Theoretical and Computational Aspects of Scattering from Periodic Surfaces: Two-dimensional Perfectly Reflecting Surfaces Using the Spectral-Coordinate Method

We consider the scattering from a two-dimensional periodic surface. From our previous work on scattering from one-dimensional surfaces (Waves in RandomMedia 8, 385(1998)) we have learned that the spectral-coordinate (SC) method was the fastest method we have available. Most computational studies of scattering from two-dimensional surfaces require a large memory and a long calculation time unless some approximations are used in the theoretical development. Here by using the SC method we are able to solve exact theoretical equations with a minimum of calculation time. We first derive (in PART I) in detail the SC equations for scattering from twodimensional infinite surfaces. Equations for the boundary unknowns (surface field and/or its normal derivative) result as well as an equation to evaluate the scattered field once we have solved for the boundary unknowns. Special cases for the perfectly reflecting Dirichlet and Neumann boundary value problems are presented as is the flux-conservation relation. The equations are reduced to those for a two-dimensional periodic surface in PART II and we discuss the numerical methods for their solution. The twodimensional coordinate and spectral samples are arranged in one-dimensional strings in order to define the matrix system to be solved. The SC equations for the two-dimensional periodic surfaces are solved in PART III. Computations are done for both Dirichlet and Neumann problems for various periodic sinusoidal surface examples. The surfaces vary in roughness as well as period and are investigated when the incident field is far from grazing incidence (“no grazing”) and when it is near-grazing. 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, and wavelength to period ratio. The results show that the SC method is highly robust. This is demonstrated with extensive computations. Further the SC method is found to be computationally efficient and accurate for near-grazing incidence. Computations are presented for grazing angles as low as 0.01o. In general we conclude that the SC method is a very fast, reliable and robust computational method to describe scattering from two-dimensional periodic surfaces. Its major limiting factor is high slope and we quantify this limitation. PART I: THEORETICAL DEVELOPMENT FOR AN INFINITE SURFACE