A CONVERGENT NUMERICAL SCHEME FOR SCATTERING OF APERIODIC WAVES FROM PERIODIC SURFACES BASED ON THE FLOQUET-BLOCH TRANSFORM

Periodic surface structures are currently standard building blocks of optical devices. If such structures are illuminated by aperiodic time-harmonic incident waves, e.g., Gaussian beams, the resulting surface scattering problem must be formulated in an unbounded layer including the periodic surface structure. An obvious recipe for avoiding the need to discretize this problem in an unbounded domain is to set up an equivalent system of quasi-periodic scattering problems in a single (bounded) periodicity cell via the Floquet-Bloch transform. The solution to the original surface scattering problem then equals the inverse Floquet-Bloch transform applied to the family of solutions to the quasi-periodic problems, which simply requires us to integrate these solutions in the quasi periodicity parameter. A numerical scheme derived from this representation hence completely avoids the need to tackle differential equations on unbounded domains. In this paper, we provide rigorous convergence analysis and error bounds for such a scheme when applied to a two-dimensional model problem, relying upon a quadrature-based approximation to the inverse Floquet-Bloch transform and finite element approximations to quasi-periodic scattering problems. Our analysis essentially relies upon regularity results for the family of solutions to the quasi-periodic scattering problems in suitable mixed Sobolev spaces. We illustrate our error bounds as well as the efficiency of the numerical scheme via several numerical examples.