Skew Incidence Half-Plane Diffraction Problems: Particular Aspects of the Factorization Problem

Abstract

Problems of diffraction at half-planes or half-plane junctions with different impedance boundary conditions for normal incidence depend upon the polarization of the incident plane electromagnetic wave: either the electric vector or the magnetic vector is parallel to the discontinuity of the boundary condition (TE and TM mode). For isotropic boundary conditions both cases can be treated independently from each other. For normal incidence but arbitrary polarization the diffraction problem can be solved by splitting up the polarization of the incident wave into the sum of the TE and the TM modes which can often be solved by the Wiener-Hopf technique.

For skew (oblique) incidence this splitting up into independent TE and TM modes is no longer possible. The Wiener-Hopf technique then leads to systems with 2 x 2 or 4 x 4 Fourier symbol matrices. These can be decoupled by polynomial similarity transformations to single equations or 2 x 2 block diagonal matrices everywhere in the complex plane except at points of the singular set. Traditionally for about 40 years these constraints have been neglected. Then the final solution of the diffraction problem involves unphysical “leaky wave” type poles that are eliminated by fixing undetermined constants in the classical Wiener-Hopf solution by the residue calculus technique. The concept of restricted range of the analytic family of polynomial transformations that are used to reduce the Fourier symbol matrix functions to simpler forms is shown to be closely related to the pole elimination procedure. As a representative example, diffraction by a two-face impedance half-plane for arbitrary angles of incidence is treated in some detail. Other examples are outlined.