Fredholm factorization of Wiener-Hopf scalar and matrix kernels

A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape

The generalized Wiener-Hopf equations for wave motion in angular regions: Electromagnetic application

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener-Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works

The generalized Wiener-Hopf equations for the elastic wave motion in angular regions

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener-Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions

Combined Spectral Methods to Study Complex Scattering Problems Formulated with the Wiener-Hopf Technique: the Semi-infinite Grounded Dielectric Slab problem

In this work we present a new combination of spectral methods that allows to study complex scattering problem in spectral domain containing abrupt discontinuities of materials. The method is applied to a formulation of problem given in terms of incomplete Wiener-Hopf (WH) equations, where, for incompleteness, we intend that some of the physical boundary conditions arising from abrupt discontinuities provide terms in the WH equations not directly related to the plus and minus unknowns of the problem. This is the case of the semi-infinite grounded dielectric slab problem where the semi-infiniteness geometry of the material provide such situation. This problem is of great interest in antennas and propagation community and studies are proposed in different papers with different methods; see references in [1]. The WH incomplete equations are obtained as in [1] starting from the application of unilateral Laplace transform to wave equations defined in three different sub-regions, see Fig. 1

The Wiener-Hopf Theory for the Scattering by an Impenetrable Polygonal Structure

The Generalized Wiener-Hopf technique and the associated Fredholm factorization method constitute powerful tools that allow to study in quasi-analytical form the diffraction by complex structures with edges. A characteristic of this technique is the possibility to break down the complexity of the diffraction problem into different homogeneous canonical subregions where the WH functional equations and their associated integral representations of Fredholm kind are deduced. The mathematical-physical model is comprehensive and it allows spectral interpretation. In this paper we consider a novel canonical scattering problem: the three face impenetrable polygon

Wiener-Hopf Solution of Diffraction by a PEC Wedge in Anisotropic Media

In this work we present our recent work on novel Wiener-Hopf(WH) formulation for the analysis and the study of scattering of wedges [1] immersed in complex materials [2-3]. We start from an original study of the perfect electrically conducting (PEC) wedge immersed in uniaxial (εx=εy≠εz,μx=μy≠μz) and biaxial (all different εi, μi=x,y,z) media illuminated by plane waves, where the wedge has an aperture angle of 2π-2γ. The uniaxial case has been studied by Felsen in [4], however generalization and exploitation of this case has not been further carried out in literature

Wiener-Hopf Analysis of the Scattering from an Abruptly Ended Dielectric Slab Waveguide

Abruptly ended dielectric slabs are important components in several areas of applied electromagnetics. For the study of these geometries, a variety of analytical methods have been proposed in the past. In this paper we formulate the problem in terms of Wiener Hopf equations and we apply the novel and effective semi-analytical solution technique known as Fredholm factorization