Scattering of electromagnetic waves by many nanowires

Electromagnetic wave scattering by many parallel to $z-$axis, thin, impedance, circular infinite cylinders is studied asymptotically as $a\to 0$. Let $D_m$ be the crossection of the $m-$th cylinder, $a$ be its radius, and ${\widehat{x}}_m=(x_{m1},x_{m2})$ be its center, $1\le m\le M$, $M=M(a)$. It is assumed that the points ${\widehat{x}}_m$ are distributed so that $$\mathcal{N}(\mathrm{\Delta })=\frac{1}{2\pi a}{\int }_{\mathrm{\Delta }}N(\widehat{x})d\widehat{x}[1+o(1)],$$ where $\mathcal{N}(\mathrm{\Delta })$ is the number of points ${\widehat{x}}_m$ in an arbitrary open subset $\mathrm{\Delta }$ of the plane $xoy$. The function $N(\hat{x})\geq 0$ is a given continuous function. An equation for the self-consistent (efficient) field is derived as $a\to 0$. A formula is derived for the effective refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nanowires embedded in the medium. Our result shows how these cylinders influence the refraction coefficient of the medium.