Abstract

The translational and boiling motions of dynamic speckles produced in the Fresnel diffraction field under illumination of a Gaussian beam are investigated in detail. The speckle motion is analyzed from the space–time cross-correlation function of speckle intensity fluctuations detected at the two points in the receiving plane. The correlation distance of time-varying speckles is compared with the translation distance of the spatial speckle pattern. The optical conditions for the translational and boiling motions of dynamic speckles are examined and expressed in a diagram. The characteristics for the correlation distance of time-varying speckle intensity fluctuations are finally verified by several experiments.

© 1983 Optical Society of America

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References

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  1. T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981). Past studies on dynamic speckles and their application to speckle metrology have been reviewed in detail in this reference paper. Therefore, a further reference on past work is omitted in this paper.
    [CrossRef]
  2. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
    [CrossRef]
  3. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1975), p. 9.
  4. P. N. Pusey, J. Phys. D 9, 1399 (1976).
    [CrossRef]
  5. J. H. Churnside, H. T. Yura, Appl. Opt. 20, 3539 (1981).
    [CrossRef] [PubMed]
  6. J. H. Churnside, H. T. Yura, Appl. Opt. 21, 845 (1982).
    [CrossRef] [PubMed]
  7. L. H. Tanner, Appl. Opt. 13, 2026 (1974).
    [CrossRef] [PubMed]
  8. L. Ronchi, A. Fontana, Opt. Acta 22, 243 (1975).
    [CrossRef]
  9. I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).
  10. N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
    [CrossRef]

1982

1981

J. H. Churnside, H. T. Yura, Appl. Opt. 20, 3539 (1981).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981). Past studies on dynamic speckles and their application to speckle metrology have been reviewed in detail in this reference paper. Therefore, a further reference on past work is omitted in this paper.
[CrossRef]

1976

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

1975

L. Ronchi, A. Fontana, Opt. Acta 22, 243 (1975).
[CrossRef]

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).

1974

Asakura, T.

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981). Past studies on dynamic speckles and their application to speckle metrology have been reviewed in detail in this reference paper. Therefore, a further reference on past work is omitted in this paper.
[CrossRef]

Churnside, J. H.

Fontana, A.

L. Ronchi, A. Fontana, Opt. Acta 22, 243 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
[CrossRef]

Iwai, T.

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
[CrossRef]

Komatsu, S.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).

Pusey, P. N.

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

Ronchi, L.

L. Ronchi, A. Fontana, Opt. Acta 22, 243 (1975).
[CrossRef]

Saito, H.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).

Takai, N.

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981). Past studies on dynamic speckles and their application to speckle metrology have been reviewed in detail in this reference paper. Therefore, a further reference on past work is omitted in this paper.
[CrossRef]

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
[CrossRef]

Tanner, L. H.

Yamaguchi, I.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1975), p. 9.

Yura, H. T.

Appl. Opt.

Appl. Phys.

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981). Past studies on dynamic speckles and their application to speckle metrology have been reviewed in detail in this reference paper. Therefore, a further reference on past work is omitted in this paper.
[CrossRef]

Appl. Phys. B

N. Takai, T. Iwai, T. Asakura, Appl. Phys. B 26, 185 (1981).
[CrossRef]

J. Phys. D

P. N. Pusey, J. Phys. D 9, 1399 (1976).
[CrossRef]

Jpn. J. Appl. Phys.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys.Suppl. 14-1, 301 (1975).

Opt. Acta

L. Ronchi, A. Fontana, Opt. Acta 22, 243 (1975).
[CrossRef]

Other

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
[CrossRef]

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1975), p. 9.

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Figures (8)

Fig. 1
Fig. 1

Coordinate system for investigating the properties of dynamic speckles produced in the Fresnel diffraction field.

Fig. 2
Fig. 2

Variations of the correlation distance Xc, given by Eq. (14), as a function of waist radius w0 of the illuminating Gaussian beam for the three values of distance R0.

Fig. 3
Fig. 3

Diagram indicating the motion type of dynamic speckles as a function of waist radius w0 of the illuminating Gaussian beam and position z of the diffuse object. In the figure, the symbols B and T indicate the boiling and translation regions of the speckle motion, respectively.

Fig. 4
Fig. 4

Experimental setup used to investigate the cross-correlation function of time-varying speckle intensity fluctuations.

Fig. 5
Fig. 5

Measured cross-correlation functions γΔI(τ) of time-varying speckle intensity fluctuations as a function of delay time τ for four different separations |X| = X of the two detecting points and three different positions z of the diffuse object.

Fig. 6
Fig. 6

Theoretical cross-correlation functions γΔI(τ) corresponding to the measured cross-correlation functions in Fig. 5.

Fig. 7
Fig. 7

Variations of the peak values of the cross-correlation functions of time-varying speckle intensity fluctuations as a function of separation |X| = X between the two detecting points for four different positions z of the diffuse object and two different beam waist radii of w0 = 0.014 and 0.022 mm.

Fig. 8
Fig. 8

Correlation distances of time-varying speckle intensity fluctuations as a function of object position z for two different beam waist radii of w0 = 0.014 and 0.022 mm.

Equations (16)

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γ Δ I ( X , τ ) = exp ( | v | 2 τ 2 / w 2 ) exp [ | X ( 1 + R ρ ) v τ | 2 / Δ x 2 ] ,
Δ x = λ R / π w ,
X = ( 1 + R / ρ ) v τ .
X T = ( 1 + R / ρ ) w .
γ Δ I ( X , τ ) = exp ( | X | 2 / X c 2 ) exp [ ( τ τ d ) 2 / τ c 2 ] ,
τ c = 1 | v | [ 1 w 2 + ( 1 + R ρ ) 2 / Δ x 2 ] 1 / 2 ,
τ d = τ c 2 ( 1 + R ρ ) v X / Δ x 2 ,
X c = [ Δ x 2 + ( 1 + R ρ ) 2 w 2 1 + ( 1 + R ρ ) 2 w 2 sin 2 θ / Δ x 2 ] 1 / 2 ,
X c = [ Δ x 2 + ( 1 + R ρ ) 2 w 2 ] 1 / 2 .
X c 2 = X T 2 + Δ x 2 .
w = w ( z ) = w 0 ( 1 + z 2 / a 2 ) ,
ρ = ρ ( z ) = z ( 1 + a 2 / z 2 ) ,
a = π w 0 2 / λ
X c = w 0 ( 1 + R 0 2 / a 2 ) 1 / 2 = w 0 [ 1 + ( λ R 0 / π w 0 2 ) 2 ] 1 / 2 .
X c min = 2 λ R 0 / π for w 0 = w ˆ 0 = λ R 0 / π .
η = x T / Δ x = τ d ( X c ) / τ c = R 0 z / a + a R 0 z ,

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