Abstract

The inverse source problem for one- and two-dimensional coherently radiating sources is studied within the framework of generalized holographic imaging. It is shown that the image field produced from a generalized hologram of a one-dimensional source contains a complete description of the source. For such sources, structure larger than one-half wavelength is accurately imaged, whereas information on smaller scales can be obtained from a linear operator applied to the image field. The image fields of two-dimensional sources are shown to contain only partial information about these sources. It is shown, however, that perfect imaging is possible for the class of sources satisfying the homogeneous Helmholtz equation. This class is shown to be identical with the class of <i>minimum energy sources</i> recently encountered in connection with the inverse problem for three-dimensional sources. For noisy images there are serious practical limitations on source detail obtainable from any imaging system; the generalized hologr m achieves near-ideal performance when the image is noisy.

© 1982 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. P. Porter, "Diffraction limited scalar image formation with holograms of arbitrary shape," J. Opt. Soc. Am. 60, 1051–1059 (1970).
  2. R. P. Porter and W. C. Schwab, "Optimum imaging. Closed holograms and optical channel capacity," J. Opt. Soc. Am. 61, 789–796 (1971).
  3. E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).
  4. R. P. Porter, "Determination of structure of weak scatterers from holographic images," Opt. Commun. 39, 362–364 (1981).
  5. R. P. Porter and A. J. Devaney, "Holography and the inverse source problem," J. Opt. Soc. Am. 72, 327–330 (1982).
  6. C. W. Barnes, "Object restoration in a diffraction limited imaging system," J. Opt. Soc. Am. 56, 575–578 (1966).
  7. N. J. Bershad, "Resolution, optical channel capacity and information theory," J. Opt. Soc. Am. 59, 157–162 (1969).
  8. G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).
  9. A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).
  10. N. Bojarski, 16 Pine Valley Lane, Newport Beach, California 92660, "A wave equation for radiating source distributions" (personal communication).
  11. R. P. Porter, "Image formation with arbitrary holographic type surfaces," Phys. Lett. A 29, 193–194 (1969).
  12. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43–64 (1961).
  13. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-II," Bell Syst. Tech. J. 40, 65–84 (1961).
  14. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III," Bell Syst. Tech. J. 41, 1295–1336 (1962).
  15. D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV," Bell Syst. Tech. J. 43, 3009–3057 (1964).
  16. D. Slepian, "Some asymptotic expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99–140 (1965).
  17. C. K. Rushforth and R. W. Harris, "Restoration, resolution and noise," J. Opt. Soc. Am. 58, 539–545 (1968).
  18. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  19. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6.
  20. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964), pp. 116–123.
  21. N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).
  22. N. N. Bojarski, "Inverse scattering," Naval Air Systems Command rep., contract N00019-73-C-0312 (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6.

1982

1981

R. P. Porter, "Determination of structure of weak scatterers from holographic images," Opt. Commun. 39, 362–364 (1981).

G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).

1977

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

1973

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

1971

1970

1969

N. J. Bershad, "Resolution, optical channel capacity and information theory," J. Opt. Soc. Am. 59, 157–162 (1969).

R. P. Porter, "Image formation with arbitrary holographic type surfaces," Phys. Lett. A 29, 193–194 (1969).

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

1968

1966

1965

D. Slepian, "Some asymptotic expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99–140 (1965).

1964

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV," Bell Syst. Tech. J. 43, 3009–3057 (1964).

1962

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III," Bell Syst. Tech. J. 41, 1295–1336 (1962).

1961

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).

Barnes, C. W.

Bershad, N. J.

Bleistein, N.

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

Bojarski, N.

N. Bojarski, 16 Pine Valley Lane, Newport Beach, California 92660, "A wave equation for radiating source distributions" (personal communication).

Bojarski, N. N.

N. N. Bojarski, "Inverse scattering," Naval Air Systems Command rep., contract N00019-73-C-0312 (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6.

Cohen, J.

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6.

Devaney, A. J.

R. P. Porter and A. J. Devaney, "Holography and the inverse source problem," J. Opt. Soc. Am. 72, 327–330 (1982).

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Harris, R. W.

Haywood, R. A.

G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).

Landau, H. J.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III," Bell Syst. Tech. J. 41, 1295–1336 (1962).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-II," Bell Syst. Tech. J. 40, 65–84 (1961).

Pollak, H. O.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III," Bell Syst. Tech. J. 41, 1295–1336 (1962).

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).

Porter, R. P.

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6.

Rope, E. L.

G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).

Rushforth, C. K.

Schwab, W. C.

Slepian, D.

D. Slepian, "Some asymptotic expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99–140 (1965).

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV," Bell Syst. Tech. J. 43, 3009–3057 (1964).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43–64 (1961).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964), pp. 116–123.

Tricoles, G.

G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).

Wolf, E.

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

Bell Syst. Tech. J.

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).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III," Bell Syst. Tech. J. 41, 1295–1336 (1962).

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV," Bell Syst. Tech. J. 43, 3009–3057 (1964).

IEEE Trans. Antennas Propag.

G. Tricoles, E. L. Rope, and R. A. Haywood, "Improved resolution in microwave holographic images," IEEE Trans. Antennas Propag. AP-29, 320–326 (1981).

J. Math. Phys.

D. Slepian, "Some asymptotic expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99–140 (1965).

N. Bleistein and J. Cohen, "Nonuniqueness in the inverse source problem in acoustics and electromagnetics," J. Math. Phys. 18, 194–201 (1977).

J. Opt. Soc. Am.

Opt. Commun.

E. Wolf, "Three-dimensional structure determination of semitransparent objects from holographic data," Opt. Commun. 1, 153–156 (1969).

R. P. Porter, "Determination of structure of weak scatterers from holographic images," Opt. Commun. 39, 362–364 (1981).

Phys. Lett. A

R. P. Porter, "Image formation with arbitrary holographic type surfaces," Phys. Lett. A 29, 193–194 (1969).

Phys. Rev. D

A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).

Other

N. Bojarski, 16 Pine Valley Lane, Newport Beach, California 92660, "A wave equation for radiating source distributions" (personal communication).

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Chap. 6.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964), pp. 116–123.

N. N. Bojarski, "Inverse scattering," Naval Air Systems Command rep., contract N00019-73-C-0312 (Naval Air Systems Command, Washington, D.C., 1973), Sec. 11, pp. 3–6.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.