Generalized Multipole Techniques for Electromagnetic and Light Scattering (Mechanics and...

GENERALIZED MULTIPOLE TECHNIQUES FOR

ELECTROMAGNETIC AND LIGHT SCATTERING

MECHANICS AND MATHEMATICAL METHODS

A SERIES OF HANDBOOKS

General Editor

J.D. ACHENBACH Northwestern University, Evanston, Illinois, U. S. A.

First Series

COMPUTATIONAL METHODS IN MECHANICS

Editors

T. BELYTSCHKO

Northwestern University, Evanston, Illinois, U. S. A.

K.J. BATHE

Massachusetts Institute of Technology, Cambridge, Massachusetts, U S. A.

m

ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Singapore • Tokyo

GENERALIZED MULTIPOLE TECHNIQUES FOR

ELECTROMAGNETIC AND LIGHT SCATTERING

Volume 4 in Computational Methods in Mechanics

Edited by

Thomas WRIEDT

IWT - Stiftung Inst. f. Werkstojftechnik Bremen, Germany

H

1999

ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Singapore • Tokyo

ELSEVffiR SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

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Contents

Chapter 1. Introduction by A. Doicu and T. Wriedt 1 References 4

Chapter 2. Review of the GeneraUzed Multipole Technique Literature by T. Wriedt 5

2.1. Point matching method 6 2.2. Extended boundary condition method (EBCM) 7 2.3. Multiple multipole method (MMP) 8 2.4. Yasuura's methods 10 2.5. Discrete sources method (DSM) 11 2.6. Method of auxiliary sources (MAS) 12 2.7. Analytical continuation of solutions of boundary problems 13 2.8. Conclusion 14

References 14

Chapter 3. The Multiple Multipole Program (MMP) and the Generalized Multipole Technique (GMT) by Ch. Hafner 21

3.1. Introduction 22 3.2. From CHA to MMP expansions 22 3.3. Matrix methods 25 3.4. Special MMP features 27 3.5. Example: Scattering at a particle on a planar structure 33

References 38

Chapter 4. Models of Electromagnetic Scattering Problems Based on Discrete Sources Method by Yu.A. Eremin, N.V Orlov, andA.G. Sveshnikov 39

4.1. Introduction 40 4.2. Mathematical models for the Helmholtz equation 42 4.3. Mathematical models for the Maxwell equations 55 4.4. Conclusion 78

References 79

Chapter 5. Singularities of Wave Fields and Numerical Methods of Solving the Boundary-Value Problems for Helmholtz Equations byA.G. Kyurkchan, A.L Sukov, andA.I. Kleev 81

5.1. Introduction 82 5.2. Basic analytical representations of wave fields 82 5.3. Singularities of a wave field and their localization 87 5.4. Utilization of the information about wave field singularities when solving the boundary-

value problems for the Helmholtz equation 95 References 108

vi Contents

Chapter 6. Yasuura's Method, its Relation to the Fictitious-Source Methods, and its Advancements in the Solution of 2D Problems by Y. Okuno andH. Ikuno I l l

6.1. Introduction 112 6.2. Formulation of a sample problem 113 6.3. Modal functions 114 6.4. An approximate solution 116 6.5. Integral representation of the solution 116 6.6. Method of solution 1: theCYM 116 6.7. Method of solution 2: the YMSP 118 6.8. Method of solution 3: the YMSSP 121 6.9. Method of numerical analysis and examples 124

6.10. Miscellanea 134 6.11. Conclusion 136 6.12. Appendix A 136 6.13. Appendix B 137 6.14. Appendix C 138

References 140

Chapter 7. The Method of Auxiliary Sources in Electromagnetic Scattering Problems by EG. Bogdanov, D.D. Karkashadze, and R.S. Zaridze 143

7.1. Introduction 144 7.2. Problem formulation 145 7.3. Construction of the solution by the method of auxiliary sources 146 7.4. Choice of auxihary parameters 150 7.5. AppUcation to particular problems 158 7.6. Conclusions 169

References 170

Chapter 8. Numerical Solution of Electromagnetic Scattering Problems of Three Dimensional Nonaxisymmetrical Bodies on the Foundation of Discrete Sources Method by A. Dmitrenko 173

8.1. Introduction 174 8.2. Perfectly conducting scatterer 175 8.3. Impedance scatterer 178 8.4. Magneto-dielectric scatterer 179 8.5. Chiral scatterer 182 8.6. Coated scatterer 184 8.7. Some ideas towards the solution of dense ill-posed linear algebraic equation systems of

discrete sources method 188 8.8. Numerical results 190 8.9. Conclusion 202

References 202

Chapter 9. Hybrid GMT-MoM Method by F. Obelleiro, J.L. Rodriguez, and L Landesa 205

9.1. Introduction 206 9.2. Formulation 208 9.3. On the location of GMT and MoM sources 214 9.4. Regularization of the GMT-MoM method 218 9.5. Conclusions 225

References 226

Chapter 10. Null-Field Method with Discrete Sources by A. Doicu 229 10.1. Introduction 230 10.2. Transmission boundary-value problem 231

Contents vii

10.3. Null-field equations 233 10.4. Complete systems of functions 234 10.5. Null-field method 246 10.6. Numerical results 248 10.7. Conclusions 251

References 252

Author Index 255 Subject Index 261

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CHAPTER 1

Introduction

Adrian DOICU

Verfahrenstechnik Universitdt Bremen Badgasteiner Str. 3 D-28359 Bremen, Germany e-mail: doicu @ iwt uni-bremen.de

and

Thomas WRIEDT

Stiftung Institutfur Werkstojftechnik Badgasteiner Str 3 D-28359 Bremen, Germany e-mail: thw@ iwt. uni-bremen. de

Generalized Multipole Techniques for Electromagnetic and Light Scattering Edited by T. Wriedt © 1999 Elsevier Science B.V. All rights reserved

2 A. Doicu and T. Wriedt

Three-dimensional problems of electromagnetic scattering have been a subject of intense investigation and research in various scientific and engineering fields such as astronomy, optics, meteorology, remote sensing, optical particle sizing or electrical engineering. These efforts have led to a development of a large number of analytical tools and modelling techniques for quantitative evaluation of electromagnetic scattering by various particles. One of the fastest and most powerful numerical tools for computing nonspherical light scattering is the Generalized Multipole Technique. This name was coined by A. Ludwig for a spectrum of related methods [1].

An elaboration of mathematical ideas forms the basis of the Generalized Multipole Tech-nique. The theoretical background was estabhsh by the Georgian mathematicians Kupradze and Vekua and independently by Yasuura. In essence the Generalized Multipole Technique can be regarded as a method of finding the solution of a boundary-value problem for a given differential equation by expanding it in terms of fundamental or other singular solutions of this equation. To be more precise let u be the exact solution of an exterior boundary-value scattering problem with the boundary condition Qu = Quo on S, where 2 is a boundary operator and UQ is an exciting field. An approximate solution of the boundary-value prob-lem can be constructed in the form of a semi-analytic construction us, that satisfies the differential equation in the exterior domain DQ in explicit form and the boundary condition in an approximate form, i.e., \\u — uo\\2,s ^ ^ for any given 5 > 0. In this context, the esti-mate \\u — U8 ||oo,Ge ^ 11 Q^s — Qi^o\\2,s shows that the approximate solution converges to the exact solution in a continuous metric and in any compact set GQ C DQ if it converges to the boundary data in strong norm on S. Essentially, the boundary-value problem simplifies the approximation problem of the boundary values of the exciting field. In the Generalized Multipole Technique the approximate solution is represented by a finite linear combination of fields of elementary sources with amplitudes determined from the boundary condition. In spite of its conceptional simplicity, some problems regarding the choice and the dis-tribution of discrete sources, the elaboration of stable numerical algorithms for amplitude determination, the evaluation of the accuracy of results and the solution of scattering prob-lems for domains with geometrical singularities, for example, have to be solved.

This book contains contributions from several authors from different scientific disci-plines addressing these problems. The present volume brings together theoretical and nu-merical results of eight research groups (from six countries) currently working in light scat-tering modelling with the Generalized Multipole Technique. The idea to publish an edited volume on the Generalized Multipole Technique arose at the 3rd Workshop on Electro-magnetic and Light Scattering which had a special focus on this subject [2]. No effort has been made to estabhsh a common notation throughout the book.

The editor of this volume intends to provide an overview of the literature on the Gener-alized Multipole Techniques in Chapter 2. The history as well as the current state of the art is reviewed.

The basic concepts of the Generalized Multipole Technique are presented by Ch. Hafner in Chapter 3. Essentially, the development and the main features of the Multiple Multi-pole Program (MMP) are outlined. The main goal of this presentation is to give a quick overview of the state of the art and to demonstrate in an example involving the scattering from a complex structure how different MMP features work together and how reliable and accurate results may be obtained.

In Chapter 4 the theoretical background of the methods are presented by Yu. A. Eremin et al. The analysis includes a description of schemes allowing the construction of complete systems of radiating and regular solutions for the Helmholtz equation, a method of analytic

1. Introduction 3

continuation of discrete sources fields into a complex plane with respect to the sources coordinates, an examination of the main properties of fields generated by discrete sources and a scheme of approximate solution construction for the transmission boundary-value problem. The basic concepts are fully presented in the acoustic case and then extended to the electromagnetic case.

For an exterior scattering problem the sequence of amplitudes is bounded, if and only if the support of discrete sources encloses the singularities of the continuation of the scattered field. In this context the localization of the singularities of the continuation of wave fields plays an important role in Chapter 5. This subject is discussed in detail by A.G. Kyurkchan et al. The authors first consider the basic analytical representations of wave fields and define the domains of existence for these representations. Next, the connection between the location of the precisely determined boundaries of these domains and the location of the singularities of the continuation of fields is established. Finally, techniques for locating the so called principal singularities of the continuation of wave fields are examined.

In Chapter 6 Y. Okuno and H. Ikuno review the fundaments of Yasuura's method for numerical solution of 2D scattering problems. Basic concepts like a set of modal func-tions, a sequence of truncated modal expansions and convergence analysis of approximate solutions in terms of modal expansions are described. The conventional Yasuura method, the method using a smoothing procedure for smooth boundaries and a singular-smoothing procedure for edged boundaries are introduced. It is noted that the smoothing procedure employed in the conventional method accelerates the rate of convergence of the approxi-mate solution. This method can be regarded as an elegant alternative of the standard tech-nique which relies on the usage of a class of modal functions (fields of discrete sources) adapted to the geometry of the scattering problem.

In Chapter 7 F.G. Bogdanov et al. give a conventional interpretation of the method of auxihary sources applied to electromagnetic scattering problems. General recommenda-tions for its implementation and illustration of its application to particular problems for a single body or a cluster of bodies made of various material, through numerical simula-tions in a wide frequency band starting from the quasi-statics up to quasi-optics are pre-sented. The general recommendations refer to the proper choice of the auxiliary surface in accordance with the location of the main singularities domain, the optimal distribution of sources and collocation points and the selection of the type of discrete sources, whereas the numerical simulations include the problem of anisotropy, chirality, and those of multiply-connected boundaries.

In Chapter 8 A. Dmitrenko synthetically reviews the mathematical formulations of the Generalized Multipole Technique with tangential electric and magnetic dipoles for per-fectly conducting, impedance, magneto-dielectric, chiral, and coated scatterers. Addition-ally some ideas for the solution of dense ill-posed systems of linear algebraic equations are given.

In Chapter 9 a hybrid technique consisting of the method of moments and the Gener-alized Multipole Technique is presented by F. Obelleiro et al. This approach includes the advantages of both methods: the Generalized Multipole Technique is efficient for large structures with smooth boundaries, whereas the method of moment is more attractive for structures with sharp edges, wires and other discontinuities. Consequently, the hybrid tech-nique allows the investigation of complex structures with less computational costs and memory requirements. In addition, an interesting method concerning the reduction of the ill-posedness of the impedance matrix generated by numerical dependencies between dif-ferent sources is presented. The ill-conditioning problem is overcome by using a Tikhonov

4 A. Doicu and T. Wriedt

regularization over the full impedance matrix, by imposing a quadratic constraint on the unknown amplitudes.

Finally, in Chapter 10 a hybrid approach relying on the basic concepts of the null-field method and the generalized multipole technique is discussed by A. Doicu. The method consists of representations of surface current densities by fields of elementary sources and formulations of null-field equations in terms of discrete sources. The system of localized and distributed spherical vector wave functions, distributed dipoles and vector Mie poten-tials are examined as examples of discrete sources.

We hope that these papers written by those presently involved in the subject, will encour-age others to join efforts in future. To help students in this field some computer programs based on the Generalized Multipole technique will be available from the web-page "List of electromagnetic scattering codes" [3].

In compiling this edited volume we gratefully acknowledge work done by the contrib-utors of the chapters. The editor would also like to thank Institut fiir Werkstofftechnik for the opportunity to devote some time for editing this volume. As no book is published without some help, we would like to take this opportunity to express our deep thanks to Gordon Videen, B.M. Nebeker and Michel Haasner for their careful reading of the original manuscript. During the preparation of this manuscript we received the most valuable help from Henning Sagehom, Ute Comberg, Roman Schuh and Jan Koser.

References

[1] A.C. Ludwig, A new technique for numerical electromagnetics, IEEE Antennas Propagat. Newslett. 31, 40-41 (1989).

[2] T. Wriedt and Yu. Eremin, Electromagnetic and Light Scattering Theory and Applications, Proc. 3rd Work-shop on Electromagnetic and Light Scattering Theory and Applications, March 16-17, 1998, Bremen, Ger-many (University of Bremen, Bremen, 1998).

[3] T. Wriedt, List of Electromagnetics Scattering Codes, http: //imperator.cip-iwl.uni-bremen~/fg01/codes2. html (1999).

CHAPTER 2

Review of the Generalized Multipole Technique Literature

Thomas WRIEDT

Stiftung Institutfur Werkstojftechnik Badgasteiner Str. 3 D-28359 Bremen, Germany e-mail: thw@ iwt uni-bremen.de


6 T. Wriedt

Scattering computations help to understand new physical phenomena or to design new particle diagnostics systems for the identification of variations in particle optical proper-ties or particle shape. Furthermore, computation of light scattering by particles plays an enormous role not only in optical particle characterization but also in astronomy, optical oceanography, photographic science, meteorology, coatings technology, to name but a few. Similar electromagnetic modelling methods are needed to investigate microwave scattering by raindrops and ice crystals or to analyse electromagnetic interference problems.

Up to now Mie scattering was an important tool to diagnose micro particles or aerosol particles in technical or natural environments. Mie theory is restricted to spherical, ho-mogeneous, isotropic and nonmagnetic particles in a nonabsorbing medium. But micro-particles are hardly ever spherical or homogeneous, thus there is much interest in more advanced scattering theories. During the last decades, scattering methods for nonspherical and nonhomogeneous particles have been developed and even some computer codes are readily available.

The Generalized Multipole Technique (GMT) is a relatively new and fast advancing method which has been developed by different research groups. Ludwig [1] coined the term generalized multipole technique for this spectrum of methods. In Mie theory and in the T-matrix method the fields inside and outside a scatterer are expanded by a set of spherical multipoles having their origin at the centre of the sphere. With the GMT method many origins are applied for multipole expansion. The coefficients of these expansions are the unknown values to be determined by applying the boundary conditions on the particle surface. The coefficients may be found by point matching, that is, fulfilling the bound-ary conditions at a discrete number of surface points, or fulfilling the boundary conditions in a least squares sense, or by a surface integral similar to the extended boundary con-dition method. Not only multiple spherical multipoles can be used for field expansion, other "equivalent sources" are also possible. The "equivalent sources" may be of any type, as long as they are solutions of the wave equation. Spherical waves, dipoles and Gabor functions have been applied as expansion functions. Therefore, other names for similar concepts have been given like Multiple MultiPole Method (MMP) [2], Discrete Sources Method (DSM) [3], Method of AuxiHary Sources (MAS), Fictitious Sources Method [4] or Yasuura method [5]. Although the GMT methods have a history of over 30 years it did not find that much interest compared to other methods. But nowadays its popularity is steadily increasing. Extensions and enhancements to the methods and computer codes are continuously being published, which broaden the scope of the methods. To help the reader the division of this review mainly follows the chapter division of this book although some other division might be more suitable and there will be some redundancy. First related methods will be reviewed. That is the Point Matching Method and the Extended Boundary Condition Method. Then the different variants of the GMT methods will be reviewed under the name given above.

2.1. Point matching method

The Point Matching Method may be considered one forerunner of the Generalized Mul-tipole Techniques. The Point Matching Method has been developed to compute microwave scattering by spheroidal rain drops by Oguchi [6]. In principle this technique can be applied to an arbitrary scatterer with a regular boundary surface. The formulation of the technique closely follows Mie theory and is based on the expansion of fields in terms of partial solu-

2. Generalized multipole technique literature 1

tions of the wave equation. Similar to Mie theory, the scattered field and transmitted field are expanded into terms of spherical vector wave functions (also called multipoles).

The expansion coefficients of the scattered field are found by satisfying the boundary conditions at the surface of the scatterer by point matching. The boundary conditions re-quire the continuity of the tangential components of the electric and magnetic field across the surface of the scatterer. If the scatterer is axisymmetric the incident plane wave is ex-panded into a complex Fourier series in the azimuthal angle. Because an axially symmetric scatterer is considered, the boundary condition can be enforced independently for each term of the Fourier series. With this collocation method the number of fitting points is the same as the number of unknown expansion coefficients. For a nonaxisymmetric scatterer a least square fitting procedure can be used on an increased number of matching points as published by Morrison and Cross [7]. The method is considered to have uncertain conver-gence and accuracy and to need much computer time [8]. The point matching method is still being used in the field of radar scattering by raindrops. Recently the method has been applied to compute microwave scattering by oblate spheroidal hydrometeors up to a size parameter of 9.42, expressed as equivolume radius by Zhang et al. [9]. The point match-ing method has apparently been reinvented by Sarkar and Halas [10]. The point matching method is restricted to scatterers only slightly deformed from a sphere.

2.2. Extended boundary condition method (EBCM)

The Extended Boundary Condition Method (EBCM) is another well-known technique which found a wide range of applications because a code for a conducting scatterer was published very early by Waterman [11]. A disk with FORTRAN code for dielectric bodies of rotation is included with the book by Barber and Hill [12]. The Extended Boundary Con-dition Method is also called Null Field Method, Schelkunoff Equivalent Current Method, Eswald-Oseen Extinction Theorem or T-Matrix Method. It is based on a series of papers by Waterman [13]. An early collection of conference papers on this method has been edited by Varadan and Varadan [14].

In this method the scattering particle is replaced by a set of surface current densities, so that in the exterior region the sources and fields are exactly the same as those existing in the original scattering problem. A set of integral equations for the surface current densities is derived by considering the bilinear expansion of the Green function. The solution of the scattering problem is then obtained by approximating the surface current densities by the complete set of partial wave solutions to Helmholtz equation in spherical coordinates. Most numerical computations based on Extended Boundary Condition Method use spherical vector wave functions as global basis functions.

The incident, transmitted and scattered field is expanded into a series of spherical vector wave functions. The expansion coefficients of the scattered field are related to the coeffi-cient of the incident field by the T-matrix (transition matrix).

The elements of the T-matrix are obtained by numerical integration. For an arbitrarily shaped particle a surface integral has to be computed. As this is computationally expen-sive, most implementations of the methods are restricted to axisymmetric scatterers. In this case line integrals have to be computed. Nevertheless, there are some papers applying the T-matrix method to arbitrarily shaped scatterers [15,16]. In the paper by Wriedt and Doicu [16] computational examples of scattering by a dielectric cube of size parameter 2 and by a dielectric spheroid of size parameter 20 are presented. Scattering by a dielectric

8 T. Wriedt

cube is computed by both the Extended Boundary Condition Method and the EBCM with Discrete Sources.

It is easy to extend the T-Matrix method to coated spheroids [17,18] to model water coated ice particles in the atmosphere. Mishchenko and Travis [19] demonstrate that us-ing extended precision instead of double-precision variables helps to improve convergence of the method up to particle size parameters of 100 (equal-surface-area-sphere size pa-rameter). A review of the current status of the T-matrix approach has been published by Mishchenko, Travis and Mackowski [20]. They conclude that recent improvements make the method applicable to particles with size parameters well exceeding 50. The standard Extended Boundary Condition Method can be applied to more deformed scatterers but there are still some problems with concarve particles and with particles having a high axial ratio such as fibres.

Although the spherical vector wave functions appear to provide a good approximation to the solution when the surface S is smooth and nearly spherical, there are some disad-vantages when this is not the case. A number of modifications of the Extended Boundary Condition Method have been suggested, especially to improve the numerical stability in computations for particles with extreme geometries. These techniques include formal mod-ifications of the standard Extended Boundary Condition Method [22,24], different choices of basis functions and the application of the spheroidal coordinate formahsm [21,23] and the use of discrete sources [16].

Multiple spherical wave expansions were used for the first time in the iterative version of the Extended Boundary Condition Method lEBCM by Iskander et al. [24]. This approach utilizes multipole spherical expansions to represent the internal fields in different over-lapping regions, rather than summing the various expansions and using it throughout the particle as in Extended Boundary Condition Method with Discrete Sources. The various expansions are then matched in the overlapping regions to enforce the continuity of the fields throughout the entire interior volume.

The strategy followed in the EBCM with Discrete Sources by Wriedt and Doicu [16] is to derive a set of integral equations for the surface current densities in the variety of auxiliary sources and to approximate these densities by fields of discrete sources. Actually, distributed sources are better suited to model complex boundaries than localized sources, since the null-field condition will be satisfied in the interior of the discrete sources support, whose form and position can be correlated with the boundary geometry. As discrete sources localized and distributed spherical vector wave functions, distributed dipoles and vector Mie-potentials can be used.

2.3. Multiple multipole method (MMP)

Obviously the Multiple Multipole Method is a very well known variant of the general-ized multipole techniques because a computer program and FORTRAN codes are easily available.

Starting from an analysis of the Point Matching Technique combined with the Circular Harmonic Analysis, this method was proposed by Hafner in 1980 [26] as an extension of these two methods. The main goal of the first MMP codes was the investigation of the in-fluence of well-known simplifications that were required for obtaining analytic solutions. In 1980-1982 the very first 2D MMP code was written for computation of guided waves on arbitrary cylindrical structures, followed by a version for scattering and one for elec-trostatics and magnetostatics and, finally, a 3D MMP code for scattering by Klaus [27].


These early MMP codes ran on a CDC mainframe and were written in Fortran-66. Later a Fortran-77 code running even on a PC, was published by Hafner and Bomholt [32] on the most common version of 3D MMP for scattering. The capability of this code was continu-ously improved by various scientists from the Zurich group: Automatic general procedure for the expansion function choice in 3D by Regli [33], thin wire expansions by Zheng and Hafner [36], dipole expansions on layered media by Novotny [37], anisotropic media by Piller [38], Parameter Estimation Technique (PET) [39,40] and Eigenvalue Solvers [40] by Hafner. In order to become windows compatible and more flexible, a new MMP version was written in Fortran-90 and C + + as a part of MaX-1 [41].

The MMP codes have a wide range of applications (electrostatics, scattering, guided waves, resonators, discontinuities). All of them work in the frequency domain and most of them are restricted to linear, isotropic media. There are no restrictions concerning the geometry of the media, except that the surface must be sufficiently smooth.

The field domain is separated into a number of subdomains, each filled with linear, homogeneous and isotropic material. In each domain a separate expansion of the field (e.g., into series of multipole fields with different origins) is made. Any choice of the unknown expansion coefficients results in a correct solution of Maxwell's equations, since each of expansion function is such a solution. The incident fields generated by sources in each domain are added to the respective expansion to obtain the total fields, which are matched at the boundaries. Generally MMP demands only one domain per material region, but "artificial" boundaries may be employed to reduce the complexity of a boundary shape or to smoothen a boundary.

This is why one may call MMP a semi-analytical method: the differential equations in each subdomain are solved analytically, whereas the boundary conditions on the bound-aries between two subdomains are fulfilled numerically by point matching. The employed extended point matching method is numerically equivalent to a Galerkin projection tech-nique and to a least squares error minimization. The resulting system of equations is overdetermined, since this proved to lead to fast and stable convergence.

In the 3D scattering case, spherical multipole expansions with Hankel functions may be applied. This implies, that the origins of these functions (the poles) have to be situated outside the respective domain, where they are used as expansion functions. The origin as well as the order of the poles are chosen, so that all surface points, which do not have to lie on a closed surface (!), are "illuminated" by at least one expansion function. The choice of the poles is done quite experimentally. A good criterion is to look for a set of poles, of which the regions of influence do not "overlap" too much, i.e., the "illumination" should be as uniform as possible.

Usually, graphic front ends are used for modelling (definition of the boundaries and of the MMP expansions) as well as for validating (error representation) and visualization of the results. For extensive information on the state of art of the MMP method the reader is referred to the latest book by Hafner [41].

As the MMP code is available with the Hafner and Bomholt book the method was also applied and extended by other researchers. An extension includes the integration of a Gaus-sian laser beam [42] which has been used to simulate the Phase Doppler Anemometry (an optical particle sizing method) with two spheres in two Gaussian laser beams [43]. Ap-plication of the MMP program ranges from light scattering by disk shaped silver halide crystals [44], microwave scattering by raindrops [49] to specific absorption rate of an ab-sorbing sphere with dipole excitation [45,46].

An important point with the Multiple Multipole Method is choosing the parameters of the expansions that are the location of the multipole expansions and its orders. This prob-

10 T. Wriedt

lem has been considered by Tudziers et al. [47,48] who devised some rules to automatically find this parameters. The method presented is based on a patch model of the scattering body.

A similar method also using spherical multipole expansions has been developed by Ludwig [1,50,51] being his Spherical-Wave Expansion Method (SPEX). Ludwig obtained good agreement between the SPEX technique and the NEC wire model for a conducting elongated scatterer [50].

The Multiple Multipole Method has recently been reinvented by Al-Rizzo and Tran-quilla [52] for scattering by highly elongated objects.

The MMP method is mainly applicable to scatterers with a smooth surface. As there are singularities near edges it will hardly be possible to approximate them by smooth fields generated by the multipole sources. To overcome this problem with nonsmooth scatterers additional surface currents have been used by Rodriguez et al. [53,54] to represent the fields near the nonsmooth regions of a two dimensional scatterer.

Another hybrid approach to overcome this problem has been presented by Pascher and Leuchtmann [55] to analyse scattering by a finite cylinder. This method is based on a combination of the Method of Lines (MoL) and the Multiple Multipole Method.

Yet another hybrid MMP methods was developed by Rouss, Jakobus and Landstor-fer [56,57] coupling the Methods of Moments (MoM) to the Multiple Multipole Method to simulate radiation of mobile telephone antenna in the vicinity of a human head model. An iterative method is used to combine these two existing codes.

2.4. Yasuura's methods

Similar GMT concepts were developed in Japan by Yasuura. In a series of papers [58] published in 1965 and 1966, Yasuura established a method of numerical solution of the boundary-value problems for the 2D Helmholtz equation in a homogeneous medium. This method is called the conventional Yasuura method (CYM) nowadays. Even from the present point of view, the papers include important and basic ideas that dominate later development, i.e., the set of modal functions, a sequence of truncated modal expansions, least-squares boundary matching, and an adjoint method for a surface current density. The CYM was applied to the problems of a tilted waveguide and a triangular grating, where the separated solutions were employed as modal functions. The numerical results proved that the CYM was effective provided that the boundaries were not strongly deformed from the coordinate axes. Yasuura presented his method with numerical examples at the XVI General Assembly of URSI held in Ottawa, Canada, in 1969 [59]. In those days there was much controversial discussion on the validity of the Rayleigh assumption. Because the infinite-series solution in terms of separated solutions employed in Rayleigh's context [60] apparently has a radius of convergence, the discussion was focused mainly on the range of validity [61-64]. From this point of view, the CYM can be understood as a modified (or justified) Rayleigh method: the infinite series has been replaced by a sequence of truncated modal expansions and, consequently, convergence of the sequence has been proven. This provides us with a practical means for the numerical solution of problems with arbitrarily shaped boundaries. Today modal expansion approaches including Yasuura's methods are accepted as standard methods for solving the boundary-value problems in computational electromagnetics. Besides, it is known that the approximation efficiency of the separated solutions is often so poor that huge numerical computation is required to obtain a solution


with accuracy. There are two alternative means to remove this difficulty. One is to employ a class of modal functions that is appropriate for the geometry of the problem, i.e., to choose the kind of poles and to locate the poles cleverly. Another way to overcome the difficulty is to equip a smoothing or a singular-smoothing procedure (SP or SSP) with the least-squares boundary matching. The SP is for problems with smooth boundaries and the SSP is for edged boundaries. Yasuura's method with the smoothing procedure (YMSP) [65,66] and the one with the singular-smoothing procedure (YMSSP) were reported by Yasuura et al. [67,68] for the first time in 1977 and 1981, respectively. Since that time the methods were applied to a wide class of 2D problems. Usually the solutions obtained by the YMSP or YMSSP converge more rapidly than the ones by the CYM. Most of the 2D problems in the resonance region can be solved by these methods. Recent development in Yasuura's methods may be found in two directions: first in the method of numerical computations and second in the selection of the set of modal functions. In solving the least-squares prob-lem formulated on the surface of an obstacle, a discretization procedure is needed because the problem is stated in a space of continuous functions. The orthogonal decomposition methods (the QR and the singular-value decomposition) are found to be well fitted for both the discretization and solution of the discretized problem. On the other hand, in view of the poor efficiency of the separated solutions, trial employment of a class of modal functions other than the separated solutions has been done in the solution of 2D and 3D problems. Yasuura's methods are extensively employed to examine the scattering from 2D and 3D obstacles [69], diffraction from metal gratings [70], and diffraction by multilayer-coated gratings.

2.5. Discrete sources method (DSM)

An other well known variant of the Generalized Multipole Techniques is the Dis-crete Sources Method. The present version of the DSM developed by the group of Y Eremin [71-73] has some features, which makes it different from other related GMT techniques. Those main differences are:

(1) It includes the explicit scheme for constructing the complete systems of DS fields; (2) It is the most effective applied to axially symmetric structures; (3) DS are deposited at the axis of symmetry or in adjoining area of a complex plane; (4) Representation for the approximate solution takes into account an axial symmetry

of a scatterer and polarization of an external excitation; (5) It enables to estimate a posterior error for the result computed. The first version of the DSM was pubHshed in 1980 by Sveshnikov and Eremin [74].

It allowed treating electromagnetic scattering from a perfect conductor under axial excita-tion (plane wave propagating along an axis of symmetry) only. Such a simplified kind of excitation allowed the approximate solution to be constructed as a combination of electric and magnetic dipoles located at the symmetry axis. In 1982 the method was extended to an analysis of a homogeneous penetrable obstacle [75]. Regular functions were employed to represent the field inside a scatterer. In 1983 the DSM theory was published [76]. The first version of DSM had some limitations associated with the geometry of the scattering ob-stacle. Particularly, it was not able to analyse an oblate obstacle. This restriction was over-come in 1983 by means of analytic continuation of the DS support into a complex plane adjoining the symmetry axis [77]. The deposition of DS at the complex plane eliminated the limitations of the original DSM scheme and enabled treating any obstacle geometry. In

12 T. Wriedt

1985 the DSM was generalized to analyse non-axial excitation [78]. Theoretical outlines of DSM along with a generic scheme allowing to generate complete systems based on DS support geometry came in 1987. Some new complete systems of DS for BVSP analysis were built [79]. In the same year an algorithm generating complete systems of DS for an obstacle located in a half-space was developed [80]. The next modification, proposed in 1993, allowed taking into consideration not only the axial symmetry of the obstacle, but also the polarization of the exciting field [81]. Recent editions of the DSM have the ben-efit of expanding the technique to a wide variety of applications [82-85]. During last few years, DSM has been applied mostly to modelling scatterers in the presence of stratified structures at a smooth substrate [86-89] to investigate surface particle counters.

2.6. Method of auxiliary sources (MAS)

The development of the Method of AuxiHary Sources is based on the works of the famous Georgian mathematicians Vekua, Kupradze and Aleksidze carried out in 1943-1967 [90-99]. The main idea of these works was the basic theorem of the completeness in L^iS) (of the infinite) set of fundamental or other singular solutions of the differential equation corresponding to the given boundary problem. These ideas found application in specific boundary problems: elasticity, hybrid, biharmonic, hydro, electrodynamics, and acoustic problems. Thus, the mathematical basis for finding solution of various boundary problems was grounded.

The implementation of MAS in the field of the applied electrodynamics is connected with the Laboratory of Applied Electrodynamics (LAE) of Tbilisi State University (TSU), starting in the beginning of 1970. This laboratory performed studies not only applying the above-mentioned ideas, but also essentially improving the efficiency of MAS and extend-ing it to solve a wide region of electrodynamic problems [100-114]. During the numerical realization of the applied problems, some problems of bad convergence and even diver-gence of the solution were encountered [100-106]. The problems arose for some geome-tries of auxiliary surface [101]. Examination of these problems did show that they could be overcome if only all the physical properties of scattered field were taken into account in the algorithm, namely, the scattered field singularities and the resonance of auxiliary surface.

By the middle of 1980, the studies of TSU and other scientific centres allowed to formu-late the main principles of constructing solutions to the diffraction problems based on the conventional MAS. The geometry and the incident field determine which auxiliary surface should by used to construct solution. The resonance of the auxiliary surface had been used to develop a highly efficient method for solving interior boundary problems. The MAS be-came one of the most powerful and efficient methods for solving applied electrodynamic problems even on the low-performance computers in the former USSR.

Recent investigations revealed that singularities are distributed as bright centres around the caustic surface and that the distance between them is half a wavelength. The localiza-tion of the singularities is used for partial representation of the scattered field for solving scattering problems associated with large objects. In this vision, the approach is similar to the mirror image method. For localization of the wave fields singularities, the functions, describing the converging and diverging waves are used. Based on this concept numerical methods for field reconstruction and inverse problems solution were suggested. Thus, the current conception of the MAS was formed. These improvements allowed to solve and optimize the key scattering and diffraction problems for open and close surfaces, isolated and set of bodies [101-108], periodic gratings [102,109], and waveguides [110-114].


The MAS is efficiently employed in TSU to solve applied 2D and 3D scattering and diffraction problems for isotropic [115,116], as well as for complex filling - anisotropic, chiral, and biisotropic bodies [117-120]. The method is employed for antenna, inverse scattering and transient diffraction problems, problems of reconstruction and visualization of singularities of scattered field [121-123], and electrostatic discharge problems [124].

A similar method also based on elementary electric dipole sources distributed on an auxiliary surface homothetic to the surface of the scatterer was developed by Dmitrenko [125,126].

Another related method is the method developed by Leviatan and coautores [127-130]. Fictitious electric currents flowing on a mathematical surface enclosed by a conducting body are used to simulate the exterior scattered field. The resulting operator equation is solved by the methods of moments [127]. This method has also been extended to compute scattering by an oblate dielectric spheroid using dipoles located in complex space [129].

An other Fictitious Current Method has been extended to compute scattering from ec-centric multilayered dielectric bodies of revolution by Na and Kim [ 131 ]. As there are two boundaries, two sets of fictitious current sources are used and the boundary value problem is solved repeatedly for each boundary until the desired degree of accuracy is reached.

2.7. Analytical continuation of solutions of boundary problems

The issue of analytical continuation of solutions of elliptical equations plays an impor-tant role in development of the GMT theory, therefore it is reviewed in this separate section. This issue has been discussed in literature for over a hundred years. Apparently, its origin are in papers by Schwartz [132], Herglotz [133], and Jeans [134]. In the two-dimensional case, explicit formulae for the analytical continuation were derived, which are based on Schwartz's principle of symmetry and related to static fields [132]. In an effort to gener-alize these studies, the reflection formulas were extended to the multidimensional case by Garabedian [135]. In succeeding years the theory for the analytical continuation of solu-tions of boundary problems evolved in the light of the so-called Rayleigh hypothesis for the extendibility of the expansion of secondary (diffraction) fields into a series in terms of metaharmonic functions up to the boundary of the scatterer. Most serious results were obtained by Petit and Cadilhac [136] and Millar [137]. They were concerned with prob-lems on the wave diffraction by periodic surfaces. Of importance were also results obtained by Millar [138] and van der Berg and Fokkema [139] were dedicated to the identification of the conditions under which the Rayleigh hypothesis is applicable to the case of wave diffraction by compact scatterers. Millar [140,141] made an attempt to use integral equa-tions for solving the problem of locaHzing the singularities of the continuation of boundary problem solutions.

Many works are dedicated to the inverse problems of the Helmholtz equation, including their relation to the analytical continuation of wave fields. Among them, Mueller [142] should be noted since he presumably for the first time treated the class of functions that represent patterns of wave fields. The results obtained by Mueller [142] were used by Colton [143] and Sleeman [144] for determining the boundary of the domain into which the wave field characterized by a given pattern can be continued. An ingenious method for the continuation of a wave field, which is based on the analysis of the asymptotic properties of the field's pattern in the domain of complex angles, was proposed in a paper by Weston etal. [145].

14 T. Wriedt

Kyurkchan [146-148] established exact boundaries of the existence domains for the expansion of wave fields in terms of metaharmonic functions as well as for the plane wave integral representations. These boundaries are defined by parameters describing the growth of a wave field pattern within the domain of its complex angles. The description of the class of functions, in which the field patterns of compact sources and the scattering patterns of bodies belong, is also given [149,150].

Additionally the representation of diffraction fields is generalized using wave potentials in the case where the carrier of potential densities is in the interior of the scatterer [151, 152].

In a series of papers Kyurkchan [153-155] developed the method of locaHzing the pri-mary singularities for the continuation of fields in the problems of wave scattering by compact bodies as well as by periodic and nonperiodic surfaces.

Stemin and Shatalov [156] obtained the solution of the problem of "sweeping of sources", that is, seeking the function g{x) whose carrier is strictly enclosed by the carrier f{x) such that the equality [// ^ = (7^(^) is correct outside carrier of/(x),(/^^^> being the solution of the equation {A + k^n{x)}U^^^'^ = f(x).

Savina [157] generalizes the reflection formula (Schwartz's principle of symmetry) on the Helmholtz equation.

Stemin and Shatalov [158,159] generalized the results of investigations into the contin-uation of the solutions of boundary problems for elliptical equations.

Finally, the current status of the problem is reviewed by Kyurkchan et al. [160]. In whole, the problem of the continuation of solutions and of localizing their singularities can be assumed to be solved theoretically. However, in regard to applicability of the corresponding theory to mathematical modeUing, we are at the very beginning of activities.

2.8. Conclusion

Over the last 30 years the Generalized Multipole Techniques have demonstrated to be ef-ficient and flexible concepts for analysis of electromagnetic and light scattering problems. Indeed its different variants have been successfully applied in a number of scientific disci-plines outside classical electromagnitics and optics. This fields include acoustics, particle sizing, photographic science, neuroscience, and astrophysics.

Acknowledgement

I would like to acknowledge Christian Hafner, Adrian Doicu, Ute Comberg, Yoichi Okuno, Yuri Eremin, Revaz Zaridze and Alexander Kyurkchan for contributing to this review.

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[ I l l ] R.S. Zaridze, D.D. Karkashadze, J.Sh. KhatiashviU, and G.Z. Akhvlediani, Approximate calculation method for dielectric waveguides with complex cross-section. Bull. Georgia Acad. Sci. 102(1), 53-56 (1981) (in Russian).

[112] R.S. Popovidi-Zaridze, D.D. Karkashadze, G.Z. Akhvlediani, and J.H. KhatiashviU, Investigation of the possibilities of the method of auxiUary sources in solution of the two-dimensional electrodynamics prob-lems, Radiotekh. Elektron. 26(2), 254^262 (1981) (in Russian).


113] R. Zaridze, D. Karkashadze, and J. Khatiashvili, Method of Auxiliary Sources for Investigation of Along-Regular Waveguids (Tbilisi State University Press, Tbilisi, 1985) pp. 1-150 (in Russian).

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119] E.G. Bogdanov and D.D. Karkashadze, Conventional method of auxiliary sources in the problems of elec-tromagnetic scattering by the bodies of complex materials, in: Proc. 3rd Workshop on Electromagnetic and Light Scattering, Bremen, March 16-17, 1998, T. Wriedt and Yu. Eremin, Eds. (Bremen, 1998) pp. 133-140.

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122] R. Zaridze, G. Bit-Babik, D. Karkashadze, R. Jobava, D. Economou, and N. Uzunoglu, The Method of Auxiliary Sources (Institute of Communications and Computing Systems, Athens, Greece, 1998).

123] R. Zaridze, R. Jobava, G. Bit-Babik, D. Karkashadze, D. Economou, and N. Uzunoglu, The method of auxiliary sources and scattered field singularities (caustics), J. Electromagn. Waves Applic. 12(11) (1998).

124] R. Zaridze, D. Karkashadze, R. Djobava, D. Pommerenke, and M. Aidam, Numerical calculation and measurement of transient fields from electrostatic discharges, IEEE Trans. Components, Packag. Manu-fact. Technol, Part C 19(3), 178-183 (1996).

125] A.G. Dmitrenko and A.I. Mukomolov, Diffraction of electromagnetic waves in a three-dimensional mag-netodielectric body of arbitrary shape, Russian Phys. J. 38(6), 617-621 (1995).

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128] B. Pomeraniec, Y Leviatan, and A. Boag, Analysis of the 5:1 benchmark case using the current-model method with an SVD-improved point matching technique, J. Electromagn. Waves Applic. 7(12), 1577-1593 (1993).

129] E. Erez and Y Leviatan, Computational analysis of scattering by penetrable oblate spheroids using a model of dipoles located in complex space, lEEProc. - Microwave Antennas Propagat. 142(3), 245-250 (1995).

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vorgeschriebenen Grenz- und Unstetigkeitsbedingungen, Monatsber den Koenig. Akad. der Wiss. zu Berlin, 767-795 (1870).

133] G. Herglotz, Ueber die analytische Forsetzung des Potential ins Innere der Anziehenden den Massen (Gekroente Preisschr der Jablonowskischen Gesellsch., Leipzig, 1914).

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[137] R.F. Millar, On the Rayleigh assumption in scattering by a periodic surface, Proc. Cambridge Philos. Soc. 65, 773-791 (1969).

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obstacle, IEEE Trans. Antennas Propagat. AP-27(5), 577-583 (1979). [140] R.F. Millar, The Singularities of Solutions to Analytic Elliptic Boundary Value Problems, Lecture Notes in

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entsialnye Uravneniya 33(8), 1123-1133 (1997) (in Russian). [160] A.G. Kyurkchan, B.Yu. Stemin, and V.E. Shatalov, The singularities of the continuation of wave fields.

Physics Uspekhi 39, 1221-1242 (1996).

CHAPTER 3

The Multiple Multipole Program (MMP) and the Generalized Multipole Technique (GMT)

Ch. Hafner

Laboratory for Electromagnetic Fields and Microwave Electronics Swiss Federal Institute of Technology Zurich CH-8092 Zurich, Switzerland e-mail: [email protected]


22 Ch. Hafner

3.1. Introduction

The Multiple Multipole Program (MMP) has been developed in the late 1970s and early 1980s starting from the Point Matching (PM) technique [1,2] in conjunction with the Cir-cular Harmonic Analysis (CHA) [3,4]. These and similar methods that had been widely criticized with mainly wrong arguments (see [4] for a list of references), although a strong mathematical background had been developed for the CHA already in 1948 by Vekua [5]. It seems that the relevant mathematical theorems were almost unknown to those who de-signed codes for computational electromagnetics in the 1960s outside the Soviet Union. Moreover, the implications of numerical approximations were not well understood by most of the code designers in those days. Obviously, the theoretical problems were overestimated and the numerical problems were underestimated. To obtain efficient and reliable codes, the numerical problems of both the CHA and the PM had to be removed. This goal was achieved by a careful analysis and a generahzation of both methods. The resulting code was called Multiple Multipole Program (MMP) and the corresponding method was called Multiple MultiPole (MMP) method [6].

In 1989, Ludwig [7] presented the SPEX code that was obviously very similar to the 3D MMP code for EM scattering. Moreover, it was recognized that several groups were working on techniques that could be considered as special cases of the MMP method. Therefore, Generalized Multipole Technique (GMT) was proposed as a new generic name.

In Zurich, several MMP codes were developed on various computers. The first MMP codes on CDC mainframes were written between 1980 and 1987 (Hafner: Guided waves, Hafner and Ballisti: 2D electro- and magnetostatics, Kley: 2D scattering, Klaus: 3D scat-tering). After 1987, several codes were written for personal computers and workstations. A relatively general version of 2D MMP for scattering and eigenvalue problems was pub-Hshed in 1990 [8], followed by the 3D MMP code for EM scattering [9]. The kernel of 3D MMP was mainly written by Bomholt, the graphic front end and the Fourier tool by Hafner. Some portions of the 3D MMP kernel and upgrades were contributed by Frohlich, Gnos, Hafner, Leuchtmann, Novotny, Regli, and Zheng. A more general and flexible 2D MMP version is contained in the MaX-1 code [10].

3.2. From CHA to MMP expansions

The Circular Harmonic Analysis (CHA) is based on a series expansion of the EM field inside each of the homogeneous domains of a given, cylindrical or 2D structure.

Field = y ^ A/tbasisA: + Error. (3.1)

The basis field functions of this expansion are obtained with the separation of variables in a polar coordinate system (r, (p) and have the form

basisjt = L{Rn{Kr) • (l)(n(p)), (3.2)

where L is a linear operator that can be derived from Maxwell equations, R denotes a Bessel, Neumann, or Hankel function, and 0 denotes a harmonic function (cos, sin, or

3. The multiple multipole program 23

exp). Since L contains only some simple derivatives, Eq. (3.2) can easily be handled ana-lytically. The CHA had been used already by Mie in 1900 [11] for computing waves on two circular wires. An extension of the CHA for 3D problems with a separation of variables in spherical coordinates leads to SPherical Expansions (SPEX) [7] that are very similar to Eq. (3.2). A strong mathematical background for the CHA in multiply-connected domains bounded by Hoelder-continuous boundaries was given by Vekua in 1948 [5]. Although one can prove the completeness of the CHA for all geometrical shapes of technical interest, it is important to recognize that the completeness of a basis and the convergence toward the correct solution are not the most important issues for numerical methods. First of all, com-pleteness in the sense that every possible solution may be expanded is not required at all. It is sufficient that the desired solution of the given problem can be approximates as accu-rately as desired. Furthermore, the required accuracy in engineering is often not very high. For quick approximations, methods with a bad convergence or even non-convergent meth-ods can therefore be much more efficient than methods with fast convergence. Essentially, the situation is similar to the well-known semi-convergent series expansions of functions.

The convergence of the CHA turns out to depend on the material properties, the fre-quency, and the geometric shape. Especially the Bessel terms may converge very slowly in non-circular domains with a strong skin effect. In general, the CHA is inefficient, except for domains with a boundary that is sufficiently close to a circle. To obtain an efficient series expansion, the CHA must be generahzed.

3.2.1. Generalization of the CHA

In electrostatics, the charge simulation technique [12] is a very old and powerful method, where the charges on the surface of a conductor are simulated by "fictitious" charges (or sources) inside the conductor. This technique can be easily generalized and applied to elec-trodynamics as well. Since only one unknown parameter is associated with each charge, one can easily correlate one matching point with each charge. Useful results can be ob-tained with a reasonable distribution of the fictitious sources along all boundaries and the corresponding matching point distribution.

From the CHA point of view, the field of an electric charge is a monopole, which is in-cluded in the expansion Eq. (3.1) as the zero order term, i.e., the CHA is more general than the charge simulation and similar techniques, but it uses less polar coordinate systems. The restricted number of polar coordinate systems required by Vekua's theorems is the most important drawback of the CHA. Technically, the generalization of the CHA by admitting more coordinate systems is almost trivial and directly leads to the standard MMP expan-sion. One can even obtain a much more general formulation without formally changing the expansion Eq. (3.1) by admitting any solution of Maxwell equations inside one or several domains as a basis field function in Eq. (3.1). Although the term MMP means Multiple MultiPole, the more general formulation is used in the MMP code. In this code, many other types of basis field functions were implemented in addition to multipole fields. How-ever, multipole expansions are still most frequently used due to their agreeable numerical behaviour. Note that (1) the first M of MMP denotes the use of multiple coordinate sys-tems, each of the associated with a multipole expansion and that (2) a multipole expansion usually contains several orders at the same time, i.e., several unknowns.

The generality of the MMP expansions causes several problems. First of all, the effi-ciency of the method and the accuracy of the results depend very much on the selection of

24 Ch. Hafner

the basis functions. For the appropriate selection of multipole expansions, several heuristic rules have been found and implemented in routines that automatically set the locations and orders of the multipoles.

The second problem caused by the generality of the MMP expansion is the high risk of obtaining relatively ill-conditioned matrices. This problem will be outlined separately.

The MMP expansion may also cause severe problems in finding an appropriate set of matching points when the PM technique shall be applied to compute the unknown coef-ficients in Eq. (3.1). To overcome these problems, PM must be generalized as outlined below.

Finally, the generality of the MMP expansion causes problems in theoretical studies of the method. For example, it is hard to reasonably define even simple terms such as the convergence. This problem will also be discussed later.

However, the generalization of the CHA led to the MMP expansion Eq. (3.1), where basisA: may be an arbitrary solution of Maxwell equation in at least one domain. When it is a solution in exactly one domain, it is called single-domain expansion. Otherwise, it is called multi-domain expansion. The practical implementation of Eq. (3.1) in the MMP code is less general because the number of available basis field functions is necessarily limited, but MMP contains a large library of useful functions for computational electromagnetics and extending this library is relatively easy.

3.2.2. Connections

All of the MMP expansions are analytic solutions of the Maxwell equations. These equations are linear and only linear material properties are admitted in the MMP code. Therefore, any linear combination of MMP expansions is again an analytic solutions of the Maxwell equations. Such linear combinations are called connections. Note that con-nections may be nested, i.e., a connection can contain other connections. Connections are often solutions of subproblems of the given problem. For example, when one considers an object that is illuminated by an antenna, the antenna without the object is a subproblem. Its field may be included in a connection. When the interaction between the antenna and the object is weak, the following procedure leads to accurate results: (1) Compute the field of the antenna without the object; (2) Enclose the resulting field in a connection; (3) Com-pute the scattered field of the object illuminated by the antenna field, i.e., the connection. For strong interactions, an iterative procedure might be used. In such a procedure, one will enclose the resulting field of step three in a connection and illuminate the antenna with this field. The resulting field is again enclosed in a connection that now illuminates the object and so on. Obviously, this leads to nested connections.

When a connection contains several expansions for several domains, the connection must be a multi-domain expansion. The proper definition of multi-domain expansions re-quires the information on the boundaries of the corresponding domains. The connection feature of the 3D MMP implementation [9] does not contain this information, whereas the MaX-1 implementation [10] allows the user to attach the boundary information to connec-tions.

Connections are very helpful for solving complex problems. Moreover, connections al-low one to approximate expansions such as line multipoles, ring multipoles, surface mul-tipole distributions, and other expansions that may be approximated by a discrete set of MMP expansions.


3.3. Matrix methods

Mathematicians prefer orthogonal bases. First of all, the term "orthogonal" requires an appropriate definition of a scalar product. The expansion Eq. (3.1) guarantees that Maxwell equations are satisfied within each domain because all basis functions satisfy Maxwell equations. Therefore, the unknown parameters must be computed in such a way that the boundary conditions derived from the Maxwell equations on all boundaries of all domains hold as accurately as possible. This means that a scalar product that is useful for the CHA must be defined on the boundaries, i.e., the CHA is a boundary method. Using such a scalar product, the CHA basis is orthogonal only for special geometries like circular ones. When the basis is non-orthogonal, one can orthogonalize it, but the corresTponding procedure is time-consuming. Therefore one usually derives a linear system of equations instead of applying an orthogonalization procedure, i.e., one uses a matrix method like in the Method of Moments (MoM) [13].

There are three interesting ways to obtain matrix equations [14,15], (1) the error min-imization technique that is based on the norm, (2) the Projection Technique (PT) that is based on the scalar product, and the (3) the Point Matching (PM) technique or collocation that is based on the sampling.

Since one can obtain a definition of the square norm from the definition of the scalar product, there is some relation between the first two techniques. For reasons of simplicity, we consider a simple Dirichlet problem for a single domain D with a closed boundary dD. This means, that the Laplace equation A / = 0 holds in the domain D and the Dirichlet boundary condition f = g holds on the boundary dD. Note that the formalism for scatter-ing and other dynamic problems is more complicated but similar. Therefore, the extension of the following for electrodynamics is straightforward. Now, let {a, b) denote the scalar product of the fields a and b on dD. Assume that the solution / is approximated by the series expansion

K

/ = / ^ + error =Y^akfk-\- error. (3.3) k=\

The projection technique projects (3.3) on a set of testing fields ti,i = 1,2,..., I, which leads to the matrix equation

MA = G + E, (3.4)

where M is a rectangular matrix with the elements mtk = ifk^U)- The parameter vector A contains the linear parameters ak. The vector G is given and contains the elements 8i = (g, U), i.e., the projections of the boundary values g on the testing fields. Finally, the error vector E shall be minimized. When M is a square matrix, i.e., when I = K, one can easily solve Eq. (3.4) in such a way that E = 0 holds. Note that this does not mean that there is no error in the result, i.e., that error = 0. In fact, the size of the error field depends very much on the testing functions. When we minimize the square norm of the error function, i.e., (error, error), we also obtain a matrix equation of the form (3.4), but now, we have mtk = (fk, ft) and gi = (g, ft). This means, that the set of testing functions is equal to the set of basis functions, which is called Galerkin's choice of testing functions. This choice is optimal in the sense that the least square norm of the error is minimized along the boundary. Note that this does not mean that the square norm of the error inside the domain D is minimized.

26 Ch. Hafner

3.3.1. Analysis of the PM technique

The main drawback of Galerkin's method is that the evaluation of the matrix elements Mik and gi is often time-consuming, because a numerical integration is required. Therefore, one often uses much simpler testing functions that allow one to evaluate the scalar products analytically. The simplest choice is ti (r) = <5(r — ?/), i.e., Dirac testing functions. For these testing functions, one simply has

(fk,ti)= [ f{r')8{r'-n)ds' = fk{ri). (3.5) JdD

The resulting matrix equation is the same as the matrix equation obtained with the PM technique. Note that PM can be applied directly without any definition of a scalar product: one simply writes down the boundary condition in a finite set of matching points on the boundaries. The PM has several disadvantages. First of all, the accuracy of the results de-pends very much on the location of the matching points and useful results are obtained only when the matching points are appropriately set. In the CHA codes published in the 1960s and 1970s, PM was used throughout. The matching points were set in such a way that the angles between neighbour points seen from the origin of the polar coordinate system were equal. This is possible, when only one polar coordinate system is required for all domains, i.e., in relatively simple cases. When the boundary is not almost a circle, the distances be-tween neighbour matching points can differ considerably, which leads to unbalanced error distributions along the boundary.

Since the appropriate distribution of the matching points depends mainly on the basis functions of the expansion and not on the geometry of the problem, one has to replace the CHA or the PM for obtaining a more general code. MMP contains both, a generalization of the CHA basis and of the PM technique.

3.3.2. Generalization of the PM (GPM)

So far, three methods for deriving matrix equations of the form Eq. (3.4) have been mentioned. It is important to note that PM and the PT are usually described in such a way that a square matrix is obtained and that all error vectors in the formalism are erased or neglected. This is possible because a square matrix equation can theoretically be solved exactly, i.e., with E = 0. Although ignoring the error terms simplifies the formalism, this procedure has an important drawback: The information on the errors is lost. When one is interested in accurate and reliable results, this loss of information should be avoided.

The matrix equations obtained by the error minimization technique are typically square and symmetric. All matrix elements are integrals that have to be evaluated numerically -at least in a general code. For boundary methods, the integrals extend over the boundaries. When these integrals are approximated numerically, one can write, for example:

tik = {fjk) = j fi{r)f^{r'-ri)ds'

dD

J

X I • ' (^J) -fk (O* - ^i) ^ in ) + Errorintegration. (3.6)


A similar approximation can be used for the elements gt. When one and the same sum-mation is used for the approximation of all mtk and for all gt, one can replace the symmetric square matrix M by a product of a rectangular matrix R and its adjoint R*:

M = R*R. (3.7)

The rectangular matrix R can be directly obtained by a Generalized PM technique (GPM). In this technique, one simply writes down the boundary conditions in more match-ing points than necessary for obtaining as many equations as unknowns, i.e., one sets I > K, where / is the number of equations and K the number of unknowns. This leads di-rectly to an overdetermined system of equations characterized by a rectangular matrix R^ R is identical with the matrix R obtained from the numerical approximation of the in-tegrals in the error minimization technique, when (1) the matching points coincide with the numeric integration points in Eq. (3.6) and (2) the equations of the GPM are properly weighted.

In the MMP code, GPM plays an even more important role than the MMP expansion itself, because the code would allow to solve much less complicated problems when the GPM would be replaced by an error minimization technique or by a projection technique. This surprising fact can be explained as follows. First of all, GPM leads to the matrix equation

R^A' = G' + E^ (3.8)

From this equation, Eq. (3.4) is obtained when Eq. (3.8) is multiplied by R * and when the equations are weighted in such a way that R = R. In complicated situations, R is that much ill-conditioned that the numerical solution of Eq. (3.1) leads to a total loss of accuracy even when double precision arithmetic is used. Nonetheless - with appropriate algorithms - Eq. (3.8) can be solved directly in such a way that (1) the square norm of E is minimized and that (2) A is computed with a sufficient accuracy.

3.4. Special MMP features

3.4.1. Weighting

The standard weighting factors of the GPM are defined in such a way that the square norm of the error function along all boundaries is minimized. Sometimes, there are some critical sections of the boundary, where a higher local accuracy is required. This may be achieved by the introduction of user-defined weights.

Once an MMP computation has been finished, the user may analyse the error distribution (mismatching) along the boundary. When the error is too high in some area, one may refine the multipole expansions and increase the weighting factors in this area. This usually leads to a locally reduced error in a second run.

Another weighting problem occurs when several boundary conditions with different or-ders of magnitude are imposed. For example, in electrodynamics, the size of the E'-field and //-field boundary conditions is considerably different when the SI system of units is applied. MMP therefore automatically adds some reasonable weighting factors.

28 Ch. Hafner

3.4.2. Fictitious boundaries

In general, weighting affects the error distribution along the boundaries and allows one to obtain reasonably balanced error distribution along the boundaries. Typically, the errors of the field are highest near the boundaries. Sometimes, the errors inside the domains, far away from the boundaries are much smaller than on the boundaries, i.e., the error distri-bution in the domains is not balanced at all. To obtain a more balanced error distribution inside the domains, one can spHt the domains by fictitious boundaries. This allows the MMP code to increase the errors along the fictitious boundaries, i.e., inside the original domains and to reduce the error along the original boundaries.

Obviously, fictitious boundaries lead to bigger MMP matrices. At the same time, these matrices become more sparse and their condition number may be reduced. Thus, fictitious boundaries may have interesting numerical benefits. Obviously, the introduction of such boundaries is a step of MMP toward finite elements.

Fictitious boundaries are frequently used to split a domain with a complicated geometry into several domains with a simple geometry, which simplifies the modeUing of the field in the resulting domains.

When multipole expansions are used to expand the field in a given domain, splitting the domain into several subdomains has another advantage. It increases the freedom of modelling. Note that all multipoles must be placed outside the corresponding domain, i.e., in the complement of the domain. Since the complements of the subdomains are bigger than the complement of the original domain, one has more space for placing the multipoles for the subdomains. This is of special importance when the complement of a domain is very small. A typical example is the scattering at a thin PEC plate. Since the field inside the plate is zero, one has only one domain to be modeled, i.e., the space outside the plate. The plate itself is the complement of this domain, i.e., the multipoles that model the scattered field must be placed inside the plate. When the plate is thin, one has almost no space where the multipoles can be set. Now, one can split the domain outside the plate into some subdomains in such a way that the problem of the multipole setting is avoided. Figure 3.1

X X X X X X X X X X

O X

Fig. 3.1. Left hand side: x indicates multipoles for modeling the scattered field at a thin plate without fictitious boundaries. Center: domain 1 is split into three parts by fictitious boundaries. Right hand side: multipoles placed along the boundary of domain 2. The multipoles along the circular part of the boundary are replaced by a Bessel

expansion in the center of the circle indicated by a square.


illustrates this. The domain 1 outside the plate is first modeled by a set of multipoles outside the domain, i.e., inside the plate. To obtain good results, the distance between neighbour multipoles should be similar to the distance of a multipole from the boundary. This distance is zero for an infinitesimally thin plate. Therefore, the number of multipoles is infinite for a plate of zero thickness. When one splits the domain 1 in three subdomains as in Fig. 3.1, one can model the field in the exterior domain 1 by a single multipole and the fields in the two inner domains by a finite number of multipoles. Since these domains have a circular boundary section, some of the multipoles may be replaced by Bessel expansions. Note that the total number of MMP expansions in Fig. 3.1 is increased when the fictitious boundaries are introduced because the plate is not very thin for graphical reasons. For sufficiently thin plates, the number of MMP expansions may always be reduced. Note that most of the multipoles in the model with fictitious boundaries are placed outside the plate.

3.4.3. Periodic problems

Gratings and antenna arrays are typical periodic problems that may be solved by differ-ent means. Often, special expansions are derived from Floquet theory. These expansions have a field with the desired periodicity. For example, one can introduce arrays of multi-poles with the desired periodicity. MMP uses a more powerful approach that is essentially based on fictitious boundaries [16]. First, the periodic structure is separated from the sur-rounding space by fictitious boundaries. The field in the surrounding space is modeled by Rayleigh expansions that have the desired periodicity. Then, an additional set of fictitious boundaries is introduced to isolate a single cell of the periodic structure. On these ficti-tious boundaries, special (periodic) boundary conditions are imposed that guarantee that the field inside the structure has also the desired periodicity. The field inside a single cell is modeled explicitly. This is done exactly as for ordinary domains.

3.4.4. Eigenvalue computation

Eigenvalue problems, for example, guided waves and resonators, have no explicit excita-tion. Therefore, such problems lead to homogeneous matrix equations of the form Eq. (3.4) or Eq. (3.8) with G = 0 or G = 0. Usually, the matrix elements of eigenvalue problems are functions of the eigenvalue e. When the matrix M in (4) is a square matrix, a non-trivial solution of Eq. (3.4) obtained from

det(M(^))=:0. (3.9)

To obtain the eigenvalues, a numerical search for the zeros of (3.9) is required. When one wants to work with Eq. (3.8) with a rectangular matrix R and G' = 0, one

encounters the problem that the determinant of a rectangular matrix R is not defined. Since the use of Eq. (3.4) should be avoided, a new algorithm for solving eigenvalue problems must be found. First, one can consider the relation between Eq. (3.4) and Eq. (3.8) that was already considered for inhomogeneous matrix equations. Since (3.8) is solved in such a way that the square norm of the error vector E' is minimized, ||E(^)|p is a function of the eigenvalue e when G = 0. This function has no zeros in general, but it may have some minims. When one numerically locates the minima and computes the field of the

30 Ch. Hafner

solutions with "eigenvalues" that correspond to these minims, one sometimes finds good approximations of the desired eigenvalues, but sometimes one detects solutions with almost zero fields, i.e., approximations of trivial solutions. To avoid trivial solutions, one has to define the amplitude of an eigenvalue field and to keep it constant during the numerical minimum search. This leads to an algorithm that is more complicated than the search of the zeros of det(M). At the same time, this algorithm has interesting advantages. In the vicinity of the zeros, the bad condition of M causes an almost statistical behaviour of det(M) that considerably disturbs the zero search algorithm. Since the accuracy of the eigenvalue is not known, one has no reasonable stopping criterion. In the MMP code one has a minimum residual search with a well defined amplitude of the mode to be searched. This algorithm does not only provide information on the accuracy of the computed EM field, but also on the accuracy of the computed eigenvalue. Therefore, the stopping criterion is obvious. Since the algorithm is based on Eq. (3.8) with G = 0, rather than on (3.4) with G = 0, one has much less problems with the ill-conditioned matrices. Therefore, the numeric search is not disturbed near the desired solution. For more information on the MMP eigenvalue solver, see [15,17].

3.4.5. Ill-conditioned matrix methods

Considering the different MMP tricks (fictitious boundaries, introduction of connec-tions, special MMP expansions, etc.) required for handling ill-conditioned matrices, one might obtain the impression, that it would be simpler and more reasonable to invent tech-niques that reduce the condition number. It is well known in the MoM that replacing entire-domain basis functions by subdomain basis functions, i.e., functions that are non-zero only in a subdomain of the given domain, allows one to drastically reduce the condition num-ber. Subdomain basis functions cannot fulfil the Maxwell equations in the entire domain exactly. Therefore, it seems that one cannot use subdomain basis functions in the MMP code. In fact, one can easily subdivide a domain into subdomains with fictitious bound-aries between the subdomains and handle the subdomains exactly in the same way as usual domains. This useful technique is quite often applied to simplify geometrically compli-cated domains of MMP models, but its main purpose is not a reduction of the condition number and it is also not an exact translation of the MoM subdomain basis concept.

Beside the difficulties in the realization of techniques that reduce the condition number, there is a much deeper, strange and surprising argument against these techniques. This argument is outlined in the following. It fully supports the MMP approach of implementing algorithms for handling relatively ill-conditioned matrices.

If one considers the consequences of replacing entire-domain basis functions by sub-domain basis functions in the MoM, one should recognize that this does not only reduce the condition number of the matrix obtained, but also the accuracy of the results. In order to achieve the same accuracy for both types of basis functions, the number of unknowns must be increased considerably when subdomain basis functions are introduced. Similar observations can be made probably for all techniques that reduce the condition number -provided that the original method is carefully designed. For example, higher order FE and FD schemes typically lead to less dense matrices with higher condition numbers, but more accurate results.


This effect has been studied extensively for the general series approximation of a given function / that is formally identical to the field approximation (3.1):

f{x) = / ^ x ) +Error(x) = f ] A^/^(x) +Error(x), (3.10) k=\

where the basis functions fk are arbitrary functions defined in the same interval as / . The most important result of this study is the following observation:

The probability of finding an accurate solution with a fixed number of unknowns is in-creased when the admitted condition number of the corresponding matrix is increased. The probability of finding accurate results with low condition numbers is extremely small.

Consequently, it is most important to provide techniques that allow working with ill-conditioned matrices if one is interested in accurate results. For this purpose, the GPM is the best method that has been found so far.

Of course, one can also increase the number of unknowns in order to obtain more ac-curate results. Whether one obtains accurate results more efficiently with lower condition numbers and more unknowns depends on many things, first of all on the convergence. This will be discussed below. Another aspect of the problem is that matrix solvers that can handle relatively ill-conditioned matrices are more time-consuming. In FE and FD codes, higher order schemes cause not only higher condition numbers, but also less sparse ma-trices, which means that both the computation time and the memory requirement become considerably higher. Therefore, high order schemes are not necessarily efficient.

The MMP code works with relatively small and dense matrices. Several techniques for reducing the condition number have been studied. All of them make the code more com-plicated. Most of them reduce the accuracy of the results so drastically that they also re-duce the efficiency when a fixed accuracy is aimed. A typical example is the multipole-beam method [18]. A simple technique that may be helpful in some cases is to omit some of the orders of some of the multipoles. For example, instead of using all integer orders 0,1, 2 , . . . , M, one can use the even orders 0, 2 ,4 , . . . , 2M. This allows one to easily re-duce the condition number without a drastic reduction of the resulting accuracy.

3.4.6. Convergence

The definition of the convergence of a method based on series approximations is quite obvious when the basis is a well ordered set of functions like in the Fourier series, where the frequency is increasing with the order of the basis function. In this case, the norm of the error term in function of the number of unknowns can be easily analysed for obtaining useful information on the convergence.

The general formulation Eq. (3.10) of a series expansion with an arbitrary basis and the MMP expansion Eq. (3.1) are not ordered at all. For a given number of unknowns K, there are infinitely many sets of basis functions. Many of them lead to very inaccurate results, others to acceptable or even excellent results. Finding the optimal basis for a given K is extremely difficult and time-consuming. In the MMP code, one has several heuristic rules for finding appropriate expansions that hopefully lead to relatively accurate results. Based on these rules, one can automatically generate useful but sub-optimal MMP expansions. This allows one to analyse the behaviour of the error as a function of K. The automatic

32 Ch. Hafrier

procedures contain several parameters that have great influence on the expansion and on the resulting condition number and accuracy.

Another problem arises from the fact that the error of a field computation is also a field and not a simple number. The exact solution is unknown in most cases. Thus, one can only estimate the error field. Moreover, the error field carries far too much information and one usually wants to compress this information in a single, real number, which requires again an appropriate definition. It is important to recognize that neither the accuracy nor the convergence of the field are uniform. Since MMP is a boundary method, it has relatively high errors near the boundaries. Moreover, the convergence of the field on the boundary considerably depends on the geometric shape of the boundary, especially on the number of continuous derivatives of the boundary. Close to an edge, where already the first derivative is discontinuous and where the field can have a singularity, one obtains both high errors and a poor convergence. Therefore, it is often preferable to smoothen the edges, for example, with small arcs or with spline approximations of the boundary.

MMP allows not only to compute an overall error number but also the error distribu-tion along all boundaries, which carries much more information that is important for the validation of the results.

However, the MMP convergence mainly depends on the geometric shape of the bound-aries and on the type of basis functions in the MMP expansion. Although the local con-vergence near some points can be slow, the overall convergence is typically fast. In simple cases, exponential convergence is obtained. Incidentally, a fast convergence is good for obtaining very accurate results (which was the original goal of MMP), but methods with a slow convergence can be much more efficient for relatively rough estimates (which are often good enough in engineering).

3.4.7. Recycling of information

MMP is a boundary method working in the frequency domain with quite dense, rela-tively small (<2500 columns), and often ill-conditioned matrices. The computation time for the direct solution of such matrices is almost proportional to the cube of the number of unknowns. Since the computation of the matrix elements is very fast, the setup of the matrix has almost no influence on the computation time, except for very small problems. Therefore, the matrix solver is the main source of computation time.

Early attempts of speeding up MMP by iterative and block-iterative matrix solvers failed - except for relatively simple cases. Since the iterative procedures are responsible for the effectiveness of time-domain codes, a deeper study of iterative techniques was per-formed [19].

In time-domain method, the knowledge of the field in the previous time point is an ex-cellent guess for the field in the actual time point, provided that the time step is sufficiently small. Therefore, an iterative method can work with one iteration per time step. The ef-ficiency of this method is based on the fact that one has a quite good knowledge of the solution from previous computations (of previous time points). Because of storage prob-lems, one usually keeps the information of one time point only, which allows a simple zero order extrapolation only. One can expect that higher order extrapolations might allow an even more accurate extrapolation, i.e., guess of the field in the actual time point. This would allow to obtain more accurate results or to increase the time steps.

In frequency-domain methods, it seems that there is no information that would allow an accurate guess of the solution. Obviously, this information is present when one is comput-


ing not only a single frequency, but also an entire frequency range - except for the first frequency. But even if one is interested in one frequency only, one often has information from previous computations that allows one to obtain a quite good guess of the unknown parameters to be computed. This is especially true when one is solving inverse scattering problems and optimization problems. Here, one has to compute many, sHghtly different models that provide excellent information that should be recycled rather than wasted. The Parameter Estimation Technique (PET) [15,19] is an interesting MMP feature for effi-ciently speeding up the MMP performance. This technique is also very promising for other matrix methods.

The PET may also be applied to eigenvalue problems. Here, the PET can considerably speed up the eigenvalue search by providing a good initial guess of the location of the minims that describe the eigenvalues.

3.5. Example: Scattering at a particle on a planar structure

For reasons of simplicity, we consider a 2D model of the EM scattering of a plane wave that is incident on a circular particle on a planar structure. Similar 3D models were studied by Novotny with the 3D MMP code [20]. Assume that the planar structure consists of a coated PEC plate with a loss-free dielectric film, whereas the particle consists of a dielectric with a permittivity that may differ from the permittivity of the film. Figure 3.2 shows the time-average of the Poynting vector field for such a structure for an incident plane wave. The angle of incidence is 45 degrees and the polarization of the electric field is perpendicular to the plane of incidence which is shown in the figure. The free-space wavelength of this computation is twice the diameter of the circular object. The width of the film is equal to the diameter of the circular object. Finally, the relative permittivity of the film is equal to 4.

Although this example looks very simple, its modeling and computation is quite tricky and requires some physical knowledge. First of all, one has open domains with infinite

\ I \ %

Fig. 3.2. Time-average of the Poynting vector field for a circular dielectric (relative permittivity 2) on a planar structure, illuminated by a plane E wave. Angle of incidence 45 degrees.

34 Ch. Hafner

boundaries. Infinite boundaries cause similar problems for MMP and other boundary meth-ods as open domains for domain methods. The discretization of such boundaries must be truncated, which may cause numerical errors. In [20], this problem was solved by the intro-duction of a new set of MMP expansions that automatically fulfil the boundary conditions on the planar structure. Such expansions correspond to non-free-space Green's functions that are sometimes used in MoM codes. The implementation of appropriate expansions leads to tricky integrals that are numerically very demanding. Moreover, the use of such ex-pansions is not very user-friendly. Therefore, we now try to obtain solutions with standard MMP expansions (multipole expansions, Bessel expansions, and harmonic expansions -the latter contain plane waves and evanescent waves as special cases) that are available in theMaX-lcode[10].

Let us start with an even simpler model that is obtained when the circular object is removed. In this case, one has a simple planar structure and one easily can obtain the resulting field analytically. The scattered field can easily be described by a reflected plane wave in free space and by a superposition of two plane waves or a harmonic expansion in the film. Since these expansions are standard expansions of the MaX-1 code, one can easily describe the analytic solution. Moreover, one can embed it in a single connection, together with the incident plane wave. This means that one can construct a connection that describes the analytic solution of the planar structure.

We can consider the circular object as a disturbance of the planar structure. What ef-fects are caused by this disturbance? Obviously, the circular object will scatter the incident field and this will cause some radiation. Now, it is important to note that the planar struc-ture can guide some waves. In the example considered here, there are two guided waves and an infinite number of evanescent waves on the undisturbed planar structure. These waves may be excited when the circular object or any other disturbance is present. It is easy to obtain analytic solutions that describe the guided and evanescent waves on the undisturbed planar structure, but it is even easier to compute these waves with the eigen-value solver of MaX-1. These solutions can then be embedded in several connections. Note that the field of high order evanescent waves decays very rapidly. Therefore, it is sufficient to compute only the lowest order evanescent waves, i.e., one can work with a finite set of connections that describe the guided waves and the lowest order evanescent waves.

Unfortunately, the energy of the guided waves propagates from minus infinity to plus infinity in the entire planar structure and not from some location near the circular object to infinity, i.e., these waves propagate through the object rather than away from the object. Similar statements hold for the evanescent waves. How can we model the waves that are "caused" by the object and propagate away from it? There are several answers to this ques-tion. The simplest method is to introduce fictitious boundaries that separate the object and the planar structure near the object from two planar sections that extend to minus infinity and plus infinity, respectively. These fictitious boundaries are indicated by white lines in Fig. 3.2. Note that they extend also to infinity and must be truncated. However, one now has a model with three sections. The field in each of these sections is modeled differently. In the first (leftmost) section, one has a first connection that describes the scattering of the incident plane wave at the undisturbed planar structure and a set of connections that de-scribe the plane waves that travel to the left as well as the corresponding evanescent waves. Note that the disturbance will also cause some radiation field in this section. The radiation field may be modeled by one or a few multipoles located somewhere near the disturbance, which is obviously outside the first section. A similar modeling may be used for the third


Q-

Fig. 3.3. MMP expansions and distribution of the relative error along the boundaries of the planar structure with the circular object. The errors are indicated by lines perpendicular to the boundaries. The maximum relative error is 1.14%, the average relative error is ten times smaller. Note that the fictitious boundaries are invisible (white lines), but the errors along them are visible. The expansions are indicated by cycles with a short radial line. Most, but not all of them are multipoles. Note that the errors along the truncated fictitious boundaries are quite small.

The truncation seems not to cause severe problems.

(rightmost) section, which is essentially symmetric to the first section. No connections that describe guided or evanescent waves may be introduced in the second (centre) section. Here, the first connection that describes the undisturbed scattering problem is combined with a sufficiently big set of multipole expansions that describe the EM field in all domains of the second section, namely, (1) free space, (2) dielectric film of the planar structure, (3) circular object. Since the object is circular, one can easily model the field inside the object by a single Bessel expansion.

To obtain the desired MMP solution, one first computes the different connections, which includes the solution of an eigenvalue problem. As soon as the connections are known, one can compute the entire model like a standard scattering problem. MaX-1 provides a script language that allows one to specify the different steps to be done in advance. As soon as the corresponding directives are properly defined, it may automatically compute the resulting field, the error distribution along the boundaries, etc. MaX-1 could also automat-ically compute the resulting field for different permittivities, different diameters or differ-ent locations of the object. Since such computations are time-consuming, one best com-putes a single model first and validates the results before one starts a more sophisticated analysis.

The visualization of the error distribution is most useful for the validation. MaX-1 can either visualize absolute or relative errors along all boundaries (see Fig. 3.3). As one can see, the errors are already sufficiently small, but not very well balanced.

To obtain a more balanced error distribution, one can refine the modeling near those parts of the boundary, where the error is relatively high. Since we already have a high accuracy, we can also use a more rough modeling near those areas where the error is small. This reduced the computation time. Figure 3.4 shows the error distribution obtained when a more rough modeling with less multipoles and lower multipole orders is used in the area to the left of the circular object. Note that the errors along the fictitious boundary

36 Ch. Hafner

Fig. 3.4. The same as in Fig. 3.3 with a more rough MMP model.

T~T 1 1 / / / / / f f f / / / / f / f / / / /

'''• J^ ^ ^

epiiloii ' 1

!»».miiii'»»-»iiinn jani»--awiMpnnj(piiip,.

Fig. 3.5. Time-average of the Poynting vector field that is obtained when the connection is omitted. The same MMP computation as in Fig. 3.2.

to the left are increased, while the errors on the fictitious boundary to the right remain approximately the same. However, one should not waste to much time for the tuning of the error distribution as soon as sufficiently good results have been obtained. One must keep in mind that a different error distribution may be obtained when one of the modeling parameters (frequency, material properties, geometry, incident wave) is modified. When we want to study the dependence of the field on some modeling parameter, we need a robust model rather than one that is tuned for a specific set of modeling parameters. Therefore, overdiscretization as in Fig. 3.3 is often reasonable.

In Fig. 3.2, the field of the connection that describes the undisturbed problem is domi-nant. Therefore, it is hard to see the effects caused by the circular dielectric. To make these effects more obvious. Fig. 3.5 shows the time-average of the Poynting vector field that is


^ ^ \ \ \ \ \ \ \ \ n / f f / / r

r \ ^ \ \ \ \ \ \ \ \ f ) / / / /

. . ^ -u, -^ " N. *S. 'N,';

Fig. 3.6. The same as in Fig. 3.5, for a relative permittivity of the circular dielectric equal to 4.

i V jX . ^ \ \ ^ x \ s T" ;

! W N X \ \ \ \ \ t . * v X , X X . W \ \ N \ ,

hvvwxx \ \ \ \ 1 1 -^'V^x.'K W X \ \ \ \ i » ^ » ^ • •v , "»v^ " N , ,

-V, *^ *>S -WV. " ^ ^ '^•Xv "^X, " ^ S . ^ ^ ' ^ ^ ^ ^ ^ ^ f c

^ ^ . -, ^ * " "" ' ' ^ H ^ • , - * . % •H, \ % ^ ^ _ ^ , , ^ . , | , „ r ^ '/

i

( $ r

f

/

J

. . . / . / / / J" / / ^

t I f f / ^ / f / / ^ /*

i t / / Z / / / * / " ^ ^ ^

. / / / / / / / ^ / ^ y ^ ^ y '

f / / / / / / ^ / ^ y ^ y ^ ^ ^

/ / / / / ^ / ^ ^ y ^ y ^ ^ ^ ^

f / / ^ ^ y ^ ^ y ^ ^ ^ ^ ^ ^

j^J^y^y^y^y^^^^ ^ ^ ^

^ ^ ^ ^ » « ^ * ( . . ^ « ^ **«*** ' ,J««^ v-*' ^ ^ ^

^ \ —- ' t \ \ • ^ ^ ^ ^ M ^ J M ' ^ ^ ^ ,

Fig. 3.7. The same as in Fig. 3.5, for a relative permittivity of the circular dielectric equal to 8.

obtained when the connection is omitted. As one can see, the object causes some radiation mainly into the left half space, and it also excites guided waves travelling to the right. The guided waves travelling to the left are much weaker.

Without any modification of the MMP expansions and of the boundaries, we now can compute the field for varying permittivity of the circular object. Note that MaX-1 can perform such an analysis automatically when appropriate directives are defined in the MaX script language. The results may be viewed as animations that give a quick overview. When a modeling parameter, for example, the permittivity, is continuously modified in many subsequent computations, one may apply the PET feature to speed up the evaluation of the MMP matrix. However, the Figs. 3.6 and 3.7 show that both the radiation pattern and the guided waves travelling to the left and right may change drastically when the permittivity of the object is modified.

38 Ch. Hafner

3.5.1. Outlook

MaX-1 contains the latest implementation of the MMP code, which is more user-friendly and more robust than previous implementations. For tricky problems, a good knowledge of the physics behind the results is still very important - not only for the analysis of the results, but also for the modeling.

The growing power of computers allows one to simulate more and more complicated structures. The most interesting aspect of such simulations is the optimal design of new devices. Optimization procedures may be linked with simulation codes such as MaX-1. Since the optimization procedure will often call the simulation code thousands of times, such a code must be highly robust, efficient, and automated. In this paper, some useful techniques that have been implemented in MaX-1 were outlined. It is hoped that these techniques give some ideas for the development of future codes.

References

[1] C.R. MuUin, R. Sandburg, and CO. Velline, A numerical technique for the determination of scattering cross sections of infinite cylinders of arbitrary cross sections, IEEE Trans. Antennas PropagaL AP-12(1), 141-149 (1965).

[2] H.Y. Yee and N.F. Audeh, Uniform waveguides with arbitrary cross-section considered by the point-matching method, IEEE Trans. Microwave Theory Techn. MTT-13(11), 847-851 (1965).

[3] J.E. Goell, A circular-harmonic computer analysis of rectangular dielectric waveguides. Bell Syst. Tech. J. 9, 2133-2160 (1969).

[4] Ch. Hafner, Beitrdge zur Berechnung der Ausbreitung elektromagnetischer Wellen in zylindrischen Struk-turen mitHilfe des Point-Matching Verfahrens, Dissertation ETH Nr. 6683 (Zurich, 1980).

[5] I.N. Vekua, New Methods for Solving Elliptic Equations (North-Holland, Amsterdam, 1967) (English trans-lation from original in Russian, 1948).

[6] Ch. Hafner and R. Ballisti, The multiple multipole method (MMP), COMPEL - Int. J. Comput. Electr Electron. Eng. 2(1), 1-7 (1983).

[7] A. Ludwig, A new technique for numerical electromagnetics, IEEE APS Magazine 31(2), 40-41 (1989). [8] Ch. Hafner, 2D MMP: Two-Dimensional Multiple Multipole Software and User's Manual (Artech House,

Boston, 1990). [9] Ch. Hafner and L. Bomholt, The 3D Electrodynamic Wave Simulator (John Wiley, Chichester, 1993).

[10] Ch. Hafner, MaX-I: A Visual Electromagnetics Platform for PCs (John Wiley, Chichester, 1998). [11] G. Mie, Elektrische Wellen an zwei parallelen Drahten, Ann. Phys. 2, 201-249 (1900). [12] H. Singer, H. Steinbigler, and P. Weiss, A charge simulation method for the calculation of high-voltage

fields, IEEE Trans. Power Appar. Syst. 93, 1660-1668 (1974). [13] R.F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968). [14] Ch. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House,

Boston, 1990). [15] Ch. Hafner, Post-Modem Electromagnetics: Using Intelligent Maxwell Solvers (John Wiley, Chichester,

1999). [16] Ch. Hafner, MMP computation of periodic structures, /. Opt. Soc. Am. 125, 1057-1067 (1995). [17] Ch. Hafner, Multiple multipole (MMP) computations of guided waves and waveguide discontinuities. Int.

J. Num. Model. 3, 247-257 (1990). [18] A. Boag and R. Mittra, Complex multipole beam approach to 3D electromagnetic scattering problems,

/ Opt Soc. Am. 11(4), 1505-1512 (1994). [19] Ch. Hafner, MMP-CG-PET: The parameter estimation technique applied to the MMP code with the method

of conjugate gradients, Appl. Comput. Electromagn. (ACES) J. 9(3), 176-187 (1994). [20] L. Novotny, Light Propagation and Light Confinement in Near-Field Optics, Dissertation ETH No. 11420

(Zurich, 1996).

CHAPTER 4

Models of Electromagnetic Scattering Problems Based on Discrete Sources Method

Yu.A. Eremin, N.V. Orlov, and A.G. Sveshnikov

Laboratory of Computational Electromagnetics Applied Mathematics and Computer Science Faculty Moscow State University Vorobyev' Hills Moscow 119899, Russia e-mail: [email protected]


40 Yu.A. Eremin, N.V Orlov, andA.G. Sveshnikov

4.1. Introduction

The Discrete Sources Method as a basis for constructing mathematical models of elec-tromagnetic wave scattering problems will be reviewed. Our main goal is to represent in detail the Discrete Sources Method's theoretical background. We will concentrate on nu-merical schemes for investigating polarized scattering by a penetrable obstacle. Computer simulation of results associated with discrimination of smooth substrate defects will be discussed.

The analysis of scattering of electromagnetic waves by local obstacles and structures has a wide variety of applications in electromagnetics, optics, computerized tomography, metrology and many other domains. This is mostly due to the development and innovation of advanced technologies, as well as the elaboration of new approaches to the interpretation of experimental data. Mathematical modelling, operating with Boundary Value Scattering Problem (BVSPs), is a common tool for such an advanced analysis. From mathematical viewpoint BVSPs are classical problems of mathematical physics. They involve the differ-ential equations (Maxwell or Helmholtz)

AU) = 0 in De^R^XWi, (4.1)

boundary condition

QU = -QU^ at a A , (4.2)

and

radiation condition at infinity. (4.3)

Here Q is a boundary operator, 3D/ is a smooth closed surface, confining the bounded simply connected region Dt, and U^ is an exciting field.

The essential feature of BVSP under consideration is that the obstacle is far away from both the primary field sources and the region of the scattered field measurement. This al-lows us to employ the Quasi-Solution (QS) concept, enabling us to avoid methods requir-ing the boundary conditions at the obstacle surface to be satisfied exactly, which obviously increases computational costs. Quasi-Solution concept means, that we represent an approx-imate solution of BVSP (Eqs. (4.1)-(4.3)) in the form of a semi-analytic construction U^ that satisfies the equation (4.1) in an explicit analytic form

/:(U^) = 0, MeDe, (4.4)

and

radiation condition. (4.5)

The boundary condition is to be satisfied approximately in the following form:

||QU^ + QU^ | | ^5 . (4.6)

4. Models based on discrete sources method 41

A numerical scheme is needed to determine U^. If BVSP (Eqs. (4.1 )-(4.3)) has an unique solution U for any external excitation, then the following estimation is valid

l|U-U'|lc(.) =o(l|2U^ + Q^X^(,D^)\ (4.7)

where d is any compact in De. The relation (4.7) means the following. In order to provide a convergence of the QS to the exact solution in a continuous metric outside the obstacle, it is sufficient to approximate the boundary condition in L^{dD) norm. Thus, the BVSP (Eqs. (4.1)-(4.3)) is reduced to solving an approximation problem at the surface of a local obstacle Dt.

The Discrete Sources Method (DSM) seems to be one of the most effective and flexible tools for QS construction. In the frame of DSM the approximate solution is constructed as a finite linear combination of the fields of dipoles and multipoles. This representation satisfies Maxwell's equations and the radiation condition at infinity. Unknown amplitudes of Discrete Sources (DS) are to be determined from the boundary condition (4.6). One of the most attractive features of DSM consists in a flexible choice of DS fields that are to be used for approximate solution construction. In the Surface Integral Equation Method a scattered field is represented on the basis of a surface current distributed over the obstacle surface. In the frame of DSM there are no limitations to a choice of DS support, which should provide fulfilling the Maxwell equations, radiation condition and yield a complete system of DS fields at the obstacle surface.

These outlines were once formulated over 30 years ago and published practically simul-taneously by Kupradze in the USSR [1] and Yashuura in Japan [2]. A.C. Ludwig suggested the generic name Generalized Multipole Technique (GMT) in his review [3]. GMT seems to be an appropriate name for most techniques, which several teams are currently working on [4-7].

DSM under examination has some features, which make it different from other tech-niques [7-9]. Those main differences are:

(1) It is applied to axially symmetric structures only; (2) It includes the explicit scheme for constructing complete systems of DS fields; (3) DS are deposited at the axis of symmetry or in an adjoining area of a complex plane; (4) Representation for the approximate solution takes into account an axial symmetry

of a scatterer and polarization of an external excitation; (5) It enables estimation of a posteriori error for the result computed. The first version of DSM was published in 1980 [10]. It allowed treating electromagnetic

scattering by perfect conductors under axial excitation (plane wave propagating along an axis of symmetry) only. Such a simplified kind of excitation allowed the approximate so-lution to be constructed as a combination of electric and magnetic dipoles located on the symmetry axis. In 1982 the method was extended to analysis of a homogeneous penetrable obstacle [11]. Regular functions were employed to represent the field inside a scatterer. In 1983 the DSM theory was generated [12]. The first release of DSM had some Hm-itations associated with the obstacle geometry. In particular, it was not able to analyze an oblate obstacle. This restriction was overcome in 1983 by means of analytic contin-uation for the DS support into a complex plane adjoining the symmetry axis [13]. The deposition of DS at the complex plane eliminated the limitations of the original DSM scheme and enabled treating any obstacle geometry. In 1985 the DSM was generalized to analyze non-axial excitation [14]. Theoretical outlines of DSM along with a generic scheme, allowing generation of complete systems based on DS support geometry came

42 Yu.A. Eremin, N.V. Orlov, andA.G. Sveshnikov

in 1987. Some new complete systems of DS for the BVSP analysis were built [15]. In the same year an algorithm generating complete systems of DS for an obstacle located in a half-space was developed [16]. The next modification, proposed in 1993, allowed taking into consideration not only the axial symmetry of the obstacle, but also the po-larization of the exciting field [17]. Last editions of the DSM had the benefit of expand-ing the technique to a wide variety of applications [18-21]. During the last few years, DSM has been applied mostly to modelling scatterers in the presence of stratified struc-tures [22-25].

4.2. Mathematical models for the Helmholtz equation

In this part of our review we will concentrate on the scattering of time-harmonic waves by local obstacles surrounded by a homogeneous medium, that is with a BVSP for the Helmholtz equation. In obstacle scattering we must distinguish between impenetrable and penetrable objects. The first case leads to Dirichlet or Neumann boundary condi-tions on the obstacle surface 9D/. The scattering by a penetrable obstacle D/ with ho-mogeneous characteristics leads to transmission conditions enforced at an obstacle sur-face [26].

We will begin with the description of a scheme allowing construction of complete sys-tems of radiating solutions for the Helmholtz equation. We will then extend the elaborated scheme to the construction of complete systems for regular solutions of the Helmholtz equation. Some new complete systems will be constructed based on the outlined schemes. Then, we will present a method of analytic continuation of DS fields into a complex plane with respect to DS coordinates. We will briefly analyze the main properties of the field originated by DS, deposited in a complex plane. Finally, we will regard a transmission scattering problem and describe a scheme of approximate solution construction based on the DSM background. We will prove a convergency of the approximate solution to the exact one. In this part we will represent the main outlines of the DSM theory in detail, to avoid repeating similar details of description of electromagnetic wave scatter-ing.

4.2.1. Construction of complete systems for radiating solutions

In this section we will represent a scheme enabling the construction of complete systems of radiating solutions for the Helmholtz equation. We will demonstrate the close relation between DS support geometry and particular kinds of corresponding complete systems. Some new systems will be constructed for application of an elaborated scheme.

Let us consider the BVSP, including the Helmholtz equation

{^^-kl)Us=^ inDe,

radiation condition at infinity, and boundary condition

Q{Us-^U^) at 8 A ,

where D/ is a simply connected bounded region. Here Q is a boundary operator cor-responding to a specific type of the exterior problem: either Dirichlet, or Neumann, or


impedance. Following DSM, we seek an approximate solution to the BVSP as a finite linear combination of elementary sources, like

U^{x) = Y.Pn^n{x)^

where the functions ^fn (x) satisfy the Helmholz equation in De and the radiation condition. If the system {QV^^l^j is complete in L?{dDi), BDi e C^, then due to the correctness relation (4.7)

Us{x)= lim U^(x)

at any compact in De. Now we start with the scheme of constructing a complete system. Let \lf(x,y,ke)hQ3. fundamental solution of the Helmholtz equation

(A + k^,)ilf(x, y, ke) = -5(x, > ), X, y € /?^

where ^{x,y) is the Dirac delta function. yj/{x,y,ke) is symmetrical with respect io x,y, and an analytic function everywhere outside the neighbourhood ofx = y [26]

if(x, y, ke) = , (4.8)

where Rxy = \x — y\. Let us examine the single-layer potential with a density v{y) e L^idDi), dDi e C^

V(x)= j if(x,y,ke)v(y)day, (4.9)

dDi

where V(jc) is known to be an analytic function in Dt.

LEMMA 1. Let y(jc) = 0 in Dt. Then vocO at dDi.

PROOF. Using properties of a single-layer potential of which the density belongs to L^idDi) [26], we have for almost all xedDi

dV, . f d}l/{x,y,ke) 1 lim—{x-hn(x))= / ' / 'v(y)dcyy + -vix) = 0. (4.10) h-^0 dn ^ ^ J on(y) 2

dDi

Here w is a unit outward normal to dDi. As the operator on the left-hand side of (4.10) is an elliptic pseudo-differential operator of the order 0, u a UQ € C(9Dj). Thus, the function

Vo(x)= / \l/ix,y,ke)vo(y)day

dDi

44 YuA. Eremin, N.V. Orlov, andA.G. Sveshnikov

is a continuous function in R^. So, Vo(x) is a solution of an exterior Dirichlet problem of the form

{A-\-kl)Vo = 0 inDe,

Vo(y) = 0, yedDi,

—^ -^jkeVo = o(r- i ) , r ^ oo, (4.11) dr

where r = |x |. Uniqueness of the solution to B VSP (4.11) implies VQ = 0 in D^. Therefore,

^ ( y ) - ^ ( y ) = vo(y) = 0, yedDi. an an

From the last relation we deduce u a UQ = 0. Lemma is proved. D

Let us represent the scheme for construction of a complete system. It consists of several steps:

(1) Choose a DS support S e Dt and a set of DS coordinates {x„}^j e S; (2) Apply to the set {xn]î the following conditions

AV(x)\ = 0 , V ^ G N (4.12) I A — A ^

(with A being a projection operator) which provide that V(x) = 0 in D/, and therefore yield u oc 0.

Let us designate the mentioned conditions as

xlfn(y):=Axlf(x,y,ke)l_^ , (4.13)

then (4.12) can be rewritten in the following form

/ irn(y)v(y)day = (Îfn, i *) 2 = 0 , Vn G N . (4.14)

dDi

The relations (4.14) represent the closeness conditions for the system {V /l j in L^{dDi). Then the following principal result is valid

THEOREM 2. The system of radiating solutions to Helmholtz equation {ifi}î'- (4.13) is complete in L^(dDi).

PROOF. It is sufficient to prove that {iA/}î is closed in L^(dDi). Functions \l/i (x) satisfy the Helmholtz equation in R^\dDi and the radiation condition at infinity. For any v e L^{dDi) the following relations

(V^n,i^*)^2=0, V / G N ^ U O C O

hold (since V/ satisfies (4.14)). Therefore, the system {V /} i is closed in L^(dDi). The theorem is proved. D

Let us demonstrate an application of the scheme discussed.


THEOREM 3. Let S consist of a single point MQ € D/, then the corresponding system

Wl^{x) = hf\ker)Pf^\Qoê)^'^'f, I e {0,N}, m=Û, (4.15)

(2) is complete at any closed smooth surface, confining MQ, where h^ (•) are spherical Han-

kel functions, PJ (-) are associated Legendre functions, and (r, 0, cp) are spherical coor-dinates of the point x with the origin in MQ.

The system {W/^} represents a system of spherical wave functions, of which completeness was estabhshed by I.N. Vekua in 1943 [27].

PROOF. Let us take MQ as an origin of the spherical coordinate system (r,0,(p), and choose a neighbourhood of MQ — do e Dt. Because V (Eq. (4.9)) is a real valued ana-lytic function in do, we can switch to its Fourier harmonic with respect to the azimuth variable (p. The Addition Theorem for the fundamental solution of the Helmholtz equation \l/(x,y,ke) has the following form [26]

oo / Xl,(x,y,ke) = -ikeJ2 J2 Ni,„h\^\ke\x\)jl{ke\y\)Pl'"kcO&e,)PJ'"\cosey)

1=0 m=-l

^ ^m(<p,-<py)^ |X|>|>'|, (4.16)

where

^ /(2/+l)( / - |m|)! ''" y Anil + \m\)\ '

and ji(-) are spherical Bessel wave functions. Substituting (4.16) into (4.9) we obtain

00

V'"(r,e) = J^fifMker)PJ"'\cosey), m€{0,N}, (4.17) l=m

where

Pf^ = af f Wil(x)v(x)da^, otf = const^0. (4.18)

dDi

We applied the necessary and sufficient conditions of vanishing of the function V{x) in do. Similar conditions can be provided by the requirement of vanishing V(x) in Mo along with all its derivatives. Let us proceed to the expansion of the function V into Fourier series with respect to (p mdo. Then the mentioned condition for vanishing of the function V in do takes the following form

Vîr.O) lim = 0 , "in^m, mG{0, N}, andany ^ € [0, TT].

Thus, we need to demonstrate that from the last relations we shall get

^ ^ = 0 , n = m , m + l , . . . (4.19)

46 YuA. Eremin, N.V Orlov, andA.G. Sveshnikov

To accomplish this we start with

j^pm JA^p\^\^ôô) = ()^ \ln^m, andany 6> G [0,7r].

By choosing n=m, letting r -> 0, and using the asymptotic for the Bessel functions in the vicinity of 0,

inM={'^j^[^Ôix')]

we obtain

)0;;^P;'"'(cos^) = o.

Because the last relation should be valid for V^ G [0, TT], )S^ = 0. Assuming n = m-\-\ and repeating the procedure described, we obtain ^^^j = 0. Thus, we have shown, that (4.19) is valid. In turn, Eq. (4.19) implies

f wl^(P)v{P)dap=0, /G{0,N}, meZ, I^\m\. (4.20)

dDi

So, the necessary and sufficient conditions for vanishing of function y(x) in A being ap-plied at point Mo have resulted in relations (4.20). For the system (4.15) conditions (4.14) have a form of closeness conditions. The theorem has been proved. D

DEmNlTlON 4. Let the DS support S represent a segment dz of axis OZ, and the set [Zn}'î ^ dz have at least one limit point zo ^ d^. We will refer to a similar DS support as ^0-

Let us prove some auxiliary statements.

LEMMA 5. Let V(x) be a function satisfying the Helmholtz equation in some area do with a wave number ke, and V"^ be its Fourier coefficient with respect to variable cp. Then in a cylindrical coordinate system (p,(p,z) there exists a limit

l i m — - ^ , VmG {0,N},

which represents an analytic function of the z coordinate. Here ^ = (p,z) is a point located in a half-plane cp = const (hereinafter we will designate such a half-plane cp = const by 4>).

PROOF. Let ô be a sphere completely contained in do. Then inside ô the function V can be represented in the form

Vix) = Yl E <Jniker)P;:'(cose)^'"r n=Om=—n


Its Fourier coefficients with respect to cp are

oo

^""C?)- I I < i " ( ^ ^ ' - ) C ( c o s 0 ) , m e {0,N}.

Let us express associated Legendre functions by a hypergeometric function [28]:

p H ( c o s . ) = 2-NI (" + I-I)' 4 1 " ^ («- |m|)! |m|! rl' l

xF( |m | - n , | m | + w + l, |m| + l , ( l - cos^) /2 ) .

Taking into account that F{\m\ — n,\m\-\-n -\- I, \m\-\-1,0) = I we evaluate the Hmit at p ^- 0 as

u „ . r : i i > = . " . ( , ) = f ^ „ " ^ M ^ , (4.21) ' n=m

where

^" (^z-|m|)!|m|! " '

Due to the asymptotic of spherical Bessel functions when z ^^ 0, we see that v^ (z) is an analytic function of variable z. Lemma is proved. D

LEMMA 6. Let DS support be COQ and v"^ (zn) = 0, Vn G N, then V^ (§) = 0 in do 0 <t>.

PROOF. Due to analyticity of v'^(z): (4.21) from v'^(zn) = 0 , Zn e COQ, it follows that i;^(z)=0,i.e.,

n=m

Using the technique similar to the proof of Theorem 3, we get from the last formula

y6^=0, n = m,m + \,K

which proves lemma. D

THEOREM 7. Let DS support be COQ, then the corresponding complete system at any am-bient closed surface dDi G C^, has the form

>'mn(-^)=/2L^H^e^^zJ^m(c<^s^^J{l,cosm(^,sinm(^}, n,meN, (4.22)

where (p,(p,z) is a cylindrical coordinate system originated in Zn,

< „ = Pl + iz,-zf,

PJ^hcose) = (2|m| - l)!!(sin6»)l'"l sm6»z„ - —'


PROOF. Let zo be a limit point of {zn}^=\, and let us choose a neighbourhood of ô ^ A of zo- Due to the results obtained at the proof of Lemmas 5 and 6, it is sufficient to find a particular kind of a functional system, having closeness conditions like

v'^{zn) = ^, H G N , m € N . (4.23)

Here functions v^ (z) are defined by (4.21). Let us examine the ring-type currents Sm:

In

In Sm(^, ^j) = ^ / V - ' y^ ^^) Qxplimcpx] d(px. (4.24)

0

It is easy to see that Fourier coefficients of the function V(x) take the following form

V'^(^)= j f(y)Smi^,^y)txp[-imcpy]day

dDi

Using the representations

lry^^ (t t \ - / ^ \ ^ (PPj) ^" '" ' (2/+|m| + l/2) . ,^ . ImSmi^^^y) - - V y Z . / , ( / + |^ |)I22/+H^l ^2/+|m| + l/2(/^l),

r» c /^ ^ \ [^\r^ (PPy) o-(2/+|m| + l/2)., .^ . Re^^(^,g,) = - ^ 2 l ^ / ! ( / + |m|)!22'+N^' ' Â^2/+H+i/2(/?l),

where /?j = p^ + p^ + (z — Zy)'^ and /«(•), Aâ() are the cylindrical Bessel and Neumann functions, respectively. From the latest relations it is easy to see that

lim " ' ^ = qmh\ÂkeR^z)Pm(^^^^z)^ qm = const.

Thus, (4.23) means closeness conditions of the functional system (4.22) and the theorem is proved. D

REMARK 8. The complete systems constructed seem to be appropriate for analysis of scattering by axial symmetric structures.

4.2.2. Regular complete systems for scattering problems

The approach developed above enables construction of complete systems for solution of an exterior scattering problem. Another conventional scheme of DSM allows using com-plete systems for interior scattering problem. For several reasons we would like to present an approach that can be used to construct complete systems based on entire functions. This makes sense because regular Bessel functions are mostly used by many researchers for representation of an internal field.


We shall consider the mathematical statement of an interior boundary value problem

(A + y^f)^/=0 in A ,

U{y) = U^{y). yedDi. (4.25)

Herelm)^/ ^ 0, 9 A e C^, U^ e C{dDi) and a A is a non-resonance surface (which means that ki does not coincide with an egenvalue of Laplace operator inside A )• Then (4.25) has a unique solution in D/.

Let us take into consideration an entire function of the form

X(x,y,k) = - ^ ^ ^ (4.26)

where Rxy = \x — y\. Analogous to (4.9) we define

U(x)= j x(x,y,k)viy)day (4.27)

dDi

with density v(y) £ L^(dDi). Similar to our previous analysis, the following statement is valid.

LEMMA 9. Let U(x) = 0 in A • Then v (xOatdDi.

PROOF. First, we will show that from the condition U(x) = 0 in A follows that V(x): (4.9)y(jc)=0in A.So, le t

U(x)= x(x,y,k)v(y)day = 0, J C G A . (4.28)

dDi

Because U(x) is an entire function, U(x) = 0 in any finite region of IZ^. Let us take a non-resonance sphere E, enclosing A inside and use the following representation

oo /

X{x,y.k) = kJ2J2 ^imji{k\x\)ji{k\y\) /=0 m=- /

X pl'^l(cos^^)P;'^'(cos^^)eJ'^^^--^^\ (4.29)

Substituting (4.29) into (4.28) and using orthogonality of spherical wave functions at E, we obtain

/ ji{k\y\)PJ"'^(cosOy)t-^'^^yv(y)day = 0, /e{0,N}, \m\^L (4.30)

dDi

Multiplying (4.30) term-by-term by

Ni„hf\k\x\)PJ'"kcose,)^'"'^', X € E,


and summing over /, m, we deduce

/ \lr(x,y,k)v(y)day = 0, jc G E.

dDi

So, the last relation implies that V(x): (4.9) satisfies the exterior homogeneous Dirichlet problem

(A + / : ^ ) y = 0 outside E,

V{y) = 0, J G S ,

dV _i —-+j / :y = o(r ^), r->(X). 9r

The last property yields y(jc) = 0 outside S, and due to analyticity ofV,V = 0,xeDe. To complete the proof of the lemma we apply the technique that has been used while proving the Lemma 1. The lemma is proved. D

A scheme for constructing a complete system corresponding to interior scattering prob-lems consists of several steps:

(1) Choose a DS support S and a set of DS coordinates {^nl^j e S, and {xn}î must have at least one limit point XQ G S;

(2) Apply at {x„}^^ the conditions

Af/WU^„=0, V ^ G N , (4.31)

which provide U(x) = 0 in Dt and therefore, via Lemma 9, v oc 0 is valid. Let us designate

Xn(y)'-=Ax(x,y,k)l^^^, (4.32)

then we can rewrite (4.31) in the following form

f Xniy)v(y)day = {xn,v'')^2=0, V ^ G N . (4.33)

dDi

Relation (4.33) represents a closeness condition for the system {Xn }î in L^(d Dt), then the following principal result is valid:

THEOREM 10. The system {x«}î- (4.32) is complete in L^(dDi).

PROOF. It is sufficient to prove a closeness of {xn}î in L^(dDi). Functions Xi satisfy the Helmholtz equation in any finite region of K^. For any v e L^(dDi) the following relation is valid

( x , - , u % = 0 , V / G N - ^ u a O ,

since Xi satisfies (4.33). Therefore, (x/l/î is closed in L^(dDi). The theorem is proved. D


REMARK 11. It is easy to see that

X{x, y, k) = lm\lf(x, y, k), (4.34)

at least, for real values of k. Then functions U(x): (4.27) and V(x): (4.9) in the vicinity of coo are analytic functions with respect to x. So, all results of Lemmas 5 and 6 remain vaHd for^(jc).

THEOREM 12, Let DS support be COQ, then a correspondent complete system at any ambi-ent closed non-resonant surface dDi € C^ has the following form

Xmnix) = jm(kRxzn)K(^^^^zn){^^^^^f^^^ sinm^}, n,meN. (4.35)

PROOF. Due to Lemma 9 and Remark 11 it is sufficient to find a specific kind of functional system associated with

lim — = u'^(zn) =0, w, m G N. (4.36)

Due to (4.34) the following relations is valid

lim Im ^^^^^%^ = qmjm(kR^,)P;:(cosO,). (4.37)

Comparison of (4.36) and (4.37) completes the proof of the theorem. D

4.2.3. Discrete sources in a complex plane

In this section we shall focus on the procedure of analytic continuation of DS fields onto a complex plane with respect to the source coordinate z„. This procedure enables selection of the DS support in agreement with locations of singularities for the scattered field continuation inside an obstacle. Similar choice of the DS support allows ensuring the boundedness for the sequence of DS amplitudes while A -^ CXD [18]. The boundedness property seems to be very important for providing stability of a numerical model based on DSM.

Let us consider the half-plane

(p = const: O = {)/ = (yo, z) I yo > 0, zeR}

and introduce the complex plane

$ = {f = ( R e | , Im§) I R e | , Imf G R}.

We assume that <J adjoins the real axis in such manner, so that Re coincides with the z-axis. Now, the functions introduced in previous sections can be expressed in terms of the complex coordinate ^, using the analytic continuation procedure, that is

Ymn(x) = h^r^\keRrj^„)P!^(cosO^^){l,cosmcp, sinmcp}, n,meN, (4.38)


where

Here the function R 7 of a complex variable § is chosen to be the branch corresponding to an arithmetical root at a positive part of the real axis. It is necessary that all functions can be expressed as single-valued analytic functions, which are properly defined for an arbitrary complex coordinate §.

DEFINITION 13. Point § G ^ is called an image for the point r] e <^, provided that R \ = 0.

LEMMA 14. For each point rj = (p,z) ^^, there exist two images § '- such that

Re|'^'^ = z and Im|'^'^ = ± p . (4.39)

PROOF. Relation R\ = 0 impHes

/?J^ = ( p - I m ? ) ( p + I m ? ) - 2 j l m ? ( z - R e ? ) + ( z - R e ? f = 0

from which we obtain (4.39). D

LEMMA 15. Let r] = (p,z) be located either outside or at the surface 9 Dt. Then the region of the proper analytic continuation represents a connected domain D^ C i of which the boundary dD{t coincides with the image of the obstacle meridian X C O.

PROOF. The boundary of the domain D^ is obviously determined by location of r] at the obstacle surface. That originates singularities distributed in accordance to (4.39). These singularities confine a connected region at the complex plane O of which the boundary coincides with the image of the meridian X. It is easy to prove that Re R r ^ 0 for any YJ located outside or at the obstacle surface, and ^ eD^. D

LEMMA 16. Each point § in the complex plane O originates the ring of singularities in the real space.

PROOF. Let us set a point § G O, then the relation (4.39) gives us the point of singularity ?7 = (p, z), satisfying (4.39)

p = | lmi ' | and z = Re|'. (4.40)

The latter means that in R^ we have the singularities distributed at the ring of radius p. D

COROLLARY 17. Relation (4.40) leads to an indistinguishability of the original complex source (Re^, Im§) and conjugate source (Re§, — Im§). Because the original source in-duces the ring of singularities (4.40), which, in turn, causes the images via (4.39) in-volving the original and conjugate sources. From this point of view, a distribution of


DS in the complex domain I>t C ^ should be symmetrical with respect to the real axis

Im§ = 0.

LEMMA 18. The functions (4.38) with the DS located at the region D^ satisfy the Helmholtz equation outside an obstacle and radiation condition at the infinity.

The proof is a consequence of Lemma 14 and asymptotic of a spherical Hankel function.

THEOREM 19. Let a set of complex DS coordinates {§«}^i d Dr be distributed via Corollary 17 have at least one limiting point inside Dt, then the systems (4.15) and (4.35) are complete in L?{dDi).

Proof can be obtained taking into account properties of analytical functions of a complex variable.

THEOREM 20. An approximate solution of an exterior {interior) boundary valued problem for the Helmholtz equation can be constructed as a finite linear combination of functions of the system (4.15) {or (4.35)) with DS located at the proper domain of the complex plane D^CO.

Proof follows directly from Lemma 18 and Theorem 19. Thus, by deposition of the DS support in the region D^ of the complex plane O ad-

joining the real axis, we can adjust the support location with singularities of the analytic continuation for the scattered field inside D/. This approach proved to be most efficient for analysis of an oblate scatterer.

4.2.4. Discrete sources method for transmission problem

In this section we will consider a transmission scattering problem for the Helmholtz equation. We will represent a scheme for construction of an approximate solution based on the systems of DS, discussed in previous sections.

Let us consider the following transmission scattering problem for the Helmholtz equa-tion

{^ + kl)Us=0 inDe,

{A-\-kf)Ui=0 in A ,

Ui-Ue = U^, dDi,

dUi dUe _ dU^ dn dn dn

dUe

(4.41)

9 A ,

-\-}keUe=o{r 0 , r ^ o o , ar

where 9 A ^ C^, U^ is an external excitation. The transmission problem (4.41) is well known to have a unique solution [26]. Now we need some auxiliary results.

LEMMA 21. Assume, that for some square integrable densities {a,b} e L^{dDi) x L^{dDi) the following relations are valid


f xlfeiM, P)a(P)dap - j ^^tf^^b(P)dap = 0 , MeDt,

dDi dDi

/ xlfiiM,P)a(P)dap- / J!^lj-^-lb(P)dap = 0 , MeDe, (4.42)

dDi dDi

where xj/giM, P) — \I/(M, P,kg). Then a = 0, b=^0 almost everywhere at dDf.

PROOF. Designating the function on the left-hand side of (4.42) as WejiM), it is clear, that We(M) is an analytic function in Dt and Wt (M) in D^. Relations (4.42) yield for their limit values

{We(Q - hnQ) - WiiQ+huQ)} = 0 , Qe dDi,

( ^ We(Q - hng) - - ^ WiiQ + hng)] = 0 , Qe dDi. [driQ driQ J

Using jump conditions for a surface potential with L^(dDi) densities we get

b(Q) + j [feiQ. P) - MQ, P)}a{P)dap - j ^^^t^zMt(P)dap=0,

dDi dDi

a(Q)^^ {xlfe-î}a(P)dap-—- / ^l^—!^biP)dap=0, (4.43) oriQ J driQ J dnp

dDi dDi

for Q e dDi and almost everywhere at 8D/. This system of integral equations is a Fred-holm type system of the second kind. From existence of its solution {a,b} e L^(dDi) x L^(dDi) and taking into account a resolvent property of the Fredholm integral equation with the smooth right-hand side, we can conclude that {a, b] a [ao, bo} G C^^'^\dDi) x (^(0,«)(SDi). The latter means that ao = 0,bo = 0 at 3 A . Hence {a,b](x 0. The lemma is proved. D

REMARK 22. Let us emphasis that the result obtained is valid even for resonance values of fc^,^

COROLLARY 23. Via Lemma 9 the system (4.43) can be rewritten in equivalent form

I MM, P)a(P)dap - j ^^t^^l^b(P)dap = 0, dDi dDi

j Xi(M, P)a{P)dap - j ^^î^lIlb(P)dap = 0, (4.44)

dDi dDi

for M G Di and with Xi given by (4.26).

THEOREM 24. Let {COQ^} be supports (possibly different) located in Di^ then for any

{a,b} € LîdDi) X LîdDi) and 8 > 0 there exist K^' = {M^^\N^^'} and {pf^l} that


yield

m=On=l

U!'(x)=J^J^pL^mn(x), (4.45)

m—On=\

SO that the following estimation holds

u!'-^f-4l2,3D^,+ dUf^ dU^ "

dn dn -b < 5. (4.46)

PROOF. TO validate (4.46) it is sufficient to prove the completeness of systems used for representation (4.45). Completeness relations associated with (4.45)-(4.46) are

/ Ymn{P)a\P)dap - f ^l!p^bHP)dap = 0, J J dnp

dDi dDi

j Xmn{P)a\P)&ap - j ^^2l^\\P)dap = 0, (4.47)

dDi dDi

n,m eN, where Ymn and Xmn are defined by (4.22) and (4.35), respectively. Properties of DS supports {COQ' } and (4.47) lead to

j ,lfe(M, P)aHP)dap - j ^-^t^^l^ b\P)daP = 0,

dDi dDi

f XiiM, P)a\P)dap - f ^^'^^'^\%P)dap = 0, (4.48) J J dnp

dDi dDi

for M e Di. Comparing (4.48) with (4.44), and applying the results of Lemma 21 and Corollary 23 we get {a,b}(x 0. The theorem is proved. D

COROLLARY 25. Approximate solution of the transmission problem (4.41) can be repre-sented inform o/(4.45).

4.3. Mathematical models for the Maxwell equations

Up untill now, we have considered scattering problems for the Helmholtz equation. In the following sections we will extend these results to scattering problems for time-harmonic electromagnetic waves. We will begin with an outline of the scheme for con-struction of complete systems for radiating and regular solutions of Maxwell equations. Based on a specific geometry of DS support, a system of lowest order distributed multi-poles will be constructed as an application of the scheme to be developed.


We will then proceed to analyze electromagnetic scattering problems of a penetrable particle. We will construct a mathematical model of the BVSP based on DSM theoretical backgrounds. We will present an approximate solution for an axially symmetric penetrable particle accounting the axial symmetry. Finally we will formulate the approximate solu-tion in a form which takes into account not only an axial symmetry but a polarization of an exciting linear polarized plane wave as well. We will give some numerical examples illustrating abilities of the enhanced model.

The last two sections of this part will be devoted to mathematical models describing polarized light scattering from features of a smooth penetrable substrate. This problem has been subject of many theoretical and experimental studies in recent years [32-37] because of its fundamental scientific importance to such practical applications, as, for example, the design of optical scanning instruments for use in semiconductor industry. Some numerical results associated with problems of substrate deffects discrimination will be represented.

4.3.1. Complete systems construction for electromagnetic scattering problems

Let us consider scattering in an isotropic homogeneous medium in R^ of an electromag-netic wave from a perfect conducting obstacle Dt. We assume the time dependence to be Qxplicot). Scattering is described by fields {E , H^} satisfying the Maxwell equations

V X H^ =ikSeEe; V x E^ = -}kiXeê in D^, (4.49)

the boundary condition

np xE^(P) = - n p xE^(P) , PedDi, (4.50)

and Silver-MuUer radiation condition at infinity

lim ( y/sÊe X JJI^Yle 1=0 , r = \M\. (4.51)

Here {E^, H^} is an exciting field, n the unit outward normal to 9D/ and Ime^, /x = 0. BVSP (Eqs. (4.49-(4.51)) is well known to have only one solution [26].

We need auxiliary results in order to describe our scheme for complete systems con-struction for the Maxwell equations.

LEMMA 26. Let a vector potential with a square integrable tangential density v e L\(dDi)

A(M) = / ê{M, P)v{P)dap (4.52)

dDi

satisfies

V X V X A(M) = 0 , MeDi, (4.53)

then V = 0 almost everywhere at dDi.


PROOF. We introduce vector functions

E(M) = -^— V X V X A(M), H(M) = — ^ V x A(M). (4.54)

By (4.53) E(M) = 0, M G A , whence, the analyticity of E(M) yields H(M), M e Dt. Using Hmit relations for V x A(M) with potential densities belonging to L^ (9 A ) [29] we get

^v(Q) - j {UQX [VQ X (u(P)iA,(e, P))]}dcrp=0, Q e 3 A (4.55)

dDi

almost everywhere on dDi. The integral equation (4.55) is the Fredholm equation of the 9 (1 a)

second kind. Since its solution exists, i.e., v e Lt:{dDi), we obtain vocvoeQ' \dDi), taking into consideration resolvent properties of the Fredholm integral equation and the smoothness of its right-hand side. Now the proof is completed as in Lemma 1. From limit relation for E(M) with density of the vector potential VQ G C-C (dDi) we re-ceive

[ n x E ] + : = lim \np xE(P-\-hnp)]=0, PedDi.

It follows that {E, H} solve homogeneous exterior BVSP for Maxwell equations (4.49)-(4.51). Therefore E = 0, H = 0 in A . Finally, from the jump relation

[n X H]+ - [n X H ] _ = UQ = 0

we conclude u a UQ = 0. Lemma is proved. D

LEMMA 21. If k is not an eigenvalue of the interior Maxwell boundary value problem and

V x V x A ( M ) = 0 , M e A , veL]{dDi), (4.56)

then u = 0 almost everywhere on dDi.

PROOF. The proof is similar to the previous lemma. Constructing the fields {E, H} ac-cording to (4.54), we get a Fredholm integral equation of the second kind for the sur-face density. Using the same arguments associated with the resolvent smoothness we ob-tain u a uo G C^ '^\dDi). The continuity relation yields [n x E]_ = 0. In this case, the fields {E, H} solve the homogeneous interior Maxwell boundary value problem, which, with our assumption for k, has a only solution E = 0, H = 0 in D/. Then from [n X H]-f — [n X H]_ = uo = 0 we obtain u a UQ = 0. The lemma is proved. Q

LEMMA 28. Let k be the same as in Lemma 27, then condition (4.56) is equivalent to

V x V x B ( M ) = 0 , M G A , veL]{dDi), (4.57)

where


B(M)= J Xi(M,P)v(P)dap, dDi

and Xi' (4.26).

PROOF. TO prove this statement it is sufficient to combine the proof of Lemmas 9 and 27. D

Now we are able to represent a scheme that allows complete systems construction for Maxwell equations. We denote by {e/} a vector basis at DS support S, which provides

{e,}. F(jc) = 0 =^ F(jc) = 0, VJC € S.

For example {e/} can be chosen as a set of Cartesian bases {Cjc, e^, e^}. Let us take into consideration a vector function W(M) = {Wt (M)}:

W(M) = J VMXVMX [ê(M, P){ei}] • v(P)dap (4.58)

with density v e LîdDt). W(M) is well known to be an analytic function off 9D/. Let us denote

E(M, 2) = VM X VM X [xlfeiM, Q){ei}]

and analyze (4.58) in A . The scheme for constructing a complete system consists of two steps: 1. Choose a DS support S and a set of DS coorditates {M„}^j e 3 , the set {M„}^j

must have at least one limit point MQ e S. 2. Apply at {M„}^^ conditions

AW(M)|^^^^ = 0 , Wne N, (4.59)

which provide W(M) = 0 in Dt and therefore yields i; a 0. Let us designate the mentioned conditions as

ifn(M):=AE(M,Q)\Q^^^, (4.60)

then (4.59) can be rewritten in the form

/ iA„(P) . i ; (P)dap = ( ^ „ , i ; V = 0 , VneN. (4.61)

dDi

The relations (4.61) represent closeness relations for the system of DS of an electric type, {V^nlî in L\{dDi). Then the following principal result is valid.

THEOREM 29. The system of ''electricaV radiating solutions of the Maxwell equations {Ânlî • (4.60) is complete on dDi,


PROOF. It is sufficient to show the closeness of the system (V^nlî in L^(dDi), which is

(V^«,^*)jr2=0, WneN=^v(xO, veL]{dDi). (4.62)

From (4.62) and properties of the constructed system {V^^j^j we deduce

j E(M, P) • v{P) dorp = 0 , MeDi.

dDi

Using the relations of the vector analysis

I WpxVpx [xlfeiM, P){e/}]. v(P)dap

dDi

= {e/} -VMXVMX f [xlfe(M, P)]viP)dap

dDi

and properties of {e/}, we finally obtain

VMy^'^My^ I [V^.(M, P){e/}]. u(P)dap = 0 , M e Dt.

dDi

From Lemma 26 we conclude that u oc 0. Then the system {V^„}^j of DS of an electric type is complete in L^ (9 Dt). The theorem is proved. D

The Theorem 29 enables construction of new complete systems of radiating solutions based on (4.58) and geometry of DS support S.

THEOREM 30. Let S consist of a single point, then the complete system associated with a similar structure ofDS support will be

WJîM) = V X V X [hf\ker)Pl"'\cosO)^'^^{ei}l

/€{0,N}, m=^17i, (4.63)

The system {W/^} represents a system of localized Cartesian spherical multipoles of the electric type.

PROOF. The proof can be achieved with Theorem 29 and using the method of Theorem 3. D

THEOREM 31. Let DS support be COQ, then the associated complete system at any closed smooth surface, confining CJOQ, has a form

YmniM) = V X V X [h^^\keR^zn)Pm(ôÔzn){^^ ôsmcp, sinm^}{e/}], n,meN, (4.64)

{Ymn} represents a system of distributed electrical multipoles of lowest order


PROOF. The proof can be achieved with Theorem 29 in combination with the method of Theorem 7. n

COROLLARY 32. The completeness of magnetic radiating solutions

yjJM) = V X [h\^\ker)Pl'^^(cosO)eJ'^^le,}], / e {0, N}, m = Tj, (4.65)

and

IJmn(M) = V X [h^^\keRrjZn)Pm(^^^^Zn){^^ÔSm(p,Smm(p}{^^^^^

n,meN. (4.66)

can be validated in the same manner as in Theorems 29-31.

Similar to our previous analysis (see Theorem 12) it is easy to prove the scheme of constructing complete systems for regular functions. The following result is obtained.

THEOREM 3 3. Ifk does not coincide with an eigenvalue of the interior Maxwell boundary value problem, then the systems of regular solutions of Maxwell equations

WL(M) = V X V X [ji{ker)Pl'^\coÔ)^'^nti]\

and

/G{0,N}, m = - / , / , (4.67)

Xm«(M) = V X V X [jm{keRr^zn)Pm^^^^^zn){^^ cosm(p, sinm(^}{e/}], n,meN (4.68)

are complete at dDt.

4.3.2. Electromagnetic scattering by a penetrable particle

Adopting the DSM outlines given in the previous parts, we now switch to applications of the method. In this section we will consider scattering by an axially symmetric homoge-neous penetrable particle Dt in a free space De. Assuming time dependence as exp{j(W }, we have a mathematical statement of the scattering problem of the form

V X H^ = jkSgEg] V XEg= -ikflgHg ITlDg, ^ = /, C, (4.69)

^ ^l(Ei(P)-Ee(P))\ ^ ^ |(EV))1 p^^r. ..^r,. ""^ 1 (H,(P) - H.(F)) ) - - '^^ X ( ( H O ( P ) ) ) ' ^ ^ ^ ^ ' (4.70)

^lim^^VÊ, X - v ^ H , " ) = 0 , r = |M|. (4.71)

Here {E^, H^} is scattered field, {E^, H^} is an exciting field, and medium parameters sat-isfy Imei.îi < 0. Then BVSP (4.69)-(4.71) has an unique solution [26].

The following result is valid:


THEOREM 34. Let {CDQ''} he possibly different DS supports located in Dt, then for any field {e , h^} e LîdDi) x L?(aA) and 8 > 0, there exist K'' = {M'^\ N'^'] and [pf^lr] such that

Ef (M) = X] E^-«Y-«^^) ' ê(^) = r ~ ^ " ê(M)^ (4.72)

and

m=On=l

M' A^

k/jie

Ef (M) = YlY.Pmn^-^n(M), Hf (M) = / " V X Ef (M), (4.73)

satisfy

m=On=\

||Ef - Ef - e%^^^^^ + ||Hf - Hf - hm2^sD.^ ^ 5. (4.74)

//^r^ {Y^nY (4.64) an J {X^„}: (4.68).

Proof can be obtained following the methods used in Theorems 24 and 31-33. This result impHes that the approximate solution of the scattering problem (4.69)-(4.71)

can be constructed in the form (4.72)-(4.74). If we deposit the DS supports {co^^} at the axis of symmetry or in an adjoing part of a

complex plane, the representations (4.72)-(4.74) leads to a form of finite sums of Fourier series with respect to ( . It allows proceeding from the surface approximation of an exciting field to the sequence of approximating problems for Fourier harmonics of the fields (excit-ing and scattered) enforced at an obstacle meridian J , for DS amplitude determination.

In case of excitation of an axially symmetric particle by linear polarized plane wave, it appears to be possible to take into account not only rotational symmetry of the obstacle but a polarization of the external excitation as well [18]. Let us introduce some functions that will be appropriate to construct the approximate solution

yairi) - h''^'>{keR,,„)P;:{cose,„) (4.75)

and

yL(ri) = jm{kiR,,^)p;;;{coso,^), (4.76)

where a = {n, m}, /i G N, m G {0, N}, ry G O and {Zn]î ^ m- Here 0^^ is a meridian angle for a local coordinate system originated in Zn- We will regard some special combinations of vector potentials

a^(M) = y^(rj){exCOsm(p — eysmm(p],

b^(M) = y^(ri){ex^^^m(p-\-eyCOsm(p},

4i^) = yL^^)^z. ^ = ej', (4.77)


here {^x^^y^ e^} is a Cartesian basis. Introduced vector potentials (4.77) are orthogonal at a smooth surface of revolution. We assume that electromagnetic fields form a dual vector G^ = {E^, H^}^ and shall have the following differential forms

/ j

Dl-kSgllg

Vx Vx

V / ^ • V x

Vx

Df / kSg/jig

V x Vx /

Supposing that the external excitation is a P-polarized plane wave propagating at an angle ô with respect to the axis of symmetry Z, then the exciting field has the form

E^ = (tx cosô + ^z sin(9o)K,

H^ = —Cj y cos OQ, y = exp{ —jke {x sin OQ — z cos (4.78)

THEOREM 35. Let {COQ'^] be the same as in Theorem 34, then for approximate solution of the scattering problem (4.69)-(4.71) in case of a P-polarized plane wave excitation (4.78), the following representation is valid

Gf (M) = Y. {p'oI>H+^'.I>lK] + Y^r^Dlcl, (4.79) a=ao n=\

where [pa, qi^rn) ^^^ J^^ amplitudes, a = [n,m}, ofo = {0,1}, K = {N, M}, g = {e,i}.

It can easily be seen that we used a combination of electric and magnetic multipoles to construct the approximate solution (4.79).

In the case of S-polarized plane wave

E^ = tyy cosô, H^ = i^x cosô + ©z sinOo)y

the following result is valid.

(4.80)

THEOREM 36. Approximate solution of the scattering problem (4.69)-(4.71) in case of S-polarized plane wave excitation (4.80) can be represented in the form

Gf(M)= X; {/;|Z)fb^+^„^£>|a|} + ^r„fZ)|cf. (4.81) a=ao n=l

The favorable numerical scheme for DS amplitude evaluation is based on a point-matching technique for Fourier harmonics of electromagnetic fields at the obstacle merid-ian J . Its application leads to over-determined linear systems for DS amplitude determina-tion. The DS amplitudes are calculated by employing matrix pseudo-inversion procedures with a preconditioned As it was established by theoretic analysis of the singular values of the matrixes involved, the over-determined linear systems provide the most stable numeri-cal scheme [18]. By computing surface residuals (left-hand side of (4.74)) at the obstacle surface in a mean square norm one can control a convergence of the approximate solution as A' -> oo.


The possibility to calculate any scattering characteristics in a far zone is based on the computation of a far field pattern. It is defined according to [26]

E^(r) exp{-j^^r} , ^ , - K _ _ ¥(0,(p)-\-o{r ), r-^ OQ.

|EO(r)| (4.82)

Note, that far field pattern F(^, (p) (4.82) has only two components 0 and ( in a far zone, so that its components are determined at the unit sphere as

F(^, if) = FeiO, (p) ee + FîO, cp) e^.

According to the approximate solution (4.79) for P-polarized exciting plane waves the far field pattern components accept the form

F^{eîp)\,^

N'

X

(-j)^+isinî ( cos(m + l)(p \ sin(m + l)(p J

l-Crcol.)!?!:)]-!:)-^!'.;-'--) where y„ = exp{—j eZ„ cos6'}.

For S-case, components of the far field pattern are

(-jr+isin'«4 ''"^7+/^,t )

fcosO - 1

The scattering cross section can be defined based on a far field pattern, namely as

a(0,(p,Go) = lim 4jtr^^^^^. (4.85) r-ôo |E^(r)|^

Back Scattering Cross Section (BSCS) can be determined as BSCS(ô) =cr(Oo,(p,Oo).

4.3.3. Computer simulation results

Scattering from a single penetrable particle, as well as from a cluster of such particles, has a wide range of current applications. Implementations in metrology (calibrations and measurements involving lasers), electromagnetic monitoring of the atmosphere (aerosols and pollution scanning, scattering from atmospheric hydrometeors), antenna applications (receiver-transmitter wave tracks, dielectric antennas) etc., are to be mentioned.

Traditionally, in optics scattering analysis has dealt with particles that are small in com-parison to the wavelength: a <X, a <^X, where a is a typical dimension of the scattering particle. That is why approximate models were preferred to the Maxwell system. Because

64

lE+03

lE+02

lE+01

lE+00

lE-01

Yu.A. Eremin, N.V Orlov, andA.G. Sveshnikov

BSCS( 0 o ) ^ k e ^

1 E ~ 0 2 I I I I I I I I I I • • I I • I I I

0 30 60 90 120 150 180

incident angle [deg]

Fig. 4.1. Back scattering cross section vs. incident angle, P-polarized source. Dielectrical (e/ = 4.5 — 2.5j) disc with semi-spherical edge, axis ratios b/a are equal to 2, 5, and 10, ka = 0.6.

E+02 1

E+01 I

E+OO 1

E - 0 1 i

E - 0 2 I

E - 0 3 •

BSCS

\

'

Oo) *ke '

P

S

0 30 60 90 120 150 180

scattering angle [deg]

Fig. 4.2. BSCS vs. incident angle of a disc with b/a = 5.0, P- and S-polarized. Permittivity and ka are the same as in Fig. 4.1.

of the range of particle sizes a <^ X, the spherical approximation for such small scatterers is valid. Note, that the spherical approximation model is not valid for bigger sized parti-cles, when a ^ X ov a > X. Another point is, that traditional refractive indices for particle scattering mostly belong to the interval s e (1.0,2.0) with a very small imaginary part, whereas problems of scattering from particles having essentially complex permittivities remain relatively complex. The following numerical examples illustrate scattering by a single particle, as well as of a cluster of particles, which are non-spherically shaped and have essentially complex permittivities.


i p (9.00

_ , J , , • I r -

0 60 120 180 240 300 360


Fig. 4.3. Scattered intensity vs. scattering angle for a cluster of three particles: prolate (3 : 1) spheroid, oblate (1 : 3) spheroid, and sphere, ka = 5, kb = 15, permittivities are 4.24 — 0.42j (prolate), 2.5 — 0.04j (oblate), 6.24 — 0.62j (sphere). Distance between the particles over the sphere radius is 0.1, incidences are 0°, 60°, and

120°.

T • 1 • 1 -

60 120 180 240 300 360


Fig. 4.4. Same cluster and ka as in Fig. 4.3, incidence is 90°, P- and S-polarized sources.

In Fig. 4.1 scattering of a disc with semi-spherical edge and permittivity si = 4.5 — 2.5 j (BSCS vs. incident angle) is shown in the incident plane (originated by semi-planes: (p = 0°, (p = 180°). Excitation is achieved by a P-polarized plane wave. Dimension pa-rameter is ka =0.6 (a being a small semi-axis of the disc). Different curves correspond to different disc axis ratios b/a = 2, 5, and 10. The same scattering characteristics for both of the P and S-polarized sources are shown in Fig. 4.2 (the axis ratio is b/a = 5.0).


Scattered intensities vs. scattering angle from the cluster of three different particles are shown in Figs. 4.3 and 4.4. Here we present the dimensionless scattered intensities

The cluster consists of a prolate spheroid (^ : a = 3 : 1), an oblate spheroid (a:b = l:3), and a sphere of radius a. The distances between these particles are equal to 0.1 a. Note that, by placing particles as close as this, we assume them to be strongly affected by the presence of each other. The oblate spheroid is flattened, and DS for it must be placed in a complex plane. For Fig. 4.3 the dimension parameter for smallest semi-axis is ka = 5 (kb = 15), and permittivities are si = 4.24 — 0.42j (prolate spheroid), £( = 2.5 — 0.04j (oblate spheroid), st = 6.24 — 0.62j (sphere). Incidences of the plane wave are 0°, 60°, and 120°. Comparison of both P- and S-polarizations is shown in Fig. 4.4 for an angle of incidence of 90° with ka = 5, kb = 15.

The examples given above demonstrate the ability of DSM to analyze scattering within a wide range of input parameters providing reasonablely, accurate values of the computed results (the residual for all examples was less than 1%).

4.3.4. Light scattering by penetrable particle on a substrate

The research described in this section addresses the problem of calculating light scat-tering by a defect of a smooth substrate. Two basic models were developed: the first one, for scattering by a particle upon a penetrable substrate, and the second one, for a defect located inside a substrate. These applications were aimed at model scattering, which takes place in wafer inspection systems. These systems are machines that are commonly used in semiconductor industry and allow monitoring of the purity of silicon wafers, detection wafer contaminants (such as smallest particles, dust, etc.) as well as micro-defects (pits, scratches, and so on).

Let {E^, H^} be a field with a time-harmonic plane electromagnetic wave of linear po-larization. The plane wave is supposed to propagate at an ^o angle with respect to a normal of an air-substrate interface surface S. We also denote two half-spaces as Do (air) and Di (substrate). Let an axially synmietric scatterer Di with a smooth surface dDt be located at the interface in such a way that its symmetry axis is parallel to an external normal of the interface E. We will introduce a Cartesian coordinate system with the Z axis in direc-tion of Do and choose its origin at the axis of symmetry. Then the BVSP can be stated as follows:

V X H^ =}k6gEg; W xEg = -jkiXgHg inDg, ^ = 0,1, /, (4.86)

»^><{S' (P! :H| (P)}=-»^^{S1?)} ' ^ ^ " ''•'''

e.^\l'lil~l'^fl\=0, Pes, (4.88) H o ( P ) - H i ( F )

radiation or attenuation conditions at infinity. (4.89)


Here {EQ, HQ} is a scattered field in Do, and medium parameters fit the conditions I m e ^ , jjig ^ 0.

We are going to construct an approximate solution to BVSP (4.86)-(4.89) on the base of DSM. For this let us represent the external excitation as a total of incident and reflected plane waves at the upper half-space (Do) and as the transmitted wave at the lower half-space (Di). Thus, it is sufficient to construct a representation for a scattered field only. The idea of DSM is to represent a scattered field as finite linear combination of fields of multipoles, so that Maxwell's equations over the regions Do, Di, and D/ will fit. It also satisfies the conditions at infinity, including the radiation condition in Do, and the atten-uation condition in D\. We also want this combination to satisfy transmission conditions for tangential components of fields at the interface surface E. Under these circumstances the solution of BVSP (4.86)-(4.89) will be reduced to an approximation of an exciting field at the particle surface via fields of multipoles, so that the determination of unknown amplitudes of DS is achieved by transmission conditions at the particle surface only.

Let us construct the representation of an approximate solution by taking into consid-eration the axial symmetry of a particle and polarization of the exciting plane wave by fulfilling the transmission conditions at the interface surface H analytically. Let us express the multipole sources of electric (e) and magnetic (h) types comprising the external field representation in such a way that the vector potentials for these DS have the form

The structure of such a form can be explained in terms of Green's tensor of half-spaces. The elements g ' are induced by elements G 'j , G22 and G^^^ , G32 of the Green's tensor ^^'^(M, P) [22]. Here, for ^ = 0 we denote

^m5(§'^-) = y^{H.Zn)-y^{^.-Zn)

00

+ /*/m(Vy''^exp{-^?o(z + Zn)}^^+"'dX; z , z „ ^ 0 ,

0 00

0

/?!,„ = P^ + {z- Znf\ H = (P, z), f e <t>, (4.90)

where the spectral functions K'^'^ and ^ are

^e ^ 2MI ^ft _ 2SI

? = 2(ei/xi -eo/u.o)

(sirio + soT]i){fi\r]o + fioril)'

rili=X^-kli, 4 i = ^ V i M 0 , i . (4.91)


Because the particle is located in the upper half-space Do, it is unnecessary to note the field representation inside the substrate Di. Nevertheless, in the following we will suppose this representation was obtained in the same way.

For representation of internal fields inside a particle we use the fields constructions dis-cussed in Section 4.3.2, so that we shall concentrate on the field representation in DQ only.

Let us regard the field of a P-polarized plane wave (where the electric vector belongs to the incident plane), propagating under an angle ô with respect to the Z axis. Then the total (incoming and reflected by a substrate) field of an electromagnetic plane wave at the upper half-space can be written as

E^ = ex cosô in - R?y2) + e sinô in + RFYI),

if = -ey(yi-\-Rpy2),

y\ = exp {-jô (^ sin Oo — z cos o)},

}/2 = exp{ -jô(-^ sin o + ^ cos o)}, (4.92)

where the Fresnel coefficient in P-case is [30]

Rp = ==. (4.93) s\ cosô + V ( 1 — sin ô)

In order to take into consideration the polarization of an external field, we will use linear combination of vector potentials similar to (4.90), (4.91).

Kg •= A^x^cosm(/>-A^^^sinm0,

âg '= ^mxg sinm0 -f A^^^ cosm0. (4.94)

Here ^ = 0 ,1; a = {m, n] is a multiple index: 0 < m < M , I ^n ^ N, where M is a maximal number of harmonics, and A is the number of multipoles, which are the same over the whole range of m. Denote ao, K: ao = {0,1}, K = {M, A } are minimal and maximal values of the multiple index. We will use vector potentials to describe dipole sources

A^^ = cr ' (§, z„)e„ or|'^ - 4'/^(§, zn)- (4.95)

According to the notation introduced above we can construct the representation for the field in Do U D\ for P-case as

Gf (M) = Y. {P« f Aa, +?a^f A ,} + J^r„DfA'„^, ? = 0,1. (4.96) a=ao n=l

Let us emphasize that the representation (4.96) fits the Maxwell equations in both half-spaces Do,i. As mentioned previously, the representation (4.96) satisfies all conditions of BVSP (4.86)-(4.89) except transmission conditions at the particle surface.


In the S-polarized case (when the E^ vector is orthogonal to the incident plane) we have

H^ = ec cosô ( n + Rsn) + e, sin6>o (/i - /^sn). (4.97)

The Fresnel coefficient for S-polarization has the form [30]

cosô — A/î/ô — sin^ô R^ = "^ (4.98)

cosô + Y î/ô — sin^ô

In this case the approximate solution to BVSP (4.86)-(4.89) gives the following repre-sentation

G f ( M ) = ^ { p „ £ » f A ^ , + ^ „ £ ) | A ^ , } + ^ r „ D f A ^ ^ , ^ = 0,1. (4.99)

The dipole term in (4.99) has such a form, because H^ belongs to the incident plane for a S-polarized source.

So, the approximate solution to BVSP (4.86)-(4.89) corresponding to different polariza-tion of exciting plane waves (4.96), (4.99) has the following properties:

(1) Fits Maxwell's equations within a particle and in both half-spaces; (2) Automatically fits transmission conditions for fields at the interface surface E; (3) Satisfies infinity conditions. Due to the completeness of the system of DS and its orthogonal property for P- and S-

polarizations at an arbitrary axially symmetric surface, the following result can be desired.

THEOREM 37. Let {E^,H^} be linear polarized plane wave: (4.92)-(4.93) or (4.97)-(4.98), then the scattered field {E^,H^} with the form (4.96) or (4.99) converges uni-formly to the exact solution of BVSP (4.86)-(4.89) {EQ, HQ} at any closed compact outside the particle as K ^ oo.

One of the basic scattering characteristics is the far field pattern, determined by (4.82). To obtain far field pattern representation let us use an asymptotic approach [31] for evalu-ation of the Weyl-Sommerfeld integrals

oo

where

7^(Xp)/(l)exp{-ô(z - Zn)]y^'^ dA

0

= Vôjôcos^ (j/:osin^)'^G„/(/:osin^) + o(r~^), r —^ oo,

iro = exp{-jô^}/^ Gn = exp{-jô^^z cos^}, r^ = p^ + z^.

Based on this formula, the asymptotes for Green's tensor elements in the far zone can be obtained. For the P-polarized source of the components of the far field pattern the following


representations are valid

F^^^\e, <p)J^y cos(m + 1)^ f J ^ s i n ^ V

N

n=l

_ l ^ s i n ^ V r 4 G ; + / c ^ G „ ) ,

F^^^\e. cp) = - J ^ V sin(m + l)cp 0-^ smoY

m=0 N

X J2[pM^n+'^'Gn)+qmnCOsO [ G ^ + {K ^ -^ sin^ 0)Gn]],

where spectral functions have the form

_^ _ jcos^-gp- - _ 2(81 - gp)

j cos0 -\-m' (j cos0 -\-m)Q8i cos0 + ^ o ^ ) '

jsicosO -\- som' m = ysin^^ — î/ô,

and G^ = exp{jfcoz„ cos^}. Let us remember that the DS amplitudes are the same in Do, i, so that for the S-polarized

case we obtain

Ff^'\e, v) = - T sin(m + 1)^ 0-^ sineY

N

X J2{PmnCOSe[G'„ + iK'-lsmê)Gn]-qmn{G'„+lc''G„)Y

n=\

F^<-^\e, <p)J^y cos(m + \)q> 0^ s i n ^ V

N

I]j/^mn(G;+/c^G^) - ^ ^ ^ c o s ^ [ G ; + (/c^-ysin2^)G„] j n=\

m=0

iV

X

n=\

+ J^ s in^ V r ^ G ^ + F ^ G , ) .

Here the same designations are used as for the P-polarized case. This shows that far field pattern components do not contain any Weyl-Sommerfeld inte-

gral. Thus, once unknown DS amplitudes have been determined it is sufficient to calculate only a combination of elementary functions to evaluate scattering characteristics.


4.3.5. Light scattering by subsurface defect

The last section is devoted to the analysis of polarized light scattering by an axially sym-metric subsurface defect. Again, we will begin with the BVSP statement, keeping the des-ignations introduced in the previous section. Let there be an axially symmetrical cladding with a smooth surface 9D, in the lower half-space D\. The symmetry axis of the cladding coincides with an external normal direction to interface E. The Cartesian system can be naturally introduced, choosing its origin at a point O = Z D E. Statement of the BVSP here has the following form

V ^Yig=]keg^g, V X E^ =-j)^/x^H^ in D^, ^ = 0 ,1 , / , (4.100)

, (EKP)-Eî(P)) np X <[ t ) = —DP X

\ni{P)-WAP)) ^ (E?(P)) 1 ' ^' ' M , P e a A , (4.101) (H?(P))

( ( E o ( P ) - E l ( / > ) ) ] _ ^ "" j (Ho(P) -Hi(P)) J - ^ ' ^ ^ ^' ^^'^^^^

radiation or attenuation conditions at infinity. (4.103)

Here {E,, H/} is the total field inside a cladding; {E^, H^} (^ = 0,1) are the scattered fields m Do,/)i,and{E^,H?} is a field of the transmitted plane wave in Di. As in the particle case, we will construct the approximate solution of BVSP (4.49)-(4.51) on the basis of DSM. We shall describe only points differing from the particle case in detail.

For a representation of the field in Di, we shall use multi-pole sources of both electric {e) and magnetic (m) type. Because the obstacle is now located in the lower half-space the components of the Green's tensor are

g'm\^H.Zn) = Yl(^,Zn)-Yi(^,-Zn)

oo

+ f Jm(^p)K'^êxp{m(z-\-Zn)}^^^"'dX, Z.ZnÔ;

0 oo

fm\(^,Zn) = /'/m(Ap)Cexp{î(z + z„)}A^+^dA,

0

yL(^) = Yl(^,Zn) = -h^r^\hR^,M - i ^ . (4.104)

Here spectral functions /c ' and ^ are

e 2/Xo h 2^0

2(siLi\ — £oMo) ^ = 7 ' w ! 7- (4-105)

When constructing the field representation inside the defect, we will use the construction developed in Section 4.3.2. In this case, the defect may be assumed to be filled with any material (not only with air). With given analogy to the particle case, we can construct an


approximate solution that would take into account not only axial symmetry, but a polariza-tion of external excitation as well. So, the approximate solution accepts one of the forms of (4.96), (4.99), but (4.104), (4.105) is substituted instead of (4.90), (4.91).

In the P-case, the field of transmitted plane waves in the lower half-space has the form

E^ = (e;c cos Of + e sin o sin Ot)Tpyi, H^ = - e ^ T^ni yi,

yi = expf —j/:i (x sin Ot — z cos Of)], sin ^ = — sin o, ni

cosOt = —Je\ - £osin^%, no,i = Vô,i/>ô,i, (4.106) n\ ^

where the transmission coefficient Tp is [31]

e\ cos 0 + ^0 V ^1 — ^0 sin' ô

For the case of a S-polarized source, where the H^ vector belongs to the incident plane, the transmitted plane wave field is

E = e^rsn,

= (e;c Y ^1 - ^0 sin^ o + e^ô sinô j ^sKi, (4.108)

and the corresponding transmission coefficient has the form

r s ^ ^"o;"^ô . (4.109)

no cos 0 + V ^1 — ^0 sin^ ô As previously stated, for scattering of particles, the developed approximate solution of

BVSP (4.100)-(4.103) fits Maxwell's equations, infinity conditions and transmission con-ditions at the interface surface S well. Furthermore, convergence takes place.

THEOREM 38. Let the exciting field {E^, H^} is represented in the form (4.106)-(4.107) or (4.108)-(4.109), then the approximate solution (4.96) or (4.99) of BVSP (4.100)-(4.103) converges uniformly to the exact one as K ^^ oo at any closed compact outside of the defect.

Before turning to the far field pattern representation for the case of subsurface defect we need expressions for fieds in the upper half-space (because only there the far field pattern makes sense). Now the corresponding Green's tensor components are

00

^m2(?' Zn) = f Jmi^pW^^ exp{-r7oz + mZn])^^^'^dX, z > 0, z„ < 0;

0 00

0

Si S\


Also, the expressions for spectral functions are the same as (4.105). As before, we will use asymptotic expressions for evaluation of the Weyl-Sommerfeld

integrals. Then the spherical components of the far field pattern will have the form for the P-case

F^^^\e, (p) = —cosO V cos(m + l)(p f — sin6> | 0 fr'n Vô y

and

m=0

X J^^niPmnOc' + 0-^qrnnK^} -—sine J^^nG'^l^^

n=\ ^^ n=\

M' Kf^^\0,(p) = — cos6>Vsin(m + l)(^f — sin6> I

and for the S-case

F^^^\0,(p) = — cos6> V s i n ( m + l)(^(—sin6> I m=0

N

X I ] Gn{Pmn{K' + f) " qmnK*" }

and

FJ<^>(6»,(p) = — cos6l Y"cos(m + \)(p\— sin0 )

where the over-lined spectral functions are

2cos^ _;, 2^0 COS K =

^ =

COS0 -\-m' £0cos0 -\-8\m' 2(£i - £0)

(cos^ + m)(socosO -\- £im)

G'^ = exp{j/:o^}, ^ = y î/ô - sin^^.

Hence, as was found for the particle case, the components of the far field pattern do not contain Weyl-Sommerfeld integrals and can be evaluated as a finite linear combination of elementary functions.


4.3.6. Numerical results and discussion

In this section, we will demonstrate the usefulness of DSM models by using them to analyze polarized light scattering by silicon substrate features of contemporary interest. We assume an exciting wavelength is equal to 0.488 iiim and corresponding to this wavelength, the silicon substrate refractive index isn= 4.37 — 0.08 j . The numerical results presented in the following are mostly for Differential Scattering Cross-sections (DSC) in the incident plane, with the unit |xm^:

We also use integrated values of the scattered intensity over a solid angle Q.

(4.110)

(4.111)

First, we will analyze the Total Integral Response (TIR, in |xm^), when Q coinsides with the upper hemisphere. It depends on both the incident angle and polarization. The results corresponding to the Si and Al (n = 0.73 — 5.93 j) spheres are shown in Fig. 4.5.

From examination of these results, one can conclude that P-polarization and the incident angle near 65° provide the highest scattered signals. We will use the 65° incidence in the following analysis of DSC (4.110).

The DSC dependence on a scattering angle is shown in Figs. 4.6-4.8. Figure 4.6 is devoted to the Si spherical particle of diameter D = 0.08 |xm, P and S polarizations re-spectively. Figure 4.7 demonstrates P-polarized results for a particle D = 0.08 iiim of three different materials: Si, W (n = 3.36 — 2.66 j) and Al. DSCs for W sphere of three different diameters (D = 0.06 iiim, D = 0.08 }xm, D = 0.11 tm) are depicted in Fig. 4.8. Basic

lE-02 3

l E - 0 3 :

lE-04 ;

lE -05 •

TI R [ |LLm^ ]

1 1 1

A l ^ p _

S i , p

A l , s ^ v

1 • 1 '-

s

0 20 40 60 80

incident angle [deg]

Fig. 4.5. TIR (Eq. (4.111), |xm^) vs. incident angle for Si and Al spheres of D = 0.08 \im. Si, P-polarized; Si, S-polarized; Al, P-polarized; Al, S-polarized. Exciting wavelength is 0.488 |xm.


scattering properties of particles on a substrate can be summarized as follows: for "small" particles the magnitude of a light scattered by a P-polarized source, uniformly exceeds one scattered by S-source (see Fig. 4.5) over the whole range of scattering angles (—90°, 90°), except for a narrow segment close to the substrate in normal direction. The dip for P-polarized scattering takes place for all particle materials (see Fig. 4.7).

Now let us analyze pit scattering properties and compare them to particle ones. The objective is to find out the scattering differences allowing differentiation of pits from parti-

l E - 0 3 J

lE-04

lE-05 i\

lE -06

lE-07 J,

l E - 0

DSC[ |Im


Fig. 4.6. DSC (Eq. (4.110), \x.vo?') vs. scattering angle for Si sphere of D = 0.08 |j,m, P-polarized and S-polarized. The incidence is -65° {OQ = 65°, (p = 180°), wavelength is 0.488 |Lim.

l E - 0 3 ^

l E - 0 4

DSC[ iLlm^]

•90 -60 -30 0 30 60 90

s c a t t e r i n g a n g l e [deg]

Fig. 4.7. DSC for particle D = 0.08 ]xm of three different materials: Si, W, and Al. Incidence is —65°, P-polarized source.


. 2 n

lE-02

lE-03

lE-04

lE-05

lE-06

DSC [ |Llm^

d = 0 . 1 1 |LLm

l E - 0 7 1—*—*—I—*—*—I—.—.—I—.—.—I—.—*—I—^—.-

-90 -60 -30 0 30 60 90


Fig. 4.8. DSC for W sphere of three different diameters: D = 0.06 |j.m; D = 0.08 M.m; D = 0.11 |j,m. Incidence is —65°, P-polarized source.

l E - 0 3

lE -04

l E - 0 5

DSC[ ILlm^]

lE-06 I

lE-07

-90 -60 -30 0 30 60 90

scattering angle [deg] Fig. 4.9. DSC for Z) = 0.180 \xm conical pit (vertex angle is 108°) compared to W sphere D = 0.08 |xm, for a

P-polarized source at normal incidence.

cles. Small surface pits in silicon substrates have sizes ranging from 0.10 |am to 0.24 |xm. They often have the form of inverted pyramids. They are aligned along crystal planes, with vertex angles that are typically about 100° [37]. We will simulate such pits as inverted axisymmetrical cones. The DSM model will be used to produce computer simulations for DSC of pits and particles. Differences shall be used to distinuguish between the two. A typical set of results is given in Figs. 4.9 and 4.10.

4. Models based on discrete sources method

. 2 1

77

l E - 0 3 DSC [ |Llm'

lE -04 i

l E - 0 5

l E - 0 6

l E - 0 7

scattering angle [deg;

Fig. 4.10. DSC for D = 0.180 \xm. conical pit (vertex angle is 108°) compared to W sphere D = 0.08 |xm, for a P-polarized source at —65° incidence.

l E - 0 3 1

l E - 0 4 i

l E - 0 5 1

l E - 0 6 1

l E - 0 7 1

l E - 0 8 •

DSC[ ILlm ]

^ — •

; V ^ ^ d = 0 . 1 8 l m

f/ ^ ( / d=0 .12 |am

'

1 1 1 1 1 1 1 1

d = 0 . 2 4 |am

~~ ~*°"'*-""'~-.

\ ' — ^ ^ \ \

\

-90 -60 -30 0 30 60 90


Fig. 4.11. DSC for D = 0.180 fj,m conical pit (vertex angle is 108°) of three different diameters: D = 0.12 fxm; /) = 0.18 |j,m; D = 0.24 |xm. Incidence is —65°, P-polarized source.

In this case, the DSC of a 0.08 |am W particle is compared to a conical pit with diameter 0.18 iLim and vertex angle 108°. This case was picked because the DSCs have about the same level in most directions and thus should be more difficult to distinguish. Differences between the DSCs of the two features (particle and pit) become visible in Fig. 4.10. The obvious dip in the particle DSC near the normal direction occurs only for a particular con-figuration (P-source and high incident angle, see Fig. 4.6). Figure 4.11 demonstrates DSCs


. 2 -DSC[ ILlm^]

lE -07 J

l E - 0 8

-..-.4=0.24 |Llm

= 0 . 1 8 |Lim

= 0 . 1 2 |Llm

— T —

0 30 60 -90 -60 -30 0 30 60 90


Fig. 4.12. DSC for D = 0.180 fxm conical pit (vertex angle is 90°) of three different diameters: D = 0.12 |xm; D = 0.18 |Lim; D = 0.24 fxm. Incidence is —65°, P-polarized source.

corresponding to different pit diameters (108° vertex angle). Figure 4.12 contains DSCs for the pits of a 90° vertex angle corresponding to the different diameters. So, placing scattered light detectors in the forward (scattering angle ^ 65°), center (near substrate normal) and back (scattering angle ^ —65°) directions provides the data needed to distinguish pits from particles. Many combinations of particle diameter, material and pit dimensions were ana-lyzed. The discriminating approach found works quite well [24]. These examples illustrate the practical use of modelling feature scattering for generation of practical instrumentation of concepts and design optimization.

4.4. Conclusion

Modem state-of-art of the Discrete Sources Method was considered. Although the DSM is a relatively new technique, it has already demonstrated it can compete well with other techniques. From our point of view the outstanding advantages of the DSM are:

(1) Strict mathematical background that includes DS support deposition and a clear scheme of complete system construction. One of the main preferences of the DSM is that there are no limitations to choice of DS support geometry (even for deposition in a complex plane). This has already enabled construction of numerical models for axial symmetries of a scatterer and polarization of an external excitation;

(2) Code reliability results from a simple numerical scheme, reducing the original BVSP to the solution of a set of one dimensional approximating problems enforced at the obstacle meridian, which are solved by a point-matching technique. One of the key advantages of the numerical model is the ability to check the obtained result errors by evaluating the boundary conditions residual;

(3) Flexibility, that allows to proceed quickly from one scattering problem solution to another more complex problem. DSM extension from one particle to a cluster consisting


of several different particles can be mentioned as a typical example for the flexibility of the method;

(4) Fast, by the simplest structure of the DS fields being constructed. Furthermore, DSM codes enable computation of basic polarizations (P/S) and the whole set of external excita-tions at once, without additional computing costs. Even for a substrate defect analysis the DSM code takes just a few seconds with a Pentium-II-350.

We have shown that DSM will be further developed and extended to new classes of scattering problems. Then further advantages will become clear.

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CHAPTER 5

Singularities of Wave Fields and Numerical Methods of Solving the Boundary-Value Problems for Helmholtz Equation

A.G. Kyurkchan

Moscow Technical University of Communication and Information Science Aviamotomaya 8a Moscow 111024, Russia e-mail: [email protected]

A.I. Sukov

Department of Applied Mathematics Moscow State Technological University ''STANKIN" Vadkowski per 3A Moscow 101472, Russia e-mail: [email protected]

and

A.I. Kleev

P.L. Kapitza Institute for Physical Problems Kosygina 2 Moscow 117334, Russia e-mail: kleev @ kapitza. ras. ru


82 A.G. Kyurkchan, A.I. Sukov, and A.I. Kleev

5.1. Introduction

Continuation of the solutions of boundary-value problems beyond the domains of their initial definition is a problem of great interest from both theoretical and practical view-points. Over the last thirty years this problem has been studied with particular intensity. The results obtained so far provide the basis for an appropriate theory. For the greater part, these results can be found in [1,2]. They are of paramount importance specifically in solv-ing both the direct and inverse problems for Helmholtz equation [1]. In this paper we will only discuss the relation between the theory for the continuation of wave fields and the methods for solving direct problems. We will consider only those methods which involve analytical representations of the fields, with special emphasis on the method of auxiliary currents. This is a consequence of the method of auxiliary currents, which best demon-strates the role of information about the singularities of the continuation of wave fields (see below).

The methods of auxiliary currents reduce the solution of, for example, an exterior boundary-value problem for the Helmholtz equation to the Fredholm integral equation of the first kind with a smooth kernel relative to an unknown function which expresses the density of the auxiliary current on a certain (auxiliary) surface situated inside the scatterer. It turns out that the computational algorithms based on the method of auxiliary currents are correct only when this auxiliary surface envelops all the singularities of the continua-tion of a wave field inside the scatterer. Next we will explain what singularities we refer to. The wave field which is a solution of the homogenous Helmholtz equation (of ellip-tical type), is a real analytical function. This function goes to zero at infinity according to the Sommerfeld radiation conditions. Therefore, the wave field must possess singular-ities, since otherwise, as can be readily shown, it would be identical to zero everywhere. It is clear that these singularities must lie inside the scatterer (in the so-called nonphys-ical domain), that is, where the auxiliary surface is located. Ignoring these singularities leads, as will be shown below, to the destruction of computational algorithms used in both the method of auxiliary currents and its most widespread version known as the method of auxihary (discrete) sources. For brevity sake, we will consider only scalar problems, with a closer look at two-dimensional ones, bearing in mind that all the basic results and conclusions are applicable to the vector case as well. The working plan is as follows. First we consider the basic analytical representations of wave fields and define the domains of existance for these representations. We next find the connection between the location of the precisely determined boundaries of these domains and the location of the singularities of the continuation of fields. Further we point the ways for locating the so-called principal singularities of the continuation of wave fields and offer a variety of examples. In conclu-sion, we give some examples of numerical solutions of boundary-value problems taking into account the information about the location of the singularities of the continuation of the solutions.

5.2. Basic analytical representations of wave fields

By a wave field we mean a function U^{r) satisfying the homogenous Helmholtz equa-tion

(A-\-k^)U^=0, (5.1)

5. Singularities of wave fields 83

the condition at infinity, for example, of the kind

hm r(—-^ikU^]=0 (5.2) r ^ o o \ dr )

(k = const is the wave number), as well as the boundary conditions defined at a certain surface S. In this case we shall restrict our consideration to only two types of the surface S, a closed surface bounding a compact body (domain D) as well as an infinite periodic surface. In the latter case the condition at infinity takes, in general, a somewhat different form [3].

If S is the boundary of a compact body, then the wave field U^(r) can be represented in the form of either of the following two expansions [4-6]:

oo n

uHr,0,,p) = J2 E «m«(-i)"+'/!?(^O/'„'"(cos0)e*'«^ (5.3) n=Om=-n

Expansion (5.3) is often referred to as the Rayleigh representation, or the series of wave harmonics (metaharmonic functions). Relation (5.4) is called the Atkinson-Wilcox series.

(2)

Here r,0,(p art the spherical coordinates of the point of observation, hn (kr) is the Hankel spherical function of second kind (of n-th order), and P^ (cos6) is the adjoined Legendre polynomial. For vector fields similar expansions can be found, for example, in [5-7].

It is seen from (5.4) that the asymptotic formula for kr -^ oois

Q-ikr / 1 \

f/'(r,.,,) = - ^ m , ) + 0 ( ^ j , (5.5)

where f (0, (p) = ao{6, (p) is the wave field pattern which can be expanded into a Fourier series as

oo n

f(e,<p) = J2 E «m„P„'"(cos0)e''«<^. (5.6) n=Om=-n

The coefficients amn in (5.6) and (5.3) are the same. Given ao = / , the coefficients ap(0, (f) in (5.4) are expressed from the recurrent relations

1 a p • [p(p - 1) + D]ap-u /7 = 1,2,..., (5.7)

2p

in which

;i2 1 a / a \ 1 a

is the Beltrami operator.

84 A.G. Kyurkchan, A.L Sukov, and A.I. Kleev

Rearranging the terms in (5.3), series (5.4) can be derived in the domain of convergence. Being a power series, it converges uniformly and absolutely in every domain lying outside the sphere

r^n (5.9)

passing through the singular point of the wave field U^(r) which is farthest from the origin. It is clear that the inequality (5.9) also determines the exact boundary of the domain of convergence of the series (5.3). Shown below is one way of finding n before solving the corresponding boundary-value problem.

In the two-dimensional case, the Rayleigh expansion can be represented as follows:

oo

U\r,<p)= J2 ani-irHJ,^\kr)c'"'^, (5.10) n=—oo

where r, ( are the polar coordinates of the point of observation and an are the Fourier coefficients of the wave field pattern f{(p) determined by the asymptotic equality

Series (5.10) is an asymptotical (for \n\ kr) power series. Therefore, it also converges absolutely and uniformly for

r>ri. (5.12)

In the two-dimensional case, the Wilcox expansion is an asymptotic series (see [1,6]). For the interior (inside the domain D) wave fields U^, series expansions in Bessel func-

tions are frequently used. For example, in the two-dimensional case such an expansion has the form

oo

U'ir,cp)^ J2 alMkr)^"'^. (5.13) n = — O O

Series (5.13) converges absolutely and uniformely in the domain [8]

r<re, (5.14)

where r is the distance to the nearest-to-the-origin singularity of the continuation of the field U^r) beyond the domain D. An approach to determining r is also explained be-low. Similar expansions take place in the three-dimensional and vector cases where exact boundaries of the domains of convergence are also represented by (5.14).

Another widely used method of the analytical representation of wave fields involves the plane wave integrals, or Sommerfeld-Weil representations, which have the form

0 0

X exp{—i/:r[sin^sinacos((^ — P) +cos^cosa]} sinadofd^S. (5.15)


In the vector case, the representation is similar. Integral (5.15) (as well as the similar inte-gral in the vector case) converges absolutely and uniformly in any closed domain lying in the half-space [10]

z>Zi, (5.16)

where zi is the z-coordinate of the singular point of the wave field U (r) farthest from the plane xy (see below). For two-dimensional wave fields one has

TT+i00

U\r,(p) = - j f(xlf)cxp[-ikrcos((p-ir)]dif. (5.17)

—ioo

For representation (5.17), the exact boundary of the existence domain is defined by the inequality

y>yi, (5.18)

where yi is the ordinate of the singular point of the field U^(r) farthest from the axis x. By revolving the coordinate system, it is possible using (5.15) and (5.17) to continue

the wave field into the domain Z) up to a convex hull B enveloping the singularities of the wave field [11]. Papers [12,13] suggest two modifications of the above-mentioned plane-wave representations, which converge in the domains R^\B and R^\B, respectively. For example, in the two-dimensional case such a representation has the form

2 -|-ioo

lj^(r,(p) = — I f((p-\-\l/)Qxp(-ikrcos\l/)d\l/. (5.19) TT J

The problems of wave scattering by periodic structures rely heavily on the wave field representation as a series in terms of plane waves (Rayleigh series). In the two-dimensional case such a representation has the form [14]:

rji, ^ 2 V^ / exp(-iw;„x)exp(-ii;„3;) U (x,y) = - > go(Wn) . (5.20)

Here b is the period of the structure described by the equation

In y = h(x) =h(x-\-b), V0n = — n + k^inO,

b

Vn = yjk^ - wl, or Vn = -i^W^ - P (5.21)

for \wn\> k, 0 is the angle of incident of the initial plane wave, and goiu^n) is the pattern of the central period [14]

b/2

gQ(u)) = — / f/x V J exp[iw;jc + ivh(x)]y/l + h'^(x)dx, (5.22)

-b/2


where

V = V^^ — w^, Imf ^ 0,

M = {U^-^U\, v = — (f/^ + ^^) (5.23)

are the densities of the potentials of the double and simple layers (magnetic and electric currents), respectively; L^^(r) is the incident field.

The coefficients of m-th spectral orders are related to goiv^m) through the following relation

R.4'-^^. (5.24)

Series (5.20) as well as integral (5.17) converge absolutely and uniformly for y > yt. In the three-dimensional case the representation is in perfect analogy to the two-

dimensional case (see, for example, [15]). The boundary of the domain of convergence is defined by the inequality (5.16).

Now we will turn our attention to the representation of wave fields through the wave potentials

U\r) = J\fi{rs) ^ ^ ^ - v(r,)G(r, r , ) l d.. (5.25)

s

Here r is the radius-vector of the point of observation; r is the radius-vector of the point of integration situated on a certain, fairly smooth surface S (a contour in the two-dimensional case); d/dris denotes the differentiation in the direction of the normal exterior to the surface 5; /x and v are the densities of the potentials of the double and simple layers, respectively; and G(r, r^) is Green's function. In the three-dimensional case we have

exp( - i / : | r - r , | ) G(r, Ts) = —— •—, (5.26)

4 7 r | r - r , |

in the two-dimensional case

G{r,rs) = ^^H^^\k\v-rs\), (5.27)

where HQ is the Hankel function of the second kind of order zero. Finally, in the two-dimensional problems of wave scattering by a periodic boundary we

have

1 ^ 1 G(r,rs) = — ^ — exp[-iwnix - Xs) - ivnly - ys\]^ (5.28)

^^^ n=-oo ^"

Representations similar in the sense hold good also in the vector case (see, for exam-ple, [5,15]) as well as for the fields U^ (r) interior to the surface S (in the domain D).


Deforming the potential density carrier S toward the interior of the domain D (or into the exterior domain R^\D, where n = 2,3, and D = D V S) WQ can analytically continue the wave field into the domain D (correspondingly, into R^\D), and it is evident that such a continuation is possible if and only if a closed surface E resulting from the deformation of the surface S encompasses all the singularities of the continuation of the wave field into the domain D (correspondingly, into R^\D).

In this situation, if the surface E (contour in the two-dimensional case) is not resonant, i.e., k^ is not an eigenvalue of corresponding interior Dirichlet's or Neumann's problem, then the wave field Hr) can be represented just as a potential of the single layer

/ t / ' ( r) = - / / ( rE)G(r , rs)d5, (5.29)

or, correspondingly, the double layer

'•(r) = / U\r)= I K(r^)^-^p^ds. (5.30)

Representations (5.29) and (5.30) are just the ones that form the foundation of the method of auxiliary currents [11,16].

5.3. Singularities of a wave field and their localization

So the relations (5.3), (5.4), (5.10), (5.13), (5.15), (5.17), (5.19), (5.20), (5.25) allow the wave fields to be analytically continued beyond the domains of their initial definition. The boundaries of the domains, into which such a continuation can be realized, pass through the singular points of the wave fields (their analytical continuations). Therefore, in order to use these representations correctly, it is necessary to localize the singularities of the con-tinuation of the wave fields apriori, i.e., before solving a corresponding boundary-value problem. A fairly full solution of this complicated problem can be found in [1,2,17] and some other works. Here we will mention relations which can be used for localizing so-called primary [17], or principal, singularities. In most cases the information about their localization is adequate for the correct usage of the above-mentioned analytical represen-tations.

To this end, we consider the wave field pattern

i r[ 9 1 , . = T / y(rs) - l^irs) — expji^r, cos(V^ - ^ , ) | d . (5.31) fW-

As already noted, we consider in more detail the two-dimensional case and mention only final results regarding the three-dimensional case.

It is readily seen that the pattern /(V^), as an entire function, is continuable into the whole of the complex plane xj/ =a -\-ip [12]. Here, if |y6| ^ oc, we can use the following asymptotic equalities:

f{a ± il^l) = f^(w^) + O(e-I^l), (5.32)


where f^(wz^) are the entire functions of exponential type of variables [1,12]

w;^=exp(|^|Tic^). (5.33)

For brevity sake, we consider the case when /^(r^) = 0 (the kind of boundary conditions on S does not affect the results). In this case, using the polar coordinates, we can write

fîw^) = fq(cp) e x p | ^ ^ ^ w^ e^^^j dcp, (5.34)

0

where

qi<P) = \ 9M p^((p) du dr p((p) dip

(5.35)

U = U^ -\- U^ is the total field, and r = p((p) is the equation of the boundary of the sur-face S.

Let us introduce the following quantities [4]:

a ± = 1 5 r ( 4 l n max | / f ( i t ;±) | l . (5.36)

The quantity ai = max(a4-, G-) will be called the power of the wave field pattern. We also define GS as

Gs = max{/z+(0), /z-(7r)}, (5.37)

where h±{y) are the indicatrices of growth of the entire functions f±(w±) [4]

/i±()/) = l l n r - l n | / f (/ê^^)|, 0 ^ y < 27r. (5.38) Rôo /? ' '

Therefore, in order to determine ai and a^, it is necessary to estimate f^(w±) for \w±\-^ oo. The way to do this is through the asymptotic estimation of corresponding inte-grals of type (5.34). This estimation can be performed, for example, using the saddle-point method.

As it can be seen from (5.34), the saddle points satisfy the following equations:

[p^(^)Tip(^)]e^^^ = 0. (5.39)

It is evident that the roots cp^ of (5.39) are complex. Therefore, the estimation involves the deformation of the initial contours in (5.34) complexifying (p. Realizing the deforma-tion we may come across singularities of the integrands in (5.34). Their singularities are, first of all, the singular points of (p±(^±) inverse to ^± = p{(p) e * [8], whose coordinates can also be determined from (5.39), as well as the singular points of ^(^) appearing when q((p) is continued to the domain of complex-valued. The question is, primarily, about the


singularities of the boundary value of U^(r) contained in q((p) when U^(r) is continued to the domain of the complex values of cp.

These singularities will be at the points corresponding to the "images" of the primary field sources, passing there along the complex characteristics [1,2]. The coordinates of the images can be determined using the Riemann-Schwarz symmetry principle [18]. That is to say, if the source coordinate is zi (for example, on the complex plane z+ = x + iy) and the image coordinate is z*, then we can write

+ 7* 71 — 7* \ ' * 0, (5.40)

< ^ 2i

where F(x, y) = 0 is the equation of the contour S, and z* is the complex conjugate of z*. Polar coordinates of a point source image (for example, in the plane z+) can be deter-

mined from the formula [8]

C;exp[-i2(^+(?;)] = r i e - i^ i , (5.41)

where ri, cpi are the polar coordinates of the source and (p+(^^) is the function defined above. For example, in the case of the circle p((p)=a

^^(cp)=ac^^, (5.42)

therefore

exp[- i2^+(f ; ) ] = ^ , (5.43)

thus, for the image, we have

from which

C = - - , (5.45)

i.e.,

n

^ = r i e - ^ ^ i , (5.44)

r * ^ | ^ ; | = - ; (^*^arg^; = (^i. (5.46)

Thus, the image of a point source with respect to a circle is obtained by transforming the inversion relative to this circle [18]. An example of determining the image relative to an ellipse is given in [8].

Keeping in mind relations (5.36)-(5.41) we can now determine ai and cxs directly using the equation of the boundary of the contour S and the coordinates of the source [1,8,12]

a ^ = max 2

(5.47)


a/' = max Re ,f I 2 exp .(.j-i)]]. TH^^-W) \^ (5.48)

where cp^ are the roots of (5.39). The maxima in (5.47) and (5.48) are to be found among those roots of (5.39) which are in correspondence with the points inside C=p when the func-tions ^^ = p (cp) Q^^^ transformating map the contour S into the contours dp on the planes

Next, we have

(7^ = m a x i a { c r ^ . ^ j , a / = m a x { a / , ^ j , (5.49)

where the coordinates (x*, y*) or (r*, ^*) of a source image are determined from (5.40) or (5.41). If the boundary of the contour S contains nonanalytical points, then the maxima in (5.47) and (5.48) should be looked for among these points as well.

The quantity a2 can be found from a relation similar to (34). That is, we can write [19]

72 = m m ^0

2 (5.50)

Here the minimum is looked for among roots of (5.39) which correspond to the points on the planes z± outside the contours C± when the mapping functions f± = p{(p)d^^^ are realized.

Quantities defined by (5.47)-(5.50) are marked with P because the primary singularities of the wave field are taken into account.

The boundaries of S satisfying the inequality

a^>G^, (5.51)

shall be called weakly nonconvex boundaries [19]. In particular, all the convex boundaries are weakly nonconvex boundaries.

In the case when 5 is a nonconvex boundary the primary singularities also "induce" other singular points [17] which makes it is difficult to localize. The method of their local-ization is, in general, a far-reaching generalization of the mirror image method [1,2]. If the boundary of 5 is a weakly nonconvex boundary, then [1,2]

a\ = a[, cr2 = cr^, a^ = o-f. (5.52)

It can be shown [1,8,10], that

2^1 2(72 2^5

'•'•^X' ''^^-F' ''^ir- ^^-^^^

Knowing as for an arbitrary orientation of the coordinate system, the convex envelope B of the singularities of a wave field can be found [8,11,16]. The set B determined from or/ will be denoted B^ .


In three-dimensional case, we have

a^ = max Oo,(po

^ ^ ^ ^ e ^ ^ o l (5.54)

erf = max Re] e ^ }, (5.55)

af = mm '2 Oo,(Po

I^PiOo^îOol (5.56)

where Y = p(0,(p) h the equation of the boundary of S in spherical coordinates and ô, <o are the roots of the equations

[p'oiO, cp) + ip(0, cp)] ^^ = 0, p;(^, ip) = 0. (5.57)

The maxima in (5.54) and (5.55) are looked for among those roots of (5.57) which on reahzing ^ = p{0, (p)^}^ turn out to be inside the contours C^ representing the section of the surface S by the plane ((p,(p -\-7t) mapped onto the plane z = r e^ . Similarly, the minimum in (5.56) is looked for among those roots of (5.57) which are mapped into points outside contours C(p.

Finally, in case of plane wave scattering by the periodic boundary y = h(x) WQ have

af = - maxRe\kh(xo) + i5'/:xo|, (5.58) 2 0,"

where XQ are the roots of the equations

h\x) = -is, s = ±l. (5.59)

Consider some examples of localizing the primary singularities of a wave field when the primary field U^ is a plane wave. In this case the source is at infinity and its image is in the origin of coordinates. Therefore, af = a^, af = af (see Eq. (5.49)).

1. Circle p((p) =a. In this case (5.39) takes the form (we consider only the function /+(u;+) since consid-

ering f-(w-) gives nothing new)

e^ôÔ, (5.60)

thus ^(ô) = 0, therefore ai = 0, (72 = oo, since there are no singular points generated by the roots of (5.39) outside the circle.

2. Ellipse

p(cp) = --=J======., (5.61) yi s^ cos^ (p

where b is the semiminor axis and s is the eccentricity. In this case (5.39) are reduced to

8^ cos(ôsin(ô - i(l - £^cos^cpo) = 0, e ' o = 0. (5.62)


Latter equations were already considered. The first equation has the roots

e^^o^ ±^(2-^2)1/2^

Substituting there roots into (5.47) and (5.48) we obtain

kb s kas a\ = 2 V T ^ 2 ' ^ . = 0 , (5.63)

where a is the semimajor axis of the elUpse. Again, as in the circle case, there are no points corresponding to the roots of (5.39)

outside the contour C which represents the ellipse mapped by the function f {(p). Therefore, here we also have (72 = oo.

Here, as it is readily seen, the set B is the segment [—as, as], i.e., an interfocal segment. 3. Multifoil

p{(p) = a ( l + r cos^(^), 0 < r < 1, ^ = 1,2, 3 , . . . (5.64)

The roots of the equation t^^ = 0 are mapped in the plane z by the function ^ = p((p) c^^ into a point at infinity (except the case when q = 1, which requires further consideration). Let us introduce the substitution Q^^ = t. Cutting the plane t through t = 0 and r = oo we obtain an one-to-one mapping (that is, we isolate the strips of width 2n on the plane cp). The equation

p'((p)-\-ip((p)=0

after the substitution takes the form

l + ^ ( l+^ ) f^ + ^ ( l - ^ ) / - ^ = 0

(5.65)

(5.66)

and has the solutions [20]

- _ l + (l + r2(^2_i)) i /2ni/^

[ riq +1) m = 0 , 1 , . . . , ^ — 1,

tin r i + (l + T ^ g ^ - l ) ) ' / ^ l ' / ^ /.(2m + l ) j r \

[ riq + l) J ^""^y q ) ' m = 0 , 1 , . . . , ^ — 1.

By applying

^=a ^ +1(^^+1+r^+i)l

the roots are mapped in the plane z+ into the points

Z\m = 2af - - ^ e x p

k

' /n 2mix\~ ?

(5.67)

(5.68)

(5.69)

Zln la^

k m = 0, 1

exp :<i

5. Singularities of wave fields

(2m + l)7r^ + •

^ ,q-\, q = 2,3,...

93

(5.70)

where

a, =

al =

ka q[q + {l + THq^-l}y^^]

q^-l

ka q[q-il + T^iq^-l)y/^]

2 q^-l

r(^ + 1) J '

(5.71)

Notice that the points zi™ he inside C+ and the points Z2m lie outside C+. Substitut-ing (5.70) into (5.47) and (5.50) allows the quantities of and af to be expressed by (5.71), and af can be represented, using (5.48), by the formula [8,20]:

cr, =CT, max 2m jr

sm- m = 0 , 1 , . . . , ^ — 1, q = 2,3, (5.72)

When ^ = 1, the function ^ — p{<p) e" maps the roots of the equation e"'' = 0 in the plane z+ into the point zi = a r / 2 and the roots of the equation p'{<p) + ipi<p) — 0 into the point Z2 = ~ ^ ( 1 ~ ^•^). Thus, for q = 1,

p KCIX p KCl 9

af = , (79 = — (1 - r ), - ; = o . (5.73)

The contour S turns out to be a weakly nonconvex one when [21]

r < 0,22145 . . . (^ = 3), etc. (5.74) ^ < y r (for^^r^l), r <-(q = 2),

For the figure under consideration, the set B^ represents [20] a regular convex ^-angle figure (except the case q = 1, when this set degenerates into a point) with vertices cpm = Imn/q {m = 0, I,..., q — I) for r = 2af /k.

4. Triaxial ellipsoid

Once the roots of the equations (5.57) [22] are found, we have

p(0o,^o)e'^» = V'c2-a2.

Thus we obtain

(Tl = , (7^=0.

(5.75)

(5.76)

(5.77)

Aligning the axis z with the middle (in size) semiaxis we will find that a = A:\/fc23c2/2, and aligning with the semiminor axis yields as = k\la^ — c'^/2. Doing in


such a way (changing orientation of the z-axis) we find that here the convex envelope of the singularities is an ellipse with the semiaxes Vfl^ — c^ and V^^ — c^ lying in the plane xy.

5. Sinusoid [23]

h{x)=acospx, p-In

(5.78)

The roots of (5.59) (for s — —\ since the case of ^ = +1 does not give new roots) are

(5.79) Xn = — In i^ 1 + Vi + a V

Xn = In i l + V T T o V , ^

ap ap i h - .

Using the function t, = exp{ip[x + i/i(jc)]}, we can map these roots in the complex plane z = cxp[ip(x + iy)] into the points

1 _ Hhj /T+^V - V l ^ ^ a/7

20 = - (

1 + y i + a V / I ^

ap (5.80)

Note that the function f (jc) maps one period of the curve h(x) into a certain closed con-tour C in the plane z, and the points lying above /z(x) are mapped inside C.

Now, from (5.80) we have

x\q = bq, ^ = 0 , dbl ,±2, . . . ,

>1 = +« 1 ^ i+yi+«v yrr^v

m ^ fl/7 ap <2/7

X2q = - ( 2 ^ + 1), ^ = 0 , ± 1 , ± 2 , . . . , y2 = - J l .

Using (5.58) we obtain

ka

'•=Y

y i + « V 1 1 + y i + a V In

ap ap ap

(5.81)

(5.82)

(5.83)

Similarly, defining cr2 for a periodic surface as

a2 = - minRe{/:/z(jco) + i /:A:o}, (5.84)

where the minimum is determined using those roots of (5.59) to which the points inside the contour C on the plane z correspond when the mapping

f =exp{i/7[x + i/i(A:)]}

takes place, we obtain

oTo = - o r . (5.85)


5.4. Utilization of the information about wave field singularities when solving the boundary-value problems for the Helmholtz equation

Discussing this question we confine ourselves primarily to the methods of auxiliary cur-rents and sources, as well as to the method of adaptive collocation. Other methods are considered in [24,25].

As was already noted, the method of auxiliary currents is based on the representations of a wave field using wave potentials (of type (5.29), (5.30)).

Let us, for definiteness sake, solve the uniform boundary-value Dirichlet problem

AU^+k^U^ = 0, (5.86)

(U^ + U^)\^ = 0, (5.87)

where S is the boundary of the scatterer and U^ is the incident field. At infinity, for the U^ condition (5.2) is satisfied. Using representations (5.29) this problem can be reduced to solving the following integral equation of first kind with smooth kernel [11,16]:

/ /(rE)G(r; r^) d = u\r), r e 5, (5.88)

where E is the auxiliary closed surface (contour) lying in the domain D. Let the surface E be such that k^ is not the eigenvalue of the uniform interior Dirichlet

problem for the domain inside (in this case it is said that the surface is nonresonant). Then equation (5.88) is solvable if and only if the D envelopes all the singularities of the wave field inside the domain D [2,11,16,26].

Indeed, if (5.88) is solvable, that is, the integrable function /(r^:) such that (5.88) is realizable exists, then, on the strength of properties of the function G(r; r^) , we can assert that function (5.29) satisfies (5.1) everywhere outside E and, therefore, has no singularities.

Conversely, let E envelope all the singularities of the wave field inside D, i.e., the field U^{r) is continuable into D up to E. Let U^ (r) be solution of the interior (regarding^the domain inside E) Dirichlet problem with a given U^(r) on E. Then the function U{r) which is equal to U^(r) outside E and U^ (r) inside E, is continuous everywhere in the space and its normal (to E) derivative undergoes a jump when going through E. We can take this jump as / (r^).

The method of auxiliary (or discrete) sources is essentially the discrete version of the method of auxiUary currents when the integral on the left of (5.88) is substituted with the sum according to the rectangle formula, and the equality of the left- and right-hand sides is satisfied at the discrete points (collocation points). In the method of auxiliary sources, the sources form a discrete set lying on a certain imaginary surface E. When increas-ing (for more exact calculations) the number of sources, they are arranged on E more densely. Therefore, in this case, it is also necessary that E envelope the singularities of the field f/Ur).

5.4.7. Method of auxiliary currents and computational modelling of wave fields

In the following, are examples of some approaches to direct diffraction problems. We will see that the singularities of the continuation of a wave field are fundamental to realizing computational algorithms based on the method of auxiliary currents.


Fig. 5.1. Geometry of the problem.

Let us analyse the pattern of the plane monochromatic electromagnetic wave

(7o = e~ ^ - ~ ^ ^ (5.89)

scattered by an ideally conducting infinite cylinder whose generatrix is parallel to the axis z and directrix S is given in the parametric form (z = 0):

x=x{t), y = y{t), rG[0;27r].

Here ifo = A:sin^o. o = A:cos o» k = In/k, X is the wavelength, and ^o is the angle of incidence (see Fig. 5.1).

Assume that the electric field vector is parallel with the generatrix and perpendicular to the plane of incidence.

Let E be the directrix of auxiliary cylinder whose generatrix is parallel with the axis z. Assume that it is given in the parametric form (z = 0)

x = x o ( 0 , y = yo(t'), r'G[0;27r], (5.90)

lies inside S (see Fig. 5.1) and satisfies all the conditions which are imposed on S. Then, if all the singularities of the wave field lie inside a domain interior to E and k^ is not an eigenvalue of the corresponding uniform Dirichlet problem for this domain, the integral equation (5.88) is valid and can be written in the form

27r

f H^^^[kR(t, t')]l(t') dt' = Q-^[mx{t)-voy{t)]^ (5.91)

where

R(t, t') = ^J[x{t)-xo{t')f + [y{t)-yoit')f, (5.92)


Here v(rO is the density of the current on the surface of the auxihary cyUnder (aux-ihary current density). In this case the expression for the scattering pattern in the polar coordinates (r, cp) (see Fig. 5.1) is given by

In

/(^) 0

Here ro(t^) and (^o(^) ^ ^ the polar coordinates of the points belonging to E. Let us divide the interval [0; 2n] by N equal parts and replace the integrals in (5.91)

and (5.94) with elementary sums in the sense of Riemann using the values of the integrands at the points t^ = t^ = 2nm/N, m = 1,2, ...,N. Requiring the equality of the left and right sides of (5.91) at the collocation points t = tn = 27tn/N, n = 1,2,..., N, WQ can reduce the initial integral equation to the system of linear algebraic equations with the square matrix of order N for unknown quantities J^ = 27TI (f^)/N as follows:

J2 K\kR{tn. t'j] Jm - c-'^^Ox{tn)-voyitn)] ^ Q. (5.95)

m=l

Once Jm have been determined, we can calculate the scattering pattern (5.94) from the expression

f((p) =Y^Jm Q^kro{C)cos[cpo{C)-<p]^ ^39^^

This simplest approach to solving the integral equation (5.91) essentially leads to the method of auxiliary (or discrete) sources and does not take into account the fact that the auxiliary current density is continuous.

Suppose that the approximation of the auxiliary current density in the interval [t^^_^;t^] is linear. Then we can write

/(O = ^ [new - 4-i) + nC-iWm -1% m = i, 2,..., iv, (5.97)

where /(r^) = / ( r^) . If, taking into account (5.97), we replace the integral in (5.91) with the corresponding integral sum, and then for the determination of I(t^), we will have, instead of the system (5.95), the system of linear algebraic equations as follows:

^(C^.,m +)^.,m+l)/(C) -e-^f"«"^^"^-^«^^^"^^ = 0 , m=0

n=0,l,...,N -I, (5.98)


where

N Oin,m — T

Z7t J H^^\kR{tn,t')]it'-C_i)dt', an,o = otn,N\ (5.99)

m-\

c Pn.m = ^ f H^^\kR{tn,t')]{C-t')dt\ (5.100)

By substituting the values of /(f^) and the relations (5.97) into (5.94), we obtain the expression for the scattering pattern in the form

This approximation takes into account the continuity but does not ensure the smoothness of the auxiliary current density at the points ^ == ?^.

Now, in the interval [^^_i; ^^] , we consider the quadratic spline-approximation of the auxiliary current density

+ a4t'-C_i){t'-C), m = l,2,...,N, (5.102)

which ensures the continuity of the first derivatives of /(?') at the nodes t' — t^. Therefore, the coefficients a^ satisfy the relations

«m +am+l = (^^^ [/(4+l) - 2/(4) + /(4-l)] .

« i + a N = ( ' ^ j [I{t[)-2l{t^) + l{t'f^_i)l m = l , 2 , . . . , A r - l , (5.103)

and the integral equation (5.91) can be represented in the form of the system of linear algebraic equations for unknown / ( 4 )

N-l

m=0

t' N ^

w = 0 , 1 , . . . , A ^ - 1 . (5.104)

In this case the scattering pattern can be calculated from (5.101) where the expressions for I{t') are chosen according to (5.102).


It can be noted that the solution of the system of algebraic equations (5.104) is essentially nothing but a perturbed solution of the system (5.98). Therefore, for sufficiently large A' , we can expect that the corresponding values of I(t^) for both systems will practically coincide.

Taking into account the above relations, let us calculate the scattering patterns corre-sponding to ideally conducting infinite cylinders with various cross-sections.

1. Elliptical cylinder. In this case the directrix S is given in the form

x{t)=acost, y(t) = bsmt, tG[0;27t]. (5.105)

As an auxiliary surface we choose the cylindrical surface characterized by E which can be represented as

xo(t^) =aocost^ yo(r^) =bosint\ t' e [0; 2n]. (5.106)

The case when the angle of incidence ^o = 7t/4, ka = 3, and kb = 1.2, can be con-sidered as a step of the computational experiment procedure without loss in generality. The quantity kao is varied for kbo = 1.08. According to the above results, the directrix E of the auxiliary cylinder must enclose the interfocal interval (2kf = l^{}id)^ — (kb)^ = 2 X 2.7495 . . . , 2 / is the interfocal distance) where the singularities of the analytical con-tinuation of the wave field are concentrated. The limiting case considered is reduced to the inequality kao > kf. The computational experiments show that, for kao € [2.73; 2.77] (i.e., when kao S ^ / ) ^^^ N = 10 and 90, the scattering patterns lie down on the curve of Fig. 5.2 with a graphic accuracy for all kinds of parameter combinations and various approaches to the realization of the method of auxiliary currents. Therefore, in deciding on an auxiliary contour E, the key role of the singularities of the analytical continuation of a wave field is open to question.

I I I

90 180 270 q> [deg]

360

Fig. 5.2. Scattering pattern of elliptical cylinder.

100 A,G. Kyurkchan, AJ. Sukov, and A.I. Kleev

ka(

Fig. 5.3. Dependence of disparity 8 on kaQ for A = 70.

no -,

80-

60-

40-

20-

0-

5

\ { E q . 9 5 )

\

\

( E q . 9 8 ) , ( E q . l 0 4 ) ^ v,, ^ ^ ^

2 , 7 3 2 , 7 4 2 , 7 5 2 , 7 6 2 , 7 7

kao

Fig. 5.4. Dependence of disparity 8 on kuQ for N ~ 90.

It is to be noted that the reliabihty of the results obtained depends primarily on the ac-curate compliance with the boundary condition between the collocation points. Therefore, as a measure of accuracy for the resulting numerical solution, we choose the difference h between the left and right sides of (5.95), (5.98), (5.104) to be maximal in magnitude, if t is varied from 0 to 27r. The dependence of h on A:ao for various N and different approaches to the realization of the method of auxiliary currents can be seen in Figs. 5.3 and 5.4.

It is readily seen that the value of A:ao = /::/ is a threshold, i.e., if /cao > kf then the solution is stable, and if A;ao < kf, the solution is unstable. The larger the iV, the more


pronounced is the threshold effect with the worst results corresponding to the method of auxiliary sources as compared with the results obtained from (5.98) or (5.104).

The behaviour of the results in Figs. 5.3 and 5.4 is invariant to the choice of the contour, only 8 varies.

2. Cylinder with the cross-section in the form of a multifoil. In this case the equation for the directrix S is more conveniently expressed in the polar coordinates:

r((p) = a(l + r cosqcp), (5.107)

where a > 0, r G (0; 1), and ^ € A . As the auxiliary surface we choose a cylindrical surface whose E can be represented as

ro((p) =ao(l -^TQCOsqip), (5.108)

where (see (5.71) and (5.73))

l + to

bo=--^, (5.110) k

c = ^[a(l^T)-bo]. (5.111)

To ensure that the method of auxiliary currents is correctly used, it is necessary for the values of p to be in interval [0; 1] since, if S ^ 1, the directrices E and S interchange their position, and if y6 < 0, the singularities of the continuation of a wave field do not lie inside E. Consider the case when ^o = ^ / 2 ; ka = 6;r = 0.2; and q =4 (quarterfoil). fi is varied and ro is supposed to be equal to 0.2.

The computational experiments show that, for fi e [—0.3,0.3] and A = 90, the scat-tering patterns corresponding to various approaches to realizing the method of auxiliary currents lie on the curve of Fig. 5.5 with a graphic accuracy.

At the same time, the behaviour of 5 as a function of ^ (see Fig. 5.6) is unambiguously indicative of the necessity to take into account the arrangement of the singularities of the analytical continuation of a wave field when choosing the auxiliary surface. As in the case of an elliptic cylinder, the threshold effect is most significant when the method of auxiliary sources is realized.

3. Cylindrical bodies with an unlimited cross-section. In the examples above we have considered cylindrical bodies with a limited cross-

section. In [27] the directrix S is given as

y = acosx, (5.112)

i.e., it is considered as a cylindrical surface whose cross-section has an unlimited cosine form. As before, the results obtained imply that the singularities of the analytical continua-tion of a wave field must be localized if we use various versions of the method of auxiliary currents, just as, incidentally, other methods utilizing the concept of the analytical continu-ation of the solution, in particular, the methods based on various analytical representations of a wave field.


f ( 9 ) I

(p [deg]

Fig. 5.5. Scattering pattern of multifoil.

- 0 , 3 - 0 , 2 - 0 , 1 0 0 , 1 0 ,2 0 , 3

Fig. 5.6. Dependence of disparity 6 on ^ for multifoil.

5.4.2. Adaptive collocation technique

Among various methods widely used for solving boundary-value problems for the Helmholtz equation we give prominence to the methods based on the representation of wave fields by series in terms of metaharmonic functions (5.3), (5.10), (5.13), (5.20), and others. As has already been mentioned, for these representations the boundaries of the do-mains of convergence are defined by the location of the singular points of the analytical continuation of a wave field. This fact is of fundamental importance if the methods under consideration are used for numerical calculations.


By way of example, let us consider the problem of diffraction by a periodic surface. Besides being of interest from the point of view of practical applications, this problem is the key to a wide variety of problems associated with diffraction. There is need for the de-velopment of simple methods allowing the diffraction characteristics of periodic structures to be effectively calculated. The most simple and physically adequate method is based on the expansion of a scattered field in terms of outgoing plane waves (metaharmonic func-tions) (5.20). Such an expansion, as well as the method for calculation of the coefficients, was proposed as long ago as by Rayleigh in his classical work [28]. More recently, the Barantsev-MMM method was reported [29,30], and it has been shown in [31] that, in the case of a plane wave normally incident upon a symmetrical structure, both the Rayleigh method and the Barantsev-MMM method give the identical systems of linear equations for the expansion coefficients. It is necessary to note that the Rayleigh method provides for the matrix elements to be determined by quadratures, which is, with the exception of some special cases, wasteful of computation.

The collocation method allows us to compute the coefficients of a scattered field much easier. This method can be effectively used for the development of application packages with minimal requirements placed upon computing facilities. However, like the Rayleigh method, the collocation method is comparatively simple only if the depth of a structure is small comparing with the period. In order to obtain the convergent algorithm for the analysis of deep structures, authors of [32] propose the number of collocation points to be significantly (by several times) increased as compared with the number of metaharmonic functions, and the rms residual of the boundary condition to be minimized. Later we will consider an alternative, more efficient approach called the adaptive collocation method. This method is based on the rational distribution of the collocation points rather than on their multiplication. In this case, as will be shown in the following, optimal choice of the collocation points is dictated by the location of the singular points of a scattered field.

It should be recognized that the basic concept of the adaptive collocation method can readily be used for solving the problems of diffraction by a smooth cylinder of arbitrary cross section.

It is important to keep in mind that the usefulness of solving any diffraction problem sig-nificantly depends upon the amount of computations performed with the aim to determine the diffraction coefficients. The adaptive collocation method [33] allows the diffraction coefficients to be computed at minimal cost. At the same time, the adaptive collocation method makes it possible to calculate the scattering by various periodic structures with a smooth profile over a fairly wide range frequencies.

The geometry of the key problem and the coordinate system are shown in Fig. 5.7. We consider the case of //-polarization when the total electromagnetic field is determined by the z-component of magnetic field. For the primary field exciting the periodic structure, this component can be written in the form:

H^ = exp{-i[27Tr}psinO - p^ cosO]}. (5.113)

Here we introduce the new dimensionless coordinates

b b


Fig. 5.7. Geometry of the problem and the coordinate system.

and the parameter

b

The total magnetic field above the corrugated surface

? = V ( ) = f{r] + «), n = 1, 2 , . . . ,

can be represented as the sum of primary and secondary fields

(5.115)

(5.116)

(5.117)

The secondary field H^ can be approximated by linear combination of the form (5.20), which can be written, in view of the notation introduced, as

No-\

where

Hi = J2 ^nexp{-i[27rr/y„ + ^i>j}, n=-Ng-\-\

yn=psmO-\- n, i^n = ^JP^ - Yn^ Imi^n < 0.

(5.118)

(5.119)

The number of terms (2Ng — 1) in (5.118) can be chosen according to the required accuracy of the solution.

In case of an ideally conducting corrugated surface the boundary condition can be writ-ten in the form

[nE]=0. (5.120)

If we substitute the fields computed by (5.113), (5.117), and (5.118) into (5.120) and equate the left- and right-hand sides at the collocation points rjm, we will obtain the system


of equations for determining unknown coefficients Rn

r(rim) In

In

Yn -T^n exip{-i[2nrjmgn -\- T^n^(r]m)]}

psinO + p cos 0

X exp{—i[27r?7^psin^ — pcos^V^(y/m)]}-

Later we will give numerical results for the sinusoidal corrugation

\j/(ri) =a(l — coslnr]),

(5.121)

(5.122)

where a = nl/b (see Fig. 5.7). It is well known (see, for example, [33,34]) that the convergence of a collocation algo-

rithm essentially depends on the distribution of collocation points. According to [34], the collocation method furnishes an adequate result when the limit (for Ng -^ oo) density of the collocation nodes a(ri) satisfies the relation

1

/ In p (r;, r]^)a (rj^) dr]^ = const, (5.123)

where p(r], r]') is the distance between the point of observation and the point of integration on contour C.

If we solve (5.123) for a particular contour we can find the coordinates Y]J of the collo-cation nodes from the equation

j = (2Ng - 1) / o(r])dr][ \ cr(r])dr] (5.124)

It must be emphasized that the convergence of an interpolation process is closely related to the location of the singular points of the analytical continuation of a scattered field with respect to the auxiliary contour C which is the boundary of the set Dc. Taking into account the previously mentioned analogy to polynomial interpolation, we can describe the conver-gence of the interpolation process in a simple and convenient manner. Consider a cylinder with a cross-section C. Let an electric charge whose surface density is proportional to the limit density of the distribution of the collocation nodes be placed on this cylinder. The convergence of the interpolation process requires that the singular point of the analytical continuation of the scattered field be outside the equipotential C of a given system of charges, which encompasses contour C. In fact, let D / be a set of points in the complex plane z having the boundary C^ It is easy to show that in this case

lim

\

\2Ng-l

sup 7 = 1

- Z j ) (5.125)


where c is the content of the set Dc' [31]. Therefore, as it is shown in [35-37], the interpo-lation process converges in Dc' uniformly for this system of nodes. The rate of convergence depends on the distance between the singular point and the equipotential. The equipoten-tials for an adaptive distribution of collocation nodes coincide with equipotentials of the charged metallic cylinder and, therefore, the regularity in Dc suffices to converge the in-terpolation process.

It should be pointed out that alternative approaches can also be used for the construc-tion of the system of collocation nodes. These approaches, however, yield the density of collocation nodes which is identical to the solution of (5.123) for Ng-^ ooAn particular, the distribution of zeroes of the Szego polynomials for contour C which is the map of the period of h{x) onto the plane z = exp[i/?(x + iy)] satisfies this condition (see the text be-fore (5.80)) [35-37]. Generally, the construction of a Szego polynomial system followed by the computing of the zeroes is a time-consuming procedure. The computational burden can be significantly reduced if we use an approximate rather than an exact distribution.

According to the computation results, the following relation conveniently approximates the exact solution of (5.124)

1 r]j = - arctg

n

1-6 i'^) (5.126)

Here parameter e depends on the parameters of a corrugated structure and is defined by parameter a (introduced earlier), and the quantity is, in fact, a function of this parameter, that is s = £(a). For 0 <a < amax ~ 0.9, we can use the polynomial approximation

6(a) = B\a + B2a^ + 83,0^ • • •. (5.127)

Calculations show that, for a sinusoidal corrugation, we can assume B\ ^ 0.983, B2 ~ 0, ^ 3 ^ - 0 . 1 7 3 .

The convergence of the solution of (5.121) for Ag ^- 00 is an important criterion for the choice of collocation nodes. Figure 5.8 is a plot of the quantity

8 = max In x (E^ ^ +E^i^)| (5.128) rie[-l/2,l/2V ^ ^'

(characterizing the fulfilment of the boundary condition (5.120)) as a function of A^ for various approaches to the choice of collocation nodes. Parameter a is chosen to be equal 0.75. The points on curve 1 are computed for the collocation nodes by (5.127) for the values of coefficients cited above. Results obtained for the "exact" distribution (5.123) fall fairly close to results obtained when using approximate adaptive nodes.

It should be recognized that attempts to construct a system of collocation nodes were repeatedly made by various authors. For example, in [38] the coordinates of collocation points are given by

I f / / /^ 1 Vi = -\ -^ P -^— + P n L (5.129) '^ 2[Ng-l ^ Ng-l ^ (Ng-l)^]

for p = 0.5. In Fig. 5.8 curves 2-3 are results for the nodes obtained from (5.15) for var-ious p. Comparing the results in Fig. 5.8 we see that the residual obtained for the distri-bution of collocation nodes as proposed in [38] (for ^ = 0.5) increases with increasing the


lE+05

lE+04

lE+03

lE+02

lE+01

lE+00

lE-01

lE-02

lE-03

5

1

1 • r

3

2

T • 1 •

5 10 15 20 25 30

Fig. 5.8. Residual versus the number of terms in the expansion for a = 0.75. 1 - collocation nodes are calculated by (5.126); 2, 3 - the nodes are calculated by (5.129) for ^ = 0.5, ^ = 0 respectively.

number of terms in the field representation. It is worth noting that the choice of fi other then that in [38] makes it possible to obtain a convergent algorithm (for a not-to-deep sinusoidal corrugation). Nevertheless, the numerical results show that, for large values of a, a residual as small as desired is not obtainable if we follow the methodology proposed in [38]. Notice that the use of collocation nodes uniformly distributed along the interval [0,1] (for ^ = 0 in (5.129)) leads to fairly fast increase in the residual (curve 3).

In a number of works (for example, in [39]) the convergence of collocation algorithms is related to the applicability of the Rayleigh hypothesis [8,24] irrespective to the distribution of collocation points. Since we have used flexible expansions [40], the applicability of the adaptive collocation method is in general, not so restrictive. Hence it does not follow, of course, that the connection between the collocation algorithms and the Rayleigh hypothesis is broken but this connection is not so direct as was thought earlier [39,41]. It is known, for example, that the Rayleigh hypothesis is valid when the singular points of the field are lying far away from the boundary of the scatterer. It is evident that, if the Rayleigh hypothesis is valid, the collocation methods converge not only for adaptive nodes, but also for other types of node distributions. It is known that we have a similar case when working on problems of polynomial interpolation of analytical functions [36].

The results cited above essentially indicate the fact that, for convergence of the solution, the completeness of the basis used [42] is inadequate [8,24]. For practical computations, the approach to calculation of expansion coefficients is of great significance. If we deter-mine the coefficients by the collocation method then the problem reduces to interpolation of the solution on the contour of the scatterer and the convergence of the computational procedure depends on the choice of collocation points. In deciding on interpolation nodes we used results related to polynomial intergration in the complex plane. This allowed us to put forward an approach to the choice of collocation points such that, as numerical ex-periments show, ensures convergence of the solution both on the boundary and in the far zone. It should be noted that when using the methodology addressed here, we "put" infor-


mation about the singular points of the analytical continuation of a scattered field into the distribution of collocation nodes.

Acknowledgements

This work was supported by the Russian Fund of Basic Researches (Project 97-02-16722) and the Target-Oriented Federal Program "Integratsiya" (Project No. 43, Sub-ject 2.1).

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[32] H. Ikuno and K. Yasuura, Improved point-matching method with application to scattering from a periodic surface, IEEE Trans. Antennas Propagat. AP-21(5), 657-662 (1973).

[33] A.I. Kleev, Diffraction by a periodic surface and the adaptive collocation method, Radiophys. Quant. Elec-tron. 33(2), 181-187 (1990).

[34] A.G. Kyurkchan, A.I. Kleev, and A.I. Sukov, The methods for solving the problems of the diffraction of electromagnetic and acoustic waves using the information on analytical properties of the scattered field, ACES Journal 9(3), 101-111 (1994).

[35] V.L. Goncharov, Theory of the Interpolation and Approximation of Functions (Gostekhizdat, Moscow, 1954) (in Russian).

[36] J.L. Walsh, Interpolation and approximation by rational functions in the complex domain. Am. Math. Soc. Colloq. 20, 1960.

[37] V.L Smimov and V.A. Lebedev, Constructive Theory of Functions of Complex Variable (Nauka, Moscow-Leningrad, 1964) (in Russian).

[38] S. Christiansen and E. Kleinman, On a misconception involving point collocation and the Rayleigh hypoth-esis, IEEE Trans. Antennas Propagat. AP-44(10), 1309-1316 (1996).

[39] L. Levin, On the restricted validity of point-matching techniques, IEEE Trans. Microwave Theory Tech. MTT-18(12), 1041-1047 (1970).

[40] C. Lancos, Applied Analysis (Prentice-Hall, Englewood Chffs, NJ, 1956). [41] P. Maystre, Rigorous vector theories of diffraction gratings. Prog. Opt. 21(11), 1-67 (1984). [42] I.N. Vekua, New Method for Solving Elliptic Equations (North-Holland, Amsterdam, 1967).

CHAPTER 6

Yasuura's Method, its Relation to the Fictitious-Source Methods, and its Advancements in the Solution of 2D Problems

Y. Okuno and H. Ikuno

Department of Electrical and Computer Engineering Kumamoto University Kumamoto 860-8555 Japan e-mail: okuno @gpo.kumamoto-u.ac.jp


112 Y. Okuno and H. Ikuno

Abstract Yasuura's methods for numerical solution of 2D scattering problems will be reviewed. Both the conventional Yasuura method and the method using a smoothing procedure will be introduced. The theory of the former proves the convergence of a sequence of approximate solutions in terms of finite modal expansions, thus supporting the validity of fictitious-source methods. The latter is described as an improved version of the former, because more rapidly converging sequences are found.

6.1. Introduction

The prototype of Yasuura's method, which today is called the conventional Yasuura method (CYM), was first published in the mid sixties [1]. Important concepts of Yasuura's methods, i.e., set of modal functions, sequence of truncated modal expansions, and an adjoint method for surface current density, were established then.

At that time there was much controversial discussion about the validity of the Rayleigh assumption. Because the infinite-series solution employed by Rayleigh [2] apparently has a radius of convergence, the discussion was focused mainly on the range of validity [3-6]. In this connection the CYM may be understood as a modified (or justified) Rayleigh method: the infinite series has been replaced by a sequence of truncated modal expansions and, consequently, the convergence has been proved. This provides us with a practical means for numerical solution of problems with arbitrarily shaped boundaries.

Today, however, modal expansion approaches such as the generalized multipole tech-nique [7], the fictitious source method [8], the equivalent source method [9], and the cur-rent model method [10] seem to be accepted as standard methods in computational elec-tromagnetics. Computer codes employing the GMT, for example, are available as well as programs based on integral equations combined with the method of moments. Under these circumstances Yasuura's theory of the CYM, together with that of Calderon [11] and Vekua [12], performs its duties to support the convergence of solutions obtained by a properly employed modal expansion approach.

Although the CYM is complete in theory, it sometimes fails to yield rapidly converging sequences of solutions. This causes a problem that we cannot find a solution with accuracy. One possibilty to avoid this problem is to employ a class of modal functions that describe the geometry of the problem appropriately, i.e., by choosing and locating appropirate poles. This was applied to the methods mentioned above. Another is to introduce a smoothing procedure (SP) in fitting an approximate solution to the boundary condition. Although the SP can only be used to solve 2D problems, it still has a wide range of application. When for example a set of separated solutions are employed as modal functions, without tak-ing notice of the approximation efficiency, Yasuura's method with a smoothing procedure (YMSP) [13,14] or with a singular-smoothing procedure (YMSSP) [15,16] leads to a se-quence of solutions that converge more rapidly than one obtained by the CYM. The YMSP is for problems with smooth boundaries, and the YMSSP is for edged boundaries.

Because Yasuura's theory is extensive we will concentrate in this article on methods to determine the field scattered by a perfectly conducting 2D obstacle: the CYM, the YMSP, and a part of the YMSSP. Reasons for this choice are: (i) the theory of the CYM includes basic ideas that dominate the entire theory; (ii) the YMSP and YMSSP are excellent ex-amples that show the advantages of Yasuura's methods. Further topics will be discussed in Section 6.10.

6. Yasuura 's method and advancements in 2D problems

^y s

113

Fig. 6.1. Geometry of the problem. SQ is an arbitrary closed region that is entirely inside S, the exterior infinite region to the contour C. The total length of C is assumed to be 1.

6.2. Formulation of a sample problem

Let us consider, for simplicity, the problem of E-wave scattering by a perfectly con-ducting cylindrical obstacle. Figure 6.1 shows the geometry of the problem, where C is a smooth closed curve and S is the exterior infinite region of C. A point inside S is denoted by r and one on C is represented by an arc-length s measured counterclockwise from a fixed point ^o- Assuming an £-wave incidence

E^(r) = u,F(r) (6.1)

we seek the scattered electric field

r ( r ) = u ,^ ( r ) , (6.2)

where u^ stands for a unit vector in z-direction and F(r) = exp[—i^r cos(^ — i)] is a plane wave with unit amplitude. Although we will confine ourselves to the method for E'-wave problems, the H-wave problems can be handled similarly.

The problem is stated as follows:

PROBLEM 1. Find the scalar function ^ ( r ) that satisfies: (Dl) The 2D Helmholtz equation in S; (D2) The 2D radiation condition at infinity; (D3) The boundary condition

^\l{s) = f{s) ( 5 G C , i . e . , 0 ^ ^ ^ 1 ) . (6.3)

Here, f{s) = — F(^) is a given boundary value.


Fig. 6.2. Relevant to the definition of modal functions. L is a smooth closed curve that is entirely inside C; D is the exterior infinite region to L; and, hence, S is a sub-region of D.

6.3. Modal functions

Here we will introduce the concept of modal functions, describe the character of a set of modal functions, and give examples.

6.3.1. Definition given in [1]

If an enumerable set of functions {<^^(r): m = 1,2,...} meets the requirements below, then {(pm (r)} is a set of modal functions defined in D, where D is an exterior infinite region of a smooth closed curve L, and vice versa:

(Ml) Each (Pfn (r) is a radiative solution of the Helmholtz equation in D. (M2) Let C be a closed curve circumscribing L and S be an exterior region bounded

by C. For any C and for any radiative wave function ^ ( r ) defined in S, there exists a sequence in terms of finite summations of <^m(r)'s

M

^M(r) = J2^rn(M)(pm(r) (M = l ,2 , . . . ) (6.4) m=l

that converges to ^ ( r ) uniformly in wider sense in S, i.e., uniformly in any closed subre-gionof S.

6.3.2. Completeness on C

Although the definition of modal functions seems to be abstract, it leads us to [1]: [3.1] Let ^ ( r ) be an arbitrary wave function in S. Then, for the boundary value ^(s)

(s e C) and for any positive number s, there is a positive integer MQ such that

M

\Y,Am(M)(pm-^<e (M>Mo). (6.5) llm=l

6. Yasuura's method and advancements in 2D problems 115

Here, the norm of a function defined on C is understood in the mean-squares sense, i.e., for square-integrable functions f{s) and g{s),

(f,g)^ I f{s)g{s)ds, 11/11 = (/ , /) ' /2. (6.6) f,s)^f

Note that the range of integration (0,1) is the contour C. Because ^ ( r ) can have any continuous function as its boundary value on C (e.g., [17]) and because the space of continuous functions is dense in H = L^(C), we have

[3.2] The set of boundary values of modal functions {(pm(s)' m = 1,2,...} is complete inH.

6.3.3. Examples

Starting with the definition, we can show that the following are sets of modal functions inD.

EXAMPLE 1. The set of radiative separated solutions

cp^(r) = H^\kr) exp(im^) (m = 0, ± 1 , ±2 , . . . ) , (6.7)

where the coordinate origin is inside D which is the complementary region of D.

EXAMPLE 2. Let L' be a smooth closed curve inside D; and let {A:H(DO} be the set of eigenvalues of the homogeneous Neumann problem inside D^ the interior finite region bounded by L^ Furthermore, let {/m(0- w = 1,2,...} be a complete set in the function space L^(V). Then the set of functions with fmitYs as double-layer density functions

(Pm(r) = - / fm(t) "^^ dt (r G D; m = 1, 2,...) (6.8) J dn V

is a set of modal functions, provided that k does not coincide with a member of {^H(DO}-

Here, R is the distance between t and r and \l/(kR) stands for the free-space Green's function. Note that this includes Example 1 and that a set of single-layer potentials can also be the modal functions.

EXAMPLE 3. Monopole functions whose poles are located on V

(Pm(r) = H^^\kRm) (r G D; m = 1, 2 , . . . , M) (6.9)

form a set of modal functions when we let M ^- oo under the condition that there is no internal resonance in D^

EXAMPLE 4. The multiple-multipole fields:

(Pmnir) = HP(kRm)tXip{mOm)

(r G D; m = 1 ,2 , . . . , M; n = 0, ± 1 , ± 2 , . . . ) . (6.10)


6.4. An approximate solution

Let a set of modal functions be {(pmi^)'- m = 1,2,...}. This can be either one of the four examples above or a set of functions defined alternatively in accordance with (Ml) and (M2). In the following, we will not pay much attention to a particular set of modal functions because the main purpose of this article is to introduce the SP. The employment of cleverly located poles, although being a powerful way to solve a wide range of problems, is a further means to improve the rate of convergence.

An approximate solution to Problem 1 is defined in the same form as in (6.4), where Am(M) means that the Am coefficient depends on the number of truncations. By the def-inition of modal functions, ^M(r) satisfies (Dl) and (D2). Hence, the coefficients should be determined so that ^M(r) approximately meets the boundary condition (D3).

We will see that the least-squares method is a rigorous means to fit the approximate solution to the boundary condition in the sense that the sequence of solutions converges uniformly to the true solution ^ ( r ) in wider sense in S.

6.5. Integral representation of the solution

It is convenient to assume the existence of Green's function that satisfies the homoge-neous boundary condition

G{r,s)=0 (reS\ seC) (6.11)

to show the convergence of the sequence of least-squares solutions. It should be noted, however, that we can prove the convergence without assuming Green's function [1]. Hence the assumption is for explanation but it is not essential.

Because the solution satisfies the conditions due to (Dl) through (D3), we have

1

xl/(r) = - f dyGir,s)f(s)ds ( r e S ) . (6.12)

0

Here, dy denotes the normal derivative at s, and the range of integration is the contour C. A similar expression holds for ^M(r)- Subtracting (6.12) from ^M(r) leads to:

1

^M(r) - ^ ( r ) = - j avG(r, s)[^Mis) - f(s)] ds (r e S). (6.13)

0

This is our starting point to establish a solution method.

6.6. Method of solution 1: the CYM

Because 9vG(r, s) is a continuous function of s provided that r is inside S, taking the absolute value of both sides of (6.13) and applying the Schwarz' inequality to the right side, we obtain

6. Yasuura 's method and advancements in 2D problems 117

\^M{r)-^{v)\^jj |a ,G(r,^) |^d^| |vI/M-/ | | ( r e S ) . (6.14)

Assuming that r is inside a closed region So that is entirely inside S (see Fig. 6.1), the first factor on the right of (6.14) is a continuous function of r and has a maximum inside SQ:

G"(So)=max reSo (i: \dyG{Y,s)rds (SoCS).

Thus we have

|vI/M(r) - vl/(r)| < G^So)||^M - / | | (r G So C S).

(6.15)

(6.16)

Because the set of boundary values of the modal functions {(pmis)'. m = 1,2,...} is complete in H ([3.1]) and because the boundary value f{s) is a member of H, there exists a sequence of boundary values of approximate solutions

^?(^), ^2^ )' •••' < ( ^ ) ' ••• (6.17)

that converges to f{s) in the mean-squares sense:

I K - / | | ^ 0 (M^cx)). (6.18)

The superscript 0, when appended to ^ M , ^m, etc., means that the quantity is obtained (or expected to be obtained) by the CYM. Reference to (6.16) finds that the corresponding sequence of approximate solutions

vl/?(r), vl/0(r), . . . , vi/0^(r), .. (6.19)

converges uniformly to ^ ( r ) in the closed region So: for any positive e, there is a positive integer Mo (So, e) such that

| ^ ^ ( r ) - v l / ( r ) | < £ ( r e S o C S ; M>Mo(So,£)). (6.20)

The sequence (6.17) may be found by the conventional least-squares method:

The CYM

Find the coefficients A^{M) (m = 1,2,..., M) that minimize the mean-square error

E% = \\^M-ff = M

Y^Am{M)(pm- f m=l

(6.21)

It is worth mentioning that:


1. The modal expansion solution to the problem of an arbitrarily shaped obstacle con-verges if the modal functions are appropriate and if the boundary matching is done by the least-squares method.

2. The least-squares matching may be understood as relaxation of the boundary condi-tion. This is because ^(s) = f(s), leads to | |^ - / | | = 0 but the converse is not always true, so that the ^%is) function, even if the E^ error is extremly small, may not be a good approximation of f(s) in uniform sense. Here, of course, the relaxation is no more than an expedient to establish a computer-aided algorithm, the CYM. In the following sections we will introduce the active relaxation - the SP - to find a sequence of solutions that converges more rapidly than one obtained by the CYM.

6.7. Method of solution 2: the YMSP

Roughly speaking, the SP means an indefinite integration of the boundary values of wave functions. The term smoothing comes from the fact that the indefinite integration increases the smoothness of a function. Application of the SP reduces unnecessary oscillation of higher space harmonics and accelerates the convergence.

6.7.1. Modification of (6.13)

Considering the fact that the d.c.-component of the boundary value (1, / ) has high in-fluence on the scattered wave, we impose the constraint

(l,^M) = (hf) (6.22)

on the boundary value of the approximate solution. Because a constant is a member of H and because {(pm(s)} is complete in H, there is an element (p^,(s) that is not orthogonal to constants: (1, cp^) 7 0. Hence, if we set M > /x, we can let ^M(S) satisfy (6.22). This equation may be written alternatively as

A^iM) = (l,<^/i)

• M

J^(h(Pm)Am(M)-(hf) .m=l

(6.23)

Here, the prime denotes the omission of the term m = fi from the summation. Insertion of (6.23) in ^M(S) — f(s) eliminates the A^ coefficient and leads to

Here,

^ M ( ^ ) - fis) = huis) - his). (6.24)

h(s) = f(s)-j^cp^(s), (6.25)

Vm (s) = (Pm (s) - 7 7 — ^ (PfM (s) (m ^ /x) (6.26)

6. Yasuura '5 method and advancements in 2D problems 119

and

M

huis) = Y^ Am{M)ylrm{s). (6.27) m=\

Note that: to find huis) is equivalent to obtain ^M{S) under the constraint (6.22). Thus, the integral representation (6.13) is modified to

1

^M(X) - vl/(r) = - j a,G(r, s)[hM{s) - h{s)\ ds (r G S) . (6.28)

6.7.2. The function space HQ

It is convenient for later discussion to introduce a function space HQ that is a subspace of H and orthogonal to constants. A function h(s) is a member of Ho if ||/? || < 00 and

1

(l,h)= I h(s)ds = 0, (6.29)

0

1

and vice versa. It is obvious that all the functions h(s), ir^is), and huis) are members of Ho.

6.7.3. A smoothing operator

Next an operator of the SP will be defined. Let

Ki(5, t)=u{s-t)-{s-t)- 1/2 (6.30)

be a kernel function and an integral transformation be defined by

1

ICf(s)=fKi(s,t)f(t)dt, (6.31)

0

where u(s) denotes the unit step function. Direct calculation shows that ICf(s) is an indef-inite integral of the HQ component of f(s) and that ICf(s) again is a member of HQ:

dICfis) - ^ = /(0)(5), (lXf) = 0. (6.32)

Here, f(0)(s) denotes the HQ component of f(s) and is defined by

/(0)(^) = / ( ^ ) - ( ! , / ) . (6.33)

120 K Okuno and H. Ikuno

Iterated application of the /C operator is defined recurrently by

ICP^^f(s)=ICilCPf)(s) (/7 = 1,2,...). (6.34)

This leads us to

1

ICPf(s) = JKp(s,t)f(t)dt (6.35) 0

with the iterated kernel

; , , „ ,„ , I-[^^(1—I)-•-•^.C)]/*^)' f" '^)- (,.36) '' \sgn{t-s)B2q^l{\s-t\)/{2q^l)\ ip = 2q^l),

where Bp(s) stands for the Bernoulli polynomial [18]. By definition, it is obvious that

= ICP-^f(s), ( l , /C^/) :=0 ( /7= l ,2 , . . . ) . (6.37) dICPfis) ^ , _ i

ds

With p = 1 in the left equation above, the right-hand side is equal to /(O) (s). The derivation of (6.36) can be found in Appendix A.

6.7.4. Method of solution

By integrating the right side of (6.28) p times by parts, we obtain

1

^M(r)-^(r) = (-l)P^^ fids)Pd,G(r,s)ICP[hM(s)-h(s)]ds ( r e S ) . (6.38)

0

Here, the perfectly integrated terms have vanished because /C/(0) = /C/(l) and (ds)Pdy x G(r, 0) = (ds)PdvG(r, 1). Following the same discussion as in Section 6.6, it can be esti-mated that

|M^M(r) - ^ ( r ) | ^ GP(So)\\lCP(hM - h)\\ (r G So C S), (6.39)

where

GP(So) = m^xJ f \(ds)PdyG{r,s)\^ds ( S Q C S ) . (6.40) reSoVJo

Thus, we have arrived at the method of solution:


The YMSP(p-th order)

Find the coefficients A^{M) (m = 1, 2 , . . . , M; m 7 /x) that minimize the mean-square

error

M

EP, = \\lCPihM-h)\\' = J2'Am(M)ICP^Ifm-ICPh m=\

(6.41)

The convergence of the YMSP solutions is guaranteed because K^his) is a member of Ho and

[7.1] the set of functions {ICP\l/m{s): m 7 /x} is complete in HQ. The proof can be found in Appendix B.

6.7.5. The significance of the SP

By expanding the given boundary value f(s) in Fourier series

f(s) = Y,fi^W(2l7tis) (6.42)

is obtained where the Fourier coefficients are given by fi = (e^^^^ , / ) . Use of eigenfunc-tion expansion of the Kp kernel function leads to an alternative representation that

E exp[2^7ri(^ — t)] Kp(s, 0 - 2 ^ r^.S^^v ' ( -4 )

Hence, if f(s) is given by (6.42), the application of the /C^ operator yields

fi {2l7ti)P

JCPfis) = J2 TJ^ oxpillnis). (6.44) £^0

Comparison of (6.44) with (6.42) finds the function of the /C^ operator: 1. To remove the d.c.-component (1, / ) = /o; 2. To reduce the l-ih Fourier coefficient by the factor l/(2£7ri)^. Hence the SP may be understood as a low-pass spatial filter that suppresses high-

frequency oscillation of the boundary values, while the constraint (6.22) makes up for the information of the d.c.-component that has been removed by the application of /C^. It is assumed that the scattering characteristics of an obstacle, if it is not electrically large, depend strongly on the Fourier components of f(s) with relatively small \l\. Naturally such components are of more importance than those of higher order.

6.8. Method of solution 3: the YMSSP

In this section we will briefly introduce the method with a singular-smoothing procedure (SSP). Although we can establish a method with a higher-order SSP, here we will only explain the first-order procedure. A brief introduction of a higher-order SSP may be found in Section 6.10. We will begin by explaining why the SSP is needed separately from the SP.


Fig. 6.3. The cross section of an edged cylinder. The arc-length s is measured from the edge point A. Both ^ = 0 and 1 correspond to A. ©Q is the jump in the direction of the tangential vector at A, and 0 = ©Q + ^ denotes the

apex angle measured in S.

6.8.1. Singularity

When dealing with a problem that involves an edged contour as shown in Fig. 6.3, the singularity of the EM field at the edge point A, i.e., s = 0 and 1 has to be taken into consideration. In our modal expansion approach, the singularity can be seen as a pole of the normal derivative of Green's function. The dominant term may be written as:

dyG{Y,s)(X l(i-^r-^ c

(6.45)

where a stands for a parameter that represents the sharpness of the contour near the edge point and is defined by

71 (6.46)

Because the range of a is (1/2,1) for the convex case shown in Fig. 6.3, the normal derivative diverges at ^ = 0, 1. This means that we cannot employ the /C operator to obtain an indefinite integral of the boundary value.

6.8.2. A smoothing operator

We define an integral transformation by

1

Cf(s) = JLi(sj)f(t)dt, 0

(6.47)


where the kernel function is given by

L\{s, t) = u(s, t).

It is obvious that

ds = f(s).

123

(6.48)

(6.49)

Furthermore, if h{s) is a (sectionally) continuous function of Ho, the indefinite integral Ch(s) has zeros of order no less than 1 at ^ = 0 and s = 1:

Ch(s) = 0(s) (s -> 0), 0(1-s) ( 5 -> l ) .

(6.50)

6.8.3. The method of solution

Going back to (6.28) and integrating the right side by parts leads to

1

^M(r)

0

G(r, s)CihM - h)(s) ds (r e S).

Note that the integrated term has vanished because of

lim dyG(r,s)C(hM-h)(s)=0

(6.51)

(6.52)

from (6.45) and (6.50). The derivative of Green's function in the integrand of (6.51) has poles of order (2 — a)

at 5 = 0 and 1 and does not belong to H. On the other hand, the indefinite integral C(hM — h)(s) has zeros of an order no less than 1 at the same points as shown above. We therefore modify (6.51) further to

1

^M(r) - ^ ( r ) = f [w(s)dsd,G{r, s)] w(s)

C(hM -h)(s) ds ( r e S ) , (6.53)

where it; (5) is a weighting function defined by

w(s) =s(l — s)

so that the two functions in the square brackets in (6.53) are members of H. Again, repeating the same procedure as in Section 6.6, it can be estimated that

(6.54)

|vI/M(r)-vI/(r)|^G^(So) I C(hM - h)

w (r G So C S) (6.55)

124 Y. Okuno andH. Ikuno

provided r is inside SQ. The positive constant G' (So) is defined by

G'(So) = maxJ /* \w{s)dsdyG{Y,s)\^ds ( S Q C S ) . (6.56) reSo V Jo

Thus, we can state the method of solution:

The YMSSP

Find the coefficients Aj^(N) (m = 1,2,..., M; m 7? /x) that minimize the mean square error

C(hM-h)f^

w

M ||2

Am{N) w w

(6.57)

Convergence is proven by: [8.1] The set of functions {>CV^^(5)/M;(5): m 7^/X} is complete in H . The proof can be found in Appendix C.

6.9. Method of numerical analysis and examples

So far we have seen the analytical side of Yasuura's methods and found that the methods may be stated as a standard form of the least-squares problem in the function space H. Since computers cannot handle continuous functions, the problem must be approximated by a problem in a finite-dimensional complex-valued vector space. In addition, it should not be forgotten that a vector in that space has only a finite number of digits of significant figures.

6.9.1. Preliminary remarks

Let us assume that we are going to solve an £^-wave problem with the CYM, for ex-ample: minimization of the squared distance \\^M — / I P - It is commonly known that the solution - the unknown coefficients - can be obtained by solving the following set of linear equations:

M

J^(cpm,(Pn)An(M) = ((pm,f) (m = 1,2,..., M). (6.58)

At first the inner products, e.g., ((pm, (Pn) (^ ,« = 1,2,..., M), must be calculated by nu-merical integration to obtain the coefficient matrix (Gram matrix) in (6.58). Then this set of equations must be solved to find the solution. Here two problems arise:

1. It is a time-consuming job to calculate the inner products, although the coefficient matrix is Hermitian. Sometimes this can be accomplished analytically. Even if this is the case, direct solution of (6.58) is not recommendable because of the next item.


2. From computational point of view, (6.58) may be understood as a normal equation, use of which squares the condition number of the problem; and consequently, (6.58) is sensitive to computational error.

In the following, we will explain the method of numerical solution, representing the practical side of the Yasuura method. The method of numerical solution may be divided into two parts:

1. Discretization of the least-squares problem stated in H; 2. Solution of the discretized problem. Taking the case of CYM for example, both parts must be solved in order to explain how

to derive numerical analysis with a computer from the theory.

6.9.2. Numerical algorithm based on the CYM

Discretization

Let us set equally spaced / ( > M) sampling points on the boundary C:

Sj=JlJ ( j = 0 , ! , . . . , / ) . (6.59)

The number of divisions / will be determined later by numerical experiments. Because C is a closed curve, both o and sj represent the same point. In some applications, equal discretization with respect to a coordinate variable may be convenient. For example, if the closed curve C is defined as r = r(^), then defining

Oj^ljir/J ( 7 = 0 , 1 , . . . , / ) (6.60)

we have J sampling points Sj = (r(Oj), Oj). We should keep in mind, however, that by going this way we implicitly employ a weighting factor ds/dO.

Then, we can define a 7-dimensional complex-valued vector by

(Pm = {(Pm(s\) (Pm(S2) • * • (pm(sj)) • (6.61)

This is a discretized form of (pm(s). Similar forms for f(s) and ^M(S) are obtained and are denoted as f and ^M •

Next, we define a 7 x M matrix by

^JM = [<P$P2 • • • (PM] (6.62)

which is usually termed Jacobian matrix. Explicitly, the matrix has the form

^JM =

(Pi(si) (P2(s$ ••• (PM{S\) (fli^l) (plisi) '" (PM(S2)

(6.63)

_(p\(sj) (p2{sj) ••• (PM{SJ)

If we denote an M-dimensional solution vector by

AM = (A1 (M) A2 (M) . • • AM {M)f (6.64)

126 Y. Okuno andH. Ikuno

then we have an approximation to (6.21) in such a form that

EIJ{KM) = J IIO/MAM - f|| (6.65)

Here and hereafter, || • || represents the EucHdian norm of a 7-dimensional complex-valued vector. The norm is defined by using an inner product

(lg) = tg = J27Jgj (6.66) 7 = 1

as

l|f|| = (f,f) ' / ' . (6.67)

Thus we have a discretized form of the CYM:

Discretized CYM

Find the solution vector AM that minimizes the mean-square error £'^y (AM).

Orthogonal decomposition methods

To solve the least-squares problem, we employ orthogonal decomposition methods [19]: the singular-value decomposition (SVD) or the QR decomposition. They have the follow-ing features:

1. The SVD can examine the behaviour of the Jacobian matrix. This is helpful in deter-mining the number of sampling points.

2. The QR decomposition takes less computation time than the SVD and solves the problem provided that no rank deficiency occurs.

Hence, we recommend the application of the SVD for formulating the discretized least-squares problem, i.e., to find a sufficient number of sampling points etc. After formulating the problem, we prefered to solve it by the QR decomposition.

The SVD

By using the SVD, we can decompose the Jacobian matrix to

<^JM = UJJ'EJMV;J^, (6.68)

Here, Ujj and VMM are unitary matrices, and * denotes Hermitian conjugation. E / M is a stack of an M x M diagonal matrix of which the elements are non-negative and a (J — M)x M zero matrix. The non-negative diagonal elements of E / M are called singular values of O/M-


Examination of the Jacobian matrix

By using the SVD, we can examine the behaviour of the Jacobian matrix. Although the rank of O / M is defined as the number of independent column vectors, this is not very convenient in practice: it is difficult to find the number. Instead, if O / M is decomposed as in (6.68), then the number of non-zero elements of S / M can be counted which leads to

rankCOjM) = rank(EyM). (6.69)

The technical terminology full rank or rank deficiency applies to the case where rank(<l>/M) = M ox rank(c|)7M) < - From a computational point of view, however, this may not be sufficient. Because we shall work with finite digits of significant figures, the singular values less than some r - which is decided mainly by the number of significant figures and by estimated magnitude of errors in the matrix elements - should be treated as zero. Thus we can find an effective rank of O / M -

QR decomposition

Assume that O J M is full rank, i.e., the set of M column vectors of O / M is linearly independent. Then, we can decompose ^JM as

^JM = QJMRMM- (6.70)

Here, Q / M is a unitary matrix and RMM is an upper triangular matrix. Two alternative ways of decomposing are known: a modified Schmidt and a Householder QR decompo-sition algorithm. Having obtained a QR decomposition of O / M , the solution to the least-squares problem may be found by solving

RMM^M = Q*jM^- (6.71)

Because RMM is triangular, (6.71) can be solved easily by backward substitution.

6.9.3. Numerical examples

Here we will show two numerical examples obtained by Yasuura's methods. A problem of plane wave scatering by a periodic deformed cylinder will be solved by the CYM and the YMSR A problem of a square cylinder will be solved by the YMSSP.

Example 1: A periodic deformed cylinder

In Fig. 6.4 the cross section of a periodic deformed cylinder is shown, which is governed by

C:rs=a{l+8cos4es). (6.72)


i y

Fig. 6.4. Cross section of a periodic deformed cylinder.

For solving this problem we chose the separated solutions (6.7) as the modal functions. Because of the periodicity with respect to Os, we can make use of polyphase wave func-tions [14] to save computer storage and computation time. We will first explain briefly how to use the polyphase functions, and then show the numerical results.

Let us decompose ^ (r) into four phase components to

where ^'^ (r) denotes the /c-th phase wave function defined by

By definition, ^'^(r) meets

^"(r, 0 + 7t/2) = exp(i/c7r/2) ^"(r, 0).

(6.73)

(6.74)

(6.75)

The modal functions are classified into four phases. According to (6.75), the members of /c-th phase modal functions are {^4w+/^(r): m = 0, ±1 , ±2 , . . . } . Hence, an approximation to the /c-th phase solution ^'^(r) may be defined as

" ^ M ^ W ^ J2 ^4m+/c(M)(/?4m+/c(r). m=-N

Here, the number of truncations in each phase is given by

A (^ = 0),

(6.76)

N' = N-l (/c = l ,2,3).

(6.77)


cond ((|)) lE + 13

J

Fig. 6.5. Results of SVD: ka = 15,5= 0.2, L = n/A, A = 10 and p = 1 (£-wave case).

1 TT-uno .

lEH

lEH

l E -

l E -

l E -

l E -

l E -

l E -

l E -

1 E -

- 0 1 :

^ 0 0 ;

- 0 1

- 0 2

- 0 3

- 0 4

- 0 5

- 0 6

- 0 7

-OR

M ( % ) , T T cond (([)) -1 1-1 , -1 O

1 r—

- — — •

^ - — " •

— 1 1 1 1—

''max

Tnin

cond ( (]))

E M '

1 1 E - M 2

: l E + 1 1

'• lE-HlO

i l E + 0 9

1 l E + 0 8

; l E - h 0 7

i l E + 0 6

: l E + 0 5

; 1E-K04

\lE+03

: 1E-H02

20 30 40 50

J

60

Fig. 6.6. Results of SVD: ka = 1 5 , 5 = 0.2, i = n/A, AT = 10 and ;J = 1 (//-wave case).

and the number of modal functions in each phase is: M' = 2N + \ iox K —0 and M' = IN for /c = 1,2,3. An approximation for ^ ( r ) is

4^

*M(r )= Y. ^'n(M)(p,n{r) (M = 8Ar + l) . (6.78) m=-4N

Having made the decomposition, we can fit the approximate solution to the boundary con-dition (i) phase by phase and (ii) on one fourth of the boundary, say, on Ci in the first quadrant. Instead of solving an (SN + l)-unknown problem, we can find the solution by


^ M . ( % )

l l l j + UZ :

lE + 01 :

lE+00 :

l E - 0 1 :

lE -02 ;

l E - 0 3 •

^ s \ ^ ^ ^

kb = 15 5 = 0,3 I = 71/4 J = 2 (2N+1)

' 1 '

. • ^ -

^ ^/^

H ^

——£lL "p="l •

P=2

^ >1

5 10 15 20 25

N

Fig. 6.7. Errors as functions of N: ka = 15, 5 = 0.3, L = n/A (norm error).

%(%)

lE+01 :

lE+00 :

l E - 0 1 :

lE -02

t^hf'ai-z'^ =5-;

" " " ' " " " • • - . . - . . • : - .

1

^jr^ l^]^ '"•""'•'"---

*'"•. '"•••.. ' *

'""-..

1 1 1

""^^ - ..., =0

^ ^ 1

'"'"'••••

P=3

1 •

10 15 20 25

N

Fig. 6.8. Errors as functions of A : ka = 15, 5 = 0.3, L = n/A (energy error).

solving 2A^-unknown problems four times, thus reducing the range of computation by four. Hence, both (i) and (ii) save much numerical computation time.

Before proceeding with numerical analysis, the boundary values of the wave functions in each phase must be modified so that the boundary values are periodic on Ci. To do this, we replace the boundary values as follows:

f{s) -> exp(-i/c^^)/(5), (6.79)


Gb/a

131

14 -

12 -

10 -

8 -

6 -

4 -

2 -

0 -

/

/

;'

A

\\

1

r

A

11

\

\

\

\

E-wave

/

;

/ A

i V'v •' / l -

- — ^ = 1 — .

8=0.3

\f\ \\ \

\ ^ I / \ I

7 ;• < *».

/ / Y

W=15°

1=45°

: l=5°

\ l=0°

0 5 10 15 20 25

ka

Fig. 6.9. Frequency dependence of backscattering cross section.

O h / a

1 /A\

1 ' '."• i J i /;; \ / v

1 .'V ; / ;v

i f • \ /• \ •

1 • •

H-wave

A A

/ U .J \

/KT" M\ ^

f M M^ W *'\ / ' •' \" V-/••

1

A 1 ^

A 1=15°

/ 4 T4,. t=5° ' A[\I }C' ^=^^°

/ ^A / \ 1=0°

16

14

12

10

8

6

4

2

0 5 10 15 20 2 5

k a

Fig. 6.10. Frequency dependence of backscattering cross section.

In Figs. 6.5 and 6.6 the results of SVD are shown: the /-dependence of the maxi-mum and minimum singular value tmax and tmin, condition number of the Jacobian ma-trix cond(O), and the mean-square error EM'- The error of the total solution (6.78) on energy balance SM is also shown as a function of J. Although the values except SM de-pend on K, only the values of the zeroth phase solution are illustrated. This is because the 7-dependence of the values are common to all phases. The figure shows that each quantity varies while J increases from M^ (= 2N + 1 = 21) to 2M\ Furthermore it can be seen that all the quantities are stable in the range / > 2M^ This means that increment of J beyond 2M' does not improve the degree of independence between the column vectors of


the Jacobian matrix. Although the results in Figs. 6.5 and 6.6 were obtained for a specific problem with the YMSP as the method of solution, the tendency is the same for other types of problems and for other methods of solution. Therefore in the following the number of sampling points is taken as twice the number of modal functions, i.e..

/ = 2M'. (6.80)

In usual applications, / given by (6.80) is sufficient: no rank deficiency due to lack of sampling points occurs.

Having found the necessary and sufficient number of sampling points, the least-squares problem can be solved by the QR-decomposition algorithm. In Figs. 6.7 and 6.8 the A^-dependence of the mean-square error in zeroth phase and the error on energy balance is shown. Note that because the errors of £^-wave problems are much smaller, only the errors of the //-wave solution are illustrated. Note further that the mean-square error E^, normal-ized by E^^ and Q^M' = ^M'/^U ^^ shown. The figure shows that the solutions obtained by the YMSP converge more rapidly than solutions by the CYM. Furthermore, the errors of higher-order SP-solutions decrease faster than those of lower-order solutions. For example, the minimum A for which the solution satisfies the energy conservation within 1% error is A = 28 for /7 = 0 (CYM) and TV = 16 for /? = 3 (YMSP). The computation time is 17.0 s (CYM) and 7.4 s (YMSP) per point on a medium-size computer Fujitsu FACOM M360R. The YMSP with /? = 2 or 3 can easily find solutions with 0.1% energy error throughout the resonance region. In Figs. 6.9 and 6.10 the frequency dependence of the backscattering cross section by the same obstacle is shown as an example for numerical calculation.

Example 2: A square cylinder

Next, results of scattering from a square cylinder of which the cross section is illustrated in Fig. 6.11 will be shown. Here, again, we will employ the modal functions defined in (6.7) and make use of the polyphase wave function to save computer storage and computation time.

•y

o

c.

X

Fig. 6.11. Cross section of a square cylinder.


To solve a problem with an jE-wave incidence, we will apply the YMSSP described in Section 6.8. For an //-wave problem, we will apply the SP - not the SSP - for first or-der smoothing. This is because the integral representation of a wave function includes the boundary value of Green's function A (r, s) for this case of polarization, where N{Y, S) stands for the Green's function satisfying dvN{r, s) = 0. The SSP is not needed until sec-ond order smoothing.

In Figs. 6.12 and 6.13 the result of numerical analysis is shown: backscattering cross sec-tion of a square cylinder is drawn as a function of normalized frequency. The £-wave prob-lem was solved by the method with first-order SSP. For the //-wave problem we employed

aja

Fig. 6.12. Backscattering cross section of a square cylinder.

Ob/a

ka

Fig. 6.13. Backscattering cross section of a square cylinder.


a method with a combination of the SP and SSR The number of truncations are / / = 20 for £-wave and 23 for H-wave, and the energy error of the solutions is less than 1%, which is sufficient to draw Figs. 6.12 and 6.13. It should be noted, however, that solutions with 0.1% energy error throughout the resonance region can be obtained.

6.10. Miscellanea

Here we will introduce a few items that were not included in the foregoing sections.

6.10.1. Supplements to the method for edged contours

Higher-order SSP

A higher-order SSP can be applied to the solution of problems with edged contours. Here we will introduce the method briefly. First, let ^M(S) satisfy

(^^ vi/^) = ( 5 ^ / ) ( ^ = 0 , l , . . . , p - l ) (6.81)

and modify ^M(S) — f(s) as in (6.24). The constraint (6.81) eliminates p unknown coefficients {A^(M) : JJL = /xi,/X2,.. .,/ip} and the functions h{s), {i/m(s) : m ^ )L6i, /X2,..., /Xp}, and hM(s) are orthogonal to {1,5, . . . , SP~^]. On the other hand, suc-cessive operation of the C operator may be represented as

1

CPf{s) = JLp(s,t)f(t)dt, (6.82)

0

where the iterated kernel is given by

Lp{s, t) = -^-rr: (s - tV'^uis - t). (6.83) (P- 1)1

Then it can be shown that the functions C^his), etc., have zeros of order no less than p 3.1 s = 0,1. Furthermore it can be proved that the set {CP'^m(s)/[w(s)y : m / /xi, /X2,..., /X/?} is complete in H. Hence, the solutions obtained by the following method converge to the true solution:

The YMSSP (a higher order)

Find the coefficients A^(M) (m = 1,2,..., M; m 7 /xi, /X2,..., fip) that minimize the mean-square error

ry \\CP(hN-h)f EP=\\^:mL n (6.84)

^ \\ wP \\

and obtain the eliminated coefficients by (6.81).


A method with a variable transformation

Another way to accelerate the convergence of solutions for a problem with an edged boundary is a combination of a variable transformation and the SP [20]. Application of a Schwarz-Christoffel-type variable transformation removes the singularity of the Green's function and allows the use of SP.

6.10.2. A locally deformed plane

The problem of scattering by a locally deformed plane can be solved by the YM [21]. Here a band-limited Fourier integral

w

^ ^ ( r ) = — / irw(h)txp[ihx-\-iK(h)y]dh, K(h) = y/k^ - h^, (6.85) Z-7t J

-W

plays the role of an approximate solution. The least-squares method yields a Fredholm second kind equation for the unknown spectrum irw(h). The convergence is supported by Plancherel's theorem. The SP can be used if necessary. This method is employed mainly to solve waveguide problems.

6.10.3. A dielectric obstacle

Scattering by a dielectric obstacle has been examined by both the CYM and YMSP [22]. Two independent sets of modal functions were employed to represent the exterior and interior solution ^^(r) and ^/ (r). The CYM may be stated as: find the coefficients Aem (M) and AimiM) that minimize the relative mean-square error

^^ = uf ^ uf ' '-'' where u is a constant depending on the polarization. The SP can be applied if the constraint is defined so that the mismatch in the boundary condition meets the constraint (6.22).

6.10.4. Methods for the surface current density

If the obstacle is made of a perfect electric conductor, a surface current flows to induce the scattered field. The CYM, YMSP, and YMSSP, respectively, have their counterpart to determine the current density. Each of the methods for current density determination is adjoint to its counterpart for the scattered field. Hence, we will term the methods for density determination ACYM [1], AYMSP [23], and AYMSSP [24]. Here we will interpret the ACYM briefly. To find an approximation to the density function j(s) = d(F + ^f)(s)/dv, an approximate density must be defined by

M

JM(s) = J2Cn(M)^M^. (6.87) n=\


The Cn coefficients are determined by the conventional least-squares technique: minimize II7M — 7II • This leads us to a simultaneous linear equation

M

J2<^rniM)(cpm,cpn) = -{F,(Pn) (« = 1, 2, . . . , M ) . (6.88)

m=l

Here, (F, cpn) = —4iexp[in(t -f n/2)] is a particular form of the reaction [25]. On the other hand, in the CYM, the minimization of the E^ error in (6.21) results in a set of linear equations (6.58), which is adjoint to (6.88) for the current density. This is an example of a structure of duality [1] between the scattered field and the current density calculation. A similar relation can be found between YMSP and AYMSP [26] and between YMSSP and AYMSSP. It should be noted that (6.88) (or (6.58)) was not employed in practice. Instead, we solved the original problem by discritization and orthogonal decomposition methods.

6.10.5. 3D problems

Also 3D problems can be solved by Yasuura's methods. Because the SP is based on integration by parts, this technique cannot be employed to solve these problems because no indefinite integral of a function on a surface of a 3D obstacle exists. Hence, to solve 3D problems one must make use of fictitious sources and apply the CYM [27]. In formulating a discretized least-squares problem, one should keep in mind that a function on the surface of an obstacle / ( ^ , 0 ) is (i) periodic with respect to 0 but (ii) aperiodic in 0. Hence, we employed the trapezoidal rule and the Gauss-Legendre rule to discretize f{6,(j)) in 0 and 0, respectively.

6.11. Conclusion

Yasuura's methods were reviewed for 2D problems. Some numerical examples were shown to prove the effectiveness of the methods.

In our opinion Yasuura's methods are satisfactory models of computational techniques in the sense that the convergence of solutions is proven and that a complete scheme of nu-merical computation is given. Although the speed of convergence of the CYM solutions is often slow, rapidly converging solutions can be obtained by applying the SP or SSP. Using a class of fictitious source fields also accelerates the convergence. Thus Yasuura's methods are rigorous and practical means for solving electromagnetic boundary value problems.

6.12. Appendix A

From (6.31), (6.34), and (6.35) we have

1

Kp{s,t)= f Kiis,v)Kp-i(vj)dv (/7 = 2,3, . . . ) . (6.89)

0


Insertion of (6.30) in (6.89) finds

Kp{t,s) = {-\)PKp{sj) (6.90)

and

1 1

j Kpis,t)ds= I Kp(s,t)dt = 0. (6.91)

0 0

We define new variables ^ and r] by

^=s-t, r] = s-\-t, (6.92)

and differentiate (6.89) with respect to § and r] to obtain

^ y ' =Kp.iis,t), I' ' =0. (6.93)

Thus we find that the Kp kernel is a function of § alone. The first equation above may be written as

- ^ ^ = Kp^,(^), (6.94)

which is to be solved for Kp(^). We can assume that the range of ^ is 0 ^ ^ ^ 1 because of (6.90). In addition, (6.91) leads us to

1

j Kp(^)d^=0, (6.95)

0

which determines the constant component of the Kp kernel. The solution to (6.94) satisfying (6.95) is given in terms of Bernoulli's polynomials [28]:

[-B2cj(^)-(-irBq(0)]/(2q)l (p = 2q), ^2^+l(§)/(2^ + l)! (p = 2q + l). Kp(^) = \^ r^'Mw;.;: :r''''^" "r~7:\.. (6.96)

Substituting {s — t) for ^ while considering (6.90), we have (6.36).

6.13. Appendix B

First, let us show that {1/ (5) : m / /x} is complete in HQ. It is obvious that each \l/m(s) is a member of HQ. Assume that there is a function f(s) that satisfies

(f,(l>m) = 0 {ym^pi). (6.97)


By definition, we have

( / , 0 m ) - ^ f ^ ( l , 0 m ) (Vm/M). (6.98) (1,0/x)

Because the fraction on the right side does not depend on m, we put

c = ^ . (6.99)

Then we find that

( / ,0^) = (c,0^) (Vm) (6.100)

for any m including m = jx. Because {(t>m{s)} is complete in H, (6.100) means that f{s) is equal to a constant almost everywhere. Thus we have found that {'\lrm{s) : m / /x} is complete in HQ.

To prove the completeness of {ICP\l/m (s) :m^ /x} we need an identity

(/C/,g) + ( / , /Cg)=0, (6.101)

which holds for arbitrary functions in H. This can be seen by direct calculation. Starting from the completeness of [il/mis) \m^ ix] and making use of (6.101), we can show the completeness of {)CP\l/m(s): m 7 /x} easily through mathematical induction.

6.14. Appendix C

Suppose there is a square-integrable function f(s) that is orthogonal to every C\lrm(s)/ w(s):

( / • ^ ) -(Vm7^/x). (6.102)

If we denote an indefinite integral of f(s)/w(s) by

<D( ) = Oi(5) -02(5) , (6.103)

where

1 f(t) -^^^dt (6.104)

0 s

we may modify (6.102) to

(O,V^^) = 0 (m^/x) (6.105)


through integration by parts. Note that the perfectly integrated term ^{s)C'ilrm{s)\\ has vanished: Let us see what happens at the upper limit ^ = 1. If we set ^ = 1 — s, where e is a small positive number, then we have

IB AtCirm{\-e)

\-e

f fit) \J l-t

dt

1/2

\Cxlfm(l-£)\

max |V^m(Ok l—e^t<l

The last expression vanishes as e tends to 0. Let us assume that <^(s) is a member of H. This will be proven later. Then, we find that

<t>(s) is a member of Ho because

(1, O) = (1, OO - (1, (D2) = (1, / ) - (1, / ) = 0. (6.106)

Hence, the completeness of the T/ ^ functions (Appendix A) implies that <^{s) should be zero in HQ. Consequently, it follows that f{s) in (6.102) must be zero in H.

Next, we will prove that ^{s) is a member of HQ. Assuming that f(s) is a real-valued continuous function of s,

1 r s

lOi -nm dt

0 "-0

d .

Integrating the right-hand side by parts, we obtain

(6.107)

|Ol||^ = / l + / 2 , (6.108)

where

h=(s- 1) / /^J l , /,=2//w/Md,d. (6.109)

Noting that 1/(1 — 0 is square integrable over the interval (0,1 — e), it can be estimated for /i that

-ii/r^/i^o. (6.110)

On the other hand, because I2 is finite as a double integral, by Fubini's theorem, we may change the order of integration to obtain

r l2=2jf(t)\j^Jf(s)ds dt. (6.111)

(=0


Application of Schwarz' inequality yields that

1

<ii/ir t=0 ^s=t

1

fis)ds dt.

Use of Hardy's inequality [29] finds an estimation for the integral with respect to t on the right-hand side:

r

J (TTT)^ J

-\2

f(s)ds dt^4\\ff. (6.112)

Consequently,

| / 2 l ^4 | | / | | 2 .

Combining (6.110) and (6.113) with (6.108), it can be estimated that

\mf^5\\ff.

(6.113)

(6.114)

Because the class of continuous functions is dense in H, the same inequality as (6.114) holds for any real-valued square-integrable function f(s). Supposing that f(s) is complex-valued, then it is obvious that an estimation similar to (6.114) can be found by decomposing f(s) into the real and the imaginary part. Likewise, it is shown that ^2(5) is a member of H. This completes the proof.

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[17] V.S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1967) (in Russian). [18] A. Elderyi, W. Magnus, F. Oberhettinger, and KG. Tricomi, Higher Transcendental Functions (McGraw-

Hill, New York, 1953). [19] C.L. Lawson and R.J. Hanson, Solving Least-Squares Problems (Prentice-Hall, Englewood Cliffs, NJ,

1974). [20] Y. Okuno, A numerical method for solving edge-type scattering problems. Radio Sci. 22(6), 941-946

(1987). [21] K. Yasuura, K. Shimohara, and T. Miyamoto, Numerical analysis of a thin-film waveguide by mode-

matching method, / Opt. Soc. Am. 70(2), 183-191 (1980). [22] M. Tomita and K. Yasuura, Numerical analysis of plane wave scattering from dielectric cylinders. Trans.

lECE Japan J69-B(2), 132-139 (1979). [23] K. Yasuura and Y Okuno, Numerical method for calculating surface current density on a two-dimensional

scatterer with smooth contour, IEEE Trans. Antennas Propagat. AP-33(12), 1369-1378 (1985). [24] Y. Okuno, An improved algorithm for calculating the current density on an edge-type scatterer, in: Proc.

Int. URSI Symposium on Electromagnetic Waves, Miinchen, 1980, p. 23IB. [25] V.H. Rumsey, Reaction concept in electromagnetic theory, Phys. Rev. 94(6), 1483-1491 (1954). [26] Y. Okuno, Yasuura's methods for calculating the surface current density on a cylindrical obstacle, Proc.

MMET '94, Kharkov, Ukraine, 1994, pp. 278-281. [27] M. Kawano, H. Ikuno, and M. Nishimoto, Numerical analysis of 3-D scattering problems using the Yasuura

method. Trans. lEICE Japan E79-C, 1358-1363 (1996). [28] L. Schwartz, Methodes Mathematiques pour les Sciences Physiques (Hermann, Paris, 1961). [29] G. Hardy, I.E. Littlewood, and G. Polya, Inequalities, Second edition (Cambridge University Press, Cam-

bridge, 1951) pp. 239-243.

CHAPTER 7

The Method of Auxihary Sources in Electromagnetic Scattering Problems

KG. Bogdanov, D.D. Karkashadze, and R.S. Zaridze

Laboratory of Applied Electrodynamics Department of Physics Tbilisi State University 3 Chavchavadze Ave. 380028 Tbilisi, Georgia email: lae @ resonan.ge


144 EG. Bogdanov, D.D. Karkashadze, and R.S. Zaridze

7.1. Introduction

A conventional Method of Auxiliary Sources (MAS) is a method of finding the solution of the boundary problem for a given differential equation by expanding it in terms of fun-damental [1-3], or other singular [4-6] solutions of this equation. The special choice of the set of expansion functions is a characteristic feature of the MAS that distinguishes it from the conventional variational methods, in which every expansion function a priori satisfies the boundary conditions, but does not satisfy the initial differential equation.

A long elaboration of mathematical ideas forms the basis of MAS. The Georgian math-ematicians V. Kupradze, M. Aleksidze and I. Vekua were the first who introduced and proved the usefulness of interchanging the differential equation and boundary condi-tions [7-9] and applied this concept to the solution of specific boundary problems. Here are only some of them: 3D problems of acoustic diffraction [10], hybrid problems for equa-tions of the elliptic [11] and parabolic type [12,13], elasticity boundary-contact problems in inhomogeneous media [14], biharmonic problems [15], general type of diffraction prob-lems for hydrodynamics and Maxwell's equations in inhomogeneous media [16], certain boundary electrodynamic problems [17,18], etc. "The common rationale of these works is a basic theorem of the completeness in L2(S) of the totality of denumerable infinite set of the particular solutions generated by the chosen fundamental" [7] or other singular solutions.

The name MAS currently used, did not appear at once. The authors themselves adhered to the names: "The Method of Generalized Fourier Series" (MGFR) [1-3,7], "The Method of Expansion in Terms of Metaharmonic Functions" (METMF) [4-6] and "The Method of Expansion by Fundamental Solutions" (MEFS) [8,9]. In earlier works, solutions of bound-ary problems were represented by the generalized Fourier series, of which the coefficients could be calculated in explicit form. However, this required an orthonormalization of the sets of fundamental solutions beforehand. From the mathematical point of view, these so-lutions were flawless. However, the performed numerical calculations [8,9,19-21] showed at once the non-optimality of preliminary orthonormalization of the set of expansion func-tions. On the other hand, expanding the solution in terms of sets of non-orthogonal func-tions in conjunction with the collocation method for determining expansion coefficients appeared to be close to optimal [22]. The sets of fundamental solutions satisfy the condi-tions above exactly [8,9]. Thus, the use of this set for construction of the solution combines the efficiency of the approach with optimality of its implementation.

In spite of the fact that the mathematical basis of the MGFR, the METMF and the MEFS was formed as early as 1953-1967 [6,24,25] and 1967 [23], the first numerical calculations performed in the field of applied electrodynamics appeared somewhat later. However, by the mid 80s the main electrodynamic boundary problems had been tested. Those were electromagnetic scattering and diffraction problems upon ideally conducting and dielectric cylinders and bodies of rotation embedded in free space or near the interface between the different media [19,21,26-39], periodic gratings [20,40], longitudinally-regular hollow, dielectric and inhomogeneous waveguides [41-43], etc.

However, the above studies revealed that the problems with algorithm convergence arise for arbitrarily chosen auxiliary surfaces in the MEFS (MAS) [19-21,44,45] or the location of unified radiation center in the METMF [26,44-48] (the so called Rayleigh hypothesis). Later, it turned out, that the reasonable approximation to the exact solution for each fixed geometry and incident wave could be reached only through the proper choice of the aux-iliary surface or arrangement of the radiation centers (the latter was pointed out in [6] and

7. Method of auxiliary sources in scattering problems 145

implemented in [27,32,38,49]). The performed studies initiated the more detailed elabora-tion of one of the fundamental properties of scattered fields, namely of their main singu-larities [44,45]. As a result, some general recommendations have been offered to construct the MAS solution of rather intricate applied problems which allow their simple numerical implementation [43-45]. Application of the MAS to the objects of complicated shape and complex filling have also been considered [49-63] (the authors do not pretend the list of cited references is complete).

This work gives a conventional interpretation of MAS applied to electromagnetic scatter-ing problems. It offers general recommendations for its implementation and illustrates its application to particular problems for a single and a set of bodies made of various materi-als, through numerical simulations in a wide frequency band starting from the quasi-static, up to the quasi-optics.

7.2. Problem formulation

Electromagnetic scattering problems are some of the most important problems in ap-plied electrodynamics. These problems arise in the study of various physical processes in antenna and waveguide theory, radiolocation, meteorology, microelectronics, defectoscopy and other branches of science and engineering. From the physical point of view, these problems appear in the investigation of propagation of waves in media with various dis-continuities (Fig. 7.1) and should then be formulated as appropriate boundary problems of electrodynamics. The main goal in the analysis of these problems is to find vectors of the secondary electromagnetic field {E/, H/, D/, B/} in each domain Df while the primary field of electromagnetic sources {E^, H^, D^, B^} are given.

To formulate the boundary problem, the constitutive equations in each domain D/, as well as the values of material parameters used, should be specified. To a certain extent

Do

D,

<S) J

>03

D3

.D4

Fig. 7.1. General geometry of the problem.

146 EG. Bogdanov, D.D. Karkashadze, andR.S. Zaridze

most of the well-known media belong to the special cases of the general four-parameter bianisotropic medium, described by the following constitutive relations [64,65]

D=:£E + iaB, H = i)6E-h/x~^B. (7.1)

In the general case, the material properties of the medium are determined by tensors of permittivity s, permeability /x and magnetoelectric admittances a and fi (hereinafter, an cxp(—icot) time dependence is assumed). When a / ^ ^ 0, then this case corresponds to the general type of anisotropic medium, while for the scalar parameters s, ix, a and P the general case of biisotropic medium is described. The latter case involves subcases of isotropic magnetodielectrics when a = j0 = 0, chiral medium when a = ^ / 0 and Tellegen medium when a = —g / 0.

In the general case, formulation of a scattering problem for a certain set of bodies with known material properties involves writing electrodynamic equations and constitutive re-lations in each of the domains Dt with different properties, as well as boundary conditions on the interfaces Stj between the neighbouring domains. However, the general scattering problem can be formally replaced by a few more simple boundary problems for sepa-rate domains D/, solution of which allows restoration of a scattered field for the whole space. Therefore, without restriction of generality, let us consider formulation and solution of the boundary problem for the arbitrary domain D specifying, besides electrodynamic equations, only the constitutive relations, the wave equation for the potential function of unknown fields and the behaviour of this function on the boundary (boundary conditions).

The boundary problem considered for the domain D surrounded by the surface S is reduced to the solution of wave equation

LU(r) = 0 (7.2)

in the domain D with constitutive relations of (7.1), where L is a wave operator, and U(r) (r e D) is an unknown potential function, determining uniquely the vectors of scattered field and satisfying on S the following boundary conditions

WU(r)l^^s = f(r^), M(r^) e S. (13)

Here W is an operator of boundary conditions, and f(r'^) is the given function determin-ing the behavior of an unknown scattered field on the boundary. For the exterior domains extending to infinity, the unknown field should also satisfy the radiation condition.

The solution to the boundary problem (7.2)-(7.3) in domains with different constitutive relations is the aim of the forthcoming analysis.

7.3. Construction of the solution by the method of auxiliary sources

7.3.1. Basic statements

In this section, a construction of the solution to the boundary problem (7.2)-(7.3) by the conventional Method of Auxihary Sources (MAS), or MEFS [8,9] is considered.

To solve boundary problem (7.2)-(7.3) in the domain D bounded by the surface S, let us enclose this domain by auxiliary surface 5" and distribute on it uniformly the set of points


S f /

Fig. 7.2. Problem and auxiliary geometry of domain D.

{r„}^j (Fig. 7.2). Considering at these points the fundamental solutions ^ ( r , r„) of wave equation (7.2) and allowing for the radiation condition, we construct the set of fundamental solutions {^(r, r„)}^j with radiation centers at the specified points. Construct now the new set of vector-functions {0(r'^, r„)}^i defined on S such that each function of this set is derived from that of the preceding one by the relation

0 ( r ^ r , ) = t 5 i ( r , r^)!^^^^, M(r^) G S. (7.4)

It can be shown [52], that for an arbitrary smooth surface S (in the Lyapunov sense) one can always find the auxiliary surface S' such that the constructed set of functions {(|)(r'^, r„)}^j is complete and linearly independent on S in the functional space L2{S). In other words, if the auxiliary surface S^ is chosen properly, any vector function being continuous on S can be expanded in terms of the first A functions of the set {O (r* , r „ )}^ j , and the expansion coefficients ensure obtaining a solution with any predesigned accuracy of the approximation as N -^^ oc.

Applying the properties of the constructed set of functions to the right-hand side of equation (7.3), we can find the coefficients of best expansion of the function f(r'^) (in the sense of L2(S)) in terms of the first N functions of the set {0(r*^, r„)}^i

f(r^) = J2^nHr\rn). (7.5) n=l

Then, the approximate solution to the considered boundary problem can be written as fol-lows

U^^\r) = ^ f l „ $ ( r , r„), reD. (7.6)

The expression (7.6) represents the expansion of an unknown field in terms of fundamental solutions of appropriate wave equation. The expansion coefficients a„ can be interpreted


as the amplitudes of auxiliary sources, the fields of which are described by the fundamen-tal solutions of wave equation (7.2) with radiation centers at the chosen points r„ of the auxiliary surface. Ît should be noted, that the properties of a specially chosen set of fundamental solutions

{^(r, r„)}^j guarantee the existence of coefficients {flnl^^j providing the best in L2{S) mean-square approximation of the constructed solution (7.6) to the true solution L^(r), as well as the convergence of the approximate solution U^^\v) to the exact solution U{Y) as A ^^ oo.

73.2. Determining the unknown coefficients and estimating the accuracy of the solution

To determine the unknown coefficients {««}^î, the optimal numerical method should be chosen, which would ensure obtaining the stable solution to the boundary problem (7.2)-(7.3) with predesigned accuracy and minimal computational resources. The accuracy of this solution can be estimated by the relative value of mean-square error (deviation) of fulfilment of the boundary conditions (7.3) on the surface S

l|f(r)llL2(5)

Among the methods for finding unknown coefficients an allowing in principle to reach any predesigned accuracy 5 with increase of the number A of auxihary sources, the fol-lowing methods have been analyzed: the method of orthogonalization, the method of least squares, the method of moments, the collocation method and others. When comparing the possibilities of these methods, in addition to the requirement of providing the minimal deviation (7.7) of approximate solution (7.6) for the same A , the complexity of numer-ical realization (simplicity of matrix elements), as well as conditionality of the obtained algebraic system was taken into consideration.

The performed numerical experiments [8,9] have revealed the advantage of the colloca-tion method for finding coefficients an because under the same conditions with this method maximal conditionality of the obtained algebraic matrix can be reached. Also, the coeffi-cients an determined by the collocation method are close to the coefficients of the best mean-square approximation of the obtained solution to the true one. Finally, the collo-cation method provides, in contrast to well-established methods of orthoganalization and least squares, the most simple matrix elements and, as a consequence, minimal computer costs.

When employing the collocation method, the boundary conditions (7.3) should be writ-ten at M points (collocation nodes) r^, m = 1 , . . . , M, where M A . As a result, the problem is reduced, generally speaking, to the determination of pseudo-solutions of the over-determined linear system of algebraic equations

N

Y,cin^T{î.rn) = fr{ri). m- l , . . . ,M, (7.8) n=\

where r is a unit vector tangential to the surface S at the point r^. Just as ordinary solu-tions appear when M = N, pseudo-solutions a„ also ensure the fulfillment of the boundary conditions (7.3) in average.


The obtained algebraic system is of the first kind, therefore it is, generally speaking, unstable with respect to small perturbations of its right-hand side and requires additional examination for providing reasonable approximation of unknown solutions to the true one. The latter depends on many factors, but above all, on the proper choice of auxiliary param-eters which are the shape and dimension of the auxiliary surface, distribution of the nodes on the boundary and auxiliary surfaces, etc. The choice of auxihary parameters, in turn, essentially depends upon the analytic properties of scattered field, namely, upon the char-acter and location of singularities outside of the basic domain of the continuously extended scattered field (we call them main singularities).

The wave fields singularities form the basis of the MAS, therefore let us consider them in more detail.

73.3. Wave field singularities in the light of the MAS

The efficiency of the MAS essentially depends on whether the analytical properties of the scattered field, namely, the character and location of its main singularities are properly taken into account. The examination of this point affects the proper choice of auxiliary surfaces, as well as the type and arrangement of the auxiliary sources. Moreover, ignoring this point leads to a weakening of convergence and even to a diverging of solution with increase of N. Therefore, analysis of the main singularities of the scattered field is an essential part of the scheme to construct a solution by means of the MAS.

As is well-known, the singularities of wave field are the points in which the field be-comes irregular (i.e., its continuity with derivatives are lost) or unbounded. In an homo-geneous medium, the field's singularities coincide with the position points of the primary field sources. Similarly, in inhomogeneous media the secondary field sources (equivalent currents and charges) are introduced on the interfaces between the different media; they are responsible for the appearance of the scattered field and for breaking its regularity on the interfaces. These points are the singularities of the scattered field, and the full description of these singularities allows determination of the scattered field in the whole space. The implementation of such an approach makes up the basis of the method of singular integral equations.

The MAS assumes the more tidy analysis of analytical properties of wave fields exam-ining the singularities of a continuously extended (together with its derivatives) scattered field L' (r) across the boundary S of its domain D. The basis for this analysis is the fact that any wave field, both scalar and vector, which is continuously extended to the whole space and satisfies the radiation condition at infinity, certainly has sources, i.e., irregular points (otherwise, it should be identically zero [66]). From this it follows, that if the wave field L^(r) is continuously extended across the boundary S of its domain D up to the boundary S' of a new domain of its continuity, then we reach the set of singular points, or singular-ities of the continuously extended wave field. These singularities of the field we call the main singularities.

Each wave field possesses a unique set of the main singularities which manifest them-selves in the form of isolated singular points, open fines and surfaces. From the physical point of view, the main singularities can be interpreted as the images of the primary sources of the wave field (including, maybe, the secondary sources induced on other interfaces) in the surface of the scattering domain. Therefore, the character, location and weight of these singularities depend on both the shape of the scattering domain, and the characteristics of


primary field sources. The complete knowledge of the main singularities of the wave field allows recovering of this field in its domain in a similar manner as with aid of usual singu-larities. However, in practice, it is impossible, as a rule, to determine all the characteristics of the main singularities, but to realize the MAS, it is not necessary to do so: it is sufficient to determine only the location domain and the character of the main singularities.

The localization of the main singularities domain is necessary for the proper choice of the auxiliary surface: the latter should envelop their location domain, because it is needed for regularity of the expansion (7.6) outside the auxiliary surface. Various techniques have been developed to localize the main singularities domain [45,62]. Some of them are based on analytical approaches, e.g., on searching of images of the given sources in the sur-face of the body [62] or on Fourier-analysis of the directional pattern of the scattered field [45]. Techniques based on half-empirical approaches have also been developed, e.g., based on the analysis of the behaviour of amplitudes and phases of the auxiliary sources while changing the shape of the auxiliary surface [45]. However, in the general case of complicated shape of boundary surface and complex excitation kind, this problem can be solved only by numerical means.

Knowing the character of main singularities, in contrast to their location, is not necessary for implementing MAS, but it allows increase in efficiency of the MAS at the expense of modifying the choice of the type of auxiliary sources. The character of main singularities is determined by the behaviour of the continuously extended scattered field in the vicinity of the singular points. Thus, the following kinds of main singularities are distinguished at the isolated singular points: the logarithmic singularity (alog |r — r'|), the pole of fz-kind ( a |r — r^|~") and essentially the singular point (the pole of oo-kind). The continuously distributed groups of singularities form the main singularities in the form of open lines and surfaces.

Thus, analysis of continuously extended scattered fields (main singularities of the field) is an essential component for constructing an algorithm of the MAS. Moreover, the contin-uous extension of scattered fields is the physical essence of the MAS. In fact, the MAS uses the replacement of the main singularities of scattered fields by the auxiliary ones, which are treated as the auxiliary sources of scattered fields. From this it follows that to increase the efficiency of the MAS, the characteristics of the auxiliary sources should coincide with the characteristics of main singularities of the scattered field.

The problem of choosing the auxiliary parameters is considered in the following section.

7.4. Choice of auxiliary parameters

For numerical implementation of the MAS, it is necessary to properly choose the aux-iliary parameters for ensuring the predesigned accuracy of approximation of boundary problem solution to the true one. The recommendations for choosing the main auxiliary parameters are given below.

7.4.1. Choice of auxiliary surface

The proper choice of the auxiliary surface plays the decisive role in numerical imple-mentation of the MAS because this choice essentially affects the degree of conditionality, stability and the rate of convergence of the solution of algebraic system (7.8) with increase of the number A of auxiliary sources. Moreover, in case of improper choice of the auxiliary surface, the computational process even diverges with increasing A .


To obtain a stable and well-conditional matrix, it is necessary to construct a system with dominating diagonal elements and maximal difference between the lines. Moreover, matrix elements should include the information concerning the scattering field's main singulari-ties. Finally, the corresponding homogeneous algebraic system should not allow nontrivial solutions. Based on these requirements, we come to the following recommendations for choosing the auxiliary surface:

(1) The auxiliary surface S^ should be equidistant from the boundary surface S, i.e., the distance d between the surfaces S and S\ measured along the normal, should be constant at each point. This ensures the approximate diagonality of the principal determinant of the system (7.8) and, therefore, the best conditionality and stability of the system (7.8) for the arbitrary A' .

(2) The surface S^ should embrace the domain of location of the main singularities of the scattered field. This follows from the specific character of the expansion (7.6) which describes, together with the unknown field in domain D, its continuous (with derivatives) extension up to the boundary S^ of the extended domain D\ Due to the smoothness of the surface S, the condition stated above is equivalent to the requirement of absence of singularities in the intermediate domain confined by the surfaces S and S\ The violation of this condition results in the divergence of the computational process.

(3) For a domain with complicated boundary 5, the distance between the surfaces S and S^ should satisfy the condition d < R^^^, where R^^ is the minimal radius of posi-tive curvature of the surface S. This condition, which is the consequence of the previous one, precludes the scattered field's main singularities connected with the geometry of the scatterer from being within the intermediate domain between the surfaces S and S\ It also indicates a way of investigating scattering problems upon bodies with non-smooth surfaces (edges), offering at edge points the finite rounding radius p^^^ and the choice of a surface S according to the requirement d < p^^^.

(4) If the domain D is an outside one, the shape and dimensions of the auxiliary surface S^ should be chosen in view of the resonant properties of the interior domain D^ bounded by this surface. That means, the shape and dimensions of the domain D^ should be such that the intrinsic frequencies of this domain do not coincide with the frequency of the pri-mary field sources. Otherwise, the intrinsic field of the domain D^ bounded by the auxiliary surface could be added to the solution of scattering problem, because this field satisfies the corresponding boundary problem. In order to avoid this effect for scattering problems, one can use a solution trial and error by modifying auxiliary surfaces. On the other hand, the in-dicated effect allows creation of the effective algorithm for finding the intrinsic frequencies and fields of complicated regions [42^4] (the analysis of this phenomenon is performed in [8,9]).

The recommendations listed above set only upper limits for choosing the distance d between the boundary and auxiliary surfaces. The optimal distance Jopt between these surfaces is connected with the number N and distribution law for the nodes on the auxiliary surface which, in turn, depend on the complexity of the described field and on the given accuracy of computations.

7.4.2. Distribution of nodes on the boundary and auxiliary surfaces

The number and distribution of the nodes {r„}^^^ on the auxiliary surface (auxiliary sources) determine the degree of possible approach (in mean-square sense) of the sought-after solution of the boundary problem to the true one (i.e., minimal possible deviation of


the obtained solution). Approximation of the boundary surface by applying the collocation method leads to an additional degradation of solution accuracy. Therefore, the number and distribution of the nodes {rf }^^^ on the boundary surface (collocation points), determine the realized accuracy (mean-square error 5) of this approximation.

When choosing the number A and distribution of auxiliary sources, one should pro-ceed from the given accuracy of approximation and the complexity of a described field on the boundary surface S. It is obvious that the larger the characteristic dimensions of the surface S (relative perimeter Lx = L/X, where X is the medium wavelength), the more complicated the field is to be described, and the more auxiliary sources must be used to describe this field under the given accuracy of computations. On the other hand, the choice of these parameters determine the optimal distance Jopt between the boundary and auxil-iary surfaces. Indeed, decrease of the distance d between the surfaces leads to increasing conditionality of the algebraic matrix (7.8). However, non-uniformity of the described field along the boundary surface caused by the closeness of radiation centers of auxiliary sources is then increased. Therefore, to ensure the same computational accuracy, more terms in the expansion (7.8) for the scattered field should be considered. Increase of d leads to the weakening of conditionality of the system matrix, but increasing uniformity of the de-scribed field along the boundary surface that needs a lower number A of auxiliary sources to describe the field with the same accuracy.

From the above considerations, it is clear that for a given solution accuracy <5 in each spe-cific case there exist an optimal relation between the relative number of auxiliary sources nx = N/Lx and the distance Jopt between the boundary and auxiliary surfaces. Also, the optimal distribution of auxiliary sources providing the minimal mean-square error should exist for each nx. In turn, the value of deviation of the obtained approximation depends on the character and coverage degree of the main singularities of the scattered field, as well as on the relation between the numbers of collocation nodes M and auxiliary sources N.

The typical dependence of the solution convergence on the relative distance kd between the boundary and auxiliary surfaces (k is a medium wavelength) is presented in Figs. 7.3, 7.4, and 7.5 for different nx and various characters and locations of main singularities of the scattered field relative to the auxiliary surface. The above results are computed for the equidistant arrangement of the same number (M = N) of nodes on the boundary and auxiliary surfaces. Figures 7.3 and 7.4 correspond to the case, when the auxiliary surface passes across the main singularity of logarithmic (Fig. 7.3) or simple pole (Fig. 7.4) kind, and Fig. 7.5 to when the logarithmic main singularity is in the middle between the boundary and auxiliary surfaces.

Figures 7.3 and 7.4 show that if the auxiliary surface envelops the main singularities of the scattered field, a fast and monoton convergence of the results with increase of the relative number nx of auxiliary sources and relative distance kd is observed. In this case, the achievable accuracy of computations is restricted only by the finite number of computer digits, and to obtain the stable solution of the corresponding algebraic system (7.8), the following procedure of "soft" regularization by Tikhonov [67] is sufficient:

ama + J2^n<^r{ri,rn) = fr{ri), m = 1 , . . . , M. (7.9) n=\

Here a is a regularizing complex parameter satisfying the condition |a| <^ |Or(r^, r^)! and providing the closeness of the solutions of the second kind equation (7.9) to the first kind one (7.8) (in our example, |a| ~ 10~^^ -^ 10"^). A more rapid (by order of magnitude)


u -

- 2 -

- 4 -

- 6 -

- 8 -

in-

N

\ \

1

^ ^ " ^ -J}x=3

N ^

kd

Fig. 7.3. Typical dependence of the solution convergence when the auxiliary surface passes across the main singularity of logarithmic kind.

n -, u

-'/-

-A-

- 6 -

- 8 -

10 -

6 ,B

^ N

r

****>. \ ^ \

— I —

•V.--- __________ nx-3

\ ^ "- ^ ^ ' " ^ "" -.

\ ^

1 1 1 1

kd

Fig. 7.4. Typical dependence of the solution convergence when the auxiliary surface passes across the main singularity of simple pole kind.

rate of the solution convergence in Fig. 7.3 in comparison to those in Fig. 7.4 is explained by presence of a stronger main singularity in the second case.

In Fig. 7.4 one can also observe the above mentioned sensitivity of the MAS to the resonance frequencies of the domain bounded by the auxiliary surface. This is shown in weakening of the solution convergence in vicinity of the resonance frequency (kd ~ 2.45). The observed resonance of auxiliary surface is easily eliminated by changing the shape

154 EG. Bogdanov, D.D. Karkashadze, and R.S. Zaridze

6,B

kd

Fig. 7.5. Typical dependence of the solution convergence with the main singularity between the boundary and auxiliary surfaces.

of the auxiliary surface (markers in Fig. 7.4 for nx = 3). With increasing N these reso-nances become so sharp {nx = 5,6,7), that their detection, should the need arise, requires construction of special algorithms.

From Fig. 7.5 it follows that the presence of any main singularity of the scattered field between the boundary and auxiliary surfaces, whatever weak it is, imposes a limit upon the increase of accuracy of solution with increasing kd. The full procedure of regulariza-tion [67] is also unable to increase the accuracy of the solution. Moreover, starting from some nx the results diverge with increasing kd. As a. result, the best accuracy obtained in Fig. 7.5 is about 8 c^ 0.03%, while in Figs. 7.3 and 7.4 where the auxiliary surface envelops the main singularities of the field, it is about 8 ~ 10~^^%.

The uniform distribution of the same number (M = N) of nodes on the boundary and auxiliary surfaces is not always optimal for obtaining minimal deviation. However, for cre-ating the general computational algorithm valid for arbitrary geometry of the domain and incident wave, only such choice of distribution of nodes permits reasonable solution of the boundary problem. Therefore, to obtain the best-conditional system matrix, we recom-mend placing the auxiUary sources at the points of crossing of the auxiliary surface by the normals drawn from the collocation nodes on the auxiliary surface.

Should a priori information about the sought solution be known, the special algorithm of distribution of the nodes, taking into account specific features of the geometry and in-cident wave, can be implemented. In particular, for complicated surfaces one can consider a non-uniform and/or unequal number (M > A' ) of nodes on the boundary and auxiliary surfaces. As stated by several authors [33,67], the case M > N leads to the improvement of conditionality of the system matrix.

Thus, the proper choice of auxiliary surface and the number of nodes on the boundary and auxihary surfaces allows a guaranteed solution of the boundary problem with pre-designed accuracy.

Further increase of efficiency of the MAS is achieved by suitable selection of potential functions and auxiliary sources.


7.4.3. Selection of the type of potential functions and auxiliary sources

By potential function, as mentioned above, we imply the function identically defining vectors of the electromagnetic field. As such a function, e.g., the following quantities might be considered: electric and magnetic electrodynamic vector potentials A^"^, Debye scalar

potentials ^"^Fl [68], Hertz vectors Z^"^ [69], spinor dyad of Hertz potential Z, spinor

dyad of electromagnetic field F [70] and, finally, arbitrary field vectors {E, H, D, B}. Each potential function satisfies its own wave equation and is related to the field vectors

in a certain way. In turn, the specific form of equations depend on the constitutive rela-tions of the medium, therefore, each potential function is characterized by its own wave operator L and own relations with field vectors.

Thus, when choosing as a potential function the electric Hertz vector (L = Z^), the wave operator in isotropic magneto-dielectric is as follows

L = W^+k^, (7.10)

where V is a Hamilton operator, and k = co^^/sjl is a wavenumber in dielectric. The unknown field vectors are then expressed via the chosen potential function in a certain way [69]

E = : V x V x f 7 , ll=-i(osV xU. (7.11)

When choosing as a potential function any field vector, e.g., U =E, the wave operator, as a rule, has a more intricate form, however the relations for field vectors in this case appear to be more simple. Thus, for isotropic magnetodielectric we obtain

L = V X V x / -k^I , (7.12)

E = U, n= WxU. (7.13) ICOfZ

In complex media such as chiral and biisotropic which distinguish the direction of polar-ization rotation (right-handed r and left-handed /) the role of Hertz vector plays the spinor dyad of Hertz potentials

U^Z= Z^

satisfying the wave equation (7.2) with operator

L = V^ + k^, (7.14)

where

kr 0 ^ ~ ' 0 -k,

156 KG. Bogdanov, D.D. Karkashadze, andR.S. Zaridze

is a wavenumber matrix, the elements of which depend on the parameters of medium by intricate manner. The field vectors are found to be connected with the potential function via relations

E = xyt / , H = |yf7, (7.15)

where x and | are two-element rows of parameters,

is a matrix differential operator with elements, and

>>'*' = V x / ± i t - / V V ± ^ , , / / ,

/ is a unit dyadic [64,65]. For description of various media, it is convenient to introduce as a potential function the

spinor dyad of electromagnetic field

satisfying the wave equation (7.2) with operator

L = V x / -kl . (7.16)

In view of relation F = y Z, the electromagnetic field vectors in this case are related to the potential function by extremely simple expressions

E = x£^, H = |£^. (7.17)

Thus, any choice of the potential function allows mathematical formulation of the boundary problem and determination of the electromagnetic field vectors. However, the choice of the potential function influences the complexity of wave operator, as well as the relations connecting the field vectors with the potential function. Therefore, the ability to determine fundamental solutions of an appropriate wave equation plays the conclusive role when choosing the potential function. This is obligatory for implementing the scheme of the MAS (this is the reason why a conventional MAS is also named as the method of expansion by fundamental solutions [8,9]).

To effectively implement the MAS it is also necessary to properly choose the type of auxiliary sources. However, the fields of the same auxiliary sources can be described by different potential functions. For example, the field of elementary auxiliary sources (el-ementary electric and magnetic currents) is described both with the help of fundamental solutions

1 e^ l'"-'''! v l / ( r - r0 = — - T (7.18)

An r - r ^


of wave equation with Helmholtz operator (7.10), and with the help of fundamental solu-tions

^ ( r - rO = V X V X T (7.19) 47r |r —r l

of wave equation with Helmholtz-like operator (7.12). In these expressions, r denotes the source point, and T a unit vector of direction of source current at this point. However, (7.18) corresponds to the choice of Hertz vector as the potential function, while (7.19) to that of the electric field vector.

Similarly, both the fundamental solutions

Z(r-r') = G(r-r')T (7.20)

of wave equation with operator (7.14) for spinor dyad of Hertz potentials, and fundamental solutions

^ ( r - rO = y G(r - r') t (7.21)

of wave equation with operator (7.16) for spinor dyad of electromagnetic field describe the field of auxiliary sources in the form of spin-vector dyad of elementary currents

r =

which are rotated in opposite directions (clockwise and counter-clockwise). Here

is the matrix of fundamental solutions of scalar Helmholtz wave equation. When solving specific problems, especially with some kind of symmetry, along with

elementary auxiliary sources, compound (combined and integrated) auxiliary sources can be used. The fields of these sources can be represented, accordingly, as the sums and in-tegrals of the fields of elementary sources. The combined sources describe the fields of finite set of elementary sources, e.g., the combinations of electric and magnetic dipoles creating the fields of heterogeneous polarization. The integrated sources describe the fields of continuous aggregates of elementary sources, e.g., the sources continuously distributed along the ring, disk or other smooth ring-supported surface. The field of latter sources can be described by Deschamps functions [71], i.e., by the cylindrical H^ (x) or spheri-cal HQ (x) Hankel functions of complex argument. The fields of metaharmonic functions (Hankel functions of higher orders) can also be referred to as integrated sources (they are convenient for description of the fields of round-cylindrical and spherical surfaces).

The choice of the type of auxiliary source is determined by the specific character of the problem to be solved, but above all, by the character of the main singularities of the scat-tered field. To achieve a higher efficiency of the MAS, the type and disposition of the aux-iliary sources should coincide with the character and location of the main singularities of


the scattered field. Thus, for logarithmic main singularity, the auxiliary sources described by Hankel functions of zero order H^ \k\r — r^\) should be chosen, while for the pole of n-th order, cylindrical and spherical Hankel functions of n-\h order should be applied. For the main singularities continuously distributed on the open lines and surfaces, the auxiliary sources described by Hankel functions of complex argument (Deschamps functions) can be used.

Application of the compound auxiliary sources enables a solution to the initial problem with minimal effort that leads finally to maximal efficiency of the employed method. How-ever, for lack of additional information concerning the character and location of the main singularities, the proper use of elementary auxiliary sources always allows the boundary problem to be solved.

7.5. Application to particular problems

In the sections above, construction of the MAS solution was clarified and recommenda-tions for choosing the auxiliary parameters to obtain the optimal solution were given. In this section, we will consider the application of the MAS for solving 2D and 3D scattering problems upon single and set of bodies made of various materials: isotropic, anisotropic and chiral.

The geometry of the problems to be studied is as follows. As 2D, cylindrical bodies with elements along the z axis and excitation in a transverse plane are considered, while as 3D, bodies of rotation excited along the z axis of rotation by plane waves are considered. Because of the symmetry of these problems, in the 2D case, the boundary conditions can be written only on the contour of cylinder cross-section, while in the 3D case the initial problem can be transformed into the problem of two circular polarized waves excitation, and the boundary conditions can be written on the semi-contour of axial section of the body.

In the calculations performed, the number of auxiliary sources are equal to the number of collocation nodes (A = M).

7.5.1. Electromagnetic scattering upon the anisotropic bodies

Anisotropic materials are widely used in microwave engineering to develop various elec-tronic devices. Therefore, the scattering problems upon the bodies of anisotropic materials are of great interest. We consider here one of the problems, connected with the forming of narrow-directional radiation with the aid of a magneto-dielectric anisotropic cylinder.

The elementary calculations show that an isotropic magneto-dielectric spheroid with semi-axis a, b, c {or elliptical cylinder with semi-axis a, b) forms a narrow-directional radiation, if only a point (linear) source is placed in a focus, and the semi-axis ratio and material parameters are related as follows: a/b = nl\ln^ — 1, where n — y/Sriir is a re-fractive medium index. The focal distance of such a body is / == \/a^ — b^, where a and b = c < a 3IC the semi-major and semi-minor axes.

This idea can be used for implementing the same process with aid of spherical or round-cylindrical anisotropic bodies. Consider here the 2D case with z-axis oriented along the element of cylinder.

If we introduce along with the reference frame (jc,y,z) the new reference frame (x\ y\ zO related new to the old one by a linear transformation (x^ = x^^f\x^\ y' = y^\Jixr\


z^ = z), then the medium is isotropic in a new frame with the refractive index n^ = ,/s^. Also field components in the new frame satisfy the usual isotropic Helmholtz equation with fundamental solutions

vI/(x y\ z') = H^'^(k^V^r^{x'-x'^)^ + {y-y^)^) (7.23)

with centers at the points (XQ, JQ, Z'). Here ^o is free space wavenumber. Expression (7.23) is rewritten in an old reference frame as follows

^{X, y, Z) = H^^^{ko^f^r^lJiyr{x - XQ)^ + ^l^r{y " Jo)^)- (7.24)

The circular region jc + j ^ = {d/2)^ in old reference frame (where J is a cylinder diameter) corresponds in a new one to an elliptical region

l^yr l^xr

with semi-axes

and focal distance /^ = {dll)^\Xyr — iixr- Therefore, if we place a linear source into a imaginary focus

(K0 1 - ^ , 0 Mr

of a circular anisotropic cylinder, then the radiation will be narrow-directional along the focal distance, if only the components of dielectric and magnetic tensors satisfy the condi-tion

l^yr EiY — , IXyr > l^xr

l^yr M'xr

being the consequence of a'/b^ = n'l\ln''^ — 1. Figures 7.6 and 7.7 show a distribution of radiating energy density produced by

anisotropic circular cylinder outside (Fig. 7.6) and inside (Fig. 7.7) the cylinder drawn after solution of the boundary problem. The material parameters of the problem are as follows: e^y = 1.6667, jjixr = 1-0, /x r = 1.5625, kod = 350. Figure 7.6 shows a narrow-directional beam behind the cylinder and an intricate interference picture around the cylinder. Fig-ure 7.7 reveals clearly a location of a real source to the left of the center of the cylinder, an imaginary focus to the right of the cylinder, also forming of a beam structure inside the cylinder.

Thus, knowing the medium type and its material parameters, the MAS algorithm can be used to solve the scattering problem and to analyze the desired physical problem.


Fig. 7.6. Distribution of the total energy density outside anisotropic circular cylinder.

Fig. 7.7. Distribution of the total energy density inside anisotropic circular cyUnder.

7.5.2. Electromagnetic scattering upon the chiral bodies

In recent years, new complex media have become of interest due to their special prop-erties and potential applications [64,65,72-77]. Among them, the so called chiral medium is of greatest interest. As was mentioned above, chiral medium can be described by three scalar material parameters, i.e., s, ji and a = p. The most significant property of this medium is its handedness, i.e., the sensitivity with respect to the rotation direction of po-larization plane of transmitted wave (clockwise or counter-clockwise). As a result, chiral medium is characterized by dyad of wavenumbers

kr,l = k\^Jl -]-rj^a'^ ±r]al (7.25)

where k = co^/FjI and r] = V/Z/^ are not the physically relevant wavenumber and wave impedance. To describe the fields in chiral medium fully, it is also necessary to determine

-50

- 6 0

-70

7. Method of auxiliary sources in scattering problems

a/7la^[dB]

161

-80

\

- N = 8

\ --N=10 E - p l a n e

—N=12 ^

1 o Compar i son

J 1 I 1 ' ' 1 —

H--plane

"^^^^^^^^^

/jP^\

\,f \ 0 45 90 135 180

S c a t t e r i n g a n g l e [deg]

Fig. 7.8. Comparison of the scattering cross-sections of a circular lossy chiral cylinder with those obtained in [72] and [74].

the rows of parameters x and ^ incoming in expressions (7.13) and (7.17)

1 1 X = \-^^r^c •ir]c

. 1 1 (7.26)

with r]c = r]/(l + r]â^) being the wave impedance of chiral medium. Then, (7.15) and (7.17) represent the decompositions of the electromagnetic field by right- and left-polarizations.

Below, some numerical results for 2D and 3D scattering upon the single chiral bodies both of canonical and complicated shape are presented and analyzed. The distinguishing feature of this case is the appearance of cross-polarized fields in external medium along with co-polarized ones because of coupling between the transverse polarizations in chiral medium.

To verify the IVIAS solution in 2D case. Fig. 7.8 shows a comparison of co-polarized and cross-polarized differential scattering cross-sections

(j(0,Oo)= lim 27tr m Hoi

(7.27)

of a circular lossy chiral cylinder of diameter d illuminated by a TE to z polarized plane wave (H^ = e ^ ) with those obtained by eigenfunction method [72] and volume inte-gral equations approach [74]. The problem is characterized by the following parameters: d = 0.3 m, M = 1.8863, Sr = 3.0 -h i0.15, /x, = 2.0 + iO.lO, a = p= 0.002, ô = 180°. Hereinafter, ô is a free space wavenumber, Sr = S/SQ, Mr = M/MO. and 0 and ô are the observation and source polar angles. We will compare here two cases for N = 12 and 14 auxiliary sources (collocation nodes).


1,5

1,0-^

0,5

0 , 0 0 1 2 3 4 5

kod

Fig. 7.9. Scattering and absorption cross-sections of a lossy chiral cylinder for large absorption.

- -scattering

1 — scattering

—absorption

1 ( f \

1 '' l 1 '

co-

cross-

total

/ \ y^*»*-

/ / / /

-T '

y 1

1

\ \

1 ' 1 •

The inset in Fig. 7.8 shows a quick convergence of MAS results with increasing A , so that for A'' = 14 they are in excellent agreement with exact ones quoted from [72]. Moreover, the MAS results for A/ = 14 are significantly more accurate than those obtained by volume integral equations approach for 763 cells [74] (the latter ones are close to the MAS results for A = 12). It should also be noted, that deviation of the MAS results for A = 14 is less than 0.1%. These reasons confirm the validity of the proposed method in chiral case and its significant advantage in comparison to well-known ones.

In order to study electrodynamic properties of 2D chiral bodies in a wide frequency range, Fig. 7.9 shows the normalized co-polarized and cross-polarized scattering cross-sections

o = lim Re /^{E(r)xH*(r)}n(r)d/

E n x H * (7.28)

01

and the total absorption cross-section

(Tabs = R e /^{[E + Eo]x[H + Ho]*}n(r)d/

|EoxH*| (7.29)

for a TE to z plane wave incident upon a lossy chiral cylinder with material parameters of Fig. 7.8 versus the non-dimensional wave parameter kod. To evaluate the influence of ab-sorption upon the scattering plots. Fig. 7.10 shows the same dependencies for considerably lesser values of dielectric and magnetic losses (Sr = 3.0 + i 0.006, /z^ = 2.0 + i 0.004).

From analysis of Figs. 7.9 and 7.10 we gather that scattering and absorption plots of the chiral cylinder are strongly modified with increasing frequency of incident wave. Besides, the cross-polarized scattering cross-section of chiral cylinder is as large as, or sometimes larger, than the co-polarized one. These plots also show the absorption maxima correspond-ing to the resonant frequencies of oscillations inside the chiral cylinder.


2 , 0 j

1,0

0, 0

1 --scattering

— scattering co-cross-

i —absoption total f

1^ J /

I / / \ \ ^' \

H • — ^ — 1 !

\

!i / p ' !1 / /I

-T ' 1 1

\ \ ' \ ,'

\ /-'

1

> / '

;' / /

/ \ /' \ /A

0 1 2 3 4 5 kod

Fig. 7.10. Scattering and absorption cross-sections of a lossy chiral cylinder for low absorption.

The presence of losses essentially affects the course of scattering and absorption plots. Thus, decrease of losses in Fig. 7.10 in comparison to Fig. 7.9 leads to a significant increase of scattering level, change of the scattering structure and redistribution of energy between the polarizations. This process intensifies with increasing wave parameter kod, because of more and more oscillations arising inside the cylinder. Decrease of the losses especially manifests itself in increasing the quality of resonances and formation of sharp peaks on the scattering and absorption plots.

To study the structure of eigen-oscillations. Figs. 7.11 and 7.12 depict the normalized co-polarized component H^ of near magnetic fields for one of the maxima in absorption plot for larger (Fig. 7.6) and lesser (Fig. 7.7) absorption. One can clearly see in Fig. 7.12 the os-cillations of whispering gallery for k^esd = 2.6686 with 3 total vibrations along the perime-ter, one vibration along the cylinder radius and maximum of the internal field magnitude of 5.99 regarding the incident field magnitude. Figure 7.11 shows the shift of the resonant frequencies (kresd = 2.692), decrease of the of internal field magnitude maxima (1.749) and destruction of resonances because of decay of the oscillations quality for the larger losses.

To verify the MAS solution in 3D case. Fig. 7.13 presents the comparison between the results for the normalized total scattering cross-section

cr(0,cp)

na^ = lim Re

r->oo 7ta

1 /^{E(r)xH*(r)}n(r)d/ 2 EQXH;

(7.30) 0"

for a plane wave incidence upon the chiral sphere of radius a versus the zenithal angle 0 in the E- and ^-planes ((p = 0 and cp = njl accordingly), calculated by the MAS and the method of integral equations [75]. We gather from Fig. 7.13 that quick convergence of algorithm with growing A is achieved, so that for A = 10 the deviation of the results is about 1%, and for A = 12 the results are indistinguishable from the accurate ones and are in excellent agreement with those quoted from [75].


Fig. 7.11. Distribution of the co-polarized component of near magnetic field in a maximum of absorption plot of a lossy chiral cylinder for larger absorption.

Fig. 7.12. Distribution of the co-polarized component of near magnetic field in a maximum of absorption plot of a lossy chiral cylinder for less absorption.

To Study the scattering properties of chiral bodies in a wide frequency range and to com-pare them to achiral case, Fig. 7.14 shows the normalized total scattering cross-section of chiral and magneto-dielectric spheres versus the wave parameter k^d. The material param-eters of chiral spheres are those in Fig. 7.13, and for magneto-dielectric are the same except a = 0. Besides, a similar curve for chiral spheres obtained by the formulae of [75] is also depicted in Fig. 7.14 for the purpose of comparison to MAS results.

Figure 7.14 reveals that for smaller k^d the quasi-static model employed in [75] is true, and the scattering cross-section of chiral sphere satisfies the Rayleigh law of scattering.

7. Method of auxiliary sources in scattering problems

a/m [dB]

165

-10

-20

-30

Cross-polarized

—N=12

—N=14

o Comparison

0 45 90 135 180

Scattering angle [deg]

Fig. 7.13. Comparison of the total scattering cross-section of a chiral sphere with that quoted from [75].

12 a/d

achir^

-chira

1 sphere o comparison

sphere

knd

Fig. 7.14. Total scattering cross-sections of the chiral and achiral spheres exposed to a plane wave versus the wave parameter.

However, with growing kod, a strong difference between the compared results arises, and to obtain the true resuhs, appHcation of a dynamical model is necessary. In the case of further increasing of kod, resonance effects similar to those of the 2D case appear. Comparison between the chiral and achiral results show, that the beginning of resonance domain is shifted to the left with growing chirality.

To analyze the scattering plots of Fig. 7.14 for chiral spheres in more detail, Fig. 7.15 presents the same plots for right- and left-polarized plane waves incidences. Similar


^D

2 0 -

1 5 -

10-

5-

0-

— c h i r a l

1 -J-• » - = ^

— d i e l e c t r i c ef

i A U T"

11 'U ! I

T 1 1 1

0 1 2 3 4 5 6

knd

Fig. 7.15. Total scattering cross-sections of the chiral and effective dielectric sphere exposed to the circular po-larized waves.

10 a/Tia^

3 ,75

d ie l ec t r i c eq. chi ra l

I I I" I I I

4 , 0 0

kod

4 ,25

Fig. 7.16. Total scattering cross-sections of the chiral and equivalent dielectric sphere exposed to the circular polarized waves.

plots when illuminating the magneto-dielectric spheres with so-called effective parameters

£ ^ = l/J = Js^qf/eq are also presented in Fig. 7.15. Here eq and /Xgq are well-known equivalent parameters of chiral medium [65] introduced as proportionality factors be-tween the right- and left-polarized wave contributions (D'"' = £eqE'*^ B''' = /XeqH ' )- Fig-ure 7.16 presents a comparison of the results for chiral and equivalent magneto-dielectric spheres.


Fig. 7.17. Distribution of cross-component of electric field inside and outside of a special shape chirolens in axial cross-section.

- 0 , 2 0 0 , 2 0 , 4 0 , 6 0 ,

Fig. 7.18. Distribution of the total cross-component and circular contributions of electric field inside and outside of a special shape chirolens along the axial line.

Upon inspection of Fig. 7.15 we conclude that chiral body, in contrast to the achiral one, exhibits the sensitivity with respect to the rotation direction of incident wave po-larization plane. Besides, each resonance is associated with a certain rotation direction. And what is more important, the chiral body behaves as a magneto-dielectric one with corresponding effective parameters (the plots for chiral and effective magneto-dielectric are almost indistinguishable in Fig. 7.15). The distinction between the effective magneto-dielectric and the equivalent one is that the former is best matched to free space by wave impedance. Thus, the effective magneto-dielectric, unlike the equivalent one, does not cause the resonance splitting resulting in the appearing thin structure in scattering plots (see Fig. 7.16).


To study the behaviour of chiral bodies in high-frequency range, Figs. 7.17 and 7.18 show the focusing process performed by a special shape non-aberrational chirolens. The 3D lens with height d = 0.6, thickness r = 0.125 and material parameters Sr = 3.0, /Xr = 1.389 and a = fi = 0.3/(1207r) is exposed to a plane wave with ^o = 400. Figure 7.17 shows a distribution of cross-polarized component of the total electric field in the axial cross-section of the lens. The same field distribution along the axis of the lens, as well as the distributions for the right-hand and left-hand circular waves contributions are depicted in Fig. 7.18 (all the calculations were performed with an accuracy of about 0.03%).

From analysis of Fig. 7.17 we gather that the chirolens forms the two bright focal spots separated in space. Figure 7.18 shows that each of the focal spots in Fig. 7.17 is created by the single handedness of incident wave. Thus, the chirolens exhibits the property of space separation of waves with opposite handedness. Comparison of field amplitudes in the centers of the focal spots with those calculated for a smaller wavenumber (ko = 200) [62] shows that the larger the wavenumber of incident wave, the stronger the focusing effect. It should be noted, that although the relative dimensions of the lens corresponds to the low limit of quasi-optics. Fig. 7.17 shows all the characteristic properties of wave propagation and focusing in optical band.

7.5.3. Electromagnetic scattering upon the sets of bodies

Let us consider, finally, the results of computer modelling of the general scattering prob-lem of Fig. 7.1. This is the problem of scattering of electromagnetic waves generated by given electromagnetic sources upon the set of bodies of complicated shape and filling. It should be noted, that problems of such kind belong to the most important boundary prob-lems, from the practical point of view. On the other hand, these problems are also the most intricate because of interference between the fields scattered by every body. However, if the method employed provides the predesigned accuracy of calculations, the possibility to solve them depends only on the computer resources.

The geometry of the 2D scattering problem to be considered is shown in Fig. 7.1. The set of bodies consists of an isotropic triangular shaped dielectric cylinder with the mate-rial parameters £r\ =3 .0 + i0.001, /x i = 1.0 + iO.O (domain Di), an elliptically shaped (b/a = 1/3) real conductor with the parameters 6r2 = 3.0 + i3.0, jiri = 1-0 + iO.O (do-main D2), an anisotropic magneto-dielectric cylinder with oval Kassini cross-section with the parameters ^ 3 = 3.0 + iO.O, /X;cr3 = 1-5 + iO.Ol, /x B = 2.0 + iO.Ol (domain D3) and an ideally conductive screen (domain D4).

This set of bodies is exposed to a TM to z polarized narrow-directional beam described by the Deschamps function (the function of Hankel of the complex argument)

E^^ = j^H^'^(^ko^(x-x^^)^ + (y-y^)^). (7.31)

Here XQ = 5.0 — i 160.0 m, y^ = 3.0 -h i 160.0 m, ko = 80 m"" is the free space wavenum-ber, Jo is a coefficient providing the unit amplitude of incident field at the origin of the reference frame.

Figure 7.19 shows the distribution of the amplitudes of the total near electric field inside and outside the bodies. The dimensions of the depicted domain correspond to 100A, X lOOX, beam width is about 15A, and dimensions of the bodies are about 20A (here


Fig. 7.19. Distribution of the co-polarized component of electric field inside and outside the set of complicated shape bodies of Fig. 7.1 with various material properties: dielectric (1), real conductor (2), anisotropic magneto-

dielectric cylinder (3) and screen (4).

A, is the wavelength in free space). One can clearly see in Fig. 7.19 the interference struc-ture of the field inside the magneto-dielectric and in free space, focusing process inside the anisotropic body and the rapid attenuation of the field inside the conductor. It should be emphasized that only sufficient accuracy of calculations (about 0.03%) allows a detailed description of the field to be obtained.

Thus, the proper use of the MAS ensures the solution of complex scattering problems with predesigned accuracy and the detailed determination of both the near and far fields.

7.6. Conclusions

In this work, we have described the conventional method of auxiliary sources (MAS) in application to 2D and 3D scattering problems upon bodies of compHcated shape and fill-ing. Next, we offered general recommendations for the proper implementation of the MAS with predesigned accuracy. Finally, we illustrated the application of the MAS to particular problems, including the problems of anisotropy, chirahty and those of multiply-connected boundaries. Far and near fields for different situations have been analyzed through numer-ical simulations in a wide frequency band starting from the quasi-static up to the quasi-optics. The efficiency of the MAS to study complex scattering problems, as well as to visuahze various physical phenomena in electromagnetic and light wave band has been demonstrated.

Acknowledgements

The authors are grateful to their colleagues, and especially to Dr. D. Tsiklauri, Dr. G. Bit-Babik and K. Tavzarashvili for technical assistance in preparing this contribution.


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the complicated shape body by the method of composition, in: Proc. VII All-Union Symposium on Diffrac-tion and Wave Propagation, Vol. 3, Rostov, 1977 (Moscow, 1977) pp. 83-85 (in Russian).


[27] R.S. Popovidi-Zaridze, D.D. Karkashadze, and J.Sh. Khatiashvili, The problem of the choice of the set of functions for solving the electrodynamic problems by the method of non-orthogonal series, in: Proc. VIII All-Union Symposium on Diffraction and Wave Propagation, Vol. 3, Tbilisi, 1981 (Moscow, 1981) pp. 83-85 (in Russian).

[28] R. Zaridze and J. Khatiashvili, Investigation of resonant properties of some open systems, in: Proceedings of Institute of Applied Mathematics of Tbilisi State University (Tbilisi, 1984) pp. 1-116 (in Russian).

[29] R. Zaridze, D. Karkashadze, G. Talakvadze, J. Khatiashvili, and Z. Tsverikmazashvih, The method of auxil-iary sources in applied electrodynamics, in: Proc. URSI International Symposium of EM Theory, Budapest, August 25-29, 1986 (Budapest, 1986) pp. 104-106.

[30] Yu.A. Eremin, A.S. Ilinski, and A.G. Sveshnikov, The method of non-orthogonal serieses in the problems of electromagnetic waves diffraction, Dokl. Akad. Nauk SSSR 247(6), 1351-1354 (1979) (in Russian).

[31] A.G. Sveshnikov, Yu.A. Eremin, and N.V. Orlov, The method of non-orthogonal serieses in the problems of electromagnetic wave diffraction, Radiotekh. Elektron. 30(4), 697 (1985) (in Russian).

[32] Yu.A. Eremin, O.A. Lebedev, and A.G. Sveshnikov, Investigation of mathematical models for space diffrac-tion problems by the method of multipole sources, Radiotekh. Elektron. 33(10), 2076 (1988) (in Russian).

[33] Yu.A. Eremin and N.V. Orlov, The method of discrete sources in the diffraction problems upon the body of rotation in dissipative half-space, Radiotekh. Elektron. 33(12), 2506 (1988) (in Russian).

[34] A.G. Kyurkchan, On the method of the auxiliary currents and sources in the problems of wave diffraction, Radiotekh. Elektron. 29(11), 2129 (1984) (in Russian).

[35] A.G. Kyurkchan, Representation of the diffraction fields by means of wave potentials and the method of aux-iliary currents in the problems of electromagnetic wave diffraction, Radiotekh. Elektron. 31(1), 20 (1986) (in Russian).

[36] A.G. Dmitrenko and A.I. IVIukomolov, On the one modification of the method of non-orthogonal serieses for solving the problems of electromagnetic diffraction on the arbitrary smooth perfectly conducting bodies, Radiotekh. Elektron. 33(3), 449^55 (1988) (in Russian).

[37] Y. Leviatan and A. Boag, Analysis of electromagnetic scattering from dielectric cylinders using a multifila-ment current model, IEEE Trans. Antennas Propagat. AP-35, 1119-1127 (1987).

[38] Ch. Hafner, Numerische Berechnung Elektromagnetischer Felder (Springer-Verlag, Berlin, 1987). [39] R. Zaridze, G. Lomidze, and L. Dolidze, Diffraction on a Dielectric Body Near the Surface of Division of

the Two Dielectric Media (Tbilisi State University Press, Tbilisi, 1989) pp. 1-80 (in Russian). [40] R. Popovidi-Zaridze and G. Talakvadze, Numerical Investigation of the Resonant Properties of Metal-

Dielectrical Periodical Structures (Institute of Applied Mathematics of Tbilisi State University, Tbilisi, 1983) pp. 1-80 (in Russian).

[41] R.S. Zaridze, D.D. Karkashadze, J.Sh. Khatiashvili, and G.Z. Akhvlediani, Approximate calculation method for dielectric waveguides with complex cross-section. Bull. Georgia Acad. Sci. 102(1), 53-56 (1981) (in Russian).

[42] R.S. Popovidi-Zaridze, D.D. Karkashadze, G.Z. Akhvlediani, and J.Sh. Khatiashvih, Investigation of the possibilities of the method of auxiliary sources in solution of the two-dimensional electrodynamics prob-lems, Radiotekh. Elektron. 26(2), 254-262 (1981) (in Russian).

[43] R. Zaridze, D. Karkashadze, and J.Sh. Khatiashvili, Method of Auxiliary Sources for Investigation of Along-Regular Waveguids (Tbilisi State University Press, Tbilisi, 1985) (in Russian).

[44] R. Popovidi-Zaridze, The Method of Auxiliary Sources, Preprint No. 14(386) (Institute of Radio-Engineering of Academy of Sciences, Moscow, 1984) pp. 1-80 (in Russian).

[45] VF. Apeltsin, R.S. Zaridze, D.D. Karkashadze, A.G. Kyurkchan, and A.I. Sukov, The method of auxiliary sources and wave field singularities, calculations of the filed out of the boundary surfaces, in: Proc. IX All-Union School on Diffraction and Wave Propagation (University of Kazan, Kazan, 1988) pp. 1-80 (in Russian).

[46] K. Yasuura and H. Ikuno, On the modified Rayleigh hypothesis and MMM, in: Proc. Int. Symposium on Antennas and Propagation, Sendai, Japan, 1971, pp. 173-174.

[47] VF. Apeltsin and A.G. Kyurkchan, Rayleigh's hypothesis and analytical properties of wave fields, Ra-diotekh. Elektron. 30(2), 193-210 (1985) (in Russian).

[48] A.G. Kyurkchan, On the analytical continuation of the wave fields, Radiotekh. Elektron. 31(7), 1294-1303 (1986) (in Russian).

[49] Ch. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Boston, 1990).

[50] Y. Leviatan, Analytic continuation consideration when using generalized formulation for scattering prob-lems, IEEE Trans. Antennas Propagat. AP-38, 1259-1263 (1990).


[51] Ch. Hafner, Multiple multipole (MMP) computations of guided waves and waveguide discontinuities, Int. J. Num. Modeling 3, 247-257 (1990).

[52] D. Karkashadze and R. Zaridze, The method of auxiliary sources in applied electrodynamics, in: Computa-tional Electrodynamics, Latsis Symposium ETH, Zurich, 1995, pp. 163-180.

[53] R. Petit, The method of fictitious sources in EM diffraction, MMET, URSI, Kharkiv, 1994, pp. 302-314. [54] Y. Okuno, A duality relationship between scattering field and current density calculation in the Yasuura

method, MMEZ URSI, Kharkov, 1994, pp. 278-281. [55] KG. Bogdanov, D.D. Karkashadze, R.G. Jobava, R.S. Zaridze, and Ph.I. Shubitidze, The method of auxiliary

sources in problem of chirality, in: Proc. XXV General Assembly of URSI, Lille, France, 1996, p. 39. [56] F.G. Bogdanov, D.D. Karkashadze, D.A. MetskhvarishviU, and R.S. Zaridze, Investigation of diffraction

properties of the single and periodical scatterers made of complex materials, in: Proc. of IEEE Semi-nar/Workshop DIPED '97, Lviv, Ukraine, 1997 (Lviv, 1997) pp. 15-17.

[57] D. MetskhvarishviU, D. Karkashadze, and R. Zaridze, Anisotropic, absorbing magnito-dielectrical bodies in the field of SHF waves, Bull. Georgia Acad Sci. 158(2), 225-228 (1998) (in Russian).

[58] Yu.A. Eremin and A.G. Sveshnikov, Simulation of light scattering by particle inside a film via discrete sources method, in: Electromagnetic and Light Scattering - Theory and Application III, Proc. 3rd Workshop on Electromagnetic and Light Scattering, T. Wriedt and Yu. Eremin, Eds. (Universitat Bremen, Bremen, 1998) pp. 83-90.

[59] R. Zaridze, G. Bit-Babik, and K. Tavzarashvih, Some recent developments in MAS for inverse and scat-tering problems on large and complex structure, in: Electromagnetic and Light Scattering - Theory and Application III, Proc. 3rd Workshop on Electromagnetic and Light Scattering, T. Wriedt and Yu. Eremin, Eds. (Universitat Bremen, Bremen, 1998) pp. 287-294.

[60] R. Zaridze, G. Bit-Babik, D. Karkashadze, R. Jobava, D. Economou, and N. Uzunoglu, The Method of Auxiliary Sources (Institute of Communications and Computing Systems, Athens, Greece, 1998).

[61] A.G. Dmitrenko and S.V. Korogodov, Electromagnetic waves scattering by the perfectly conducting body with chiral coating, Izv. Vyssh. Uchehn. Zaved. Radiofiz. 41(4), 495-506 (1998) (in Russian).

[62] F.G. Bogdanov and D.D. Karkashadze, Conventional method of auxiliary sources in the problems of elec-tromagnetic scattering by the bodies of complex materials, in: Electromagnetic and Light Scattering -Theory and Application III, Proc. 3rd Workshop on Electromagnetic and Light Scattering, T. Wriedt and Yu. Eremin, Eds. (Universitat Bremen, Bremen, 1998) pp. 133-140.

[63] F.G. Bogdanov, D.D. Karkashadze, and R.S. Zaridze, Propagation in and scattering by biisotropic objects of complicated shape, in: Proc. 7th International Conference on Complex Media (Bianisotropic '98), Braun-schweig, Germany, June 2-6, 1998, A. Jacob and J. Reinert, Eds. (Braunschweig, 1998) pp. 133-136.

[64] A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994) pp. 1-536. [65] I.V. Lindell, A.H. Sihvola, A.A. Tretyakov, and A.J. Viitanen, Electromagnetic Waves in Chiral and Bi-

isotropic Media (Artech House, Boston, London, 1994). [66] V.D. Kupradze, The Main Problems in the Mathematical Theory of Diffraction (GROL, Leningrad, Moscow,

1935). [67] A.N. Tikhonov and V.Ya. Arsenin, The Methods for Solving Non-Correct Problems (Nauka, Moscow, 1986)

(in Russian). [68] M. Bom and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1968). [69] L.A. Veinshtein, Electromagnetic Waves (Radio i Svyaz, Moscow, 1988) (in Russian). [70] R. Penrose and W Rindler, Spinors and Space-Time, Vol. 1 (Cambridge University Press, Cambridge, 1986). [71] A.A. Izmestyev, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13(9) (1970). [72] C.F. Bohren, Scattering of electromagnetic waves by an optically active cylinder, J. Colloid Interface Sci.

66(8), 105-109 (1978). [73] A. Lakhtakia, V.K. Varadan, and V. V. Varadan, Scattering and absorption characteristics of lossy dielectric,

chiral, nonspherical objects, Appl. Opt. 24(23), 4146-4154 (1985). [74] M.S. Kluskens and E.H. Newman, Scattering by a chiral cylinder of arbitrary cross section, IEEE Trans.

Antennas Propagat. AP-38(9), 1448-1455 (1990). [75] R.G. Rojas, Integral equations for the scattering by a three-dimensional inhomogeneous chiral body, /.

Electromagn. Waves Applic. 6(5/6), 733-749 (1992). [76] J.C. Monzon, Scattering by a biisotropic body, IEEE Trans. Antennas Propagat. AP-43(11), 1273-1282

(1995). [77] H. Cory, Chiral devices - an overview of canonical problems, / Electromagn. Waves Applic. 9(5/6), 805-

829 (1995).

CHAPTER 8

Numerical Solution of Electromagnetic Scattering Problems of Three Dimensional Nonaxisymmetrical Bodies on the Foundation of Discrete Sources Method

Anatoli Dmitrenko

Tomsk State University Siberian Physical and Technical Institute Revolution Sq. 1 634050 Tomsk, Russia e-mail: [email protected]


174 A. Dmitrenko

8.1. Introduction

During the last twenty years, together with the integral equation method, the auxiliary sources method has been used for solution of electromagnetic scattering problems. In the Western literature this method is named Generalized Multipole Technique or Multiple Mul-tipoles Method, the researchers from Russia often use the name Discrete Sources Method (DSM). The first to publish the theoretical foundations of this method was Kupradze in the former USSR [1]. The attractiveness of DSM is based on its conceptional simplicity. An unknown scattered field in a domain under consideration is presented as a finite linear combination of fields of some fictitious discrete sources, placed outside this domain. It is assumed that these sources radiate into the infinite homogeneous medium with the same electromagnetic parameters as the domain under consideration. Such linear combination usually satisfies Maxwell equations and additional radiation conditions, if necessary. The coefficients of the combination are found as a solution of the system of linear algebraic equations, obtained by imposing the boundary conditions in some norm. Thus, if the set of discrete sources is suitably chosen, the solution of the scattering problem is reduced to the solution of the corresponding system of linear algebraic equations. The dimension of this system is determined by the number of discrete sources and by the number of boundary conditions approximation points on the surface of the scatterer. In 2D scattering problems the boundary conditions are imposed on the contour line of a scatterer. This decreases the dimension of the corresponding system of linear algebraic equations. For axisymmetric scatterers it is also possible to use the approximation of the boundary conditions on the full surface of a scatterer, rather than the approximation of the boundary conditions on the generated line of the surface of a scatterer, thus decreasing the dimension of the system of linear algebraic equations to be solved [2]. If the scatterer is a nonaxisymmetrical 3D body (an arbitrary shaped body), it is necessary to approximate the boundary conditions on the full surface of the scatterer. This will lead to a large dimension linear algebraic equations system, which is ill-posed as a rule.

In 1985 at the Siberian Physical & Technical Institute of Tomsk State University inves-tigations began, directed at the appplication of DSM for numerical solution of electromag-netic scattering problems of 3D nonaxisymmetrical bodies. At first, the scattering problem for a perfectly conducting body was considered. As discrete sources, elementary electric dipoles were chosen. These dipoles were placed within the body on a discrete set of points of the auxiliary surface homothetic to the surface of the body. In any one of the selected points two dipoles with unknown moments were placed. Each of these dipoles was oriented tangentially to the auxiliary surface. The unknown dipole moments were defined as a solu-tion of the system of linear algebraic equations obtained by using the boundary conditions according to the collocation method. The chief difficulty we encountered was the solution of the resulting linear algebraic equations system. The different ways of its solution were investigated. By means of computer experiments we found that a really reasonable way is the transition of the problem of solution of linear algebraic equations system to the equiva-lent problem of minimization of the residual function. As a rule, we solved the last problem by the conjugate gradient method. Subsequently, the ideas mentioned above were extended towards impedance, magneto-dielectric, chiral and coated bodies.

In this paper main results of performed investigations will be presented. In Sec-tions 8.2-8.6 the full mathematical formulations of developed variants of DSM for per-fectly conducting, impedance, magneto-dielectric, chiral and coated scatterers will be given. In Section 8.7 some ideas for the solution of dense ill-posed systems of linear al-

8. Scattering on nonaxisymmetrical bodies 175

gebraic equations of DSM will be reported. In Section 8.8 the possibilities of the writ-ten software package will be briefly described, and some results of computational experi-ments will be presented that were performed to estimate the method parameters influence on the solution accuracy, and the adequacy of the obtained results, and also the chiral-ity parameter influence on scattering properties of the different shaped bodies and coat-ings.

Some results concerning this topic were published in Russian in [3-8] and in English in [9-16].

8.2. Perfectly conducting scatterer

The geometry of the problem is shown in Fig. 8.1. Let a perfectly conducting body Dt, bounded by the smooth surface 5'sc, be located in an infinite homogeneous medium De with the permittivity Se and permeability /x . We choose a point O within D, to be the origin of a Cartesian coordinate system Oxyz> The system is excited by a time-harmonic field {Eo, Ho} (time dependence exp(—itt>r))- It is necessary to find the scattered field {E , H^} mDe.

The field {E , H^} must satisfy the Maxwell equations

V X Ee = icoiXeUe, V X H^ = -ICOSeEe in D^, (8.1)

the boundary conditions

n x ( E , + E o ) = 0 on^sc, (8.2)

and the radiation conditions

{ V ^ E , ; V M ; H , } X R / / ^ + { V ^ H , ; - V ^ E , } = 0 ( / ? - ^ ) , R^OO, (8.3)

where n is the unit normal to the surface ^sc, R is the distance between the body Di and the observation point and a x b is a vector product.

(Ee,He}

(Eo,Ho}

Fig. 8.1. Geometry of the problem for a perfectly conducting scatterer.

176 A. Dmitrenko

The solution of problem (8.1)-(8.3) is constructed in the following way. Inside the scatterer we introduce an auxihary surface 5 ,aux which is homothetic to

the surface of the scatterer. This means that <S' ,aux is determined by the transformation OM' = KeOM, where OM' is the distance from point O (the centre of homothety) to point M' on 5' ,aux along the half line OA, OM is the distance from point O to point M on the surface S^c along the same half line OA. We denote the coefficient Ke as homothetic coefficient; ^^ < 1 if an auxiliary surface lies inside the scatterer. Let us choose a discrete set of points {Mn,e}nL\ ^^ » ,aux and place in any point two indepen-dent fictitious elementary electric dipoles which are oriented tangentially to Se^mx- Let us represent the unknown field {E^,H^} in De as a sum of fields of above-mentioned dipoles:

. Ne 10)

Ef(M) = - 2 ^ V x ( V x n „ , , ) , n=\

Ne

Hf(M) = - L ^ V x n „ , „

Tine = fe{M,Mn,e)vV^ ^n,e ^n,e . ^n,e n,e n,e . n,e n,e nyr ^ r^ /o A\

Pr = Pri + Pr2 = Prl ^rl + PT2 ^12' ^ ^ ê, (8.4)

where V^ (M, Mn,e) = exp(iê/?^^)/(47r/?^^) is the fundamental solution of the Helm-holtz equation for region De, ke = coêe/jie, R^^ ^^ ^^ distance from point Mn,e to point M in De', / "f, p^2 ^^^ unknown components of the vector dipole moment p"'^ along the directions e"j^, e^2 selected in the tangential plane to 5' ,aux in the point Mnê-The field (8.4) satisfies equations (8.1) and radiation conditions (8.3). In order to satisfy the boundary conditions (8.2), it is necessary to define in a suitable manner the unknown components p^'^, p^2 (^ = 1, • • •, Ne) of dipole moments. For this purpose let us use the collocation method. Let Mj (j = 1,2,..., L) be collocation points on ^sc- Then we find the following system of linear algebraic equations with a complex matrix of 2L x INe dimension for the unknown dipole components:

n " X Ef^" = -n^' X E^, 7 = 1,2,..., L, (8.5)

where n^ is the unit normal to the surface sc in the collocation point Mj, E^ '- and EQ are the values of scattered and exciting fields in point Mj and L is the number of collocation points.

It should be noted that different solutions of system (8.5) exist. The correct choice of the solution determines the success in the numerical solution of a scattering problem. Some ideas on this subject will be expounded in Section 8.7. Here we state that the problem of solution of system (8.5) is considered as a problem of minimization of the residual function, i.e.,

cD = X]k'x(Ef''+E^)r- (8-6) 7=1

After the solution of system (8.5) has been determined, any scattering characteristic may be calculated from Eq. (8.4). In particular, for field scattering components in the far zone


we have expressions

Ee,<p(M) = - J^HeMM) - £ ^ ^ ^ 1 ^ D^(e, ip) + 0(/?-2), (8.7)

where DeiO, cp) and D^(^, (p) are determined as

Ne

n=l

X {(cos^cos(^cosQf"'^ +cos^sin(^cos)S"'^ — sinOcosy^'^)p^'^

+ (cos cos ( cos 0 2' +cos^sin(^coS)62'^ — sin^cosy2'^)Pr2^}' Ne

D^(^,^) = ^G«,e(6>,(^){(-sin(^cosa"'^+cos(^coSi6"'^)/7"f n=\

+ (—sin ( cos 0^2' +COS(^COS)S2'^)/^r2^}'

Gn,e(^^^) = exp{—iê(sin^cos(^x„,e + sin^sin^y„,e +cos^z„,^)}. (8.8)

In expressions (8.7)-(8.8) R is the distance from the origin O of a Cartesian coordinate system to observation point M, 0 and (p are angular spherical coordinates of point M, a"'^, pn,e^ ^n,e ^ ^ ^j^^ angles between the direction e^^'^ and the x-, j - , and z-axes, respectively; 0^2' , ^2'^, }/2' are those for direction 6 2 ; x„,^, j„,^, z„,^ are Cartesian coordinates of point M„,e.

The variant of DSM explained above gives a possibility to control the accuracy of the ob-tained solution. The relative boundary conditions residual value on the set of intermediate points is used as a measure of solution accuracy, i.e.,

L'

Q = (CDVOO)'/', Ô=J2 I"" ^ ^0 1 (8.9) m=l

where O is the value of the function (8.6) on the set of points Mm (m = 1,2,..., LO which are intermediate to the collocation points My (7 = 1,2,..., L), OQ is the corresponding value of exciting field on the the same set points.

According to theoretical foundations of DSM [1,2], a functional system used for field representations must be able to approximate the boundary conditions on the surface of a scatterer in the square norm. For the system introduced above, of tangentially oriented electric dipoles, the following theorem is valid.

THEOREM 1. Let Ssc ^ A^^'^\ and the set [Mn^^î be dense on the auxiliary surface Se,mx nd the surface Se^mx be a nonresonant surface, then for any f € LJî^S) and 5 > 0 there exist such Ne(S) and {p"f, P^2^n=l' ^^^^

l|nxEf-f|L.(S.)<^'

178 A. Dmitrenko

where E^ is defined by Eq. (8.4), L2(5'sc) is the space of vector fiinctions which lie in the tangential plane to Ssc, and whose components belong to L2(Ssc)-

The schemes for proving theorems, which are similar for different types of scatterers, were described in detail in [2]. These schemes are valid for the functional systems used in this paper.

8.3. Impedance scatterer

Let an impedance body Di, bounded by the smooth surface ^sc, be located in an infinite homogeneous medium De with the permittivity Se and permeability /x . The body is ex-cited by a time-harmonic field {EQ, HQ}. It is necessary to find the scattered field {E , H^} in De.

The unknown field {E , H^} must satisfy the Maxwell equations

V X E^ = ico/jLeB^e, V X H^ = —icoeeEe in De, (8.10)


n x ( E ^ - h E o ) - y 7 [ n x ( n x ( H ^ + Ho)) ]=0 on 5sc (8.11)


{Vi7E,; V M T H , } X R/R + { V A ^ H , ; - V ^ E , } = 0(R-^), R->oo, (8.12)

where rj is the surface impedance of a scatterer. In the general case, ij isa. tensor of second rank of which the components are continuous on ^sc- Here we consider the simpler case where r] is an isotropic scalar function of coordinates Os, cps on ^sc-

As for a perfectly conducting scatterer, we introduce 5' ,aux = KeSsc (Ke < I) as an auxiliary surface inside the scatterer, and represent the unknown scattered field {E , H^} in De as a sum of fields of fictitious elementary electric dipoles, which are placed on the surface Se,mx in points {Mn,e}nLi ^^^ ^^ oriented tangentially to Se,mx'

. Ne ICO '

Ef(M) = ^J]Vx(Vxn„,,), ^ n=l

J Ne

Hf(M) = — ^ V x n „ , „

Pr = P r l +Pr2 = Px\ ^ 1 + ^r2 ^r2 ' ^ ^ ^^- (8.13)

The field (8.13) satisfies equations (8.10) and the radiation conditions (8.12). Using the boundary conditions (8.11) according to the collocation method, we get the following system of linear algebraic equations with a complex matrix of dimension 2L x INe for


unknowndipole components p^'j^, p^2 (n = 1,..., Ne):

= -nJ X E^'^' + r]J [nJ x (n ' x H^'^)], 7 = 1, 2 , . . . , L, (8.14)

where E^ '^, H^ ' and EQ, HQ are the values of scattered and exciting fields in point Mj and L is the number of collocation points.

The solution of system (8.14) is found via minimization of the function:

CD = ^ | n ^ ' X (Ef'^ +E^) - ^^[n^' x (n ' x (H^^ +H^) ) ] | . (8.15)

After the solution of system (8.14) is determined, any scattering characteristic may be calculated from Eq. (8.13). In particular, for field scattering components in far zone the expressions (8.7)-(8.8) are valid.

Control of the solution accuracy is realized via calculation of the value:

Q = {0'/<t>of\ L'

<t>o = J2 r "" ^0 ~''"'["'" " ("*" ^ ''o)] m=\

(8.16)

where O is the value of function (8.15) on a set of intermediate points M^ (m = 1,2,..., LO and OQ is the value of the corresponding norm of the exciting field on the same set of points.

The mathematical consistency of above-presented numerical method is obtained from the validity of the next theorem:

THEOREM 2. Let Ssc ^ A^^'^\ and the set {M^,^}^^ be dense on the surface Se,mx (^^d the surface »S' ,aux be a nonresonant surface, then for any f e L2 (-Ssc) and 5 > 0 there exist such Ne(S) and {p^f, P^2^n=i' ^^^^

inxE,^ - r , [nx(nxHf) ] - f | | , . ( , ^^^^5 ,

where E^ and H^ are defined by (8.13).

8.4. Magneto-dielectric scatterer

The geometry of the scattering problem is shown in Fig. 8.2. A homogeneous magneto-dielectric body Di with permittivity st and permeability /x/, bounded by a smooth sur-face S^c, is located in an infinite homogeneous medium De with parameters Se, jjie- The body is excited by a time-harmonic field {EQ, HQ}. It is necessary to find the scattered field

180 A. Dmitrenko

D.

^ e . ^ i .

\ {Ee.He}

{EQ^HQ}

Fig. 8.2. Geometry of the problem for a magneto-dielectric scatterer.

{E^, H^} in De. Further, the field {E , H^} is defined in D^, and the field {E/, H/} exists inside the body Di. These fields must satisfy the Maxwell equations

V X Ee = icofieUe, V X H^ = —icoSeEe in De,

V X E/ = ia;/x/H/, V x H/ = -icoSiEt in D/,

(8.17)

(8.18)


n X (Ei - E^) = n X Eo, n x (H/ - H^) = n x HQ on ^sc, (8.19)


{ V ^ E , ; V i ^ H , } X R/R + {VP^H^; - V ^ E , } = 0(R-^), R -^ oo(8.20)

for field {E^,He} in D^.

Similar to the perfectly conducting and impedance scatterers, we represent the scattered field {Ee,H^} in De as the sum of fields of fictitious elementary electric dipoles which are placed on the auxiliary surface 5' ,aux in points {Mn,e}nL\ ^^^ ^^^ oriented tangentially to kJ ,aux*

Ne

Ef(M) = ^ ^ V x ( V x n „ , , ) , " n=l

, Ne

Hf(M) = — ^ V x n „ , „

^n,e ^n,e , n,e n,e n,e , n,e n,e »>f ^ n Pr = P r l + P r 2 = Prl ^ r l + / ^ r 2 ^r2 ' ^ ^ ^e- (8.21)

To represent the field {E/, H/} inside the body A , we introduce a second group of fic-titious elementary electric dipoles. These dipoles are placed on a second auxiliary surface

1 "^^

n=l


5'/,aux = KiSsc, which encloses the body A (Kt > 1), in the points {Mn,i}^L,i and are oriented tangentially to / aux- Then we have the representations

Ef^(M) = ^ ^ V x ( V x n „ , , ) , ^ n=\

Ni

nnJ=xlfi(M,Mn,i)v''/

l^'=lî+lî=P:^<l-^Pri<i^ ^Â, (8.22)

where ^j/i (M, Mnj) = exp(i^/ R^^ )/{AnR^j^) is the fundamental solution of the Helmholtz equation for region D/, ki = coêiî, ^^ is the distance from point M„,/ to point M in Di\ p^'l, p^2 ^^^ unknown dipole moments along the directions e"'/, e^2 selected in the tangential plane to 5'/,aux in point Mnj. The fields (8.21) and (8.22) satisfy Eqs. (8.17) and (8.18), respectively. Moreover, the field (8.21) satisfies the radiation conditions (8.20). Using the collocation method on Eq. (8.19), we get the following system of linear algebraic equations with a complex matrix of dimension 4L x (2Ne + 2Ni) for unknown complex constants p^f, p^^'^ (n = l,2,..., Ne), p'l'l, p'li (« = 1, 2 , . . . , A ,):

n ^ x ( E f ' ^ - E f ^ ) = n ^ x E ^ ,

n ' X (Hf' " - H f ^ = n ' X H^, 7 = 1,2,..., L. (8.23)

The solution of the system (8.23) is found via minimization of the function:

L

CD = ^[ |n^- X (Ef' " -Ef '^) -n^' x E^|'

7=1

+ ^ |n^- X (Hf'^' - H f ^ - n " x H ^ | ' j . (8.24)

After the solution of system (8.23) has been determined, any scattering characteristic may be calculated from Eq. (8.21). In particular, for field scattering components in far zone the expressions (8.7)-(8.8) are valid.

Control of the solution accuracy is realized via calculation of the value

Q = (<I>7<I>o)'/

L'

Oo = ^ { | n - x E - | ' + (/x./^.)|n- x H - | 2 J , (8.25) m=\

where O is the value of function (8.24) on a set of intermediate points Mm (m = on (5sc and OQ is the value of the corresponding norm of exciting field on

the same set of points. For representations (8.21)-(8.22) the next theorem is vahd:

182 A. Dmitrenko

THEOREM 3. Let Ssc e A^^'^\ and the sets {Mn,e}^^x and {M„,/}^^ be dense on the surfaces 5e,aux and 5'/,aux respectively, and the surface Se,m\ be a nonresonant surface. Then for any {fi,f2} € L\{S^c) x L\{S^c) and 5 > 0 there exist such Ne(8), Ni(8) and f n,e n,e^Ne r n,i nJ^Ni .j .

n x ( E f - E f ) - f i n x ( H f - H f ) - f 2 < 5 ,

where Ef, Hf and Ef", Hf are defined by (8.21) and (8.22).

8.5. Chiral scatterer

Let us consider the scattering problem for a body prepared from isotropic chiral mate-rial. The difference of isotropic chiral material from isotropic magneto-dielectric material in respect to their macroscopic electrodynamics, appears in the form of the constitutive relations: the vectors of electric and magnetic inductions are connected by both the electric and magnetic fields. Different forms of constitutive relations exist for chiral media [17]. In this paper we will use the constitutive relations in the form

D,- = 8i(Ei -\-pVx E/), B/ = fii(Hi + ^V X H/), (8.26)

where vectors D/, B/ are the vectors of electric and magnetic inductions respectively, vec-tors E/, H/ are the vectors of electric and magnetic fields, si, fit are the permittivity and permeability of chiral medium, and )S is a chirality parameter.

If the constitutive relations are chosen in form (8.26), the mathematical formulation of the scattering problem for a chiral body Dt, bounded by the surface ^sc, and located in an infinite homogeneous medium De with parameters Se, /Jie, is

V X E^ = icofiel^e, V X H^ = -icoSeEe in De, (8.27)

V x E , = i a ) / x / ( H / + ^ V x H , ) ,

V X H,- = -ia)8i(Ei -\-pVx E/) in A , (8.28)

n X (Ei - E^) = n X Eo, n x (H,- - H^) = n x HQ on Ssc, (8.29)

{ V ^ E , ; V M 7 H , } X R/R + {Vi^H,; - V ^ E , } = 0(R-^), R->oo, (8.30)

where {E^, H^} is the unknown scattered field in De, and {E/, H/} is the unknown field inside the chiral body.

As for a magneto-dielectric scatterer, for construction of a solution to the prob-lem (8.27)-(8.30) we introduce two auxiliary surfaces Se,mx = KeSsc and St^^ux = KtSsc which are homothetic to the surface Ssc The surface Se,a.ux with the homothetic coefficient i g < 1 is placed inside D/, and the surface 5'/,aux with the homothetic coefficient Kt > 1 is placed outside Di. As for the scatterers considered above, the scattered field {E , H^} in De is presented as a sum of fields of the tangentially oriented elementary electric dipoles with


unknown moments ;?"'j^, p^^^ (^ = 1,2,..., A^ ), placed on the auxiliary surface 5' ,aux

points {M„,^}^^i: in

Ef(M) = ^ ^ V x ( V x n , , , ) , ^ n=\

^ Ne

Hf(M) = — ^ V x n „ , „ n=l

n,e

I n=l

Ni

Pr = P^f + Vl^2 = P^t < f + P^l <t^ M ^ ^e. (8.31)

The field {E , H/} inside the chiral body Di is presented as a sum of fields of analogous dipoles which are placed on the surface 5/,aux in the points {M^,/}^^^. Using Green's func-tion [17] for the infinite homogeneous chiral medium with the parameters £/, /x/, yS, we have the representations

Erw = ^ | E | v x [ v x « > + n2)]

+vx[vx«>-n2)]), n**; = irf\M, M„j)p"/, n g = ,Af (M, M„,,-)p«-', ^n,i ^n,i , ^n,i n,i n,i . n,i n,i n/r ^ r\ /o ION Pr = P r l + P r 2 = PT\^X\ + PT2^T2^ M e Di, (8.32)

where ki = co^st /x/; yi =ki/(l—kiP) and y2 = ki / (I-\-ki fi) are the wave numbers of the left- and right-polarized waves, V . ' (M, Mnj) = exp(iyi,2/^)J^')/(47r/?^') are the scalar Green's functions, R^^ is the distance from point Mnj on / aux to point M in A ; /^^'/, p^'^ (n = 1, 2 , . . . , A /) are unknown dipole moments and e 'j and 6 2 are non-coUinear directions selected in the tangential plane to / aux in the point Mnj.

The fields (8.31) and (8.32) satisfy equations (8.27) and (8.28), respectively. Moreover, the field (8.31) satisfies the radiation conditions (8.30). To satisfy the boundary condi-tions (8.29), it is necessary to define the unknown constants p^'^, p^2 (^ = 1,2,..., A^ ), ^ r / ' ^r2 (^ = 1 2 , . . . , A /) in a suitable manner. As usual, for this purpose we use the collocation method. Let Mj (j = 1,2, . . . ,L ) be collocation points on Ssc- Then we get the following system of linear algebraic equations with a complex matrix of dimen-sion 4L X (2Ne + 2Ni) for unknown constants ;7"f, p'^'2 (n = l,2,..., Ne), /?"'/, p'^'2 (^ = 1,2,...,A^,):

n ^ x ( E f ' ^ - E f ' ^ ) = n ^ ' x E ^ ,

n ' X (Hf' " - H f ^ ) = n ' x H^, 7 = 1,2,..., L, (8.33)

184 A. Dmitrenko

where Ef '^ Hf'• ' and Ef '^ Hf'• ' are the values of fields (8.31) and (8.32) in point Mj on the surface of the chiral body.

The solution of the system (8.33) is found via minimization of the function

+ ^ |n ' X (Hf' ^ - Hf'^) - n ' x H^f 1. (8.34) ^e J

After the solution of system (8.33) has been determined, any scattering characteristic may be calculated from (8.31). For field scattering components in the far zone the expres-sions (8.7)-(8.8) are valid.

Control of the solution accuracy is realized via calculation of value

Q = (<D7<I>o)' ', L'

<Do = 2 _ ^ j | n ' " x E g ' | V ^ | n ' " x H ^ | ^ [ , (8.35)

where ^' is the value of function (8.34) on a set of intermediate points Mm (m = 1,2,..., LO on sc and OQ is the value of the corresponding norm of the exciting field on the same set of points.

For representations (8.31)-(8.32) the next theorem is vaHd:

THEOREM 4. Let Ssc e A^^'"\ and the sets {Mn,e}^^i ^^^ {^n,i]^=i he dense on the surfaces 5 ,aux cind 5'/,aux respectively, and the surface Se,mx he a nonresonant surface, then for any {fi,f2} G L2(*S'sc) x L2(5'sc) and 5 > 0 there exist such Ne(S), Ni(8) and f n,e n,e^Ne t n,i nJ^Nt .j . ^Prl ' PT2 }„=!' {Pr\ ' Pr2 UU' ^f^*

n x ( E f - E f ) - f i n x ( H f - H f ) - f 2

<5,

where Ef, H^ andEf^, Hf are definedby (8.31) and (8.32).

8.6. Coated scatterer

The geometry of the scattering problem is shown in Fig. 8.3. A perfectly conducting body D enclosed by a coating Dt, is located in an infinite homogeneous medium De with the permittivity Se and permeability fie. It is presumed that the coating is prepared of isotropic chiral material, and its constitutive parameters are: 8i, /x/, )S; if chirality parameter ^ is equal to zero, the chirality coating is a magneto-dielectric one. The body D is bounded by the surface S, and the coating Z)/ is bounded by the surface Sc- It is supposed that surfaces S and Sc have an arbitrary shape, but the surface Sc is homothetic to S around the origin O of a Cartesian coordinate system. The structure is excited by a stationary


i . . : ^ ^ {Ee,He}

{Eo,H,}

Fig. 8.3. Geometry of the problem for a coated scatterer.

electromagnetic field {EQ, HQ} (time dependence exp(—ICDO)- It is necessary to find the scattered field {E , H^} in D^.

The fields {E^,H^} in De and {E/,H/} inside the coating must satisfy the Maxwell equations

V X Eg =ia)ljielie, V xlle =—icose^e in D^, (8.36)

V X E,- = icofXi(Hi + ^ V X H / ) ,

V X H/ = -icosi(Ei +PV X Ei) in A , (8.37)


Uc X (Ei -Ee) =ncx Eo, n^ x (H/ - H ^ ) = iic x Ho on Sc, (8.38)

nxEi =0 on S, (8.39)


{ V ^ E , ; V M I H , } X R/R + { V A ^ H , ; - V ^ E , } = OiR'^), R~^oo, (8.40)

where n^ and n are the normal unit vectors to Sc and S, respectively. The solution to the problem (8.36)-(8.40) is constructed in the following way. Let us

(1) ^ ( i ) ( M) ri2), introduce three auxihary surfaces Se aux = KeS, 3}'^, = K}'^S and S}Z, = K^S which are homothetic to the surface S. The surface Se,aux with the homothetic coefficient Ke is placed inside region DUD/ , the surface 5 ! ^ ^ with the homothetic coefficient K^^^ <lis

placed inside D and the surface S^ ^^ with the homothetic coefficient K^ > I is placed in De. Let us represent the scattered field {E , H^} in De as a sum of fields of fictitious elementary electric dipoles which are placed on the surface Se,mx in the points {Mn,e}nLi and are oriented tangentially to »Sg,aux*

186 A. Dmitrenko

E,^(M)-g^Vx(Vxn„,,),

Ne

Hf(M) = — V V x n „ , ,

n,e Tk^^^ _L rfc"' n'^'^ cP^'^ P r i + Pr2 = Prl ^ r l + /^r2 "^rl ^ ^ ^ ^ ^ • (8.41)

The field {E/, H^} inside the coating Dt is given by the sum of fields of analogous dipoles ^(1) r.(2) 1^1 placed on the auxiliary surfaces ^ ^ux ^^^ * /aux ^ ^^^ points {Mnj,i}^^i and {Mn,i,2}nLi •

• i V i

Ef (M) = -1 ico

N2

E | v X [V X (n :,), + nS,)] + V X (y^n^l -nnS,)) .n=l

+ ^ j v X [V X (n^,, + nS,,)] + V X (nni:>, - ^nS,))

Hf(M) 1

A 2

+ E h X (n<>, + nnS,^) + V X [V X (n<;,',, - n^.^)]) «=1

nlXf = V- ' ^ ^ ' M,,u)P?'^'\ n; X2' = fi\M, Mnj,2)pr^\ ^n,i,l -j.n,i,l i^rij,! _ nj,l n,i,l , n,i,l n,i,l Pr — P r l ^ P r 2 — Prl *^rl ^ " r2 * r2 '

r^nJ,2 n,i,2 n,i,2 vr'=Kr+K2=Pr n,i,2 nj,2 + Pr

n,i,2 n,i,2 ^z2 MeDi, (8.42)

where

xl/> \M,Mn,i,i) = 4nR

ki=0)^6ifli, yi,2 =

,n,i,l M ki

irr(M,M.,,2) = ^ ' ' ' ' ' ' ^ ' ^ 47cR

,n,i,2 M

iTkiP'

R^^'^ and /?^''^ are the distances from the points Mn,i,\ on s\^l^^ and Mn,i,2 on s\^l^^ to

the point M in A , P^'/ ' \ / ^ '' (n = 1,2,..., iVi) and /7^'/'^ ^^2''^ (AZ = 1, 2 , . . . , 7V2)

are unknown dipole moments, e^'/' , 6^2' and e"'/' , 6^2' are non-coUinear directions ,(1) (2) selected in the tangential planes to S) i^^ and S) 1,^ in the points M„ /1 and Mni2, Ni and

N2 are the numbers of dipole points on SJ^^^ and S^^^^, respectively. The fields (8.41) and (8.42) satisfy the equations (8.36) and (8.37), respectively. More-

over, the field (8.41) satisfies the radiation conditions (8.40). To satisfy the boundary con-ditions (8.38)-(8.39), it is necessary to define the unknown dipole moments in a suitable


manner. Let Mcj (j = 1, 2 , . . . , Li) and Mj (j = 1,2,..., L2) be collocation points on Sc and S, respectively. Then we get the following system of linear algebraic equations with a complex matrix of dimension {4L\ + 2L2) x (2Ne + 2N\ + 2N2) for unknown dipole moments p^f, 77 2 (n = 1, 2 , . . . , A ,), /7^'/'\ ^^2''^ (n = 1,2,..., A/ i), /?^'/'^

/7^2'''(^ = 1.2,...,iV2):

n ^ x ( H f ' ^ - H f ^ ) = n ^ x H ^ , j = 1,2,..., Li,

n ' X Ef' " = 0 , 7 = 1,2,..., L2. (8.43)

The solution of the system (8.43) is found via minimization of the function

<D 1

+

+

1 ^

/ ^ £ , j

: (Ef' ^ -

X ( H f • •

xEf'i

-EfO-

- H ^ )

2

l ie X

-n^' xHÎ

(8.44)

After the solution of system (8.43) has been determined, any scattering characteristic may be calculated from (8.41). For field scattering components in the far zone, the expres-sions (8.7)-(8.8) are valid.

Control of the solution accuracy is realized via calculation of the value

Q = (07O0) 1/2

*»=5l <xE™|^ + ^|n™xH2'|^[, (8.45)

where O is the value of function (8.44) on the set of intermediate points on the surfaces Sc and S, and OQ is the value of the norm of exiting field on the set of intermediate points on Sc.

For representations (8.41)-(8.42) the next theorem is valid:

T H E O R E M S . Let S e A^^'^\ and the sets {M„,^}^p {M„,/,i}^^ and {M„,/,2}î be

dense on the auxiliary surfaces Se^mx, S\ l^^ and S\ ^^ respectively, and the surfaces 5' ,aux

and •S -aux ^^ nonresonant surfaces, then for any {fi,f2,f3} G L^2^S) x U2{Sc) x U2^Sc)

and8>0 there exist such Ne(8), Nii8), N2(8) and {p^t.P^^fnU^ IPr'/'^'/^r2'^}îi' f «,/,2 n,i,2^N2 >7 .

n x ( E f - E ^ ) - f 2 n x ( H f - H f ) - f 3 inxEf-fi|L.c.) +

where Ef, Hf and Ef, Hf are defined by (8.41)-(8.42)

188 A, Dmitrenko

8.7. Some ideas towards the solution of dense ill-posed linear algebraic equation systems of discrete sources method

The chief difficulty we encountered in our first efforts using the DSM for numerical solution of three dimensional nonaxisymmetrical scattering problems, was the solution of systems of linear algebraic equations. In this subsection for abbreviation purposes, we will denote the system of linear algebraic equations as

Ap = f, (8.46)

where A is a complex matrix, p is an unknown vector of dipole moments, and f is a known vector consisting of exiting field components.

Different methods of solution of system (8.46) were tested. First, we turned to the well-known Gaussian elimination method (GEM), but quickly

observed that GEM yields the correct solution only for individual values of parameter Ke. For the majority of values of Ke the derived solution did not make any physical sense. To understand this we investigated in detail the dependence of the boundary condition residual value (8.9) on the homothetic coefficient Ke for different forms of bodies. A typical result is curve 1 shown in Fig. 8.4. It refers to the scattering problem of a perfectly conducting ellipsoid with the semi-axes kea = 1.28, keb = 1.5, keC = 1.72. In the coordinate system of Fig. 8.1, the semi-axes kea, keb, and keC were directed along the x-, y-, and z-axes, respectively. The ellipsoid was excited by a plane wave propagating along the z-axis so that the electric field EQ was oriented along the jc-axis. The parameters of the method were chosen as Ne = 30, L = 30. The dipole and collocation points were allocated on the corresponding surfaces in the same manner: in each semi-section (p = const to be separated on the angular distance A^ = 60°, 5 dipole (collocation) points were evenly distributed, and the system of linear algebraic equations with square complex matrix of dimension 60 X 60 was solved by GEM.

As can be seen in Fig. 8.4, the dependence of the residual value Q on Ke includes a series of minima. The positions of these minima depend on the geometrical parameters of the body and on the quantities and allocations of dipole and collocation points. In the vicinity of the minima the small variations of Ke lead to small variations of Q. The analysis shows that the solutions related to the minima are the physically correct solutions. In the intervals between the minima small variations of Ke lead to large variations of Q- A visible feature of the obtained solution is the nonstability. In these intervals the values of Q can attain very large magnitudes, and the obtained solutions make no physical sense.

Thus, the GEM cannot be recommended for solution of systems of linear algebraic equations arising in DSM. There are only a few method parameters for which the GEM can yield the correct solution. For more complete scatterers (for example for a magneto-dielectric scatterer) there is a problem: all regular structures of Q on Ke are de-stroyed.

After the GEM had been tested, we turned to the singular value decomposition method (SVDM) [18]. This was quite logical since up to the time of our numerical experiments (1986) the SVDM had already been used [19] for solution of systems of linear algebraic equations of DSM. The results obtained by SVDM for the scattering problem described above under the same conditions, are shown in Fig. 8.4 by circles. In interval Ke > 0.15 the results obtained by SVDM coincide with those obtained by GEM. As we can observe in Fig. 8.4, the SVDM gives acceptable results only for small values of parameter Ke, and also the obtained values of the residual value Q are large enough.


Fig. 8.4. Dependencies of boundary condition residual value versus the homothetic coefficient for a perfectly conducting ellipsoid with semi-axes kga — 1.28, keh = 1.5, keC — 1.72. Curve 1: results obtained by GEM for a system of dimension 60 x 60. The circled markers: results obtained by SVDM for the same system. Curve 2: results obtained by CGM for a system of dimension 60 x 60. Curve 3: results obtained by CGM for a system of

dimension 120 x 60.

Naturally, the observed trouble phenomena are connected with ill-condition of the solved system. The analysis detects that condition number cond A of a system of dimension 60 x 60 lies in interval 10^ < cond A < 10^ (dependent on the value of parameter Ke). The question was: how to overcome the existing difficulties?

We turned our attention to iterative methods. Iterative methods have several advantages in comparison to direct methods of solution of system (8.46). The chief advantage is that iterative methods do not accumulate round-off errors. Also, iterative methods may be real-ized without keeping the full matrix of solved systems in the computer memory, therefore they may be used for solutions of systems of linear algebraic equations of extremely high dimensions. But the attributes of matrix A in (8.46) do not allow application of a majority of iterative methods directly to the system (8.46), therefore we looked at the problem of solution of system (8.46) as a problem of minimization of the function

0= | |Ap-f | |^^ , (8.47)

which is the square of Euclidian norm of residue of the system (8.46). It is well known that the problem of minimization of function (8.47) is equivalent to the problem of solving the normal system

A*Ap = A*f, (8.48)

where A* is the transpose conjugate of A, but we looked at the problem of minimization of function (8.47) as a problem of minimization of a square function of many variables. The solution of the last problem can be obtained by using any iterative procedure, for example, the coordinate descent method. It is also well known that the condition number of matrix

190 A. Dmitrenko

A* A is equal to the squared condition number of matrix A, but if any iterative procedure is used, it influences the rate of convergence of the iterative procedure.

We investigated the efficiency of different iterative methods for solution of the prob-lem of minimization of (8.47) and found that the conjugate gradient method (CGM) is a reasonable method. Beginning in the year 1986, we used CGM for the solution of linear algebraic equation systems of DSM. All numerical results presented in this paper were obtained by using CGM. For the scattering problems described in this subsection, the de-pendencies of Q on Ke, obtained by using CGM, are shown in Fig. 8.4 as curves 2 and 3. Curve 2 shows results obtained under the same conditions as results presented by curve 1 and circled markers, and obtained by GEM and SVDM. Curve 3 shows results obtained by CGM for an overdetermined system of dimension 120 x 60. In this case the allocation of dipole points was preserved in the former manner, but the collocation points were allocated so that in each semi-section cp = const to be separated on the angular distance A(p = 36° were 6 evenly distributed collocation points, L = 60. As can be observed, there exists an interval of Kg in which the solution of system (8.46) is stable, if the CGM is used. The width of the stability interval increases with "reinforcing" of the overdetermination of the solved system of linear algebraic equations.

8.8. Numerical results

The numerical methods described in Sections 8.2-8.6 were realized with the software package. The input data of the package were the exciting field, the geometry of the scat-terer, the constitutive parameters st, fit, p (or impedance rf), the thickness of coating (for coated scatterer), and also the method parameters: the homothetic coefficients of the aux-iliary surfaces and the numbers of the dipole and collocation points. The problem of min-imization of the corresponding function was solved by CGM. As a rule, the initial values of unknown dipole moments were chosen to be equal to zero. The interruption of the con-jugate gradient iterative process was executed upon the condition

{^k-^M)/^k^^, (8.49)

where O^ is the value of minimized function after executing the A:-th iteration, and O^+i is the value after executing the {k + l)-th iteration. By using this package a large set of computer calculations was performed. These calculations were directed at the investiga-tion of the influence of method parameters on solution accuracy, detection of adequacy of the results obtained and some known results, and also at the research of the influence of geometric and electrodynamic parameters on scattering properties of different shaped bodies. In this section some typical and interesting results will be presented.

8.8.1. Some particularities of the conjugate gradient method

The numerical experiments detected some general trends in the behaviour of the conju-gate gradient iterative process for different types of scatterers. One such trend is that the highest rate of convergence of CGM takes place for the first iteration steps and decreases rapidly with the number of executed iteration steps. Therefore, a small number of iteration steps can be used in the conjugate gradient iteration process, thus reducing computer time.

1 . 2

0 . 9

0 . 6

0 . 3

0 . 0


Fig. 8.5. Dependencies of values of minimized function versus iteration number k. Curve 1: results for a perfectly conducting ellipsoid with kga = 3.0, keb = 2.0, keC = 1.0. Curve 2: results for a dielectric cube with s = 0.2X,

6i/£e=4.0.

As typical examples, Fig. 8.5 shows the plots of O / O Q versus the number of iteration steps for a perfectly conducting ellipsoid (curve 1) and dielectric cube (curve 2). Here O is the value of function (8.6) or function (8.34) after executing the corresponding iteration, OQ is the initial value of function (8.6) or function (8.34) obtained under the condition that initial values of dipole moments are chosen to be equal to zero. The perfectly conduct-ing ellipsoid has the semi-axes: kea = 3.0, keb = 2.0, keC = 1.0. In the coordinate system of Fig. 8.1, the semi-axes kea, keb, and keC are directed along the x-, y-, and z-axes, re-spectively. The ellipsoid is excited by a linearly polarized plane wave propagating along the z-axis so that the electric field EQ is oriented along the x-axis. The parameters of the method are: Ke = 0.5, A^ = 121, L = 242. The dipole points are allocated on the aux-iliary surface Se so that in each semi-section (p = const to be separated on the angular distance Acp = 32.72° are 11 evenly distributed dipole points. The collocation points are allocated on S as in the above-mentioned semi-sections, so in semi-sections are placed be-tween them. In each semi-section cp = const the collocation points are allocated in the same manner as the dipole points. The dielectric cube has the parameters: side length s =0.2X, permittivity st/Se = 4.0 and permeability /x/Z/x^ = 1.0. The cube is excited by a plane wave propagating perpendicular to the face of the cube. The method parameters for the di-electric cube are: Ne = Nt = 54, L = 150, Kg = 0.4, Kt = 4.0. The dipole and collocation points are allocated only on the faces of the auxiliary and scattering cubes. On each face of the auxiliary cube, 9 dipole points are evenly distributed, on each face of the scattering cube, 25 collocation points are placed evenly as well. As can be seen from Fig. 8.5, it is sufficient to execute 20 iterations to obtain the solution for a perfectly conducting ellipsoid, and 30 iterations for the solution the dielectric cube.

The second particularity observed of the conjugate gradient iterative process is the nonuniformity of the rate of convergence. With the scale of Fig. 8.5 this nonuniformity can hardly be seen. To illustrate the nonuniformity of the rate of convergence of the con-jugate gradient iterative process, in Fig. 8.6 the dependence of AO = (c^j^ — <t>k+i) on the

192 A. Dmitrenko

AO

0 . 0 6

0 . 0 4

0 . 0 2 H

0 . 0 0

Fig. 8.6. Dependence of A4> versus iteration number k for dielectric cube with s = 0.2 X, Ei/Se = 4.0.

number of iteration steps is shown. These results relate to the scattering problem described above for a dielectric cube. As can be seen, the rate of convergence of the iterative process changes rapidly with the number of iteration steps, but on average there is a trend to a decreasing rate of convergence with increase of the number of executed iteration steps.

8,8.2. Influence of the auxiliary surfaces positions on solution accuracy

The numerical experiments show that the positions of the auxiliary surfaces have an influence on the accuracy of the solution of the scattering problem under consideration. These effects for perfectly conducting and dielectric scatterers are illustrated in Figs. 8.7, 8.8, and 8.9.

In Fig. 8.7 the dependence of the residual value (8.9) on the position of the auxiliary surface Se for perfectly conducting scatterers is shown. We recall that the position of Se is characterized by parameter Ke. Curve 1 refers to the scattering problem for a perfectly conducting ellipsoid with semi-axes kea = 3.0, keb = 2.0, keC = 1.0, described in subsec-tion 8.8.1. The method parameters are chosen in the same manner: A^ = 121, L = 242. The auxiliary surface is an ellipsoid with semi-axes Kekea, Kekeb, KekeC. Curve 2 refers to a cylinder of length kel = 5.0. The cross section of the cylinder is an ellipse with semi-axes kea = 1.0, keb = 0.5. The end faces of the cylinder are smoothed by caps of height keh = 0.5. The cylinder is excited by a plane wave propagating along the semi-axis kea so that the electric field EQ is oriented along the cylinder axis. The auxiliary surface is a cyhnder with smoothed end faces and the geometrical parameters Kekel, Kekea, Kekeb, Kekeh. The numbers of dipole and collocation points are chosen as A^ =96, L = 192. The dipole points are allocated on the auxiliary cylinder in the following manner: 60 points on the cylindrical piece and 18 points on each cap. The collocation points are distributed on the surface of the cylinder so that on the cylindrical piece there are 120 points and 36 points are on each cap. Curve 3 refers to the cube with side length s = 0.75 X (keS = 4.112) and broadside plane wave incidence. The auxiliary surface is a cube with side KekeS. The dipole and collocation points are allocated on the faces of the cubes only. On each face


1 . 0

0 . 8

0 . 6

0 . 4

0 . 2

0 . 0

Q

0 0 .2 0.4 0 .6 0,

Fig. 8.7. Dependencies of boundary condition residual value on the homothetic coefficient Kg for perfectly con-ducting scatterers. Curve 1: results for an ellipsoid with semi-axes kga = 3.0, kgb = 2.0, kgC = 1.0. Curve 2: results for a cylinder of length kgl = 5.0 and of an elliptic cross section with semi-axes kga = 1.0, keb = 0.5.

Curve 3: results for a cube with a side length of 0.75 A.

K,

Fig, 8.8. Dependencies of the boundary condition residual value versus the position of inner auxiliary surface, if the position of outer auxiliary surface is fixed: Ki = 4.0. Curve 1: results for ellipsoid with semi-axes kga = 3.0, keb = 2.0, kgC =1 .0 and permittivity Si/Se = 4.0. Curve 2: results for cylinder of length kgl = 5.0, permittivity of 8i /Se = 4.0 and of elliptical cross section with semi-axes kga = 1.0, kgb = 0.5. Curve 3: results for cube with

side length of 0.75 X and permittivity et/se = 4.0.

of the auxiliary cube, 9 dipole points are evenly distributed; on each face of the scattering cube, 81 collocation points are evenly placed, so that Ne = 54, L= 486.

As can be seen, for different geometrical forms, the same interval 0.2 < Ke < 0.5 exists in which the smallest boundary condition residual values can be attained. In domain Ke <

194 A. Dmitrenko

0.2 the increase of the residual value is explained by a degrading of the condition number of the linear algebraic equations system connected with condensation of dipole points. In domain A' > 0.5 the increase of the residual value is explained by discreteness of sources.

Figures 8.8 and 8.9 illustrate the influence of the positions of auxiliary surfaces on the solution accuracy for dielectric scatterers. The plots in Fig. 8.8 represent the boundary con-dition residual value (8.25) as a function of position of the inner auxiliary surface, if the position of the outer auxiliary surface is fixed: Kt = 4.0, and the plots in Fig. 8.9 repre-sent the boundary condition residual value (8.25) as a function of the position of the outer auxiUary surface, if the position of inner auxiliary surface is fixed: Ke = 0.4. The curves 1 in Figs. 8.8 and 8.9 refer to the scattering problem for dielectric ellipsoid with semi-axes kea = 3.0, keb = 2.0, keC =1 .0 and permittivity et/Se = 4.0. The ellipsoid is excited by a linear polarized plane wave propagating along the ^^c-semi-axis, so that the electric field is oriented along the A:^a-semi-axis. The method parameters are: Ne = Ni = 64, L = 128. The dipole points are allocated on the auxiliary surfaces Se and St so that in each semi-section (p = const to be separated on the angular distance A(p = 45°, 8 dipole points are evenly distributed. The collocation points are allocated on S in the semi-sections mentioned above and in semi-sections placed between them; in each semi-section (p = const the col-location points are allocated in the same manner as the dipole points. Curve 2 in Figs. 8.8 and 8.9 refers to the dielectric cylinder with geometric parameters kel = 5.0, kea = 1.0, keb = 0.5, permittivity si/Se = 4.0, which is smoothed by elliptic caps end faces. The cylinder is excited by a plane wave propagating along the A:^«-semi-axis, so that the elec-tric field Eo is oriented along the cylinder axis. The numbers of dipole and collocation points are chosen as A^ = Ni = 64, L = 128. The dipole points are allocated on the auxil-iary cylinder in the following manner: 40 points on the cylindrical piece and 12 points on each cap. The collocation points are distributed on the surface of the cyUnder, so that on the

1 .0 Q

0.8 j

0.6-1

0.4

0 .2

0.0

J \

J '

0 2 4 6

Ki

8 10 12

Fig. 8.9. Dependencies of the boundary condition residual value versus the position of outer auxiliary surface, if the position of inner auxiUary surface is fixed: Ke = 0.4. Curve 1: results for ellipsoid with semi-axes kga = 3.0, keb = 2.0, keC =1.0 and permittivity et/Se = 4.0. Curve 2: results for cylinder of length kgl = 5.0, permittivity of Ei/Se = 4.0 and of elliptical cross section with semi-axes kea = 1.0, keb = 0.5. Curve 3: results for cube with

side length of 0.75 k and permittivity e,- /se = 4.0.


cylindrical piece there are 80 points and 24 points on each cap. Curve 3 in Figs. 8.8 and 8.9 refers to a cube with side length s = 0.75 X and broadside plane wave incidence. The dipole points and collocation points are allocated on the faces of the cube only. On each face of the auxiliary cube, 9 dipole points are evenly distributed, on each face of the scattering cube, 25 collocation points are also evenly placed, so that A^ = Nt = 54, L = 150.

As can be seen in Fig. 8.9, the position of the outer auxiliary surface in interval 2.0 < Ki < 10.0 has little influence on the boundary condition residual value, i.e., solu-tion accuracy. The curves in Figs. 8.7, 8.8, and 8.9 show that the dependence of boundary condition residual values versus the position of the inner auxiliary surface are similar for perfectly conducting and dielectric scatterers.

Analogous investigations were performed for coated scatterers. It was found that the

optimal values of homothetic coefficients Kg, K\ \ and K\ \ yielding the least bound-

ary condition residual values, he in the intervals 0.3 < Ke < 0.6, 0.3 < - ^ < 0.5 and

3.0</^P<5.0.

8.8.3. Comparisons

The validity of results produced by the developed variants of DSM was verified by com-parison to results obtained by other authors. Below, some comparisons relating to perfectly conducting, dielectric and coated bodies will be presented.

Figure 8.10 shows the results of a comparison of the backscattering cross section fre-quency dependence for a perfectly conducting cube with side length s, excited by a lin-early polarized plane wave, with that published in [20]. The solid curve represents the results of paper [20], the circles correspond to results obtained by the method presented in Section 8.2, with Ke = 0.5, A^ = 54, L = 486. The dipole and collocation points are distributed as described in subsection 8.8.2.

G/X^[dB]

10 12

Fig. 8.10. Backscattering cross section frequence dependence for a perfectly conducting cube. Solid curve: results of paper [20]. Circled points: results of the method presented in Section 8.2, with Kg = 0.5, Ng = 54, L= 486.

196 A. Dmitrenko

l E e l / I E e l :

0 30 60 90 120 150 180

e [deg]

Fig. 8.11. Comparison of normalized far fields for 0.2 X dielectric cube of £, /«« = 4.0. Curve 1: results obtained by the presented method with Ke = 0.4, Ki = 4.0, Ne = Nt = 54, L = 150. Curve 2: results of paper [21].

G/X^ [dB]

0 30 60 90 120 150 180

0 [deg]

Fig. 8.12. Radar bistatic cross sections of a coated perfectly conducting sphere of radius kga = 1.957. Parameters of coating: keac = 2.475, st/se = 2.0, /z/Z/Xe = 1- Parameters of method: Ke = 0.5, KJ^^ = 0.4, KJ^^ = 5.0, Ne = Ni = N2 = 40, Li = L2 = 80. Curve 1: results produced by Mie solution. Curve 2: results produced by

the method presented in Section 8.6.

Figure 8.11 shows the results of comparison of the normalized scattered field for a 0.2 A, dielectric cube of st/Se = 4.0, excited by a linearly polarized plane wave, with that pub-lished in [21]. Curve 1 shows the results obtained by the method presented in Section 8.4, with Ke = 0.4, Ki = 4.0, Ne = Nt = 54, L = 150. The dipole and collocation points are


distributed as described in subsection 8.8.1. The results of the comparison of radar bistatic cross sections for a coated sphere of radius

kea = 1.957 are presented in Fig. 8.12. The dielectric coating has the parameters: keac = 2.475, Si/Se = 2.0. Curve 1 shows the results produced by Mie solution, and curve 2 shows the results produced by the method presented in section 8.6. The parameters of the method are: Ke = 0.5, K^^^ = 0.4, /^f ^ = 5.0, Ne = Ni=N2 = 40, Li = L2 = 80. The dipole points are allocated on the auxiliary surfaces Se, S^ , and S^ so that in each semi-section (p = const to be separated on the angular distance A(p = 45°, 5 dipole points are evenly distributed. The collocation points are allocated on S and Sc as in the above-mentioned semi-sections, meaning in semi-sections placed between them; in each semi-section cp = const the collocation points are allocated in the same manner as the dipole points.

As can be seen, there is good agreement between the results obtained by our methods, and results obtained by other authors. Here it is relevant to remark that the results presented do not exhaust work directed towards testing of numerical methods. As a rule, we obtained true results for scattering characteristics in the far zone, even if the boundary condition residual values Q exceeded 20, 30 or more percentages.

8.8.4. Influence ofchirality on scattering properties of different shaped bodies and coatings

With the numerical methods presented above and the developed software package, it is possible to solve a large set of applied problems of practical importance. In particular, we used this package to investigate the scattering properties of homogeneous chiral bodies, and of structures covered by a chiral shell. It is well known [22] that one of the distinct manifestations of "chirality" is the existence of a cross-polarized component in the field scattered by the chiral object. Therefore we analyzed the co-polarized scattering field com-ponent, and cross-polarized scattering field component. The radar bistatic cross sections (BCS) for the co- and cross-polarized scattering field components are given by the expres-sions

aee= Hm 47TR^^-^^, ae^= lim AnR^^-^^ (8.50)

in which Ee^e and Ee,(p are the field components (8.7)-(8.8) in the spherical coordinate system.

Below, some results concerning the influence of chirality on scattering properties of bodies of different shape, and of structures covered by chiral shells, are presented.

Figures 8.13 and 8.14 show the modification of BCS towards "reinforcing" of the chiral-ity parameter of the scatterer. The results refer to a sphere of radius ker = 1.5,^^ = 2n/X, where A is a wavelength in outer space. The following parameters of the sphere are used: Ei/se = 4.0, iJiil[le = l-O- The sphere is excited by a linearly polarized plane wave propa-gating along the z-axis of the Cartesian coordinate system with its origin in the centre of the sphere; vector EQ of the plane wave is directed along the x-axis (see Fig. 8.2). Figure 8.13 concerns the co-polarized field component. Fig. 8.14 concerns the cross-polarized field component. The results are given for a x-z-plane, the angle 0 is the angle between the prop-agation vector and the direction of observation in x-z-plane. The curves 1, 2, 3, and 4 refer to the cases when kifi = 0.1, 0.3, 0.5, and 0.0, respectively. The parameters of the method

198 A. Dmitrenko

GQQ/X^ [dB]

-10-^

- 3 0 0 30 60 90 120 150 180

0 [ d e g ]

Fig. 8.13. Radar bistatic cross sections for the co-polarized scattering field component of a sphere of radius kgr = 1.5 and of permittivity et/se =4 .0 versus the observation angle 0 for different values of the chirality

parameter fi. Curves 1, 2, 3, and 4 refer to the cases when ki^=0.l, 0.3, 0.5, and 0.0, respectively.

GQ^/X^ [dB]

0 30 60 90 120 150 180

0 [deg]

Fig. 8.14. Radar bistatic cross sections for the cross-polarized scattering field component of a sphere of radius kgr = 1.5 and of permittivity si/Se = 4.0 versus the observation angle 6 for different values of the chirality

parameter )8. Curves 1, 2, 3, and 4 refer to the cases when ki^ = 0.1, 0.3, 0.5, and 0.0, respectively.

are: Kg = 0.3, Kt = 3.0, Ne = Nt = L = 60. The dipole and collocation points are allo-cated on the corresponding surfaces in the same manner: in each semi-section cp = const separated by the angular distance Acp = 36° there are 6 evenly distributed dipoles (collo-cation points). Under these conditions the magnitude of the error function (8.35) does not

8. Scattering on nonaxisymmetrical bodies

CSQQIX^ [dB]

199

0 30 60 90 120 150 180

9 [deg]

Fig. 8.15. Radar bistatic cross sections for the co-polarized scattering field component versus the observation angle 9 for ellipsoids with the constitutive parameters Si/Se = 4.0, /x/Z/Xe = 1.0, ki^ = 0.3, but different ratio of

semi-axes. The curves 1, 2, and 3 refer to the cases when 5 = 0, 0.2, and 0.3, respectively.

Ge^p/X^ [dB]

-10

0 30 60 90 120 150 180

0 [deg]

Fig. 8.16. Radar bistatic cross sections for the cross-polarized scattering field component versus the observation angle 6 for ellipsoids with the constitutive parameters si/se = 4.0, At//Ate = 1.0, kip = 0.3 but different ratio of

semi-axes. The curves 1, 2, and 3 refer to the cases when 8 = 0, 0.2, and 0.3, respectively.

exceed 4%. Comparison of curves 1-4 reveals that chirality of the scatterer considerably influences the form of the scattering diagrams. It is interesting to note that a larger chirality parameter does not always lead to an increase in the level of the cross-polarized scattering field component (compare curves 1 and 3).

200 A. Dmitrenko

CQQ/X^ [dB]

0 30 60 90 120 150 180

e [deg ]

Fig. 8.17. Radar bistatic cross sections for co-polarized scattering field components of spherical structure, contain-ing a perfectly conducting core of radius kga = 1.5 and shell of radius kgac = 1.8, permittivity Si/sg = 3.0+i 1.5, and permeability fii/iXe = 1-0 versus the observation angle 0 for different values of the chirahty parameter ^.

Curves 1, 2, 3, and 4 refer to the cases when ^ = 0.0, 0.01 k, 0.02 k, and 0.03 k, respectively.

Figures 8.15 and 8.16 show the influence of the shape of a scatterer on BCS if the chirality parameter is fixed. They refer to ellipsoids with constitutive parameters st/Se = 4.0, /x///x^ = 1.0, kiP = 0.3 but different ratio of semi-axes. Semi-axes kea, kbb and keC are directed along the jc-, y- and z-axes of a Cartesian coordinate system with the origin in the centre of the ellipsoid and defined in the following manner: kea = (1 - 8)kb\ keb = 1.5; ^^c = (1 + b)kb. Such a definition of semi-axes allows a change in the shape of the scatterer by changing 5 only. If (5 = 0, the scatterer is a sphere. The ellipsoids are excited by a plane wave propagating along the z-axis so that the electric field EQ is oriented along the x-axis. The parameters of the method were chosen to be the same as for results shown in Figs. 8.13 and 8.14. Curves 1, 2, and 3 refer to the cases when 5 = 0 , 0.2, and 0.3, respectively. The results are given for the jc-z-plane. As can be seen in Fig. 8.16, the amplitude of the cross-polarized scattering field component in directions close to the forward scattering direction (^ = 0°) is more sensitive to the form of the chiral scatterer than that of the co-polarized field component. This effect can be used for the detection of forms of chiral objects. However, the cross-polarized scattering field component decreases significantly in the directions close to the backscattering direction {6 = 180°).

In some papers, for example, in [22], it is suggested that chirality intensifies the mecha-nisms by which electromagnetic energy is absorbed inside a body. This effect might have practical importance for raising the efficiency of electromagnetic energy absorbing coat-ings. To inspect this, we calculated the BCS of differently shaped, perfectly conducting bodies covered by absorbing coatings with increased chirality parameter.

Some results on this theme are presented in Figs. 8.17 and 8.18, where the radar bistatic cross sections in .x:-z-plane, i.e., E-plane, of coated spheres and ellipsoids are shown. For a spherical structure the radius kea of a perfectly conducting core is 1.5, and the radius

8. Scattering on nonaxisymmetrical bodies

GQQ/X^ [dB]

201

-15 0 30 60 90 120 150 180

e [deg]

Fig. 8.18. The same as in Fig. 8.17 but for ellipsoidal structures containing a perfectly conducting core of semi-axes kea = 2.0, keb = 1.2, kgC = 0.8, and a shell of semi-axes kgac, kebc, kgCc = l.2(kea, kgb, kgc).

keUc of the shell is 1.8. For an ellipsoidal structure the semi-axes of a perfectly conducting core are: kea = 2.0, keb = 1.2, keC = 0.8 and semi-axes of the shell are: keac, kebc, keCc = l.2(kea, keb, kec). In the coordinate system of Fig. 8.3 the ellipsoidal structure is placed so that semi-axes kea, keb, and keC are directed along the x-, y- and z-axes, and is excited by a plane wave propagating along the z-axis, so that the electric field EQ is oriented along the X-axis. The permittivities and permeabilities of spherical and elliptical shells are identical: Si/Se = 3.0-hil.5,/x//^6^ = 1.0. The curves 1,2, 3, and 4 refer to the cases when y6 =0.0, 0.01 k, 0.02 A, and 0.03 A, respectively. For both structures the parameters of the method are: Ke = 0.3, K^^^ = 0.4, /^f ^ = 5.0, Ne = Ni = N2 = 40, Li = L2 = 80. The dipole and collocation points are allocated on the corresponding surfaces in the same manner as those for a coated sphere in subsection 8.8.3. Under these conditions, the magnitude of the error function (8.45) does not exceed 8% for spherical structures, and 16% for elliptical structures.

The results presented in Figs. 8.17 and 8.18 and other results lead to the following con-clusions. Integrally, the trend to a decrease of BCS is seen, if the magnitude of the chirality parameter ^ is increased. The influence of the chirality parameter is not reduced only by the decrease of BCS. The presence of chirality leads to a redistribution of electromag-netic energy in outer space. For this reason, the magnitude of scattered field can increase significantly in some scattering directions. This effect takes place for a spherical struc-ture, for example. As seen in Fig. 8.17, the growth of the chirality parameter fi to 0.03 A leads to an increase of the backscattering cross section (direction 0 = 180°) by 7 dB in comparison to the spherical structure with an ordinary magneto-dielectric shell (^ = 0.0). Thus, in view of the decrease of radar backscattering cross sections, the presence of chi-rality may be ill-posed and for special cases, the rationality of chiral materials must be explored.

202 A. Dmitrenko

8.9. Conclusion

In this chapter, variants of DSM for numerical solution of electromagnetic scattering problems of three dimensional nonaxisymmetrical bodies of different physical nature were presented. The main ideas of the developed variants are: the auxiliary surfaces are chosen as homothetic to the surface of a scatterer; elementary electric dipoles tangentially oriented to the auxiliary surfaces as discrete sources are chosen, and a linear algebraic equations system of the collocation method is solved via minimization of the residual functional by the conjugate gradient method. It is evident that the tangentially oriented elementary mag-netic dipoles (or combinations of tangentially oriented electric and magnetic dipoles) can be used as discrete sources, and for minimization of residual function any other iterative method may be used. Rigorous mathematical formulations of the variants of DSM for per-fectly conducting, impedance, magneto-dielectric, chiral and coated scatterers were given.

Some results concerning the particularities of the conjugate gradient iterative process and of the influence of the auxiliary surface positions on solution accuracy were described. These results facilitate the use of presented variants of DSM. Results related to the influ-ence of numbers of dipole and collocation points on solution accuracy were not described in this paper, but show that an increase in the number of dipole and collocation points leads to an increase of solution accuracy. The mutual allocation of dipole and collocation points, and mutual orientation of dipole directions and boundary condition setting directions also influence the solution accuracy. For obtaining highest solution accuracy, we recommend allocation of the dipole and collocation points as described in Section 8.8.

Our software package can be used to solve a large range of problems. As an exam-ple, some results concerning the influence of chirality on scattering properties of different shaped bodies and coatings were presented.

Hopefully, this chapter has demonstrated the posibilities of DSM in the numerical solu-tion of electromagnetic scattering problems of three dimensional nonaxisymmetrical bod-ies of different physical nature.

References

[1] V.D. Kupradze, About approximate solution of mathematical physics problem, Russ. Math. Surv. 22(2), 59-107 (1967) (in Russian).

[2] Yu.A. Eremin and A.G. Sveshnikov, Method of Discrete Sources in Electromagnetic Scattering Problems (M. MSU, Moscow, 1992) (in Russian).

[3] A.G. Dmitrenko and A.I. Mukomolov, On the nonorthogonal series method modification for solving elec-tromagnetic scattering problems for arbitrary shaped smooth perfectly conducting bodies, Radiotekhnika i Elektronika 33(3), 449-455 (1988) (in Russian) (This Journal is translated into English as Journal of Communications Technology and Electronics).

[4] A.G. Dmitrenko and A.I. Mukomolov, On the development of the numerical method for solving of three-dimensional vector scattering problems, Radiotekh. Elektron. 35(2), 438-441 (1990) (in Russian).

[5] A.G. Dmitrenko and A.I. Mukomolov, Modification of the auxiliary sources method for solving of three-dimensional vector scattering problems, Radiotekh. Elektron. 36(5), 1032-1036 (1991) (in Russian).

[6] A.G. Dmitrenko and A.I. Mukomolov, Numerical method for solution of electromagnetic scattering prob-lems of three-dimensional arbitrary shaped magneto-dielectric bodies, Radiotekh. Elektron. 40(6), 875-880 (1995) (in Russian).

[7] A.G. Dmitrenko and A.I. Mukomolov, Numerical method for solution of electromagnetic scattering prob-lems of three-dimensional chiral bodies, Radiotekh. Elektron. 43(8), 910-914 (1998) (in Russian).

[8] A.G. Dmitrenko and S.V. Korogodov, Numerical method for solution of electromagnetic scattering prob-lems of perfectly conducting bodies covered by magneto-dielectric coating, Radiotekh. Elektron. 43(10), 1157-1162 (1998) (in Russian).


[9] A.G. Dmitrenko and A.I. Mukomolov, Numerical analysis of electromagnetic scattering by three-dimensional arbitrary shaped bodies on the foundation of discrete sources method, in: Proc. 15th URSI International Symposium on Electromagnetic Theory, St. Petersburg, Russia, May 23-26, 1995, pp. 473-475.

[10] A.G. Dmitrenko and A.I. Mukomolov, Diffraction of electromagnetic waves in a three-dimensional magneto-dielectric body of arbitrary shape, Russian Phys. J. 38(6), 617-621 (1995).

[11] A.G. Dmitrenko, Numerical analysis of electromagnetic scattering by three-dimensional arbitrary shaped bodies of different physical nature, in: Electromagnetic and Light Scattering, Proc. 1st Workshop on Elec-tromagnetic and Light Scattering - Theory and Applications, T. Wriedt et al., Eds. (Universitat Bremen, Bremen, 1996).

[12] A.G. Dmitrenko, Numerical method for solving of electromagnetic scattering problems of three-dimensional arbitrary shaped magnetic dielectric bodies, in: Proc. VI International Conference on Mathe-matical Methods in Electromagnetic Theory, MMET '96, Lviv, Ukraine, September 10-13, 1996, pp. 84-87.

[13] A.G. Dmitrenko, A.I. Mukomolov, and V.V. Fisanov, Scattering of electromagnetic waves on a magneto-dielectric with chiral properties, Russian Phys. J. 39(8), 781-785 (1996).

[14] A.G. Dmitrenko, A.I. Mukomolov, and V.V. Fisanov, Electromagnetic scattering by three-dimensional arbi-trary shaped chiral objects, in: Advances in Complex Electromagnetic Materials, NATO ASI Series: 3. High Technology, Vol. 28 (Kluwer, Dordrecht, 1997) pp. 179-188.

[15] A.G. Dmitrenko and S.V. Korogodov, Electromagnetic scattering by three-dimensional arbitrary shaped perfectiy conducting bodies with a magneto-dielectric coating, Proc. 2nd Workshop on Electromagnetic and Light Scattering, Theory and Applications, Moscow, Russia, May 27-28, 1997, Yu. Eremin et al., Eds. (Moscow Lomonosov State University, 1997) pp. 105-108.

[16] A.G. Dmitrenko, Discrete sources method for solution of electromagnetic scattering problems of three-dimensional arbitrary shaped bodies, in: Electromagnetic and Light Scattering, Proc. 3rd Workshop on Electromagnetic and Light Scattering - Theory and Applications, T. Wriedt et al., Eds. (Universitat Bremen, Bremen, 1998).

[17] A. Lakhtakia, VK. Varadan, and V.V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Veriag, Berlin, 1989).

[18] C.L. Lawson and R.J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).

[19] A.G. Sveshnikov, Yu.A. Eremin, and N.V Orlov, Investigation of some mathematical models in diffraction theory by nonorthogonal series method, Radiotekh. Elektron. 30(4), 697-704 (1985) (in Russian).

[20] M.G. Cote, M.B. Woodworth, and A.D. Yaghjian, Scattering from the perfectly conducting cube, IEEE Trans. Antennas Propagat. AP-36(9), 1321-1329 (1988).

[21] T.K. Sarkar, E. Arvas, and S. Ponnapalh, Electromagnetic scattering from dielectric bodies, IEEE Trans. Antennas Propagat. AP-37(5), 673-676 (1989).

[22] A. Lakhtakia, VK. Varadan, and V.V. Varadan, Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects, Appl. Opt. 24(23), 4146-4154 (1985).

CHAPTER 9

Hybrid GMT-MoM Method

R Obelleiro, J.L. Rodriguez, and L. Landesa

E. T. S.E. Telecomunicacion Departamento Tecnoloxias das Comunicacions Universidade de Vigo Campus Universitario s/n 36200 Vigo, Spain e-mail: [email protected]


206 F. Obelleiro, J.L. Rodriguez, and L. Landesa

9.1. Introduction

The problem of electromagnetic scattering by obstacles has been subject to extensive research for many years due to its great importance in different application areas. These efforts have lead to a great variety of methods that were chosen according to the size and the properties of the object.

Worth mentioning, among others, is the Method of Moments (MoM) [1] which entails the formulation of an integral equation for the induced surface current in the case of a per-fectly conducting scatter, or the induced polarization current for a dielectric scatter. These currents are expanded using basis functions with unknown coefficients, which are deter-mined solving a matrix equation obtained throughout the application of suitable testing functions. Finally, the scattered fields can be obtained from the knowledge of the induced surface or polarization currents.

On the other hand, the equivalent source methods, such as the Generalized Multipole Technique (GMT) [2^] and the multifilament current method [5-8], have been found to be a viable alternative to techniques in which ordinary surface formulations amenable to conventional moment method solutions are used. In these kind of formulations, the scat-tered field is represented as a linear combination of the fields due to a set of fictitious ele-mentary sources (particular solutions to Maxwell's equations whose fields are analytically derivable), which do not lie on the body surface but rather are retracted some distance away from it. If the location and the number of these sources are chosen appropriately, their com-plex amplitudes can be adjusted in order to satisfy the boundary conditions on the scatter surface, thus, the scattered fields can be obtained straightforwardly without integrating over surface currents. The multifilament current method can be understood as a particular case of the GMT, in which only monopoles (filaments of constant current for two-dimensional case) are used. Nevertheless, both methods were developed independently so that we will refer to them as different methods in the following.

All above mentioned methods provide a significant reduction of the number of unknowns with respect to the conventional MoM solution. This is the reason why they are often prefer-able to MoM, although the size of scattering objects is still restricted to a few wavelengths, due to computational costs and storage requirements associated with these kind of for-mulations. Also, due to ill-conditioning associated with numerical dependencies between sources, accuracy is very dependent on the location of the fictitious sources.

Recently efforts have been made in order to enable the application of these methods to larger bodies. In accordance to the Impedance Matrix Location (IML) method proposed by Canning [9-11], directive sources and testing procedures were employed to obtain gen-eralized impedance matrices with a banded structure. In [12,13] directivity was achieved by positioning point sources into the complex space; a similar solution was applied in the Complex Multipole Beam Approach (CMBA) [14,15], where the scattered fields were pro-duced by a set of multipole sources located in complex space; finally, in [16,17] the direc-tivity was achieved choosing arrays of fictitious sources and testing points with radiation and receiving directional patterns.

Both the Method of Moments (MoM) and the Generalized Multipole Technique (GMT) represent well-established numerical techniques, extensively used for solving computa-tional electromagnetic problems. Even though there are some common foundations in both methods [18], each of them is more attractive for a certain kind of problem; namely, the GMT can be applied in a more efficient manner to structures with a smooth surface, whereas the MoM is more attractive for metallic structures with sharp edges, wires, and other discontinuities.

9. Hybrid GMT-MoM method 207

In the last years, the use of hybrid techniques combining advantages of the single meth-ods has been found to be a very flexible alternative for analyzing electromagnetic radiation and scattering problems (a review can be found in [19]). These kind of techniques allow the investigation of more complex structures yielding accurate results with less computa-tional costs and memory requirements. Following this tendency, some hybrid approaches combining fictitious sources with MoM have been developed by different authors [20-27].

In [20], a hybrid method combining filamentary current sources with a set of on-surface pulse functions (i.e., MoM basis) is presented. In this case, MoM basis were used to negoti-ate the field behavior near edges while monopoles represented the smooth field constituent on the boundary of the scatter. Later on, in [21], rooftop basis functions, were also included as an additional type of source in the GMT formulation. Furthermore, a hybrid technique combining the Complex Multipole Beam Approach (CMBA) with the MoM was presented in [22]. In this work, the complex multipole beams were used to model the fields in the vicinity of smooth portions of the scatter surface, while MoM basis were considered as ad-ditional sources to account for the fields in the vicinity of edges and non-smooth regions. In recent years [23-26] GMT and MoM were directly coupled for two-dimensional problems, leading to a single system of linear equations (including both GMT and MoM matrices and sources coefficients) by means of generalized point-matching principle. The same idea was presented in [27], where the solution was obtained by an iterative process, which, somehow models the electromagnetic coupling between multipole and MoM sources. This solution avoids the inversion of the whole GMT-MoM matrix, and also allows an easy combi-nation of already existing MoM and GMT codes. Finally, closely related to the hybrid combination of fictitious sources method and MoM, is the use of specialized expansion functions [28-30] to simulate the singularity of the fields and the physical surface currents.

As we previously mentioned, perhaps the main difficulty of the above mentioned meth-ods is the adequate selection of the number of sources and their locations, which has a great influence on the accuracy of the method. Some guidelines concerning these topics may be found in [5,20,4,31] whereas the location of the MoM basis in the hybrid GMT-MoM method is treated in [24,26].

Further drawbacks common for all these fictitious source methods and also related with the source location problem are numerical dependencies that usually appear between dif-ferent sources and can be associated to ill-conditioned problems. This drawback becomes even more important for hybrid combinations with MoM, where a number of sources are concentrated near edges or other discontinuities to simulate the singular behavior of the fields, because this concentration of sources inherently generates ill-conditioning and hence provides potentially unstable solutions.

The impedance matrix ill-conditioning has been addressed in [32] for the current fil-ament method, using the Singular Value Decomposition (SVD) technique [33]. In other works, the SVD [34,35] and a convergent minimum-norm solution [36] were used to avoid matrix ill-conditioning produced by resonant frequencies of the EFIE integral equation.

In [37,38], the hybrid GMT-MoM method was investigated concerning the reduction of its ill-posedness and consequently its high dependence on the choice of number and loca-tion of sources. The ill-conditioning problem was overcome by using a Tikhonov regular-ization [39] over the full GMT-MoM impedance matrix, by imposing a quadratic constraint on GMT and MoM coefficients. The proposed technique improves the results significantly with respect to the original GMT-MoM solution.

This chapter is organized as follows: In Section 9.2, the hybrid GMT-MoM method will be outlined briefly and the notation established. Section 9.3 will be devoted to a descrip-tion of the algorithm used to determine the location of the MoM basis. In Section 9.4, the


regularization process will be presented. In all these sections results will be shown, illus-trating the capabilities of the GMT-MoM method. Finally, conclusions will be presented in Section 9.5.

9.2. Formulation

Let us consider an infinite perfectly electric conducting (PEC) cylinder of arbitrary cross section, S, of which the axis is to be parallel with the z-axis of a rectangular coordinate system, as depicted in Fig. 9.1. The unbounded region surrounding the cylinder, VQ, is free space (^0. /> o)» is denoted as the region interior to S. The excitation is due to an external source with harmonic time dependence ej^^ that will be assumed for all fields and currents and will not be mentioned in the rest of the chapter. This geometrical restriction will be considered in order to keep the following formulation clear, so that it can be extended to arbitrarily shaped 3D problems.

Our goal is to determine the current distribution over the surface S and the scattered field in Vb- In order to do this, we solve the equivalent problem shown in Fig. 9.2, in which the field scattered by the conducting body E^ is set up by a superposition of the fields radiated by: (i) A set of elemental sources (multipole expansions) of adjustable ampHtudes, placed

over the contour T in the region originally occupied by the cylinder V, which has been replaced by free space.

Source / Incident wave

E=E +E

^o'K

Fig. 9.1. Scattering from a PEC. General problem.

Source / Incident wave

Fig. 9.2. Scattering from a PEC. Equivalent problem.


(ii) A set of subdomain MoM basis functions placed near the geometrical discontinuities (i.e., comers, edges or sharp curvature changes) on S, in order to account for the sin-gularities of the field in these regions.

At an observation point r in the region VQ, we have:

N P M

r (r) = J^Y1 ^npK^'^'^ir, r„) + c E i ^ Cr, r,), (9.1) n=l p=\ m=l

where E^'^^^ and E^^""^ are the scattered fields due to GMT and MoM sources, Cnp and Cm are their corresponding coefficients, and r„ and Vm are the position vectors for the n-th multipole origin and the m-th MoM basis, respectively. Index m refers to the MoM basis functions, whereas n denotes the multipole origins, and p the multipole orders. Detailed expressions for E ' ^^^ and E^^""^ can be found in [1] and [40].

On the perfectly conducting boundary S, the scattered field must satisfy:

n X E'(r) = - n X E^"^(r) at r e 5, (9.2)

where n is a unit vector outward normal to S, E " is the incident field (due to the excitation) and E^ is the scattered field.

The GMT and MoM source coefficients (cnp and Cm) can be obtained solving (9.2) via Generalized Point Matching [4], by imposing (9.2) at a set of At selected testing points on S, which leads to the following system with At equations and Ns = M-\-NP unknowns:

r A p M

fit X ^ ^CnpK^ '^ ' ' ( r„ r„) + J2 CmE^^^°^(rt, r^) \_n=l p=l m=\

= -lit X E^"^(rt) Vrt, t = l,...,N\ (9.3)

This expression can be rewritten in matrix form as follows:

, rGMT rGMT r^MoMl I ^ Z C = [Z^^T ^MoM] .

C MoM V, (9.4)

where bold italics are used to denote the following matrices: Z - full impedance matrix of dimensions NtX Ns, which is split into the GMT submatrix

^GMT of dimensions NtX NP, and the MoM submatrix Z ^ ° ^ of dimensions NtX M. C - full coefficients vector of dimensions N^ x 1, which contains the GMT and MoM

coefficient vectors {C^^^, NP x 1 and C^^^, M x 1). V - excitation vector due to the incident plane wave (A t x !)•

In the same way as in conventional GMT, the number of matching points must be greater than the number of unknowns (A s PN + M) in order to obtain accurate results [4]. Thus, (9.4) is solved in a least-square sense:

m m { | | Z . C - y | | 2 } . (9.5)

Both normal equations or QR factorization methods can provide the solution of the matrix equation above [33], nevertheless, QR factorization may be preferable when the al-


gorithm [24,26] described in the next section is to be applied, to obtain the optimal location of the MoM sources.

Once C has been found, it is straightforward to calculate the scattered field and other re-lated quantities of interest. The surface induced current Js, for example, is not immediately available as it is in conventional MoM, however, it can easily be derived from

Js(r) = n X [ff(r) + ff^'^Cr)] at r G S, (9.6)

where H ' is the incident, and H^ is the scattered magnetic field, obtained by:

N P M

ff (r) = I ] E "P Kp^'^i^^ r„) + J ] c„H^^°^(r, r , ) , (9.7) n=\p=\ m=\

where H^^°'^ and H^'^°'^ are the scattered magnetic fields due to GMT and MoM sources of which the expressions are described in detail in [1] and [40].

Some numerical results, obtained by TE and TM polarizations, are presented below in order to illustrate the effectiveness and accuracy of the hybrid GMT-MoM approach. These results were compared with the conventional GMT method and with a reference solution obtained by an ordinary MoM discretization of the CFIE [41]. We used the same number of unknowns for the hybrid and GMT approaches, and a higher density of basis functions for the MoM reference solution. Several approximate rules for GMT sources location and the maximum order of the expansions derived from [4] were used for both the GMT and the hybrid approach.

The error in the induced current density on S is defined as:

2|nxHinc(r)|max

where Jref is the induced current density obtained by the MoM reference solution. For simplicity reasons, we only calculated the boundary condition error at the interpolation points. Two kinds of errors were considered: the peak error §p and the root mean square (RMS) error §nns.

The following results were obtained for a square conducting cylinder with 44 A, perimeter (Fig. 9.3), illuminated by a plane wave incident at an angle of 25° from normal to one of the faces. Figure 9.3 shows the exact location of multipoles and MoM sources used both for the GMT and the hybrid approach.

Figure 9.4 shows the magnitude of the TM surface currents | Js| obtained by GMT and the hybrid GMT-MoM method compared with the MoM reference solution | Jref | (obtained with 440 basis functions lO/X located on the testing points {rt}). The surface currents are normalized by the incident magnetic field Iff ' l. Table 1 shows the error values calculated with 440 interpolation points {rt}.

Figure 9.5 shows results for the same square cylinder when illuminated by a TE plane wave. The number of unknowns and their distribution are exactly the same as in the pre-vious case. Table 2 shows the error values calculated with the same set of interpolation points.

It can be seen that the hybrid method leads to a significant error reduction with respect to the GMT solution.


l a

a

MoM region

n ^. . . .®. . . .0. _ . 0 . . . 0

0.98 X ® ®

• Multipolar • ^ origins @

0. . . .0. . . .0. . . . 0 . . . 0

•0.82?i

r Fig. 9.3. Square cylinder illuminated by an incident plane wave. Localization of GMT and MoM sources. Both GMT and GMT-MoM methods use 176 sources (4/A) distributed as shown. GMT (A = 16, P = 11); hybrid

GMT-MoM (M = 64, iV = 16, P = 4). Both results were obtained with A t = 440 testing points (10/A.) [23].

^ ^ ^ Reference GOO Hybrid • • • GMT

kjoôOGoQoQJ

2^

\ , O § 6 S Q $ 5 ^

^^î^-^yX-^'OCf . iVyî^^nAnnnrvl

Distance along perimeter

Fig. 9.4. Normalized surface current magnitudes. TM polarization [23].

The higher error obtained for TE polarization is due to the rapidly varying current mag-nitude which appears in this case. On the other hand, GMT cannot predict the singular behavior of the current in the vicinity of the illuminated edges for TM polarization. Thus, the GMT errors for the TM case are dominated by the contribution of these regions.

It must be pointed out that the higher errors associated with the conventional GMT solution can be reduced significantiy by an optimization of the location of the GMT sources [4,20]. However this would mean going to far as we were mainly interested in improving the GMT solution by including additional MoM basis, without any concern for optimization processes carried out with the GMT sources location (that could be consid-ered anyway).


A B C D A

"•"— Reference o o o Hybrid

11 22 33 44

Distance along perimeter

Fig. 9.5. Normalized surface current magnitudes. TE polarization [23].

Table 1 Errors for TM polarization

^rms

GMT 24.2% Hybrid 0.8%

speak

57.6% 5.7%

Table 2 Errors for TE polarization

^rms

GMT 13.9% Hybrid 2.3%

speak

49.9% 12.5%

Finally, Figs. 9.6 and 9.7 show the bistatic scattering cross section defined by

|E^(r)|^ or(r)= lim litr——.—^, (9.9)

where r = rr. To illustrate the accuracy of the hybrid method, we compared its numerical results with a reference MoM solution. As could be expected of current distributions, a very good agreement was achieved in both states of polarization. It is worth mentioning that the same scattered field could be obtained with the GMT alone, so GMT-MoM improvement is only significant when the current distribution or near field quantities are of interest.

With respect to computational costs, let us now compare the computational cost of the hybrid method with that of the conventional MoM solution in order to show the computa-tional efficiency of the GMT-MoM method. The comparison was established in terms of

9. Hybrid GMT-MoM method

90°

213

Fig. 9.6. Bistatic cross section. TM polarization [23].

270°

Fig. 9.7. Bistatic cross section. TE polarization [23].

FLOPS, as defined in [33]. The Gaussian elimination, used to solve the system of equations for the moment method (MoM), implies IN^/S FLOPS [33], where At is the number of equations. The overdetermined system of equations for the hybrid approach was solved in a least square sense using normal equations, so that the computational cost is NtN^ + A s /3 FLOPS [33], where Nt is the number of equations and As the number of unknowns (mul-tipoles and MoM basis). For the parameters used in the previous examples, A t = 440 and Ns = 176, the relative cost to solve the system of equations was 27.2%. Moreover, an addi-tional reduction was obtained in the matrix filling process, due to the recurrence properties of the functions involved in the calculation of multipole expansions [40,42].


filamentary currents

^ ^ • 1 V •

• • •

V V

* • • • •

•J:J

Fig. 9.8. Schematic location of filaments in a square cylinder.

9.3. On the location of GMT and MoM sources

As was previously mentioned, the main difficulty of the GMT as of other fictitious source methods, is to find the adequate location and number of multipole sources. This step is very important because it significantly influences the accuracy of the results ob-tained, i.e., a wrong selection of the location of the fictitious sources inherently generates ill-conditioning and hence potentially unstable results. Some work concerning these topics, can be found in literature: • In [5] and [6], Y. Leviatan and A. Boag present an extensive study of the behavior of the

filamentary current method for different locations of the sources (in this case filamentary currents or monopoles) when dealing with smooth scatters.

• In [20], the location of the sources is also considered, but here the problem is applied to scatters with edges that must be taken into account by placing additional sources in the proximity of the edges. However, as the influence of the edge does not disappear abrubtly, the authors have found it better to locate sources very near the edges and then gradually place more sources further from S, as shown schematically in Fig. 9.8, where filamentary current sources are used to simulate the scattering by a square cylinder.

• In [4], some rules are presented trying to automate location and selection of the number of multipoles in the GMT. These rules are very useful to avoid numerical dependencies associated with wrong location and selection of multipole expansions. They are based on a correlation of the distances between the multipole origins and the distances of the oringins to the boundary of the scatterer. Here, the concept of region of influence is introduced as a powerful tool to guide the user to the selection of appropriate multipole locations.

• Finally, in [31], a practical scheme for placing the sources in \hQ filamentary current method is presented. This scheme utilizes a packing concept for the regions of influence of the sources, of which the locations are further described according to the radius of curvature of the scatterer. The GMT-MoM method shows the same problems associated with number and location

of sources. In this work, the multipoles or GMT sources were placed following the rules recommended in [4]. With respect to the location of the MoM basis, it is clear that they must be concentrated in the region of the edges or other discontinuities in S. But the ex-


tension of these MoM regions becomes a critical parameter that must be chosen carefully: too large MoM regions can generate ill-conditioning and hence potentially unstable results, whereas too small MoM regions might not be able to consider the influence of the edge, which is known not to diminish abruptly.

In the previous section, only simple geometries analyzed with the hybrid GMT-MoM method, and the MoM basis functions were placed directly in the proximity of the geo-metrical discontinuities. For more complex scatterers, the location of MoM basis is not as evident and even depends on the excitation, so that it becomes a very difficult task which furthermore has a great influence on the final accuracy of the method.

In this section we will present an automatic algorithm [24,26], which will allow broader application of the hybrid approach to bodies of arbitrary shape, regardless of the complex-ity of either the excitation or the shape of the scatterer. Both the location and the extension of the regions covered were determined by MoM basis with the GMT-MoM approach.

As in the previous section, the original problem of a perfectly conducting cylinder of arbitrary cross section, illuminated by an incident wave, was solved by setting up an equiv-alent problem for the region surrounding the scatterer (Figs. 9.1 and 9.2). In this region, the scattered field was defined as the sum of fields due to a set of fictitious sources located within the region originally occupied by the scatterer, and a set of MoM basis functions located on sections of the surface, determined by the algorithm described below.

The first step of the algorithm consists of finding an initial GMT solution [4] by enforc-ing the £-field boundary condition on a set of matching points {rt} on S. This process leads to a matrix equation which was solved to initially guess the multipole coefficients Cnp:

[Z^MT] . [CGMT] = y (9.10)

As in the GMT-MoM method described in the previous section, the number of match-ing points must be greater than the number of unknowns, and the solution of the matrix equation is obtained in a least-square sense.

Once the first approach for the GMT coefficients Cnp is obtained, the residual error in the electric field boundary condition is calculated at the same set of matching points {rtj. For convenience reasons, this quantity was normalized with respect to the magnitude of the incident field, as follows:

u, | E i i i c ^ g s , G M T | ^ ^ b c ^ I _ 1 ^9 J j ^

where E ' ^ is the incident field and E^'^^^ is the initial approach of the GMT scattered field, obtained by:

A p

n=l p=\

At first glance, one can suppose that the local maxima of the residual error lS.E^^ could serve as an indication for worst modelled regions by GMT sources, and consequently for the location of the MoM basis. But, due to its rapidly varying behavior, it is more conve-

216 F. Ohelleiro, J.L. Rodriguez, and L. Landesa

nient to define a related magnitude, namely the integrated error AEq^, which stands for the mean error along a small section of the surface:

j=t-q

(9.12)

where 2q + I matching points are considered to determine the integrated error at each point Ft. Thus, those points {rt} of which the integrated error AE^^ exceeds a fixed thresh-old level, are selected as the location of the MoM basis for the hybrid GMT-MoM ap-proach. In some cases, the threshold level is selected according to the maximum number of desired unknowns. Practice shows that better behavior is obtained when {2q + 1) is chosen to cover an extension close to a wavelength being a good compromise between accuracy and computational cost.

Once number and location of the MoM basis are selected, the GMT-MoM method can be applied exactly as explained in Section 9.2. Once again, a matrix equation is obtained for the E-field boundary condition at a set of points on S, but now for both the multipole and MoM basis coefficients. It is convenient to use the same set of points {rt} as in the GMT solution, because the GMT matrix has already been calculated for them.

It is important to note that the use of the QR factorization to solve the initial overde-termined equation (9.10) allows taking advantage of this initial GMT solution, because the QR factorization of the whole system (obtained when both GMT and MoM sources are considered) can be updated [33] on the basis of the previous one [26], so that GMT matrix factorization does not need to be repeated. Nevertheless, this technique requires the 2-matrix of the initial solution to be stored, which in some cases may be an decisive drawback.

The performance of the proposed algorithm is illustrated now for a two-dimensional (2D) test geometry illuminated by an incident transverse magnetic (TM) plane wave, shown in Fig. 9.9. The distribution of the sources for both GMT and the hybrid method are shown in Fig. 9.10. _ _

In order to determine the location of the 72 MoM basis, the integrated error AE\^{q = 4) depicted in Fig. 9.11, was considered. With these parameters the algorithm selected a threshold level u = 0.302 and located 72 additional MoM basis functions near comers B, C, E and F as shown in Fig. 9.11.

\J^^'

Fig. 9.9. L-shape geometry illuminated by an incident transverse magnetic (TM) plane wave.

9. Hybrid GMT-MoM method

E^ • 1

;i o

i.

217

_ j ;

•^

origins f"^

'A Testing points

Fig. 9.10. Localization of GMT and MoM sources. Both GMT and GMT-MoM methods use 162 sources (4/X) distributed as shown. GMT (A = 18, P = 9); hybrid GMT-MoM (A = 18, P = 5, M = 72). Both results were

obtained with A t = 440 testing points (10/-^).

Number of . MoM basis I

B C D E F A

Distance along perimeter (t)

Fig. 9.11. Integrated error ^^'fiq = 4) [24].

The improvement of the hybrid approach to conventional GMT was again illustrated by solving both methods with the same number of unknowns and comparing them to a reference solution obtained from MoM. The validity check was provided by evaluating the error in the current density, as defined in (9.8).

Figure 9.12 shows the normalized surface current obtained by GMT and the hybrid GMT-MoM method, and compares them to the MoM reference solution. The error levels of the GMT and the hybrid method are summarized in Table 3.

Once again, it can be seen that the hybrid method produces considerably lower errors than GMT, demonstrating the accuracy of the selection of MoM basis derived by the pro-


T3 2

O

o O o MoM Reference

Hybrid method

GMT

B C

Distance along perimeter (t)

Fig. 9.12. Normalized surface current distribution [24].

Table 3 Errors for TM polarization (automatic algorithm)

speak

GMT Hybrid

14.1% 1.4%

80.2%

posed algorithm. Even though only a TM polarization problem was shown, the method is general, moreover, the same algorithm can be applied to other hybrid approaches [20-26].

But, this automatic algorithm may lead to source distributions of which the Z matrix in (9.4) has a high condition number, which inherently could generate unstable results. In order to overcome this problem, a regularization procedure, called Tikhonov regularization, can be applied as will be described in the following section.

9.4. Regularization of the GMT-MoM method

In this section, the hybrid GMT-MoM method will be investigated from the perspec-tive of reducing its ill-posedness and consequently its high dependence on the choice of number and location of both GMT and MoM sources. The ill-conditioning is overcome by using a Tikhonov regularization [39] on the full GMT-MoM impedance matrix, by impos-ing a quadratic constraint on GMT and MoM coefficients. This technique was proposed in [37,38] and has provided significant improvements with respect to the original GMT-MoM solution.


First of all, we will present two methods for solving a linear least-square system of the form:

A x = b ( A G C ^ ^ " , J C G r ^ \ beC'^''^), (9.13)

when its solution presents difficulties due to its ill-posedness. The main feature of such ill-posed problems is the fast decay to zero of the singular values of A and, consequently, the large condition number of A. Conventional methods for solving the linear least-squares problem of (9.13), such as normal equations or QR decomposition, usually lead to so-lutions dominated by perturbation errors (such as non-exact evaluation of A, errors in measurements for b or those associated to finite machine precision), so that the problem becomes ill-posed [39].

So-called regularization methods are needed to obtain meaningful solution estimations for such ill-posed problems. The purpose of regularization is to incorporate further infor-mation about the desired solution in order to stabilize the problem and thus to find a more efficient and stable solution.

One of the most popular regularization techniques is the TSVD (Truncated-SVD), that uses the SVD to obtain a stable solution of the problem [39]. Thus, the SVD of the matrix A can be written as:

A = Ui:V^ = Y^Uiaivf, (9.14) i=\

where U = (ui,..., Un) e C^^^ and V = (vi,.. .,Vn) e C"^" are orthonormal matrices and X is a diagonal matrix H = diag(ori,..., or„) where the at are the singular values of A in decreasing order. The superscript 'H' denotes the conjugate transposition operator.

The solution of (9.13), in accordance to (9.14), can be expressed as:

A (ufb)vi

^ ^^

From the last expression, it can be clearly seen that a small singular value cxi, compared with cTi, makes difficulties due to the amplification of errors in A and b. To overcome this problem, TSVD computes the rank-^ matrix

Aj = ^w/or/i;P, k^n, (9.16) i=\

which is a truncation in the finite series of (9.14). This matrix Ak, avoids the influence of ihcn — k smaller singular values. So that the new equation system for the TSVD technique is

Ak'Xk = b. (9.17)


This equation can be computed in the minimum-norm least-squares sense, for example by using the previous SVD decomposition:

Xk = T ^ ^ ^ ^ . (9.18)

The TS VD method is useful when there is a well-determined gap between the large and small singular values of A; in this case, the k parameter can be chosen in order to avoid these small singular values [39].

Another regularization technique, which is more suitable when the singular values decay gradually to zero, is the Tikhonov regularization [39,43], which can be formulated as:

min{ | |A-x-6 | | ^ + A^||L.jc||^}. (9.19)

It is equivalent to apply a quadratic constraint of the form ||L • x ||2 < of to the original least-squares problem of (9.13). The a parameter depends on the regularization parameter A in a nonlinear way. The matrix operator L should be a banded matrix, e.g., an identity matrix, a diagonal weighting matrix, or a derivative operator. It should incorporate information concerning the physical problem, when available.

Alternatively, the Tikhonov formulation of (9.19) can be expressed as a new least-squares problem:

mm X G1)-Q||;

which can be efficiently solved by a bidiagonalization of A and simple Givens rotations over XL matrix (supposing that this matrix has a banded structure). This leads to a bidiag-onal system of equations that can be quickly solved by back substitution. Similar to (9.15), the Tikhonov regularization solution can also be formulated as:

E -{ufb)vi a} fi = ^ ^ , (9.21)

,=1 - ^}^^

forL = I (notice that when L ^ / , the Eq. (9.19) can be transformed to the so-called standard form with aV = I [39]). This expression shows that the Tikhonov regularization considers all singular values, but each of them is weighted by a filter factor fi instead of truncating the matrix as in TSVD.

It must be pointed out that the selection of the A-parameter is not obvious, and some-times it may require the computation of several A-solutions, each of which can be easily obtained by back substitution in 0(n) FLOPS. When the regularization parameter X is too small the problem becomes underregularized and the solution is dominated by perturbation errors; on the other hand, too large values of k provide solutions which are dominated by regularization errors. So, it is interesting to select the regularization parameter which bal-ances both regularization and perturbation errors. There are different approaches to select the A-parameter, from which we have chosen the L-curve criterion due to its stabihty and its illustrative notion [39,44,45].


loglllx

(0 1

•;= c E o 0 "-5 "D 5 ^ -Q c 1-0 2

U

•43 t 1 ^ CD 0 0-CO > 1

n 1

• " n * *

Corner [ ( } i optimum)

- ^

Solution dominated ^ ^ by regularization errors

-

— •

\og\\Ax-b i

Fig. 9.13. L-curve of an ill-posed problem.

The L-curve criterion is a relatively new method, based on a parametric plot of the reg-ularized solution ||L • x||2 versus the residual norm ||A • x — ft||2, in a log-log scale, with A, as the parameter. In ill-posed problems, the L-curve plot has a characteristic L-shape appearance, with a sharp comer between vertical and horizontal parts. The vertical part corresponds to a underregularized solution while the horizontal part corresponds to a high regularized solution. The purpose of the L-curve criterion is to calculate the regulariza-tion parameter by choosing a point of this curve at the comer between the horizontal and vertical parts. By doing this, a A parameter that balances both the regularization and the perturbation errors is selected. Figure 9.13 shows the characteristics of a typical L-curve.

It is important to notice that a closed expression for the curvature of L-curve can be obtained when the Tikhonov regularization is used; so that the selection of the corner is performed by a simple maximum curvature search [39].

From the GMT-MoM method point of view, it must be pointed out that the use of the above mentioned regularization tool helps avoiding one of the main drawbacks for users of the GMT and GMT-MoM methods, namely, its great dependence on source location, that inherently generates ill-conditioning and hence potentially unstable results [20]. It was found that both GMT and MoM coefficients grow without limit when the problem becomes ill-posed. In order to overcome this drawback, the Tikhonov regularization was selected instead of TSVD for the following reasons: (i) it is more stable with respect to the selection of the regularization parameter; (ii) it obtains better results (a few tenth percent), due to the gradual decay to zero of the singular values; and (iii) it implies a minor number of FLOPS.

The appHcation of the Tikhonov regularization (9.19) to (9.21) leads to the following expression:

minlyZC c ^ vwl + xHL^cfA (9.22)

Matrix operator L was selected to be the identity matrix L = / , in order to provide a physical constraint for the GMT and MoM coefficients. But, due to the different order


Incident plane wave

Multipolar expansions

^ 1 • i a

MolVI basis ^ ^ ^ functions

Fig. 9.14, Test geometry. R = 2.92A, a = 45°, ^ = 0°, A = 72 multipole sources located in 24 origins, M •-0 , . . . , 40 pulse basis functions [38].

of magnitude between MoM and GMT coefficients, a new parameter y must be included in L:

-[""«"' rL.\ that can be interpreted as a combination of two independent operators (one for MoM and the other for GMT sources) by a Sobolev norm [39].

An interesting consequence of the apphcation of the quadratic constraint to the coeffi-cients is that the radiated power of each source alone is controlled, and therefore the cou-pling between them is somehow reduced. Furthermore, the GMT-MoM method becomes less dependent on the location of the sources as the source coupling becomes weaker, which was found to be a very important factor to provide a significant improvement for the accuracy of the solution.

Some numerical results will be presented concerning convergence and stability of the proposed hybrid solution, together with its dependence on number and location of MoM sources. For illustrative purposes we will consider the case of a two-dimensional PEC body, illuminated by TM and TE plane waves, as shown in Fig. 9.14, which also contains a description of the location of the GMT and MoM basis.

The results for TM polarization are shown in Figs. 9.15 and 9.16. Figure 9.15 shows the average error of the induced current density as a function of the number of MoM sources (M) located near the comer of the geometry. This error is defined as

^=mean{^(rt)}, (9.24)

where ^(rt) is the error defined in (9.8). From Fig. 9.15, it can be seen that the GMT-MoM method improves the solution obtained by conventional GMT (M = 0) for a range of values of M. Nevertheless, for larger values of M the problem becomes ill-posed, and therefore the solution is not accurate. Overall, the Tikhonov regularization provides a uniformly convergent solution, achieving better results as the number of MoM basis in-creases.

The above mentioned behavior is better illustrated in Fig. 9.16, where the condition number is plotted as a function of M. It can be seen that the Tikhonov regularization stabilizes the problem, keeping its condition number in an acceptable range of values.

The same results for TE polarization are plotted in Figs. 9.17 and 9.18, showing the same behavior previously commented for the TM polarization, although the condition number


Conventional GMT-MoM Regularized GMT-MoM

0 8 16 24 32 40

Number of MoM basis funcitons

Fig. 9.15. TM polarization. Plot of average current error as a function of number of MoM sources (M) [38].

o O

10

E

§ 10

Conventional GMT-MoM Regularized GMT-MoM

8 16 24 32


Fig. 9.16. TM polarization. Plot of the condition number as a function of number of MoM basis (M) [38].

presents weak oscillations for large values of M. This does not lead to any significant inconvenience as can be shown in the mean error plot.

Finally, the L-curve for the Tikhonov regularization (TM case, M = 32), is shown in Fig. 9.19. This L-curve illustrates the compromise between minimizing the two quantities involved in the regularization problem, namely, the residual norm and the solution norm, showing how these quantities depend on the regularization parameter A. For the particular problem considered here, the L-curve has a clearly sharp corner, of which the position


• Conventional GMT-MoM -0 Regularized GMT-MoM

0 8 16 24 32 40


Fig. 9.17. TE polarization. Plot of average current error as a function of number of MoM sources (M) [38].

10

10

§ 10

o O

• - - • Conventional GMT-MoM 0 0 Regularized GMT-MoM

0 8 16 24 32 40


Fig. 9.18. TE polarization. Plot of the condition number as a function of number of MoM basis (M) [38].

(marked in Fig. 9.19) corresponds to a regularized solution in which the perturbation error and the regularization error are balanced.

The main drawback of the proposed regularization method is the selection of the regu-larization parameter X which is not obvious. The adequate selection of this parameter is a very important task, for which we chose the L-curve criterion that was found to be very efficient and straightforward to apply.


10" 10'

Residual Norm II Z C - F I L

Fig. 9.19. L-curve corresponding to the previous example with M = 32 MoM basis functions and TM polariza-tion [38].

9.5. Conclusions

A complete formulation for the hybrid GMT-MoM method was presented in this chap-ter. The application of the hybrid method was illustrated for two-dimensional problems for both TE and TM polarizations showing the accuracy and computational efficiency of the hybrid approach, although the method is general and the same formulation can be extended to a general case. Also, an automatic location algorithm was described, which allows the determination of location of MoM basis for the hybrid GMT-MoM method. This algorithm enhances the scope of application of the hybrid approach to complex bodies of arbitrary shape, without mindering the complexity of either the excitation or the geometry of the scatterer. Finally, the application of regularization tools in the hybrid GMT-MoM method was discussed. Here studies were focused on convergence and accuracy aspects of the so-lution, examining its dependence on location and number of GMT and MoM sources. As expected the regularization significantly improves the conventional GMT-MoM solution and furthermore reduces the condition number associated with the problem. Finally, the use of this regularization tool avoids one of the main drawbacks for users of the GMT-MoM method, namely, its high dependence on sources location.

Acknowledgements

This work was partially supported by the Comision Interministerial de Ciencia y Tec-nologia (CICYT), Project Ref. TIC97-0821-C02-01, and by the European INTAS Program Proj. Ref. INTAS-96-2139.


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conducting and homogeneous material bodies - theory and numerical solution, IEEE Trans. Antennas Prop-agat. AP-36(12), 1722-1734 (1988).

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[10] EX. Canning, Improved impedance matrix locaHzation method, IEEE Trans. Antennas Propagat. AP-41(5), 659-667 (1993).

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[12] E. Erez and Y. Leviatan, Analysis of scattering from structures containing a variety of length-scales using a source-model technique, / Acoust. Soc. Am. 93(6), 3027-3031 (1993).

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[14] A. Boag and R. Mittra, Complex multipole beam approach to electromagnetic scattering problems, IEEE Trans. Antennas Propagat. AP-42(3), 366-372 (1994).

[15] A. Boag and R. Mittra, Complex multipole-beam approach to three-dimensional electromagnetic scattering problems, /. Opt. Soc. Am. 11(4), 1505-1512 (1994).

[16] Y. Leviatan, Z. Baharav, and E. Heyman, Analysis of electromagnetic scattering using arrays of ficticious sources, IEEE Trans. Antennas Propagat. AP-43(10), 1091-1098 (1995).

[17] Z. Baharav and Y. Leviatan, Scattering analysis using ficticious wavelet array sources, /. Electromagn. Waves Applic. 10, 1683-1697 (1996).

[18] C. Hafner, On the relationship between the MoM and the GMT, IEEE Antennas Propagat. Mag. 32(12), 12-19 (1990).

[19] G.A. Thiele, Overview of selected hybrid methods in radiating system analysis, Proc. IEEE 80(1), 66-78 (1992).

[20] S. Eisler and Y Leviatan, Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model, lEEProc, PartH 136(12), 431-438 (1989).

[21] C. Hafner, J. Waldvogel, J. Mosig, J. Zheng, and Y Brand, On the combination of MMP with MoM, Appl. Comput. Electron. Soc. J. 9(3), 18-27 (1994).

[22] A. Boag, E. Michelssen, and R. Mittra, Hybrid multipole-beam approach to electromagnetic scattering problems, A/7p/. Comput. Electron. Soc. J. 9(3), 7-17 (1994).

[23] J.L. Rodriguez, F. Obelleiro, and A.G. Pino, A hybrid multipolar-expansion-moment-method approach for electromagnetic scattering problems, Microwave Opt. Tech. Lett. 11(4), 304-308 (1996).

[24] F. Obelleiro, J.L. Rodriguez, and A.G. Pino, An automatic location algorithm of MoM basis in the hybrid GMT-MoM method, Microwave Opt. Tech. Lett. 13(12), 327-329 (1996).

[25] F. Obelleiro, J.L. Rodriguez, and A.G. Pino, Hybrid GMT-MoM method for solving electromagnetic scat-tering problems, IEEE Trans. Magn. 33(3), 1424-1427 (1997).

[26] J.L. Rodriguez, Desarrollo de Metodos Eficientes para el Estudio de Problemas de Dispersion Electromag-netica, PhD thesis (E.T.S.I. Telecomunicacion, Universidad de Vigo, 1997).

[27] U. Jakobus, H.-O. RuoB, and EM. Landstorfer, Analysis of electromagnetic scattering problems by an iterative combination of MoM with GMT using MPI for the communication. Microwave Opt. Tech. Lett. 19(9), 1 ^ (1998).


[28] P. Leuchtmann, MMP modeling technique with curved hne multipoles, Appl. Comput. Electron. Soc. J. 9(3), 69-78 (1994).

[29] P. Leuchtmann, The construction of practically useful fast converging expansions for the GMT, in: 1989 IEEE APS Int. Symposium (1989) pp. 176-179.

[30] Y. Shifman, M. Friedmann, and Y. Leviatan, Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method, lEE Proc, Part H 144(8), 235-240 (1997).

[31] K.I. Beshir and I.E. Richie, On the location and number of expansion centers for the generaUzed multipole technique, IEEE Trans. Electromagn. Compat. 38(5), 177-180 (1996).

[32] A. Boag, Y Leviatan, and A. Boag, On the use of SVD-improved point matching in the current-model method, IEEE Trans. Antennas Propagat. AP-41(7), 926-933 (1993).

[33] G.H. Golub and C.F.V. Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, MD, 1989).

[34] F.X. Canning, Singular value decomposition of integral equations of EJVl and applications to the cavity resonance problem, IEEE Trans. Antennas Propagat. AP-37(9), 1156-1163 (1989).

[35] F.X. Canning, Protecting EFIE-based scattering computations from effects of interior resonances, IEEE Trans. Antennas Propagat. AP-39(11), 1545-1552 (1991).

[36] T.K. Sarkar and S.IVI. Rao, A simple technique for solving E-field integral equation for conducting bodies at internal resonances, IEEE Trans. Antennas Propagat. AP-30(11), 1250-1254 (1982).

[37] L. Landesa, F Obelleiro, J.L. Rodriguez, and M.R. Pino, Stable solution of the GlVIT-MoM method by Tikhonov regularization J. Electromagn. Waves Applic. 12(12), 1447-1448 (1998).

[38] L. Landesa, F Obelleiro, J.L. Rodriguez, and M.R. Pino, Stable solution of the GMT-MoM method by Tikhonov regularization, in: Progress in Electromagnetics Research, PIER 20, J.A. Kong, Ed. (EMW Pub-Hshing, Cambridge, MA, 1998) Ch. 3.

[39] PC. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion (SIAM, Philadelphia, 1998).

[40] L.H. Bomholt, A Computer Code for Electromagnetic Scattering Based on the GMT, PhD thesis (Swiss Federal Institute of Technology, Zurich, 1990).

[41] J.R. Mautz and R.F Harrington, A combined-source solution for radiation and scattering from a perfectly conducting body, IEEE Trans. Antennas Propagat. AP-27(7), 445-454 (1979).

[42] M. Abramowitz and LA. Stegun, Eds., Handbook of Mathematical Functions (Dover Publications, NY, 1970).

[43] A.N. Tikhonov and V.Y Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, DC, 1977). [44] PC. Hansen and D.P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems,

SIAM J. Sci. Comput. 14, 1487-1503 (1993). [45] M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT 36, 287-301 (1996).

CHAPTER 10

Null-Field Method with Discrete Sources

A. Doicu

Verfahrenstechnik Universitdt Bremen Badgasteiner Sir. 3 D-28359 Bremen, Germany e-mail: doicu @ iwt. uni-bremen. de


230 A. Doicu

10.1. Introduction

Three-dimensional problems of electromagnetic scattering have been a subject of intense investigation and research. These efforts have led to a development of a large number of an-alytical tools and modelling techniques for quantitative evaluation of electromagnetic scat-tering by various particles. One of the fastest and most powerful numerical tools for com-puting nonspherical light scattering using spherical vector wave functions expansions is the null-field method [1-3]. In the null-field method the particle is replaced by a set of surface-current densities, so that in the exterior region the sources and fields are exactly the same as those existing in the original scattering problem. A set of integral equations for the surface-current densities is derived by considering the null-field condition for the total electric field inside the particle. The solution of the scattering problem can be obtained by approximat-ing the surface-current densities in the mean square norm by the complete set of tangential single-spherical-coordinate vector wave functions of the internal problem. A number of modifications to the null-field method have been suggested, especially to improve the nu-merical stability in computations for particles with extreme geometries (prolate and oblate spheroids with large aspect ratio). These techniques include the following: formal modifi-cations of the single-spherical-coordinate-based null-field method [4-6], different choices of basis functions [7,8] and the application of the spheroidal coordinate formahsm [9].

In recent years the discrete-sources method has become an effective means for solv-ing a wide variety of boundary-value problems in scattering theory [10]. Essentially, this method entails the use of a finite linear combination of fields of elementary sources to construct the solution. The discrete sources are placed on a support in a region exterior to the region where the solution is required. Unknown discrete-source amplitudes are de-termined from the boundary conditions at the particle surface. We note here that the main concept is to eliminate singularities in the singular integral equation by shifting the surface of sources relative to the surface of integration as proposed by Kupradze [11]. He proved the completeness and the linear independence of a system of fundamental solutions of the Helmholtz equations when their poles are distributed on a closed surface in a nonphysical region. Kersten [12], and Miiller and Kersten [13] established the completeness of various systems of vector functions which have poles located on auxiliary surfaces. Another com-plete system of functions having singularities distributed on a portion of a straight line was discussed by Eremin and Sveshnikov [10].

Discrete sources were used in the iterative version of the null-field method [5,6,8]. This approach utilizes multipole spherical expansions to represent the internal fields in different overlapping regions, rather than summing the various expansions and using them through-out the particle as in the discrete-sources method. The various expansions are matched in the overlapping regions to enforce the continuity of the fields throughout the entire interior volume.

The aim of our contribution is to describe various formulations of the null-field method with discrete sources for the transmission boundary-value problem. The goal of the theo-retical development is to derive a set of integral equations for the surface-current densities which guarantee the null-field condition inside the particle. Actually, sufficient conditions should be imposed to ensure that the total electric field becomes zero inside the particle. The remainder of the analysis then consists of approximating the surface-current densities by the complete system of fields of elementary sources. The choice of the discrete-sources support plays an important role. Any support must satisfy an important property: the dis-appearance of the analytic vector function of real variables on the discrete-sources support must lead to the disappearance of this function throughout the analytical region. For the

10. Null-field method with discrete sources 231

discrete-sources support, we can use a point, a curve, a surface, etc. Consequently, the dis-crete sources are the system of locaHzed and distributed spherical vector wave functions, magnetic and electric dipoles, and vector Mie potentials.

The organization of our chapter is as follows. We begin by formulating the transmission boundary-value problem in the classical setting of continuous and Holder-continuous func-tions. We then proceed to prove some fundamental lemmas which enable us to construct complete systems of vector functions on the particle surface. After that we consider the general null-field equations for the transmission boundary-value problems. We discuss the existence and the uniqueness of the solution, and show the equivalence of these equations with a system of boundary-integral equations. The main part of our analysis consists of completeness results and formulations of null-field equations in terms of discrete sources. Finally, we present the numerical scheme of the null-field method with discrete sources and give some numerical results.

10.2. Transmission boundary-value problem

Scattering of electromagnetic waves by dielectric obstacles can be formulated in terms of the following transmission problem.

Transmission Boundary-Value Problem: Let EQ, HQ be an entire solution to the Maxwell equations representing an incident electromagnetic field. Find the vector fields, Es.Hs G C^{Ds)(^C{Ds) and Ei, Hi e C^(Di) nC(Di) satisfying Maxwell's equations

V X Ei =ikoiîlli, V X H/ = -jkoSiEi in D/,

kt =ko^StiXt, t = sj, ko = co^£ofô, (10.1)

and two transmission conditions on the interface

n X E/ — n X E^ = n x EQ,

n X H/ - n X H^ = n X Ho. (10.2)

In addition, the scattered field Es, Hs must satisfy the Silver-Muller radiation condition

^ x H ^ + E ^ = o ( - ), a s x ^ o o (10.3) X \X

uniformly for all directions x/x. Let Rtksj > 0 and Imksj > 0. Then there exists a unique solution to the transmission

boundary-value problem [14]. For the scattering problem, the boundary values are the re-strictions of an analytic field EQ, HQ to the boundary and therefore they are as smooth as the boundary. In our analysis we assume that the surface S fulfills sufficient smoothness requirements such that eo, ho G C^^^j(S), where eo = n x Eo and ho = n x Ho. Conse-quently, the scattered field solution E^, H^ belongs to C^'^(Ds) and the internal field solu-tion E/, H/ belongs to C^'"(A). It is noted that C^^^îS) is the space of all a in C^^^(S) with Holder-continuous surface divergence V • a.

Let us consider the vector potential with integrable density a

- / A«(x)= / aiy)gix,y,k)dSiy), (10.4)

232 A. Doicu

where g stands for the Green function. For analysing the completeness of different systems of vector functions in C^^îS) we shall make use on the results of following lemmas.

LEMMA 39. Let Dt be a bounded domain of class C^ and consider the vector potential Aa with density a e C^^jîS) that satisfies

V x V x A = 0 inDi. (10.5)

Then at almost any point on S = dDi we have a = 0.

LEMMA 40. Assume the bounded set Dt is the open complement of the unbounded domain Ds of class C^ and let A^ be the vector potential with density a G C^^îS). Ifk is not an eigenvalue of the interior Maxwell boundary-value problem, k ^ cx(Di), and

V x V x A « = 0 inDs, (10.6)

then at almost any point on S = d Di we have a = 0.

The above lemmas were proved by Eremin et al. in Chapter 3.

LEMMA 41. Assume the bounded set Dt is the open complement of the unbounded do-main Ds of class C^ and let A.1'^ and A^'' be the vector potentials with square integrable tangential densities e and h:

/ A'/(x) = j e(y)g{x,y,ksj)dSiy),

s

A'^\x) = j h(y)g(x, y, ksj) dS(y). (10.7)

s

If

V X A ' + ^ - V X V X Ai = 0, X € A , koSs

V x A ^ + - ^ V x V X A | , = 0, xeDs, (10.8)

then at almost any point on S = dDi we have e = h = 0.

PROOF. The jump relations for the vector potential with square integrable densities give

lim L J n X (V X Al(-Xn))] + ^ [ n x (V x V x Al(-Xn))]

• Si[n X (V X A^,(+Xn))] - ^ [ n x (V x V x A|,(+Xn))]

•i e + Msej - Si Q e + M,ej + ^ ( ^ . - Ôh = 0, (10.9) 2


where M and V are the magnetic and the electric dipole operators defined in [15,16]. Then, we find that

i {8s + ^/)e - {SsMs - SiMi)t ~-j-(Ps- Vi)h = 0. (10.10)

Taking the curl of the left hand sides of (10.8) and proceeding as above we get

l (lis + lii)h- (fisMs - iiiMi)h+ ^(Vs -Vi)e = 0. (10.11)

The system of integral equations consisting of (10.10) and (10.11) is Miiller's system of equations for the surface densities e and h [14]. This system is a Fredholm system of integral equations of the second kind. Consequently, the smoothness of the resol-vent gives (e, h) ^ (eo, ho) e Ctan(*S') x Ct^niS). Since Mullers's system is uniquely solv-able, i.e., there are no continuous non-zero solutions of the homogeneous system, we get ( e , h ) - ( e o , h o ) = 0 . D

10.3. Null-field equations

Let Es,lls ^CHDs)nC(Ds) and E/,H/ e C\Di) H C(Di) solve the transmission boundary-value problem. The Stratton-Chu representation theorem [15,16] gives the null-field equations in the interior and exterior region; that is,

Vx jes(y)g(x,y,ks)dS(y)

s

+ - ^ V x V x [hs{y)g{x,y,ks)dS(y) = 0 f o r x e A , (10.12)

s

and

^x jei(y)g(x,y,ki)dS(y)

s

+ - ^ V x V x fhi(y)g(x,y,ki)dS(y) = 0 f o r x e D , , (10.13)

s

where e = n x E^, h^ = n x H^, e/ = n x E/ and h/ = n x H/. Taking into account the boundary conditions on the particle surface: e + eo = e/ and h^ + ho = h/, we rewrite Eqs. (10.12) and (10.13) as

V X I [eKy) - eo(y)]g(x, y,/:,) d5(y)

s

+ ^ V X V X /* [h,(y) - ho(y)]g(x, y, ks)dS(y) = 0, x G A , (10.14)

234 A. Doicu

and

^x fei(y)g(x,y,ki)dS(y)

s

+ J _ V x V x [hi(y)g(x,y,ki)dS(y) = 0, xeD^, (10.15)

s

respectively. In this context, we can state the following formulation of the general null-field equation for the transmission problem.

General null-field equations for the transmission problem: Assume the bounded set Di is the open complement of the unbounded domain Ds of class C^ and let n denote the unit-normal vector to the boundary S = dDi directed into the exterior of Di. Given EQ, HQ as an entire solution to the Maxwell equations find continuous tangential fields e and h satisfying the set of integral equations

V X y [e(y) - eo(y)]g(x, y, k^) dS(y)

s

+ ^ V X V X J [h(y) - ho(y)]g(x, y, k^) dS(y) = 0, x G A,

s

^x je{y)g(x,y,ki)dS(y)

s

+ J _ V x V x [h(y)g(x,y,ki)dS(y) = 0, XGD,. (10.16)

s

The existence of continuous solutions to the general null-field equations follows from the existence of solutions to the transmission problem. From Lemma 3 we see that the solution is unique in the space H = C^^j^iS) x C^^^(S). In addition we mention that it is possible to establish the equivalence between the general null-field equation and a system of boundary-integral equations of the second kind. This equivalence enables us to prove that the solution b e l o n g s t o C l " ^ ( 5 ) x C l « / 5 ) .

10.4. Complete systems of functions

In this section we present complete and linear independent systems of functions for the transmission boundary-value problem and formulate the null-field equations in terms of these functions. We begin our presentation with fundamental results on the completeness of the localized spherical vector wave functions. After that we construct complete systems of functions using distributed sources. We start with the systems of spherical vector wave functions and vector multipoles distributed on a portion of a straight line. Our analysis is based on the addition theorem for spherical wave and vector wave functions. The next results concern the completeness of the system of magnetic and electric dipoles and the system of vector Mie potentials having the singularities distributed on auxiliary closed and open surfaces. These systems are especially suitable for analyzing the scattering character-istics of nonaxisymmetrical particles.


10.4.1. Localized spherical vector wave functions

We begin by defining our notation. The independent solutions of the vector wave equa-tions V X V X X — k^X = 0 can be constructed as

Mmnikx) = Vumnikx) X X and ISimnikx) = - V X M^„(^x), (10.17) k

where in spherical coordinates the Umn are the spherical wave functions. The specific form of the spherical vector wave functions are

j m — - T : ^ e — e^ eJ' ' , smO dO

Nl,l(kx) = \n(n + 1 ) ^ 4 ^ Pfl(cos^)e,

[krzn(kr)Y [dPJT^cosO) ^''"'

kr

\krzn(kr) + kr dO

e . + j m ^ ^ L J ^ e J } d - ^ (10.18) sm^ ^JJ

where (e^, e^, e< ) are the unit vectors in spherical coordinates. The superscript' 1' stays for the regular spherical vector wave functions while the superscript ' 3 ' stays for the radiating

1 3

spherical vector wave functions. It is useful to note that for w = m = 0 we have MQQ =

The expansion of gl in terms of spherical vector wave functions is well known in the literature [1-3]. It is given by

n=l m=-n

+ N^_^^ (ky)Nl^^ (kx)] + Irrotational terms, (10.19)

for y > X, and

_ .. oo n

^(x,y,^)!= J - ^ J2 D^„[Ml^„iky)Mi„{kx) n=\ m=-n

+ N L ^ „ (^y)N^„ (kx)] + Irrotational terms (10.20)

fory < X. Here, Dmn is a normalization constant:

2^ + 1 (n-\m\)l Dmn = - —^- (10.21)

4n(n-\-l)(n + \m\)\ ^ ^

We are now in the position to construct complete systems in jC^^^iS).

THEOREM 42. Let S be a surface of class C^. Then (a) the set of tangential components of radiating vector spherical functions

{ n x M ^ „ , n x N ^ „ } m€Z,n^max(l,|m|)'

236 A. Doicu

and, for k ^ cr(Di), (b) the set of tangential components of regular vector spherical wave functions

[n X M ^ „ , n X ^mn\meZ,n^max{l,\m\)

are complete in jC^^^iS),

PROOF. For proving (a) we consider the vector field

J S="-V xW xAa k

(10.22)

with a G C^^ji(S). Let D[ be a sphere enclosed in D, and let x e D^. Then, using the

identity a^ = a • gl and expansion (10.19) we get

j^2 oo n (F

^W = - - E E ^-M / a(y)-M _ „(/:y)d (y) n=lm=-n I L c

UUkx)

+ / a (y ) .N3_ , r(ky)dS(y) Kjkx) (10.23)

Using the orthogonality of vector spherical wave functions on the spherical surface 5''* = 9D[, and taking into account the identity a = —n x (n x a), which holds almost every-where on S, we see that the set of equations

j[nx a(y)] • [n x Mi^(ky)]dSiy) = 0,

s

j[nx a(y)] • [n x Ni,(ky)]dS(y) = 0, (10.24)

with n = 1, 2 , . . . , and —n < m < n, leads to f = 0 in D[, and conversely. Since £ is an analytic function in Dt we conclude that (10.24) implies £ = 0 in Dt. In this context application of Lemma 39 gives a '^ 0. Hence, n x a* ~ 0 and (a) is proved. For proving (b) in the case k ^ cr(Di) we use the same technique. Actually, we derive an analogous set of equations implying £ = 0 in D^ by replacing the radiating spherical vector wave functions by the regular spherical vector wave functions. D

Let us consider the general null-field equations for the transmission problem given by (10.16). Then, we have the following result:

THEOREM 43. Let Dt be a domain of class C^ with boundary S. If the pair (e, h) solve the set of null-field equations

I {[n(y) X (e(y) - eo(y))] • [n(y) x M^„(^,y)]

+L— [n(y) X (h(y) - ho(y))] • [n(y) x N ^ f e y ) ] dS{y) = 0,


j j [n(y) X (e(y) - eo(y))] • [n(y) x N^„(^.y)]

s

+ j . /^ [n(y) X (h(y) - ho(y))] • [n(y) x M^C^.y)] j d5(y) = 0,

n[n(y)xe(y)]-[n(y)xMi,„(A:,y)]

s

+j M[n{y) x h(y)] • [n(y) x NJ^„(^,y)]) d5(y) = 0, V ^i J

n[n(y)xe(y)].[n(y)xNL(/:,y)]

s

+ j /^[ii(y) X h(y)]. [n(y) x M^C/^/y)]} d5(y) = 0, (10.25) V ^i J

for m eTj, n^ max(l, |m|), then (e, h) solve the general null-field equations (10.16), and conversely.

PROOF. The proof is similar to that of Theorem 42 by using expansions (10.19), (10.20) for the electromagnetic fields

and

^^=^ ^ ^^-- ^ i : " ^ ^ ^-^0' ^^=j%i; ^ ^- ^''-''^

,- = V X A^ + - ! - V X V X Aj,, Hi = V X £i. (10.27)

D

10.4.2. Distributed spherical vector wave functions

Let us construct complete systems of vector functions using lowest-order vector spher-ical wave functions. Consider a set of points {Zn}^\ located on a segment F^ c Oz. We define the set of distributed vector spherical wave functions as

1 ,|m|+/ A/"! (^x) = N^';,,,+,[^x - z.es)], (10.28)

where m € Z, ^ = 1, 2, . . . , and / = 1 if m = 0 and / = 0 if m 7 0, then we can state the following theorem:

00

THEOREM 44. Let {Zn]n=i = ^z- Replace in Theorem 42 the localized spherical vector wave functions M '„ and N^'„, m eZ, n^ max(l, \m\) by the distributed spherical vector wave functions Aimn ^nd Mmn ^ nt eZ,, n = 1,2,..., respectively. Then, the resulting systems of vector functions are complete in Cl.^^{S).

238 A. Doicu

PROOF. For proving (a) we have to show that for any a G C^^îS), the set of equations

ya*(y ) . [nxA<^„( / :y ) ]d5 (y )=0 ,

s

j a*(y) • [n x Af^îky)] dS(y) = 0, m G Z, n = 1,2,..., (10.29)

impHes a '^ 0 on 5. For a fixed azimuthal mode m we assume F^ c D[ and use the addition theorem for spherical vector wave functions to rewrite the closeness relations as

fmizn) = y*a*(y).{nxM^,^,^^[/:(y-z„e3)]}d^(y)

- E -4:; l '«l+'(-z„)/a*(y). [n x M^,(^y)] d5(y) «'^max(l,|m|) ^

+ C ' ' " ' ^ ' ( - ^ « ) / a * ( y ) • [n X N^„,(A;y)]d5(y) ^0, (10.30)

and

gmiZn) = / a * ( y ) • {n X N^, |m|+/[^(y-^«e3)]}d5(y)

- E <'^'^'(-^n) I a*(y) • [n x ^^ky)] dS(y)

-Zn)jii*(y) • [n X M^„,(^y)]dS(y) - 0 . (10.31)

«'^max(l,|m|)

•^m,|m|+/._ ' mn' ^

Here, ^ and B are the translation addition coefficients. If the translation is along the z-axis they can be expressed as

AZ:'izo) = 2f-"D, mn' 71 n

X

0

Jt

X

0

xeJ^ôcos^g.j^^^^ (10.32)


with Dmn' being the normahzation constant (10.21) and zo the distance between the origins of the original and translated coordinate system. Since fm and gm are analytic functions

and {z^l^i is dense in r „ we find that fm{z) = gm(z) = 0, Vz G T,. Then, friz) = gly!\z) = 0, Vz G r^, where the superscript denotes the derivative of order n with respect to z. In particular, /^"^(0) = gln^(0) = 0 for n = 0 , 1 , . . . . That means

fi'HO) = Y. A-Ja*(y).[nxM^,,(^y)]d5(y)

+ B:.,/a.W.[„xNL.(WldSW

and

where

and

= 0, ^ = 0 , 1 , . . .

g(^\0) = ^ A-, I a*(y) • [n x N .C^y)] d5(y)

i,/a-W.[„xML.(W]dS(,)

«'^max(l,|m|)

0, n = 0 , 1 , . . . ,

A m

dz' z=0

= 2f'-i-i-^D^,.(-j/cr

X (cos^)"sin)0d^

Bm _ .^„gm,|m|+/ ( - Z ) '

dz' z=o

n

0

X (cosySfsin^dig.

Integration by parts and application of the well known formula

(10.33)

(10.34)

(10.35)

(10.36)

P,i^l(cos^) = (2|m|-l)!!(sin^) \m\ (10.37)

240

yields

A. Doicu

\m _ I ^nn' — 1

2r'-^"'^Dmn'(-ikr\m\(n + 1 + |m|)(2|m| - l)!!/^";^,, m / O , (10.38)

and

B[ " 10, m = 0.

Here, I^, is the integral

; . = / < C = / (cos^r(sin^)l'"lp]r'(cos)g)sin)6d^, (10.40)

where n = 0 , 1 , . . . and n^ > max(l, |m|). Note, that for m = 0 and n = 0 we set by con-vention /^j , = 0. The integral / ^ , can be computed as follows. Define

(10.41) /mnKx) = J eJ '"^ (sin ) " P„ (cos ^) sin ^ d;6

0

for m = 0 , 1 , . . . and n^ ^ max(l, |m|). For m = 0 we find that 7on' is the standard integral

hn'ix) = j ^'''''^Pn'{0OSP)sixiP&P = lf jn'{x). (10.42)

0

Integration by parts and the recurrence relation

dP^^(cos^) sin^S

dyS = -mcoS|SP„'7(cos^)

+ sinjS (n' + m)(n' - m + l)P'?~'(cos/S) (10.43)

gives

Hence,

Jmn'{x) = (n' - m + l ) (n ' + m)^- l ,« ' ( Jc) •

y + m , ixm("'+'«)!7n'(A^) ^„'W=2f+'"(-ir^ {n'—m)\ x"

On the other hand we see that

, _._„rd"/|^K(A:)1

(10.44)

(10.45)

(10.46)


for n = 0 , 1 , . . . and n' > max(l, \m\). Therefore, using the series expansion of spherical Bessel functions we ultimately find that

^""'-^ J ( n ' - | m | ) ! k\i2n' + 2k+iy. ^^M-n'+iml- (10.47)

The coefficients / ^ , are different from zero if, for a given azimuthal mode m and a given pair of integers (n, n'), there exists an integer ^ = 0 , 1 , . . . such that 2k = n — n' -\- \m\. Consequently, A^^, = 0, for n' > w + 1 and m = 0, while A^ , = 0 for w > n + |m| and m 7 0. Similarly^B^^, = 0 for n' > n + |m| - 1 and m / 0. Thus, from (10.29) we obtain

Ja*(y).[nxM^„,(^y)]d5(y) =0, s

/ 'a*(y). [n x N^„,(/:y)]d5(y) = 0 , m G Z, n'^ max(l, |m|), (10.48)

s

and due to the completeness of the localized spherical vector wave functions we conclude that a ~ 0. In an analogous manner we can prove the rest of the theorem by using the addition theorem for spherical vector wave functions and the completeness of the system {n X M^„, n X N^„}mGZ, i max(i,|m|) in Aln(^)- •

Formulations of null-field equations in terms of distributed vector spherical functions are given by the following theorem:

oo

THEOREM 45. Let [zn)n=\ — ^z- ^^^ ^he pair (e, h) solve the set of null-field equa-3 1 3 1

tions (10.25) in which the localized spherical vector wave functions M^„ and N^^, m G Z, 3 1

n > max(l, \m\) are replaced by the distributed spherical vector wave functions Mmn ^^d J^mn^ m eJj, n = 1,2,..., respectively. Then, (e, h) solve the general null-field equa-tions (10.16), and conversely. PROOF. The proof is in a manner analogous to Theorem 44. We use the addition theorem for vector spherical functions and Theorem 43. D

10.4.3. Distributed electric and magnetic dipoles

Let V be an arbitrary vector in R^, and define the vector functions

1 m(x, y,v) = -^ [v(x) x Vyg(x, y, k)],

n(x, y,v) = -VyX m(x, y, v). (10.49)

It is noted that for x 7 y and arbitrary a, the following identities hold

[Vx X (a(y)^(x, y, k))] • v{x) = [a(y) x Vyg{x, y, k)] • u(x)

= -a(y) • [u(x) X Vyg(x, y, k)] = -k^y) • m(x, y, v) (10.50)

242 A. Doicu

and

[V^ X V^ X (a(y)g(x, y, k))] • v(x) = {V^ x [a(y) x V^^(x, y, k)]} • i;(x)

= [-(a(y) • V;,)V,g(x, y, k) 4- a(y)V^ • (V,g(x, y, k))] • v(x)

= [(a(y) • V^) Vy^(x, y, k) - a(y)A^^(x, y, /:)] • v(x)

= [{v{x). V^)V^g(x, y, k) - v(x)Ayg(x, y, k)] • a(y)

= _{V^ X [v(x) X V^^(x, y, k)]}. a(y) = -^^aCy) • n(x, y, v). (10.51)

Let S be an arbitrary smooth surface and {x„}^j a sequence of points on 5 . For x ^ 5 we define the system of magnetic and electric dipoles distributed on the surface S by

Mni(kx)=m(Xn,X,Tni), « = 1,2, . . . , / = 1,2, (10.52)

and

Mni(kx) = n(x„, X, Tni), « = 1,2,..., / = 1,2, (10.53)

respectively. Here, T„I and T„2 are two tangential linear independent unit vectors at the point x„. In view of (10.50) and (10.51) we have

ja(y)g(x,y,k)dS(y)\ •Tni = -k^j

s -'x=x« s

2i(y)'Mni(ky)dS(y) (10.54)

and

V X V X J a(y)^(x, y, k) dS(y) • Xnt = -k^ I a(y) • Mni(ky) dSiy). (10.55)

We denote by At^ • and A/ - the system of magnetic and electric dipoles having the origins {x~}^j on a smooth surface S~ = dD^ enclosed in D/, i.e., Df C A . Analogously, we denote by M^i and Af^- the system of magnetic and electric dipoles having the origins {x+}^j on a smooth surface S~^ = dD^ enclosing A , i.e., D/ c D^. Complete systems in C^^n(^) are given by the following theorem:

THEOREM 46. Let Dt be a domain of class C^ with boundary S. Let the sequence {x~}^j be dense on a smooth surface S~ enclosed in Di, and let the sequence {x^}^^ be dense on a smooth surface 5" enclosing Dt. Then (a) the sets of tangential components of radiating electric and magnetic dipoles

{nx-^m}„=l,2 ,=1,2 ««^ {nx-^m-1=1,2,...,/=1,2

and, for ks ^ cr(Di), (b) the sets of tangential components of regular electric and magnetic dipoles

{n X A/;l. }^ j 2,..., i=l,2 «"^ {" ^ ^ni }„=l,2,..., i=l,2

are complete in >C an(* )-


PROOF. Consider (a). Define the electromagnetic field

f = i v x V x A « (10.56) k

with a G Cl^^{S). For x~ G 5"" we have

£{x-). r 7 = -]k^ j a(y) • M^,(ky) dS(y). (10.57)

Then, the set of equations

j [n(y) X a(y)] • [n(y) x Af^ (ky)] dS(y) = 0 (10.58)

s

with n = 1,2,... and / = 1,2, gives S(x~) • T ^ = 0. Since r~^ and T ^ are two tangential linear independent unit vectors on S~, it follows that n(x~) x £(x~) = 0. We then use the assumption that {x~}^j is dense on S~ to obtain n x £ = 0 on S~. Since ks ^ cr{D^), we conclude that f = 0 in Dr . Then, by the analytic continuation procedure we get f = 0 in Di. Finally, application of Lemma 39 gives a ~ 0. Thus, n x a* ^ 0 and the proof is finished. The second part of (a) can be proved by repeating the above reasons for the magnetic field

W = V x A . (10.59)

For proving (b) we see that the closeness relations written for the systems

|n X Afli I 1 o • 1 o and In x MII ] . ^ . , ^ I ^^^^=1,2,..., 1 = 1,2 I ni in=l,2,..., i=\,2

imply n X £ = 0 on 5" and n x H = 0 on 5'" , respectively. Hence, the fields £ and H vanish in Ds. Consequently, application of Lemma 40 gives a ~ 0. Note that no restrictions are imposed on the surface S'^ to overcome irregular frequencies. D

Formulations of the null-field equations in the variety of electric and magnetic dipoles are given by the following theorem.

THEOREM 47. Let Dt be a domain of class C^ with boundary S. Let the sequence {x~}^j be dense on a smooth surface S~ enclosed in Dt, and let the sequence {x^}^j be dense on a smooth surface S^ enclosing Dt. Assume ks ^ a{D^), S~ = dDJ~ and let the pair (e, h) solve any sets of null-field equations

(a)

n [ n ( y ) X (e(y) -eo(y))] • [n(y) x Ml^iksy)]

+ j J—[n(y) X (h(y) - ho(y))] • [n(y) x Af^^iksy)] \ dS(y) = 0,

244 A. Doicu

y*j[n(y)xe(y)].[n(y)xAî,(/ : ,y)]

s

+J , / ^ [ n ( y ) X h(y)]. [n(y) x M^ {kty)]) d5(y) = 0, (10.60)

for w = 1,2,... and i = 1,2, and

(b)

y*{[n(y) X (e(y) - eo(y))] • [n(y) x KmiksY)]

s

+ j , / ^ [ n ( y ) X (h(y) - ho(y))] • [n(y) x Mi^^ (/:.y)]) dS(y) = 0,

y{[n(y)xe(y)] . [n(y)xAC.. ( / : .y)] s

+ j / ^ [ n ( y ) X h(y)] • [n(y) x A^^„,(/:,y)]) d5(y) = 0 , (10.61) V î J

/or « = 1,2,... flnJ i = 1,2. Then, (e, h) ô/v^ ?/z general null-field equations (10.16), and conversely.

PROOF. The proof is in a manner analogous to Theorem 46. We consider the electromag-netic fields defined by (10.26) and (10.27), and make use of identities (10.54) and (10.55). D

10.4.4. Distributed vector Mie potentials

Let us consider the set of functions

< n(x) = ^(x„,x,/:), n = l,2,..., x^Xn (10.62)

having singularities {x„}^j distributed on the auxiliary surfaces. We use the notations (p~(x) and ^(x) to designate the system of functions (10.62) having poles {x~}^^ and {x+}î distributed on the auxihary surfaces S~ = dD^, D7 c A , and 5+ = dDf, Di c Df, respectively. Miiller and Kersten [13] proved that the systems {(p~}^^^ and [^n}^=\ ^^ complete in L^(5) and therefore can be used to approximate any solution of the Helmholtz equation.

Let us extend the above systems to the electromagnetic case. Define the system of vector functions

M];\kx) = ^ V X (^„^(x)x) = i V^±(x) x x (10.63)

and

Mîkx) = ^ V X M];^{kx) = 4 V X (V^^(x) x x). (10.64)

a , 3 _ . ./-i,3 The functions Mn and Mn' are called vector Mie potentials.


The following theorem states the completeness of the vector Mie potentials in C^^in^S):

THEOREM 48. Let Di be a domain of class C^ with boundary S. Let the sequence {x~}^^ be dense on a smooth surface S~ enclosed in Di, and let the sequence {x^}^j be dense on a smooth surface S'^ enclosing Di. Assume ks ^ p{DJ~), S~ = dDf and re-

3 1 3 1

place in Theorem 2 the localized spherical vector wave functions M^'„ and N^«, m eZ, n > max(l, \m\) by the distributed vector Mie potentials Mn cind Nn' , n = 1, 2 , . . . , respectively. Then, the resulting systems of vector functions are complete in >C ajj(5').

PROOF. We prove only (a). It has to be proved that the closeness relations

s

ja%y)'[n(y)xXl(ky)]dS(y) = 0 , p = l , 2 , . . . (10.65)

f-

3 mn give a ~ 0. Let us consider the radiating spherical multipoles M^„ and N:

M L ( ^ X ) = V M ^ „ ( X ) X X ,

N L ( ^ X ) = ^ V X {Vui,(x) X x) (10.66)

and choose an interior surface 5" which is parallel to 5* and encloses S . For fixed indices — 1(X)

pfp=i

^^ ^j X.XXWV*. wombinaUv x... ^ ^p •

m and n and {x^ }^j G 5 , we can approximate the radiating spherical wave functions

w^„ by linear combinations of (p~. Therefore, there exists the sequence

1 ^ UN(x) = -Y^a^(p-(x) (10.67)

k p=i

such that

lim UN(x) = ui^(x) (10.68)

uniformly in closed subsets of D | . Note that the coefficients ap depend on the indices m

and n. In fact the derivatives of w^^ can also be approximated uniformly in closed subsets

of Ds by the derivatives ofuM- Then, defining

N N

MNiy) = J2''p^V^y^ ^ ^ Ar;v(y) = ^<Arj(/:y), (10.69) p=i p=\

we see that

lim MN(y) = M^^(ky) and lim A/'iv(y) = A^l(/:y) (10.70)

246 A. Doicu

uniformly on S. Therefore, multiplying the first A equations in (10.65) by a^, summing the resulting expressions, and letting N ^^ OOWQ arrive at

/ a*(y).[n(y)xML(/:y)]d5(y) = 0,

s

j a*(y). [ii(y) x N^C^y)] dS(y) = 0. (10.71)

s

Since m and n are arbitrary we see that (10.71) hold for m € Z and n ^ max(l, |m|). Finally, the completeness of spherical vector wave functions in >C an( ) gives a ~ 0.

For proving the completeness of the regular vector Mie potentials we proceed analo-gously. We choose a homothetic exterior surface S" and construct the approximates M.N and XN by replacing in Eq. (10.69) M^p and A/^ by M], and Af^, respectively. Since by construction UN = (l/k) J2^=i ^p^^ approximate w^„ uniformly in closed subsets of of, we see that limAr- oo MN = M^„ and limA/ oo J^N = ^mn uniformly on S. D

We conclude this section by noting a theorem which establishes the null-field equations in terms of vector Mie potentials.

THEOREM 49. Let Dt be a domain of class C^ with boundary S. Let the sequence {x~}^^ be dense on a smooth surface S~ enclosed in Dt, and let the sequence {x^}^j be dense on a smooth surface 5" enclosing Dt. Assume ks ^ p(D^), S~ = dD^ and let the pair (e, h) solve the set of null-field equations (10.25) in which the localized vector spherical functions M^^ and N^'^, m eZ, n^ max(l, |m|) are replaced by the distributed vector Mie potentials Mn and Mn' , w = 1, 2 , . . . , respectively. Then, (e, h) solve the general null-field equations (10.16), and conversely.

10.5. Null-field method

Within the analysis of the preceding sections we are now well prepared to discuss the null-field method with discrete sources for the transmission value problem. Essentially, this method consists of the projection relations

(e-eo,nxnxvl/^^*) + / h - h o , - j / ^ n x i i x < D ^ * \ = 0,

( e -eo ,nxnx(D^*} + ( h - h o , - j / ^ n x n x x i ^ , ^ = 0,

(e,n X n X vl/i*} + /h, H y ^ n X n X Oj*) = 0,

( e , n x n x O j * ) + / h , - j / ^nxnx^^^* \===0 , y = l , 2 , . . . (10.72)

with the approximations e v and IIA^ of the form

e^, = f ] < ( n x * i ) + & ; r ( n x O ,


^ f ^ a ; r ( n x O + ^ ; r ( n x * i ) . (10.73)

The set {^y' , ^\'^}v=\,2,... represents the sets of locaHzed and distributed spherical vector wave functions, magnetic and electric dipoles, or vector Mie potentials. The regular func-tions ^^ and Oj, depend on the wavenumber kt corresponding to the interior region, while the radiating functions ^l and Oj have a ks dependence.

When the localized spherical vector wave functions are used as basis and testing func-tions, the projection method (10.72)-( 10.73) is identical to the scheme obtained in the frame of the single-spherical-coordinate-based null-field method. From (10.73) we see that the surface-current density e// is written as a linear combination of elementary sources which generate the interior field. The form of h v is dictated by a physical consideration: the surface-current densities e and h are the tangential components of the interior electric and magnetic fields E/ and H/ = — 0/(^0^/))V x E/, respectively. The above represen-tations for CA and \i^ reduces the computation effort, since the null-field equations are identically satisfied in the exterior of S. Indeed, taking into account that ^ ^ and 0|^ are

regular in D/, V x ^ ^ = ^1^)1^ ^^^ ^ ^ ^]i= î^ji^ ^ ^ ^ Green's theorem applied in the region D/, it follows that the null-field equations are identically satisfied in Di. Con-sequently, the amplitudes of discrete sources which produce the surface-current densities will be computed by using the null-field equations in the interior region.

Once the surface densities are determined, the approximate solution for the scattered field can be obtained using the representation theorem. Restricting x to lie outside a cir-cumscribing sphere S^ and using the vector spherical harmonics expansion of the Green's function we get

E f ( x ) = E E D^n[flMi„{ksX) + g!Â(k,x)l (10.74) m=-mmax ^^max(l,|m|)

where

fl = — [fe^Cy) • ^-mniksy) + j , / ^ h ^ ( y ) . Ml^îksy)) dSiy),

s 2

'"-?/( e^(y). Ml^îksy) + j /— h^(y) • Nl^^(ksy) ] dS(y). (10.75)

Most numerical computations based on the null-field approach use spherical vector wave functions as global basis functions. Although these wavefunctions appear to provide a good approximation to the solution when S is smooth and "near" a spherical surface, they are disadvantageous when this is not the case. In the single-spherical-coordinate-based null-field method the surface-current densities are obtained by imposing the null-field condi-tion inside the maximal inscribed sphere. By analytic continuation the total field is zero throughout the entire interior volume. However, in a numerical scheme we guarantee that the residual field tends to zero inside the maximal inscribed sphere but, in general we can-not conclude that this field converges to zero within the whole interior volume. If instead of locaHzed spherical vector wave functions we use distributed sources, it is possible to

248 A. Doicu

extend the applicability range of the single-spherical-coordinate-based null-field method. The explanation lies in the fact that distributed sources are better suited to model complex boundaries than localized sources. Essentially, the null-field condition is satisfied in the interior of the discrete sources support, whose form and position can be correlated with the boundary geometry.

10.6. Numerical results

Computer programs using the null-field method with discrete sources have been devel-oped. To check the accuracy of the proposed methods we consider particles of refractive

l E - 0 1 ,

lE-02 ;

l E - 0 3

lE-04 :

lE -05 :

lE -06

DSCS

\

I

• Mie 1

—NFM " — I — 1 — 1 — 1 — 1 — 1 — 1 — 1 — | —

s - p o l a r i z e d

p - p o l a r i z e d

ka=1.0 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 —

0 30 60 90 120 150 180 (a)

lE+00 ^

l E - 0 1 .

lE-02 ;

lE -03 :

lE -04 :

lE -05 :

lE -06

DSCS

^^>fc|5j s - p o l a r i z e d

p - p o l a r i z e d \ jT

• Mie —NFM ka=2 .0

— . — . — 1 — . — 1 — , — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 —

0 30 60 90 120 150 180 (b)

Fig. 10.1. Plots of the normalized DSCS patterns for a spherical particle with size parameter (a) ksa = \ and (b) /c fl = 2. The angular scattering pattern is determined in the ( = 0° plane for ofpoi = 0° and a^Q\ — 90°. The

plots correspond to the Mie solution and the null-field method with dipoles and Mie potentials.


index M = 1.5 + 0 j whose morphology gradually increases in complexity. For the particle surface we consider a parametric representation in the Cartesian coordinate system Oxyz. Specifically, the normalized differential scattering cross-section (DSCS) is evaluated for plane-wave incidence over the azimuthal planes ip = 0° and (p = 90°. The incident wave is linearly polarized and the polarization direction encloses an angle ofp with the X-axis.

The first objective is to demonstrate the validity of the null-field method with distributed dipoles and vector Mie potentials by considering a spherical scatterer. In this case we choose S~ and S~^ to be spherical surfaces of radii a_ and «+, respectively, concentric with S. In accordance with the guidelines given in [17], it is found that for a sphere of ra-dius a selections of a- between 0.2 a and 0.6 a and a^ greater than 2a have a comparable

l E + 0 1 j

l E + 0 0 :

l E - 0 1

l E - 0 2 ,

l E - 0 3

l E - 0 4

l E - 0 5 :

l E - 0 6

l E - 0 7

l E - 0 8

DSCS • l o c a

— d i s t

l i z e d s o u r c e s

r i b u t e d s o u r c e s

\ / p h i = 0 \

p h i = 9 0

ka=0.6 kb=0.8 kc=1.0

—1—'—'—1—'—'—1

0 30

DSCS

60 90 120 150 180 (a)

T U J.

+ 00

- 0 1 "

- 0 2

- 0 3

- 0 4

- 0 5

• l o c a l i z e d s o u r c e s

— d i s t r i b u t e d s o u r c e s

^ ^ ^ X ^ ph i=

p h i = 0 \ ^

— 1 — I — 1 — 1 — • — 1 — . — i — 1 — 1 — i — 1 —

= 90

ka=l kb=l kc=2

2 6 0

0 30 60 90 (b)

120 150 180

Fig. 10.2. Plots of the normalized DSCS patterns for dielectric ellipsoids with size parameter (a) kga = 0.6, ksb = 0.8 and kgC = 1 and (b) kga = 1.2, kgb = 1.6 and kgC = 2. The normalized DSCS is evaluated in the azimuthal planes (p = 0° and (p = 90°. The plots are computed by using the single-spherical-coordinate-based

null-field method and the null-field method with dipoles and Mie potentials.

250 A. Doicu

rate of convergence over the number of sources. Results for the problem of plane wave scattering by a sphere with a size parameter of (a) ksa = \ and (b) ksa = 2 are shown in Fig. 10.1. The angular scattering pattern is determined in the plane (p = 0° for a^ = 0° and Qfp = 90°. Good agreement with the exact Mie series solution was obtained for 44 sources distributed on auxiliary surfaces in case (a), while 58 sources are required in case (b).

In the next examples we consider particles without rotational symmetry. We verify the accuracy of the null-field method with discrete sources distributed on auxiliary surfaces by using the single-spherical-coordinate-based null-field method as reference. Let us consider a dielectric ellipsoid whose surface is described by

a^ b^ c^

and choose the size parameters to be: (a) ksa = 0.6, ksb = 0.8 and ksC = 1 and (b) ksa = 1.2, ksb = 1.6 and ksC = 2. The plots in Fig. 10.2 represent the normalized DSCS evaluated in the azimuthal planes (p = 0° and (p = 90°. In line with the criteria for spherical scatterers the auxiliary surfaces 5" and 5+ are chosen to be homothetic to S, with a homothetic ratio of 0.5 and 2, respectively. It is found that 92 sources are required to obtain an agreement with the single-spherical-coordinate-based null-field method in case (a), while 112 sources are necessary in case (b).

Results for the problem of plane wave scattering by a dielectric cube are shown in Fig. 10.3. The cube size parameter is ksl = 2, where / is the side length. Evaluation of the DSCS is done in the plane cp = 0° for ofp = 0° and a^ = 90°. The selected auxiliary surfaces are taken to be spherical surfaces of radii a- = 1/4 and a+ = 3//2, respectively. Good agreement with the single-spherical-coordinate-based null-field method solution is obtained for 92 poles.

lE + 01 ,

lE+00 :

l E - 0 1 '

lE-02 :

lE -03 :

lE-04 :

lE -05

DSCS • l o c a l i z e d sources — d i s t r i b u t e d

sources

' * • • *~M.l; -*-*-»~»> . . s - p o l a r i z e d

p - p o l a r i z e d \ /

V k l = 2 . 0 — 1 — 1 — 1 — 1 — 1 — I — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — , —

0 30 60 90 120 150 180


Fig. 10.3. Results for the problem of plane wave scattering by a dielectric cube. The cube size parameter iskgl = 2. Evaluation of the DSCS is done in the plane (p = 0° for apoi = 0° and apoi = 90°. The plots are computed by using the single-spherical-coordinate-based null-field method and the null-field method with dipoles and Mie

potentials.

10. Null-field method with discrete sources

DSCS

251

l E -

l E -

l E -

l E -

l E -

l E -

l E -

l E -

l E -

-01 J

-02 :

- 0 3 ;

- 0 4 ;

- 0 5 ;

-06

-07 \

- 0 8 ;

-09

P-

ka

- p o l

=20 , - " — 1 —

a r i z e

kb= = 1 —r-

d \

.0

^ ^'

—1— • -

-po 1

T -

• MMP

—NFM

a r i z e d

- 1 — . — 1 — 1 — 1 —

0 30 60 90 120 150 180


Fig. 10.4. Plots of the normalized DSCS patterns for an spheroidal particle with a size parameter kga = 20 and an aspect ratio a/b = 20. The DSCS is computed in the (p = 0° plane for ofpoi = 0° and apoi = 90° by using the

multiple multipole method and the null-field method with lowest-order multipoles.

In Fig. 10.4 we consider a spheroidal particle with a size parameter of ksa = 20 and an aspect ratio a/b = 20. In this case we use a set of 30 lowest-order multipoles located on the particle's symmetry axis. For comparison we have plotted the DSCS computed by using the multiple multipole method [18], since this type of particle cannot be handled by the single-spherical-coordinate null-field method. The angular scattering pattern is determined in the (p = 0° plane for ofpoi = 0° and ofpoi = 90°. These results clearly demonstrated that no significant differences exist between the scattering diagrams. The superiority of the lowest-order multipole-based null-field method over the standard method lies in the fact that the matrix formulation includes Hankel functions of low orders which lead to a better conditioned system of equations.

10.7. Conclusions

In this section we summarize the basic features of the null-field method with discrete sources.

1. In the conventional null-field method (single-spherical-coordinate-based null-field method) the surface-current densities are generated by the system of spherical vector wave functions with a single origin, while the null-field condition is imposed inside the maximal inscribed sphere. Conversely, in the null-field method with discrete sources the surface-current densities are produced by fields of elementary sources, while the null-field condi-tion is imposed inside an auxiliary region (support of discrete sources) whose form and position can be correlated with the particle geometry. In this context it is noted that one of the great merits of the discrete-sources method consists in the possibility of computing the scattering characteristics from particles with complex geometries for which the con-ventional null-field method fails. The discrete-sources-based null-field method retains this advantage over the conventional approach.

252 A. Doicu

2. As discrete sources we use the following: localized and distributed spherical vector wave functions, distributed dipoles and vector Mie potentials. In the multiple spherical-coordinate-based null-field method the matrix formulation includes Hankel functions of low order which results in a better conditioned system of equations compared to that ob-tained in the single-spherical-coordinate-based null-field method. The use of lowest-order spherical multipoles is most effective for axisymmetric particles. By using a system of vec-tor spherical functions distributed along the axis of revolution it is possible to reduce the problem of the surface-current densities approximation to a sequence of one-dimensional problems relative to the Fourier harmonics of the surface currents. In contrast, the null-field methods with electric and magnetic dipoles and vector Mie potentials are suitable for the analysis of particles without rotational symmetry. The position of the poles where the null-field condition is imposed should be correlated with the singularities of analytic continuation of the scattered field inside Dt. In this context, for analyzing particles with complex geometries it is necessary to use the analytic continuation of the solution into the complex plane as described in [19].

3. In the discrete-sources method the amplitudes of the fictitious sources which generate the internal and the external fields are computed by approximating the incident field on the particle surface. In contrast, in the null-field method the values of the discrete sources which produce the surface-current densities are computed by using the null-field condition of the total electric field within Dt. Since, the discrete sources which generate the surface-current densities produce the internal field, the present approaches exhibit the potential for providing a substantial saving in terms of the number of unknowns relative to the discrete-sources method.

4. For numerical implementation one considers a finite number of discrete sources. Con-sequently, one obtains an approximate solution to the scattering problem. Since the rate of convergence of the numerical scheme depends on the location of the discrete sources with respect to 5, an a posteriori error estimation of the approximate solution must be given. In the discrete-sources method one uses as internal criterion the differences between the boundary values of the fields on the particle surface. In the null-field method one can choose as error estimation the residual of the total electric field on spherical surfaces with shifted origins enclosed in D/ [20].

References

[1] RC. Waterman, New formulation of acoustic scattering, J. Acoust. Soc. Am. 45,1417-1429 (1969). [2] RC. Waterman, Symmetry, unitarity and geometry in electromagnetic scattering, Phys. Rev. D 3, 825-839

(1971). [3] RW. Barber and S.C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singa-

pore, 1990). [4] A. Bostrom, Scattering of acoustic waves by a layered elastic obstacle immersed in a fluid: an improved

null field approach, / Acoust. Soc. Am. 76, 588-593 (1984). [5] M.F. Iskander and A. Lakhtakia, Extension of the iterative EBCM to calculate scattering by low-loss or

loss-less elongated dielectric objects, Appl. Opt. 23, 948-953 (1984). [6] M.F. Iskander, A. Lakhtakia, and C.H. Dumey, A new procedure for improving the solution stability and

extending the frequency range of the EBCM, IEEE Trans. Antennas Propagat. AP-31, 317-324 (1983). [7] R.H.T. Bates and D.J.N. Wall, Null field approach to scalar diffraction: I. General method; II. Approximate

methods; III. Inverse methods, Philos. Trans. R. Soc. London A 2H1, 45-117 (1977). [8] A. Lakhtakia, M.F. Iskander, and C.H. Dumey, An iterative EBCM for solving the absorbtion characteristics

of lossy dielectric objects of large aspect ratios, IEEE Trans. Microwave Theory Tech. MTT-31, 640-647 (1983).


[9] R.H. Hackman, The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordi-nates, J. Acoust. Soc. Am. IS, 35^5 (1984).

[10] Y.A. Eremin and A.G. Sveshnikov, The Discrete Sources Method in Electromagnetic Diffraction Problems (Moscow State Univ. Publ., Moscow, 1992).

[11] V. Kupradze, On the approximate solutions of problems in mathematical physics, Russian Math. Surveys 22, 58-108 (1967).

[12] H. Kersten, Die Losung der Maxwellschen Gleichungen durch voUstandige Flachenfeldsysteme, Math. Meth. Appl. Set 7, 40-45 (1985).

[13] C. Miiller and H. Kersten, Zwei Klassen voUstandiger Funktionensysteme zur Behandlung der Randwer-taufgaben der Schwinkungsgleichung AM + k^u = 0, Math. Methods Appl. Sci. 2, 48-67 (1980).

[14] C. Miiller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).

[15] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley-Intersience, New York, 1983).

[16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

[17] Y. Leviatan, Am. Boag, and Al. Boag, Generalized formulations for electromagnetic scattering from per-fectly conducting and homogeneous material bodies - theory and numerical solutions, IEEE Trans. Anten-nas Propagat. AP-36, 1722-1734 (1988).

[18] C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Boston, 1991).

[19] A. Doicu and T. Wriedt, EBCM with multipole sources located in the complex plane. Opt. Commun. 139, 85-98 (1997).

[20] T. Wriedt and A. Doicu, Comparison between various formulations of the extended boundary condition method, Opt. Commun. 142, 91-98 (1997).

Author Index

Abramowitz, M. 47, 80 [28]; 213, 227 [42] Aidam, M. 13,19 [124] Akhvlediani, G.Z. 12,18 [111]; 18 [112]; 144,

151, 777 [41]; 777 [42] Al-Rizzo, H.M. 10,16 [52] Aleksidze, M.A. 12,18 [93]; 18 [94]; 18 [98];

18 [99]; 41, 79 [1]; 144, 146, 148, 151, 156, 170[l];170[2]-170m;170[9]

Apeltsin, V.F. 144, 145, 150, 777 [45]; 777 [47] Apeltzin, V.F. 84, 88-90, 93, 95, 107,108 [8];

109 [24] Ar,E. 13,20[145] Arsenin, V.Y. 152, 154, 772 [67]; 220, 227 [43] Arvas,E. 196, 205 [21] Audeh, N.F. 22, 38 [2] Aydin, K. 8, 75 [18]

Baharav, Z. 6, 13,14 [4]; 19 [130]; 206, 226 [16]; 226 [17]

Bakhvalov, N.S. 144,170 [22] Ballisti, R. 22, 38 [6] Barantsev, R.G. 103, 109 [29] Barber, RW. 7, 75 [12]; 230, 235, 252 [3] Bashaleishvili, M.O. 12,18 [97]; 144,170 [7] Bates, R.H.T. 8, 10, 75 [21]; 77 [63]; 112,140 [5];

230, 252 [7] Beshir, K.I. 207, 214, 227 [31] Bit-Babilc, G. 13,19 [121]; 19 [122]; 79 [123];

145, 772 [59]; 772 [60] Boag, A. 13,19 [127]; 19 [128]; 31, 38 [18]; 41,

79 [5]; 112, 140 [10]; 144, 777 [37]; 206, 207, 214, 218, 226 [14]; 226 [15]; 226 [22]; 226 [5]; 226 [6]; 226 [7]; 226 [8]; 227 [32]

Bogdanov, F.G. 13,19 [117]; 19 [118]; 79 [119]; 79 [120]; 41, 79 [6]; 145, 150, 168, 772 [55]; 772 [56]; 772 [62]; 772 [63]

Bohren, C.F. 83,108 [7]; 160-162, 772 [72] Bomholt, L. 6, 9,14 [2]; 75 [32]; 22, 24, 38 [9];

209, 210, 213, 227 [40] Bom, M. 68, 69, 80 [30]; 155, 772 [68] Bostrom, A. 8, 75 [22]; 230, 252 [4] Bowman, J.J. 13, 20 [145] Brand, Y. 9, 76 [36]; 207, 218, 226 [21] Burchuladze, T.V. 12,18 [97]; 144, 770 [11];

770 [7]

Cadilhac, M. 10, 13, 77 [61]; 79 [136]; 112, 740 [3]

Calderon, A.R 112,140 [11]; 144, 770 [24] Canning, F.X. 206, 207, 226 [10]; 226 [11];

226 [9]; 227 [34]; 227 [35] Christiansen, S. 106, 107, 709 [38] Colton, D. 13, 20 [143]; 42, 43, 45, 53, 56, 57, 60,

63, 80 [26]; 80 [29]; 83, 88,108 [4]; 233, 253 [15]; 253 [16]

Cook, L.IVI. 56, 76, 80 [37] Cooper, J. 9, 76 [45] Cory,H. 160, 772[77] Cote, JVI.G. 195, 203 [20] Courant,R. 89, 705[18] Cross, M.-J. 7, 75 [7]

Dahl, H. 9, 76 [42]; 76 [43]; 76 [44] Djobava, R. 13, 79 [124] Dmitrenko, A.G. 13, 79 [125]; 79 [126]; 144, 145,

777 [36]; 772 [61]; 175, 202 [3]; 202 [4]; 202 [5]; 202 [6]; 202 [7]; 202 [8]; 203 [10]; 203 [11]; 203 [12]; 203 [13]; 203 [14]; 203 [15]; 203 [16]; 203 [9]

Doicu, A. 7, 8, 75 [16]; 56, 80 [36]; 252, 253 [19]; 253 [20]

Dolidze, L. 12,18 [109]; 144, 777 [39] Domanski,Z. 144, 770[13] Dumey, C.H. 8, 75 [24]; 230, 252 [6]; 252 [8]

Economou, D. 13, 79 [122]; 79 [123]; 145, 772 [60]

Eisler, S. 207, 211, 214, 218, 221, 226 [20] Elderyi, A. 120, 747 [18] Eremin, Yu. 2, 4 [2]; 6, 11, 12,14 [3]; 77 [71];

77 [72]; 77 [73]; 77 [74]; 77 [75]; 77 [76] 77 [77]; 77 [78]; 77 [79]; 77 [80]; 77 [81] 77 [82]; 77 [83]; 77 [84]; 77 [85]; 77 [86] 77 [87]; 77 [88]; 18 [89]; 41, 42, 51, 61, 62, 67, 78, 79 [10]; 79 [11]; 79 [12]; 79 [13]; 79 [14]; 79 [15]; 79 [16]; 79 [17]; 79 [18]; 79 [19]; 79 [20]; 79 [21]; 79 [7]; 79 [8]; 79 [9]; 80 [22]; 80 [23]; 80 [24]; 80 [25]; 95, 709 [26]; 144, 145, 154, 777 [30]; 777 [31]; 777 [32]; 777 [33]; 772 [58]; 174, 177, 178, 188, 202 [2]; 203 [19]; 230, 253 [10]

Erez, E. 13, 79 [129]; 206, 226 [12]; 226 [13] Evers, T. 9, 76 [42]

256 Author index

Felsen, L. 69, 72, 80 [31]; 108 [9] Fisanov, V.V. 13,19 [126]; 175, 203 [13]; 203 [14] Fiser, O. 9, 7(5 [49] Foldcema, J.T. 13, 20 [139] Friedmann, M. 207, 227 [30]

Gallett, LN.L. 10,17 [63]; 112,140 [5] Garabedian, R 13,19 [135] Gegelia, T.G. 12,18 [97]; 144, 770 [7] Gnos,M.75[34];76[35] Goell, J.E. 22, 38 [3] Golub, G.H. 207, 209, 213, 216, 227 [33] Goncharov, V.L. 106,109 [35] Grishina, N.V. 12,18 [89]; 42, 80 [25]

Hackman, R.H. 8, 75 [23]; 230, 253 [9] Hafner, C. 6, 8, 9,14 [2]; 75 [26]; 75 [29]; 75 [30];

75 [31]; 75 [32]; 16 [36]; 16 [39]; 16 [40]; 16 [41]; 22, 24, 25, 29, 30, 32-34, 38 [10]; 38 [14]; 3S [15]; 35 [16]; 38 [17]; iS [19]; 38 [4]; 35 [6]; 35 [8]; 38 [9]; 41, 79 [4]; 112,140 [7]; 144, 145, 777 [38]; 777 [49]; 772 [51]; 206, 207, 209-211, 214, 215, 218, 226 [18]; 226 [21]; 226 [4]; 251, 253 [18]

Hajny, M. 9,16 [49] Halas, N.J. 7, 75 [10] Haake, M. 220, 227 [45] Hansen, RC. 207, 218-222, 227 [39]; 227 [44] Hanson, R.J. 126, 7^7 [19]; 188, 203 [18] Hardy, G. 140, 747 [29] Harrington, R.F. 25, 38 [13]; 206, 209, 210,

226 [1]; 227 [41] Herglotz, G. 13,19 [133] Heyman, E. 6, 13,14 [4]; 19 [130]; 206, 226 [16] Hill, S.C. 7, 75 [12]; 230, 235, 252 [3] HMeman, E.D. 56, 80 [33] Hombach, V. 9, 76 [45] Huffman, D.R. 83,108 [7]

Dcuno, H. 6,11, 75 [5]; 77 [65]; 77 [66]; 77 [69]; 103, 709 [32]; 112, 128, 136, 747 [13]; 241 [14]; 747 [27]; 144, 777 [46]

Ilinski,A.S. 144, 777[30] Iskander, M.F 8, 75 [24]; 230, 252 [5]; 252 [6];

252 [8] Itakura, T. 10,16 [58]; 41, 79 [2]; 112, 114, 116,

135, 136, 740 [1]; 144, 770 [25] Izmestyev, A.A. 157, 772 [71]

Jakobus, U. 10,16 [56]; 16 [57]; 207, 226 [27] James, J.R. 10, 77 [63]; 112, 740 [5] Jeans, J. 13, 79 [134] Jobava, R. 13, 79 [117]; 79 [122]; 79 [123]; 145,

772 [55]; 772 [60] Johnson, B.K. 56, 80 [34]

Karkashadze, D. 12,13, 75 [103]; 75 [105]; 75 [106]; 75 [108]; 75 [111]; 75 [112];

79 [113]; 79 [115]; 79 [116]; 79 [117]; 79 [118]; 79 [119]; 79 [120]; 79 [122]; 79 [123]; 79 [124]; 41, 79 [6]; 144, 145, 147, 150, 151, 168, 770 [20]; 770 [26]; 777 [27]; 777 [29]; 777 [41]; 777 [42]; 777 [43]; 777 [45]; 772 [52]; 772 [55]; 772 [56]; 772 [57]; 772 [60]; 772 [62]; 772 [63]

Kawano, M. 6, 11, 75 [5]; 77 [69]; 136, 747 [27] Kersten, H. 12, 75 [106]; 75 [111]; 75 [112]; 144,

145, 151, 777 [27]; 777 [41]; 777 [42]; 777 [43]; 230, 244, 253 [12]; 253 [13]

Khatiashvili, J. 12, 75 [106]; 75 [107]; 75 [108]; 75 [111]; 75 [112]; 79 [113]; 144, 145, 151, 777 [27]; 777 [28]; 777 [29]; 777 [41]; 777 [42]; 777 [43]

Kim,H.-T. 13,79[131] Klaus, G. 8, 75 [27] Kleev, A.I. 95, 103, 105, 709 [25]; 709 [33];

709 [34] Kleinman, E. 106, 107, 709 [38] Kluskens, M.S. 160-162, 772 [74] Kopaleishvili, V. 12, 75 [100]; 75 [101] Korogodov, S.V. 145, 772 [61]; 175, 202 [8];

203 [15] Kress, R. 42, 43,45, 53, 56, 57, 60, 63, 50 [26];

50 [29]; 83, 88, 705 [4]; 233, 253 [15]; 253 [16]

Kupradze, V.D. 12, 75 [92]; 75 [93]; 75 [94]; 75 [95]; 75 [97]; 41, 79 [1]; 144, 149, 770 [1]; 770 [12]; 770 [16]; 770 [2]; 770 [23]; 770 [3]; 770 [7]; 772 [66]; 174, 177, 202 [1]; 230, 253 [11]

Kyurkchan, A.G. 14, 20 [146]; 20 [147]; 20 [148]; 20 [149]; 20 [150]; 20 [151]; 20 [152]; 20 [153]; 20 [154]; 20 [155]; 20 [160]; 82-90, 92-95, 101, 105, 107, 705 [1]; 705 [10]; 705 [11]; 705 [12]; 705 [13]; 705 [14]; 705 [15]; 705 [16]; 705 [19]; 705 [20]; 705 [21]; 705 [22]; 705 [6]; 705 [8]; 709 [23]; 709 [24]; 709 [25]; 709 [27]; 709 [34]; 144, 145, 150, 777 [34]; 777 [35]; 777 [45]; 777 [47]; 777 [48]

Lakhtakia, A. 8, 75 [24]; 146, 156, 160, 772 [64]; 772 [73]; 182, 183, 197, 200, 203 [17]; 203 [22]; 230, 252 [5]; 252 [6]; 252 [8]

Lancos, C. 107, 709 [40] Landesa, L. 207, 218, 222-225, 227 [37]; 227 [38] Landstorfer, F.M. 10,16 [56]; 16 [57]; 207,

226 [27] Lawson, C.L. 126, 747 [19]; 188, 203 [18] Lebedev, O.A. 144, 145, 777 [32] Lebedev, V.A. 106, 709 [37] Leuchtmann, R 10, 75 [28]; 16 [35]; 16 [55]; 207,

227 [28]; 227 [29] Leviatan, Y. 6,13, 74 [4]; 79 [127]; 79 [128];

79 [129]; 79 [130]; 41, 79 [5]; 112,

Author index 257

140 [10]; 144, 145, 171 [37]; 171 [50]; 206, 207, 211, 214, 218, 221, 226 [12]; 226 [13]; 226 [16]; 226 [17]; 226 [20]; 226 [5]; 226 [6]; 226 [7]; 226 [8]; 227 [30]; 227 [32]; 249, 253 [17]

Levin, L. 107,109 [39] Lindell, I.V. 146, 156, 160, 166,172 [65] Littlewood, J.E. 140,141 [29] Loan, C.F.V. 207, 209, 213, 216, 227 [33] Lomidze, G. 12,18 [109]; 144,171 [39] Loncki, S.B. 56, 76, 80 [37] Ludwig, A. 2, 4 [1]; 6, 10, 74 [1]; 16 [50]; 76 [51];

22, 23, 38 [7]; 41, 79 [3]; 206, 226 [2]; 226 [3]

Mackowski, D.W. 8, 75 [20] IVIagnus, W. 120,141 [18] IVlanenkov, A.S. 95,109 [25] IVlarcuvitz, N. 69, 72, 80 [31]; 708 [9] Martin, A. 9, 76 [46] IVlartin, O J.F. 9, 76 [38] Martinez-Burdalo, M. 9, 76 [46] Masel, R.I. 103,109 [30] Matsuda,T. 11,77[70] Mautz, J.R. 210, 227 [41] Maystre, R 107,109 [41] Mazanek, M. 9, 76 [49] Merrill, R.R 103, 709 [30] Metskhvarishvili, D. 13,19 [116]; 19 [118]; 145,

772 [56]; 772 [57] Metz, H.J. 9, 76 [44] Michelssen, E. 207, 218, 226 [22] Mie, G. 23,38[11] Millar, R.F. 10, 13, 77 [62]; 77 [63]; 77 [64];

20 [137]; 20 [138]; 20 [140]; 20 [141]; 112, 140 [4]; 140 [5]; 140 [6]

Miller, W.H. 103, 709 [30] Mirianaslivili, M.M. 12, 78 [102]; 144, 770 [19] Mishchenko, M.I. 8, 75 [19]; 75 [20] Mittra, R. 31, 38 [18]; 206, 207, 218, 226 [14];

226 [15]; 226 [22] Miyamoto, T. 135, 747 [21] Monzon,J.C. 160, 772 [76] Morrision, J.A. 7, 75 [7] Mosig, J. 9, 76 [36]; 207, 218, 226 [21] Mtiulishvili, K.A. 12, 78 [105]; 144, 770 [26] Mueller, C. (MuUer, C.) 13, 20 [142]; 230, 231,

233,244, 253 [13]; 253 [14] Mugnai, A. 7, 75 [8] Mukomolov, A.I. 13, 79 [125]; 79 [126]; 144,

777 [36]; 175, 202 [3]; 202 [4]; 202 [5]; 202 [6]; 202 [7]; 203 [10]; 203 [13]; 203 [14]; 203 [9]

MuUin, C.R. 22, 38 [1]

Na,H.-G. 13, 79[131] Nebeker, B.M. 56, 80 [33] Newman, E.H. 160-162, 772 [74]

Nishimoto, M. 6, 11, 75 [5]; 77 [69]; 136, 747 [27] Novotny, L. 9, 76 [37]; 33, 34, 38 [20]

O'Leary, D.R 220, 227 [44] Obelleiro, E 10, 76 [53]; 76 [54]; 207, 210-213,

215, 217, 218, 222-225, 226 [23]; 226 [24]; 226 [25]; 227 [37]; 227 [38]

Oberhettinger, E 120,141 [18] Oguchi, T. 6, 75 [6] Okuno, Y. 11, 77 [67]; 77 [68]; 77 [70]; 112, 135,

136,141 [15]; 747 [16]; 747 [20]; 747 [23]; 747 [24]; 747 [26]; 145, 772 [54]

Orlov, N.V. 11, 12, 77 [73]; 77 [80]; 77 [81]; 77 [82]; 77 [83]; 77 [84]; 77 [85]; 77 [86]; 77 [87]; 77 [88]; 41, 42, 51, 61, 62, 67, 78, 79 [16]; 79 [17]; 79 [18]; 79 [19]; 79 [20]; 79 [21]; 79 [9]; 80 [22]; 80 [23]; 80 [24]; 144, 154, 777 [31]; 777 [33]; 188, 203 [19]

Paatashvili, L.A. 144, 770 [10] Paatashvili, L.R144, 770[18] Pascher, W. 10, 76 [55] Peden, I.C. 7, 75 [15] PenroscR. 155, 772[70] Peterson, B. 8, 75 [17] Petit, R. 10, 13, 77 [61]; 79 [136]; 83, 708 [3];

112, 740 [3]; 740 [8]; 145, 772 [53] Pierce, K.G. 56, 76, 80 [37] Piller, N.B. 9, 76 [38] Pino, A.G. 10, 76 [53]; 76 [54]; 207, 210-213,

215, 217, 218, 226 [23]; 226 [24]; 226 [25] Pino, M.R. 207, 218, 222-225, 227 [37]; 227 [38] Piskorek, A. 144, 770 [13] Polishiuk,I.M. 144, 770[17] Polya, G. 140, 747 [29] Pomeraniec, B. 13, 79 [128] Pommerenke, D. 13, 79 [124] Ponnapalli, S. 196, 203 [21] Popovidi, R.S. 144, 770 [20] Popovidi-Zaridze, R. 12, 78 [100]; 78 [101];

78 [102]; 78 [103]; 78 [104]; 78 [105]; 78 [106]; 78 [110]; 78 [112]; 79 [114]; 144, 145, 151, 770 [21]; 770 [26]; 777 [27]; 777 [40]; 777 [42]; 777 [44]

Rao, S.M. 207, 227 [36] Regli, R 9, 75 [33] Richie, J.E. 207, 214, 227 [31] Rindler,W. 155, 772[70] Rodriguez, J.L. 10, 76 [53]; 76 [54]; 207,

210-213, 215-218, 222-225, 226 [23]; 226 [24]; 226 [25]; 226 [26]; 227 [37]; 227 [38]

Roek,Z. 144, 770[13] Rogovin,K. 206, 226 [11] Rojas, R.G. 160, 163-165, 772 [75] Rozenberg, V.I. 12, 77 [82]; 77 [84]; 42, 51, 61,

62, 79 [18]; 79 [20]

258 Author index

Rukhadze, J.A. 144,170 [14] Rumsey, V.H. 136,141 [25] RuoB, H.-O. 207, 226 [27] Ruoss, H.-O. 10,16 [56]; 16 [57]

Salonen, E.T. 7, 75 [9] Sandburg, R. 22, 38 [1] Sarkar, D. 7, 75 [10] Sarkar, T.K. 196, 203 [21]; 207, 227 [36] Savina, T.V. 14, 20 [157] Scheider, J.B. 7, 75 [15] Schiavoni, A. 9,16 [45] Schwartz, H.A. 13,19 [132] Schwartz, L. 137, 747 [28] Shatalov, V. 14, 20 [156]; 20 [157]; 20 [158];

20 [159]; 20 [160]; 82, 84, 87-90, 93, 95, 108 [1]; 108 [2]; 108 [17]; 705 [22]

Shefer,G. 144,770[15] Shen, J.J. 56, 76, 80 [37] Shifman, Y. 207, 227 [30] Shigesawa, H. 112,140 [9] Shimohara, K. 135, 747 [21] Shubitidze, Ph.I. 13,19 [117]; 145, 772 [55] Sihvola, A.H. 146, 156, 160, 166, 772 [65] Singer, H. 10,16 [48]; 23, 38 [12] Sleeman, B.D. 13, 20 [144] Smimov, V.I. 106,109 [37] Starr, G.W. 56, 80 [33] Stegun, I.A. 47, 80 [28]; 213, 227 [42] Steinbigler, H. 23, 38 [12] Stemin, B. 14, 20 [156]; 20 [157]; 20 [158];

20 [159]; 20 [160]; 82, 84, 87-90, 93, 95, 108 [1]; 108 [2]; 108 [17]; 70S [22]

Stover, J.C. 12, 77 [88]; 42, 78, 80 [24] Strom, S. 8, 75 [17] Strutt (Lord Rayleigh), J.W. 10,16 [60]; 103,

709 [28]; 112,140 [2] Sukov, A.I. 101, 103, 105, 106,109 [27];

109 [31]; 709 [34]; 144, 145, 150, 777 [45] Sveshnikov, A.G. 6, 11, 12,14 [3]; 77 [71];

77 [72]; 77 [73]; 77 [74]; 77 [75]; 77 [76]; 77 [78]; 77 [81]; 41, 42, 79 [10]; 79 [11]; 79 [12]; 79 [14]; 79 [17]; 79 [7]; 79 [8]; 79 [9]; 95, 709 [26]; 144, 145, 777 [30]; 777 [31]; 777 [32]; 772 [58]; 174, 177, 178, 188, 202 [2]; 203 [19]; 230, 253 [10]

Talakvadze, G. 12,18 [108]; 18 [110]; 144, 777 [29]; 777 [40]

Taubenblatt, IVI.A. 56, 80 [32] Tavzarashvili, K. 13, 79 [121]; 145, 772 [59] Tervonen, J.K. 7, 75 [9] Thiele, G.A. 207, 226 [19] Tikhonov, A.N. 152, 154, 772 [67]; 220, 227 [43] Tomita, M. 135, 747 [22] Tran, T.K. 56, 80 [32] TranquiUa, J.M. 10,16 [52] Travis, L.D. 8, 75 [19]; 75 [20]

Tretyakov, A.A. 146, 156, 160, 166, 772 [65] Tricomi, EG. 120,141 [18] TsverikmazashviH, Z. 12,18 [102]; 18 [103];

18 [104]; 18 [108]; 144, 770 [19]; 770 [20]; 770 [21]; 777 [29]

Tudziers, Ch. 10,16 [47]; 16 [48]

Uzunoglu, N. 13, 79 [122]; 79 [123]; 145, 772 [60]

van den Berg, RM. 13, 20 [139] Varadan, V.K. 7, 75 [14]; 160, 772 [73]; 182, 183,

197, 200, 203 [17]; 203 [22] Varadan, VV 7, 75 [14]; 160, 772 [73]; 182, 183,

197, 200, 203 [17]; 203 [22] Veinshtein, L.A. 155, 772 [69] Vekua, I.N. 12,18 [90]; 18 [91]; 75 [96]; 22, 23,

38 [5]; 45, 80 [27]; 107, 709 [42]; 112, 141 [12]; 144, 770 [4]; 770 [5]; 770 [6]

Velline, CO. 22, 38 [1] Videen, G. 56, 80 [35] Viitanen, A.J. 146, 156, 160, 166, 772 [65] ViUar, R. 9,16 [46] Vladimirov, VS. 115, 747 [17]

Wainstein, L.A. 103, 106, 709 [31] Waldvogel, J. 9,16 [36]; 207, 218, 226 [21] Wall, D.J.N. 8, 75 [21]; 230, 252 [7] Walsh, J.L. 106, 107, 709 [36] Waterman, RC. 7, 75 [11]; 75 [13]; 75 [25]; 230,

235, 252 [1]; 252 [2] Weiss, R 23, 38 [12] Weston, V.H. 13, 20 [145] Wilcox, C.H. 83, 86, 705 [5] Wiscombe, W.J. 7, 75 [8] Wolf, E. 68, 69, 50 [30]; 155, 772 [68] Woodworth, M.B. 195, 203 [20] Wriedt, T 2, 4,4 [2]; 4 [3]; 7-9, 75 [16]; 16 [42];

16 [43]; 16 [44]; 56, 50 [36]; 252, 253 [19]; 253 [20]

Yaghjian, A.D. 195, 203 [20] Yasuura, K. 10, 11,16 [58]; 16 [59]; 77 [65];

77 [66]; 77 [67]; 77 [68]; 41, 79 [2]; 103, 709 [32]; 112, 114, 116, 128, 135, 136, 740 [1]; 747 [13]; 747 [14]; 747 [15]; 747 [16]; 747 [21]; 747 [22]; 747 [23]; 144, 770 [25]; 777 [46]

Yee, H.Y. 22, 38 [2]

Zaridze, R. 12, 13, 75 [107]; 75 [108]; 75 [109]; 75 [111]; 79 [113]; 79 [115]; 79 [116]; 79 [117]; 79 [118]; 79 [120]; 79 [121]; 79 [122]; 79 [123]; 79 [124]; 41, 79 [6]; 144, 145, 147, 150, 151, 777 [28]; 777 [29]; 777 [39]; 777 [41]; 777 [43]; 777 [45]; 772 [52]; 772 [55]; 772 [56]; 772 [57]; 772 [59]; 772 [60]; 772 [63]

Author index 259

Zhang, W. 7, 15 [9] Zhou, D.-Q. 11,77 [70] Zheng, J. 9,16 [36]; 207, 218, 226 [21] ZoUa, F. 112,140 [8]

Subject Index

3D MMP code, 22

accuracy, 177 acoustic diffraction, 144 addition theorem, 45, 238, 241 aerosol particles, 6 aerosols, 63 analytical continuation, 13, 41, 247 angular spherical coordinates, 177 anisotropic materials, 158 anisotropic medium, 9, 146 antenna, 29, 63 Atkinson-Wilcox series, 83 auxiliary cube, 191 auxiliary sources method, 174 auxiliary surface, 150, 174, 176, 180, 186 azimuthal mode, 238, 241

Barantsev-MMM method, 103 basis functions, 230 Bernoulli polynomial, 120 Bessel expansions, 34 Bessel functions, 22, 46 - cylindrical Bessel functions, 48 - regular Bessel functions, 48 - spherical Bessel functions, 47, 241 Bessel wave functions - spherical Bessel wave functions, 45 biisotropic medium, 146 boundary conditions, 174, 175, 177, 178, 180 boundary operator, 40 boundary-integral equations, 234 boundary-value problem, 40, 82

caps, 192 Cartesian coordinate system, 175, 177 charge simulation technique, 23 chiral, 174, 182 -chiralbody, 160, 167 -chiral cylinder, 162 - chiral shell, 197 -chiral sphere, 163 chirality, 197 circle, 91 Circular Harmonic Analysis, 8, 22

circumscribing sphere, 247 closeness relations, 238, 245 cluster, 66 co-polarized scattering cross-sections, 162 coated scatterers, 174 coated spheroids, 8 coating, 184 - coated bodies, 174 - coated sphere, 197,200 collocation, 25 collocation method, 7, 103, 144, 148, 152, 181, 183 collocation technique - adaptive collocation technique, 102 complete systems, 234 completeness, 230, 241, 245, 246 complex matrix, 181, 187 Complex Multipole Beam Approach, 206 complex plane, 11, 51 complex space, 13 computer simulation, 40 computerized tomography, 40 conducting, 174 conducting body, 175 Conjugate Gradient Method, 174, 190, 202 connections, 24 constitutive relations, 182 continuation, 82 conventional Yasuura method, 112 convergence, 31, 190 corrugated surface, 104 cross-polarized, 197 cross-polarized scattering cross-sections, 162 cube, 191, 250 current model method, 112 cylinder, 96, 101, 144, 168, 192, 194 cylindrical obstacle, 113

Debye scalar potentials, 155 Deschamps function, 158, 168 dielectric cube, 7, 196 dielectric obstacles, 231 Differential Scattering Cross-sections, 74 diffraction problems, 95 dipole moments, 181 dipoles, 174, 176

262 Subject index

Dirac delta function, 43 Dirac testing functions, 26 Dirichlet boundary condition, 25, 42 Dirichlet problem, 44, 50, 87, 95 disc, 65 discrete sources, 174, 246 Discrete Sources Method, 11, 40, 174 distributed electrical multipoles of lowest order, 59 double layers, 86 DSM, 174, 177

edged boundaries, 112 edges, 211 electric dipoles, 157, 234, 242, 247 electric fields, 182 electrodynamics, 145 electromagnetics, 40 electrostatics, 22, 23 elementary electric dipoles, 174 elementary sources, 157, 230 ellipse, 89, 91, 192 ellipsoid, 188, 191, 200, 250 elliptic, 144 elliptic pseudo-differential operator, 43 elliptical cylinder, 99, 158 equivalent sources, 6 - equivalent source method, 112 Eswald-Oseen extinction, 7 Euclidian norm, 126, 189 evanescent waves, 34 expansion coefficients, 144 Extended Boundary Condition Method, 7 extended precision, 8 external excitation, 61

far field pattern, 63 far zone, 63, 176 fictitious boundaries, 28 Fictitious Current Method, 13 fictitious discrete sources, 174 fictitious elementary electric dipoles, 176, 178, 180 fictitious elementary sources, 206 filamentary current method, 214 Floquet theory, 29 focal spots, 168 Fourier coefficients, 48 Fourier harmonic, 45, 252 Fourier series, 121, 144 Fourier-analysis, 150 Fredholm integral equation, 57, 82 Fresnel coefficient, 68 functional systems, 178

Galerkin's method, 26 Gaussian eUmination, 213 - Gaussian elimination method, 188 Gaussian laser beam, 9 Generalized Multipole Technique, 41, 112, 174, 206

Generalized Point Matching, 209 geometrical discontinuities, 215 Givens rotations, 220 gratings, 29 Green's function, 34, 86, 115, 116, 183 Green's tensor, 71 Green's theorem, 247

Holder-continuous functions, 231 Holder-continuous surface divergence, 231 half-plane, 51 half-space, 42 Hamilton operator, 155 Hankel function, 22, 86, 252 - cylindrical Hankel functions, 157 - spherical Hankel functions, 45, 83, 157 harmonic expansions, 34 harmonic function, 22 Helmholtz equation, 40,42, 55, 82,114, 157,176,181,

230, 244 Hermitian matrix, 124 Hertz vectors, 155 homogeneous penetrable obstacle, 41 homothetic, 176 - homothetic coefficient, 176 - homothetic ratio, 250 hybrid combination, 207 hybrid method, 212 hydrid MMP, 10 hydrometeors, 63 hypergeometric function, 47

ill-condition, 189 ill-conditioned matrices, 30 ill-conditioning, 206 images, 89 impedance, 174 impedance body, 178 Impedance Matrix Location, 206 integral equation method, 174 interface, 231 intermediate points, 177 irrotational terms, 235 iterative methods, 189

Jacobian matrix, 125, 127 jump relation, 57

L-curve criterion, 220 Laplace equation, 25 Laplace operator, 49 least squares, 148 left polarized waves, 183 Legendre functions - associated Legendre functions, 45, 47 Legendre polynomial, 83 lens, 168

Subject index 263

linear algebraic equations, 181, 188 linear combination, 230 localized Cartesian spherical multipole, 59 localized spherical vector wave functions, 241, 247 lowest-order, 237 Lyapunov, 147

magnetic dipoles, 157, 234, 242, 247 magnetic fields, 182 magneto-dielectric, 146, 174, 179 magnetostatics, 22 matching points, 209 matrix factorization, 216 MaX-1, 37 Maxwell equations, 22, 40, 175, 178, 180 mean square norm, 230 metaharmonic functions, 83, 157 Method of Auxihary (discrete) Sources, 82, 146 Method of Auxihary Sources, 12, 144 Method of Expansion by Fundamental Solutions, 144 Method of Expansion in Terms of Metaharmonic Func-

tions, 144 Method of Generalized Fourier Series, 144 Method of Moments, 10, 148, 206 methods of auxiliary currents, 95 metrology, 40 micro-defects, 66 Mie, 23

- Mie potentials, 244, 246 - Mie scattering, 6 - Mie series solution, 250 - vector Mie potentials, 234, 247 minimization, 179, 181 modal expansion, 112 monopole, 23, 206 multifilament current method, 206 multifoil, 92 Multiple Multipole Method, 8, 22, 174 Multiple Multipole Program, 22 multipole expansions, 34

Neumann boundary conditions, 42 Neumann function, 22, 48 Neumann's problem, 87 non-axial excitation, 41 nonaxisymmetrical, 174, 188, 234 nonaxisymmetrical bodies, 174 nonconvex boundaries, 90 nonspherical light scattering, 230 normal derivative, 122 normal equations, 209, 213 normalized differential scattering cross-section, 249 Null Field Equations, 243 Null Field Method, 7, 230, 246

oblate spheroid, 66 optics, 40 optimization processes, 211

orthogonal decomposition method, 126 orthogonaUty, 49, 236 orthogonalization, 148 orthonormalization, 144 overdetermined system, 190

P-polarized, 62 parabolic type, 144 Parameter Estimation Technique, 9 particle diagnostics system, 6 particle shape, 6 particles, 230 penetrable obstacle, 40 perfect conductors, 41 perfectly conducting body, 174 perfectly electric conducting, 208 periodic deformed cylinder, 127 permeability, 175, 179 permittivity, 175, 179 Phase Doppler Anemometry, 9 pits, 66 planar structure, 33 plane wave, 188 Point Matching, 22, 25 - Point Matching Method, 6 Pointing vector, 36 poles, 244 polyphase wave functions, 128 projection operator, 44 Projection Technique, 25 prolate spheroid, 66 pseudo-solutions, 148

QR decomposition, 127, 219 QR factorization, 216 Quasi-Solution, 40

radar bistatic cross sections, 197, 200 radar scattering, 7 radiating spherical vector wave functions, 236 radiation conditions, 175, 176, 178, 180 raindrops, 7 Rayleigh, 103 - Rayleigh expansions, 29 -Rayleigh hypothesis, 13, 107 - Rayleigh method, 10 - Rayleigh representation, 83 recurrence relation, 240 region of influence, 214 regular functions, 41 regular spherical vector wave functions, 236 residual value, 177 resonance, 153 Riemann, 97 Riemann-Schwarz symmetry, 89 right-polarized waves, 183 rotational symmetry, 61

264 Subject index

S-polarized, 62 saddle-point method, 88 scattered field, 196 scatterer - axisymmetric scatterers, 174 Schelkunoff Equivalent Current Method, 7 Schwarz' inequality, 116 scratches, 66 segment, 237 semi-axes, 192 semiconductor industry, 66 series expansion, 241 series of wave harmonics, 83 silicon wafers, 66 silver halide crystals, 9 Silver-MuUer radiation, 231 simple layers, 86 single-layer potential, 43 Singular Value Decomposition, 188, 207 singular values, 219 singular-smoothing procedure, 11, 112 singularity, 51, 82, 122, 149 sinusoidal corrugation, 105 smooth surface, 246 smoothing operator, 119 smoothing procedure, 11, 112 Sobolev norm, 222 software package, 190 solution accuracy, 179, 184 solution of systems, 188 Sommerfeld-Weil representations, 84 spectral functions, 67 SPEX, 10, 22 sphere, 66, 197, 249 spherical coordinates, 235 spherical scatterer, 249 spherical vector wave functions, 230, 234, 246, 247 spherical wave functions, 235 spheroid, 158 spheroidal coordinate formalism, 230 spheroidal particle, 251 spinor dyad of electromagnetic field, 155 spinor dyad of Hertz potential, 155 square conducting cylinder, 210 square cylinder, 132 square function, 189

square norm, 25 stability, 51 Stratton-Chu representation theorem, 233 substrate, 56, 66 subsurface defect, 71 support, 230 surface impedance, 178 Surface Integral Equation Method, 41 surface-current densities, 230 system matrix, 152

T-Matrix Method, 7 Tellegen medium, 146 tensor, 178 Tikhonov, 152 Tikhonov regularization, 207, 218, 220 time-harmonic field, 178, 179 total absorption cross-section, 162 Total Integral Response, 74 total scattering cross-section, 163 translation, 238 translation addition coefficients, 238 transmission value problem, 246 Truncated-SVD, 219 truncations, 116

unique solution, 231 uniqueness, 44

vector dipole moment, 176 vector Mie potentials, see Mie vector multipoles, 234 vector potential, 155, 232 vector space, 124

wafer contaminants, 66 wafer inspection systems, 66 wave numbers, 183 wave operator, 146 wave potentials, 86 wavelength, 63 weighting function, 123 Weyl-Sommerfeld integral, 69, 70, 73

Yasuura's method, 10, 112

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