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We present a fast spectral Galerkin scheme for the discretization of boundaryintegral equations arising from two-dimensional Helmholtz transmission problemsin multi-layered periodic structures or gratings. Employing suitablyparametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorouslyestablish the well-posedness of both continuous and discrete problems, andprove super-algebraic error convergence rates for the proposed scheme. Throughseveral numerical examples, we confirm our findings and show performancescompetitive to those attained via Nystr\"om methods.

Fast Solver for Quasi-Periodic 2D-Helmholtz Scattering in Layered Media