Joint Analyticity of the Transformed Field and Dirichlet-Neumann Operator in Periodic Media

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this thesis we establish rigorous analytic results necessary for the proper numerical analysis of a class of High–Order Perturbation of Surfaces/Asymptotic Waveform Evaluation (HOPS/AWE) methods for numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two–layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. Additional analysis is performed in order to analyze any finite number of perturbation parameters. This result provides hypotheses under which a rigorous numerical analysis is performed. We extend our recently developed HOPS/AWE algorithm to utilize a stabilized numerical scheme. An implementation of this algorithm is described, validated, and utilized in a sequence of challenging and physically relevant numerical experiments.