Degrees of freedom for scatterers with circular cross section
JOSA, Vol. 71, Issue 3, pp. 250-258 (1981)
http://dx.doi.org/10.1364/JOSA.71.000250
Acrobat PDF (1221 KB)
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
Sort: Year | Journal | Reset
References
- E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).
- G. Toraldo di Francia, "Some recent progress in classical optics," Riv. Nuovo Cimento 1, 460–484 (1969).
- D. Dändliker and K. Weiss, "Reconstruction of the three-dimensional refractive index from scattered waves," Opt. Commun. 1, 323–328 (1970).
- A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19, 1526–1531 (1978).
- B. J. Hoenders, "The uniqueness of inverse problems," in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 41–82.
- In order to obtain, at least in principle, a unique solution to the inverse scattering problem, an infinite number of scattering experiments would be needed (see Ref. 4).
- A. W. Lohmann, "Three-dimensional properties of wave-fields," Optik 51, 105–117 (1978).
- A. W. Lohmann and D. P. Paris, "Binary Fraunhofer holograms generated by computer," Appl. Opt. 5, 1739–1748 (1967).
- W. H. Lee, "Computer generated holograms: techniques and applications," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, pp. 121–232.
- See Ref. 13 and references therein.
- G. Toraldo di Francia, "Degrees of freedom of an image," J. Opt. Soc. Am. 59, 799–804 (1969).
- A. W. Lohmann, "The space-bandwidth product," IBM Tech. Rep. RC438 (1966).
- M. Bertero, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, and G. A. Viano, "The inverse scattering problem in Born approximation and the number of degrees of freedom," Opt. Acta 27, 1011 (1980).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 9.
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
- W. Lukosz, "Equivalent-lens theory of holographic imaging," J. Opt. Soc. Am. 58, 1084–1091 (1968).
- J. R. Shewell and E. Wolf, "Inverse diffraction and a new reciprocity theorem," J. Opt. Soc. Am. 58, 1596–1603 (1968).
- D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43–64 (1961).
- H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65–84 (1961).
- D. Slepian and E. Sonnenblick, "Eigenvalues associated with prolate spheroidal wave functions of zero order," Bell Syst. Tech. J. 44, 1745–1760 (1965).
- F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Chap. 3.
- M. Bendinelli, A. Consortini, L. Ronchi, and B. R. Frieden, "Degrees of freedom, and eigenfunctions, for the noisy image," J. Opt. Soc. Am. 64, 1498–1502 (1974).
- G. J. Buck and J. J. Gustincic, "Resolution limitations of a finite aperture," IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
- G. Cesini, G. Guattari, G. Lucarini, and C. Palma, "An iterative method for restoring noisy images," Opt. Acta 25, 501–508 (1978).
- M. Bertero, C. De Mol, and G. A. Viano, "Resolution beyond the diffraction limit for regularized object restoration," Opt. Acta 27, 307–320 (1980).
- Actually, for finite kR, some of the first few eigenvalues can be slightly greater than 1. However, for sensible values of σ_{0/}/σ_{ω}, Eq. (22) involves high-order eigenvalues µ_{n} < 1.
- P. De Santis and C. Palma, "Degrees of freedom of aberrated images," Opt. Acta 23, 743–752 (1976).
- V. Blažek, "Sampling theorem and the number of degrees of freedom of an image," Opt. Commun. 11, 144–147 (1974).
- H. Gamo, "Matrix treatment of partial coherence" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 189–332.
- F. Gori and G. Guattari, "Shannon number and degrees of freedom of an image," Opt. Commun. 7, 163–165 (1973).
- F. Gori and G. Guattari, "Degrees of freedom of images from point-like-element pupils," J. Opt. Soc. Am. 64, 453–458 (1974).
- D. Slepian, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case," Bell Syst. Tech. J. 57, 1371–1430 (1978).
- 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.
- 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 J_{kR} has vanishingly small values. In other words, the behavior of a typical eigenfunction Φ_{n} for ρ < n/k is somewhat reminiscent of an evanescent wave.
- The similarity between the present analysis and that relating to the Culgoora radioheliograph is to be noted. See Refs. 36–39.
- J. P. Wild, "A new method of image formation with annular apertures and an application in radio astronomy," Proc. R. Soc. London Ser. A 286, 499–509 (1965).
- J. P. Wild et al., "The Culgoora radioheliograph," Proc. Inst. Radio Electron. Eng. Aust. spec. iss. 28, 277–384 (1967).
- F. Gori and G. Guattari, "Effects of coherence on the degrees of freedom of an image," J. Opt. Soc. Am. 61, 36–39 (1971).
- T. W. Cole, "Quasi-optical techniques of radio astronomy," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV, pp. 189–244.
- F. Gori and G. Guattari, "Eigenfunction technique for pointlike- element pupils," Opt. Acta 22, 93–101 (1975).
Cited By |
Alert me when this paper is cited |
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
« Previous Article | Next Article »
OSA is a member of CrossRef.