Numerical determination of scattered field amplitudes for rough surfaces

Abstract

In theory, when an incident plane wave strikes a perfectly reflecting periodic surface, the resulting scattered field is comprised of a discrete spectrum of plane waves. Upon applying Dirichlet boundary conditions to the surface, one can construct what is referred to as a spectral-coordinate (SC) formalism for the scattered amplitudes. A Fredholm integral equation of the first kind is involved, and the integration is performed over a single surface period. Since the Rayleigh approximation is not utilized in the construction of this formalism, one may use this method to determine the exact scattered field above the highest surface excursion. The problem will be approached numerically by directly discretizing the mixed SC representation, then solving the system using a pseudoinverse SVD technique. It is very important to note that the scattered amplitudes are obtained without constraining the value of normalized energy. This particular approach is unique. It differs from others in which the discretizations are implemented entirely in coordinate space or entirely in spectral (i.e. Bragg) space. It is thus an additional computational tool designed for cases when a mixed representation is appropriate. Although this numerical scheme has been developed for arbitrary periodic surfaces, the results presented in this paper are restricted to sinusoidal surfaces. Particularly interesting features of this approach are the high level of accuracy attained for near-grazing incident fields and the maintenance of stability even for badly conditioned systems of equations.