A method for creating materials with a desired refraction coefficient

It is proposed to create materials with a desired refraction coefficient in a bounded domain $D\subset \R^3$ by embedding many small balls with constant refraction coefficients into a given material. The number of small balls per unit volume around every point $x\in D$, i.e., their density distribution, is calculated, as well as the constant refraction coefficients in these balls. Embedding into $D$ small balls with these refraction coefficients according to the calculated density distribution creates in $D$ a material with a desired refraction coefficient

A recipe for making materials with negative refraction in acoustics

A recipe is given for making materials with negative refraction in acoustics, i.e., materials in which the group velocity is directed opposite to the phase velocity. The recipe consists of injecting many small particles into a bounded domain, filled with a material whose refraction coefficient is known. The number of small particles to be injected per unit volume around any point $x$ is calculated as well as the boundary impedances of the embedded particles

Completeness of the set of scattering amplitudes

Let $f\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\subset \R^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $\alpha\in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A(\alpha')=A_q(\alpha')=A_q(\alpha',\alpha,k)$ approximates $f(\alpha')$ with arbitrary high accuracy: \|f(\alpha')-A_q(\alpha')_{L^2(S^2)}\|\leq\ve where \ve>0 is an arbitrarily small fixed number. This means that the set $\{A_q(\alpha')\}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials"