CONFIGURATIONAL FILTERING AND SAMPLING
METHODS IN THE COMPUTATION OF
FREQUENCY-DOMAIN ELECTROMAGNETIC FIELDS
R.W.C. van der Veeken
TR diss
1549
<
'I-«I
( ^ j
CONFIGURATIONAL FILTERING AND SAMPLING
METHODS IN THE COMPUTATION OF
FREQUENCY-DOMAIN ELECTROMAGNETIC FIELDS
R.W.C, van d e r Veeken
<^
^
(ijj &o,
■V&\
9 Prometheusplein 1 ^ ) 26to ZC DELFT "7/TR diss
1549
CONFIGURATIONAL FILTERING AND SAMPLING
METHODS IN THE COMPUTATION OF
FREQUENCY-DOMAIN ELECTROMAGNETIC FIELDS
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan
de Technische Universiteit Delft, op gezag van
de Rector Magnificus, prof. dr. J.M. Dirken,
in het openbaar te verdedigen ten overstaan
van een commissie aangewezen door het
College van Dekanen op dinsdag 9 juni 1987
te 16.00 uur door
R e n a t u s Willem Clemens van d e r Veeken
elektrotechnisch ingenieur
STELLINGEN behorende b i j het p r o e f s c h r i f t :
"Configuratlonal F i l t e r i n g and Sampling Methods i n the
Computation of Frequency-Domain Electroraagnetlo Field3"
CONTENTS p a g e
ACKNOWLEDGEMENTS
1 . GENERAL INTRODUCTION 1
2 . ANALYTICAL SOLUTIONS FOR ELECTROMAGNETIC RADIATION AND
SCATTERING IN SOME ELEMENTARY MULTI-LAYER CONFIGURATIONS 11
2 . 1 . I n t r o d u c t i o n 12 2 . 2 . Formulation of the source and the s c a t t e r i n g problem 14
2 . 2 . 1 . The f i e l d problem in the ( x , y ; k )-domain 22 2 . 2 . 2 . The f i e l d problem in t h e (k ,k ;z)-domain 23
x y
2.3. Electromagnetic fields generated by some elementary
sources in free space 27 2.3.1. One-dimensional source problems 32
2.3.2. Two-dimensional source problems 34 2.3-3. Three-dimensional source problems 40
2.3.4. Numerical results 48 2.4. Scattering in a spherical multi-layer configuration 58
2.4.1. Spherical layer 58 2.4.2. Boundary conditions 66 2.4.3. The homogeneous sphere in free space 68
2.4.4. Incident fields 71 2.4.5. Numerical results 74 2.5. Electromagnetic fields in cylindrical multi-layer
configurations 81 2.5.1. Cylindrical layer 81
2.5.2. Circular cylindrical layer 88 2.5.3. Boundary conditions 91 2.5.4. Two-dimensional scattering by a circular cylinder 95
configurations 106 2.6.1. Homogeneous plane layer 106
2.6.2. Boundary conditions 116 2.6.3. Scattering in two half-spaces 121
2.6.4. One-dimensional scattering 123
2.6.5. Numerical results 125
2.7. Conclusions 128
SPECTRAL-DOMAIN FILTERING AND SAMPLING APPROACH TO THE NUMERICAL SOLUTION OF THREE-DIMENSIONAL ELECTROMAGNETIC
SOURCE PROBLEMS 129 3.1. Introduction 130 3.2. Integral representation of the source problem 132
3.3- Systematic discretization of the integral representation
by spectral-domain filtering and sampling 137 3.4. The three-dimensional free-space Green's function
filtered by spherically symmetric filters 145
3.4.1. Unit-step spherical filter 149 3.4.2. Cosine-squared spherical filter 150 3.4.3. Bessel spherical filter, a non-band-limited
filter 153 3 . 5 . Numerical r e s u l t s 155
3 . 6 . C o n c l u s i o n s 161
SPECTRAL-DOMAIN FILTERING AND SAMPLING APPROACH TO THE NUMERICAL SOLUTION OF ELECTROMAGNETIC SOURCE PROBLEMS OF
LOWER DIMENSIONALITY 163
4 . 1 . I n t r o d u c t i o n 164 4 . 2 . Formulation of the two source problems 166
4 . 2 . 1 . The (k ,k ;z)-domain source problem 169 x y
4 . 2 . 2 . The ( x , y ; k )-domain source problem 171 4 . 3 . S y s t e m a t i c d i s c r e t i z a t i o n of t h e i n t e g r a l
r e p r e s e n t a t i o n s by s p e c t r a l - d o m a i n f i l t e r i n g and
sampling 174 4 . 4 . The one-dimensional (k ,k ;z)-domain G r e e n ' s function
x y s p e c t r a l l y f i l t e r e d by symmetric f i l t e r s 181 4 . 1 . 1 . U n i t - s t e p f i l t e r 183 4 . 4 . 2 . Cosine-squared f i l t e r 186 4 . 4 . 3 . Bessel f i l t e r , a n o n - b a n d - l i m i t e d f i l t e r 18? 4 . 4 . 4 . Numerical r e s u l t s 188 4 . 5 . The two-dimensional ( x , y ; k )-domain G r e e n ' s function
s p e c t r a l l y f i l t e r e d by c i r c u l a r symmetric f i l t e r s 193 4 . 5 . 1 . U n i t - s t e p c i r c u l a r f i l t e r 198 4 . 5 . 2 . C o s i n e - s q u a r e d - l i k e c i r c u l a r f i l t e r 200 4 . 5 . 3 . Bessel c i r c u l a r f i l t e r , a n o n - b a n d - l i m i t e d f i l t e r 203 4 . 5 . 4 . Numerical r e s u l t s 204 4 . 6 . C o n c l u s i o n s 211
LEAST-SQUARES ITERATIVE TECHNIQUES IN THE SOLUTION OF
ELECTROMAGNETIC SCATTERING BY PENETRABLE OBJECTS 213
5 . 1 . I n t r o d u c t i o n 214 5 . 2 . I n t e g r a l - e q u a t i o n formulation of the s c a t t e r i n g problem
and a root-mean-square error c r i t e r i o n 216 5 . 3 . I t e r a t i v e m i n i m i z a t i o n of the r o o t - m e a n - s q u a r e e r r o r 220
5 . 1 . The i n i t i a l guess and t h e f i r s t i t e r a t i o n s t e p 228 5 . 5 . S e l e c t i o n of the v a r i a t i o n a l f u n c t i o n s and enumeration
of s i g n i f i c a n t i t e r a t i v e t e c h n i q u e s 230
5 . 6 . Numerical r e s u l t s 235
5 . 7 . C o n c l u s i o n s 243
CONFIGURATI0NAL FILTERING AND SAMPLING METHODS IN THE NUMERICAL SOLUTION OF THREE-DIMENSIONAL ELECTROMAGNETIC
6.4. Spatial-domain filtering and spectral-domain sampling 269
6.5. Numerical solution method and results 277
6.6. Conclusions 298
CONFIGURATIONAL FILTERING AND SAMPLING METHODS IN THE NUMERICAL SOLUTION OF ELECTROMAGNETIC SCATTERING BY
ONE-AND TWO-DIMENSIONAL OBJECTS 299
7.1. Introduction 300 7.2. Formulation of the two scattering problems 302
7.2.1. One-dimensional objects 307 7.2.2. Two-dimensional objects 311 7.3. Spectral-domain filtering and spatial-domain sampling 316
7.4. Spatial-domain filtering and spectral-domain sampling 323
7.5. Generation of test solutions 329 7.6. Numerical solution method and numerical results 333
7.7. Conclusions 373
APPENDICES 375
A. THE SPATIAL FOURIER TRANSFORM AND SOME OF ITS PROPERTIES 376
B. THE THREE-DIMENSIONAL FREE-SPACE GREEN'S FUNCTION AND SOME
OF ITS SPECTRALLY FILTERED COUNTERPARTS 379
C. THE TWO-DIMENSIONAL (x,y;k2)-D0MAIN FREE-SPACE GREEN'S
FUNCTION AND SOME OF ITS SPECTRALLY FILTERED COUNTERPARTS 385
D. THE ONE-DIMENSIONAL (k ,k ;z)-D0MAIN FREE-SPACE GREEN'S
E . SCALAR POTENTIALS 398
F . SAMPLING THEOREM FOR THREE-DIMENSIONAL, BAND-LIMITED
FUNCTIONS 1401
G. ESTIMATES OF THE GLOBAL DEVIATION OF THE SPECTRALLY FILTERED ELECTROMAGNETIC FIELD FROM THE EXACT SOLUTION IN
THREE-DIMENSIONAL SOURCE PROBLEMS 403
H. ESTIMATES OF THE GLOBAL DEVIATION OF THE SPECTRALLY FILTERED ' ELECTROMAGNETIC FIELD FROM THE EXACT SOLUTION IN
SOURCE PROBLEMS OF LOWER DIMENSIONALITY 1)08
I . ORTHOGONALITY PROPERTIES OF VARIATIONAL FUNCTIONS IN
ITERATIVE SCHEMES 417
REFERENCES 425
SAMENVATTING 431
Many people have contributed to creating t h i s t h e s i s . To a l l of these
people the author wishes to express h i s sincere thanks. Special thanks
are due to the members of the Laboratory of Electromagnetic Research for
their kind help and for their constructive criticism. The assistance of
various services of the Faculty of E l e c t r i c a l Engineering and the Delft
University of Technology is appreciated.
/
C h a p t e r 1
In this thesis the computation of time-harmonic electromagnetic fields is investigated. Electromagnetic fields are widely used in technical
applications such as radar, optical communication systems [e.g. Suematsu (1983)], radio, television, telephony and telegraphy, hyperthermia as cancer therapy [e.g. Van den Berg et al. (1983)], electromagnetic compatibility and geophysical prospecting [e.g. Wannamaker et al. (1984)]. In the electromagnetic modeling of many of these engineering problems we can distinguish four important field problems, viz. direct source, direct scattering, inverse source and inverse scattering
problems. In a direct source problem the electromagnetic field excited by a specified source, e.g. an antenna or a light source, in some embedding medium has to be determined. The solutions of these source problems are used in scattering problems. In a direct scattering problem one likes to know the distribution of an electromagnetic field in a configuration consisting of a specified source and a specified scattering object, e.g. a human body or a lens, in some embedding medium. The embedding medium can be free space. But compound background media such as half spaces or multi-layer plane slabs may also occur. The solutions of the direct source and scattering problems are frequently used in the other two field problems, which are inverse ones. In inverse source problems one tries to determine the source distribution from the knowledge of an
electromagnetic field in some spatial region. In inverse scattering problems the electromagnetic properties of an object have to be recovered.
The mathematical tools that are at our disposal for solving these field problems are analytical techniques, approximating methods and numerical techniques. Combination of all available solution methods seems to be of growing importance since no single method can cover all possible problems. The application of analytical methods is restricted to
configurations of a rather simple geometry [e.g. Bowman (1969)]. Approximating methods, such as asymptotic methods, geometrical optics
CHAPTER 1 : GENERAL INTRODUCTION
-3-[e.g. Born and Wolf (1965)] and geometric theory of diffraction -3-[e.g. Hansen (1981)], are employed when the configurations are large compared to the wavelength associated with a wave field in the embedding medium. In practice however, complicated structures with dimensions of a fraction of the wavelength up to a few wavelengths may occur, for which analytical and asymptotic methods fail to produce satisfactory answers.
Consequently, we need other solution schemes. Since the advent of digital computers, numerical methods have been developed for solving these problems. They have a more general applicability. In the computational modeling of electromagnetic fields two important techniques are employed: methods based on the solution of the partial differential equations, e.g. finite-element methods [e.g. Mur and De Hoop (1985)], and methods based on the solution of integral equations. The former one seems to be the only versatile method in the case of fields radiated by sources in strongly inhomogeneous structures.
In this thesis we introduce and investigate analytical and numerical methods for solving frequency-domain electromagnetic source and
scattering problems. We limit ourselves to direct field problems. The electric and magnetic current sources may be distributed arbitrarily. The scattering object is assumed to be linear, isotropic, time-invariant and locally reacting. Furthermore, it is penetrable and its shape can be arbitrary. Free space is assumed as the embedding medium. Engineering problems which are inherently three-dimensional are frequently modeled as problems of one or two dimensions. In this way the enormous analytical and/or computational effort associated with the solution of the three-dimensional problems can be avoided. In many cases the solution of the reduced problems gives adequate answers and allows adequate parameter studies for the technical configuration at hand. Therefore, besides the three-dimensional source and scattering problems we will also analyze in great detail and solve the problems of lower dimensionality.
The basic tools of our analysis are the integral representations of the frequency-domain electromagnetic field in both the source and the scattering problem. Green's functions play an important role in the
kernel of these integral representations. The integral representations are elegantly obtained with the aid of spatial Fourier transformations . [De Hoop (1977)]. Applying these transformations to Maxwell's equations
in complex form, manipulating the field relations in the spectral domain and applying the inverse Fourier transformations we are led to the integral representations. We will derive and employ domain-type and boundary-type integral relations as well as combinations of the two. Furthermore, in this thesis we will introduce a more general domain-type integral representation; the other two are then special cases. In order to solve the scattering problems integral equations are derived from the domain-type integral representations by locating the point of observation inside the object region. The field distribution in the object domain is obtained by solving these equations. The field outside the object domain is computed by subsequently substituting the latter solution into the integral representations. In order to solve the equations we will employ iterative techniques based upon a global root-mean-square error criterion [Van den Berg (1981))]. With these methods a considerable reduction in memory requirements and computation time is achieved. In this thesis we will introduce and investigate a general least-squares iterative procedure in which either one, two or three variational functions are applied in each iteration step. Many different iterative schemes for solving scattering problems are obtained from this general scheme. These schemes will be tested for applicability, to the scattering problems.
Unfortunately, we cannot implement the continuous integral
representations on digital computers. We have to discretize them in one way or another. Consequently, approximations must be made. In the
standard technique of the method of moments [e.g. Kantorovich and Krylov (1961)), Harrington (1968)], the unknown functions in the integral
relations are expanded into a suitable sequence of expansion functions. Subsequently, the integral relations are weighted using a suitable set of weighting functions. These expanding and weighting procedures are
CHAPTER 1 : GENERAL INTRODUCTION
-5-In this thesis we propose an alternative, new discretization method. We reverse the order of the expansion and weighting steps in the
classical discretization method. We first apply a weighting procedure to the field relations. This procedure consists of filtering in the spectral domain. It leads to new relations in which unknown, band-limited field quantities and given, band-limited source and/or object quantities occur. From Shannon's sampling theorem we know that a band-limited function is completely determined by its values at appropriately chosen sample locations. Therefore, we subsequently apply an expansion procedure in accordance with this sampling theorem. Thus, with our configurational filtering and sampling method we project the integral representations for the actual field onto systems of linear algebraic relations for the spectrally filtered field in a straightforward manner. We will show that this method results in simple, efficient numerical schemes without much analytical effort. Furthermore, the singularity problems associated with Green's functions in the kernel of the integral relations [e.g. Lee et al. (1980)] are solved in a natural way. In our approach we have to compute the spectrally filtered Green's functions and their derivatives with respect to the spatial coordinates. This must generally be done numerically. However, in this thesis we present three types of filters, viz. unit-step filters, cosine-squared filters and Bessel filters, for which the filtering of the one-, two- and three-dimensional Green's function can be carried out analytically. In order to investigate the influence of the spectral-domain filtering on the field solution we will also estimate the global discrepancies between the spectrally filtered, electromagnetic field quantities and their exact counterparts.
The discretization of the integral representations for the
electromagnetic field in the scattering problem is more difficult than in the source problem. A main complication is that multiple convolution integrals occur in both the spatial- and the spectral-domain field representations. This is in contrast with the source problem where convolutions only occur in the spatial domain. Consequently, residual terms will inevitably show up during the process of weighting and
sampling. However, in our approach these residual terms can be traced in a qualitative manner. In the source problem residual terms will not occur. In addition, it turns out that by invoking a total discretization,
i.e. discretization in both the spatial and the spectral domain, we obtain systems of linear algebraic equations in terms of periodic, discrete Fourier transforms. Very efficient fast Fourier transform (FFT) algorithms are employed to compute these sums. Consequently, an
additional, substantial saving in computation time is achieved in the iterative solution of the scattering problem. As a result, we are able to solve the scattering by objects of three dimensions on sequential
computers.
The actual computer modeling of the electromagnetic source and scattering problems involves several steps, viz. the formulation of the problem, its projection onto a problem suitable for a numerical
implementation, the actual programming of the computer code and, finally, the running of the programs. Each step in the numerical implementation of the field problems may result in discrepancies between the computed and the exact solutions. In order to assess these errors and to debug the computer programs we need test solutions, preferably those that can be found analytically. Therefore, in this thesis a great deal of attention is given to the analysis of some elementary source and scattering problems for which solutions in terms of well-known functions are available. In the process of solving some elementary source problems we have discovered an interesting phenomenon. There are a number of possible time-harmonic sources in free space which do not radiate at all. This means that an observer outside the source regions shall never perceive them. These so-called non-radiating sources only exist in fixed points in the frequency domain. Besides the source problems some elementary
scattering problems are analyzed in detail. The scatterers investigated are the sphere consisting of a number of homogeneous, concentric layers, the circular cylinder consisting of a number of homogeneous, concentric layers and the plane slab consisting of a number of homogeneous, parallel layers. In the solution of these elementary scattering problems we will
CHAPTER 1 : GENERAL INTRODUCTION
-7-employ with great advantage field representations in terms of two vector potentials and in homogeneous, source-free regions in terms of two scalar potentials. The analysis results in simple systems of linear algebraic equations which can be solved either analytically or numerically.
The analysis for the solution of all of these source and scattering problems with one-, two- and three-dimensional configurations is rather elaborate. However, all of the analytical details for the actual numerical implementations have been provided in this thesis. At every stage of our analysis numerical results are presented. These results will be compared with analytical solutions, where possible. In our analysis we employ the International System of Units (SI) for all quantities. We will omit the units in our discussions.
In Chapter 2, analytical solutions are derived for some elementary source and scattering problems. The non-radiating sources are encountered here. It will be shown that by combining integral representations in terms of two vector potentials and representations in terms of two scalar potentials, which hold for homogeneous source-free regions, we obtain manageable boundary value problems for the elementary scattering problems. Only systems of linear algebraic equations need to be solved. Numerical results are presented for several source and scattering configurations.
In Chapter 3, the numerical solution of three-dimensional source problems is investigated. The current sources may have an arbitrary distribution. The configurational filtering and sampling method is introduced. In the source problem this method consists of spectral-domain filtering followed by spatial-domain sampling. As a result, the integral representations for the frequency-domain electromagnetic field are straightforwardly projected onto systems of linear algebraic relations. Analytical solutions are obtained for the spectral-domain filtering of the three-dimensional free-space Green's function by unit-step spherical filters, cosine-squared spherical filters and Bessel spherical filters.
In Chapter H, the numerical solution of source problems of lower dimensionality is discussed. The problem is formulated partially in the spatial domain and partially in the spectral domain. Only one or two spatial coordinates are involved. In this way, we obtain general results that can be used in more complicated field problems, e.g. in guided wave problems. One and two-dimensional source problems are special cases in this approach. The spectral-domain filtering of the one- and
two-dimensional free-space Green's function is carried out analytically for several types of filters.
In Chapter 5, the general least-squares iterative scheme for solving scattering problems is introduced. It is discussed on the basis of the solution of the domain-type integral equations for the electromagnetic scattering problem. This scheme involves either one, two or three
variational functions in each iteration step. A number of schemes can be constructed in this approach. Well-known least-squares methods, viz. a method of steepest descents and a conjugated gradient method
[Van den Berg (1984)], turn out to be special cases. Orthogonality properties are derived for the various variational functions. Different schemes are compared with each other numerically in some one- and two-dimensional scattering problems. Some results of this chapter are applied in the solution of the scattering problems discussed in Chapters 6 and 7. In Chapter 6, the scattering by three-dimensional penetrable objects is examined. An extended domain-type integral representation is proposed. Special cases are the classical domain-type and the boundary-type
integral representations for the frequency-domain electromagnetic field scattered by three-dimensional objects in free space. The domain-type integral relations are projected onto systems of linear algebraic
relations with the aid of configurational filtering and sampling methods. In the scattering problem, filtering and sampling is applied in both the spatial and the spectral domain. As a result, we obtain systems of linear algebraic relations in terms of periodic discrete Fourier transforms. By employing a least-squares iterative technique and Fast Fourier Transform (FFT) algorithms the three-dimensional scattering of frequency-domain
CHAPTER 1 : GENERAL INTRODUCTION
-9-electromagnetic waves by a penetrable object of bounded extent is solved numerically. Results are compared with exact solutions wherever possible.
Finally, in Chapter 7 the scattering problems of lower dimensionality are examined. Numerical results are obtained for one- and two-dimensional scattering problems. A comparison with analytically known solutions is made wherever possible. Furthermore, a simple method is proposed for generating test solutions.
C h a p t e r 2
ANALYTICAL SOLUTIONS FOR ELECTROMAGNETIC
RADIATION AND SCATTERING IN SOME
ELEMENTARY MULTI-LAYER CONFIGURATIONS
Elementary time-harmonic electromagnetic source and scattering problems for which exact solutions in terms of well-known functions can be derived are investigated systematically. Both field problems are formulated in terms of integral representations. The elementary source problems considered here involve source distributions consisting of Dirac's impulse function and bounded functions. The interesting phenomenon of non-radiating sources is discussed. The scattering of electromagnetic waves excited by specified volume current sources, which may be located anywhere in space, is analyzed for multi-layer spherical configurations, multi-layer cylindrical configurations and multi-layer plane slab configurations. All scattering obstacles are penetrable. For the latter two problems the analysis is carried out partially in the spectral domain. It is shown that the application of integral representations in terms of vector potentials together with representations in terms of scalar potentials yields manageable boundary value problems. Only simple systems of linear algebraic equations need to be solved. Numerical results are presented.
2.1. INTRODUCTION
Basic problems in electromagnetic theory are radiation and scattering. In radiation or source problems one wants to calculate the electromagnetic field excited by specified sources. In scattering problems one has to determine the electromagnetic field distribution in space, in which a specified scattering object and a specified source are located.
In the forthcoming analysis we will investigate those elementary, time-harmonic source and scattering problems for which a field solution in terms of well-known functions can be derived. Although a numerical treatment of field problems is common practice now, the availability of analytical solutions is still imperative. First of all, they provide insight into typical difficulties and can lead to improvement of
numerical schemes. Secondly, these solutions are indispensable in testing and debugging the implementations of numerical schemes which are intended to solve more complicated field problems. Thirdly, the elementary
configurations can be used as backgrounds. This is done in, for example, geophysical prospecting and in guided-wave optics as well, where multi layer slab configurations are used as background media.
In the source problems to be considered the source distributions are simple in a certain system of coordinates. We will employ right-handed, orthogonal Cartesian coordinates, cylindrical coordinates and spherical coordinates. The scattering configurations to be investigated are respectively the multi-layer sphere, the multi-layer circular cylinder and the multi-layer slab. We will use solutions from the elementary source problems as incident fields.
In Section 2.2 we will derive integral representations of the source type for the field in a general multi-layer scattering configuration situated in free space and we will formulate the two field problems. In Section 2.3 some typical, elementary source problems will be analyzed.
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS -13" .AHO
Here, we will encounter some classes of non-radiating sources.
Section 2.1) deals with scattering by a multi-layer sphere in free space.
In particular, the scattering of waves by a homogeneous sphere in free space will be analyzed in detail. The following section handles the scattering by a multi-layer circular cylinder. We will go into detail in the two-dimensional scattering by a homogeneous circular cylinder in free space. In Section 2.6 we will examine the multi-layer plane slab
configuration. In particular, the one-dimensional scattering in a multi layer slab structure and the scattering in a double half-space will be considered. Various numerical results will be presented.
2.2. FORMULATION OF THE SOURCE AND THE SCATTERING PROBLEM
We consider the scattering of time-harmonic electromagnetic waves by a penetrable object in free space. The configuration is given in Fig. 2.1. The scattering object consists of N layers, i.e. V1 , ..., VN # Each layer
V., where j = 1, ..., N-1 , is bounded by two closed non-intersecting surfaces, i.e. S and S.+ . Layer V is enclosed by only one closed
surface, i.e. S . Each surface S. (j = 1, .... N) is assumed to be sufficiently regular, i.e. the unit vector n. along its normal, pointing away from V , is a piecewise continuous vector function of position. We
Fig. 2.1. The volume current sources in a general multi-layer scattering configuration.
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-15-allow the layers to be shift invariant in either one or two dimensions. To specify the position in space we employ the coordinates {x,y,z} with respect to an orthogonal Cartesian reference frame and three mutually perpendicular base vectors U ,± ,i_ } of unit length each. As
x y z position vector we have
£ - *i
x +y i
y + zi
z• ( 2 . 2 . D
The time coordinate is denoted by t. We shall present our analysis in the complex frequency or s-domain, where s is the time-Laplace-transform parameter. Since in physical reality only causal phenomena are of
interest, we take this Laplace transform for any transformable space-time function f = f(r,t) to be
f(r,-s) - ƒ" exp(-st) f(r,t) dt , (2.2.2)
where tn is the instant at which the sources that generate the field are
switched on. Obviously, f is defined in some domain s. < Re{s} < • in the complex s-plane (Re{...} denotes the real part), and f can be be
reconstructed from f by the standard inversion procedure [Liu and Liu (1975)]. In particular, we are interested in the complex steady-state representations of sinusoidally oscillating fields of angular frequency <u which follow upon taking s = -iu, where i denotes the imaginary unit. This corresponds to the complex exponential time dependence exp(-iut). Since in the remainder we only consider the iu-domain quantities, we will, from here on, omit the circumflex over the symbol indicating this and replace s by -iu. We take s to be purely imaginary in the sense that Rets} approaches zero from positive values. Thus we have Im{w) > 0, however infinitesimally small. The time-harmonic, steady-state component f (r,t) of angular frequency u in a space-time function f(r,t) is obtained from its iu-domain representation by applying
f (r,t) = Re{ f(r,iu) exp(-iwt) } . (2.2.3)
The electromagnetic properties of the medium in layer V. (j = 1, '..., N) differ from those of the surrounding medium in domain V.. Each
medium is assumed to be homogeneous, linear, isotropic and locally reacting. The properties of the medium in V. are specified by its constant, scalar permittivity e - e. and its constant, scalar
permeability y » y.. For the embedding medium in V we have E = en and
y = y0, which are both constants. Losses are included when e and/or y .
(j = 0 , 1, .... N) are taken to be complex quantities (Im{e.} > 0 and/or Im{y.} > 0 ) . These constitutive parameters may depend on s =■ -ito. This dependence must be in accordance with the Kramers-Kronig causality relations.
The electromagnetic field is described by the electric field strength
E = E(r,iu) and the magnetic field strength H = H(£,iu>). These field
quantities satisfy Maxwell's equations in complex form, i.e.
(2.2.4)
(2.2.5)
in which <J ' » J_y' (j^.ioi) are the specified volume-source densities. The
sources may be located anywhere in space and may have an arbitrary distribution. The V-operator is defined by V = i 3 * i 8 + i 3 . — - -x x -y y — z z
Next we derive integral representations for the electromagnetic field in each layer, e.g. V (j = 1 N-1) (Fig. 2.2). This representation will be used as a basic building block in our construction of a
representation for the electromagnetic field in the entire 8 . In order to derive the integral representations for the field quantities in region V. we introduce the spatial Fourier transform over the bounded domain V , i.e. f(k) - J7/v f(r) exp(-ik-r) dV(r) , (2.2.6) j V V i h l X X 1 ol H E h .' + -Ie i u E ioiy •m = E = H = . Te' - Im mtr.
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 1 7
-F i g . 2 . 2 . The i n c i d e n t f i e l d e x c i t e d by a volume c u r r e n t source in V. and the two s c a t t e r e d f i e l d s in t h e l a y e r V. enclosed by the two
s u r f a c e s S. and S .+. .
i n which k = k i + k i + k i belongs t o ffi . Some of the p r o p e r t i e s of — x-x y—y z—z
t h i s s p a t i a l F o u r i e r t r a n s f o r m a r e d i s c u s s e d in Appendix A. We assume t h a t the f i e l d and the s o u r c e q u a n t i t i e s a r e well behaved in t h e e n t i r e V. and on S . and S . and t h a t they can be s p a t i a l l y F o u r i e r t r a n s f o r m e d , r e s u l t i n g i n f u n c t i o n s which a r e well behaved in the e n t i r e J<-domain. The F o u r i e r i n t e g r a l s a r e understood in t h e sense of
g e n e r a l i z e d f u n c t i o n s . We apply ( 2 . 2 . 6 ) t o ( 2 . 2 . 4 ) and ( 2 . 2 . 5 ) . As a r e s u l t , we obtain ik x H + Int E » J , ik x E - iwy . H = - J , ( 2 . 2 . 7 ) ( 2 . 2 . 8 ) . i n which j e , m( k ) m - e , m( k ) + ~em{k) _ j e . m ( ) < )
" - -v - -s - - s
j + 1-( 2 . 2 . 9 )
with
j |, m( k ' ) - / /s j |, m( r ) e x p ( - i k - r ) d S ( r ) , ( 2 . 2 . 1 0 )
where the surface-source densities of electric and magnetic current, viz. e m
J_ and J_, respectively, are
j|(r) - - n(r) x H(r) , (2.2.11)
J™(r) - n(r) x E(r) , (2.2.12)
with S = S., S . and n = n , n . , respectively. The occurrence of the surface current contributions (2.2.10) is discussed in Appendix A. If we next take the scalar and vector products of both (2.2.7) and (2.2.8), respectively, with ij< and combine the resulting equations, we obtain relations for E and H, i.e.
|(k) = iuy ne - ik(ik-n^)/iü)e - i^xiï1" , (2.2.13)
H(k) = iwe Sm - ik(ij<-jin,)/iüm1 + ijs*!® , (2.2.11)
in which (Cf. (2.2.9)) n!'m(l<) = ff'™(lc) + rt^,m,(k) - Ü ?, m .(k) , (2.2.15) with 5®|J(k) - G (k) Jy,m(k) , (2.2.16) and He'mAk) - G.(k) J**m(k) , (2.2.17) — o , J — J — — £> —
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 1 9
-with S = S . , S .+. and in which
G . ( k ) = 1 / ( k - k - k2) , ( 2 . 2 . 1 8 )
J J
2 2 2 with k. = to e.iJ.. We have assumed that k. has a nonzero imaginary part in
J j J o J order to ensure that k'k t k. for all real values of k.
Before we apply the Fourier inversion theorem to (2.2.13) and (2.2.14) we investigate the spatial-domain functions which correspond with (2.2.16) and (2.2.17). With the results derived in Appendix A it is easily found that the vector quantities U?'m (Cf. (2.2.16)) correspond
" * J with spatial-convolution integrals in r-space, i.e.
n ^ j ( r ) - Jffy G (r-r') 4 ' % ' ) dV(r') , (2.2.19)
with r 6 I and in which the free-space Green's function G. satisfying the three-dimensional scalar Helmholtz equation is given by (see (B.6) in Appendix B)
G.(r-r') = exp(ik .|r-r'| ) / 4Ti|_r-j2'| , (2.2.20) 2 1 /2
where k. = (io e.u.) with Im{k.} > 0. Note that G. decays exponentially J J J J J as |r-£'| approaches infinity. Application of the Fourier inversion theorem to (2.2.17) yields (Cf. (2.2.10) - (2.2.12))
n^'j(r) - ;/s G..(r-r') j|,m(r') dS(r') , (2.2.21)
with r e a3 and S = S., S . Because (2.2.19) and (2.2.21) hold for
Q J J '
£ 6 a the transformation of (2.2.13) and (2.2.14) to the spatial domain is rather straightforward. The factors ij< in (2.2.13) and (2.2.14) can now be identified with V-operators in £-space. Applying the Fourier inversion theorem to (2.2.13) and (2.2.14) and identifying three different components in the total field {E,H}, we arrive at the desired
i n t e g r a l r e p r e s e n t a t i o n for t h e e l e c t r o m a g n e t i c f i e l d in domain V., i . e . [De Hoop (1977)] E * ( r ) + E^ . ( r ) - E* . ( r ) - { 1 , V2, 0 } E ( r ) , ( 2 . 2 . 2 2 ) - j - - S j . j - - sJ + 1. J -B*(r) + H* . ( r ) - H° . ( r ) -. { 1 . l/t,0) H(r) , ( 2 . 2 . 2 3 ) - j - - S j . j - - sJ + 1. J -when r 6 {V , S. or S . e l s e w h e r e } .
In ( 2 . 2 . 2 2 ) and ( 2 . 2 . 2 3 ) we have i n t r o d u c e d the i n c i d e n t f i e l d {E,,H*} defined by
E j ( r ) - iuy n® - V(7-IIy )/iwe - M v . j • ( 2 . 2 . 2 1 )
H j ( r ) - iooej nm j - VCV-II^jJ/iuUj + Vxn^( J for r 6 B3, ( 2 . 2 . 2 5 )
and two s c a t t e r e d f i e l d s , i . e . {E° .,H^ .} and {E^ ,H^ . } , which -Sj.j -Sj.j SJ+1,J J,j+1,j
are defined by
El (r) = im n® . - V(7-n® ,)/iue. - VxII™ , (2.2.26)
H^ ,(r) = iue, nm . - V(V«l£ .)/iuu, + Vxii® . for r € K3, (2.2.27)
with S - S., S (see Fig. 2.2). Note that (2.2.21) - (2.2.27) hold for all £ 6 B , while the total electromagnetic field in (2.2.22) and
(2.2.23) is only found inside the region V. and on the surfaces S. and S.+.. The incident field is known in principle since the volume sources
are specified. The computation of the incident field in the entire r-space is in fact the electromagnetic radiation or source problem. A detailed discussion about some elementary source problems is presented in Section 2.3- We note that each of the three constituents of the total electromagnetic field in V. satisfy Maxwell's equations (2.2.1) and
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-21-(2.2.5) for £ 6 H with e = e. and u = v but with different right-hand sides for the two scattered fields.
At this stage we have derived the integral representation for the electromagnetic field in the layer V., which is bounded by the two closed surfaces S. and S.+. (j » 1, — , N-1). However, the result can also be
applied to the fields in regions V. and V,,, which are both bounded by only one closed surface. When a region extends to infinity we can omit the contributions (Cf. (2.2.21), (2.2.26) and (2.2.27)) from the surface currents (Cf. (2.2.11) and (2.2.12)) at infinity due to the exponential decay of the Green's function in (2.2.20) (Boundary condition at infinity). We assume that there are no sources at infinity. In the case such a region is of bounded extent we also omit one of the scattered fields as given by (2.2.26) and (2.2.27). Then we only have left the contribution from the enclosing surface.
A representation for the electromagnetic field in the entire B is now found by simply adding all the field expressions for regions
VQ, V , VN (Cf. (2.2.22) - (2.2.27)). We obtain
N N
E(r) = E E*(r) -iu E ƒƒ„ n.(r')xH(r') [u.G.Cr-r')
J-0 J J-1 j J J J N " U- 1G<_1(!l-r')]dS(r') + (1/iu) V{7« E ƒƒ„ n.(r')xH(r') J-1 j 1 j = 1 Sj J [G.(r-r')/c - G (r-r')/e ] dS(r')} J J J ' J ' N - Vx E ƒƒ„ n.(r')xE(r')[G.(r-r')-G. .(r-r')]dS(r'), (2.2.28) ~ j-1 sj -J ~ " " J ~ ~ J 1 " " N N H(r) - E H,(r) + lu E ƒƒ„ n.(r')xE(r') [e.G.(r-r') J-0 ~J " J-1 bJ ~J " " ~ J J " " N - e. .G. ,(r-r*)]dS(r') - (1/iw) V{V- Z ƒƒ„ n.(r')xE(r') J 1 J-1 - - - - j = 1 Sj -J
[G (r-£')/y - G (r-r')/y ] dS(r')} N
- Vx E ƒƒ n.(r')xH(r')[G.(r-r')-G (r-r')]dS(r'), (2.2.29)
J-1 j J J J
for r € a3. The incident fields {E1,^1} (j - 0, 1 , N) are given by
(2.2.19), (2.2.24) and (2.2.25). Locating jr at the surface S , where j = 1, ..., N, and taking the vector products of (2.2.28) and (2.2.29), respectively, with the corresponding unit vector n. along its outward normal, we obtain a system of two coupled boundary integral equations with n.xE and n.xH (j = 1 , ..., N) as unknown and {_E.,jl.} as given quantities. After solving these integral equations we can compute the field distribution anywhere in space by substituting the surface currents (Cf. (2.2.11) and (2.2.12)) into (2.2.28) and (2.2.29) and using (2.2.28) and (2.2.29) again.
A modification in the formulated field problem is preferable when the scattering configuration is shift invariant in either one or two
dimensions. The volume sources may still be of finite extent. Then, the surface integrals in (2.2.26) and (2.2.27) (or in (2.2.28) and (2.2.29)) require integrations over domains of infinite extent. This causes
practical problems.
2.2.1. THE FIELD PROBLEM IN THE (x,y;k )-D0MAIN
Therefore, when the configuration is independent of the z-coordinate (Fig. 2.3) we carry out a one-dimensional spatial Fourier transformation with respect to z (Cf. (2.2.6)) to (2.2.22) - (2.2.27) (or (2.2.28) and
(2.2.29)) and arrive at a field representation in the (x,y;k )-domain. Subsequently, this field problem is solved and the £~domain field
solution is found by applying the inverse Fourier transform. We note that when the sources are also independent of the z-coordinate, we are dealing with a two-dimensional electromagnetic field profcletufi?
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 2 3
-F i g . 2 . 3 . A two-dimensional, c y l i n d r i c a l c o n f i g u r a t i o n with i t s g e n e r a t r i x a l o n g t h e z - d i r e c t i o n .
look for (complex) v a l u e s of k for which t h e homogeneous i n t e g r a l e q u a t i o n s i n the ( x , y ; k )-domain (Cf. ( 2 . 2 . 2 8 ) and ( 2 . 2 . 2 9 ) ) have a nonzero s o l u t i o n . This i s a problem i n guided-wave a n a l y s i s .
2 . 2 . 2 . THE FIELD PROBLEM IN THE (k ,k ;z)-DOMAIN x y
When the c o n f i g u r a t i o n i s s h i f t i n v a r i a n t i n two dimensions, say in both t h e x- and the y - d i r e c t i o n ( F i g . 2.14), we c a r r y out a two-dimensional
s p a t i a l F o u r i e r t r a n s f o r m a t i o n with r e s p e c t t o both x and y (Cf. ( 2 . 2 . 6 ) ) t o ( 2 . 2 . 2 2 ) - ( 2 . 2 . 2 7 ) (or ( 2 . 2 . 2 8 ) and ( 2 . 2 . 2 9 ) ) . As a r e s u l t , we o b t a i n a f i e l d problem i n t h e (k ,k ; z ) - d o m a i n . Next, t h i s problem i s solved. In
x y
the case where only k = k = 0 contributes to the inverse Fourier J x y
integrals, we have in fact a one-dimensional field problem. In guided-wave analysis one may be interested in (complex) values of k or k for
x y which the homogeneous (k ,k ;z)-domain integral equations have nonzero
L.
SOURCES
Mf^^^^+^^^^^^lf™r'
i
! " ^ f l ^ * ^ W ^ IB^ W r t " W ^ ^ ^ ^ ^ ^ W ^ W ^ ^ P ^ r T W ^ ^ W " ^,W v.-.l 1.1 ■•■■•■I I-' ■»Fft»TfTT»nw?ffn»nTW?w?KW?TT!rTnw!Tmwfnnff!wnw?w^^Mm
Fig. 2.1. A one-dimensional planar slab with its boundaries parallel to the plane z = 0.
solutions. The final step is to transform the field solution in the (k ,k ;z)-domain to r-space. x y
Unfortunately, for general configurations the electromagnetic
scattering problem under consideration cannot be solved analytically and one has to resort to numerical techniques. However, there exist a few configurations for which a more extensive analysis is possible and for which solutions in terms of well-known functions do exist. We mention the objects consisting of spherical layers, circular cylindrical layers and parallel plane slabs. These three scattering structures will be the subject of our investigations in Sections 2.1, 2.5 and 2.6. In each of these problems we employ relations (2.2.22) - (2.2.27) (or (2.2.28) and (2.2.29)) or their counterparts which are spatially Fourier transformed with respect to either one or two coordinates. Though the three
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-25-scattering configurations have essentially a one-dimensional structure, the analysis of the electromagnetic field in these is still extensive and complex. Instead of only working with (2.2.22) - (2.2.27) (or (2.2.28) and (2.2.29)), all of which contain two vector potentials, it turns out to be advantageous to employ also field representations in terms of two scalar potentials, which only apply to source-free regions in r-space. The latter are discussed in Appendix E. It will be shown that by using an expression for the free-space Green's function (Cf. (2.2.20)) in terms of a suitable, complete set of expansion functions we can deduce expressions for these scalar potentials from (2.2.22) - (2.2.27) in terms of well-known functions. Relations between the various expansion coefficients in the resulting expressions for the different scalar potentials are determined by imposing the N boundary conditions at the surfaces S , S , — , SN, i.e.
(nxE)j_,(r) - (nxE)j(r) = 0 , (2.2.30)
(nxH) (r) - (nxH).(r) = 0 when r 6 S., (2.2.3D
with n = n. and j - 1, 2 N. The subscripts j and j-1 in (2.2.30)
and (2.2.31) refer to the electromagnetic field strengths inside the regions V. and V , respectively. We approach S. from within V. and V- ,, respectively. We have assumed that the volume-source distributions
e in
£ ' are not concentrated in thin sheets on the boundary surfaces. In this way, we obtain systems of linear algebraic equations for the expansion coefficients. These systems can be solved either analytically or numerically. Accordingly, we obtain the solution of the
electromagnetic field problem. This concludes our formulation of the radiation or source problem and of the scattering problem.
In the following section we will analyze some basic incident fields, i.e. we will solve some elementary radiation or source problems. In successive sections we will discuss the r-domaln fields in a spherical multi-layer configuration, the (x,y, k )-domain fields in a cylindrical
m u l t i l a y e r c o n f i g u r a t i o n and the (k ,k ;z)domain f i e l d s in a m u l t i -x y
l a y e r p l a n e s l a b c o n f i g u r a t i o n . F i e l d e x p r e s s i o n s in terms of well-known f u n c t i o n s w i l l be derived and numerical r e s u l t s w i l l be p r e s e n t e d .
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-27-2.3. ELECTROMAGNETIC FIELDS GENERATED BY SOME ELEMENTARY SOURCES IN FREE SPACE
In this section we present a survey of a number of basic radiation problems. Representations of time-harmonic electromagnetic fields excited by typical, elementary volume current sources will be derived in terms of well-known functions. Starting with one-dimensional problems we will gradually increase the complexity of- the analysis and end up with source problems of three dimensions. In addition, it will be shown that there exist at least nine classes of frequency-domain volume-current sources, which do not radiate at all, i.e. the field outside the source region is exactly zero. This interesting phenomenon has consequences in frequency-domain inverse-source analysis and inverse-scattering problems.
In order to define and classify the different sources to be
considered, we employ three different systems of coordinates. First of all, we use the Cartesian coordinates {x,y,z}, already introduced in Section 2.2. In the second place, we employ the cylindrical coordinates
Fig. 2.5. The cylindrical coordinates {p,cf>,z} and the corresponding unit vectors i , i, and i .
Fig. 2.6. The spherical coordinates {r,9,<)>} and the corresponding three
unit vectors i , i. and i,.
— r — ö —q>{p,<t>,z} defined by (Fig. 2.5)
x = p cos(<|>) ,
y = p sin(<j>) ,
z = z ,
(2.3.1)
where 0 S <j> < 2IT, 0 Ê p < », and three mutually perpendicular base
vectors {i , i^.i } of unit length each and defined by
-p -$ -z
_i = cos(<j>) j. + sin(<|>) _i ,
p x y
i, = - s i n U ) i + cos(((i) i ,
—<p —x —y
iz=iz •
(2.3.2)
In the third place, we employ the spherical coordinates {r,e,$} defined
by (Fig. 2.6)
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-29-y = r sin(9) sin(i(>) ,
z = r cos(e) , (2.3.3)
where 0 £ 4> < 2TT, 0 £ 9 £ i r , 0 £ r < » , and three mutually perpendicular base vectors U >ifl>i.J o f u n i t length each, given by
i " sin(6) cos(ij)) _i + sin(e) sin(<(>) _i + cos(9) j. ,
i = cos(6) cos(<|>) i + cos(e) sin(c))) i - sin(e) i , —9 x y z
i = - sin(((>) i + cos((|i) i . (2.3.4) — $ —x —y
The various source distributions, which we will examine, are summarized e m in Tables 2.1 arid 2.2. The current source distributions J„ and J_ are
assumed to have a common scalar function depending on position. In Table 2.1 the scalar functions are Dirac's impulse functions, while those in Table 2.2 are functions which remain bounded in the entire r_-space.
e m
The vectorial factors in specific £„ and £„ are constant vectors in one of the three coordinate systems. Furthermore, we distinguish between sources that excite one-, two- or three-dimensional electromagnetic fields.
The electromagnetic field generated by a specified current source in free space is described by (Cf. (2.2.24) and (2.2.25))
E(r) = ium n® - V(v-n^)/iu)e - Vxn
m, (2.3.5)
H(r) = iwe n™ - V(V-n™)/iuu + Vxjl^ for r € B3 , ( 2 . 3 . 6 )
in which (Cf. ( 2 . 2 . 1 9 ) )
n.®'m(r) - ƒ " ƒ " ƒ " G ( r - r ' ) J . ^ ' V ' ) d V ( r ' ) , ( 2 . 3 - 7 )
— y — —oo - c o - c o — — — y — —
with (Cf. (2.2.20)) G(r-r') = exp(ik|r-r' | )/4ir|r-r'| , where k= ( w2e u )1 / 2
with Im{k} > 0. In the following part we will evaluate the vector
Table 2.1. Elementary surface current sources. c o n s t a n t components e,m i n
i
v Jx. Jy> JZV V
jz
Jr» Jg» J^ d i m e n s i o n a l i t y of t h e exci 1 2plane sheet line
y
circular
cylindrical
sheet
(P
ted f i e l d 3point
ring
C ^
spherical
sheet
r -Z^J
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-31-Table 2.2. Elementary volume current sources of bounded extent.
constant components
Te,m m J„
dimensionality of the excited field
v v
Jz
plane slab
circular
cylinder
sphere
V V
J2
circular cylinder pill box
with a linear profile
ir. J0. J ,
sphere with a
linear profile
d i s t r i b u t i o n s in j2-space (Cf. ( 2 . 3 . 5 ) and ( 2 . 3 . 6 ) ) f o r each source ' d i s t r i b u t i o n given i n Tables 2.1 and 2 . 2 .
2 . 3 . 1 . ONE-DIMENSIONAL SOURCE PROBLEMS
If the source d i s t r i b u t i o n s a r e independent of both t h e x- and the y - c o o r d i n a t e (9 j f 'm( r ) ■= 0 and 9 j f /m( r ) = 0) we f i n d four s e p a r a t e
J x—V — — y—V — —
f i e l d s o l u t i o n s , i . e . ( i )
E ( z ) - iwu nf. t d l j , H ( z ) - iue II™ + d nf. , ( 2 . 3 - 8 ) x V,x z V,y y V,y z V,x
while the o t h e r f i e l d components remain z e r o , ( i i )
H ( z ) - iue nm - d nf. , E ( z ) = iu>u II!! - d nm , ( 2 . 3 - 9 )
x V,x z V,y y V,y z V,x while the other field components remain zero, (iii)
Hz(z) = Jm z/iuu , (2.3-10)
while the other field components remain zero and (iv)
Ez(z) - J^ z/ia)E , (2.3.11)
while the other field components remain zero. In (2.3.8) and (2.3.9) we have the vector potentials
Iv'm(z> = - C 0,(2-2') iu'm(z') dz' , (2.3.12)
with the one-dimensional free-space Green's function (see (D.5) in
Appendix D)
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-33-= (i/2k) exp(ik|z-z'|) . (2.3.13)
From (2.3.8), (2.3.9), (2.3.12) and (2.3.13) we conclude that the waves described by (2.3.8) and (2.3.9) are plane waves in the source-free regions of £-space. Note that H and E only differ from zero in the
m e
source regions in which J„ ^ 0 and J„ i4 0, respectively. This means
' ' s m s m
that the one-dimensional sources specified as J ' = J ' (z)i are - —V V,z —z n o n - r a d i a t i n g .
In t h e case of the c u r r e n t p l a n e s h e e t source (Table 2.1) defined by
4 '
m( z ) = / '
m6(z) , (2.3.14)
i n which j _e , m are c o n s t a n t v e c t o r s with r e s p e c t t o p o s i t i o n and 6(z) i s
D i r a c ' s impulse f u n c t i o n , the v e c t o r p o t e n t i a l s II ' (Cf. ( 2 . 3 - 1 2 ) ) r e d u c e t o
n ^, m( z ) = ie , m G ^ z ) . ( 2 . 3 . 1 5 )
The corresponding four separate field distributions are found by substituting (2.3.11) and (2.3.15) into (2.3.8) - (2.3.11).
When we are dealing with a current plane slab source (Table 2.2) defined by
±e,m when OS, |z| < a/2
Jy'm(z) = , (2.3.16)
0 when |z| > a/2 e m
in which a is a positive constant and J_ * are constant vectors with respect to position, the vector potentials Jl ' (Cf. (2.3.12)) turn out to be (see (D.15) and (D.18) in Appendix D)
2e'm[ c o s ( k | z | ) e x p ( i k a / 2 ) - 1 ] / k2 when | z | < a/2
n ^ 'm( z ) - , ( 2 . 3 . 1 7 )
J .e , ma j0( k a / 2 ) ( i / 2 k ) e x p ( i k | z | ) when | z | > a/2
in which J0( z ) = s i n ( z ) / z , i . e . the s p h e r i c a l Bessel f u n c t i o n of the
f i r s t kind and of zero order [Abramowitz and Stegun (1972), Chapter 1 0 ] . Note t h a t the v e c t o r p o t e n t i a l s II ' in ( 2 . 3 - 1 7 ) a r e c o n t i n u o u s and d i f f . e r e n t i a b l e a t | z | - a / 2 . The f i e l d d i s t r i b u t i o n i n z - s p a c e i s o b t a i n e d by s u b s t i t u t i n g ( 2 . 3 . 1 6 ) and ( 2 . 3 . 1 7 ) into_ ( 2 . 3 . 8 ) - ( 2 . 3 . 1 1 ) . We observe t h a t t h e r e s u l t i n g f i e l d r e p r e s e n t a t i o n s in ( 2 . 3 . 8 ) and ( 2 . 3 . 9 ) contain a common f a c t o r j (ka/2) for |zj > a / 2 . This f a c t o r h a s ( r e a l ) zeros a t ka/2 = nir (n - ±1, ±2, . . . ) , which means t h a t we have found a second c l a s s of n o n - r a d i a t i n g sources in f r e e s p a c e . For a f i x e d r e a l frequency UJ we find an i n f i n i t e number of s l a b t h i c k n e s s e s for which t h e corresponding c u r r e n t p l a n e slab s o u r c e s do not r a d i a t e a t a l l , v i z . for a = a with a = 2irnc/(i), in which c i s the v e l o c i t y of a wave in t h e
n n ' embedding medium (k = u / c ) . An observer l o c a t e d o u t s i d e the c o r r e s p o n d i n g
s l a b source domains w i l l never p e r c e i v e them.
Next, we i n v e s t i g a t e s o u r c e s which e x c i t e f i e l d s of two d i m e n s i o n s .
2 . 3 . 2 . TWO-DIMENSIONAL SOURCE PROBLEMS
In Tables 2.1 and 2.2 we have included four sources which g e n e r a t e t w o -dimensional e l e c t r o m a g n e t i c f i e l d s : the l i n e s o u r c e , the c i r c u l a r c y l i n d e r s o u r c e , the c i r c u l a r c y l i n d r i c a l s h e e t source and t h e c i r c u l a r c y l i n d e r source with a l i n e a r r a d i a l p r o f i l e . The s o u r c e f u n c t i o n s a r e assumed t o be independent of the z - c o o r d i n a t e (3 <J ' (r) - £ ) . As a consequence, t h e e l e c t r o m a g n e t i c f i e l d (Cf. ( 2 . 3 . 5 ) - ( 2 . 3 . 7 ) ) s e p a r a t e s i n t o two mutually independent s o l u t i o n s , i . e . t r a n s v e r s e magnetic (TM) waves (E i* 0, H = 0 ) given by
E ( x . y ) = iuy nf. - 3 Hm + 3 nm
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 3 5
--,"1 ^ / * „"I . -N „m \ / , . ^ rr^
H
x(x,y) - i .
en;
>x- 3
xO
xn';
iX♦ 3
yn
v > y) / i . u ♦ 3
yn=
>z,
Hy( x , y ) " i u e ïïV,y " y3xnv , x + V v . y ^ " V v . z ' ( 2"3-l 8 )
and t r a n s v e r s e e l e c t r i c (TE) waves (H £ 0, E = 0 ) given by
m p p H ( x , y ) - iue II„ + 3 51,, - 3 II „ z J V,z x V,y y V,x E ( x , y ) = iuu nf, - J (3 n j + 3 nf )/iuie - 3 1 ° , x J V,x x x V,x y V,y y V,z E ( x , y ) = iuu nf. - 3 (3 nf. + 3 nf )/iu)E + 3 n™ , ( 2 . 3 . 1 9 ) y J V,y y x V,x y V,y x V,z in which l*,m{x,y) - ƒ"„ƒ"„ G2( x - x ' , y - y ' ) J y, m( x \ y ' ) d x ' d y ' , ( 2 . 3 . 2 0 )
with the two-dimensional G r e e n ' s f u n c t i o n ( s e e (C.8) in Appendix C):
G2( x - x \ y - y ' ) = ƒ_„ G ( r - r ' ) d z '
= ( i / 4 ) H ^1 )( k R ) , ( 2 . 3 . 2 1 )
in which H!: is the Hankel function of the first kind and of zero order [Abramowitz and Stegun (1972), Chapter 9] and R = {(x-x') + (y-y') } U 0 ) .
In the case of the current line source (Table 2.1), defined by
J v 'm( x , y ) = ie , m 6(x)6(y) , ( 2 . 3 . 2 2 )
Q [f]
i n which j_ ' a r e c o n s t a n t v e c t o r s with r e s p e c t t o p o s i t i o n , the vector
e m
p o t e n t i a l s _n ' reduce t o
Substitution of (2.3.23) into (2.3.18) and (2.3.19) yields the field 'distribution in the whole (x,y)-space.
The current circular cylinder source (Table 2.2) is defined by .e,m
j^ when 0 £ p < a
J^'m(x,y) - , (2.3.24)
0 when p > a
where J_ ' are constant vectors with respect to position and p = ( x2+ y2)1 / 2 (£ 0 ) . Using results from Appendix C (Cf. (C.18) and
(C.20)), we find
ie,m[(iira/2k)H1(1)(ka)J0(kp)-(1/k2)] when p<a
n®,m(x.y) - , (2.3.25)
j _e , m (i¥a/2k)J1(ka)H^l)(kp) when p>a
in which J. and J. are Bessel functions of the first kind and H!; and (1 )
IC are Hankel functions of the first kind [Abramowitz and Stegun (1972), Chapter 9 ] . It can be shown that _ n ' is continuous and differentiable with respect to p at p = a. The field distribution in (x.y)-space is found by substitution of (2.3.25) into (2.3.18) and (2.3.19). From the resulting field expressions we conclude that there exists a third class of non-radiating sources. When J.(ka) ■= 0 (J. has an infinite number of real zeros) the field outside the circular cylindrical source-region (Cf. (2.3.24)) remains zero, and consequently one is unable to perceive this source (see Section 2.3.4).
In order to analyze the other two sources in Tables 2.1 and 2.2 that excite two-dimensional fields, we now employ the cylindrical coordinates {p,<t>,z} (Cf. (2.3.1)) and use another representation for the
two-dimensional Green's function (Cf. (2.3.21)), i.e. [Stratton (1941), Section 6.11]:
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 3 7 -t r1 ' ( k p ' ) J (kp) when p'£p 00 m m G2( x - x ' , y - y ' ) = ( i / 4 ) Z exp[ im( W ) ] , . m =~ " H( 1 )( k p ) J ( k p ' ) when p'Sp m m (2.3.26)
i n which J and H a r e B e s s e l and Hankel f u n c t i o n s , r e s p e c t i v e l y , of m ra
t h e f i r s t kind and of o r d e r m [Abramowitz and Stegun (1972), Chapter 9 ] . The c u r r e n t c i r c u l a r c y l i n d r i c a l s h e e t s o u r c e i s defined by
J v, m( * . y > - L J p *mip + j j ' % + l l ' \ l 6(p-a)/2wp , (2.3.27)
6 m 6 m 6 m
in which a is a positive constant and j ' , j ' and j ' are complex p <p z
constants. The unit vectors i and ix are found in (2.3.2). We substitute -P -<t>
(2.3.26) and (2.3.27) into (2.3.20) and change to the polar integration variables {p',<(>'}. The integration with respect to p' is straightforward. We find
_e,m, , , ,/ 0, , 2 i rr. e , m . .e,m . .e,m. -, _ r. ., , m
nv' (x,y) = (i/8ir) !Q [ J I ,+ J ' 1 ,* Jz' lz] I exp[im((t>-<j>')]
H (ka)J (kp) when p < a
( ( d * ' . (2.3.28) H( 1 )(kp)J (ka) when p > a
m m
It should be noted that the unit vectors i . and i., depend on <j>' -P — 9
(see (2.3.2)). The remaining integral can be evaluated by applying the orthogonality property
I** exp[i(m-n)9')] d?' = 2 * 6 ^ , (2.3.29)
in which m and n are integers and 6 is Kronecker's symbol. As i„ is mn ' — z independent of <j> we easily find the z-component of JL.' . Only the term m = 0 contributes. We obtain
je.m ( i / / ( ) H ^1 )( k a ) J0( k p ) when p < a
n ^ ( x , y ) - - ( 2 . 3 - 3 0 )
j ® 'm ( i / t ) J0( k a ) H ^1 )( k p ) when p > a
Using (2.3.29) It can be derived that
(1/2ir)/Qlr{exp[im(.(.-<(>,)]+exp[-im(<(l-<).')]}J.p,d<j.,= ip( «m l +«m_1) .
(1/2Tr)/2*{exp[imU-*M]+exp[-im(<|>-<|>,)]H(j)Id<|>,= ^ ( « ^ + «m_1) • (2.3-31)
Applying (2.3-31) to (2.3.28) and using (2.3-30), we arrive at _e,m, •, „e,m, \, . r .e.m. ^ .e,M, -,
n
v' (x,y) - n
v > z( x , y H
z♦ [
Jp' i
p♦ j^ i^]
(i/i»)HJ1'(ka)J1(kp) when p < a
{ I . (2.3-32) (i/J4)J (ka)H( 1 )(kp) when p > a
The field representations are found by substituting (2.3-30) and (2.3-32) into (2-3-18) and (2.3-19). Note that here as well we encounter non-radiating sources. In (2.3-30) a factor J.(ka) and in (2-3-32) a factor J.(ka) appear for p > a, both of which have an infinite number of (real) zeros. Thus we have found the fourth and fifth class of non-radiating sources.
The last two-dimensional source which we investigate is a current circular cylinder source with a linear radial profile (Table 2.2) defined by
(p/a)[jt,mi + j ?, mi j when p < a e m p -p $ - *
J®' (x,y) = , (2.3.33) 0 when p > a
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS
-39-, ^-39-, -39-, ■ -39-,^1 L. J .e,m .e,ni ,
in which a is a positive constant and j and j ' are complex P *
constants. In (2.3.33) the linear profile is chosen to accommodate a field solution in terms of well-known functions. We substitute (2.3.26) and (2.3.33) into (2.3.20) and change to the polar coordinates {p',<|>,}.
First we carry out the integration with respect to $'. We apply (2.3.29) and (2.3.31). As a result, we obtain
n*'m(x,y) - [j;.-lp ♦ j;'%](i,/2a)
H|1 ^ k p ' ) ^ (kp) when p' > p
ƒ* P'2 dp' . (2.3.31)
J ^ k p ' j H ^ C k p ) when p' < p
By applying integral formulas and Wronskians for Bessel functions [Abramowitz and Stegun (1972), 9.1.16, 11.3.20 and 11.3.24] we can evaluate the integrals. We arrive at
„e,m. , r .e,m. .e,m. ,,. ._, ,
V <
x
-y> - cjp i p
+
V i^dna/zk)
[H^1)(ka)J1(kp) + (2ip/Trka2)] when p < a
{ } • (2.3.35) J2(ka)H(1)(kp) when p > a
The vector potentials _n ' are continuous at p = a. Substitution of (2.3.35) into (2.3.18) and (2.3.19) yields the field distributions in (x.y)-space. Note the factor J.(ka) in (2.3.35), which has an infinite number of real zeros. This means that we have found a sixth class of non-radiating sources.
This concludes our investigations into the elementary sources which generate two mutually independent two-dimensional electromagnetic fields. Next we examine six sources that excite three-dimensional fields.
2 . 3 - 3 . THREE-DIMENSIONAL SOURCE PROBLEMS
Tables 2.1 and 2.2 c o n t a i n s i x volume-current source d i s t r i b u t i o n s which g e n e r a t e t h r e e - d i m e n s i o n a l e l e c t r o m a g n e t i c f i e l d s , v i z . the p o i n t s o u r c e , the sphere s o u r c e , the r i n g s o u r c e , the p i l l - b o x s o u r c e , the s p h e r i c a l s h e e t source and a sphere s o u r c e with a l i n e a r r a d i a l p r o f i l e . The wave f i e l d s e x c i t e d by t h e s e s o u r c e s a r e d e s c r i b e d by ( 2 . 3 . 5 ) - ( 2 . 3 . 7 ) . In the following d i s c u s s i o n we w i l l e v a l u a t e the vector p o t e n t i a l s in ( 2 . 3 . 5 ) and ( 2 . 3 . 6 ) for each s p e c i f i c source d i s t r i b u t i o n .
The c u r r e n t p o i n t source (Table 2.1) i s defined by
J ^ ' V ) = j6' "1 « ( x ) 6 ( y ) 5 ( z ) , ( 2 . 3 . 3 6 )
in which j_ ' are constant vectors with respect to position. Substituting (2.3.36) into (2.3.7) it can easily be found that the integrals in
(2.3.7) reduce to
JIv'"1^ " ie'm G (^) * (2.3.37)
The current sphere source is given by
je , m when 0 £ r < a
iv'V) = , (2.3.38) 0 when r > a
e m
where a is a positive constant, j_ ' a r e constant vectors with respect to
position and r = |r|. Using results from Appendix B (see (B.19) and (B.21)), we find
je,m[ia2h^1)(ka)j0(kr)-1/k2] when r < a
J I v ' V ) - - (2.3.39) J.e,m i a ^ C k a J h ^ C k r ) when r > a
CHAPTER 2 : ELEMENTARY SOURCE AND SCATTERING PROBLEMS 4 1
-i n wh-ich j and j . and -i\^ and h a r e s p h e r -i c a l Bessel f u n c t -i o n s of t h e f i r s t and the t h i r d k i n d , r e s p e c t i v e l y [Abramowitz and Stegun (1972), Chapter 1 0 ] . I t can be shown t h a t t h e vector p o t e n t i a l s JI ' (r_) a r e c o n t i n u o u s and d i f f e r e n t l a b l e with r e s p e c t t o r a t r = a. We observe i n ( 2 . 3 - 3 9 ) t h a t for r > a, n ^ 'm c o n t a i n a f a c t o r j . t k a ) , which has an
i n f i n i t e number of r e a l z e r o s . This means t h a t we have a g a i n found a c l a s s of n o n - r a d i a t i n g s o u r c e s , t h e seventh v a r i a n t ( s e e S e c t i o n 2 . 3 . 4 ) .
Next, we i n v e s t i g a t e t h e r i n g source and t h e p i l l - b o x s o u r c e . Although no simple s p a t i a l - d o m a i n f i e l d s o l u t i o n s i n terms of well-known f u n c t i o n s can be d e r i v e d , we analyze t h e s e s o u r c e s as they a r e of g r e a t p r a c t i c a l i n t e r e s t i n guided-wave problems. I n t h e a n a l y s i s we employ t h e c y l i n d r i c a l c o o r d i n a t e s { p , $ , z } .
The c u r r e n t r i n g s o u r c e i s s p e c i f i e d by
J * 'm( r ) - [ j *, mi * f ' m i , + J ®, mi J ó ( z ) 6 ( p - a ) / 2 i r p , ( 2 . 3 . 4 0 )
—V — p —p cp —9 Z —%
in which a i s a positive constant and j ' , j . ' and j ' are complex P <f> z
constants. The unit vectors i and i . are found in (2.3.2). In order to —p — *
em
derive expressions for the vector potentials Jl ' in (2.3.7) we apply a spatial Fourier transformation with respect to z defined by
f(k ) - ƒ f(z) exp(-ik z) dz for k 6 B . (2.3.41) z —<° z z
Substitution of (2.3.40) into (2.3.7) and application of (2.3.41) yields
S®,nl(x,y;k2) = - O - » G2(x-x',y-y';kz)J®,m(x',y';kz)dx'dy' , (2.3.42)
in which
J ^r a( x , y;kz) - [ j j 'mip ♦ i l ' \ + j ^miz] 5(p-a)/2TrP , (2.3.43)
