Electromagnetic Scattering by Many Small Inhomogeneities

Content of the research project. We are concerned with the time-harmonic electromagnetic waves propagating in a medium which consist of a large number of small inhomogeneities arbitrary distributed in a background medium. We will derive the asymptotic expansions of the electromagnetic waves scattered by this large collection of inhomogeneities . Our aim is to understand (and quantify at the maximum extent) the influence of these small inhomogeneities in reflecting these electromagnetic waves. This study will allow us 1. to considerably reduce the complexity in computing these electromagnetic waves. 2. to apply our results to the design of new engineering materials with desired reflectivity properties (as metamaterials) and to cloak given objects against incident electromagnetic waves. Hypotheses. We model our collection of inhomogeneities in terms of their number M, their maximum diameter a and the minimum distance between them d (with eventually the jumps of the coefficients, modeling the material, across the inhomogeneities). These are the main characteristics that influence the scattered electromagnetic waves. We assume no periodicity in distributing the inhomogeneities (in contrast to what is done in the homogenization theory). This allows us to consider more general configurations than it was done in the previous literature. Methods. We base our analysis on the asymptotic expansion techniques and the main tool is the integral equation method. The emphasize here is on taking into account all the parameters describing the collection of the small inhomogeneities. Originality of the project. With the derived asymptotic expansions: 1. We can reduce the computation of the scattered fields, by a large number of small inhomogeneities, to either inverting an algebraic system or solving a scattering problem by a single extended scatterer. This reduces tremendously the computation complexity. 2. We can characterize the equivalent media. This characterization has potential applications in the theory of materials. For acoustic waves, we can mathematically generate any refractive index by perforating a given background acoustic medium with small holes of impedance type. In particular, (1) we can approximately cloak a given index of refraction and (2) we can switch a given refractive index from positive to negative values (acoustic metamaterials) by appropriately choosing the surface impedances. These results, concerning the acoustic waves, are derived with explicit error estimates. These error estimates are useful in (1) testing the accuracy of the algorithms for computing the scattered fields and in (2) designing new engineering materials. To the best of our knowledge, these are the most general and precise results published in the literature. The corresponding results for the electromagnetic waves are expected and they will be the outcome of this project proposal.

Understanding the interaction of light with matter and estimating the effect of the later on the propagation of the former is of fundamental importance in many applications. In this project, we studied the effect of a cluster of small particles on the propagation of the electromagnetic waves in the linear and time-harmonic regime. We extracted the dominant part of the electromagnetic field that is scattered, or that goes through, the cluster. This dominant field is characterized by the number of small particles, their minimum distance and the contrast of their electric permittivity and eventually magnetic permeability. As these last parameters are at our disposal, we can tune them to control this dominant electromagnetic field. In particular, we can assemble the cluster of small particles so that it behaves as a new material that generates electromagnetic fields with desired properties. We analyzed mathematically these questions by providing useful approximations of the electromagnetic field and quantified how the parameters, of the cluster of particles, mentioned above enter into considerations in the design of new electromagnetic materials. In particular, this cluster can be distributed in volumetric domains or on surfaces with high generality. Potential applications of such designs in material sciences, i.e. metamaterials, and imaging sciences are immediate.