Efficient high-order evaluation of scattering by periodic surfaces: vector-parametric gratings and geometric singularities

We introduce a highly accurate and efficient solver for problems of scattering by vector-parametric, possibly non-smooth one-dimensional periodic surfaces. For grating-diffraction problems in the resonance regime (heights and periods of the order of the wavelength of radiation) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. Our algorithm can also produce, in reasonable computing times, solutions for highly challenging TE and TM scattering problems - defined by very deep multi-valued scattering surfaces and high frequencies of radiation - including cases in which non-smooth bounding surfaces enclose open cavities.