PLANE WAVE SPECTRA IN GRATING THEORY: II. SCATTERING BY AN INFINITE GRATING OF IDENTICAL CYLINDERS

Abstract

The scattering of a scalar plane wave by an infinite grating of identical elements is analyzed by methods of Part I. Representations for the field outside the grating, and between the elements, are given. It is shown that the discrete plane-wave spectrum for the exterior field, in general, may be continued analytically into the grating; in particular, this exterior field representation for a grating of circular cylinders is valid everywhere other than on the plane containing the axes of the cylinders. Field representations within the corrugations of a reflection grating are also discussed; it is inferred that analytic boundary data prescribed on an analytic boundary surface permit continuation of the exterior solution into the grating. An equation is derived which relates the scattering function of an element in the grating to its form for an isolated cylinder. From this relation it is shown that the many-body scattering function has branch points in the plane of complex angle of incidence at the Rayleigh wavelengths, and is two-valued in a suitable neighborhood of each branch point. Elsewhere, including the neighborhood of grazing incidence, the scattering function is analytic with respect to the angle of incidence. It is shown, for a grating of arbitrary elements, that the scattered field for grazing incidence takes an especially simple form.