Method of field expansions for vector electromagnetic scattering by layered periodic crossed gratings

In many applications of scientific and engineering interest the accurate modeling of scattering of linear waves by periodic layered media plays a crucial role. From geophysics and oceanography to materials science and imaging, the ability to simulate such configurations numerically in a rapid and robust fashion is of paramount importance. In this contribution we focus upon the specific problem of vector electromagnetic radiation interacting with a multiply layered periodic crossed diffraction grating. While all of the classical methods for the numerical simulation of partial differential equations have been brought to bear upon this problem, we argue here that in this particular context a high-order perturbation of surfaces approach is superior. In particular, we describe how the method of field expansions can be extended to the fully vectorial and three-dimensional scattering problem in the presence of multiple layers. With specific numerical experiments we will show the remarkable efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.