Distribution of particles which produces a desired radiation pattern: ETOPIM7

Abstract

If $A_q(\beta ,\alpha ,k)$ is the scattering amplitude, corresponding to a potential $q\in L^2(D)$, where $D\subset {\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$, 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 $\epsilon$. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles $D_m\subset D$, $1\le m\le M$, distributed in an a priori given bounded domain $D\subset {\mathbb{R}}^3$. 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 $A({\alpha }^{\prime },\alpha )$, ${\alpha }^{\prime },\alpha \in S^2$, at a fixed $k>0$, arbitrarily close in the norm of $L^2(S^2\times S^2)$ to an arbitrary given scattering amplitude $f({\alpha }^{\prime },\alpha )$, corresponding to a real-valued potential $q\in L^2(D)$, i.e., corresponding to an arbitrary given refraction coefficient in $D$.

References in corpus (1)

• Electromagnetic wave scattering by many small particles

Cited by in corpus (5)

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