Electromagnetic Scattering by Complex Interfaces

Scattering or boundary value problems for complex interfaces have at least two different origins: I. They are approximate models for penetrable obstacles where the material of the interior part of the obstacle is so highly reflecting and does allow only week absorption. Hence the transmission conditions across the interface of the obstacle are replaced by surface impedance type boundary conditions. II. Artificial materials, distributed along the surface of the obstacle, are used to produce more (or less) scattering by the obstacle. These artificial materials are modeled by coefficients appearing in the `modified` transmission conditions. A complex interface is characterized by its location, its geometry and the material distributed in/on it. We are interested in studying the inverse problems associated to these two situations for some models described by the Maxwell systems. We divide our study into two interdependent parts: a.) Inverse problems. In this case, we are given exterior measurements (boundary or scattering measurements) and we want to reconstruct the unknown complex interface. b.) Design problems. In this case, we know the (shape of the) obstacle and we want to design it in such a way to be more (or less, if needed) visible from exterior measurements. We are interested by designs using surface or body coatings. In particular, we wish to understand and explain to what extent the already proposed methods (including sampling, probing and music algorithm) are practically accurate. To study this accuracy issue, we propose to use the asymptotic expansions of the corresponding indicator functions with respect to the used point sources. Precisely, we wish to provide qualitative information on the behavior of the mentioned methods and their explicit links to the complex interface`s parameters, i.e. the geometry and the material distributed in/on it. Providing these links will give answers to the problems a) and b). The use of asymptotic expansions requires the complex interfaces to be smooth enough. To handle the cases when the complex interfaces are rough (i.e. Lipschitz interfaces and only bounded surface coefficients), we propose another approach based on the use of layer potential techniques to transform the inverse problem to the inversion of invertible integral equations stated on the interfaces.

The main goal of this project is the localization and characterization of hidden (or inaccessible) targets from remote signals (or reactions) they can deliver after exciting them by incident waves. We considered acoustic, elastic as well as electromagnetic waves. We focused on the design and the use of the so-called direct methods. These methods provide direct links between the measured signals and some geometric (as the shape) and material properties of the targets (as their jumps across the surfaces of the targets). This means that we can read some of these properties directly from the signals. The obtained results are mathematically justified and quantified. These results are applicable in the following areas:Medical imaging: detection of the locations and estimation of the sizes of early (or some geometrical features of advanced) tumors, as breast cancers for instance, using electrical signals collected on an accessible surface surrounding the investigated part of the body, the breast for instance.Submarine and geophysical prospections: in particular the use of acoustic or elastic waves for minerals detection (as oil, gaz, etc.) and sediment classifications in geology.Material engineering: (1) checking and detecting defects in used materials, as cracks in bridges, buildings etc., (2) design of materials with desired scattering properties as refraction indices, permeability and permittivity modulus.Few mathematical highlights as outcomes of the project: Elastic Lame model: it is known that any elastic body wave is a superposition of two fundamental waves called the shear and the pressure waves. We prove that any of these two fundamental waves is enough for the detection purpose. It is the first time that this result is rigorously proved. Electromagnetic model: the enclosure method, based on the use of complex geometrical optic solutions, was left unjustified for the Maxwell system. We justified it by proving the appropriate estimate. To derive this estimate, we proved the corresponding Groeger-Meyers's Lp estimate for the Maxwell system.Acoustic model: the Foldy-Lax approximation for the scattered waves by many scatterers. We gave sufficient, but general, conditions on the number M of the scatterers, their size a and the minimum distance between them d under which this approximation is valid. Immediate applications of these approximations are in (1) imaging by allowing very close targets and (2) the theory of metamaterials. This second field is a new direction of research that we will develop more as a next step.