Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

Abstract

We discuss the scattering of acoustic or electromagnetic waves from one dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE-polarization). The theoretical development is for both infinite surfaces and periodic surfaces, the latter equations derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either first kind (CC1) or second kind (CC2). Mixed representation equations or SC type are solved as well as SS equations fully in spectral space. Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime. The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest but did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence doesn't occur then try SS and finally CC.

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

“…A vast literature has thus developed in this area: an excellent survey of early approaches to this problem is given in the classical reference [1]; other reviews and contributions, spanning many decades, include [2][3][4][5][6] among many others. Significant progress has emerged from these efforts.…”

“…A case study is presented, for example, in which the depth of the scattering surface is 40 times its period and for which the period equals 20 times the wavelength (in particular, the period equals 800 times the wavelength); our solver produced the solution for this problem with full ten digits accuracy. The closest example we found in the literature is given in [3]: there a surface of depth equal to eight times its period is presented, for which the solution was produced with an accuracy of three digits. Other rather extreme cases are considered in the present text, including cases involving high frequencies and small grazing angles.…”

“…The aforementioned contribution [3] includes a survey of modern integral equation methods that provides an excellent frame of reference for our methods and results. The most robust mathematically sound approach considered in that reference is a certain collocation method (referred to as CC in [3]).…”

“…The most robust mathematically sound approach considered in that reference is a certain collocation method (referred to as CC in [3]). The essentials of the CC method are related to the present approach.…”

“…The essentials of the CC method are related to the present approach. The present work differs from the CC and related methods of [3] in some fundamental ways, however: the CC method uses the kernel decomposition K͑x,xЈ͒ = − 1 2 ln͉x − xЈ͉ + K ͑x,xЈ͒, ͑1͒…”