Second-Order Perturbative Solution of Scattering From Two Rough Surfaces With Arbitrary Dielectric Profiles

Analytical expressions for the cross-polarization components of the normalized radar cross sections (NRCS) of three-dimensional stratified media with two rough surfaces are presented. In a recent paper, the zeroth- and the first-order scattering solutions of a two-rough-surface structure with arbitrary dielectric profiles were presented using the small perturbation approach based on the extended boundary conditions (EBC). In this study, the proposed method is extended to obtain the second-order solutions including cross-polarized fields. Unlike copolarizations, the main contribution of cross-polarizations is determined by the second-order solutions. A direct extension of the recently proposed formulation leads to difficult and lengthy manipulations; hence, first a compact recursive matrix formulation for the unknown surface fields of an arbitrary order is derived. Then, for the second-order fields, the matrix system is efficiently and straightforwardly solved. Finally, the cross-polarized scattered fields and the normalized radar cross sections are derived. The final solutions are given in simple, compact, and closed-form expressions that make the computations very easy. The derived expressions are validated against known solutions of special cases, available in the literature. The results are shown to be exactly consistent with existing ones, both analytically and numerically.