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 ou r previous work on scattering from one-dimensional surfaces (1998 Waves Ran dom Media 8 385) we have learned that the spectral-coordinate (SC) method w as the fastest method we have available. Most computational studies of scat tering from two-dimensional surfaces require a large memory and a long calc ulation time unless some approximations are used in the theoretical develop ment. By using the SC method here we are able to solve exact theoretical eq uations with a minimum of calculation time. We first derive in detail (part I) the SC equations for scattering from two -dimensional infinite surfaces. Equations for the boundary unknowns (surfac e field and/or its normal derivative) result as well as an equation to eval uate the scattered field once we have solved for the boundary unknowns. Spe cial cases for the perfectly reflecting Dirichlet and Neumann boundary valu e problems are presented as is the flux-conservation relation. The equations are reduced to those for a two-dimensional periodic surface i n part II and we discuss the numerical methods for their solution. The two- dimensional 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 pa rt III. Computations are performed for both Dirichlet and Neumann problems for various periodic sinusoidal surface examples. The surfaces vary in roug hness as well as period and are investigated when the incident field is far from grazing incidence ('no grazing') and when it is near-grazing. Extensi ve 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 o f the azimuthal angle of incidence, the polar angle of incidence and the wa velength-to-period ratio. The results show that the SC method is highly robust. This is demonstrated with extensive computations. Furthermore, the SC method is found to be comp utationally efficient and accurate for near-grazing incidence. Computations are presented for grazing angles as low as 0.01 degrees. In general, we co nclude 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 slopes and we quantify this limitation.