Efficient calculation of the Green’s function for electromagnetic scattering by gratings

Abstract

We consider both the spatial domain and spectral domain forms of the Green's function, appropriate in the electromagnetic diffraction of a plane wave incident in the xy plane on a singly periodic structure, or grating, oriented along the x axis. For the spectral domain form, we exhibit for an obliquely incident plane wave, cubically convergent forms for the Green's function and both its x and y derivatives. We compare the spatial and spectral forms of the Green's function, and so establish expressions from which grating lattice sums can be efficiently evaluated for normal incidence. We use these lattice sums in an alternative expression for the Green's function, which we show to be computationally faster than the accelerated spectral domain expressions for the Green's function if knowledge of this function at several points is required, for small values of y.