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A boundary integral equation method for the 3-D Helmholtz equation inmultilayered media with many quasi-periodic layers is presented. Compared withconventional quasi-periodic Green's function method, the new method is robustat all scattering parameters. A periodizing scheme is used to decompose thesolution into near- and far-field contributions. The near-field contributionuses the free-space Green's function in an integral equation on the interfacein the unit cell and its immediate eight neighbors; the far-field contributionuses proxy point sources that enclose the unit cell. A specialized high-orderquadrature is developed to discretize the underlying surface integral operatorsto keep the number of unknowns per layer small. We achieve overall linearcomputational complexity in the number of layers by reducing the linear systeminto block tridiagonal form and then solving the system directly via block LUdecomposition. The new solver is capable of handling a 100-interface structurewith 961.3k unknowns to $10^{-5}$ accuracy in less than 2 hours on a desktopworkstation.

Robust fast direct integral equation solver for three-dimensional quasi-periodic scattering problems with a large number of layers