Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional transmission surfaces using the spectral-coordinate method

Abstract

We consider the scattering from and transmission through a two-dimensional periodic surface. We use the spectral-coordinate (SC) method for all the computations. It was the fastest method for one-dimensional problems and proved optimal for scattering from two-dimensional surfaces where computation time can prove to be excessive. In particular, we can avoid approximation methods and solve the exact equations. The SC equations are derived for an infinite surface and reduced to coupled equations for a periodic surface which are solved numerically for the two boundary unknowns. Solutions of the SC equations for various periodic sinusoidal surface examples are studied. The surfaces vary in roughness and period. Extensive computations are included in terms of the maximum roughness slope which can be computed using the method with a fixed maximum error as a function of azimuthal angle of incidence, polar angle of incidence, wavelength-to-period ratio, density ratio and wavenumber ratio. Examples of reflectionless interfaces as a function of density and wavenumber are presented. Particular attention is paid to the case of near-grazing incidence. As a result of these extensive computations we conclude that the SC method is stable and robust (a) over the entire incident azimuthal variability, (b) over a 50-fold change in value of the wavenumber ratio and (c) as the density parameter varies over two orders of magnitude. In addition, SC works very well under extreme near-grazing conditions even for very rough surfaces with large slopes over a very broad parameter range in density and wavenumber. Spectral-based methods can thus play an important role in the description of the scattering from two-dimensional periodic surfaces.