The direct and inverse scattering problem for cracks∗

We consider the scattering of an electromagnetic time harmonic plane wave by very thin obstacles. In this lecture we will restrict ourselves to the scattering of an Epolarized electromagnetic field by a perfectly conducting infinite cylinder having an open arc in R as cross section. We further allow the possibility that one side of the cylinder is coated with a dielectric material. This leads to the direct scattering problem for the Helmholtz equation outside the open arc (the crack) with Dirichlet or mixed Dirichlet-impedance boundary conditions on the crack. For the sake of presentation we treat in details only the Dirichlet case. We first show that the direct scattering problem is well posed. This is done by using an integral equation approach. We also discuss in this context the singular behavior of the solution at the tips of the crack. We then continue with the inverse problem of determining the crack from a knowledge of the scattered wave measured far away from the crack. We will discuss here three methods for solving the inverse problem: Newton’s method developed by Kress, the factorization method developed by Kirsch and Ritter (both methods for only the Dirichlet case) and the linear sampling method developed by Cakoni and Colton which can be used for more general type of the boundary conditions (without requiring a-priori knowledge of these conditions). ∗Lecture Notes: Winter School on Direct and Inverse Scattering Problems, 27-31 January 2003, INRIA, Paris.