Date(s) - 30/09/2014
11 h 00 min - 12 h 00 min
This is a joint work with Patrick Joly (the forward problem) and Laurent Bourgeois (the inverse problem). We are interested in propagation media which are a local perturbation of an infinite (or very large) periodic waveguide. We consider first the forward problem, computing the solution of the time harmonic wave equation in such media. We investigate the question of finding artificial boundary conditions to reduce the actual numerical computations to a neighborhood of the perturbation. More precisely, by revisiting the Floquet Bloch theory, we propose a method for constructing Dirichlet-to-Neumann (DtN) operators. We will see in particular that the characterization of these DtN operators is linked to the definition of the physical or “outgoing” solution of the problem, definition which is not obvious in periodic media. This difficulty will be treated using the limiting absorption principle. We consider then the linear sampling method (LSM) to solve the inverse medium problem in a locally perturbed periodic waveguide. The periodic waveguide is known and our aim is, from scattering data, to recover the defect within such periodic waveguide. This method is based on the far field of the Green function in a periodic waveguide. Such analysis enables us to derive a modal formulation of the LSM in which the incident fields are formed by the Floquet modes.