Abstract

In this course I will focus on the time-harmonic wave equation (or Helmholtz equation) in unbounded periodic media. One of the difficulties of the Helmholtz equation in an unbounded domain is that the associated problem is not always well posed in a classical framework. In general, one must impose a behavior at infinity, called radiation condition, or derive a transparent boundary condition to reduce the problem to a bounded domain. This is a difficult and still open question for general periodic media, although the answer is now clear for one-dimensional, closed or open periodic waveguide problems. In such cases, the framework is to use the limiting absorption principle. I will explain this approach for one-dimensional problems :  (1) by adding some dissipation (i.e. an imaginary part of the frequency), one returns to a classical L2 framework (2) one can build transparent boundary conditions based on the Dirichlet-to-Neumann (DtN)coefficient, by taking advantage of the periodic structure of the medium (3) one can study the limit of the DtN coefficient when the dissipation goes to 0. I will show some numerical results, explain the extension to closed periodic waveguides and finally highlight the difficulties for more general problems.