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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 71, Iss. 3 — Mar. 1, 1981
  • pp: 250–258

Degrees of freedom for scatterers with circular cross section

F. Gori and L. Ronchi  »View Author Affiliations


JOSA, Vol. 71, Issue 3, pp. 250-258 (1981)
http://dx.doi.org/10.1364/JOSA.71.000250


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Abstract

The inverse scattering problem for a scatterer that is independent of one spatial coordinate is considered in the Born approximation. Assuming a circular cross section for the scatterer, the number of degrees of freedom of the field scattered over the full angular range (2π) is evaluated in the presence of noise. This is obtained by making use of the eigenfunction technique and determining the pertinent eigenfunctions and eigenvalues. The finiteness of the number of degrees of freedom leads us to introduce finite sampling techniques. The results also apply to the dual problem of synthesizing wave field structures starting from computer-generated holograms. The evaluation of the number of degrees of freedom for a series of scattering experiments with different illuminating waves is also outlined. Throughout the paper the similaritý between the present problems and the problem relating to the degrees of freedom of coherent images formed through thin-ring pupils is exploited.

© 1981 Optical Society of America

Citation
F. Gori and L. Ronchi, "Degrees of freedom for scatterers with circular cross section," J. Opt. Soc. Am. 71, 250-258 (1981)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-71-3-250


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References

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  26. Actually, for finite kR, some of the first few eigenvalues can be slightly greater than 1. However, for sensible values of σ0/ω, Eq. (22) involves high-order eigenvalues µn < 1.
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  33. The reader should be cautioned that the function G (and Ĝ as well) has different physical dimensions according to whether the scattering or the synthesis problem is considered. For the sake of simplicity, we use the same symbol in both cases.
  34. It could be observed that, moving to a cylinder of radius smaller than R, the same field oscillation takes place over an arc smaller than λ. This would seem to lower the resolution limit. But this is untrue because, for ρ < R, the Bessel function JkR has vanishingly small values. In other words, the behavior of a typical eigenfunction Φn for ρ < n/k is somewhat reminiscent of an evanescent wave.
  35. The similarity between the present analysis and that relating to the Culgoora radioheliograph is to be noted. See Refs. 36–39.
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  40. F. Gori and G. Guattari, "Eigenfunction technique for pointlike- element pupils," Opt. Acta 22, 93–101 (1975).

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