Abstract

The dynamic statistical properties of laser speckle produced in the far-field diffraction region from a diffuse object moving longitudinally along the optical axis under illumination of a Gaussian beam are investigated theoretically and experimentally. Although the spatial structure of speckle patterns varies in time owing to the longitudinal motion of the object, the time-varying speckle intensity detected at the center of the far-field diffraction plane is found to follow a statistically stationary variation. The autocorrelation function and power spectral density of the speckle intensity variations are studied with relation to the illuminating condition of the Gaussian beam.

© 1980 Optical Society of America

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  1. V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).
  2. V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).
  3. L. E. Estes, L. M. Narducci, and R. A. Tuft, “Scattering of light from a rotating ground glass,” J. Opt. Soc. Am. 61, 1301–1306 (1971).
    [Crossref]
  4. N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
    [Crossref]
  5. E. Jakeman, “The effect of wavefront curvature on the coherent properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
    [Crossref]
  6. I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).
  7. E. Jakeman and J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel region by a random-phase screen in uniform motion,” J. Phys. A 9, 785–797 (1976).
    [Crossref]
  8. P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D 9, 1399–1409 (1976).
    [Crossref]
  9. I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
    [Crossref]
  10. N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
    [Crossref]
  11. N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).
  12. N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossings of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
    [Crossref]
  13. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [Crossref]
  14. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, edited by E. Wolf (North-Holland, Amsterdam, 1976), pp. 1–46.
  15. N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
    [Crossref]
  16. A. Lohmann and G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt Commun. 17, 47–51 (1976).
    [Crossref]
  17. A. Lohmann and G. P. Weigelt, “Speckle methods for the display of motion paths,” J. Opt. Soc. Am. 66, 1271–1274 (1976).
    [Crossref]
  18. Y. Dzialowski and M. May, “Correlation of speckle patterns generated by laser point source-illuminated diffusers,” Opt. Commun. 16, 334–339 (1976).
    [Crossref]
  19. Y. Dzialowski and M. May, “Speckle pattern interferometer,” Opt. Commun. 18, 321–325 (1976).
    [Crossref]
  20. N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field,” Appl. Opt. 17, 3785–3793 (1978).
    [Crossref] [PubMed]
  21. N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field and its application to vibration analysis,” Proceedings of the 1978 International Optical Computing Conference, London, Digest of Papers,pp. 126–131 (unpublished).
  22. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.
  23. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958) pp. 370–414.

1980 (2)

N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
[Crossref]

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossings of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[Crossref]

1979 (1)

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

1978 (1)

1977 (1)

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[Crossref]

1976 (6)

E. Jakeman and J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel region by a random-phase screen in uniform motion,” J. Phys. A 9, 785–797 (1976).
[Crossref]

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D 9, 1399–1409 (1976).
[Crossref]

A. Lohmann and G. P. Weigelt, “Speckle methods for the display of motion paths,” J. Opt. Soc. Am. 66, 1271–1274 (1976).
[Crossref]

A. Lohmann and G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt Commun. 17, 47–51 (1976).
[Crossref]

Y. Dzialowski and M. May, “Correlation of speckle patterns generated by laser point source-illuminated diffusers,” Opt. Commun. 16, 334–339 (1976).
[Crossref]

Y. Dzialowski and M. May, “Speckle pattern interferometer,” Opt. Commun. 18, 321–325 (1976).
[Crossref]

1975 (2)

E. Jakeman, “The effect of wavefront curvature on the coherent properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

1974 (1)

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

1971 (1)

1970 (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).

1969 (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).

Anisimov, V. V.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).

Asakura, T.

N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
[Crossref]

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossings of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[Crossref]

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field,” Appl. Opt. 17, 3785–3793 (1978).
[Crossref] [PubMed]

N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field and its application to vibration analysis,” Proceedings of the 1978 International Optical Computing Conference, London, Digest of Papers,pp. 126–131 (unpublished).

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).

Bendat, J. S.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958) pp. 370–414.

Dainty, J. C.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, edited by E. Wolf (North-Holland, Amsterdam, 1976), pp. 1–46.

Dzialowski, Y.

Y. Dzialowski and M. May, “Correlation of speckle patterns generated by laser point source-illuminated diffusers,” Opt. Commun. 16, 334–339 (1976).
[Crossref]

Y. Dzialowski and M. May, “Speckle pattern interferometer,” Opt. Commun. 18, 321–325 (1976).
[Crossref]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.

Estes, L. E.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

Iwai, T.

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossings of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[Crossref]

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).

Jakeman, E.

E. Jakeman and J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel region by a random-phase screen in uniform motion,” J. Phys. A 9, 785–797 (1976).
[Crossref]

E. Jakeman, “The effect of wavefront curvature on the coherent properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

Komatsu, S.

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[Crossref]

I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Kozel, S. M.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).

Lohmann, A.

A. Lohmann and G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt Commun. 17, 47–51 (1976).
[Crossref]

A. Lohmann and G. P. Weigelt, “Speckle methods for the display of motion paths,” J. Opt. Soc. Am. 66, 1271–1274 (1976).
[Crossref]

Lokshin, G. R.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.

May, M.

Y. Dzialowski and M. May, “Speckle pattern interferometer,” Opt. Commun. 18, 321–325 (1976).
[Crossref]

Y. Dzialowski and M. May, “Correlation of speckle patterns generated by laser point source-illuminated diffusers,” Opt. Commun. 16, 334–339 (1976).
[Crossref]

McWhirter, J. G.

E. Jakeman and J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel region by a random-phase screen in uniform motion,” J. Phys. A 9, 785–797 (1976).
[Crossref]

Narducci, L. M.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.

Pusey, P. N.

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D 9, 1399–1409 (1976).
[Crossref]

Saito, H.

I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Sutanto, T.

N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
[Crossref]

Takai, N.

N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
[Crossref]

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossings of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[Crossref]

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field,” Appl. Opt. 17, 3785–3793 (1978).
[Crossref] [PubMed]

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field and its application to vibration analysis,” Proceedings of the 1978 International Optical Computing Conference, London, Digest of Papers,pp. 126–131 (unpublished).

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.

Tuft, R. A.

Ushizaka, T.

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).

Weigelt, G. P.

A. Lohmann and G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt Commun. 17, 47–51 (1976).
[Crossref]

A. Lohmann and G. P. Weigelt, “Speckle methods for the display of motion paths,” J. Opt. Soc. Am. 66, 1271–1274 (1976).
[Crossref]

Yamaguchi, I.

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[Crossref]

I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Phys. A (2)

E. Jakeman, “The effect of wavefront curvature on the coherent properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

E. Jakeman and J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel region by a random-phase screen in uniform motion,” J. Phys. A 9, 785–797 (1976).
[Crossref]

J. Phys. D (1)

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D 9, 1399–1409 (1976).
[Crossref]

Jpn. J. Appl. Phys. (2)

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

N. Takai, T. Sutanto, and T. Asakura, “Laser speckles produced by the longitudinal motion of a diffuse object under Gaussian beam illumination,” Jpn. J. Appl. Phys. 19, L75–L78 (1980).
[Crossref]

Jpn. J. Appl. Phys. Suppl. (1)

I. Yamaguchi, S. Komatsu, and H. Saito, “Dynamics of speckles produced by a moving object and its applications,” Jpn. J. Appl. Phys. Suppl. 14-1, 301–306 (1975).

Opt Commun. (1)

A. Lohmann and G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt Commun. 17, 47–51 (1976).
[Crossref]

Opt. Acta (1)

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[Crossref]

Opt. Commun. (3)

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Velocity measurement of the diffuse object based on time-differentiated speckle intensity fluctuations,” Opt. Commun. 30, 287–292 (1979).
[Crossref]

Y. Dzialowski and M. May, “Correlation of speckle patterns generated by laser point source-illuminated diffusers,” Opt. Commun. 16, 334–339 (1976).
[Crossref]

Y. Dzialowski and M. May, “Speckle pattern interferometer,” Opt. Commun. 18, 321–325 (1976).
[Crossref]

Opt. Spectrosc. (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “Space-time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. 27, 258–262 (1969).

Radio Eng. Electron. Phys. (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, “The spectral properties of the random intensity field produced through the scattering of coherent radiation at a moving diffuse surface,” Radio Eng. Electron. Phys. 15, 458–462 (1970).

Other (6)

N. Takai, T. Iwai, T. Ushizaka, and T. Asakura, “Zero -crossing study on dynamic properties of speckles,” J. Opt. (to be published).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. XIV, edited by E. Wolf (North-Holland, Amsterdam, 1976), pp. 1–46.

N. Takai and T. Asakura, “Dynamic statistical properties of vibrating laser speckle in the diffraction field and its application to vibration analysis,” Proceedings of the 1978 International Optical Computing Conference, London, Digest of Papers,pp. 126–131 (unpublished).

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958) pp. 370–414.

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Figures (10)

FIG. 1
FIG. 1

Schematic diagram for the formation of time-varying speckles in the far-field diffraction region from a diffuse object moving longitudinally along the optical axis under illumination of a Gaussian beam.

FIG. 2
FIG. 2

Electric signal-analyzing system for counting the number of peaks of time-varying speckle intensity variations.

FIG. 3
FIG. 3

Experimental setup for investigating time-varying speckle intensity variations produced from longitudinal object motion.

FIG. 4
FIG. 4

Photographic records of the speckle patterns obtained by changing the distance z of a diffuse object measured from the waist position of the illuminating Gaussian beam. In this case, the Gaussian beam used for illumination is produced by a lens of focal length f = 55 mm and has a waist width of w0 = 12 μm.

FIG. 5
FIG. 5

Photographic records of the speckle patterns obtained by changing the distance z of a diffuse object. In this case, the Gaussian beam is produced by a lens of f = 100 mm and has a waist width of w0 = 22 μm.

FIG. 6
FIG. 6

Intensity variations of time-varying speckles produced by the longitudinal motion of a diffuse object and detected at the center of the far-field diffraction plane under various waist widths of the Gaussian beam used for illumination.

FIG. 7
FIG. 7

Accumulated number of peaks of time-varying speckle intensity variations obtained under various waist widths of the Gaussian beam.

FIG. 8
FIG. 8

Dependence of the number of peaks per mm of time-varying speckle intensity variations on the waist width of the illuminating Gaussian beam given by w 0 2.

FIG. 9
FIG. 9

Normalized autocorrelation function γI(0; τ) of time-varying speckle intensity variations obtained experimentally for various waist widths w0 of the illuminating Gaussian beam.

FIG. 10
FIG. 10

Correlation length τc of time-varying speckle intensity variations as a function of the waist width of the illuminating Gaussian beam given by w 0 2.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

V ( x , t ) = E 0 ( ξ , z ) exp [ i ϕ ( ξ ) ] exp ( i k ξ x R ( t ) ) d ξ ,
z ( t ) = υ t
R ( t ) = R 0 υ t ,
E 0 ( ξ , z ) = w 0 w ( z ) exp ( i k z ) exp ( | ξ | 2 w 2 ( z ) ) exp ( i k | ξ | 2 2 ρ ( z ) ) ,
w ( z ) = w 0 ( 1 + z 2 / a 2 ) 1 / 2
ρ ( z ) = z ( 1 + a 2 / z 2 ) ,
a = π w 0 2 / λ
Γ ( x ; t 1 , t 2 ) = V ( x , t 1 ) V * ( x , t 2 )
Γ ( x ; t 1 , t 2 ) = E 0 [ ξ 1 , z 1 ( t 1 ) ] E 0 * [ ξ 1 , z 1 ( t 1 ) ] × exp { i [ ϕ ( ξ 1 ) ϕ ( ξ 2 ) ] } exp × [ i k ( ξ 1 R ( t 1 ) ξ 2 R ( t 2 ) ) x ] d ξ 1 , d ξ 2 .
Γ ( x ; t 1 , t 2 ) = E 0 ( ξ 1 , η 1 , z 1 ( t 1 ) ) E 0 * ( ξ 2 , η 2 , z 2 ( t 2 ) ) × exp { i [ ϕ ( ξ 1 , η 1 ) ϕ ( ξ 2 , η 2 ) ] } × exp [ i k ( ξ 1 x + η 1 y R 0 z 1 ( t 1 ) ξ 2 x + η 2 y R 0 z 2 ( t 2 ) ) ] × d ξ 1 d η 1 d ξ 2 d η 2 ,
exp { i [ ϕ ( ξ 1 , η 1 ) ϕ ( ξ 2 , η 2 ) ] } Δ S δ ( ξ 1 ξ 2 ) δ ( η 1 η 2 ) ,
Γ ( x , z 1 , z 2 ) = Δ S E 0 ( ξ , η , z 1 ) E 0 * ( ξ , η , z 2 ) × exp [ i k ( 1 R 0 z 1 1 R 0 z 2 ) ( ξ x + η y ) ] d ξ d η ,
E 0 ( ξ , η , z 1 ) E 0 * ( ξ , η , z 2 ) = w 0 2 w ( z 1 ) w ( z 2 ) exp [ i k ( z 1 z 2 ) ] × exp [ ( 1 w 2 ( z 1 ) + 1 w 2 ( z 2 ) ) ( ξ 2 + η 2 ) ] × exp [ i k 2 ( 1 ρ ( z 1 ) 1 ρ ( z 2 ) ) ( ξ 2 + η 2 ) ] .
Δ I ( x , t ) = I ( x , t ) I ( x , t )
Δ I ( x , t 1 ) Δ I ( x , t 2 ) = Δ I ( x , z 1 ) Δ I ( x , z 2 ) = I ( x , z 1 ) I ( x , z 2 ) | Γ ( x ; z 1 , z 2 ) | 2 .
| Γ ( x ; z 1 , z 2 ) | 2 = w 0 4 Δ S 2 w 2 ( z 1 ) w 2 ( z 2 ) × | exp [ ( 1 w 2 ( z 1 ) + 1 w 2 ( z 2 ) ) ( ξ 2 + η 2 ) ] × exp { i k 2 [ ( 1 ρ ( z 1 ) 1 ρ ( z 2 ) ) ( ξ 2 + η 2 ) ] } × exp [ i k ( 1 R 0 z 1 1 R 0 z 2 ) ) ( ξ x + n y ) ] d ξ d η | 2 .
exp ( i α ξ 2 ) exp ( β ξ 2 ) exp ( i p ξ x ) d ξ = π ( α 2 + β 2 ) 1 / 4 exp ( β p 2 x 2 4 ( α 2 + β 2 ) ) × exp { i [ α p 2 x 2 4 ( α 2 + β 2 ) 1 2 tan 1 ( α β ) ] } ,
α = k 2 ( 1 ρ ( z 1 ) 1 ρ ( z 2 ) ) ,
β = 1 w 2 ( z 1 ) + 1 w 2 ( z 2 ) ,
p = k ( 1 R 0 z 1 1 R 0 z 2 ) .
| Γ ( x ; z 1 , z 2 ) | 2 = π 2 w 0 4 Δ S 2 w 2 ( z 1 ) w 2 ( z 2 ) ( α 2 + β 2 ) × exp ( β p 2 ( x 2 + y 2 ) 2 ( α 2 + β 2 ) ) .
α = π ( z 2 z 1 ) ( z 1 z 2 a 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
β = π a ( z 1 2 + z 2 2 + 2 a 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
α 2 + β 2 = ( π λ ) 2 ( z 1 z 2 ) 2 + 4 a 2 ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
w 2 ( z 1 ) w 2 ( z 2 ) = ( w 0 2 / a 4 ) ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) .
| Γ ( x ; z 1 , z 2 ) | 2 = λ 2 a 4 Δ S 2 [ ( z 1 z 2 ) 2 + 4 a 2 ] × exp ( λ a ( z 1 2 + z 2 2 + 2 a 2 ) p 2 2 π [ ( z 1 z 2 ) 2 + 4 a 2 ] ( x 2 + y 2 ) ) .
γ ( 0 ; z 1 , z 2 ) = | Γ ( 0 ; z 1 , z 2 ) | 2 / | Γ ( 0 ; 0 , 0 ) | 2 .
γ ( 0 ; z 1 , z 2 ) = γ ( 0 ; z 1 z 2 ) = 4 a 2 ( z 1 z 2 ) 2 + 4 a 2 .
γ ( 0 ; τ ) = 4 a 2 υ 2 τ 2 + 4 a 2 = 1 1 + ( τ / τ c ) 2 ,
τ c = 2 a / υ = 2 π w 0 2 / υ λ .
Φ ( f ) = γ ( 0 ; τ ) exp ( 2 π i f τ ) d τ = π 2 a exp ( | f | / Δ f ) ,
Δ f = υ / 4 π a = λ υ / 4 π 2 w 0 2 .
N 0 = 1 2 π ( γ İ ( 0 ; 0 ) γ İ ( 0 ; 0 ) ) 1 / 2
N 0 = ( f 4 Φ ( f ) d f f 2 Φ ( f ) d f ) 1 / 2 ,
N 0 = ( 0 f 4 exp ( f / Δ f ) d f 0 f 2 exp ( f / Δ f ) d f ) 1 / 2 .
0 x b exp ( c x ) d x = Γ ( b + 1 ) / c b + 1
N 0 = 2 3 Δ f = 3 λ υ / 2 π 2 w 0 2 .
h = ( 3 λ / 2 π 2 ) υ = 0.55 × 10 4 mm 2 / s ,
s = 2 π / λ υ = 9 . 93 × 1 0 3 s / mm 2 ,