Diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface

Abstract

Summary: When a cylindrically curved perfectly conducting concave surface is terminated abruptly, an incident whispering gallery (WG) mode undergoes diffraction. This phenomenon is studied here for the canonical problem of a thin boundary and an edge that is parallel to the cylinder axis. To confine attention to diffraction by a single edge, it is assumed that the boundary extends to infinity in an infinitely extended angular space that is equivalent to placing along some radial plane intersecting the boundary a perfect absorber for angularly propagating waves. The problem can then be phrased as a functional equation problem of the Hilbert type which is solved exactly. An asymptotic approximation in the high frequency limit \(ka\gg 1\), where \(k\) is the wavelength and \(a\) the radius of curvature, is obtained in explicit form and phrased so as to exhibit the diffraction coefficient for edge-diffracted rays and the launching coefficients for creeping waves on the convex side as well as WG modes on the concave side. These quantities are of interest for placing the results within the context of the geometrical theory of diffraction (GTD) and are relevant to the analysis of curved two-dimensional reflectors of finite width, where WG modes excited by diffraction of the incident field at one edge provide illumination of the other edge. The induced surface current density function is also derived. The asymptotic results, including an observed shift of the shadow boundary relative to the direction of incidence of the WG mode, are explained completely in terms of half-plane diffraction of one of the modal ray congruences, thereby confirming via this canonical problem the validity of the GTD for diffraction of modal fields. For the limiting case \(a\rightarrow\infty\), all expressions are shown to reduce to those for the classical half-plane problem.