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Periodic surface structures are nowadays standard building blocks of opticaldevices. If such structures are illuminated by aperiodic time-harmonic incidentwaves as, e.g., Gaussian beams, the resulting surface scattering problem mustbe formulated in an unbounded layer including the periodic surface structure.An obvious recipe to avoid the need to discretize this problem in an unboundeddomain is to set up an equivalent system of quasiperiodic scattering problemsin a single (bounded) periodicity cell via the Floquet-Bloch transform. Thesolution to the original surface scattering problem then equals the inverseFloquet-Bloch transform applied to the family of solutions to the quasiperiodicproblems, which simply requires to integrate these solutions in thequasiperiodicity parameter. A numerical scheme derived from this representationhence completely avoids the need to tackle differential equations on unboundeddomains. In this paper, we provide rigorous convergence analysis and errorbounds for such a scheme when applied to a two-dimensional model problem,relying upon a quadrature-based approximation to the inverse Floquet-Blochtransform and finite element approximations to quasiperiodic scatteringproblems. Our analysis essentially relies upon regularity results for thefamily of solutions to the quasiperiodic scattering problems in suitable mixedSobolev spaces. We illustrate our error bounds as well as efficiency of thenumerical scheme via several numerical examples.

A Convergent Numerical Scheme for Scattering of Aperiodic Waves from Periodic Surfaces Based on the Floquet-Bloch Transform