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IET Electromagnetic Waves Series 1

Geometrical Theoryof Diffraction forElectromagnetic Waves

Third Edition

Graeme L. James


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IET ElEcTromagnETIc WavEs sErIEs 1

Series Editors: Professor P.J.B. ClarricoatsProfessor E.D.R. Shearman

Professor J.R. Wait

Geometrical Theoryof Diffraction for

Electromagnetic Waves

Third Edition


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Other volumes in this series:

Volume 1 Geometrical theory of diffraction for electromagnetic waves, 3rd edition G.L. James

Volume 10 Aperture antennas and diffraction theory E.V. Jull

Volume 11 Adaptive array principles J.E. HudsonVolume 12 Microstrip antenna theory and design J.R. James, P.S. Hall and C. WoodVolume 15 The handbook of antenna design, volume 1 A.W. Rudge, K. Milne, A.D. Oliver

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

and P. Knight (Editors)Volume 18 Corrugated horns for microwave antennas P.J.B. Clarricoats and A.D. OliverVolume 19 Microwave antenna theory and design S. Silver (Editor)Volume 21 Waveguide handbook N. MarcuvitzVolume 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 and E.M. Purcell

(Editors)Volume 26 Spherical near-eld antenna measurements J.E. Hansen (Editor)Volume 28 Handbook of microstrip antennas, 2 volumes J.R. James and P.S. Hall (Editors)Volume 31 Ionospheric radio K. DaviesVolume 32 Electromagnetic waveguides: theory and applications S.F. MahmoudVolume 33 Radio direction nding and superresolution, 2nd edition P.J.D. GethingVolume 34 Electrodynamic theory of superconductors S.A. ZhouVolume 35 VHF and UHF antennas R.A. Burberry Volume 36 Propagation, scattering and diffraction of electromagnetic waves

A.S. Ilyinski, G. Ya.Slepyan and A. Ya.SlepyanVolume 37 Geometrical theory of diffraction V.A. Borovikov and B.Ye. KinberVolume 38 Analysis of metallic antenna and scatterers B.D. Popovic and B.M. KolundzijaVolume 39 Microwave horns and feeds A.D. Olver, P.J.B. Clarricoats, A.A. Kishk and L. ShafaiVolume 41 Approximate boundary conditions in electromagnetics T.B.A. Senior and

J.L. VolakisVolume 42 Spectral theory and excitation of open structures V.P. Shestopalov and

Y. ShestopalovVolume 43 Open electromagnetic waveguides T. Rozzi and M. MongiardoVolume 44 Theory of nonuniform waveguides: the cross-section method

B.Z. Katsenelenbaum, L. Mercader Del Rio, M. Pereyaslavets, M. Sorella Ayza andM.K.A. Thumm

Volume 45 Parabolic equation methods for electromagnetic wave propagation M. Levy Volume 46 Advanced electromagnetic analysis of passive and active planar structures

T. Rozzi and M. FarinaiVolume 47 Electromagnetic mixing formulae and applications A. SihvolaVolume 48 Theory and design of microwave lters I.C. HunterVolume 49 Handbook of ridge waveguides and passive components J. HelszajnVolume 50 Channels, propagation and antennas for mobile communications

R. Vaughan and J. Bach-AndersonVolume 51 Asymptotic and hybrid methods in electromagnetics F. Molinet, I. Andronov

and D. BoucheVolume 52 Thermal microwave radiation: applications for remote sensing

C. Matzler (Editor)Volume 502 Propagation of radiowaves, 2nd edition L.W. Barclay (Editor)


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Geometrical Theoryof Diffraction for

Electromagnetic Waves

Third Edition

Graeme L. James

The Institution of Engineering and Technology


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Published by The Institution of Engineering and Technology, London, United Kingdom

Third edition © 1986 Peter Peregrinus LtdReprint with new cover © 2007 The Institution of Engineering and Technology

First published 1976Second edition 1980

Third edition 1986Reprinted 2003, 2006, 2007

This publication is copyright under the Berne Convention and the Universal CopyrightConvention. All rights reserved. Apart from any fair dealing for the purposes of researchor private study, or criticism or review, as permitted under the Copyright, Designs andPatents Act, 1988, this publication may be reproduced, stored or transmitted, in anyform or by any means, only with the prior permission in writing of the publishers, or in

the case of reprographic reproduction in accordance with the terms of licences issuedby the Copyright Licensing Agency. Inquiries concerning reproduction outside thoseterms should be sent to the publishers at the undermentioned address:

The Institution of Engineering and TechnologyMichael Faraday HouseSix Hills Way, StevenageHerts, SG1 2AY, United Kingdom

www.theiet.org

While the author and the publishers believe that the information and guidance givenin this work are correct, all parties must rely upon their own skill and judgement when

making use of them. Neither the author nor the publishers assume any liability toanyone for any loss or damage caused by any error or omission in the work, whethersuch error or omission is the result of negligence or any other cause. Any and all suchliability is disclaimed.

The moral rights of the author to be identied as author of this work have beenasserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data James, Graeme L.Geometrical theory of diffraction for electromagnetic waves.—3rd edn—(IEE electromagnetic waves series; v. 1)1. Electric conductors 2. Electromagnetic waves—Diffraction3. Electromagnetic waves—ScatteringI. Title II. Series537.5’34 QC665.D5

ISBN (10 digit) 0 86341 062 6ISBN (13 digit) 978-0-86341-062-8

Reprinted in the UK by Lightning Source UK Ltd, Milton Keynes


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Contents

Page

Preface

v H

1 Introduction 1

2 Electromagnetic fields 7

2.1 Basic equations 7

2.1.1 Field equations 7

2.1.2 Radiation from current distributions 9

2.1.3 Equivalent source distributions 12

2.1.4 Form ulation for scattering 15

2.1.5 Scalar po tentials for source free regions 16

2.2 Special functions 19

2.2.1 Fresnel integral functions 19

2.2.2 Airy function 22

2.2.3 Fock functions 23

2.2.4 Hankel functions 28

2.3 Asym ptotic evaluation of the field integrals 30

2.3.1 Method of stationary phase 30

2.3.1 .1 Single integrals 31

2.3.1.2 Double integrals 37

2.3.2 Method of steepest descent 40

3 Canonical problems for GTD

43

3.1 Reflection and refraction at a plane interface 43

3.1.1 Electric polarisation 44

3.1.2 Magnetic polarisation 47

3.1.3 Slightly lossy media 47

3.1.4 Highly cond ucting media 50

3.1.5 Surface impedance of a plane interface 51

3.2 The half-plane 52



3.2.3 Edge condition 63

3.3 The wedge 63


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Contents vi

3.3.1 Electric polarisation

65

3.3.2

Magnetic polarisation

73

3.4 Circular cylinder 74



3.4.3 Transition region 86

3.4.4

Sources on the cylinder 90

4 Geometrical optics

96

4.1 Geometrical optics method 96

4.2 Ray tracing 100

4.3 Higher order terms 110

4.4

Summ ary 113

5 Diffraction by straight edges and surfaces 117

5.1 Plane wave diffraction at a half-plane 117

5.2 Plane wave diffraction at a wedge 124

5.3 Oblique incidence 129

5.4 GTD formu lation for edge diffraction 132

5.5 Higher order edge diffraction terms 137

5.6 Physical optics approxima tion 146

5.7 Com parison of uniform theories 155

5.8 Multiple edge diffraction 159

5.9 Diffraction by an impedance wedge 167

5.10 Diffraction by a dielectric wedge 176

5.11 Summary 176

6 Diffraction by curved edges and surfaces

186

6.1 Plane wave diffraction around a circular cylinder 186

6.2 GTD formulation for smooth convex surface

diffraction 195

6.3 Radiation from sources on a smooth convex surface 203

6.4 Higher order terms 210

6.5 Diffraction at a disco ntinu ity in curva ture 211

6.6 Curved edge diffraction and the field beyond a

caustic 222

6.7 Evaluating the field at caustics 230

6.8 Summary 233

7 Application to some radiation and scattering problems 242

7.1 Geometrical optics field reflected from a reflector

antenna 242

7.2 Rad iation from a parallel-plate waveguide 250

7.3 Waveguide with a splash plate 261

7.4 Edge diffracted field from a reflector antenna 271

7.5 Rad iation from a circular ape rture with a finite flange 282

Index

290


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Preface

This is the third edition of a book that first appeared in 1976. Over the

past decade there have been significant advances in the Geometrical

Theory of Diffraction (GTD ) and related topics, and hence a substantial

revision of the previous editions was necessary in order to bring the text

up to date. To this end new material has been included in most chapters,

with Chapter 1 being entirely rewritten. In this introductory chapter a

concise survey of GTD and its association with related techniques is

given. Chapter 2 gives the basic equations in electromagnetic theory

required in later work, the special functions which are to be found in

diffraction theo ry, and a section on th e asym ptotic evaluation of

integrals. In Chapter 3 the formal derivations of the solutions to the

canonical problems that have formed the basis of GTD are given. The

laws of geometrical optics are developed in Chapter 4 from the

appropriate canonical problem, and in Chapter 5 high-frequency

diffraction by straight edges and surfaces is considered. Chapter 6 is

concerned with the application of GTD to curved edges and surfaces.

To conclude, a number of worked examples are given in Chapter 7 to

demonstrate the practical application of the GTD techniques developed

in the earlier chapters.

The purpose of this book, apart from expounding the GTD method,

is to present useful formulations that can be readily applied to solve

practical engineering problems. It is not essential, therefore, to under-

stand in detail the material in Chapters 2 and 3, and many readers will

want to treat these chapters as a reference only. At the end of Chapters

4 to 6 a summary is provided which gives the main formulas developed

in the chapter.

I wish to acknowledge the assistance of colleagues at the CSIRO

Division of Radiophysics in the preparation of this third edition, in

particular Miss M. Vickery and Dr. T.S. Bird. Further, I am indebted

to the following people for their contribution to specific sections:

Drs. D.P.S. Malik, G.T. Po ulton and G . Tong.

G.L. James

October 1985


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Chapter 1

Introduction

The Geom etrical Theory of Diffraction (GTD ), conceived by J.B.

Keller in the 1950s and developed continuously since then, is now

established as a leading analytical technique in the prediction of high-

frequency diffraction phenomena. Basically, GTD is an extension of

geometrical optics by the inclusion of additional diffracted rays to

describe the diffracted field. The concept of diffracted rays was

developed by Keller from the asymptotic evaluation of the known

exact solution to scattering from simple shapes, referred to as the

canonical problems for GTD . This rigorous mathem atical founda tion,

and the basic simplicity of ray tracing techniques which permits GTD

to treat quite complicated structures, are the main attractions of the

method.

In this book we will be concerned with GTD and its applications to

electromagnetic wave diffraction. Our main interest, apart from

expounding the techniques of GTD, is to develop useful formulations

that can be readily applied to solve practical engineering problems.

The approach taken is to begin with the solutions to canonical

problems, from which we develop the GTD method to treat more

complicated structures. In this way the laws of geometrical optics are

derived from the canonical problem of plane wave reflection and

refraction at an infinite planar dielectric interface. The GTD methods

for various diffraction phenomena which follow are then seen to be

obtained by a natural extension of this approach to other canonical

problems. For example, the half-plane and wedge solutions form the

basis of the GTD formulation for edge diffraction. Similarly, the GTD

formulation for diffraction around a sm ooth conv ex surface is developed

from the canonical problem of scattering by a circular cylinder.

In many instances the exact solution to the canonical problem

does not exist or is in a form not readily amenable to an asymptotic


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2 Introduction

evaluation. Often, however, diffracted rays may be extracted from an

asym ptotic evaluation of an approximate solution. Of course, these

rays will only be as good as the initial approximation. This procedure

is used here to obtain a uniform GTD solution to diffraction by a

discontinuity in curvature from the physical optics approximation

to the induced currents. It is also used to illustrate the important

relationship between GTD and the physical optics solution to the

half-plane.

With the solution to the canonical problem, whether exact or

approxima te, it is usually necessary to take an asym ptotic exp ansion

in order to determine some general laws regarding the behaviour of

both the geometrical optics and diffracted rays for the simple shape

under study. Since many complex bodies are made up of simple shapes,

we can determine by the GTD method an approximate solution to the

high-frequency radiating or scattering properties of a body by applying

laws appropriate to the individual shapes (which go to make up the

body) and then sum the various contributions. This procedure is

dependent on the local nature of high-frequency diffraction.

The classical paper on GTD is that of Keller (1962), although some

earlier work was published in Keller (1957a), Keller etal (195 7b) , and

Levy and Keller (1959). In these papers, diffraction coefficients derived

from the canonical problem are multiplied with the incident ray at the

point of diffraction to produce the initial value of the field on the

diffracted rays. These coefficients are non-uniform in the sense that

they are invalid in certain regions (such as the so-called transition

regions which will be defined later). Since this early work appropriate

integral functions for the transition regions have been developed to give

uniform solutions for quite general problems in edge and convex

surface diffraction.

Over the past decade there has been the tendency to attach labels

to the various theories that have been proposed to improve the basic

GTD. Table 1 lists the major ray optics methods currently in use. Note

that the term GTD is sometimes used to refer only to the Keller non-

uniform solution, whereas here we use it in the more usual context to

embrace all of the techniques listed in the table.

The UTD and UAT are two independent theories developed to

provide uniform diffraction coefficients. The term 'slope-diffraction'

refers to a higher-order diffraction form ulation wh ich is depen den t

on the first derivative of the incident field. An early example of slope-

diffraction given by Keller (1962) was, like his leading diffraction

term, non-uniform. Subsequent workers have developed uniform

slope-diffraction terms which are sometimes referred to as modified


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Introduction 3

Table 1. Ray optics methods

Major references

in the tex t

GO Geometrical Optics 4.1

GTD Geometrical Theory of Diffraction 5.4, 6.2

UTD Uniform Geometrical Theory of Diffraction 5.4, 5.7, 6.2

UAT Uniform Asymptotic Theory 5.7

MSD Modified Slope-Diffraction 5.5

slope-diffraction. All of these techn iques are discussed in the sections

indicated in the table.

Since its inception, GTD has proved its usefulness in solving many

and varied practical problems. Indeed GTD is capable of providing

accurate solutions to many complex problems difficult to solve by

other means. Its relatively simple formulation coupled with its ability

to give immediate insight into the mechanism involved in high-frequency

diffraction gives it wide appeal. This appeal is enhanced by the remark-

able accuracy that can be achieved when using it. By accounting for

multiple ray diffracted effects,

GTD can

make possible accurate solutions

for objects of less than a wavelength in size. The most widespread use

of GTD has been in solving problems relating to waveguides and

reflector antennas. Some of the earliest examples are to be found in

the work of Kinber (see Kinber, 1961, 1962a, 19626). More recently,

GTD has been successfully applied to treate quite complex structures.

A

good example is the analysis by Burnside

etal.

(1980) of the radiation

patte rn of an an tenna mo unted on an aircraft. A comprehensive survey

of the numerous applications of GTD up to 1980 is given in pt III of

Hansen (1981). Since this review, GTD has continued to be applied

to the traditional areas of waveguides and reflector antennas (a major

source of published work is to be found in the IEEE transactions on

Antennas and Propagation) as well as some more novel applications

such as propagation prediction over hilly terrain (Chamberlain and

Luebb ers, 198 2; Luebbers, 1984) and the effect ofa shoreline on ground

wave propagation (Jones, 1984).

GTD cannot be used in all circumstances. In common with all ray

techniques, it predicts infinite values for the field at congruencies or

caustics of the rays. In some cases edge diffraction caustics can be

overcome by the use of equivalent edge currents deduced from the

GTD formulation. In other cases, particularly where caustics lie in

illuminated regions, it is necessary to resort to more general integral


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4 Introduction

formulations to evaluate the field. The most well-known and often used

fomulations are the classical methods of aperture field and physical

optics approximation. Other more accurate integral equation methods

are sometimes used, especially in those circumstances where they can

test the accuracy of the various GTD formulations. A num ber of

integral equa tion m ethods often used in association w ith GTD are listed

in Table 2. (As with ray tracing method s they are commonly abbreviated

as shown in the table).

The first three techniques in Table 2 we have already alluded to and

they will be discussed further in the text as indicated. The Physical

Theory of Diffraction (PTD ) was developed by Ufimtsev (19 62 ), and

in concept it has close parallels with the development of GTD. Basically,

PTD is an extension of physical optics by an additional current derived

from the appropriate canonical problem. Fock (1946) originally

suggested this method for diffraction around a smooth convex surface,

and Ufimtsev extended the concept to include any shape which deviates

from an infinite planar metallic surface. In practice however, Ufimtsev

only applied this method to edge diffraction. The major difficulty of

PTD,

as is the case with all integral equation methods, is that the

resultant integrals are not always easily evaluated. Nevertheless, it does

find application (as with integral equation methods in general) to those

regions where the GTD method fails. A comparison between these

two high-frequency methods is to be found in Knott and Senior (1974),

Ufimtsev (1975), and Lee (1977).

In some cases (notably the half-plane problem), solutions for the

scattered field can be represented in terms of the Fourier transform

(or the spectrum) of the induced surface current distribution. This

correspondence between the scattered field and the currents flowing

on the surface of an obstacle was expressed as a general concept by

Mittra

et al

(19 76 ) as the Spectral Theory of Diffraction (STD).

Initially STD was applied to problems involving half-planes and has

Table 2 Integral equa tion m etho ds

ECM

—

PO

PTD

STD

MM

Equivalent Current Method

Aperture Field

Physical Optics

Physical Theory of Diffraction

Spectral Theory of Diffraction

Moment Method

Major references

in the tex t

6.7

2.1,3

2.1.4,5.6

—

—

-


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Introduction 5

subsequently been applied to plane-wave diffraction at a wedge

(Clarkowski et al., 1984). As with PTD the resulting integrals are not

always readily evaluated. The main use of STD to date has been to

provide accurate and sometimes exact solutions with which to test

and compare various approximate asymptotic methods. It has so far

been limited to analysing simple geometries, and it would not appear

to be readily amenable to treat more complex scattering bodies.

The Moment Method (MM) (Harrington, 1968) is a well-known

numerical method for analysing field problems associated with obstacles

of small dimensions. In certain applications the combination of MM

with asymptotic techniques such as GTD can yield solutions which

neither method can achieve effectively alone. For a concise overview

of these hybrid methods and their applications, the reader is referred

to the paper by Thiele (1982). It is likely that we shall see an increase

in the practical application of combined GTD-MM and other hybrid

techniques as they continue to be developed.

A number of general reviews of asymptotic methods in diffraction

theory are available, in particular those by Kouyoumjian (1965),

Borovikov and Kinber (1974), Knott (1985), and Keller (1985). These

papers, which are largely non-mathematical, provide additional insight

into GTD and associated asymptotic methods.

References

BOROVIKOV, V.A., and KINBER, B.Ye. (1974): 'Some problems in the

asymptotic theory of diffraction', Proc. IEEE, 62, pp. 1416-143 7.

BURNSIDE, W.D., WANG, N., and PELTON, E.L. (1980): 'Near-field pattern

analysis of airborne antennas',

IEEE Trans.,

AP-28, pp. 318-32 7.

CHAMBERLIN, K.A., and LUEBBERS, R.J. (1982): 'An evaluation of Longley-

Rice and GTD propagation models*, ibid., AP-30, pp. 1093-1 098.

CLARKOWSKI, A., BOERSMA, J., and MITTRA, R., (1984): 'Plane-wave

diffraction by a wedge - a spectral domain approach ', ibid., AP-32, pp . 20 -

29.

FOCK, V.A. (1946): 'The distribution of currents induced by a plane wave on the

surface of a conductor', J. Phys., USSR, 10, pp. 130-1 36.

HANSEN, R.C. (Ed.) (1981 ): 'Geom etrical theory of diffraction', (IEEE Press).

HARRINGTON, R.F. (1968): 'Field Computation by Moment Methods',

(Macmilian, New York ).

JONES, R.M. (1984) : 'How edge diffraction couples ground wave modes at a

shoreline',

Rad.

Sci., 19 , pp. 959- 965 .

KELLER, J.B . (1957a): 'Diffraction by an aperture', /.

AppL Phys.,

28, pp . 42 6-

444.

KELLER, J.B., LEWIS, R.M., and SECKLER, B.D. (1957b): 'Diffraction by an

aperture II\ ibid, 28 pp. 570 -579 .

KELLER, J.B. (1962): 'Geometrical theory of diffraction',/. Opt. Soc. Am., 52,

pp. 116-130.


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6 Introduction

KELLER, J.B. (1985): 'One hundred years of diffraction theory',

IEEE Trans.,

AP-33, pp. 123-12 6.

KINBER, B.Ye. (1961): 'Sidelobe radiation of reflector antenna', Radio

Eng.

and

Electron.

Phys.,

6 , pp. 545-558.

KINBER, B.Ye. (1962a):

4

The role of diffraction at the edges of a paraboloid in

fringe radiation', ibid., 7, pp. 79 -86 .

KINBER, B.Ye. (1962b): 'Diffraction at the open end of

a

sectoral horn ',

ibid.,

7,

pp . 1620-1623.

KNOTT, E.F., and SENIOR, T.B.A. (1974): 'Comparison of three high-frequency

diffraction techniques', Proc. IEEE, 62, pp. 1468-14 74.

KNOTT,

E.F.

(1985): * A progression of high-frequency RCS prediction techniques',

ibid.,

73, pp. 252 -264.

KOUYOUMJIAN, R.G. (1965): 'Asymptotic high-frequency methods', ibid.,

53 , pp. 864-876.

LEE, S.W. (1977): 'Comparison of uniform asymptotic theory and Ufimtsev's

theory of electromagnetic edge diffraction', IEEE Trans., AP-25, pp. 162-170 .

LEVY, B.R., and KELLER, J.B. (1959) : 'Diffraction by a smooth object',

Comm.

Pure Appl. Math., 12, pp. 159 -209 .

LUEBBERS, R.J. (1984): 'Propagation prediction for hilly terrain using GTD

wedge diffraction', IEEE Trans., AP-32, pp. 951-95 5.

MITTRA, R., RAHMAT-SAMH, Y., and KO, W.L. (1976): 'Spectral theory of

diffraction',

Applied Physics,

10, pp. 1-1 3.

THIELE, G.A. (1 982): 'Overview of hybrid methods which combine the moment

method and asymptotic techniques',

Proc. SPIE Int. Soc. Opt. E ng. (USA), 35% ,

pp.

73 - 79 .

UFIMTSEV, P.Ya. (1962): 'The method of fringe waves in the physical theory of

diffraction', Sovyetskoye radio, Moscow. Now translated and available from the

US Air Force Foreign Technology Division, Wright-Patterson, AFB, Ohio, USA.

UFIMTSEV, P.Ya. (1975): Comments on 'Comparison of three high-frequency

diffraction techniques', IEEE Proc, 63, pp. 1734-17 37.


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Chapter 2

Electromagnetic fields

In this chapter we begin with the field equations and representations

of the electromagnetic field which will be required in later developm ents.

Our main concern is with radiation and scattering from current distri-

butions, and use of potentials, both scalar and vector, to describe the

field. This is succeeded by a section on special functions which appear

in diffraction theo ry, namely the Fresnel integral, Airy, Fock and

Hankel functions. Fresnel integral functions have an essential role in

edge diffraction phenomena while both Airy and Fock functions are

used in describing diffraction around a convex surface. The discussion

of the Hankel functions is concerned with their asymptotic behaviour

for large values of argument. These various functions are not always

well tabulated, and in some instances it is necessary to resort to

numerical evaluation.

An asymptotic evaluation of the field integrals is given in the last

section. The solutions will prove to be useful in extending the methods

of GTD and in clarifying associated asymptotic methods such as the

physical optics approximation.

2.1 Basic eq uati ons

2.1.1 Field equations

Maxwell's equations

for time-harmonic electromagnetic fields, with the

time dependence exp (Jcjt) suppressed, are

- V x E = jcoB + M

VxH = / O J / > + / + /

C

where

E

is the

electric field intensity, H

is

the

magnetic field intensity,

B is the magnetic flux density, D is the electric flux density, M is the

magnetic source current, J is the electric source current, J

c

is the


17/311

8


electric current density, and to is the angular frequency. The four field

vectors and the electric current density are related by constitutive

equations,

which for linear isotropic media reduce to

B = /xff; /> = €£ ; J

c

= oE (2)

where /i, e, a are the

permeability, permittivity,

and

con ductivity

of the

medium. Eqn. 1 can now be written as

co

We shall consider solutions to eqn. 3 in those materials which are not

only isotropic and /mear (where JU and eare scalars and independent of

the field intensity) as imposed by eqn. 2, but also homogeneous (where

H

and e are independent of position).

Solutions to eqn. 3 can be expressed in terms of a magnetic vector

potential A

and an

electric vector potential F such

that

E = ~ V x F / c o / L 4 + r

/toe

(4)

# = Vx4--/toeF4- V(V-F)

/to/i

where 4 , F are determined from the inhomogeneous Helmholtz

equations

V

2

A+k

2

A

= - /

(5)

V

2

F + k

2

F = - 3 / ; *

2

= t o V

In the calculation of energy flow in the electromagnetic field, we make

use of the

Poynting vector S

which defines the intensity of energy flow

at a point. For time-harmonic fields

S = Re£V

w

' x Retfe**"

and is an instantaneous function of time containing both the average

power flow and pulsating reactive power. Hie radiating power flow at a

point is obtained from the time average

(S)

of the Poynting vector

(6)

where the asterisk denotes the complex conjugate. For the com plex

power

P

leaving a region we have


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Electromagnetic fields 9

P

=

j f f ExH*-ds

(7)

^ 8 ^

2.1.2 Radiation from current distributions

In solving eqn.

5

for the vector potentials due

to

the source c ur ren ts/

an d

M

we use the Green's function formulation such that

A

(r)

= f

J(r')G(r,r)dV

(8)

F(r) = f

M(r')G(r

f

r')dV

where the prime denotes the source co-ordinates, the volume V

9

is the

region bounded by the surface S' containing the sources as illustrated in

Fig. 2.1, and G(r

9

r) is th e Green's function. This function is th e

solution

to

the inhom ogen eous Helm holtz equation for a unit current

source, and its exact nature depends on the conditions under which the

solution is obta ined. For a source radiating into an unbound ed med ium

the solution must satisfy the radiation condition. If it is assumed that

the medium is slightly lossy, as is the case for all physical me dia, then

the radiation condition simply m eans that

all

fields excited

by the

source must vanish at infinity.

For 3-dimensional current distributions

in

unbounded media, the

appropriate Green's function is

--'-tf|r-V|)

H7r|r — r I

The electromagnetic field

is

obtained by substituting eqn .

8,

with the

Green's function of eqn. 9, into eqn. 4. Using the vector expansions

V x (B\l>) = i//V x B — B x Vi//

V- ( /? i / / ) = i//V B + B *Vi//

and noting that the source currents

/

and

M

are not functions

of

th e

vector r in eqn. 8 , eq n.

4

for the electroma gnetic field b ecom es

E(

r

) = f M(r) x VG -jcofjJ(r')G + — /(/•') • W G

\dV

'

J

V

'{ /coe J

H

(

r

)

= f

Sir )

x VG

~JueM{r)G +

—

M(r

J

v

> [ JOOfi

The operator V acting on the G reen's function yields

( 1 0 )


19/311

10 Electromagnetic fields

X

z

^ >

s'

7

f-f'

t

Fig.

2.1 Source distribution contained within a volume

V

VG

= — (/ * + —IGjft;

RR

= r - r

/•we =

and a similar expression exists for the quantity M * W G . Subst itut ion

into eqn. 10 gives

4

H{r)

= /*j (/( /) x

R

-JyA

Mr')-{M(r')

'R}A]\

G(r,r')dV

This solution for the electromagnetic field is applicable only in a

source-free region (w hich , as formulated in eq n. 1 1, is the region

external to the volume

V'

containing the sources). In radiation problems

this source-free region is conveniently divided up into three overlapping

zones;

namely, the near zone, the Fresnel zone, and the far zone. In the

near zone where R is small, no approximations in the evaluation of eqn.

11 are permissible and we must include all higher order terms in

R .

For

the

Fresnel zone, R

is sufficiently large for the field to be given, to a

good approximation, by the leading term in eqn. 11. Beyond the

Fresnel zone we have the far zone where r >

r

max

and the following

approximations are made in the first term of eqn. 11. From Fig. 2.1


20/311

Electrom agnetic fields 11

Fig.

2.2 Co-ordinate orientation

/? — r

—

r cos ̂ (for phase terms)

R ^ r (for amplitude terms)

and eqn. 11 for the far zone simplifies to

exp(jkr'cos$)dV'

•}r]

02)

r rL

In the far zone we see that the electric and magnetic field components

E and H are perpendicular to each other and to the direction of

propagation.

In evaluating field quantities we shall use the rectangular

(x,y,

z),

cylindrical (p, 0,z),and spherical

(r,O,


21/311

12 Electrom agnetic fields

where H ̂ is the zero-order Hankel function of the second kind. In

solving eqn. 10 for the Fresnel zone we can take the asymptotic

expansion of the Hankel function

H™ (kct) ~ / j j M

j

v

exp

{-jka)

for ka

> v

(15)

so that

the

substitution into eqn. 10 yields for the Fresnel zone

exp(-/W)

/(P')x

P-J\r\

IMP) -

where -4' is the area enclosing the sources and

PP = P - P '

In the far

zone

where

p

>

p

ma x

this

reduces further

to

e xp {/A:p'cos(0 —

)}dA'

( 1 7 )

~ /P)px£(p);

*2 7 . i

Equivalent source distributions

In formulating radiation problems it is often convenient to replace the

actual sources of the field with an equivalent source distribution. For

example, in Fig.

2.3a

the volume

V*

con tains the sources of a field

E

x

,H

X

and the evaluation of this field in the source free region external

to the volume

V'

is given by eqn. 10. If we now specify a source-free

field E

2

,H

2

internal to V, and maintain the original field external to

V* as in Fig.

23b,

then on the bounding surface

S'

one finds there must

exist surface currents

* The interpretation of y/ j in the above equation s and through out this boo k is

that it is taken to be equivalent to exp

(jn/4).


22/311

i i bi s'


13

I1.H

Fig.

2.3 Equivalent source distribution

M

s

=

(E

x

-E

2

)xn

(18)

to account for the discontinuity in the tangential components of the

field. T hese equ atio ns are applicable for a dis con tinu ity in bo th field

and media at the surface S\ When the surface currents J

8

and M

8

are

zero, they state the boundary conditions for electroma gnetic fields, in

that

the tangential com ponents of the electric and magn etic field are

continuous across a change in medium p rovided the conductivity is

finite. For a perfect electric conductor i = o is infinite, and a surface

conduction current exists while the tangential electric field vanishes.

If the region within the volume in Fig. 23b has infinite cond uctivity

then eqn. 18 becomes

0 = f

l

x « ;

J

8

= / ix

H

x

(perfect electric cond uctor) (1 9)

Similarly we can mathematically define (although it has no physical

meaning) a perfect magn etic condu ctor such that the tangential mag-

netic field vanishes at its surface. If the region with

V'

contains this

magnetic conductor, eqn. 18 becomes

M

s

=

E

x

x

h

0 =

h

x

H

x

(perfect magnetic condu ctor) (2 0)

Returning to the surface currents of eqn. 18, we can now substitute

these sources into eqn. 10 and perform the integration over S' to obtain

the field E

2

,H

2

internal to V\ and E

X

,H

X

external to V\ Thus the

equivalent source distribution of eqn. 18 has produced the same field

external to the volume as the original sources contained within it. If

these sources were external to V\ then the resultant field

E

X

,H

X

within the volume would be produced by eqn. 20 where

n

is now the

inward

normal from

V\

and £ 2 , # 2 is the external field to it . Thus,

having knowledge of the field bounding a source free region allows u s

to determine the field within that region.

Since the specification of the

field

E

2

,H

2

is arbitrary, it is of ten advan tageous to cho ose a null field

for the region containing the actual sources. The surface currents now

become


23/311

14


antenna

2 * 0

Fig. 2.4 Infinite plane for aperture-field method

/ j

=

n x

H

i \

M

9

—

E \

x

w

With a null field chosen for E

2

, # 2 we are at liberty to surround S'

with a perfect conductor (electric or magnetic) to remove one of the

equivalent source distributions. In such cases the remaining currents

radiate in the presence of a perfectly conducting obstacle and the

electromagnetic field cannot be determined from eqn. 10 as the

medium is no longer homogeneous.

If the surface

S*

divides all of space into tw o regions by coincid ing

with the z = 0 plane with the sources contained in the half-space z < 0,

then the perfectly conducting infinite plane at

z

= 0 can be replaced by

the image o f the surface currents. This gives a current distribution of

either J

8

= 2it x

H

or

M

s

=

2E

x

n

(depending on the conductor used)

radiating into an unbounded homogeneous medium. The field for z > 0

can then be calculated using eqn. 10. This is the basis of the aperture

field method

to ev aluate the radiated field from an aperture antenn a.

The z = 0 plane lies either in the aperture plane of the antenna or at

some position in front of it, as shown in Fig. 2.4. We then have three

possible solutions for the field in the z> 0 space. From eqn . II for

kR>\

J

9

=

In

x

H M

t

= 0

E

x

= -/* J y f e j {/. - (/, •

R)A}GdS'

Hi = jkj J.xfiGdS'

J.

= 0,

M

8

=

2Exn

E

2

=

-jk j M

8

xfiGdS'

H

2

= -7*J /(*)

{M.-(M.

(2la)

(21b)


24/311


J

8

=

n*H

y

M

8

=

Exn

+E

2

)

(21c)

These equations can also be applied to 2-dimensional distributions

when the similarity between eqns. 11 and 16 is observed.

Provided that the fields are kno wn exa ctly along the z = 0 plan e,

then identical results will be yielded from the three parts of eqn. 21. It

is preferable to use either eqn. 21a or 216 since eqn. 21c requires both

the electric and magnetic fields tangential to the chosen plane. In

practice, it is necessary to approximate these fields, so the three formu-

lations will, in general, yield differing results. It must be decided from

the type of approximations made, or imposed on for a given problem

which is the most suitable equation to use.

2.1.4 Form ulation for scattering

We now consider formulation of the field resulting from source currents

radiating in the presence of an obstacle, as illustrated in Fig. 2.5. The

region is homogeneous outside the obstacle with source currents/, M

as sho wn . The presence o f the obstacle creates inho m ogen eity w here

the constitutive parameters

[i

and e are functions of position. If we now

define these parameters in the homogeneous region outside the obstacle

as \i

x

, €i , we can rewrite M axwell's equa tions as

-VxE

=

joo^H

( 2 2 )

V x f f = / c £ ' + / '

where

M\f

are the mo dified source currents

r

= ycj(e - e , ) £ + / ;

M

1

=

/CO(JLI -H^H + M

(23 )

These modified source currents radiate in the unbounded homogeneous

region defined by Hi

f

e

i9

and the total field exterio r to the obsta cle and

the original source distribution J

f

M

is determined from eqn. 1 1. The

total field £*,// is given by the sum of an incident field

E\H

l

produced

by the original source currents J>M, and a scattere d field E*,H*

produced by

Jscatt^scatt

throughout the obsta cle, where

Jscatt

= / w (e - € , ) £ ; A f^ tt =

JG>(\x-yL

X

)H

( 2 4 )

If more than one obstacle is present, the process is repeated to yield the

field in the region exterior to the obstacles and original source

distribution.

Substitution of eqn. 23 into eqn. 10 will lead to two simultaneous


25/311


sources obstacle

Fig.

2.5 Sources radia ting in the presence of an obstacle

integral equations which will determine the field within the obstacle.

For a non-magnetic obstacle, M

Katt

will be zero , and the solution is

reduced to solving a single integral equation for the electric field E

within the obstacle. Even so, this may lead to formidable computational

difficulties if an attempt is made to solve this problem numerically. As

a first approximation we may replace

E, H

in eqn. 24 by the known

incident field E^H

1

. With E,H calculated by this approxim ation we

may generate an iteration process by substituting these values into eqn.

24 and recalculating

ad lib.

When the obstacle is perfectly conducting, the magnetic current

Mscatt

wiH be zero, and the electric current

J

K

an

reduces to a surface

current J

8

given by n x H as discussed in the previous section. If the

obstacle is large and the surface is smoothly varying with large radii of

curvature compared to the wavelength, the surface current may be

approximated by assuming that each point on the surface behaves

locally as if it were part of an infinite ground plane. The tangential

component of the magnetic field H at the surface of a ground plane is

given by tw ice the incident tangential field due to th e ima ge. Over the

illuminated portion of the surface S' we approximate the surface

current by

J

8

= In x H

l

( 25 )

and the resultant scattered field E

8

,H* is determined from eq n. 2\a,

where the magnetic field H is replaced by H

l

and the integration is

taken over the illuminated region of

S'.

This is the basis of the

physical

optics approximation and it is used extensively in electroma gnetic

scattering problems. Note that, as illustrated in Fig. 2.6, the surface

current is assigned a value only over the illuminated region of the

surface. In the shadow the surface current is taken to be zero. Thus it is

to be expected that this method will be unsuitable for predicting the

electromagnetic field in the deep shadow of the obstacle where these

neglected currents will be the main producer of the field.

2.1.5 Scalar potentials for source-free regions

In evaluating the field w ithin a ho m og ene ou s source-free region using

eqn.

4, the potential integral solution of eqn. 8 requires knowledge of


26/311

Electromagn etic fields 17

source

source

Fig.

2.6 Physical optics approximation for a perfectly conducting obstacle

a Original problem

b

Physical optics current distribution

the field bounding the region. An alternative approach is to seek

solutions to the Helmholtz equations in eqn. 5 for the potentials within

the source-free region. The potentials may then be chosen inde-

pendently of the actual sources external to the region producing the

field. Thus, if we choose A -zu

a

and F = zu

f

then the field equation s

o f eqn. 4 become

-juezu, 4- — V

7̂ M

and the scalar potentials u

ai

u

f

satisfy the scalar Helm holtz equa tion

/we

\dz)

= 0

(27)

For 2-dimensional fields independent of the z-direction, the above

equations simplify so that

( 2 8 )

and the remaining field components are given by

= z x VH

2

\ juyH = z x VE

Z

(29)

The simplest electromagnetic field solution to this equation is the plane

wave, which has some useful properties that we will exploit later in ray-

tracing techniques. As an example, consider an electromagnetic field in

a homogeneous medium having z-components propagating in the

n-

direction, as shown in Fig. 2.7, such that

E

g

= E

o

e x p { - / A ; ( x c o s 0 + > > s i n 0 ) } ; H

Z

= 0 (30a)


27/311

18


Fig.

2.7 Plane wave propagation in a homogeneous medium

and from eqn. 29 the other field components are

H = - / | - | O ? c o s 0- i s i n 0 ) £

2

(30b)

From these equations we note a fundamental property o f

a

plane wave

in that the electric and magnetic fields are orthogonal to each other and

to the direction o f propagation. Also, from the Poynting vector defined

by

eqn. 6, we see

that

the

average flow

o f

energy

is

in

the

direction

of

propagation. This latter statement, however,

is

true only

for

isotropic

media.

Eqn. 30 a is an elemental wave function which satisfies the scalar

Helmholtz equation of eqn. 27 for 2-dimensional fields. By super-

position, a linear sum of elemental wave functions can also represent a

solution

to

eqn. 27 ,

so

that

w e may

have

u = 2a

n

exp{-jk(xcos

n

+y sin

n

(0) and

P

n

(p)

may

be of

the form

M

(0) : cosw0, sin«0, exp(± /w0)


28/311


19

P

n

(p)

:

J

n

(kpl

N

n

(kp), //<

)

(*p) ,

H«\kp)

When 3-dimensional field solutions are required, then eqn. 33 becomes

a double summation with the addition of the z variable. For different

co-ordinate systems

we

must also

use the

appropriate functions.

Solutions to eq ns.

31

- 3 3 , however, are all that we shall require.

2.2 Special functions

2.2.

1 Fresnel integral functions

The

Fresnel integral

is defined for real argum ents as

F±(x)

=

I cxp(±jt

2

)d t

(34)

Jx

When the argum ent is zero

^/(tr\

I in\

(35)

and for large argum ents its asym ptotic solution is

where

A useful pro perty is

±

J

j 1 - F

±

(x) (37)

A function called

the

modified Fresnel integral

which

we

shall

use

extensively is given by

)

(38a)

I

m

s

0

tf

±

(-*) = exp(+jx

2

)-K

±

(x) (38c)

Another form of A_(x) is required in the rigorous solution for edge

diffraction. Beginning with the well-known result


29/311

20


we obtain the relationship

/ { - I e x p {-ja

2

t)dt = exp {—

t(v

2

-\-ja

2

)}dvdt

Jkp

V

\t ho

-

Interchanging the integration order, and performing the

t

integration,

the right-hand side becomes

£

ex

P

{-kp(v

2

+fa

2

)}

v

2

+ja

2

Now by simple substitution

f

7(T

3

kpN\ t

and we arrive at

(39)

In many engineering design applications the approximation to the

modified Fresnel integration in James (1 979) can be used. For positive

real values of

x

the approximation is

K

±

(x)

« J exp [±/(arctan

(x

2

+ 1.5* + 1)

- T T / 4 ) ]

l y / n x

2

+x + l x > 0

When the argument is zero, eqn. 40 retrieves the exact solution and for

large values of the argument it reduces to the asymptotic value given

by the first term in eqn. 38a. The greatest differences occur where x

is around 2.0, where the errors in the amplitude and phase of the

function peak at 8% and 2% respectively.

Fig. 2.8 plots the amplitude and phase of the modified Fresnel

integral for the exact, approx imate and a sym ptotic solutions. It is seen

that for

x

- 3 the asymptotic expansion is a good approx imation to the

function.

In problems involving multiple edge diffraction

we

sometimes require

the generalised Fresnel integral function described by

L±(x,y)

=

U(pc)K

±

(y)-G

±

(pc,yy, y >

0 (41)


30/311


21

where

U(x)

is the unit step fun ction,

K

±

(y)

the modified Fresnel

integral described above,

and G± (x,y) is the

two-argument Fresnel

integral function

of

Clemmow

and

Senior (19 53 ).

For x>0

this

function is described by

^ f L t Q * .

X

> 0 (42)

For small values of* and

>>

> 0

G

±

(x,y) * i

exp(+jx

2

){K

±

(y)- [(1 + />

2

)

tai

±jxy]/ir} (43)

When x

=

0, y

=£

0 this equation reduces to

G

±

(0,y) = \K

±

(y)

From eq n. 42 we have

G

±

(x,

0)

= 0 for x > 0 (44)

For small values of

y/x

G±(x, y) « j ; /x U ; - ̂ exp (T /ir/4)*T

±

(x) (45)

When A: = 7 the function simplifies to

G

±

(* ,x) = \Ki(x) (46)

The asymptotic expansion of

eqn.

42 for large V *

2

+ y

2

is

G±(*,^) ~ y exp (± /7r/4)/(2N /^(x

2

+

y

2

))K

±

(x)

(47)

For negative values of

x

G±(-x,y) = -G

±

(x,y)

(48)

For other values ofG±(x,

y)

not covered

by

the various approximations

in eqns. 43-48

we

need to numerically evaluate

the

function

as

given

by eq n. 42 . An alternative expression useful

for

numerical evaluation

is

G

±

(x,

y)

= J exp (+

f(x

2

-y

2

J)Kl(y)

-

; f exp (+

jx

2

)

exp (± ft

2

)

( 4 9 )


31/311


05

arg K

+

(X )

arg K.(X)

4-0 5-0

Fig.

2.8 Modified Fresnel integral

K

±

[

x

)

exact

approximate

asymptotic

2 2 . 2

Airy function

The

Airy function

for complex arguments i s given in integral form by

(50)

where

Cis the

contour

in the

complex f-plane shown

in Fig. 2.9a. We

shall

use a

different form

of the

Airy integral

to

that given

in

eqn.

5 0.

With the substitution T? = t exp

(/7r/3)

in eqn. 50, we obtain the relation-

ship

Ai

rexpl-'—

=

where

w i ( r )

*

and

the

contour T

is

shown

in

Fig. 2.9ft.

When the argument is zero

Ai(0)

=

0-355

(52)


32/311


23

t- plane

Fig.

2.9 Contours for the Airy integral

and

for

large arguments

its

asymptotic solution

is

| a r g z | < 7 r

A i ( z ) - );

n


33/311


Table 2.1 Airy function zeros and associated values

n

1

2

3

4

5

6

7

8

9

10

Ai(-a,

2-338

4-088

5-521

6-787

7-944

9023

10-040

11009

11-936

12-829

r,) = 0


34/311


t - p l a n e

Fig.

2.10 Contour for the Fock functions

/•(*)~-2/xexp|'—

«(x)~2exp -r-

(57c)

We shall make use of related functions named by Logan (1959) as the

Pekeris

carot functions

which are defined for x > 0 by the integrals

P(x) =

v(t)

=

To obtain an expression valid for all x we need to change the contour

path F. In general F can be any contour which begins at infinity within

the sector — it


35/311


S-plane

Fig.

2.11 Contour for the Pekeris carot functions

;57r\

ril

exp

T

x>0 (58c)

For a large negative argument the functions are given as

(5Sd)

Associated functions are the

Pekeris functions

defined by

p(x)

=

p(x)'

l

(59a)

q(x)

=

q(x)

When the argument is zero

p(0)

= 0-354 exp |

#

f

q(0) = -0 -3 0 7 exp r-7

(59*)

Note that the Pekeris carot functions tend to infinity as the argument

tends to zero.


36/311


Another Pekeris carot function given

by

Logan (1969) that we will

use is the two-argument function V

2

(x, y). Here, we relate this function

to

V(x>

y)

where

V(pc,y) =

-exp(/ir/4)K

2

(x,j/)

(60a)

As with eqn. 58a,

by

change of variable and integrating by parts we get

an expression valid for all x given by

f (£exp ( /

n/3)-y

2

exp (-/ir /3))exp

(fr exp(/TT/6))

X

JL (exp (- /i r/3) Ai'(?) - j exp (/TT/3)Ai(f))

2

(60Z?)

Evaluating this equation for x

>

0 by the method of residues yields

^

y

n l 6

2

? / 5 / 6 ) )

(60c)

where a

n

are the

roots

of

the equation Ai'(— a

n

) +y exp (—

jn/3)Ai

( - 5

n

)

=

0, and

For a large negative argument an asymptotic evaluation of eqn.

60b

yields

(60.)

The associated Pekeris function is given by

V(x,y) =

vipcy)*-^-

(61)

Finally, from

the

definition

of V(x,y)

we see that

V(x,

0)

= q(x)

and

F( x ,oo) =p(

X

) .

In evaluating

the

various Fock functions

the

reports

by

Logan

(1959) are an invaluable reference. However the function V(x,y) is not

well tabulated for general values of y. Some results are given in James

(1980)

for

real values

of

y, otherwise

it

will

be

necessary

to

evaluate


37/311

28


eqn.

606 numerically where

x

is small and revert to eqns. 60c and

60d

when x is sufficiently large.

2.2.4

Hankel

functions

The

Hankel

functions can be defined in terms of Bessel functions of the

first and second kind as

(62)

= ^ (63)

For large values of argument we may replace the functions in eqn. 62

with their asymptotic expansions. We begin with the

uniform

asymp totic

expansions, see Jones (1964 ).

which have the Wronskian relation

|argz|


38/311


sec~

l

(—

z) =

7T —

(66rf)

(66e)

We shall now give the approx imations to eqn. 6 4 for various ranges of v.

Initially we must assume that these equations have restrictions on arg v

and arg z as in eqn. 64. The continuation formulas of eqn. 66 can then

be used to determine if they have a wider validity. In particular, we

shall require the ranges argz=O, — f -plane where f


39/311


from which we deduce

=

2

1 / 3

r e -

inserting this value for £ into the expressions in eqn . 64 yields

/ 2 \

1 / 3

=

(J'-X)I-I

, v^

(69)

— In — for

For the condition

\v\>x

then the value of £ is now given approxi-

mately as

and

Substituting into eqn. 64, and using the appropriate Airy function

asymptotic expansion in eqn. 53 , we get

- ,

\v\>x

(70a)

and using the continuation formulas in eqn. 66

Y , \v\>x (70*)

2.3 Asymptotic evaluation of the field integrals

2.3.1 Method of stationary phase

When the electromagnetic field is determined from an equivalent source

distribution over a surface, the resultant integrals may be written in the

form


40/311


/ = J J f(x,y)exp{jkg(x,y)}dxdy (71)

where S is the surface over which the equivalent sou rces are formulated.

Usually

the

functions f(x,y)

and

g(x,y)

are

such

as to

preclude

an

analytical evaluation of eqn. 7 1 , and numerical integration has to be

used.

In

man y in stan ces, provided that k

is

large,

an

asymptotic

evaluation of the integral is possible dependent on the behaviour of the

phase function g(x,,v)-

In general,

k

will be complex , but we will assume in this section that

the medium

is

on ly slightly lossy

so

that

we

have

arg

A:

— 0.

This

assumption simplifies

the

procedure

and

will

be

adequate

for

most

of

the applications to be discussed later. We begin by considering the

asym ptotic behaviour of single integrals.

IS.LI Single integrals

For a 2-dimensional equivalent source distribution, eqn. 71 reduces to

/ = f f(x)exp{jkg(x)}dx

(72)

Jb

where a, b

are the

limits

of the

sources.

To

assist

in the

asymptotic

evaluation we rewrite this equation as

- b

- f(x)exp{jkg(x)}dx

i J a

f(-x)exp{jkg(-x)dx

(73)

and consider first the term /

0

. If

k

is large then the phase o f /

0

will vary

rapidly, and provided that

f(x)

is a sm oothly varying func tion, the

value of

I

o

will tend to zero as

k

tends to infinity. If however, the phase

function

g(x)

has points where it is stationary, i.e.

g(x)

= 0 , then the

exponential term exp

{jkg(x)}

will not vary rapidly in the vicinity of

these points, and the major contribu tions to /

0

will be from those

regions where

g(x)

= 0. By expanding the functions in /

0

around these

points we obtain an asym ptotic evaluation of the integral. Such points

are called stationary phase points

and the

procedure

is

known

as the

method of stationary phase.

A

first order

stationary phase point at

x

=

x

0

is defined by

g(x

0

)

= 0 , £"(*o)

¥*

0. Expanding the phase term


41/311


g(x) in a Taylor series about x

0

, and retaining only the firs t two non-

zero quantities, we have

£ < * ) - # ( * o ) + * ' ' ( * o ) ~ ; s = x -x

0

Over the region described by this equation it is assumed that f(x) is

slowly varying and can be approximated by f(x

0

).

The equation for /

0

now becomes

/

0

-/(x

0

)exp{M^o)}J°[

o

exp/V(x

o

)yU (74)

The integral in eqn. 74 can be rewritten as

which is in the form of the Fresnel integral with zero argument as given

by eqn. 35. Substituting this equation, after reducing it by the terms of

eqn. 35 , into eqn. 74 gives

J

*

If /(*o)

=

0> then the next higher order term in the asymptotic

expansion, obtained by expressing/(x) as a Taylor series about x

0

, is

proportional to k~

$n

f

n

(x

0

). In this case any end-point contribution, to

be discussed below , will be the leading term in the asymptotic evaluation

of eqn. 73.

If

ff"(*o)"*O,

then eqn. 75 will fail, and it is necessary to consider

an additional term in the expansion of the phase function g(x)

so that eqn. 74 now becomes

Cxo) j +

b' (*o)

j l

I

*

The integral in this equation can be expressed in terms of the Airy

function, Ai(x), given earlier (see eqn. 54) for real arguments as


42/311


from which, by change of variable

t

=

(kk\g'"(x

o

)\)

m

(a +

^ 7 ^ ) ,

get \

g

^

Xo

''

k , „

3

^

2

\

r 2

i

1 / 3

,

/

0

^ 2nfexp(jkg) exp ( / - fe )

J

(g )"

2

J

777^7 A i(f)

we

where

/,\2/3

(76)

-For large values of f the asymptotic expression for the Airy function

given by the first line of eqn. 53 reduces eqn. 76 to eqn. 75. When

f = 0, i.e.

g"(x

Q

)

= 0, we have a

second order


x

= x

0

. This can be consider ed as the conflue nce of tw o nearby first-

order stationary phase points.

If g"(x

0

) -> 0 then eq n. 76 beco me s invalid. As before we could

consider an additional higher order term in the expansion of

g(x).

The

resultant integral, however, cannot be expressed in terms of known

functions and in such a situation it is necessary to solve the original

integral numerically.

It remains now to evaluate the integrals in eqn. 73 which possess a

finite limit in the form

la

=f°°

f(x)exp{jkg(x)}dx

(77)

Jot

For stationary phase points removed from the end point at x = a, we

evaluate their contribution to I

a

from either eqn. 75 or eqn. 76 , as just

discussed. This will give the leading term in the asymptotic evaluation.

The next term is given by the contribution from the end point at

x

=

a.

Writing eqn. 77 as

and solving by parts, we get for the end-point contribution

) (78)

The contribution from the upper limit has been removed by tacitly

assuming that the medium is slightly lossy, giving k a small imaginary

component. Note that this term is of the order

k~

vl

greater than the

stationary phase value for a first order point.

We may proceed in the same way to obtain the next higher order

term for the end-point contribution as


43/311

34


_ i _ A«yw-A«)f(«)

e x p 0

^

a ) } +

0(

r>) (79)

and so

on . In

fact successive evaluation yields

the

asymptotic series

^ ^ ^ ) (80)

where

m ( ) r m l W

f o r

and

In some instances both

/ ( x ) and #'(*)

have

a

first order zero

at x = a.

For this case applying L'Hopital's rule

to

eqn.

78

gives

When a first order stationary phase point approaches the

end

point at

x = a, it is

necessary

to

consider

the

coupling effect between

the

two points. Thus, when

x

Q

approaches

a, by

expanding

g(ct)

in

a

Taylor

series about x

0

,

we

have

*(* o) -£() « ~ i( ) ;

s = x

- a

and

on

using

the

sign information

in

eqn.

82,

this

m ay be

written as

^)^(a)±|e,j|ir'(a)l

+

j ^ » l j ; i r » * 0 (83)

where

c,

=

sgn(a-x

0

)


44/311


35

Consider first the case where e

t

is positive and the stationary phase

poin t is just outside the integral. Substitution o f eqn. 83 into eqn . 77

with the assumption, as before, that

f(x)

is a slowly varying function,

then

j^ (a )| J Idx;

By the change of variable

this equation reduces to

e ,>0 , g (a)*0

where

The integral in eqn. 84 is the Fresnel integral

F±(v)

and has been

defined earlier by eqn. 34. Consider now the case when e

x

is negative

and the stationary phase point has moved inside the integral. Eqn. 77

may now be written as

}dx-[ f(x)exp{jkg(x)}dx\

J-oo

e , < 0

The first integral will yield the stationary phase contribution, 7

0

, as in

eqn. 75 at

x =x

0

.

The second integral is expanded about

x

— a, using

eqn. 83 as before, to give

±jk -

By the change of variable

••-j

this equation becomes

y |^(a)|J \dx\


45/311

36


so that the solution for I

Q

when e

t

<

0

is

given by

e i < 0 , g(c

Combining this result and that of

eqn.

84

e

I

/(«)exp{

W

«),/^}j{^}^W;

(86a)

where

ei = sgn(a-x

o

)»

and (/(x) is the unit step function, i.e., ( 7 = 1 for x>0 and zero

otherwise.

When

the

Fresnel integral argument

vis

large, the leading term

in the

asymptotic expansion of the Fresnel integral as given by eqn. 36

together with the sign information in eqn. 82 reduces eqn. 86a to

/« ~ U(-€

X

) /

0

- ~ Q& exp

{/*£(


46/311


Unfortunately such a function is not so easily generated as, for example ,

the Fresnel integral and comp lete Airy function. For a discussion on

the incomplete Airy function see Levey

and

Felsen (1 96 9) .

2.3.1.2 Double integrals

For 3-dimensional equivalent source distributions, we must use the

double integral

/

= f [

f(x,y) exp{jkg(x

9

y)}dx dy

(88)

j

J

S

where S contains -the sources. As before, the major contribution of th e

integration

for

large

k

occurs when

g{x,y)

is

stationary,

i.e.,

V^(JC,.V)

= 0 . For a stationary phase point at

(x

o

,yo)

we make the

substitutions

s

x

=

x~-x

Oi

s

2

=

y-y

0

and using the notation

bxbybz

gxyz

the expansion of g{s

x

, s

2

) in a Taylor series about the stationary point

a t ( 0 ,0 ) b e c ome s

+ . . . } ; where

g

= ^ ( 0 , 0 ) (89)

For a first-order stationary phase point we retain terms up to the

quadratic form only. In order to evaluate the resultant integrations, we

introduce a change of variables for s

{

, s

2

to u

t

, u

2

such that g«

u

= 0 .

This simply involves a co-ordinate rotation and allows us to treat each

integration (for a first-order stationary phase point) independently.

Using matrix notation

where

T

denotes the transpose of the matrix (i.e. , the interchange of

rows and columns) and 5 and G

g

are

L

S

2 j [gsih 8s

2

8

2

\

The relationship between s and the new variables u can be written as

[

c o s 0 s i n 0 l [Mil

, u = (90)

—sin0 cos0J [w

2

J


47/311


Now for

Mi,

u

U 2

(0,0) (95)

where

P

u

n

(»)

= J

^ P J ^ T ^ S u n W n O O }

du

n

\

H

=

1,

2

This integral was evaluated earlier in eqns. 74 and 75 as


48/311


I

1

* „ _ f_ i-.-xl I

fgfi\

Combining eqns. 95 and 96, and using eqn. 93, we may write /oo in the

compact form

A>0 — ,

/ r * i \

(97)

1/

\kg+^{^(g:

iU i

)+sg

n

(g:

iU i

)}\j

There may, of course, be more than one stationary phase point within

S,

and provided that these points are well separated, the asymptotic

evaluation of the integral is given by the sum of the individual contri-

butions as in eqn. 97. A different angle of co-ordinate rotation will, in

general, be required for each point.

We now consider the situation where the integration of eqn. 88 has

finite limits. It will be found convenient to express the integral in terms

o f angular variables, £, # , so that

-1.1

( 98 )

where we no te that the £ integration has no endp oint c ontrib ution , and

the only non-zero endpoint contribution from the

\p

integration is at

the upper limit ^ t ( | ) . For an asymptotic evaluation we put this

equation in the form

/ =f°°

r

iU2)

f(u)exp{/kg(u)}du

(99)

J-ooJ-oo

which may be rewritten in accordance with eqn. 73 as

£

o pa(u

2

) poo poo poo poo

oo J -oo J-oo J-o o J-oo Ja(u.,)

(100)

Any stationary phase points within

I

Oa

will cancel with those in /

O

o to

ensure that the total solution will not give a contribution outside the

limits of integration. Thus we need consider only the endpoint contri-

bution of /

O tt

. On using the solution of eqn. 86b for the u

x

integration

in /

O a

when the endpoint is isolated from any stationary phase points


49/311

40


\

]

x^[ikg{a{u

2

\u

2

})du

2

(101)

t

h

U

2

f

This integral can now be evaluated for stationary phase points at

u

2

= u

e2

determined when£y

2

{cc(u

e2

),u

e2

}= 0

to

give

C2 } = 0

: '"

a ( W e j )

' 0 0 2 a )

Note that, in general, a is a function of

u

2

.

When

the


(0 ,

0) approaches the end po int,

which alternatively means

| a | - * 0 ,

we

must

use the

formulation given

in eqn. 86 for the endpoint contribution, so that

I

0OL

now becomes

Ate

~

±

̂

/exp(/**)exp(*/i>

N\k\g

(1026)

where

i s

It

is

important

to

note that differentiation with respect

to

the variable

u

2

in eqn. 102 must encom pass th e endpoint function a(u

2

). Also, as for

eqn. 86, we use the asymptotic solution in eqn. 102# when the Fresnel

integral argument

v >

3 0.

2. i.

2 Method of steepest descent

The integrals solved asymptotically

in the

previous section were

for

functions involving real variables. A more general integral which

we

will

sometimes be required to solve involves functions of a complex

variable. For our purposes

it

will suffice

to

consider single integrals

of

the form

= f

J

f(z)exp{jkg(z)}dz

(103)

c

where / ( z ) and g(z) are regular functions of

the

complex variable z

along

the

integration path

C,

which

has

its endpoints

at

infinity, and it

is still assumed that arg

k

— 0.

The phase term g(z) in

eqn. 103 may be

written

as

g(z)

=

u(x,y)+jv(x,y),

where z = *+/> , (104)


50/311

Electrom agnetic fields 41

y

Fig.

2.12 Path C in the complex z-plane

and

M,

v are real functions which satisfy the Cauchy-Riemann equations

IT

= IT' r = - ? (

1 0 5

>

OJC

dy qv ox

Substituting eqn. 104 into eqn. 103 gives

I - f(z)exp(jku)exp(— kv)dz

c

Qearly, the magnitude of this integral will change most rapidly along

bv

the path where — i s a maximum. Similarly the phase will change most

rapidly where — is a maximum .

From Fig. 2.12 we have

bv bv bx bv by bv bv

bu bu bu

bC bx by

The values of 0 corresponding to the maximum of these functions are

determined by

3

(bv\ bv bv

M

bv by bv bx

— I—I =

s

i

n

0 -f — co s0 = — + = 0

bS \bC) bx by bx bC by bC

be\bC)

"

bx

bu bu by bu bx

Upon employing the Cauchy-Riemann equations of eqn. 105 we get

— = 0 for a maximum change in v

bv

— = 0 for a maximum change in u

bC


51/311


In other words, along paths of constant phase the amplitude of

exp{jkg(z)}

is changing mo st rapidly, and along paths of constan t

amplitude the phase of

exp{jkg(z)}\s

changing mo st rapidly.

In line with the method of stationary phase discussed in the previous

sect ion , the major con tribu tion to the integral in eqn . 103 for large A:

will

occur in the vicinity of stationary, or saddle points where ^'(z

o

) = O.

The

method of steepest descent

attem pts to deform the original

contour C in eqn. 103 into a path which, while passing through the

saddle point, gives the most rapid decay in the magnitude of ex p {jkg{z)\

Thus the requirement for a steepest descent p ath through a saddle p oint

at z = z

0

is that along the path

w(v,7) = constant ,

v(x,y)>v(x

Ot

y

o

)

( 1 0 6 )

It is not always possible to determine the complete steepest descent

path easily, and it is common practice to expand g(z) in a Taylor series

about the saddle points to obtain an asymptotic evaluation of the

integral. This procedure is the same as that carried out in the previous

section, and for the cases considered there, gives the results in the same

form. Note that the method of stationary phase emerges from the

above argument as the special case when we choose the path through

the saddle point such that

V

= constant.

We have no need of further discussion of this approach, but for more

details the reader is referred to Chapter 4 of Felsen and Marcuvitz

( 1973) .

References

CLEMMOW, P.C., and SENIOR, T.B.A. (1953): 'A note on a generalized Fresnel

integral', Proc. Camb. Phil Soc, 49, pp. 570-57 2.

FELSEN, L.B., and MARCUVITZ, N. (1973): 'Radiation and scattering of

waves', Prentice-Hall.

FOCK, V.A. (1946): The field of a plane wave near the surface of a conducting

body', /. Phys.

t

10, pp . 39 9-4 09. Also see FOCK, V.A. (1965): 'Electromagnetic

diffraction and propagation problems', (Pergamon).

JAMES, G.L. (1 979): 'An approximation to the Fresnel integral',

Proc. IEE E,

67 ,

pp . 677-678.

JAMES, G.L. (19 80): 'GTD solution for diffraction by convex corrugated surfaces',

IEEProc, 127, Pt. H, pp. 257- 26 2.

JONES, D.S. (1964 ): 'The theory of electromagnetism', (Pergamon), pp . 3 5 9 -

363.

LEVEY, L., and FELSEN, L.B. (1969): 'On incomplete Airy functions and their

application to diffraction problems', Radio Set., 4, pp. 959-9 69.

LOGAN, N.A. (19 59): 'General research in diffraction the ory ', Missiles and Space

Division, Lockheed Aircraft Corporation, Report LMSD-288087, Vol. 1, and

Report LMSD-288088, Vol. 2.


52/311

Chapter 3

Canonical problems for GTD

The principle canonical problems which formed the basis of GTD are

formally derived in this chapter. All the problems are 2-dimensional and

more general formulations are developed from them in later chapters.

Thus we develop the methods of geometrical optics from reflection and

refraction of a plane wave at an infinite plane dielectric interface. The

high frequency behaviour of the half-plane and wedge solution is the

starting poin t for a GTD edge diffraction formu lation. Although the

half-plane can be considered as a special case of the wedge where the

wedge angle is zero, it merits separate considerat

book - go - geometrical theory diffraction for em - graeme

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