Creating materials with a desired refraction coefficient

A method is given for creating material with a desired refraction coefficient. The method consists of embedding into a material with known refraction coefficient many small particles of size $a$. The number of particles per unit volume around any point is prescribed, the distance between neighboring particles is $O(a^{\frac{2-\kappa }{3}})$ as $a\to 0$, $0<\kappa <1$ is a fixed parameter. The total number of the embedded particle is $O(a^{\kappa -2})$. The physical properties of the particles are described by the boundary impedance ${\zeta }_m$ of the $m-th$ particle, ${\zeta }_m=O(a^{-\kappa })$ as $a\to 0$. The refraction coefficient is the coefficient $n^2(x)$ in the wave equation $[{\mathrm{\nabla }}^2+k^2{n}^2(x)]u=0$.