Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies

A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a → 0, where a is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a^2−κ, where κ ∈ [0, 1) is a parameter which can be chosen by an experimenter as he/she wants. The boundary impedance of a small particle is assumed to be of the form ζ = ha^−κ, where h = const, Reh ≥ 0. The scattering amplitude for a small perfectly conducting particle is proportional to a³, and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a ≪ d ≪ λ, where d is the minimal distance between neighboring particles and λ is the wavelength. The distribution law for the small impedance particles is $N ( Δ ) ∼1/ a 2 − κ ∫ Δ N ( x ) dx$ as a → 0. Here, N(x) ≥ 0 is an arbitrary continuous function that can be chosen by the experimenter and $N ( Δ )$ is the number of particles in an arbitrary sub-domain Δ. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a → 0 and a differential equation is derived for the limiting field. On this basis, a recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material.