Analysis of electromagnetic diffraction by wedges with the method of edge functions

Kurzfassung

The presence of edges in a scattering surface may considerably influence the behavior of electromagnetic fields interacting with the surface. This influence manifests itself with field singularities at the edges and specific edge-diffracted contributions in the far-field. Analysis of those important effects is complicated by the fact that all attempts to derive an exact and general solution of Maxwell’s equations for canonical wedge-shaped configurations other than a perfectly conducting wedge have failed so far, unless the direction of incidence of an incoming wave is restricted to the plane perpendicular to the edge. The goal of this paper is to demonstrate that the newly established method of edge functions (MEF) allows efficient and accurate analysis of both near- and far- fields for a broad variety of canonical configurations that may involve, for example, several adjacent arbitrarily-angled non-metallic (impedance, dielectric, isotropic, anisotropic) wedges illuminated by an obliquely-incident and arbitrarily polarized plane electromagnetic wave. In the framework of MEF the total field is represented as a superposition of the so-called edge functions (EFs) that are particular solutions of Maxwell’s equations, satisfying the edge conditions at the edge and boundary conditions on the wedge faces. Since the EFs are not required to meet radiation conditions, they are much easier to construct than a global solution of the boundary value problem. For a very wide class of wedge-shaped configurations, the EFs can be analytically derived in the form of series of Bessel functions multiplied by trigonometric functions and certain constant coefficients. The coefficients are found from recurrent systems of linear algebraic equations of finite order, resulting from the boundary conditions on the wedge faces. The zero-order system is always homogeneous, implying that in order to obtain a non-trivial solution, one has to choose the index of the lowest-order Bessel function to be a zero of its determinant. The characteristic equation is shown to have an infinite number of roots, leading to an infinite set of linearly independent solutions – the edge functions. The total field is then sought by superimposing the EFs with unknown weighting coefficients to be chosen so as to make the superposition comply with the radiation conditions (A.V. Osipov, 2001, PIERS 2001, Proceedings, p. 260). MEF can be seen as an improved and extended version of an approach introduced in (R.H.T. Bates, 1973, Int. J. Electronics, vol. 34, no. 1, p. 81) and later applied to the analysis of two-dimensional diffraction of an E-polarized electromagnetic wave by a dielectric wedge (T.S. Yeo, D.J.N. Wall, and R.H.T. Bates, 1985, J. Opt. Soc. Am. A, vol. 2, no. 6, p. 964). In order to determine the weighting coefficients, an integral equation technique (the null-field method) was used. The approach turned out to fail for an H-polarized field because in this case the EFs are built up from Bessel functions of fractional order, which drastically complicates analytical evaluation of the boundary integrals, on which the Bates’ method relies. By contrast, in the framework of MEF the weights are determined directly from the radiation conditions, i.e. without resorting to integral equations. Because of the specific structure of MEF solutions, which are linear combinations of Bessel functions, the method is particularly well suited for the analysis of the field behavior near the edge where the Bessel functions can be replaced with their small-argument series expansions. This leads to representation of fields in the form of power series with respect to the distance to the edge – the so-called Meixner’s series (J. Meixner, 1972, IEEE Trans. AP, vol. 20, no. 4, p.442). MEF is shown to recover the exact order of the field singularity at the edge predicted by Meixner’s asymptotic procedure but, in contrast to the latter, make analysis of higher-order terms much easier since the complete analytical structure of the solution is already known, thus eliminating the need to solve recurrent systems of differential equations as required by the Meixner method. One further advantage of the MEF approach is that it provides a complete description of the electromagnetic field near the edge, which accounts for the incident field and the conditions at infinity. An MEF solution cannot be directly applied to the far-field analysis because of the slowing convergence of the EF series as the distance from the edge increases. This problem can be overcome by employing a useful relation between the diffraction coefficient and the total field, which can be derived from a uniform asymptotic representation of the edge-diffracted field. This permits accurate determination of the diffraction coefficient based on the knowledge of the total field at a finite distance from the edge (A.V. Osipov, 2001, PIERS 2001, Proceedings, p. 260). The potential of MEF for the field analysis is illustrated by the example of diffraction of an obliquely incident plane electromagnetic wave by a dielectric wedge and a wedge with impedance boundaries.