Abstract

The dynamic statistical properties of laser speckle produced in the far-field diffraction region by a diffuse object moving in an arbitrary direction of three-dimensional space under illumination of a Gaussian beam are investigated theoretically and experimentally. It is found that the time-varying speckle intensity detected at the center of the far-field diffraction plane is a stationary random process with respect to time. The dependence of the autocorrelation function of the time-varying speckle-intensity fluctuation on both lateral and longitudinal components of the object velocity is studied in some detail by evaluating numerically the resultant equation of the time-varying speckle-intensity correlation. To confirm the theoretical results, an experiment has been performed. Good agreement between the theoretical and experimental results is obtained for the autocorrelation function of the time-varying speckle-intensity fluctuation that is due to three-dimensional translation of the diffuse object.

© 1982 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 application of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
    [Crossref]
  10. N. Takai, 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]
  11. N. Takai, Sutanto, and T. Asakura, “Dynamic statistical properties of laser speckle due to longitudinal motion of a diffuse object under Gaussian beam illumination,” J. Opt. Soc. Am. 70, 827–834 (1980).
    [Crossref]
  12. J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).
  13. T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [Crossref]
  14. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. I, pp. 23–24.
  15. E. R. Pike, “Photon correlation velocimetry,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins and E. R. Pike, eds. (Plenum, New York, 1977), pp. 246–343.
  16. T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
    [Crossref]

1981 (2)

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[Crossref]

T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
[Crossref]

1980 (3)

N. Takai, 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, Sutanto, and T. Asakura, “Dynamic statistical properties of laser speckle due to longitudinal motion of a diffuse object under Gaussian beam illumination,” J. Opt. Soc. Am. 70, 827–834 (1980).
[Crossref]

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

1977 (1)

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

1976 (2)

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]

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.

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[Crossref]

T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
[Crossref]

N. Takai, 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, Sutanto, and T. Asakura, “Dynamic statistical properties of laser speckle due to longitudinal motion of a diffuse object under Gaussian beam illumination,” J. Opt. Soc. Am. 70, 827–834 (1980).
[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.

Iwai, T.

T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
[Crossref]

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 application 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).

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.

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.

Ohtsubo, J.

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

Pike, E. R.

E. R. Pike, “Photon correlation velocimetry,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins and E. R. Pike, eds. (Plenum, New York, 1977), pp. 246–343.

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,

N. Takai, 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, Sutanto, and T. Asakura, “Dynamic statistical properties of laser speckle due to longitudinal motion of a diffuse object under Gaussian beam illumination,” J. Opt. Soc. Am. 70, 827–834 (1980).
[Crossref]

Takai, N.

T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
[Crossref]

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[Crossref]

N. Takai, 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, Sutanto, and T. Asakura, “Dynamic statistical properties of laser speckle due to longitudinal motion of a diffuse object under Gaussian beam illumination,” J. Opt. Soc. Am. 70, 827–834 (1980).
[Crossref]

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]

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.

Yamaguchi, I.

I. Yamaguchi and S. Komatsu, “Theory and application 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. Phys. (1)

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[Crossref]

J. Opt. Soc. Am. (2)

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, 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. Acta (2)

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

T. Iwai, N. Takai, and T. Asakura, “The autocorrelation function of the speckle intensity fluctuation integrated spatially by a detecting aperture of finite size,” Opt. Acta 28, 1425–1437 (1981).
[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).

Optik (1)

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

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 (2)

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

E. R. Pike, “Photon correlation velocimetry,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins and E. R. Pike, eds. (Plenum, New York, 1977), pp. 246–343.

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Figures (7)

Fig. 1
Fig. 1

Optical arrangement for the formation of time-varying speckles produced at the far-field diffraction plane by a diffuse object moving in an arbitrary direction of three-dimensional space under illumination by a Gaussian beam.

Fig. 2
Fig. 2

Normalized autocorrelation functions γΔI(τ) of the time-varying speckle-intensity fluctuation for five different waist widths of the illuminating Gaussian beam with wo = (a) 2.5, (b) 5.0, (c) 10.0, (d) 25.0, and (e) 50.0 μm.

Fig. 3
Fig. 3

Variations of the correlation length of the time-varying speckle-intensity fluctuation as functions of the lateral component |v| and the longitudinal component |v||| of the object velocity for three different waist widths of the illuminating Gaussian beam with wo = (a) 2.5, (b) 5.0, and (c) 10.0 μm.

Fig. 4
Fig. 4

Comparison between the variations of the correlation lengths of the time-varying speckle-intensity fluctuation calculated from Eqs. (34) and (37) as functions of the direction angle θ = 75–90° of the moving object for three different waist widths of the illuminating Gaussian beam with wo = (a) 2.5, (b) 5.0, and (c) 10.0 μm.

Fig. 5
Fig. 5

Mechanical system used to move a diffuse object, together with the optical components.

Fig. 6
Fig. 6

Normalized autocorrelation functions γΔI(τ) of the time-varying speckle-intensity fluctuation obtained (A) experimentally and (B) theoretically for two different waist widths wo = 1.9 μm and wo = 6.4 μm of the illuminating beam as a function of the direction angle θ of the moving object: θ = 0° (curve a), 82.5° (curve b), 87.5° (curve c), and 90° (curve d).

Fig. 7
Fig. 7

Experimental results for the variations of the correlation length of the time-varying speckle-intensity fluctuation for three different waist widths of the illuminating Gaussian beam with wo = (a) 1.9, (b) 4.0, and (c) 6.4 μm. The experimental values are indicated by three different symbols, and the theoretical results are added by the solid curves for comparison.

Equations (38)

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V ( x ; t ) = - E o ( ξ , z ) exp [ i ϕ ( ξ - v t ) ] exp [ i k ξ · x R ( t ) ] d ξ ,
R ( t ) = R o - z ,
z = z ( t ) = v t
E o ( ξ , z ) = w o w ( z ) exp { i [ k z - ψ ( z ) ] } × exp [ - ξ 2 / w 2 ( z ) ] exp [ i k ξ 2 / 2 ρ ( z ) ] ,
w ( z ) = w o ( 1 + z 2 / a 2 ) 1 / 2
ρ ( z ) = z ( 1 + a 2 / z 2 ) ,
ψ ( z ) = tan - 1 ( z / a ) ,
a = π w o 2 / λ
Δ I ( x ; t ) = I ( x ; t ) - I ( x ; t ) ,
Γ Δ I ( x ; t 1 , t 2 ) = Δ I ( x ; t 1 ) Δ I ( x ; t 2 ) = Γ v ( x ; t 1 , t 2 ) 2 ,
Γ v ( x ; t 1 , t 2 ) = V ( x ; t 1 ) V * ( x ; t 2 ) .
Γ v ( x , t 1 , t 2 ) = - E o ( ξ 1 , z 1 ) E o * ( ξ 2 , z 2 ) × exp { i [ ϕ ( ξ 1 - v t 1 ) - ϕ ( ξ 2 - v t 2 ) ] } × exp [ i k ( ξ 1 R 1 - ξ 2 R 2 ) · x ] d ξ 1 d ξ 2 ,
exp { i [ ϕ ( ξ 1 - v t 1 ) - ϕ ( ξ 2 - v t 2 ) ] } Δ S δ [ ξ 1 - ξ 2 - v ( t 1 - t 2 ) ] ,
Γ v ( x ; t 1 , t 2 ) = Δ S - E o [ ξ + ½ v ( t 1 - t 2 ) , z 1 ] × E o * [ ξ - ½ v ( t 1 - t 2 ) , z 2 ] × exp { i k [ ξ + ½ v ( t 1 - t 2 ) R 1 - ξ - ½ v ( t 1 - t 2 ) R 2 ] · x } d ξ ,
Γ Δ I ( x ; t 1 , t 2 ) = Δ S 2 w o 4 w 1 2 w 2 2 exp [ - 1 2 α ( α 2 - β 2 ) ( t 1 - t 2 ) 2 v 2 ] × | - exp ( i q u 2 ) exp ( - α u 2 ) × exp { - i [ ( β q α - p ) ( t 1 - t 2 ) v - κ x ] · u } d u | 2 ,
α = 1 w 1 2 + 1 w 2 2 ,
β = 1 w 1 2 - 1 w 2 2 ,
p = k 2 ( 1 ρ 1 + 1 ρ 2 ) ,
q = k 2 ( 1 ρ 1 - 1 ρ 2 ) ,
κ = k ( 1 R 1 - 1 R 2 ) ,
- exp ( i A η 2 ) exp ( - B η 2 ) exp ( - i C x η ) d η = π ( A 2 + B 2 ) - 1 / 4 exp [ - B C 2 x 2 4 ( A 2 + B 2 ) ] × exp { - i [ A C 2 x 2 4 ( A 2 + B 2 ) - ½ tan - 1 ( A B ) ] } .
Γ Δ I ( x ; t 1 , t 2 ) = π 2 Δ S 2 w o 4 w 1 2 w 2 2 ( α 2 + q 2 ) × exp [ - 1 2 α ( α 2 - β 2 ) ( t 1 - t 2 ) 2 v 2 ] × exp [ - α 2 ( α 2 + q 2 ) | ( β q α - p ) ( t 1 - t 2 ) v - κ x | 2 ] .
α = π a ( z 1 2 + z 2 2 + 2 a 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
β = π a ( z 2 2 - z 1 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
p = π ( z 1 + z 2 ) ( z 1 z 2 + a 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
q = π ( z 2 - z 1 ) ( z 1 z 2 - a 2 ) λ ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
w 1 2 w 2 2 = w o 4 a 4 ( z 1 2 + a 2 ) ( z 2 2 + a 2 ) ,
κ = 2 π λ ( 1 R o - z 1 - 1 R o - z 2 ) .
R π w 2 / λ
z R o .
κ 2 π ( z 1 - z 2 ) λ R o 2 .
Γ Δ I ( x ; t 1 , t 2 ) = λ 2 a 4 Δ S 2 4 a 2 + ( t 2 - t 1 ) 2 v 2 × exp [ - ( 2 π a λ ) ( t 2 - t 1 ) 2 v 2 4 a 2 + ( t 2 - t 1 ) 2 v 2 ] × exp { - 2 π a ( t 2 - t 1 ) 2 v 2 λ R o 2 [ 4 a 2 + ( t 2 - t 1 ) 2 v 2 ] × [ - 2 ( t 2 + t 1 ) v · x + 2 a 2 + ( t 1 2 + t 2 2 ) v 2 x 2 R o 2 ] } .
Γ Δ I ( 0 ; t 1 , t 2 ) = λ 2 a 4 Δ S 2 4 a 2 + ( t 2 - t 1 ) 2 v 2 × exp [ - ( 2 π a λ ) ( t 2 - t 1 ) 2 v 2 4 a 2 + ( t 2 - t 1 ) 2 v 2 ] .
γ Δ I ( τ ) = Γ Δ I ( τ ) Γ Δ I ( 0 ) = 1 1 + τ 2 / τ c 2 exp ( - τ 2 / τ c 2 1 + τ 2 / τ c 2 ) ,
τ c = w o / v
τ c = 2 π w o 2 / λ v .
γ Δ I lat ( τ ) = exp ( - τ 2 / τ c 2 ) .
γ Δ I log ( τ ) = 1 1 + τ 2 / τ c 2 .