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

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green's function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green's function diverges 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, becoming unwieldy when obstacles have high aspect ratio. In this paper, we bypass both problems by means of 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 expanding the linear system to enforce quasi-periodicity. Summing nearby images directly leaves auxiliary densities that are smooth, hence easily represented in the Fourier domain using Sommerfeld integrals. Wood's anomalies are handled analytically by deformation of the Sommerfeld contour. The resulting integral equation is of the second kind and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls are handled easily and automatically. We include an implementation and simple code example with a freely-available MATLAB toolbox.