Embedding formulae for diffraction problems on square lattices

Abstract

We develop embedding formulae for all possible diffraction problems with Dirichlet scatterers on square lattices using the Wiener–Hopf perspective. The embedding formula expresses solutions for arbitrary plane wave incidence in terms of a finite set of auxiliary problems, eliminating the need to re-solve boundary value problems for each incidence angle. First, we derive explicit embedding formulae for canonical geometries on a square lattice, including the half-plane, the finite strip and the right-angled wedge. We then generalize the method through an operator-based approach, obtaining an embedding formula for arbitrary configurations of obstacles on a square lattice. This general embedding formula is a key difference from the continuous setting, where this is currently not possible. To validate the theory, we performed numerical experiments that confirm the results derived using the embedding formula. The results highlight the efficiency and generality of the Wiener–Hopf approach in discrete diffraction theory, with potential applications in inverse problems and other areas of physics and mathematics.