Shifted equivalent sources and FFT acceleration for periodic scattering problems, including Wood anomalies

Highlights: • First high-order accelerated solver for periodic scattering including Wood anomalies. • The methodology greatly reduces the number of shifted Green function evaluations. • Additional acceleration is obtained by means of a dual spatial/spectral approach. • Efficient solution of highly challenging practical 2D scattering problems. • A three-dimensional version of this approach has been found equally effective. -- Abstract: This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain "shifted equivalent sources" which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme (2014) [4]). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications.