We present a solver for plane wave scattering from a periodic dielectric grating with a large number M of inclusions lying in each period of its middle layer. Such composite material geometries have a growing role in modern photonic devices and solar cells. The high-order scheme is based on boundary integral equations, and achieves many digits of accuracy with ease. The usual way to periodize the integral equation-via the quasi-periodic Green's function-fails at Wood's anomalies. We instead use the free-space Green's kernel for the near field, add auxiliary basis functions for the far field, and enforce periodicity in an expanded linear system; this is robust for all parameters. Inverting the periodic and layer unknowns, we are left with a square linear system involving only the inclusion scattering coefficients. Preconditioning by the single-inclusion scattering matrix, this is solved iteratively in O(M) time using a fast matrix-vector product. Numerical experiments show that a diffraction grating containing M=. 1000 inclusions per period can be solved to 9-digit accuracy in under 5 minutes on a laptop.
Lai, J., Kobayashi, M., & Barnett, A. (2015). A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions. Journal of Computational Physics, 298, 194–208. https://doi.org/10.1016/j.jcp.2015.06.005