Abstract
Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter . We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair . This abstract result allows us to derive explicitly a representation for the limiting absorption solution as a sum of a decaying part (along the axis of the cylinder) and a finite sum of propagating modes.
Highlights
Periodic non-absorbing surface structures allow surface waves that propagate along the structure without decaying
Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem
We prove that there is a so-called limiting absorption solution to the associated scattering problem
Summary
Periodic non-absorbing surface structures allow surface waves that propagate along the structure without decaying. This solution is, in a certain topology, limit of the unique solutions to a family of coercive problems with artificial complex-valued wave numbers This limiting absorption solution consists of two parts that we determine via the Floquet– Bloch transform: The first part belongs to H1 in any cylinder of finite radius and the second part is made up of surface waves or propagative modes. This limiting absorption solution u(1) can be explicitly expressed by the (generalized) Fourier transform in terms of Hankel functions of the first kind Both parts, the decomposition of the field u into a decaying field u(1) along the axis and a propagating field u(2) and the particular form of u(1) outside of the cylinder allows the formulation of a radiation condition which we carry out in the second part of this paper. In the appendix we prove several properties of Hankel functions with complex arguments and solve an exterior boundary value problem for the Helmholtz equation with the use of the Fourier transform
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