The WienerHopf Method in Electromagnetics
2: Academy of Sciences of Turin, Turin, Italy
This advanced research monograph is devoted to the WienerHopf technique, a functiontheoretic method that has found applications in a variety of fields, most notably in analytical studies of diffraction and scattering of waves. It provides a comprehensive treatment of the subject and covers the latest developments, illustrates the wide range of possible applications for this method, and includes an extensive outline of the most powerful analytical tool for the solution of diffraction problems. This will be an invaluable compendium for scientists, engineers and applied mathematicians, and will serve as a benchmark reference in the field of theoretical electromagnetism for the foreseeable future.
Inspec keywords: computational electromagnetics; integral equations
Other keywords: computational electromagnetics; WienerHopf method
Subjects: Monographs, and collections; Electromagnetic waves, antennas and propagation; Function theory, analysis; Electric and magnetic fields; Integral equations (numerical analysis); General electrical engineering topics; Classical electromagnetism
 Book DOI: 10.1049/SBEW503E
 Chapter DOI: 10.1049/SBEW503E
 ISBN: 9781613530016
 eISBN: 9781613530313
 Page count: 384
 Format: PDF

Front Matter
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Part 1: Mathematical Aspects
1 Forms of WienerHopf equations
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This chapter discusses the different forms of WienerHopf equations. Besides the classical WienerHopf equations (CWHE), different functional equations may be classified as modified WH equations and generalized WH equations.
2 The exact solution of WienerHopf equations
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The singularities of G(a) and [G(a)]1 are called structural singularities, and they always have an important physical meaning. To illustrate this point, let us consider some typical WH geometries, as shown in Fig. 1 of the Preface. These geometries may be considered as the junction of two (or more) waveguides, or as a single waveguide in which geometrical discontinuities are present. The singularities of G(a) and of its inverse [G(a)]1 are related to the propagation constants of the modes of the involved waveguides. For instance, in the halfplane problem we are dealing with free space with propagation constant k. This implies that in the continuous modes have propagation constants (k2  a2)^{1/2}. In this case the singularitiesof G(a) and [G(a)]1 are the branch points a 1/4 +k. Singularities constituted by poles are instead present when we deal with closed waveguides.It is difficult to obtain additive decomposition in the scalar and matrix it can be solved by using Cauchy integration approach.WienerHopf equations
3 Functions decomposition and factorization
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This chapter is divided into three parts. The first part deals with decomposition of complex functions. The second part covers factorization formula, factorization of meromorphic functions and factorization of kernels involving continuous and discrete spectrum. The third part deals with decomposition in wplane, evaluation of the plus functions and minus functions, use of difference equation for function decomposition and WH equation as difference equation.
4 Exact matrix factorization
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The central problem in solving vector WienerHopf equations is the factorization of a n x n matrix. Even though this problem has been considerably studied in the past, up to now a general method to factorize a n x n matrix is not known. A discussion of significant advances, achieved in the last few years, appears in Buyukaksoy and Serbest (1993). In this paper, the most interesting ideas for obtaining explicit matrix WienerHopf factorization were outlined.
5 Approximate solution: The Fredholm factorization
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This chapter discusses the approximate solution of the Fredholm factorization.
6 Approximate solutions: Some particular techniques
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All the powerful methods (e.g., iterative, moment) developed in functional analysis can be effectively used for solving WH equations or for obtaining factorized matrices. Furthermore, the observation that it is possible to factorize matrices with rational elements suggests approximating the kernels with rational approximants.

Part 2: Applications
7 The halfplane problem
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The study the electromagnetic propagation in stratified planar regions (Fig. 1) is very important since it provides all the fundamental concepts needed for the study of wave propagation. There are very excellent books devoted to this topic (Felsen & Marcuvitz, 1973, chapter 5; Brekhovskikh, 1960). In this book, the author examine some aspects of the problem that are not considered in the cited works. For instance, the author introduce planar discontinuities in some sections of the stratified regions and develop a unified theory for this problem that is based on the spatial Fourier transforms of the fields and on the WienerHopf formulation.
8 Planar discontinuities in stratified media
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8.1 The planar waveguide problem 8.1.1 The Fpolarization case Let us consider the planar waveguide shown in Fig. 1A. This structure can be studied via the circuit representation indicated in Fig. 1B. In particular, the PEC walls of the waveguide are simulated by the current generators A1, A2 , and the slab d < y < 0 in the physical structure is equivalent to the P twoport introduced in Fig. 3a of chapter 7.
9 WienerHopf analysis of waveguide discontinuities
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The MarcuvitzSchwinger formalism (Felsen & Marcuvitz, 1973) provides the general representation of electromagnetic fields in an arbitrary waveguide filled with a homogeneous isotropic medium. By assuming that the longitudinal axis of the waveguide is the axis ^z, we have the following representation of the transverse electromagnetic field in the TM and TE modes: Et(p,z) = Σ ν'(z)e'i(p) + ΣVi''(z)e''i(p) i i Σ Ht(p, z) = i Σ Ii'(z)h' i(p) +i (1) V'' i (z)h''i (p) where the modal voltages and currents Vi, Ii specify the zdependence and ei(p) and hi(p) = z^ x ei(p) are the transverse eigenvectors of the guide. The index i is twofold and can be discrete or continuous depending as to whether the transverse cross section of the guide is bounded or unbounded. The transverse eigenvectors are related to the transverse eigenfunctions through e'i(p) =  1' νtfi(p), h'' (p) =  1' νtyi(p) (2) ti ti where ν2t fi(p) + (ti')2fi(p) = 0, ν2t yi(p) + (t'' i )2yi(p) = 0 (3) The functions fi(p) and yi(p) provide the TM and TE modes. On the contour g that limits the transverse section of the waveguide it must be fi(p)Ig = 0, @yi(p) =where v is the normal to the contour y. If t0i 1/4 0, the TM mode becomes a TEM mode and a different normalization in the first of eq. (2) must be used. We recall that the TEM mode can occur only if the transverse section of the waveguide is not simply connected.
10 Further applications of the WH technique
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The WienerHopf technique is a mathematical technique widely used in applied mathematics.It was initially developed by Nobert Wiener and Eberhard Hopf as a method to solve system of integral equations, but has found wider use in solving 2D partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex analytical properties of transformed functions. Typically, the standard Fourier transform is used.

Back Matter
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