Scattering of EM 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 $a\to 0$, 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 $a^{2-\kappa}$, where $\kappa\in [0,1)$ 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 $\zeta=ha^{-\kappa}$, where $h=$const, Re$h\ge 0$. The scattering amplitude for a small perfectly conducting particle is proportional to $a^3$, it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions $a\ll d\ll \lambda$, where $d$ is the minimal distance between neighboring particles and $\lambda$ is the wavelength. The distribution law for the small impedance particles is $\mathcal{N}(\delta)\sim\int_{\delta}N(x)dx$ as $a\to 0$. Here $N(x)\ge 0$ is an arbitrary continuous function that can be chosen by the experimenter and $\mathcal{N}(\delta)$ is the number of particles in an arbitrary sub-domain $\Delta$. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as $a\to 0$ and a differential equation is derived for the limiting field. On this basis the recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material.