Many numerical problems arising in modern photonic and electromagnetic applications involve the interaction of linear waves with periodic, piecewise-homogeneous media. Boundary integral equations are an efficient approach to solving such boundary-value problems with high-order convergence. In the case of plane-wave scattering from an array (grating), the standard way to periodize is then to replace the free-space Green's function kernel with its quasi-periodic cousin. However, a major drawback is that the quasi-periodic Green's function fails to exist for parameter families known as Wood's anomalies, even though the underlying scattering problem remains well-posed.

We bypass this problem with a new integral representation that relies on the *free-space* Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell strip, while enforcing quasi-periodicity with an expanded linear system. The result is a 2nd kind scheme that achieves spectral accuracy, is immune to Wood's anomalies, avoids lattice sums, and reuses existing scattering codes. A doubly-periodic version provides similar benefits for the robust solution of the eigenvalue (band structure) problem for Bloch waves in a photonic crystal. We show two-dimensional examples achieving 10-digit accuracy with only a couple of hundred unknowns.

Joint work with Leslie Greengard (NYU).