The Floquet-Bloch Transform and Scattering from Locally Perturbed Periodic Surfaces

TL;DR

This paper employs the Floquet-Bloch transform to convert surface scattering problems for the Helmholtz equation on periodic surfaces into bounded domain problems, revealing how solution decay relates to Bloch transform smoothness.

Contribution

It establishes the mapping properties of the Floquet-Bloch transform between weighted Sobolev spaces, enabling analysis of scattering problems on perturbed periodic surfaces.

Findings

Transform reduces unbounded domain problems to bounded domain formulations.

Decay of solutions is characterized by the smoothness of the Bloch transform.

Provides a rigorous mathematical framework for analyzing surface scattering from perturbed periodic structures.

Abstract

We use the Floquet-Bloch transform to reduce variational formulations of surface scattering problems for the Helmholtz equation from periodic and locally perturbed periodic surfaces to equivalent variational problems formulated on bounded domains. To this end, we establish various mapping properties of that transform between suitable weighted Sobolev spaces on periodic strip-like domains and coupled families of quasiperiodic Sobolev spaces. Our analysis shows in particular that the decay of solutions to surface scattering problems from locally perturbed periodic surfaces is precisely characterized by the smoothness of its Bloch transform in the quasiperiodicity.