Distribution of particles which produces a "smart" material

If $A_q(\beta, \alpha, k)$ is the scattering amplitude, corresponding to a potential $q\in L^2(D)$, where $D\subset\R^3$ is a bounded domain, and $e^{ik\alpha \cdot x}$ is the incident plane wave, then we call the radiation pattern the function $A(\beta):=A_q(\beta, \alpha, k)$, where the unit vector $\alpha$, the incident direction, is fixed, and $k>0$, the wavenumber, is fixed. It is shown that any function $f(\beta)\in L^2(S^2)$, where $S^2$ is the unit sphere in $\R^3$, can be approximated with any desired accuracy by a radiation pattern: $||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon$, where $\epsilon>0$ is an arbitrary small fixed number. The potential $q$, corresponding to $A(\beta)$, depends on $f$ and $\epsilon$, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles $D_m\subset D$, $1\leq m\leq M$, distributed in an a priori given bounded domain $D\subset\R^3$. The geometrical shape of a small particle $D_m$ is arbitrary, the boundary $S_m$ of $D_m$ is Lipschitz uniformly with respect to $m$. The wave number $k$ and the direction $\alpha$ of the incident upon $D$ plane wave are fixed.It is shown that a suitable distribution of the above particles in $D$ can produce the scattering amplitude $A(\alpha',\alpha)$, $\alpha',\alpha\in S^2$, at a fixed $k>0$, arbitrarily close in the norm of $L^2(S^2\times S^2)$ to an arbitrary given scattering amplitude $f(\alpha',\alpha)$, corresponding to a real-valued potential $q\in L^2(D)$.Comment: corrected typo