A high-order perturbation of surfaces (HOPS) approach to Fokas integral equations: Vector electromagnetic scattering by periodic crossed gratings

The accurate simulation of linear electromagnetic scattering by diffraction gratings is crucial in many technologies of scientific and engineering interest. In this contribution we describe a High-Order Perturbation of Surfaces (HOPS) algorithm built upon a class of Integral Equations due to the analysis of Fokas and collaborators, now widely known as the Unified Transform Method. The unknowns in this formalism are boundary quantities (the electric field and current at the layer interface) which are an order of magnitude fewer than standard volumetric approaches such as Finite Differences and Finite Elements. With detailed numerical experiments we show the efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.