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.