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

When an incident Herglotz wave function scatteres from a periodic Lipschitz continuous surface with Dirichlet boundary condition, then 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 H for this scattering problem: We first prove conditions guaranteeing that incident Herglotz wave functions propagating towards the periodic structure have traces in H on the periodic surface. Second, we show that the solution to the scattering problem can be decomposed by the Bloch transform into its 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, to prove new characterizations of the Rayleigh coefficients of each of the periodic components. A corollary of our results is that under the conditions mentioned above the operator mapping densities to the restriction of their Herglotz wave function on the periodic surface is always injective; this result generally fails for bounded surfaces.