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 similarity 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

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References

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  1. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  2. G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).
  3. D. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
    [CrossRef]
  4. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
    [CrossRef]
  5. 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.
    [CrossRef]
  6. 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).
  7. A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).
  8. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 5, 1739–1748 (1967).
    [CrossRef]
  9. 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.
  10. See Ref. 13 and references therein.
  11. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, “The space-bandwidth product,” (1966).
  13. 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).
    [CrossRef]
  14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 9.
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  16. W. Lukosz, “Equivalent-lens theory of holographic imaging,” J. Opt. Soc. Am. 58, 1084–1091 (1968).
    [CrossRef]
  17. J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
    [CrossRef]
  18. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  19. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
    [CrossRef]
  20. D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1760 (1965).
    [CrossRef]
  21. F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Chap. 3.
  22. 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).
    [CrossRef]
  23. G. J. Buck and J. J. Gustincic, “Resolution limitations of a finite aperture,” IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
    [CrossRef]
  24. G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
    [CrossRef]
  25. M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
    [CrossRef]
  26. Actually, for finite kR, some of the first few eigenvalues can be slightly greater than 1. However, for sensible values of σ0/σw, Eq. (22) involves high-order eigenvalues μn< 1.
  27. P. De Santis and C. Palma, “Degrees of freedom of aberrated images,” Opt. Acta 23, 743–752 (1976).
    [CrossRef]
  28. V. Blažek, “Sampling theorem and the number of degrees of freedom of an image,” Opt. Commun. 11, 144–147 (1974).
    [CrossRef]
  29. H. Gamo, “Matrix treatment of partial coherence” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 189–332.
  30. F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
    [CrossRef]
  31. F. Gori and G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
    [CrossRef]
  32. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
    [CrossRef]
  33. The reader should be cautioned that the function G(and G˜ 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.
  36. 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).
    [CrossRef]
  37. J. P. Wild and et al., “The Culgoora radioheliograph,” Proc. Inst. Radio Electron. Eng. Aust. spec. iss. 28, 277–384 (1967).
  38. F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).
    [CrossRef]
  39. 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.
  40. F. Gori and G. Guattari, “Eigenfunction technique for point-like-element pupils,” Opt. Acta 22, 93–101 (1975).
    [CrossRef]

1980 (2)

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).
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
[CrossRef]

1978 (4)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

1976 (1)

P. De Santis and C. Palma, “Degrees of freedom of aberrated images,” Opt. Acta 23, 743–752 (1976).
[CrossRef]

1975 (1)

F. Gori and G. Guattari, “Eigenfunction technique for point-like-element pupils,” Opt. Acta 22, 93–101 (1975).
[CrossRef]

1974 (3)

1973 (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

1971 (1)

1970 (1)

D. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

1969 (3)

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
[CrossRef] [PubMed]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

1968 (2)

1967 (3)

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 5, 1739–1748 (1967).
[CrossRef]

G. J. Buck and J. J. Gustincic, “Resolution limitations of a finite aperture,” IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
[CrossRef]

J. P. Wild and et al., “The Culgoora radioheliograph,” Proc. Inst. Radio Electron. Eng. Aust. spec. iss. 28, 277–384 (1967).

1965 (2)

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).
[CrossRef]

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1760 (1965).
[CrossRef]

1961 (2)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 9.

Bendinelli, M.

Bertero, M.

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).
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
[CrossRef]

Blažek, V.

V. Blažek, “Sampling theorem and the number of degrees of freedom of an image,” Opt. Commun. 11, 144–147 (1974).
[CrossRef]

Buck, G. J.

G. J. Buck and J. J. Gustincic, “Resolution limitations of a finite aperture,” IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
[CrossRef]

Cesini, G.

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Cole, T. W.

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.

Consortini, A.

Dändliker, D.

D. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

De Mol, C.

M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
[CrossRef]

De Santis, P.

P. De Santis and C. Palma, “Degrees of freedom of aberrated images,” Opt. Acta 23, 743–752 (1976).
[CrossRef]

Devaney, A. J.

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

Frieden, B. R.

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 189–332.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

Gori, F.

F. Gori and G. Guattari, “Eigenfunction technique for point-like-element pupils,” Opt. Acta 22, 93–101 (1975).
[CrossRef]

F. Gori and G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).
[CrossRef]

Guattari, G.

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

F. Gori and G. Guattari, “Eigenfunction technique for point-like-element pupils,” Opt. Acta 22, 93–101 (1975).
[CrossRef]

F. Gori and G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).
[CrossRef]

Gustincic, J. J.

G. J. Buck and J. J. Gustincic, “Resolution limitations of a finite aperture,” IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
[CrossRef]

Hoenders, B. J.

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.
[CrossRef]

Landau, H. J.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Lee, W. H.

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.

Lohmann, A. W.

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).
[CrossRef]

A. W. Lohmann, “The space-bandwidth product,” (1966).

Lucarini, G.

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Lukosz, W.

Palma, C.

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

P. De Santis and C. Palma, “Degrees of freedom of aberrated images,” Opt. Acta 23, 743–752 (1976).
[CrossRef]

Paris, D. P.

Pasqualetti, F.

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).
[CrossRef]

Pollak, H. O.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Ronchi, L.

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).
[CrossRef]

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).
[CrossRef]

Shewell, J. R.

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1760 (1965).
[CrossRef]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1760 (1965).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 9.

Toraldo di Francia, G.

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).
[CrossRef]

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
[CrossRef] [PubMed]

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Chap. 3.

Viano, G. A.

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).
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
[CrossRef]

Weiss, K.

D. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Wild, J. P.

J. P. Wild and et al., “The Culgoora radioheliograph,” Proc. Inst. Radio Electron. Eng. Aust. spec. iss. 28, 277–384 (1967).

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).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (4)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1760 (1965).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. J. Buck and J. J. Gustincic, “Resolution limitations of a finite aperture,” IEEE Trans. Antennas Propag. AP-15, 376–381 (1967).
[CrossRef]

J. Math. Phys. (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Acta (5)

F. Gori and G. Guattari, “Eigenfunction technique for point-like-element pupils,” Opt. Acta 22, 93–101 (1975).
[CrossRef]

P. De Santis and C. Palma, “Degrees of freedom of aberrated images,” Opt. Acta 23, 743–752 (1976).
[CrossRef]

G. Cesini, G. Guattari, G. Lucarini, and C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “Resolution beyond the diffraction limit for regularized object restoration,” Opt. Acta 27, 307–320 (1980).
[CrossRef]

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).
[CrossRef]

Opt. Commun. (4)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

D. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

V. Blažek, “Sampling theorem and the number of degrees of freedom of an image,” Opt. Commun. 11, 144–147 (1974).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Optik (1)

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

Proc. Inst. Radio Electron. Eng. Aust. spec. iss. (1)

J. P. Wild and et al., “The Culgoora radioheliograph,” Proc. Inst. Radio Electron. Eng. Aust. spec. iss. 28, 277–384 (1967).

Proc. R. Soc. London Ser. A (1)

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).
[CrossRef]

Riv. Nuovo Cimento (1)

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

Other (14)

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.
[CrossRef]

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).

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.

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.

A. W. Lohmann, “The space-bandwidth product,” (1966).

F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Chap. 3.

The reader should be cautioned that the function G(and G˜ 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 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.

The similarity between the present analysis and that relating to the Culgoora radioheliograph is to be noted. See Refs. 36–39.

H. Gamo, “Matrix treatment of partial coherence” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 189–332.

Actually, for finite kR, some of the first few eigenvalues can be slightly greater than 1. However, for sensible values of σ0/σw, Eq. (22) involves high-order eigenvalues μn< 1.

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.

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Figures (6)

Fig. 1
Fig. 1

The optical system model. Two equal converging lenses (focal length f) image the plane x, y onto the plane x′, y′ through a thin-ring pupil.

Fig. 2
Fig. 2

Eigenvalues μn as a function of n for two values of kR.

Fig. 3
Fig. 3

Object function.

Fig. 4
Fig. 4

Computer-generated display of the principal part for the object of Fig. 3.

Fig. 5
Fig. 5

Pictorial representation of (a) the unilluminated scatterer and (b) its spectrum; (c) the scatterer illuminated by a plane wave propagating in the x direction and (d) the corresponding spectrum sampled along a circular array; and (e) the scatterer and (f) its sampled spectrum for a different illuminating plane wave.

Fig. 6
Fig. 6

Arrangement of the spectral samples taken during M = 12 successive scattering experiments.

Equations (73)

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n 2 ( r ) = 1 + 1 k 2 F ( r ) ,
u ( r ) = 1 4 π V F ( r ) u i ( r ) r - r exp ( i k r - r ) d 3 r ,
G ( r ) = F ( r ) u i ( r )
u ( s ) = Q G ˜ ( s / λ ) .
G ˜ ( ν ) = A G ( ρ ) exp ( = 2 π i ν · ρ ) d 2 ρ ,
Q = exp [ i ( π 4 + k ρ ¯ ) ( 1 8 π k ρ ¯ ) 1 / 2 ] ,
Φ n ( ρ , θ ) = C n J n ( k ρ ) exp ( i n θ ) μ n = π k R 2 [ J n 2 ( k R ) - J n - 1 ( k R ) J n + 1 ( k R ) ] } ,             n = 0 , ± 1 , ± 2 , ,
C n = { π R 2 [ J n 2 ( k R ) - J n - 1 ( k R ) J n + 1 ( k R ) ] } - 1 / 2 ,             n = 0 , ± 1 , ± 2 , .
N dof 2 k R + 1.
G ( ρ , θ ) = n - b n Φ n ( ρ , θ ) + G o ( ρ , θ ) ,             ρ R ,
b n = A G ( ρ ) Φ n * ( ρ ) d 2 ρ ,
G p ( ρ , θ ) = n - b n Φ n ( ρ , θ ) ,             ρ R ,
u ( θ ) = n - B n exp ( i n θ ) ,
B n = ( i ) - n Q ( 2 R μ n / k ) 1 / 2 b n .
ū ( θ ) = u ( θ ) + W ( θ ) .
ū ( θ ) = n - ( B n + W n ) exp ( i n θ ) ,
G ¯ ( ρ , θ ) = n - ( b n + w n ) Φ n ( ρ , θ ) + G o ( ρ , θ ) ,
W n = ( i ) - n Q ( 2 R μ n / k ) 1 / 2 w n .
G est ( ρ , θ ) = n - N N ( b n + w n ) Φ n ( ρ , θ ) .
= A G p ( ρ ) - G est ( ρ ) 2 d 2 ρ ,
G ( ρ ) G * ( ρ ) = σ 0 2 δ ( ρ - ρ ) , W ( θ ) W * ( θ ) = Ω w 2 δ ( θ - θ ) 2 π ,
( μ n ) 1 / 2 σ w / σ 0
σ w 2 = Ω w 2 k 2 R Q 2 .
μ n σ w / σ 0 .
N dof A S ( 0 ) ,
N dof π 2 k R .
Ψ n ( ρ , θ ) = M λ l - exp [ i ( n + l M ) ( θ + π 2 ) ] J n + l M ( k ρ ) ,             n = 0 , 1 , 2 , , M - 1 ,
μ ¯ n = π 2 k R l - [ J n + l M 2 ( k R ) - J n - 1 + l M ( k R ) J n + 1 + l M ( k R ) ] ,             n = 0 , 1 , 2 , , M - 1.
Ψ n ( ρ , θ ) D n Φ n ( ρ , θ ) ,             μ ¯ n μ n ,             0 n M - 1 2 Ψ n ( ρ , θ ) D n - M Φ n - M ( ρ , θ ) ,             μ ¯ n μ n - M ,             M - 1 2 < n M - 1 } ( 0 ρ R ) ,
t ( ν ) = K δ ( ν - a ) ,
S ( ρ ) = t ( ν ) exp ( 2 π i ν · ρ ) d 2 ν = 2 π a K J 0 ( 2 π a ρ ) ,
A Φ ( ρ ) S ( ρ - ρ ) d 2 ρ = μ Φ ( ρ ) ,
S ( ρ - ρ ) = n - μ n Φ n ( ρ ) Φ n * ( ρ ) ,
A Φ n ( ρ ) Φ m * ( ρ ) d 2 ρ = δ n m ,
2 π a K J 0 { 2 π a [ ρ 2 + ρ 2 - 2 ρ ρ cos ( θ - θ ) ] 1 / 2 } = n - μ n Φ n ( ρ , θ ) Φ n * ( ρ , θ ) .
J 0 [ ( x 2 + y 2 - 2 x y cos α ) 1 / 2 ] = n - J n ( x ) J n ( y ) exp ( i n α ) .
Φ n ( ρ , θ ) = C n J n ( 2 π a ρ ) exp ( i n θ ) ,
C n = [ A J n 2 ( 2 π a ρ ) ρ d ρ d θ ] - 1 / 2 .
μ n = 2 π a K C n 2 .
C n = { π R 2 [ J n 2 ( 2 π a R ) - J n - 1 ( 2 π a R ) J n + 1 ( 2 π a R ) ] } - 1 / 2 ,
μ n = 2 π 2 a K R [ J n 2 ( 2 π a R ) - J n - 1 ( 2 π a R ) J n + 1 ( 2 π a R ) ] .
μ n + 2 μ n ,
Φ ˜ n ( ν , φ ) = C n exp ( i n φ ) ( i ) n δ ( ν - a ) 2 π a .
Φ ˜ n ( A ) ( ν , φ ) = Φ ˜ n ( ν , φ ) * R J 1 ( 2 π R ν ) ν ,
Φ ˜ n ( A ) ( ν , φ ) = C n R ( i ) n 2 π a 0 2 π 0 exp ( i n α ) δ ( ξ - a ) × J 1 { 2 π R [ ν 2 + ξ 2 - 2 ν ξ cos ( α - φ ) ] 1 / 2 } [ ν 2 + ξ 2 - 2 ν ξ cos ( α - φ ) ] 1 / 2 ξ d ξ d α ,
Φ ˜ n ( A ) ( ν , φ ) = C n R exp ( i n φ ) 2 π ( i ) n 0 2 π exp ( i n β ) × J 1 [ 2 π R ( ν 2 + a 2 - 2 ν a cos β ) 1 / 2 ( ν 2 + a 2 - 2 ν a cos β ) 1 / 2 d β .
0 R η J 0 ( w η ) d η = R J 1 ( w R ) w .
Φ ˜ n ( A ) ( ν , φ ) = C n exp ( i n φ ) ( i ) n - m 0 2 π 0 R η J m ( 2 π ν η ) J m ( 2 π a η ) × exp [ i ( n + m ) β ] d η d β = 2 π C n exp ( i n φ ) ( i ) n × 0 R η J - n ( 2 π ν η ) J - n ( 2 π a η ) d η .
Φ ˜ n ( A ) ( a , φ ) = exp ( i n φ ) ( i ) n C n .
exp ( i n φ ) ( i ) n C n K δ ( ν - a )
J 2 n ( x ) = ( 2 π x ) 1 / 2 ( - 1 ) n cos ( x - π 4 ) ,
J 2 n + 1 ( x ) = ( 2 π x ) 1 / 2 ( - 1 ) n sin ( x - π 4 ) ,
C n = ( π a R ) 1 / 2
K = 1 2 R .
μ n = π 2 k R [ J n 2 ( k R ) - J n - 1 ( k R ) J n + 1 ( k R ) ] .
n - μ n = A S ( 0 ) = π 2 k R .
Φ n ( ρ , θ ) = C n J n ( k ρ ) exp ( i n θ ) ,             n = 0 , ± 1 , ± 2 , ,
C n = { π R 2 [ J n 2 ( k R ) - J n - 1 ( k R ) J n + 1 ( k R ) ] } - 1 / 2 ,             n = 0 , ± 1 , ± 2 , .
Φ ˜ n ( A ) ( 1 λ , θ ) = exp ( i n θ ) ( i ) - n C n .
γ p = m = 0 M - 1 c m exp ( i m p Δ φ ) ,             p = 0 , 1 , , M - 1 ,
Δ φ = 2 π M .
c m = 1 M p = 0 M - 1 γ p exp ( - i m p Δ φ ) ,             p = 0 , 1 , , M - 1.
c n = δ n m ,             n , m = 0 , 1 , , M - 1 ,
Ψ ˜ n ( ν , φ ) = δ ( ν - a ) p = 0 M - 1 exp ( i n p Δ φ ) δ ( φ - p Δ φ ) = δ ( ν - a ) exp ( i n φ ) p = 0 M - 1 δ ( φ - p Δ φ ) ,
p = 0 M - 1 δ ( φ - p Δ φ ) = M 2 π l - exp ( i l M φ ) .
Ψ n ( ρ , θ ) = 0 2 π 0 Ψ ˜ n ( ν , φ ) exp [ 2 π i ν ρ cos ( θ - φ ) ] ν d ν d φ = M a 2 π l - 0 2 π exp [ i ( n + l M ) φ + 2 π a ρ i cos ( θ - φ ) ] d φ .
Ψ n ( ρ , θ ) = M a - l exp [ i ( n + l M ) ( ϑ + π 2 ) ] J n + l M ( 2 π a ρ ) .
t ( ν , φ ) = K δ ( ν - a ) K p = 0 M - 1 δ ( φ - p Δ φ ) ,
Ψ n ( A ) ( ν , φ ) = K K a M q n δ ( ν - a ) exp ( i n φ ) × p = 0 M - 1 δ ( φ - p Δ φ ) ,
q n = l - 1 C n + l M 2 ,
μ ¯ n = K K a M q n .
K = 2 π M .
μ ¯ n = π 2 k R l - [ J n + l M 2 ( k R ) - J n - 1 + l M ( k R ) J n + 1 + l M ( k R ) ] ,             n = 0 , 1 , , M - 1.