Wave scattering by many small bodies and creating materials with a desired refraction coefficient

Abstract—Electromagnetic wave scattering by many small particles is studied. An integral equation is derived for the self-consistent field E in a medium, obtained by embedding many small particles into a given region D. Here D is a bounded domain in which small inclusions (particles) are embedded, and E is the electric field in the resulting medium. The derivation of this integral equation uses a lemma about convergence of certain sums. These sums are similar to Riemannian sums for the integral equation for the field E. Convergence of these sums is essentially equivalent to convergence of a collocation method for solving this integral equation. By choosing the distribution law for embedding the small particles and their physical properties one can create a medium with a desired refraction coefficient. This coefficient can be a tensor. It may have desired absorption properties. 1