Scattering of Herglotz waves from periodic structures and mapping properties of the Bloch transform

Abstract

When an incident Herglotz wave function scatters from a periodic Lipschitz continuous surface with a Dirichlet boundary condition, the classical (quasi-)periodic solution theory for scattering from periodic structures does not apply, since the incident field lacks periodicity. Relying on the Bloch transform, we provide a solution theory in H1 for this scattering problem. First, we prove conditions guaranteeing that incident Herglotz wave functions propagating towards the periodic structure have traces in H1/2 on the periodic surface. Second, we show that the solution to the scattering problem can be decomposed by the Bloch transform into periodic components that solve a periodic scattering problem. Third, these periodic solutions yield an equivalent characterization of the solution to the original non-periodic scattering problem, which allows, for instance, new characterizations of the Rayleigh coefficients of each of the periodic components to be shown. A corollary of our results is that under the conditions mentioned above the operator that maps densities to the restriction of their Herglotz wave function on the periodic surface is always injective; this result generally fails for bounded surfaces.

The Floquet–Bloch transform and scattering from locally perturbed periodic surfaces

“…) is the Λ * -periodic Dirac distribution at the origin, i.e., δ Λ * , ψ = ℓ∈Z 2 ψ(Λ * ℓ). The arguments proving Theorems 6 and 7 in [LN15] then show that the Bloch transform of v g equals…”

A High Order Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

“…The Floquet-Bloch based methods were used in [Coa12, HN16, LZ17a, LZ17c, LZ17b] for the numerical solutions in (locally perturbed) periodic scattering problems with non-periodic incident fields. For theoretical results, see [LN15] and [Lec17]. There is also another way to solve the locally perturbed periodic problems, by studying the operators defined on the transfparent edge of a periodic cell.…”

A Floquet--Bloch Transform Based Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

“…It builds up a relationship between a non-periodic problem and a family of quasiperiodic problems reduced to one single period. With this transform, the scattering problems from periodic surfaces and non-periodic incident fields have been discussed in [LN15] and [Lec16]. Based on these theoretic results, a numerical scheme has been developed to solve these kinds of scattering problems in [LZ16].…”