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 $O(a^{\frac{2-κ}{3}})$ as $a\to 0$, $0<κ<1$ is a fixed parameter. The total number of the embedded particle is $O(a^{κ-2})$. The physical properties of the particles are described by the boundary impedance $ζ_m$ of the $m-th$ particle, $ζ_m=O(a^{-κ})$ as $a\to 0$. The refraction coefficient is the coefficient $n^2(x)$ in the wave equation $[\nabla^2+k^2n^2(x)]u=0$.

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