ( 0 ) _ l r + 2jTsinyZ - x_Rh{0) Wo - 1 + r 2 _ 2 r c o s y ô (7.23)

where Rh(0) is the reflection coefficient at 9 = 0, T and y are the magnitudeand phase of Rh(0), respectively. To simplify eqn. 7.23 we can use the followingfacts: first, the ohmic losses are very small (F is close to 1); secondly, as d<^ A0

(A0 being the skin depth), y ^ y 0 , (where y0 is the phase of the reflectioncoefficient corresponding to perfectly conducting structure). According to


Weinstein (1969£), at kd4, 1 we have

2kd\n2y0 ~ + n

n

Then for ^ we obtain the approximate expression

j< " n \2kd\n2

where v(k) is defined by eqn. 5.35.Thus, for an arbitrary magnetically polarised field the boundary condition is

(Ey + Z"Hz)\x = o = 0 (7.25)

Combining eqns. 7.15 and 7.25 we obtain an anisotropic impedance-boundarycondition for an arbitrary polarised field

Jtx x E)\x = o = I fe x (4 x H)}\x = 0 (7.26)where % is an impedance tensor having the form

7]

There can be a different approach to the problem of calculating £e'h in eqn. 7.26.It is based on the conformal-mapping method. Depine and Brudny (1990) havestudied the problem of wave diffraction by periodic surface x = g(z) [g{z + d) =g(z)], where the Leontovich impedance-boundary condition is imposed. Theyused the conformal mapping relating w = z + jx with U= X + jY in such a waythat the region x>g(z) was transformed into the upper halfplane Y>0. Thenintroducing the function

(Hy{x(X, Y),z(X, Y)}, electric polarisationF(X, Y) = \

lEy{x(X, Y), z(X, Y)}, magnetic polarisation

they wrote the boundary condition on the surface x = g(z) as

^F~ (7'27)

for magnetic polarisation and

, (7.28,

for electric polarisation. The function s(X) is expressed by

dw

Eqns. 7.27 and 7.28 are the boundary conditions of the third kind with periodiccoefficients which are to be fulfilled at Y = 0.

These results, as we shall illustrate later, can be used for the calculation ofZe"h- If the depth and period of the structure are small compared with thewavelength, the field in the grooves is quasistatic. Then, through averagingeqns. 7.27 and 7.28 over the period, we obtain the following formulae for real


parts of £«•*:

s{x)dx (7.29)o

, , _ , , (7-30)

The calculation of the reactive components of 2ê'h is carried out in the frameworkof the model of perfect conductivity. This problem has been considered by manyauthors (see, for example, Nefedov and Sivov, 1977), so we do not touch uponit here. It is worth noting that this averaging is in fact equivalent to the use ofthe two first terms in the perturbation series in £ s . Eqns. 7.29 and 7.30, unlikeeqns. 7.22 and 7.24, are applicable for structures of an arbitrary shape and notvery deep (D <^ X).

Let us consider the following examples. For the rectangular-groove gratingshown in Figure b.\b the conformal mapping function is expressed in the form(Lavrentyev and Shabat, 1987)

TTl ^ d - l rC0s(7T^/^)~|U{w) = - cos I ^,_^tJ, I (7.31)

Using eqns. 7.29 and 7.31, we obtain

which at kD < 1 coincides with eqn. 7.22. The integral in eqn. 7.30 diverges ifthe function U(w) is given by eqn. 7.31. This confirms once again the inapplica-bility of the infinitely thin-surface model (when used directly without modifi-cations) for calculating the power loss in the case of magnetic polarisation.

Depine and Brudny (1990) considered the cycloid-shaped periodic surfacedescribed in a parametric form by the equations

( 7 - 3 2 )

where D is the groove depth.The conformal-mapping function in this case, according to the work by

Depine and Brudny (1990), has the form

(7.33)

Substituting eqn. 7.33 into eqns. 7.29 and 7.30 we can obtain the followingexpressions for Re £e'h:

(7.35)


where p = nD/d, p1 = (3 + p2)112 and K(q) and E(q) are the complete ellipticintegrals of the first and second kinds, respectively.

From eqns. 7.34 and 7.35 it follows that at />->l Re ^ e - > 4 Re Zsln a n dR e ^ - » o o . However, it is necessary to point out that at p ~ 1 eqn. 7.35 isinvalid, since for p = 1 eqn. 7.32 defines a surface with infinitely thin ridges.

Though the boundary condition given by eqn. 7.26 has been obtained for aplane infinitely wide comb-shaped structure, it can also be used for curvilinearcomb surfaces, if the main curvature radii significantly exceed the period anddepth of the structure, which can then be considered as locally plane. Thisboundary condition can be also used for finite corrugated surfaces if auxiliaryconditions, describing the field behaviour at the ends of the structure, areintroduced. These conditions should be formulated so that fictitious field sourceswould not emerge on edges.

The situation described is rather typical when using equivalent boundaryconditions of different types: these conditions are derived rigorously for a certainkey problem, simplified to the maximum. Then, on the basis of physical senseand intuition, we extrapolate the equivalent boundary conditions to a certainclass of much more complicated problems to fundamentally simplify their math-ematical formulation. In this case such small-scale details of the field structureare not considered. However, we must underline that they do not, to any notabledegree, affect the field-integral characteristics (e.g. phase and attenuationcoefficients etc.).

In our further considerations, we shall describe a simple analytical theory ofeigenmodes in corrugated waveguides and resonators, based on approximateboundary conditions developed.

7.3 Surface waves in finitely conducting comb-shapedstructures

A metal comb-shaped structure (Figure b.\b) is the simplest guiding systemcapable of supporting a surface wave. Dispersion properties of such slow-wavestructures in an approximation of perfect conductivity have been examined quiteextensively; the most rigorous theory for this case had been developed byWeinstein (1969^). It is of interest to develop such a theory which would allowus to take into consideration energy losses in the metal.

Let us consider a 2-dimensional (infinite in the direction of they axis) comb-shaped structure, the ridges of which are considered infinitely thin. Suppose alsothat kd<^\. For our purposes only electrically polarised fields are of interest:there are no surface modes of opposite polarisation. The field components areexpressed by

E jwe0 8z

COEO OX

(7.36)


From eqns. 7.15 and 7.36 we can obtain the following boundary condition forthe function ij/(x, z)'-

ze(7.37)

The surface-wave field can be written as

. K) (7-38)where y = y' + jy" = (k2 + #2)1 /2, X being an unknown transverse complex wavenumber ( lim / 2 < 0 ) . Substituting eqn. 7.38 into eqn. 7.37 we come to the

formula

(7.39)

Taking into consideration eqn. 7.22, we have

s'm2kD\

cos (kD) I d \ 2kD J

It is easy to determine the surface-wave attenuation coefficient y" from eqn. 7.39.For convenience, let us introduce the dimensionless slowing factor jV= vphjc (vph

being the surface-wave phase velocity). According to the elementary theory byWeinstein (1969a), in an approximation of perfect conductivity JV= cos kD. Thismakes it possible to connect the attenuation coefficient y" directly with theslowing factor jV. Neglecting the terms above the first order of Re £s, we obtain(Slepyan, 1981)

Zs j^{l+yd(pN+cos''N)} (7.40)

where/?- (1 - jV 2 ) 1 / 2 .Using eqn. 7.40, we can show that the surface-wave attenuation grows with

the increase of retardation. This can be explained by the fact that while JV isincreasing the wave is 'pressed' to a large degree to the surface, and this leadsto an increase in induced currents. On approaching the lower boundary of thetransmission band (jV"-»l) losses in walls, according to eqn. 7.40, disappear(y"->0), This is a result of the Rayleigh diffractional anomaly dealt with inSection 5.5.

While approaching the upper boundary of the transmission band (JV—>0) thelosses, as follows from eqn. 7.40, increase infinitely. However, in a certain narrowneighbourhood of the band-edge frequency, corresponding to the conditionJV= 0, the perturbation technique, and eqn. 7.40 based on it, are inapplicable.More accurate analysis shows that at .;Y->0 y" increases sharply while remainingfinite.

7.4 TM modes in plane comb-shaped waveguides

Let us examine propagation and attenuation properties of TM modes in 2-dimensional corrugated waveguides (Figure 7.1). The theory described is basedon equivalent impedance boundary conditions obtained in Section 7.2. The


assumptions made are similar to those in Sections 3.4 and 7.2. The mathematicalformulation of the problem is to find a solution of the 2-dimensional Helmholtzequation satisfying the boundary condition given by eqn. 7.37 at x = +A.

We present the eigenmode as

(7.41)

where x ~ X + jx" is the unknown transverse wave number and y = (k2 — x2)1/2,the upper line in eqn. 7.41 corresponding to the modes symmetrical in x, andthe lower line to modes antisymmetrical in x. The field components are deter-mined from eqn. 7.36.

Substituting eqn. 7.41 into eqn. 7.37, we obtain the following dispersionequations for X-

J(7.42)

IXIXITIXla.

Figure 7.1 Geometry of plane comb-shaped waveguide

a Original structureb Impedance model (arrows indicate the formation of an eigenmode

according to Brillouin's concept)


where £' + j£" = jkA£ejW0. The roots of eqn. 7.42 located near the real axiscorrespond to fast (spatial) modes, the roots located near the imaginary axis toslow (surface) ones.

Eqn. 7.42 for fast modes has a concrete physical sense. It corresponds to theBrillouin concept according to which fast eigenmodes are formed as a result ofconsecutive reflections of two plane partial waves from the upper and lowercomb-shaped surfaces.47 In this case we do not take into consideration theinteraction of corrugated structures through higher space harmonics, excited atevery reflection of partial plane waves and distorted by an opposite reflectingsurface. These effects, however, are essential only for closely located combstructures,48 which are not of great interest, but show an approximate characterof the equivalent boundary condition given by eqn. 7.15. Thus a theory, devel-oped here, is applied at considerably large A.

Considering the term £" describing the losses as a small perturbation, we cansimplify eqn. 7.42. Let % ~ /(0 ) + jd where /(0) is the transverse wave number ofa corresponding waveguide without losses. Then S is expressed by the formula

^C'W-r-xfoiAT1 (7.43)For X(0) we have real transcendental equations

-Z(0)Atan{x(0)A}l

X(o)A cot{x(0)A} j

an approximate solution of which does not appear to be too difficult to obtain.Classification of modes in the waveguide under consideration can be given in

an approximation of perfect conductivity on the basis of eqn. 7.44. The qualitat-ive trend of dispersion curves for symmetrical and antisymmetrical modes isshown in Fig. 7.2a and b. The cutoff frequencies determined by the conditiony(^c i ) = 0 c a n be found from eqn. 7.44, assuming that X(O) = kcl. Then it ispossible to solve the equations exactly; the roots of these equations are expressedby

_ (nit/(A + D) for symmetrical modes

\{n ~ i)7i/(A + D) for antisymmetrical modes

where n is the mode index (n = 1, 2, . . .).At k> kcl the mode is a propagating one, and at k < kcl an evanescent one.In the frequency range determined by the condition

kcl <k<kt

the given mode is a fast one; at k — kt it transforms into a slow (surface) one.Transformation frequencies of fast waves into slow waves are determined fromthe condition lim y(kt) ~ kt and can be obtained from eqn. 7.44, assuming that

#(0) = 0. For antisymmetrical modes we obtain the transcendental equation

ktA tan{ktD) = 1

4 7 In the presence of losses, Brillouin's angle Sn, formally introduced by the relationship #„ =cos" 1 (#„/£), is a complex number and does not have a direct sense of the angle between the x axisand the normal to the front of a partial plane wave.4 8 For example, such as exp[ -{ (27 i /< / - k sin S)2 - k2}1/2A] ~ 1.


t'A

0 KC1 X/2D Tl/d Kt 31/d 0 %l2D Ka TL/D d%ld

Figure 7.2 Dispersion diagrams for plane comb-shaped waveguide

a Odd modesb Even modes

For symmetrical modes the corresponding equation can be solved exactly. Theroots of the latter are

kt = nnlD,n = <d, 1, 2, . . .

At kt < k < kc2 there exist surface waves in a waveguide. When k increases thesewaves are more 'pressed' to the surface, and, at kc2 = n(n + i ) / A undergo acutoff. For antisymmetrical modes the mth frequency band Sk in which a surfacewave exists grows when A increases at a constant k. If A is constant, dk growswhen n increases, tending asymptotically to the value Sk = nj2D. For symmetricalmodes Sk = nj2D independent of A and a band number.

Among symmetrical modes one mode is a slow one throughout the range ofits existence; there are no antisymmetrical modes of such type.

Simple approximate relationships for the attenuation coefficients of TMmodes in a plane corrugated waveguide can be given at | tan (kD) | > 1(|Re Ze\ ^ | I m Ze\)- This case, as is shown by calculations made for the keyproblem (see Section 5.5), correspond exactly to minimal attenuation and,hence, are of the greatest practical interest.

To begin, let us consider a transverse resonance for the TM2 0 mode49 on thecutoff frequency. Supposing that in eqn. 7.42 X = > w e obtain a characteristic

49 When the attenuation is minimal (k n/2D), the TM0 0 mode is a surface one (k < n/D) or doesnot exist at all (k > n/D). That is why at k ~ n/2 the TM2 0 mode is the lowest symmetrical spatialmode of E-type.


equation for the wave number k = k' —jk", which, owing to the presence oflosses, becomes a complex one. Solving this equation and taking into consider-ation the smallness of losses, we obtain for the (^-factor of transverse resonancethe formula

where CL(O) = 7^W/o/2 Re J?s is the (^-factor of a similar resonance in a parallel-plate waveguide at the same frequency. From eqn. 7.45 it follows that theQ;factor of a plane corrugated waveguide on a cutoff frequency is less than thatin a parallel-plate one.

For the attenuation coefficient of the TM2 0 mode in a corrugated waveguideit is possible to obtain from eqns. 7.43 and 7.44 the formula

where y{'0) is t n e attenuation coefficient of the E20 mode in a corresponding

smooth-wall waveguide (at the same propagation coefficient) expressed as

/(U) W0{(2kA)2-n2}1/2n

From eqn. 7.46 it follows that TM modes in a corrugated waveguide have lowerattenuation than in a smooth-walled waveguide. The larger values of the ratioy"ly'(O) correspond to the larger values of kA (to larger distances between theoperational frequency and the cutoff frequencies).

We wish to emphasise that eqn. 7.46 is suitable only for fast modes (k<kt).However, it cannot be used for a fast wave in close vicinity to the frequency oftransformation, when x'~%". In this case, as mentioned in Section 7.2, thesimplified formula for the equivalent surface impedance £e given by eqn. 7.22cannot be used.

We can give a clear physical interpretation to the results obtained in the lightof the Brillouin concept. In fact, at a cutoff frequency partial Brillouin wavesare incident normal to the comb-shaped structure, and their absorption (inaccordance with the results of Section 5.5) is larger than that in a smoothsurface. Far from cutoff frequencies, the incidence of Brillouin waves is close toa grazing one and this leads to an opposite situation: energy losses of partialwaves in every single reflection act are considerably smaller than in the case ofa smooth surface. Note that the angle of incidence S is determined by $ =cos~1{n/(2kA)}. For our assumptions the absorption coefficient a introduced inSection 5.5 can be expressed as

which means that the ratio / '($) /7('o)($) 1S directly determined by the valueof a:

(9) (7.47)

Minimal attenuation takes place when k is somewhat less than 7C/2Z), i.e. in the


region where the surface wave exists. In this case, on various irregularities of awaveguide (which are always present in real waveguides), a transformation ofa spatial wave into a surface one is possible. In this connection attenuationcharacteristics can become worse as a surface wave attenuates rather strongly(see Section 7.3). To avoid this phenomenon it would be reasonable to choosethe parameters of a corrugated waveguide so that k >> TT/2Z), when there is nosurface wave.

7.5 Attenuation in waveguides with azimuthal corrugation

Figure 7.3 shows the cross-section of the waveguide under study and defines thenotation. Supposing that ka^> 1, it is posible to apply the impedance boundarycondition given by eqns. 7.22, 7.24 and 7.26 to curvilinear surfaces. The otherassumptions correspond to those accepted in Sections 5.5 and 7.2.

The impedance boundary conditions at r = a are

(7.48)

Let the dependence of the modes on z be exp(jyz). Then the azimuthal compo-

(ii)

Geometry of circular waveguide with azimuthal corrugation

(i) Original structure(ii) Impedance model (arrows indicate the formation of the 'whispering-

gallery' mode according to Brillouin's concept)


nents of field E^ and H^ can be expressed in terms of the longitudinal ones as

jy dEz jcofio 8HZ

n * X (7.49)jcoe0 dEz jy dHz

* X2 dr rx2 d(j)

where y = (k2 ~ %2)1/2, X being the transverse-wave number. The problem con-sidered, unlike those in Sections 7.3 and 7.4, cannot be reduced to a scalar one:the angularly dependent TM and TE modes do not exist separately in such awaveguide.

The longitudinal components of the field can be written as

Ez = CeJmUr)exp(jm<t)) )

where Ce and Cm are unknown amplitude coefficients. Substituting eqn. 7.50into eqn. 7.49 and using eqn. 7.48 we obtain for Ce and Cm a homogeneoussystem of two linear equations. Equating the determinant with zero, we have acharacteristic equation for ^, which can be written in the form

{Mm(iS)-a{Mm(iS) + r 1 }=/» m ( i?) (7.51)

where jS = xa, Mm(P) = J'm{P)lPJm{P) (a prime stands for Bessel's-functionderivative with respect to the argument), pm(P) = m2{(ka)2 - P2}j(ka)2/?4,t=JZel(Woka) and Z=jZhkalW0.

We are most interested in the EHwn modes,50 for which the effect of abnormallysmall attenuation may occur. In this case eqn. 7.51 can be presented in a moreconvenient form:

whereTo solve eqn. 7.52 we apply the perturbation method, considering the right-

hand side of the latter as a small perturbation, and the equation

Mm(P)=C (7.53)

to be of a zero-order approximation.It is essential that eqn. 7.53 should be solved not in a complex plane but on

the real axis /?. For this many standard techniques of numerical solution can beused; one of the most effective is the method of differentiation with respect tothe parameter (Modenov and Slepyan, 1984). Let us present the solution ofeqn. 7.52 as /? = P(O)+jP(X\ where /?(0) is the solution of eqn. 7.53 and /?(1) is asmall correction. We are interested in the real part of /?(1), which determines the

50 When D —• 0, the EHmM and HEmn modes are transformed into the Hmn modes and Emn modes ofa smooth-walled waveguide, respectively.


eigenmode attenuation:

where Mm (j8) = dMm (j8)/d/? and <f + j f ' = f.The attenuation coefficient is expressed from eqn. 7.54 by

1 ( 0 ) 1 "ni(kn\2 — R2 ^l2

Using the anisotropy of the impedance of a corrugated surface it is easy tospecify the contribution of field components with different polarisations in thesolution obtained: the augend in eqn. 7.54 corresponds to the electrically pola-rised component, and the addend to the magnetically polarised component.

This form of the perturbation technique is applicable when the followingcondition is fulfilled:

l - 1 (7.55)

When the condition expressed by eqn. 7.55 is not true, the absolute value of thesecond factor on the left-hand side of eqn. 7.51 can be small and, as a conse-quence, the right-hand side of eqn. 7.52 will not be a small perturbation. Thelatter is indicative of the proximity of/? sought to a D-multiplicity point of theEHmn and HEmn modes (see Section 3.7). Thus, for convenience, eqn. 7.51 canbe rewritten as

Mm(P) ' j {M2m(P) [ J

and now the right-hand side of this equation can be considered as a smallperturbation. Let /? = /?(0) +i/?(1). Neglecting the terms above the first order ofjS(1) in both factors on the left-hand side of eqn. 7.56, we obtain a quadraticequation for /?(1). The solution of the latter results in

m _ r + ?+[(r#4£C;U/W] (751]Pl>2 2Mm(p{0))

l • 'where q— — j(£C' ~^~ 1)C an(^ t n e P m s sign corresponds to the EHmw modes, theminus sign to HEmn modes. Eqn. 7.57 is similar to eqn. 3.67 obtained by meansof another method for a waveguide with an arbitrary shape of cross-section.However, there is the following difference between them: for cases consideredin Section 3.7 the coupling of the D-multiplicity modes results from the finiteconductivity of the walls, whereas in our case it is from azimuthal periodicityof the surface (when, even neglecting the losses, £, £' 0 and the coupling of theD-multiplicity modes remains).

Let us consider now eigenoscillations in a 2-dimensional resonator which is across-section of the waveguide under study. Equations for complex eigenfrequ-encies can easily be obtained from eqn. 7.51, assuming that fi = ka. Then wehave

TJMJZ1Jm{ka) Wo


t^=~k (7-59)Note that in this system separate TEmn and TMmn oscillations do exist. TheTEmn oscillations are described by eqn. 7.58 and the TMmw oscillations byeqn. 7.59; we shall be dealing below with the TEmn oscillations alone. For ouranalysis we can apply the perturbation method in a form similar to that usedabove. Let us suppose that k = k'—jk" and, taking into account the fact thatk" 4, k', from eqn. 7.58 we obtain

and

Eqns. 7.60 and 7.61 are valid under the condition Re Ze <t |Im £e\. Nowassuming that |tan(k'D)\ > 1, which does not contradict the previous conditiondue to the smallness of Re^/H 7 ^ , we can obtain from eqn. 7.60,51 in a zero-order approximation,

k'a*vmn (7.62)

Combining eqn. 7.61 with eqn. 7.62, we have

K ' }O^ 2k" 2 R e ^ s ( l +Djd)

It is of interest to compare the obtained value of Q^n with the Qjfactor ofcorresponding resonance QJo) in a 2-dimensional smooth-walled circular res-onator for the same resonance frequency. The relative Q -factor is

QTo") _ AU(1 + Did)

From eqn. 7.64 it follows that the 'whispering gallery' modes (n= 1, m > 1) arecharacterised by the highest relative (^-factor. The ratio QJlQ)lQJnn increases withincreasing m and diminishes when n increases at fixed m. At m — 0 the (^-factorof a corrugated system is lower than that of a smooth-walled one. Thus, anadditional rarefaction of the eigenmode spectrum occurs which is of a purelydissipative nature and is not connected with the radiation effects.

The physical meaning of the results obtained becomes quite clear when usingthe Brillouin interpretation of this class of fields. In fact, the TEmn mode in acircular waveguide can be presented as a superposition of travelling (in azimuthdirection) plane waves undergoing 2m reflections on the path / = 2na. The angle

51 The quantity k! determined by eqn. 7.60 is a cutoff wave number of the waveguide underconsideration.


of incidence 9 is determined by the formula (Weinstein, \969a)

$mn = coS-l{(\-m2lLi2

mn)112} (7.65)

Owing to the curvature of the guiding surface the reflected waves do notabandon the latter, returning to it to undergo a new reflection etc. [seeFigure 7.3(ii)]. This interpretation, being essentially geometro-optical, is evermore precise for larger m and n. Qualitatively, however, it also suits the lowermodes as well.

It is easy to see from eqn. 7.65 that 'whispering gallery' modes are character-ised by angles of incidence close to grazing angles; that is why the relativeQ;factor is especially high for them ([Q/1™]"1 ~ cos23mn). For the angular sym-metric modes (m = 0) #n0 = 0, which corresponds to normal incidence.

To reach the maximum (^-factor it is necessary to set kd 7i/2, i.e. dja ~ 7r/2vmn.Thus, the optimum number of combs is determined by the formula JVopt =4{vmn}, where {a} denotes the greatest integer in a.

A more precise determination of k' and Q/1" can be achieved by proceedingto a numerical solution of eqn. 7.60. In this case the method of differentiationwith respect to the parameter, described by Modenov and Slepyan (1984), isthe most effective one. Figure 7.4 shows xmn as a function of Dja with m and nas parameters. It can be seen that Xmi < m- This means that for any values ofDja the first root of eqn. 7.60 describes the surface mode (in the radial direction).This mode is not described by the approximate solution of eqn. 7.60, given byeqn. 7.62. The roots with n = 2 and 3 correspond to 'whispering gallery' modesfor which approximate eqns. 7.62 and 7.63 are valid (in this case, for Z)-»0,Xm+1,« "* ftmn)' With increasing ofD/a spatial modes are transformed into radiallysurface ones and this transformation takes place at Xmn ~ m-

It is noteworthy that the optima in Q resonance frequencies determined byeqn. 7.62 are located near the multiplicity points of the TMmn and TEmn modes.However, here the multiplicity is trivial. The reason for this lies in a differentpolarisation structure of these modes: the TEmn modes have only field components//z, Er and E^, the TMmn modes the components Ez, Hr and H^.

7.6 Projection method for calculation of propagation andattenuation coefficients of corrugated waveguides witharbitrary shapes of cross-section and corrugation

The structure under consideration in this Section is a good model of a flexiblecorrugated waveguide with smoothly shaped slots. Owing to low attenuation ofthe H E n mode and also high operating characteristics they are widely used asfeeders in stationary and mobile radio-relay, space and tropospheric-scattercommunication facilities.

Designing such waveguides is impossible without an accurate estimation oftheir electrodynamic characteristics. This is a rather complicated problembecause for such structures the method of separation of variables and otheranalytical techniques cannot be applied. Angular dependent modes in suchwaveguides are hybrid ones (i.e. having all six nonzero components of theelectromagnetic field), owing to which a solution of full vector problem forMaxwell's equations is needed. The considerable depths of the slots are of the


= 15

o

Figure 7.4 Cutoff characteristics for various modes in circular waveguide with azimuthalcorrugation

a n—\b n = 2c n = 3

most practical interest (it is just when the effect of abnormally low attenuationis observed). Therefore, methods considering corrugation as a small perturbationof the smooth-wall waveguide surface are inefficient. The best approach seemsto be the projection method with co-ordinate transformation suggested byIlyinsky and Sveshnikov (1968). Based on the latter, a general numerical methodsuitable for calculation of propagation and attenuation characteristics of corru-gated waveguides with arbitrary shapes of cross-section and corrugation has


been developed by Alkhovsky et al. (1975, 1986). Their analysis will now bedescribed. Note that this method puts no limitations on the waveguide geometri-cal parameters, and in particular on the depth of the slots.

The equation for the waveguide surface X in cylindrical co-ordinates r, (/>, zhas the form

r=R(<p,z) (7.66)

where R((j), z) is a periodic function of z with the period d. Let us assume thatthe function R((f), z) is continuously differentiable with respect to both variables.The family of surfaces given by eqn. 7.66 allows us to consider circular, ellipticaland double-ridge corrugated waveguides, which are most important for appli-cations mentioned above.

The mathematical formulation of the problem under study is to find thevector functions E, H satisfying:

(i) homogeneous Maxwell's equations inside region 3) with the surface S;(ii) the Leontovich impedance-boundary condition on Z;(iii) Floquet's condition of pseudoperiodicity.

The approach described would not be simplified to any significant degree ifwe now started considering lossless corrugated waveguides. This is why, withinthe framework of this method, it is more convenient for calculation of theattenuation characteristics to use the direct solution of the impedance problemthan the energy-perturbation technique (see Section 7.1).

Let us reduce the original problem for a periodic waveguide to an equivalentone for a smooth-walled waveguide with anisotropic periodically inhomogeneousmagnetodielectric filling. Using the substitution of variables

p = rlR((/>9z) (7.67)

we map the region 3) into the circular cylinder G of unit radius. The co-ordinatesp, (j), z will, generally speaking, be nonorthogonal. Maxwell's equations forcovariant components of vectors H and E can be rewritten as

1 r\T4 AH a1!2

p d(j)

dz

dp

1 dEz

p d<j)

dEp

dz

d(pEt)

dz

dp

dHp

d(j>

dE+

dzdEz

dp

dEp

,,13 ;

,1/2

dp

(7.68)


Here glj are elements of the metric tensor expressed by

=_L_rR2(4>,z)l

1 +R2(<t>,z)\

8R(4>,z)

jdR(4>,z)\ P

{dR(<j>,z)\ dz n

1 3 _s

pR3((t>,z

pR{(j), z) dz

1

P2R2((f>,z)

s 3 3 = l

and gll2 = PR2(<t>,z).Introducing the orthogonal cylindrical co-ordinates with the basis {ip, i^, iz]

inside G one can consider eqn. 7.68 as Maxwell's equations for the field in aregular periodically filled circular waveguide

curl Eo = j(ojuH0

curl Ho = —j(osE0

(7.69)

where

The tensors of 'conventional' permittivity and permeability are

Pg

12

22

r13/p

0

1/p

The boundary conditions on the walls are written as

(7.70)

where

ZsZs [ W

(ggn)ll2l -i

13,12

Let Eo = Et + £"zzz, i / 0 = Ht + Hziz. The transverse components JE, and £ f aredetermined through the incomplete Galerkin's method. The basis functionse" (p, 0), hn

t (p, (/>) are the transverse components of normal modes in the regularwaveguide G and expressed by

VpÔ^ x iz 2 = 1 , 'magnetic' functions

Ycx^n * = 2, 'electric' functions

VÔ^ 2=1 , 'magnetic'functions

x Vp< Of i = 2, 'electric' functions

where O^(p, </)) and <^^(p, (/>) are eigenfunctions of Dirichlet and Neumann


problems for a circle of unit radius (multi-index n corresponds to two indicesp and /; p = 0, + 1 , ±2, . . ., oo, / = 1, 2, . . . , oo)

DnM (p, <i>) = *Jf (p, 4>) = J V f t , J p (nplP)ex

The functions g" and /t" constitute the complete orthogonal sets and satisfy thenormalising relationship

where S± is the cross-section of the regular waveguide G.The transverse components of fields £ f , //"f are sought in the form of finite

sums:52

m = l

N2(7.71)

where N= Nx + jV2 and 4m ( ;), ^m (^), Cm (z) and Z)m (4:) are unknown functions.The approximate expressions for the longitudinal components E^ and H% canbe obtained from eqns. 7.68 and 7.71. The coefficients in the expansions forE^ and /Tf can be found from the following system of integral relations:

^% d/= i ENzh\% d/ -zh\%

(curl £,w), (curl*?*) d/

(curl E%-j(O(iH%)thn2* da

S i

22{J( u h% dl

(7.72)

52 The superscript jV denotes an approximate representation of the corresponding field componentas a superposition of TV basis functions.


(curl H% +j<mEl)ten2t do = 0

where C is the perimeter of the area S±. Eqns. 7.72 are chosen in such a waythat approximate (for any jV) and accurate solutions of the boundary-valueproblem given by eqns. 7.1, 7.69 and 7.70 will satisfy one and the same energyrelation resulting from Pointing's theorem

(ExH*)nodS+{exp(-2y"d) - 1} (E x H*)n0 dSJs

(eo\E\2-iio\H\2)dV = >

where Zd is the surface of the corrugated waveguide corresponding to a unitcell, and Vd and S are the volume and input cross-section of the latter, respect-ively. According to the work by Sveshnikov (1963) it can be shown that anapproximate solution E$, HQ converges to the accurate one when jV andJV2-»oo.

Substituting eqn. 7.71 and corresponding expressions for E^ and / / f intoeqn. 7.72 and performing some transformations, we obtain the following systemof ordinary differential equations for the functions Am(z), Bm(z), Cm(z) andA.' .KJ

AC N2

m=l m = l

m = l

(7.73)

The general formulae for the coefficients of the system given by eqn. 7.73 aregiven in Appendix 3. For circular corrugated waveguides R^R((j)) and theequations for unknown functions with different azimuthal indices are indepen-dent. The coefficients a^ , jSj^, y ^ and 3^ {n = {pj},m = {/>, r}) are expressedin a closed form

(l=r)P2R'(z)

co{n2pl-p

2)R(z)

2p2[l+{R'(z)


l / 2

>co(fi2pr-p

2)1/2R(z)

2v2plR\z)

CO V2 _

(l=r)

{R'(z)}2]z)}2]m

coR{z)

( 2 ) = - V ( Dnm imn

?<3) = ,

(3) = (2) (4) =inm

A,(z)

B,(z)

C,(z)

D,(z)

M

= ED?

The coefficients in eqn. 7.73 are periodical functions of z\ hence solutions ofeqn. 7.73 can be found by Hill's method. Presenting a solution in the form

2mn\

we reduce eqn. 7.73 to the homogeneous system of linear algebraic equations oforder 2(JV1 + JV2) (2M+ 1).

The nontrivial solutions of the linear algebraic system obtained can be foundon condition that its determinant should be equal to zero. The propagation andattenuation coefficients, i.e. real and imaginary parts of y, for a fixed k aredetermined from this determinantal equation. The numerical results obtainedby means of the projection method are presented in Sections 7.7 and 7.8.

Another version of the numerical method for solving the problem consideredwas suggested by Ilyinsky and Tupikov (1989). In this version, after performingthe co-ordinate transformation, defined by eqn. 7.67, the azimuthal and longi-tudinal components of the field sought are presented in the form

an N

o ( _ v"1

wwN ( ~ l~t"Oj</>,z n=-I

where C^tZ(p) and D^fZ(p) are unknown functions. The radial components areexpressed in terms of azimuthal and longitudinal ones by eqn. 7.68. Instead ofeqn. 7.72 a set of projection relations is written over the region 0 < (j> < 2TT,0 < z < d for arbitrary 0 < p < 1. This set results in a system of linear differential


equations for C^,tZ(p) and Z)JfZ(p), which should be complemented by theboundary conditions at p = 0 and p = 1. They are obtained from the boundarycondition for the field on C, given by eqn. 7.71. To solve numerically theboundary-value problem arising , one of the shooting-method modifications canbe used (see Appendix 1).

7.7 Propagation characteristics of circular corrugatedwaveguides

In this Section we shall present the results of theoretical and experimentalinvestigations of dispersion characteristics of modes in circular sine-corrugatedwaveguides. The theoretical results were obtained by means of the projectionmethod (Section 7.6).

The waveguide surface was described by eqn. 7.66; the function R(z) wastaken in the form

R(Z)= J l + ?cosFpH (7.74)

where a is the mean radius of the corrugated waveguide, q = I/a and / and d arethe amplitude and period of corrugation, respectively.

In Figure 7.5 one can see calculation results for dispersion characteristics ofthe TM0 1 mode at a = 18.52 mm, d= 8 mm and / = 15 mm. The three 'electric'functions (jV2 = 3) and three space harmonics (M = 1) are taken into accountin the basis.

The data obtained have been compared with experimental results. To providehigh experimental accuracy a resonance method of measuring the eigenfrequenc-ies of a waveguide short-circuit have been used. The resonance frequencies weremeasured by means of a heterodyne wavemeter. The errors in measurementsdid not exceed 0.01%. The mode type and resonance number were determinedby the use of an absorbing body inserted into the waveguide. As the excitersused in the experiment allowed for a minimal coupling, the error due to couplingdid not exceed that of the wavemeter. The quality of the inner surface of thecorrugated pipe was the main factor determining the accuracy of the experiment.The section of the circular corrugated waveguide under study was manufacturedusing electrolytic deposition of copper onto the mandrel. To process the mandrela cutter with a sine profile was used, with dimensions controlled by means of amicroscope. The manufacturing accuracy of the waveguide inner surface was+ 20 jum. As Figure 7.5 shows, there is good agreement between the calculatedand experimental data, which points to high efficiency of the calculation tech-nique used.

Figure 7.6 shows the dispersion characteristics for the TEOn modes (n =1, 2, 3, 4, 5) calculated by Alkhovsky and Ilyinsky (1979) for a = 34.9 mm, d =5.5 mm and q = 0.225. The broken line corresponds to the dispersion curve forthe H0l mode at the same values of a and d but with a different q (q — 0.0552).The sharp rise in the dispersion curves for frequencies n e a r / = 28 GHz is relatedto the fact that a corrugated waveguide is characterised by stop and transmissionbands following one another. The results presented here correspond to the firsttransmission band.


2.40 2.41 2.42 2.43KCL

Figure 7.5 Dispersion curves for TM01 mode in corrugated waveguide

See text for dimensions1 Calculation2 Experiment3- Calculation for smooth-wall waveguide (q = 0)

2.44

A calculation has been performed of the lower boundary frequencies53 of thetransmission bands and the dispersion characteristics for the E H n and H E n

modes in lower transmission bands at a = 18.512 mm, d— 8 mm and q = 0.0806.Tables 1A and 7.2 present the results of the calculation of the E H n and H E n

modes cutoff frequencies for various sets of basic functions and M=2.

53 These frequencies are defined by the condition Re y(k0) = 0 and are similar to the cutoff frequen-cies of a smooth waveguide. In the light of this, we will also call them cutoff frequencies. Using theprojection method (Section 7.6) we can determine these frequencies from the determinantal equationfor y, considering it as an equation for A:o at y = 0, £s = 0-


600

400

I

200

I

70 75 20 25frequency, GHz

Figure 7.6 Dispersion characteristics of some modes in circular corrugated w

See text for dimensions1 TE01 2 TE02 3 TE0 3

TE01 (? = 0.0552)4 TE0 5 TEn

T a b l e 7.1 Results of calculation of EH11-mode cutoff wave number obtained for varioussets of basis functions

N, N2 k0 (cm"1)

3489

131515192020

3388

121415191920

1.01091.01061.02211.02401.02601.02661.02671.02741.02771.0274


Table 7.2 Results of calculation of HEn-mode cutoff wave number obtained for varioussets of basis functions

N, A^ kQ (cm-1)

3 3 2.0649

5 5 2.0657

8 7 2.0658

8 8 2.0658

From these Tables we see fast inner convergence for the H E n mode andslower convergence for the E H n mode. These facts about the convergence aregeneral in nature as far as this calculation technique is concerned. Tables 7.3and 7.4 show the results of calculating the cutoff wave numbers of the firsttransmission band for E H n and H E n modes at JV1 = JV2 — 3 and various setsof space harmonics.

Figure 7.7 illustrates the trend of the convergence with respect to the numberof basis functions for the E H n mode at M = 1. The even values of JV correspondto equal numbers of'electric' and 'magnetic' functions (JVl = JV2), whereas theodd ones correspond to numbers of 'magnetic' functions exceeding those of'electric' functions by one. In Figure 7.8 are shown the theoretical and exper-imental dispersion characteristics of E H n and H E n modes for frequencies nearthe cutoff frequency.

The results discussed above were obtained for a waveguide with a shallowcorrugation, though, as mentioned previously, the projection method describedin Section 7.6 has no limitations in q.

In Figure 7.9 the values of f0 = kocj2n for various modes are shown asfunctions of the corrugation depth. These data were obtained at a = 28.925 mmand d= 20 mm; 21 basis vector functions and seven space harmonics were takeninto account (Alkhovsky et al., 1979).

Figure 7.10 shows the dispersion characteristics for a waveguide with deepsine corrugation calculated by means of the projection method. The waveguideparameters are a = 28.925 mm, ^ = 2 0 m m and / = 8.925 mm. The curves inFigure 7.10 are obtained for the same sets of basic functions and space harmonicsas those in Figure 7.9. The experimental data in the paper by Alkhovsky et al.(1979) are marked with circles. As can be seen from Figure 7.10, there is good

Table 7.3 Results of calculation of EHn-mode cutoff wave number obtained for varioussets of space harmonics

M

Mcrrr1)

Table 7.4

M

Mem"1)

1

1.0126

2

1.0109

Results of calculation of HE^-mode cmsets of space harmonics

1

2.0667

H

2

2

3

1.

T wave

.0649

4

0105 1.0105

number obtained for various

3

2.0649


83

82

81

80

= 0, theory

o=0.08, experiment

S JO /5 20

Figure 7.7 Propagation coefficient of EH11 mode against number of basis functions

a= 18.512 mm, d=Smm, k= 1.3 cm"1, M= 1

agreement between the theoretical and experimental results. A sharp rise in thedispersion curves for the TM0 1, E H n and EH2 i modes is due to their transform-ation into surface modes (compare with Section 7.4).

In Figure 7.11 the dispersion characteristics of a circular waveguide with astep corrugation calculated by Clarricoats et al. (1975#) using the partial-regionsmethod, are shown for comparison. Their shapes correspond to those inFigure 7.10.

We confine our discussion to the most typical examples of the calculation ofpropagation characteristics, the main goal of this book being absorption analysis.More complete numerical results and their applications for the design of reflector-antenna feeds are described by Clarricoats and Olver (1984), and those forflexible waveguides by Alkhovsky et al. (1986).

7.8 Attenuation characteristics of circular corrugatedwaveguides

A detailed study of the attenuation characteristics of circular corrugatedwaveguides with step corrugation has been carried out by Clarricoats et al.(1975<2, \915b). In this Section the results obtained for waveguides with smoothcorrugation are presented and compared with those obtained by Clarricoatset al. The experimental data are also given. The computational algorithm isbased on the projection method described in Section 7.6.


0.6

0.6

\ OA

^ 0.2

0.8

0.6

OA

0.2

1.05 115 125it, cm"1

(i)

2.03 2.12 2.16ft, cm*1

(ii)

Figure 7.8 Dispersion characteristics of EHX x and HEX x modes

a = 18.512 mm, d— 8 mm

2 q = 0.08062theory

x x x experiment(i) EH n(ii) HE n

In Figure 7.12 the attenuation coefficient of the E H n mode in a circularwaveguide with sine corrugation54 is presented as a function of the number ofbasis functions. In this Figure the value of the attenuation coefficient of theT E U mode in a smooth-wall circular waveguide with radius equal to the meanradius of the corrugated waveguide (all other parameters being the same) isshown by a horizontal line for comparison. Figure 7.13 illustrates the innerconvergence of the projection method with respect to the number of spaceharmonics.

In Figure 7.14 the frequency dependence of the attenuation coefficient of theH E n mode in a sine-corrugated circular waveguide (solid line) and similardependence for the T E n mode in a smooth-walled circular waveguide with aradius equal to the mean radius of corrugated waveguide (broken line) areshown. The experimental results from the work by Alkhovsky et al. (1978) areshown by crosses. The conductivity of the wall material is determined experimen-tally by measuring the Qjfactor of a cylindrical resonator manufactured fromthe same copper band as the waveguide under examination. The agreementbetween theoretical and experimental results confirms the high accuracy of theprojection method.

Figures 7.12-7.14 give the results for waveguides with shallow corrugations{q4, 1). In such waveguides the attenuation is higher than in smooth-walled

*The equation of the surface of a sine-corrugated waveguide was given by eqns. 7.66 and 7.74.


6.0

5.0

3,0

W 12.0 2l,mmFigure 7.9 Cutoff frequencies of various modes as functions of the slot depth

ones, which corresponds to the results obtained for the model problem inSection 5.5. From the physical point of view this can be explained by an increasein the surface current path while the resonance effect of the field 'being forcedout' from the periodic structure (see Section 5.5) is not observed at such smalldepths of corrugation.

The results of calculations for deep corrugations (Alkhovsky et al., 1979) areshown in Figure 7.15. The circles represent the experimental data. The wave-guide was made by means of the electroforming technique. Measurements werecarried out through the resonance method. Both theory and experiment pointto the fact that there can be an abnormally small attenuation of the H E n mode.

Physically, this phenomenon can be understood using the Brillouin conceptwhich, while being more complicated in this case than for a plane waveguide,is still valid. The H E U mode can be expressed as a spectrum of plane waves ofboth polarisations, the reflection of which from the round walls of the waveguideat each point is similar to that from the plane corrugated surface. The waves ofelectric polarisation dominate in this spectrum of which the resonance characterof the dependence of y" on the corrugation depth is indicative. In the angularspectrum of this mode the near-grazing plane waves are prevailing when itsfrequency is distant from cutoff. The absorption of such waves at correspondingd and / in the corrugated surface is very low. At f->fo the maximum of the


740

T 700

^ 60

20

frequency, GHz8

Figure 7.10 Dispersion characteristics of corrugated waveguide

1 TM01 2 EH n 3 EH21 4 HEn6 TM02 7 HE21° o o experimental results

5 TE0

angular spectrum shifts towards the region of smaller angles of incidence andthe loss at each reflection increases considerably.

Attenuation in circular corrugated waveguides of a non-sine shape of corru-gations has also been investigated on the basis of the projection method describedin Section 7.6. The equation of the waveguide surface was given in the form(Alkhovsky et aL, 1987)

where

= R0-l\cos{nzld)\v

v= -ln2/ln{cos(7w/2)}

(7.75)

(7.76)

s being a preset numerical parameter.Figure 7A6a shows the shapes of the slots described by eqns. 7.75 and 7.76

for different values of s. In Figure 7A6b the attenuation coefficient of the H E U

mode as a function of s is presented for Ro = 47.85 mm, / = 17.85 mm, d —20 mm, / = 7.5 GHz and o0 = 5 x 107 S/m. As can be seen from Figure 7.16&there is an opportunity for a considerable decrease in attenuation throughoptimisation of the corrugation shape. Table 7.5 compares the theoretical andexperimental data on attenuation for circular waveguides with non-sine-shapecorrugations. There is a good agreement between calculations and experiments.The data in Table 7.5 correspond to frequencies where the attenuation of theH E n mode is minimal.

Figures 7.17 and 7.18 show the calculation results for a circular waveguidewith a step corrugation obtained by Clarricoats et al. {1915a). These results aresimilar to those mentioned previously.

In Figure 7.19 the frequency dependence of attenuation of the TE01 mode in


Table 7.5

f(GHz)

7.58.6

Comparison of theoretical and experimentalwith non-sine-shaped corrugations

(mm)

47.8549.97

/(mm)

17.8515.22

d(mm)

20.0016.98

s

0.50.21

results for

Theory

y(dB/m)

0.01490.0071

corrugated waveguides

Measurementy

(dB/m)

0.01270.0062

corrugated waveguide with sine corrugation is shown (Alkhovsky and Ilyinsky,1979). Broken lines correspond to those of smooth-walled waveguides with R =R1 and R = R2 {R\,2 ~ a ± U t n e inner and outer radii of the corrugated wave-guide respectively).

From Figure 7.19 one can see that the attenuation of the TE01 mode in acorrugated waveguide is higher than that in a smooth-walled waveguide ofcomparable size. From the physical point of view this result can be explainedusing the Brillouin concept. In the expansion of the TEOi mode in a spectrumof plane waves there are only those of magnetic polarisation (E is parallel to thestructure ridges). For such waves the losses in a corrugated surface exceed thosein a smooth one (compare with Section 5.5), which results in the growth ofattenuation.

We have described only some of the numerical results illustrating the appli-cation of the projection method to study the phenomenon of abnormally lowattenuation of the H E n mode in corrugated waveguides. Important informationon attenuation characteristics of elliptical, rectangular and double-ridge corru-gated waveguides is given in the book by Alkhovsky et al. (1986). Physicalpeculiarities of corrugated structures were used to develop flexible corrugatedwaveguides, which are used as highly effective microwave-band transmissionlines manufactured in many countries.

7.9 Millimetre-waveband high-quality corrugatedresonators

In this Section we describe how the effect of abnormally low dissipation inperiodic structures is used to develop high-quality resonators at millimetrewaveband. The material presented is based on work by Luk et al. (1988),Rodionova and Slepyan (1989) and Rodionova et al. (1990). In addition totheir high quality, resonators of this type are characterised by a wide range offrequency retuning, small overall dimensions and weight, and a high level ofparasitic-mode suppression. Such resonators can be used for frequency stabilis-ation of millimetre-waveband generators, material parameters measurement,phase detection etc.

In Figure 7.20 a photograph of this resonator in knock-down form is shown.Figure 7.21 shows a sketch of an experimental model. The resonator is a hollowcylinder on the inner side surface of which there is a periodical structure (1).End covers (2, 3) are smooth and can be either plane or spherical with a radius


12

0 2frequency, GHz

Figure 7.11 Dispersion characteristics of circular waveguide with step corrugation(Clarricoats et aL, 1975a)

rx = 25 mm, d= 10 mm, b = 2 mm, /^/ro = 0.7141 EH n 2 TM01 3 HE n 4 EH216 TM02 7 HE31 8 EH31

5 TE0

of curvature of about 3b (one is made mobile for retuning over a frequencyrange). The excitation of the resonator was implemented with a thin plug (4)which was fed from a standard rectangular waveguide through a waveguide-to-coaxial transition.

To provide thermal stability, the resonator body was made of superinvar; the


0.0Mr

0.055

•I 0.050

10.045

0.040

= 0.806

ro 15N7+N2

20

Figure 7.12 Attenuation coefficient of EHn mode in corrugated waveguide againstnumber of basis functions

a= 18.512 mm, </=8mm, k = 1.3 cm"1, M= \, ao = 5x 107 S/m

0.056

0.055

0.05d0 1

MFigure 7.13 Attenuation coefficient of EHxl mode in corrugated waveguide against

number of space harmonics

a= 18.512mm^=8mm^= 1.3 cm"1, Nx = JV2 = 2, o0 = 5 x 107 S/m


52 5A 5.6 58frequency, GHz

Figure 7.14 Attenuation characteristics of EHll mode in corrugated waveguide

JV1 = 11, N2= 10, M= 1a = 19.1 mm, d= 7.19 mm, a0 = 5.05 x 107 S/m1 ? = 02 ^ = 0.1063x x x experiment

0.12

-£aos

?: 0.04

5 6 7 8•frequency, GHz

Figure 7.15 Attenuation characteristics of HE11 mode in corrugated waveguide

NX=N2 = 5, M=2a = 38.925 mm, d = 20 mm, / = 8.925 mm

theoryo o o o experiment

inner surface was coated with a layer of copper with a layer of silver on top ofit by electroforming. After compacting, the silver coating was polished to 6-7surface finish class for the comb-shaped surface and to 10-12 surface finish classfor the smooth one.

The TMOmn oscillations are the working ones. According to the information


0.25 0.5 0.752/da

0.1' 0.2 0.3 OA 0.5S

Figure 7.16 HE11-mode attenuation in circular corrugated waveguide with non-sine-shape corrugations

a Shapes for corrugationsb Attenuation coefficient against s

0.1

t-O

§0.0/

, 5 7 9frequency, GHz

Figure 7.17 Attenuation characteristics of circular waveguide with step corrugation(Clarricoats et ai, 1975a)

rl = 25 mm, d= 10 mm, b = 2 mm, 7^0 = 0.714, a0 = 5.8 x 107 S/m1 TMQI 2 EH U 3 H E U 4 TE0 1 5 HE21

6 TM0 2 7 HE12 8 HE31 9 EH12


\

rrZ

iff

"WWW

MMMrr,40 60 80

radius rr,mm100

Figure 7.18 Attenuation in circular waveguide as a function of radius (Clarricoatset al, 1975a)

Corrugated waveguide

5 H E n

Smooth-wall waveguides

1 HE1 1(/2 = r1)2 TE n (R = r0)3 TE01 (R=r1)4 TE0 1 (R = r0)

given in Section 7.4, a considerable decrease in energy absorption in a comb-shaped surface may occur at small Brillouin angles, compared with a smoothsurface, i.e. at n^>m.55 To obtain a higher Q -factor due to lower losses in theside surface, their contribution to the total losses should be dominant. This willbe the case when L ^ b (in our case Ljb c± 2-4, b = 26.35 mm).

Filtration of parasitic modes is carried out by means of a narrow (A ~ 1 mm)slot in the diametrical plane of the immobile end wall [Figure 7.21 (8)). For

55 As is shown in Section 7.8, the Brillouin concept for circularly symmetric structures can still beused. In this case the Brillouin angle is determined by the relation $m = cos"1 {Xm/hm)> where Xmand hm are transverse and longitudinal wave numbers of the TMOm modes in a circular waveguideof radius b.


3k 0.04

%0.02

0.01

5 10 15frequency, GHz

20

Figure 7.19 Attenuation characteristics of TE01 mode in circular waveguide

a = 34.9 mm, d = 5.5 mm, q = 0.0552

1 Corrugated waveguide2 Smooth-wall waveguide (R = Ri)3 Smooth-wall waveguide (R = R2)

the TMOmn oscillations, such a slot is practically nonradiating while for the othertypes of oscillation the slot is transparent (as for azimuthal slots in circularwaveguides for TEOn modes). The radioabsorbing elements are inserted into thepost-plunger space to suppress parasitic resonances. The method of resonatorexcitation hindering the generation of all modes except TMOmn and the effectof a comb structure giving rise to stop bands and increasing absorption of modeswith large radial indices are also essential to filtrate spurious oscillations.

The general structure of the eigenoscillation spectrum was examined throughthe use of an automatic SWR and attenuation meter. The spectral-line widthat a level of 3 dB and SWR in resonance was evaluated in the manual retuningoperation. A more accurate spectral-line-width measurement was performed bymeans of an electronic frequency meter and SWR measurement, with a precisionmeasuring line as recommended by Valitov (1963).

Figure 7.22 shows the resonator spectral characteristics (solid line for a res-onator with a slot, broken line for a resonator without a slot). As can be seenfrom Figure 7.22, the slot considerably enhances the selection of the oscillations,only the TMOmn oscillations having a high Q;factor. Their resonance frequencies


Figure 7.20 Cylindrical corrugated resonator in 'knock-down^ form

are slightly shifted due to the slot; the widths of spectral lines remain the samedespite the presence of the latter. This is confirmed by the fact that in one ofthe experimental models a slit was shifted 1 mm from the diameter line parallelto it, which did not change the eigenoscillation-spectrum structure (solid line inFigure 7.22), increasing the widths of spectral lines 2-2.5 times.

The widths of spectral lines, shown in Figure 7.22 (solid line) are within thelimits of 0.5-1.2 MHz which corresponds to unloaded (^-factors of the order of105 (taking into account the differences in values of SWR). Thus, the resonatordescribed has a high unloaded (^-factor and, at the same time, a rarefiedeigenoscillation spectrum. The spectrum density and (^-factor of the corrugatedresonator without a slot are approximately the same as for an open resonatorwith spherical mirrors having a volume an order of magnitude greater. Theresonator with a slot can compete, as far as spectrum rarefaction is concerned,with resonators operating at 'near-boundary' oscillations (Nefedov and Rossiisky,1978) and has a much higher (^-factor.

The resonator has such characteristics in a wider frequency range than thatin Figure 7.22 (up to / = 37.5 GHz). Within these limits, the resonator can beretuned by moving one of the end walls. When changing the operational mode,coupling readjustment (by moving the excitation plug) is not needed.

Assuming the end walls to be plane, we can estimate the resonator character-istics. The effect of the comb-shaped surface can be described mathematicallyby impedance-boundary conditions generalised for the case of finite conductivity(see Section 7.2). In this approximation a resonance frequency for the TMOmn

oscillations is determined as the mth root of the equation

(7.77)JAP) c )


fe 7

Figure 7.21 Sketch of cylindrical corrugated resonator

1 Periodic structure2 Fixed end cover3 Movable end cover4 Thin plug5 and 6 Radioabsorbing elements7 Waveguide-to-coaxial transition8 Filtering slot

where p = {(2nfbjc)2 - {mibjL)2}112, c being the velocity of light. Eqn. 7.77 caneasily be solved numerically, for example, by the method of differentiation withrespect to the parameter (Modenov and Slepyan, 1984).

In this case, we obtain the following formula for the (^-factor:

, 2c Re Zs f / D\ Bnc ( f nc \2)1^ WonfL\_ \ dJ2bf\ \2LfJ jJ l ;

where fi = Re ^s i /Re Zsi> Zsi anc^ ^S2 a r e surface impedances of the end andside surface materials, and D and d are the depth and period of the comb-shaped structure.


34.06 34.82frequency, Gtfz

Figure 7.22 Experimental frequency characteristics of corrugated resonator

with slotwithout slot

Introducing different surface impedances Zsi>Zsi (then /?7^1) we canapproximately take into account the difference in the surface finish class ofpolishing the comb and smooth surfaces.

The results obtained through this technique and the experimental data atm = 1, 2, n = 15 . . . 25 are of the same order of magnitude.

7.10 Radiation from a corrugated horn

The scope for using corrugated surfaces is not confined to low-loss waveguidesand high-quality resonators. Another important field of application is antennaengineering, in particular horn corrugated feeds for reflector antennae(Clarricoats and Olver, 1984). The approximate methods are based on theHuygens-Kirchhoff principle commonly used for the calculation of such feeds.The accuracy of these methods, however, cannot easily be estimated. Apart fromthis they are not applicable for calculating near-field characteristics. The devel-opment of rigorous methods, based on a complete solution of electrodynamicproblems, is therefore of great interest. To formulate one such method we useGalerkin's incomplete method with semi-inversion in boundary conditions. Orig-inally this technique was applied to solve the problem of diffraction on acorrugated surface with complex shaped grooves (Kopenkin et al., 1987, 1988).

In this Section we consider the problem of wave radiation from an open-ended plane irregular waveguide [Figure 7.23(i) and (ii)] having a profiledescribed by an arbitrary twice differentiable even function a(z)> This general


formulation of the diffraction problem permits calculation of a wide class ofhorns, in particular profiled corrugated horns.

Consider the field in the E-plane of a horn, assuming the metal surfaces tobe perfectly conducting (extension of this approach to include the case of finitelyconducting surfaces would not be difficult). Much of the material presented isbased on the work by Kopenkin et al. (1989).

Let the waveguide be excited by an arbitrary superposition of TM modes.Then an original diffraction problem is reduced to the following boundary valueone:

dn= 0

(7.79)

(7.80)

where Z is the total surface of conducting bodies.The solution of eqns. 7.79 and 7.80 is sought in a class of functions satisfying

the condition of the finiteness of the energy integral

) I (*2|.A|2-|gradxzlAI2)ckck

over any finite region in the plane xOz including an edge.The incident field can be presented as {z < 0)

I nn {x~a(0)}

KZZZZZLa(o)

8

D 4 ^

— L

0) (ii)

Figure 7.23 Geometry for wave radiation from open-ended plane irregular waveguide

(i) Original structure(ii) Modified structure


where Sn are given coefficients, yn(0) = [{nnj2a(0)}2 - k2]1/2, Im yn < 0 andRe yn > 0, a prime here and later meaning that the summation is taken withrespect to odd values of n. The scattered field \j/ — \jj° should satisfy the radiationcondition at r= (x2 + £2)1/2-» oo.

It follows from symmetry that \j/{ — x, z) = ~ *A(*> z)> After the function ij/(x, z)satisfying the conditions mentioned above has been found, the electromagneticfield can be expressed by

E=— curl(/>) H=L\j/(J0£0

As in Section 5.8, let us consider a modified structure shown in Figure 7.23(ii),passing to the limit A-»0 only in the final equations. At the initial stage themethod of partial regions is used. Division into partial regions is shown inFigure 7.23(ii). The peculiarity of this problem is in the fact that the structureunder study includes the open regions si, ^ , where the field cannot be describedin the form of series but should be written as Fourier's integrals. The procedureof field matching for open regions was developed by Mittra and Lee (1971);later we shall show how it can be combined with Galerkin's incomplete methodapplied to express the field in the irregular partial region 3) (in the regions <€and 8 the field is expanded in series in terms of eigenmodes of correspondingregular waveguides).

Therefore, in the regions si and $ the field sought can be presented in thefollowing form (Mittra and Lee, 1971):

Region <s/: ^ > L + A ( - oo < x < oo)foo

\l/{x,z)= A{ot) sin (xx exp{-{(X2-k2)1/2{z-L-A)} da (7.81)Jo

Region 3S\ \x\ > a{L) {z < L + A)

cos{iS(* - a(L))} exp{( iS 2 - k2)l>2 (z - L - A)} dj?

(7.82)

where A {a) and B(fi) are unknown functions. In the course of integration overthe real halfaxis in eqns. 7.81 and 7.82 we get round the branching point a,/} = k from below.

As to the rest of the partial regions, the field there is expressed in a mannersimilar to that in Section 5.8:

Region (€\ \x\ < a(L) {L< z < L + A)

*(*, Z) = I ' tn(x, L) [Tn exp{yn(L) (z - A - L)}n

+ Qnexp{-yn(L)(z-A-L)}] (7.83)

Region S: \ x\<a{0) {z < 0)

t(x, z) = t°(x, z) + X' ilfn(x, 0)Rn exp{yn(0)z} (7.84)n

Region B: \x\ < a{z) (0<z<L)


il/(x,z) = Y,' p»(zWn(x,z) (7.85)n

In eqns. 7.83-7.85 Tn, Qn and Rn are unknown coefficients, Pn(z) are unknownfunctions and i//n(x, z) ~ cos[nn{x — a(z)}l2a(z)], yn{L) being obtained from theexpression for yn(0) through the replacement: a(0) ->a(L).

According to Galerkin's incomplete method we substitute eqns. 7.79 and 7.80in the region Q) by a set of equivalent projection relationships

a(z) # ckY2~) 1/2

:=«U) I \ d V Jwhere m = 1, 3, . . . .

Substituting eqn. 7.85 into eqn. 7.86 we obtain an infinite system of ordinarydifferential equations for P= {Pn(z)}

( 7 . 8 7 )

where H{z) and B(z) are known matrix functions, elements of which arecalculated in a manner similar to that in Section 5.8.

The boundary conditions for the system described by eqn. 7.87 are obtainedas a result of matching of i// and dip/dz at the interfaces between the partialregions (z — 0, z = L, Z = L + A). The continuity of the tangential-field compo-nents across z = 0 implies

Sn + Rn = Pn(0) (7.88)

(Rn ~ Sn)yn(0) = ^ + £ ' ^nm(0).Pm(0) (7.89)

where the elements of the matrix < (0) are

a(0) = (da/dz) \z= +o- Excluding Rn from eqn. 7.89 we obtain a boundary con-dition for eqn. 7.87 at z — 0:

^ ^ + {£(0) - r(0)}P(0) = -2r (0) , s (7.90)

d^

where S = {Sn} is a given vector and F(0) is a matrix with the elements Tnm (0) =-5m«yn(0).56

Similarly, at z = L we obtain

Tnexp{-)-n(I)A}+Q.nexp{yn(L)A} = Fn(L) (7.91)

yn (L) [Tn exp{ - y n (L)A} - &, exp{yn (L)A}]dPn(L)

dz• + I ' Lm(L)Pn,(L) (7.92)

56 In fact, eqn. 7.90 is the radiation condition and shows that there are no incoming (fromZ= — oo) waves in the scattered field.


The expressions for the matrix elements £nm(L) are easily obtained from theexpressions for £wm(0) through the replacements: a(0) ->a(L), a'(0)->d(L) and

The field matching at z = L + A, as our next stage, is performed using theanalytical solution of the problem of radiation from an open-ended plane regularwaveguide (Mittra and Lee, 1971). Let us present the coefficients Tn andfunctions A (a) and B(fi) in the form of the series

r n=I 'Q.mAn m (7.93)

( }

where Anm is a coefficient of transformation of the mth incident mode into thenth reflected one for the open end of a regular waveguide with a width of2a(L)and Am((x), Bm(fi) are spectral amplitude functions in eqns. 7.81 and 7.82corresponding to the case of excitation of an open-ended regular waveguide witha width of 2a(L) by the mth eigenmode of unit amplitude. The coefficients Anm

and functions Am(a) and Bm(f5) can be determined in a closed form using thefactorisation method (Weinstein, 1969£) or a variant of the residue-calculustechnique modified for the case of open structures (Mittra and Lee, 1971). Bothmethods, naturally, produce identical results:

M+{jyn{L)}M+{jym(L } (7.95a(L)yn{L){yn(L) + ym(L)}

ljym {L)ly " P ^ W } M + (a) M + {^{L >} (7.96)

where y = (a2 — k2)111. The special function M+(a), obtained as a result offactorisation, is expressed in terms of infinite products:

M+(a) = [cos{ka{L)}f/z exp r w In

•'•I])11 v + • /n rexPi r1,3,... [ jyn{L)) I nn J

where C is Euler's constant. The function Bm(f}) is expressed similarly toeqn. 7.96.

Use of the expressions for Tn, A{OL) and B(fi) given by eqns. 7.95 and 7.96 isequivalent to the fulfilment of the continuity conditions for \j/ and di/z/dz at theZ = L + A plane. Note that this variant of field matching at the interfaces betweenthe partial regions $4, $ and jtf, %> requires an extraction and analytical inver-sion of the operator, corresponding to the problem of radiation from the open-ended plane regular waveguide of a width of 2a(L). In fact, this is a procedureto regularise the 2-point boundary-value problem obtained similarly to thatdescribed in Section 5.8.

To obtain the boundary condition for eqn. 7.87 at Z = L, it is necessary to


exclude from eqn. 7.92 the unknown coefficients Tn and Q^n using eqns. 7.91and 7.93. At the limit A—>0, we obtain the required boundary condition in theform

^ Q = -2T(L)(I+A)-lP(L) (7.97)

where T(L) is a matrix with the elements Tnm(L) = Snmyn(L), and A is a matrixwhose elements are expressed by eqn. 7.95.

Thus the original diffraction problem has been reduced to a 2-point linearboundary-value problem for the system of ordinary differential equations ofinfinite order (eqns. 7.87, 7.90 and 7.97). To find a numerical solution, weshould reduce this problem to that of the finite order JV. We can then use oneof the shooting-method modifications (see Appendix 1). The convergence of anapproximate solution to the accurate one at JV-> oo can be proved rigorously(as was done by Kopenkin et al., 1987).

For the field in the far zone we can obtain the following expression using thesaddle-point method:

kr-- + ka(L))4

where the radiation pattern F(9) is expressed by

= (1 -cos3) 1 / 2 M + (-A:cos3)

(7.98)

The vector of coefficients Q= {Q,m} in eqn. 7.91 is expressed in terms of thesolution of eqns. 7.87, 7.90 and 7.97 in the form

As in other cases when this method is used, the number JV, sufficient for theinner convergence, should exceed the number of the eigenmodes propagating ina plane waveguide of the maximum width by 2-3 units.

To monitor the accuracy of the approximate solution, we keep checking onhow the active power-balance and reciprocity relationships are fulfilled. One ofthe forms of the reciprocity relationships can be written as

where R^ is the complex amplitude of the/?th reflected mode at the incidenceof the <7th mode (Sq= 1, Sn = 0, n^q). The numerical experiments have shownthat for such JV the relative deviation from these relationships does not, onaverage, exceed 10~4-10~5.

To apply the technique described in this Section for examining corrugatedhorns we used the following function for the horn profile:

a(z) = <i(0) + aoZ + Po sin(2^//) (7.100)

where a0, /?0 and / are constant values.In Figure 7.24 the normalised radiation patterns are shown when the horn is

excited by the TM0 1 mode (S1 = 1, Sn = 0, n^\). The lengths of irregular


sections and sizes of the apertures are the same for all curves, the only differencebeing in the number of the slots defined by the ratio ijL. The radiation patterns,as we note, are of a complicated, sharply oscillating nature. The peaks aresharper for larger ratios L\l. Physically this means that the higher modes arepredominant in the process of the formation of the radiation pattern. Thesehigher modes are intensively excited due to fundamental-mode transformationwhich occurs with the horn profile nonmonotonic.

When the period of corrugation is small compared with the wavelength, useof the surface-impedance model of a corrugated horn appears to be more suitable(Clarricoats and Olver, 1984). A rigorous solution of the corresponding diffrac-tion problem, not based on the physical optics approximations, can easily beobtained within the framework of the approach described above.

To conclude with let us note that Galerkin's incomplete method with semi-inversion in the boundary conditions is a universal and highly effective newmethod applicable to various problems of mathematical physics. The advantagesof this technique arise from its being a combination of powerful analytical

0

Figure 7.24

^ 8, degreesNormalised radiation patterns for corrugated horns

Fo = max {F(S)} for L\l = 2.25

a(0)=0.5>l, a0 = 0.4, 0O = O.ULjl= 2.25L// = 4.251/1 = 6.25

90


methods (for example, the Wiener-Hopf method) and the general methods ofthe theory of smoothly inhomogeneous media and smoothly irregular structures.Thus, this method can be applied, for example, to the problems of wave radiationfrom open-ended plane and circular waveguides which are irregular not becauseof their variable cross-section but because of the inhomogeneity of the mediuminside. In some special cases (for example, when a waveguide is 'slowly' irregular)this method allows us to obtain approximate analytical solutions using the WKBtechnique or Langer's method (Felsen and Marcuvitz, 1973, Jones, 1986). Anexample of this can be found in the work by Ilyinsky et al. (1987).

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Appendix 1

Shooting method and its modifications

By means of Galerkin's incomplete method, a number of diffraction prob-lems can be reduced to the resolution of a system of ordinary differentialequations

^-A(t)y=f(t) (Al.l)

with boundary conditions

By(0) = b (A1.2)

Dy{T)=d (A1.3)

where 0 < t < T,y,f, b and d are vector columns of m, m, m — r, r components,respectively, and A, B and D are matrices of the orders m x m, {m — r) x m, r x m.The ranks of the matrices B and D are m — r and r.

To solve this boundary-value problem the shooting technique is used(Godunov, 1961). Let y0 be an arbitrary solution of eqn. A1.2, andjv1, j ; 2 ,. . . , yr an arbitrary system of r linearly independent solutions of the systemBy = 0. Then the general solution of eqn. A1.2 can be written in a form

y=yo+ £ Cnyn (A1.4)

where Cn are arbitrary coefficients. Let us assume that the functionsSoW>gi(t)>&2{t)> • • • >grW have been determined which are solutions of theinitial value problems

258 Shooting method and its modifications

where 1 <n<r.Then the solution of the boundary-value problem given by eqns. A 1.1—A 1.3

can be written in the form

y{t)=go(*)+ t Cngnit)

where Cn are constant coefficients determined from eqn. A 1.3 through solvingthe system of linear algebraic equations

Thus, for solving the 2-point boundary-value problem it is necessary to solver+ 1 initial-value problems given by eqns. A1.5 and A1.6 and to determine rconstant coefficients Cn from eqn. A 1.7.

In the course of computation of vector functions {gn(t)} it is important totake into account the fact that the solution of the initial-value problem mayincrease exponentially due to the fact that the matrix A (t) at any t has eigenval-ues with both a positive and negative real part. Thus, initial-value problems areweakly stable with respect to errors of initial data, discretisation and roundofferrors. Some components of gn may increase considerably and their values maygo beyond the limits of the computer's set of floating-point numbers. In somecases the effect of 'flattening' of the system {gn(t)} in the shooting process isessential: as a result of the shooting process, the vectors {gn(t)} may be so closeto linearly dependent that eqn. A1.7 cannot be solved numerically. At a suffic-iently large length Tand comparatively high order of the system m, these renderthe conventional shooting method inapplicable in practice.

Because of this, modification of the shooting method is required which wouldincrease its stability against errors and allow us to cancel, or at least ease, theselimitations on m and T. This can be obtained by orthogonalisation and normalis-ation of the system of solutions {gn} in the course of shooting. Let us examine ageneral scheme of modified shooting method with orthogonalisation (Bykov andIlyinsky, 1979).

The segment of integration (0, T) is divided into P parts by the points ts:

0 = to< '•• <tp=T

Let us cons ider the m a t r i x G of the o r d e r m x ( r + 1):

G= (gi> • • • >gr,go)

Let gsn(t) (n = 1, 2, . . , r) and gs

0 (t) be the solutions of the system

and eqn. A 1.1 satisfying the initial conditions y(ts) = gn and y(ts) =

respectively.

Shooting method and its modifications 259

Let us define the operation V on the matrix G in the form

Vig!, • • • ,gr,gO)={g\(ts+l), • • • ,gr(tS+l),gS0(ts+l)}At every point ts from the matrix G, as a result of orthogonalisation process Ws,we obtain a matrix Qj.

In the process of orthogonalisation the vectors gx, . . . ,gr,go are transformedlinearly by means of some complex nonsingular matrix Hs:

The vector g0 is transformed according tor

qo=go~ Z hgi

Specific choice of the matrices H and x determines the type of orthogonalisation.Solution of the 2-point boundary-value problem given by eqns. A 1.1-A 1.3 is

obtained by successive application of operations Ws and Vs to the matrix

o = {yi, - • • ,yr,yo)

Assuming that G0(t0) = W0(Y0), we obtain successively

To satisfy the boundary condition given by eqn. A1.3 the algebraic system hasto be solved

go + Z Cmgm= 1

where (gp,...,gP,gp

l)=Gp(tp)Let us dwell here upon the problem of choice of the matrix H in eqn. A 1.8.

First, we consider the simplest method using normalisation of the vectorsS1 ' ' * ' '6r*

rjn - ^mn _ . ,

\gm\

The normalisation method (NM) allows us to avoid the phenomenon of overflowbut not 'flattening' of the system of vectors {^m}.

The well known orthogonal shooting method (OSM) (Godunov, 1961) con-sists of orthogonalisation of the vectors g1, . . . , gr according to the Gram-Shmidt procedure, after which the vectors qt satisfy the conditions

(<|i> 9n) = &in (I < i < T, \ < n < r)

This method, to a certain extent, prevents the 'flattening' of the system

Application of OSM in many cases allows us to avoid disadvantages typical


of the conventional shooting method without orthogonalisation. However, bymeans of OSM it is impossible to solve problems characterised by a large lengthof the integration interval and high order of eqn. A 1.1 simultaneously. Toeliminate these limitations, a directed-orthogonalisation method (DOM) hasbeen developed (Bykov and Ilyinsky, 1979).

Let us describe a method of choosing the matrix H suggested by Bykov andIlyinsky. At t= ts we determine in some way all eigenvalues Xp = ap +7j3p andeigenvectors ep (\\ep\\ = 1) of the matrix A and divide them into two groups:

{e~, lp }: Ae~ = k~e~ (a ~ < 0)

The number of the vectors ep is represented by m +, m~ = m — m +, and sum-mation with respect to the indices p corresponding to these two groups by S +

and E~. Because of completeness of the system e1, . . . , em, we have

so that qn will be in the form

1 f r

i ) 0 = i

Assuming that m+ > r, we can uniquely determine the matrices //(n) by theconditions

After solving this algebraic system for / / ^ we can find q1, . . . , qr. The vectorq0 is expressed as

It is worth noting the following specific feature of DOM: every vector gt istransformed in such a manner that only one component e* increases with theincrease in t. After application of the operation Vs, the vector gt contains allincreasing components ep . The orthogonalisation step ts+l — ts should be suchthat the initial component e* still dominates.

Numerical testing of various versions of the shooting method was carried outby Bykov and Ilyinsky (1979, 1982), Ilyinsky and Bykov (1980), and Kuraevand Slepyan (1990). We shall present here some results from these papers. Thefollowing diffraction problems were considered:

(i) Scattering by an inclined dielectric plate with complex permittivity e2 ina parallel-plate waveguide. To the left of the plate is a vacuum (e = 1),and to the right a medium with a complex permittivity 63. Excitation isby the TE10 mode incoming from the left.

(ii) Scattering of E-polarised plane wave by a periodic structure of circulardielectric bars.

Both problems can, by means of Galerkin's incomplete method, be reducedto boundary-value problems for systems of ordinary differential equations. For


problem (i) the system has the form

dt2 n " ^ mî n

where Cn(t) are unknown functions, Tn = {{nnja)2 ~ k2}112L, L = a cot a + dcoseca, 0 is the waveguides' width, and d and a are the plate's thickness and inclinationangle, respectively,

3 ~ B2)Inm{Xl (0) + (£2 ~ \)?>nm(kL)2

{-to<t<0)

3 c > 2 I ± n m l ^ 1 1 \ / / \ 2 / nm \*2 \ ) /

where

— n)7CX I . | ( m + ;s i n < - > sin<

nx 1 . (2mnx\— sin Ia 2m \ a J (m =

Xt (/) = /L tan a + a, x2 (J) = Z> tan a -+• « 4- d sec a and /0 = (1 + a cosBoundary conditions for eqn. A 1.10 are

dCM(0)

( A l . l l )

w h e r e n 3 ) = {{nnja)2 - k2s3}1/2L a n d qn= - 2Snl Tx {\<n< JV) .

For problem (ii) we obtain

d/2 (A1.12)

where Tn = (a2 - k2)lj2L, (Xn = 2nnjd~\- k sin 3, 9 is the angle of incidence, d isthe grating period, and L is the cylinder diameter. In this case the functions\nm{t) are expressed by

[m-n)nL{\ - (2/4- 1)2}1/21

[m — n)n


£ being the relative permittivity of the cylinder material. Boundary conditionsfor eqn. A 1.12 have the form

dt

dCn(-l)

dt-rncn(-i) =

(A1.13)

, q n n 0 0

Problem (i) is a mathematical model of a waveguide high-power matchedload. The use of water (e3 = 74.0 +78.14) for absorbing the electromagneticenergy requires a special choice of £2? & and d to reduce SWR (in the calculationsbelow e2 — 6.8 + 7O.OO68). The difficulties in solving this problem result from asharp jump in permittivity and considerable absorption in water.

DOM, when applied to the problems described, as shown by numericalinvestigation, is stable against numerical errors at arbitrary length of the irregu-lar domain and the order of the system of ordinary differential equations. Forillustration, Table A 1.1 presents the magnitude and phase of the reflectioncoefficient Rx for different values of the discretisation error 3 at the step (the4th-order Runge-Kutta method was used). The order of the system m = 2jVandall ather parameters of the problem are fixed (a = 45°, <//tf = 0.1, % = kaftn =0.75, P = 1 8 , JV= 10).

When using OSM we have to sum large numbers in eqn. A 1.4, which leadsto considerable roundoff errors. They are especially significant when deter-mining the coefficients R^ with large ^-indices, as the values of the corres-ponding components gn

m are large. This is why near the solvability boundary(see Figure A 1.1) OSM results in erroneous values starting with^Jv5^Jv-i>^Jv-25 . . . etc. When using DOM, the coefficient C1 is obtained closeto 1 and thus the vector g1 is the main part of the vector y (other vectors g{

play the role of small corrections). This reveals a specific feature of DOM, wheneach basis vector of the system is responsible only for one growing componentof the solution to be found.

Table A 1.1 The influence of discretisation error 3 on magnitude and phase of reflectioncoefficient R±

3 Magnitude of R, Phase of R, (deg)

10~1 0.3142412640 57.764294

3 x 10~2 0.3141518220 57.799514

10~2 0.3141462330 57.800933

10~3 0.3141457931 57.803196

10~4 0.3141454484 57.803415

10"5 0.3141454191 57.803419

10"6 0.3141454130 57.803429

10"7 0.3141454129 57.803429

10~8 0.3141454129 57.803430


N

15

TFigure A l . l Approximate borderline of the stability zone of the orthogonal shooting

method

To obtain a high accuracy the number of modes N under consideration shouldbe chosen sufficiently large. For example, at % < 1 we choose N such that amongthe normal modes of the third region there would be 1—2 modes greatly attenuat-ing at distances of the order of 0.1 ka. At x > 1> when the length of irregularityincreases, jV should be chosen with a large reserve (3-5 evanescent modesdepending on ka and T= Lja).

As regards the problem under consideration, OSM does not have such stab-ility. Stability against errors is found in the shared region in Fig. Al. l . Outsidethis region stable values of Rx and other coefficients were not obtained throughOSM by means of step reduction in the Runge-Kutta method or increasing theorthogonalisation process number P. Certainly, the diagram in Fig. Al.l is onlysketchy: it is influenced by floating-point-number system of the computer usedand peculiarities of solving linear algebraic systems given by eqns. A 1.7.

Table A 1.2 presents the values of \R\\ obtained by means of various versionsof the shooting method for different jV at 5 = 0.01. In fact the system given byeqns. A1.10 and Al . l l was solved at JV<28, A > 0.025, A = 27c/Re T[3).

Figure A 1.2 shows the frequency dependencies of |/?i | for problem (i), calcu-lated through DOM. Note that the curves have peaks near the points of higherwaves transition from the evanescent to the propagating. As for Wood's anomal-ies in periodic structures, they can only be found using the rigorous approach.

Let us dwell on another modification of the shooting method, effective for awide range of diffraction problems (Kuraev and Slepyan, 1990). It is based ona synthesis of the shooting method and that of iteration, and can be looked uponas a particular case of the general projection-iteration technique (Luchka,1980). A similar idea was used by Litvinenko (1972) for the solution of infinitesystems of linear algebraic equations and Fredholm integral equations of the


Table Al .2 Comparison of\R1\ calculated by means of various versions of the shootingmethod

N DOM NM OSM

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

second kind. On the basis of the boundary-value problem given by eqns. A1.12and A1.13 we shall illustrate the advantages of this method. This formulationwill be used as being more convenient than the standard one (see eqns. A 1.1-A1.3).

We shall now describe the algorithm. Let C(t) = {Cn(t)} denote the solutionof the boundary-value problem given by eqns. A 1.12 and A 1.13 of the order2Af+ 1, M< X Elements of the vector function C(t) = {Cn(t)} at M< \n\ < Ncan be approximately determined through solving the ordinary differentialequations:

0.3276670.3545770.3835160.3751380.3431150.2979150.2558600.2469150.2806350.3141460.3313230.3393950.3434200.3455130.3467650.3476010.3483120.3489610.3495730.3501430.350659

0.3276650.3545720.3835030.3752220.3431120.2979090.2558560.2469230.2806520.3142100.3353150.3696910.4522890.372816-

----

0.3276640.3545730.3835040.3751350.3431110.2979120.2558540.2467590.2808120.3134000.3411230.9615580.4656500.218452-

----

with the boundary conditions given by eqns. A1.13. The functions /j,O)(0 aredetermined according to

/«°»(/)=- £ Km{t)ym (Al.15)m= ~M

The solutions of the 2-point boundary-value problems given by eqns. A 1.13-


0.8

0.6

0 4

0.2

00.5 0.9 1.3 1.7 2.1 2.5

Ka/OLFigure A1.2 Magnitudes of reflection coefficients against normalised frequency for the

waveguide matched load

A 1.15 can be written as

1

w ( 7 - u)}du (A1.16)

Introducing a vector function ze;(1) = y(t) ~y(t), we can easily show that w(1)

is the solution of the 2-point boundary-value problem

m= -N

dt

dt

(A1.17)

(A1.18)


where

M<\m\<N

Let w{1) (t) represent the solution of the 2-point boundary-value problem, givenby eqns. A1.17 and A1.18, of order 2M + 1. As for eqns. A1.14-A1.16 theelements of the vector function w^\t) at Af<|^|<jV* can be determinedapproximately according to the formulae

/!,1)(0=/!.1)- 1 KAt)^M<\m\<N

For the vector function w^(t) = w(1\t) — zc)(1) (t) it is also possible to formulatea 2-point boundary-value problem similar to that given by eqns. A 1.17 andA 1.18 and find its approximate solution w^2)(t). It is thus possible to constructan iteration process at each stage of which the corrections to the approximatesolution of the original 2-point boundary-value problem are calculated:

y(t)=y{t)+w(1){t) + w(2){t)+ "- .

Convergence of the iteration process when M is not very small results from theconvergence of Galerkin's incomplete method proved by Ilyinsky and Sveshnikov(1969).

Thus the suggested iteration scheme allows us to reduce the solution of the2-point boundary-value problem for a system of ordinary differential equations tothat of a family of similar problems of a lower order. Then the order of the lastones is such that the simplest modifications of the shooting method can be applied.

Table A 1.3 presents the transmission coefficients Tn = Cn{— 1) for a gratingmade by cylindrical dielectric bars. The values Tn were obtained by means ofdirect solution of the 2-point boundary-value problem A 1.12 and A 1.13 throughthe DOM, and Tn with the aid of the iteration method of successive corrections.The 2-point boundary-value problems for determining the vector-functionsze;(l)(/) were solved using the OSM (jV=3, M=0, iteration number = 4). The

Table A1.3 Comparison of transmission coefficients calculated by means of variousversions of the shooting method

3 0.00092 + y'0.00025 0.00092 + y'0.000252 0.00270 + y'0.00069 0.00264 + y'0.000671 0.01145 + y'0.00268 0.01145 + y'0.002680 0.93847 + y'0.33535 0.93847 + y 0.335341 0.00773 + y'0.00186 0.00777 + y0.001862 0.00215 + y'0.00056 0.00215 + y'0.000553 0.00078 + y'0.00022 0.00075 + y'0.00021


numerical results in Table A1.3 correspond to S = 30°, //</= 0.2388, L/d= 0.2,e = 4. As seen from this Table the results obtained through different methodscoincide completely.

Computational experiments have shown that for the convergence of theiteration process it is quite sufficient to assume M = max(M+, M_), M± —[(1 + sin S)rf//], where [a] is the greatest integer in a.

Appendix 2

Expressions for current-densitydistributions in a microstrip line with

a strip of finite thickness

The density of the transverse current on the side surface of the strip is

•i _ 2 f °° J°^(e~~ 1) coth(jMi)h y n Jo 1 Te{<*)Tfl{<x.)Plp2

x{coth(P2d2)-tanh(P2y)};(a) cosh(i?2/)

x

J ) r , ( a ) ca 2 /z2

a2 . h2 1 ^ . 1 . / a P r

2

The density of the longitudinal current on the side surface of the strip is

t | i81tanh(i82j)coth()81</1)'i , cosh()S2j)

x{l-tanh(jS2j)coth(i?2flf2)}

J s i n v 2 y a

Expressions for current-density distributions 269

The density of the longitudinal current on the lower screen is

jS2 coth(j?2^2

Si cothf/Ji dx) {eh2 a2 ,2 i u2\rr fM\ ~ u / /? a l / ? 2 r / « \ T - / - . \ ( ! \ '(oi2 + h2)Tt(a

l2) oth(s-l)

cos (ax)

The density of the transverse current on the lower screen is

!) [ e

1 1

~ 2 t2

a2

sin(a^)x —^—da

The function Pz(oc) in eqn. 3.88 is expressed by

hLn (a) + hL12 (a) — ( P + s)ockL13 (a)

A2 1 ( e - A 2 ) 2

A2AZ,21(a) + A7 , 2 2 (a ) - (P+ l)a*Z,23(a)

L31 (a) + AL32 (a) - (A2 + l)aJtZ,33 (a)

{/?3(a)-JV-4(a)}2f (a2 + ^f) (A2-A2g)rft]-^12 v^j = = l ~n 2—o I*

2 [pi coth(pldi) cosh ( P i ^ i ) J

270 Expressions for current-density distributions

N __ {RX{OL) sinh(j82/) + # 2 (a) cosh(/?2/)}2

^21 (a) ^

Pi sinh2 {/?2(rf2-0)

g) -R3(«) cosh(/?2<) - j?4(a) si

2^ (h2-k2){d2-t)

L23 (a) = {/?! (a) sinh(j520 + R2(a) cosh(P2t)}

x {^(a) - R 3 ( a ) cosh(P2t) - £ 4 ( a ) si

\{Rl{a)+2R2{a)}smh (2P2t) + R, (a)R2 (a) {cosh (2P2t) - 111}1

- 1}

={*i(a) sinh(/?20 +/22(a) c

x {R3(a) cosh(j820 +^ 4 ( a ) sinh(j820} ~ R2(<x)R3(oi)

| Mr7",(a)cosh(j82/) Tt(a) cosh(p2t)

a2

-£(a) T, (a) j 17J (a) cosh (fi21)

( > Tt(a) cosh(p2t)

r,(a) cosh(020 1jJ?r,(a)

Expressions for current-density distributions 271

A2

1 1

7;(a)cosh(iS2/)

Appendix 3

General formulae for the coefficientsnm) !nm) unm

In this Appendix the general formulae for the coefficients in eqn. 7.73 are given.These formulae are applicable for corrugated waveguides with arbitrary shapesof the cross-section and corrugation and have the form

'nm ro I ) "lp\g "lp ~"~ Pg

Jc

C (I.,(1) _ _ • . ) y S i n * / ~ 1 1

JS± I P

Jc

13

CO

12 um \n2t)

General formulae for the coefficients «<£, j8<'i, y<» , 5J" 273

}c

- [ g13

CD I v

v (2) = -Ynm I 2

Si

r /„12 /m

—(f^ Jc

Jsx (. P

jsL I P

S± I P

n2<f>)

7> 1 ) V-^ -n* ( -11 m \ n 12 mJ60 \ e2p\g elp^ Pg el<f>

I P

274 General formulae for the coefficients a ^ , jBjfi, yjjj,,

/J(3)_ J l ) * ,,(3) _ ^(2)* ,;(4) _ 5(2)*

where

Index

Abnormally small dissipation 3, 123,151, 165

Analytical regularisation 13Associated waves 50, 58, 59Attenuation coefficient 37, 59, 61, 62,

70, 74Autocollimation 174

BCS theory 25Boundary conditions

additional 21equivalent 201impedance 6, 81, 83, 124, 135, 213Leontovich 19third kind 9

Brillouin concept 208,211,237

Characteristic equation 86, 88, 117Coaxial line 62Comb-shaped structure 135Complex waves 54, 197, 199Conformal mapping 31, 204Cutoff

frequency 53, 60, 210, 211, 223waveguide 63,64

Degenerate modes 86Determinantal equation 86Dispersion

anomalous 54normal 54spatial 6, 21

Dispersion characteristics 43, 53, 223Dispersion equation 57, 68, 208Dissipation power 27,138,141,167,

180

Echelette 149Edge condition 32, 72Eigenfrequency 109Eigenoscillation 83, 90Eigenvalue 10,84Energy-perturbation method 37, 64,

81,83, 107, 120, 167Extraordinary plane wave 185

Factorisation 132,244Finitely meromorphic operator-

function 117Floquet-Bloch theorem 197, 200Floquet's condition 125, 157, 197

harmonics 136, 138Fredholm

alternative 17system 129

Galerkin's method 11, 104, 106incomplete 12, 42, 169, 219, 242,

257Generalised scattering-matrix

technique 189Green's function 158,160

Helmholtz equation 33, 55, 123, 157,167,207

Hertz vectors 70Hilbert space 117, 139, 140Hill's

method 22theory 198

Ill-posed problem 15,171Impedance

characteristic 79,80

276 Index

equivalent 134modified 65Leontovich 22,67, 129surface 18

Inclined comb-shaped structure 149Integral equation 12,158,159

Jordan canonical form 47, 50Jordan lemma 57, 92

Kernel 12,20,159Krylov-Bogolyubov method 12, 160

London's model 25Lorentz lemma 40, 199

Matched load 262Maxwell's equations 4, 38, 90, 94, 218,

219homogeneous 83,92

Meixner method 33Meixner's singularity 28Metric tensor 218Microstrip line 8

generalised 65Modes multiplicity

diagnolised (D) 47, 214Jordan (J) 47, 57, 199trivial 61

Moment method 11,67Monodromy matrix 198Multiplier 198

Neimann's series 140Norm 39,63Normal oscillations 83

Operatorconvolution matrix 13differential 7Fredholm 117Helmholtz two-dimensional 8Maxwell's 7non-self-adjoint 10

Penetration depth 19, 26effective 22, 26, 30

Perturbation method 64, 118, 213Pippard concept of'ineffectness' 21Pippard's model 25Poisson summation formula 160Polarisation

electric (H) 27, 123, 145, 149, 165magnetic (E) 27, 129,233

PolynomialsChebyshev 68Gegenbauer 73

Potentialdouble-layer 158simple-layer 158

Projection method 216

Q-factor 83, 88, 89eigen 94magnetodielectric 96of the free oscillation 93ohmic 96, 107unloaded 96

Radiation pattern 163, 245Rayleigh diffraction anomaly 144, 207Reflection coefficient 22, 132, 141Regularising substitution 14, 117, 126Relative convergence 16, 116, 126, 168Residue-calculus technique 14

modified 116, 171, 182Resonator

cavity 1, 84conical 99corrugated 236cylindrical 2,97cylindrical with coaxial insert 114high-quality 1metal-dielectric 4ring-dielectric sapphire 1spherical 99spheroidal 109superconducting 1with dielectric slabs 111

Semi-inversion 14, 171Shooting method 171, 245, 157

with direct orthogonalisation 258with iteration method of successive

corrections 263,265with normalisation 258with orthogonalisation 260

Skin effect 6,18anomalous 6, 20

Superconductivity 24SWR 121,238,239,262

Tikhonov's regularisation method 15Trapezoidal-groove grating 173, 175Truncation 11, 126, 168Truncation order 16,173

Index 277

Waveguides low-loss 3, 196circular 2, 61 rectangular 3, 62corrugated 3, 216 superconducting 1double-ridge 2 with rounded corners 2elliptical 3 Wiener-Hopf method 130,246flexible 3, 235 Wood anomalies 165, 177, 263

Propagation, scattering and dissipation of electromagnetic wavesThis book describes new, highly effective, rigorous analysis methods for electromagnetic wave problems. Examples of their application to the mathematical modelling of micros trip lines, corrugated flexible waveguides, horn antennas, complex-shaped cavity resonators and periodic structures are considered.

Special attention is paid to energy dissipation effects. Various physical models and methods of analysis of dissipation are described and approximate formulas and computer-based calculation results for dissipation characteristics are given and compared with experimental data. Ways of decreasing dissipation in waveguides and resonators are discussed.

The book will be of interest to physicists and engineers working on the theory and design of microwave and millimetre-wave components and devices. Designers in microwave engineering will find here all the information they need for choosing the correct waveguide (resonator) for a stipulated dissipation characteristic. The numerical algorithms and formulas can be directly applied to CAD systems. The book is also relevant for students of electromagnetism and microwave circuits.

A. S. Ilyinsky is a Professor in the Department of Computational Mathematics and Cybernetics and is Head of the Computational Electrodynamics Laboratory at Moscow State University. He obtained the Doctor of Physical and Mathematical Sciences degree in 1974. His research interests lie in the mathematical modelling of electromagnetic wave scattering, irregular waveguides, microstrip theory, antennas and antenna arrays.

G. Ya. Slepyan is a leading research scientist at the Institute of Nuclear Problems at Belarus State University, Minsk. In 1988 he obtained the Doctor of Physical and Mathematical Sciences degree from Kharkov State University. His research areas include electromagnetic field theory, microwave engineering, free-electron lasers, radiation and scattering in nonlinear media.

A. Ya. Slepyan is a senior research scientist at the Minsk Radioengineering Institute. He obtained the Candidate of Physical and Mathematical Sciences degree in 1985 from the Belarus State University. His research interests are in the scattering and diffraction of electromagnetic waves, mathematical modelling and CAD of antennas and microwave circuits.

Peter Peregrinus Ltd.The Institution of Electrical Engineers,Michael Faraday House,Six Hills Way, Stevenage, Herts. SGI 2AY,United Kingdom

ISBN 0 86341 283 1

Printed in the United Kingdom

ISBN 978-0-86341-283-1

Documents

Propagation, scattering, and dissipation of electromagnetic waves