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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
3.2.1 Electric polarisation 53
3.2.2 Magnetic polarisation 61
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
3.4.1 Electric polarisation 75
3.4.2 Magnetic polarisation 83
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
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8
Electromagnetic fields
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 )
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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
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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,
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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).
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i i bi s'
Electromagnetic fields
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
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14
Electromagnetic fields
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)
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Electromagnetic fields 15
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
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16 Electromagnetic fields
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
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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)
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18
Electromagnetic fields
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)
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Electromagnetic fields
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
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20
Electromagnetic fields
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)
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Electromagnetic fields
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 )
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22 Electromagnetic fields
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)
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Electromagnetic fields
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
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24 Electromagnetic fields
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
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Electromagnetic fields 25
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
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26 Electromagnetic fields
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.
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Electromagnetic fields 27
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
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28
Electromagnetic fields
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|
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Electromagnetic fields 29
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
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30 Electromagnetic fields
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
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Electromagnetic fields 31
/ = 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
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32 Electromagnetic fields
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
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Electromagnetic fields 33
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
stationary phase point at
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
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34
Electromagnetic fields
_ 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
)
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Electromagnetic fields
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\
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36
Electromagnetic fields
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
{/*£(
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Electromagnetic fields 37
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
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38 Electromagnetic fields
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
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Electromagnetic fields 39
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
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40
Electromagnetic fields
\
]
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
stationary phase point at
(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)
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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
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42 Electromagnetic fields
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.
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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