Abstract

The first- and second-order space-time correlation functions of the optical field and the intensity, respectively, are calculated for the case of a rotating symmetrical curved surface with Gaussian roughness, scattering laser light into the far-field zone from a small illuminated spot. The incident beam is either collimated or spherical. The diameter of the illuminated area is small compared with the radii of curvature. The mean surface is approximated by a torus with two principal radii of curvature. The sphere, the cone, and the flat surface are special cases. Optical phase variations linear and quadratic in the surface coordinates are included in the calculation. The surface is assumed to be very rough. It is found that the correlation function of the intensity depends on the radius of the incident beam, the local surface velocity, and the radii of curvature in the illuminated region. Curvature decreases the decay time of the correlation functions. This is confirmed by experiments carried out with the flat disk, the sphere, and the cone. The temporal cross-correlation function of the outputs of two detectors in the speckle field obtained with an incident spherical wave is very sensitive to normal displacements of the surface.

© 1976 Optical Society of America

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References

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  1. V. A. Fock, "Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965).
  2. T. Hagfors, "Relations Between Rough Surfaces and Their Scattering Properties as Applied to Radar Astronomy," in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), p. 187.
  3. N. George, "Wavelength Sensitivity of Speckle from an Extended Object," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.
  4. D. L. Fried, "Statistics of Scattering from a Moderately Rough Surface," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb-24–26, 1976.
  5. J. E. Pearson, Effects of Speckle in Adaptive Optical Systems, 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.
  6. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Wavesfrom Rough Surfaces (MacMillan, New York, 1963).
  7. R. J. Glauber, "Coherent and Incoherent States of the Radiation Field," Phys. Rev. 131, 2766–2788 (1963).
  8. B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).
  9. J. W. Goodman, "Statistical Properties of Laser Speckle Patterns," in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).
  10. Reference 6, p. 18.
  11. Reference 6, p. 26, Eq. (1).
  12. I. S. Sokolnikoff, TensorAnalysis (Wiley, New York, 1951).
  13. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 499.
  14. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958).
  15. Reference 14, p. 129.
  16. E. Madelung, Die Mathenmatischen Hilfsmittel des Physikers (Springer, Berlin, 1957), p. 80.
  17. Reference 6, p. 27.
  18. Reference 6, p. 88, Eq. (59).
  19. R. Pitlak, "Laser Spot Size for Single Element Lenses, Electro-Optical Systems Design, Sept. 1975, p. 30–31.
  20. W. Mendenhall and R. L. Schaefer, Mathematical Statistics with Applications (Duxbury, North Scituate, Mass., 1973).
  21. Reference 13, p. 446.

13, Reference

Reference 13, p. 446.

14, Reference

Reference 14, p. 129.

6, Reference

Reference 6, p. 27.

Reference 6, p. 26, Eq. (1).

Reference 6, p. 18.

Reference 6, p. 88, Eq. (59).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Wavesfrom Rough Surfaces (MacMillan, New York, 1963).

Bendat, J. S.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958).

Bertolotti, M.

B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 499.

Crosignani, B.

B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Di Porto, P.

B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Fock, V. A.

V. A. Fock, "Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965).

Fried, D. L.

D. L. Fried, "Statistics of Scattering from a Moderately Rough Surface," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb-24–26, 1976.

George, N.

N. George, "Wavelength Sensitivity of Speckle from an Extended Object," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.

Glauber, R. J.

R. J. Glauber, "Coherent and Incoherent States of the Radiation Field," Phys. Rev. 131, 2766–2788 (1963).

Goodman, J. W.

J. W. Goodman, "Statistical Properties of Laser Speckle Patterns," in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).

Hagfors, T.

T. Hagfors, "Relations Between Rough Surfaces and Their Scattering Properties as Applied to Radar Astronomy," in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), p. 187.

Madelung, E.

E. Madelung, Die Mathenmatischen Hilfsmittel des Physikers (Springer, Berlin, 1957), p. 80.

Mendenhall, W.

W. Mendenhall and R. L. Schaefer, Mathematical Statistics with Applications (Duxbury, North Scituate, Mass., 1973).

Pearson, J. E.

J. E. Pearson, Effects of Speckle in Adaptive Optical Systems, 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.

Pitlak, R.

R. Pitlak, "Laser Spot Size for Single Element Lenses, Electro-Optical Systems Design, Sept. 1975, p. 30–31.

Schaefer, R. L.

W. Mendenhall and R. L. Schaefer, Mathematical Statistics with Applications (Duxbury, North Scituate, Mass., 1973).

Sokolnikoff, I. S.

I. S. Sokolnikoff, TensorAnalysis (Wiley, New York, 1951).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Wavesfrom Rough Surfaces (MacMillan, New York, 1963).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 499.

Other

V. A. Fock, "Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965).

T. Hagfors, "Relations Between Rough Surfaces and Their Scattering Properties as Applied to Radar Astronomy," in Radar Astronomy, edited by J. V. Evans and T. Hagfors (McGraw-Hill, New York, 1968), p. 187.

N. George, "Wavelength Sensitivity of Speckle from an Extended Object," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.

D. L. Fried, "Statistics of Scattering from a Moderately Rough Surface," 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb-24–26, 1976.

J. E. Pearson, Effects of Speckle in Adaptive Optical Systems, 1976 Asilomar Conference on Speckle Phenomena in Optics, Microwaves, and Acoustics, sponsored by the Optical Society of America, Pacific Grove, Calif., Feb. 24–26, 1976.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Wavesfrom Rough Surfaces (MacMillan, New York, 1963).

R. J. Glauber, "Coherent and Incoherent States of the Radiation Field," Phys. Rev. 131, 2766–2788 (1963).

B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

J. W. Goodman, "Statistical Properties of Laser Speckle Patterns," in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).

Reference 6, p. 18.

Reference 6, p. 26, Eq. (1).

I. S. Sokolnikoff, TensorAnalysis (Wiley, New York, 1951).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 499.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958).

Reference 14, p. 129.

E. Madelung, Die Mathenmatischen Hilfsmittel des Physikers (Springer, Berlin, 1957), p. 80.

Reference 6, p. 27.

Reference 6, p. 88, Eq. (59).

R. Pitlak, "Laser Spot Size for Single Element Lenses, Electro-Optical Systems Design, Sept. 1975, p. 30–31.

W. Mendenhall and R. L. Schaefer, Mathematical Statistics with Applications (Duxbury, North Scituate, Mass., 1973).

Reference 13, p. 446.

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