A pseudo-spectral longitudinal expansion in a spectral domain integral equation for scattering by periodic dielectric structures

We explore several options to introduce a pseudo-spectral expansion along the longitudinal direction in a spectral-domain integral equation for scattering by periodic dielectric structures. To this end we first simplify the integral equation to the formulation for a one-dimensional dielectric slab and consider the computational efficiency, convergence, and conditioning of several schemes. One scheme has been implemented in the domain integral equation and it exhibits exponential convergence with respect to the number of expansion functions in the longitudinal direction.