# A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an efficient approach for the scattering of acoustic and electromagnetic waves from periodic arrays (gratings) of obstacles. The standard way to periodize is to replace the free-space Green's function kernel with its quasi-periodic cousin. This idea has two drawbacks: i) the quasi-periodic Green's function diverges (does not exist) for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, thus becomes unwieldy when obstacles have high aspect ratio.

We bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, and adds auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. The result is a (slightly larger) 2nd kind system that achieves spectral accuracy, is immune to Wood's anomalies, avoids lattice sums, handles large aspect ratios, and couples to existing scattering code. By careful summing of neighboring images, obstacles may intersect the unit cell walls.

If time, we will discuss a similar robust new approach to the photonic crystal band structure eigenvalue problem.

Joint work with Leslie Greengard (NYU).