Download Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.) PDF

By Peter W. Hawkes (Ed.)

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Extra resources for Advances in Electronics and Electron Physics, Vol. 75

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In this case, the uniqueness of the reconstruction off is assured, since f is an analytic function thanks to the bounded support off. We notice that we are again dealing with the problem of reconstructing a function from limited knowledge of its Fourier transform. The same mathematical problem must be solved if one applies Rytov approximation (Devaney, 1981). The basic point is the introduction of the complex phase function $(r) = In u(r) - ik(s,, r). mdu~V(r). (57) It is obvious now that if one takes the 2D Fourier transform of exp[ik(~,,r)]$'~)(r)over any plate of constant z that does not intersect the scatterer, one again gets values off on the surface of the Ewald sphere.

A general discussion of the uniqueness of the solution of the problem of inverse diffraction is given by Hoenders (1978) both in the scalar and in the vector case. The uniqueness of the problem of inverse diffraction from far-field data was proved by Devaney and Wolf (1974) for electromagnetic fields. Here we sketch the treatment of inverse diffraction from far-field data and the treatment of inverse plane-to-plane diffraction. 1. Then we denote by f(s) the field distribution on C,. The direct problem is the determination of the solution u(r,s)of Eq.

1979). Since we consider a bounded object, situated, for instance, inside the sphere of radius a, f(r) is zero for r > a. Then the total field u (incident plus scattered) is a solution of the wave equation Au + k2u = k’f(r)u, (53) where k = 2n/A is the wave number (in free space) of the incident radiation. r)- (k2/4n) s G(r - yr)eik(so,r‘lf(r’)dr’. s. of Eq. (54) (Born approximation of scattered wave). s. ,t,,,(kz - t: - 5,2)”’). It follows from Eq. ( 5 5 ) that knowledge of ijLB)({, z) is equivalent to knowledge off on the surface of the Ewald sphere with centre ks, and radius k and, therefore, from the mathematical point of view, this problem is analogous to the inverse source problem discussed in Section II,B.

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