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This paper introduces a fast algorithm, applicable throughout theelectromagnetic spectrum, for the numerical solution of problems of scatteringby periodic surfaces in two-dimensional space. The proposed algorithm remainshighly accurate and efficient for challenging configurations including randomlyrough surfaces, deep corrugations, large periods, near grazing incidences, and,importantly, Wood-anomaly resonant frequencies. The proposed approach is basedon use of certain `shifted equivalent sources' which enable FFT acceleration ofa Wood-anomaly-capable quasi-periodic Green function introduced recently (Brunoand Delourme, Jour. Computat. Phys., 262--290, 2014). The Green-functionstrategy additionally incorporates an exponentially convergent shifted versionof the classical spectral series for the Green function. While thecomputing-cost asymptotics depend on the asymptotic configuration assumed, thecomputing costs rise at most linearly with the size of the problem for a numberof important rough-surface cases we consider. In practice, single-core runs incomputing times ranging from a fraction of a second to a few seconds sufficefor the proposed algorithm to produce highly-accurate solutions in some of themost challenging contexts arising in applications.

Shifted Equivalent Sources and FFT acceleration for Periodic Scattering Problems including Wood Anomalies