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Distribution of particles which produces a desired radiation pattern ETOPIM7.pdf

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Distribution of particles which produces a desired radiation pattern ETOPIM7

a r X i v : m a t h - p h / 0 6 0 6 0 5 6 v 2 2 6 J u n 2 0 0 6 Distribution of particles which produces a desired radiation pattern Alexander G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA phone: 785-532-0580, fax: 785-532-0546 Abstract If Aq(β, α, k) is the scattering amplitude, corresponding to a potential q ∈ L 2(D), where D ? R3 is a bounded domain, and eikα·x is the incident plane wave, then we call the radiation pattern the function A(β) := Aq(β, α, k), where the unit vector α, the incident direction, is fixed, and k 0, the wavenumber, is fixed. It is shown that any function f(β) ∈ L2(S2), where S2 is the unit sphere in R3, can be approximated with any desired accuracy by a radiation pattern: ||f(β) ? A(β)||L2(S2) ?, where ? 0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ?. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles Dm ? D, 1 ≤ m ≤ M , distributed in an a priori given bounded domain D ? R3. The geometrical shape of a small particle Dm is arbitrary, the boundary Sm of Dm is Lipschitz uniformly with respect to m. The wave number k and the direction α 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(α′, α), α′, α ∈ S2, at a fixed k 0, arbitrarily close in the norm of L2(S2 × S2) to an arbitrary given scattering amplitude f(α′, α), corre- sponding to a real-valued potential q ∈ L2(D), i.e., corresponding to an arbitrary given refraction coefficient in D. Key words: nanotechnology, ”smart” materials, inverse scattering PACS: 03.04.Kf Email address: ramm@ (Alexander G. Ramm). Preprint submitted to Elsevier Science 7 February 2008 1 Introduction It is proved that one can distribute in a given bounded region D, small in comparison with the wavelength in D, acoustically soft particle