Abstract

Dynamic properties of three-dimensional speckles formed in free space on a diffuse object by an illuminating Gaussian beam have been investigated by evaluation of the space–time cross-correlation function. It has been found that the major axis of three-dimensional static speckles is always directed to the center of the beam spot on the object but that dynamic speckles are directed to a position other than the center according to the velocity of the object. Moreover, the motion of dynamic speckles is composed of two types: shifting and tilting.

© 1993 Optical Society of America

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References

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  1. R. K. Erf, ed. Speckle Metrology (Academic, New York, 1978).
  2. B. Eliasson, F. M. Mottier, “Determination of the granular radiance distribution of a diffuser and its use for vibration analysis,” J. Opt. Soc. Am. 60, 559–565 (1971).
    [CrossRef]
  3. C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produce longitudinal motion of the diffuse object-sensitivity and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
    [CrossRef]
  4. Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
    [CrossRef]
  5. L. Leushacke, M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990).
    [CrossRef]
  6. G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  9. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986).
    [CrossRef]
  10. I. Yamaguchi, S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
    [CrossRef]

1990

1988

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produce longitudinal motion of the diffuse object-sensitivity and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

1986

1977

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

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

1971

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Dzialowski, Y.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Eliasson, B.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Joenathan, C.

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produce longitudinal motion of the diffuse object-sensitivity and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Kirchner, M.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Komatsu, S.

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

Leushacke, L.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

May, M.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Mottier, F. M.

Narayanamurthy, C. S.

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produce longitudinal motion of the diffuse object-sensitivity and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Shaw, R.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Stoffregen, B.

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Weigelt, G. P.

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Yamaguchi, I.

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

Yoshimura, T.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Commun.

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produce longitudinal motion of the diffuse object-sensitivity and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Optik

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other

R. K. Erf, ed. Speckle Metrology (Academic, New York, 1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (5)

Fig. 1
Fig. 1

Gaussian-beam illumination geometry and formations of three-dimensional speckles in free space.

Fig. 2
Fig. 2

Cross-correlation functions between the z0 and the (z0 + ε) planes as functions of γ. Each solid curve was obtained for R0a under ε = 0, 5, 10, 15, 20, 25, 30, and 35 mm.

Fig. 3
Fig. 3

Cross-correlation functions for z0 = 350 mm as a function of ε. Dashed curve, standard-position R0a; solid curve, position R0b.

Fig. 4
Fig. 4

Schematic representation of three-dimensional speckles in motion. Solid curves, structure of dynamic speckles after τ; dashed curves, structure of speckles at τ = 0.

Fig. 5
Fig. 5

Dynamic behavior of three-dimensional speckles in the pure boiling condition.

Equations (19)

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E ( R , t ) = E 0 ( q , t ) exp [ i ϕ ( q v t ) ] K ( q , R ) d q ,
E ( R 0 , t 0 ) E * ( R 1 , t 1 ) = E 0 ( q , t 0 ) E 0 * ( q + v τ , t 0 + τ ) · K ( q , R 0 ) K * ( q + v τ , R 1 ) d q ,
I 0 I 1 / ( I 0 I 1 ) = 1 + g ( R 0 ; r , τ ) .
g ( R 0 ; r , τ ) = | E ( R 0 , t 0 ) E * ( R 1 , t 1 ) | 2 | E ( R 0 , t 0 ) | 2 | E ( R 1 , t 1 ) | 2 .
K ( q , R ) = [ 1 / ( i λ ) ] z exp ( i k | R q | ) / | R q | 2 .
g ( R 0 ; r , τ ) = | E 0 ( q , t ) E 0 * ( q + v τ , t + τ ) exp { i k [ | R 0 q | | R 1 ( q + v τ ) | ] } | 2 d q | E 0 ( q , t ) | 2 d q | E 0 ( q + v τ , t + τ ) | 2 d q .
E 0 ( q , t ) = ( w 0 / w ) 2 exp ( | q | 2 / w 2 ) × exp { i [ ω 0 t k l b π | q | 2 / ( λ ρ ) ψ 0 ] } ,
g ( R 0 ; r , τ ) = 1 / [ 1 + ( ɛ / l z ) 2 ] exp [ ( | v | τ / w ) 2 ] × exp [ ( 1 / r s 2 ) | r + ɛ v τ / ( 2 z 0 ) V 0 τ ( ɛ / z 0 ) ( R 0 + V 0 τ ) | 2 ] ,
l z = 4 z 0 ( z 0 + ɛ ) / ( w 2 k ) , r s = 2 ( z 0 + ɛ ) [ 1 + ( ɛ / l z ) 2 ] 1 / 2 / ( w k ) , V 0 = v ( 1 + z 0 / ρ )
g ( z 0 ; r x y , τ ) = exp ( τ 2 / τ T 2 ) exp [ ( 1 / r s ) 2 | r x y V 0 τ | 2 ] ,
τ T = w / | v | , r s = 2 z 0 / ( w k ) ,
g ( R 0 ; r , 0 ) = 1 / [ 1 + ( ɛ / l z ) 2 ] exp [ ( 1 / r s ) 2 | r ( ɛ / z 0 ) R 0 | 2 ] .
γ = ɛ ( x 0 / z 0 ) = ɛ m x , δ = ɛ ( y 0 / z 0 ) = ɛ m y .
g ( z 0 ; ɛ , 0 ) = [ 1 + ( ɛ / l z ) 2 ] 1 = { 1 + [ w 2 k ɛ / 4 z 0 ( z 0 + ɛ ) ] 2 } 1 .
L z = 4 z 0 2 / ( w 2 k 4 z 0 ) .
L = 4 z 0 ( x 0 2 + y 0 2 + z 0 2 ) 1 / 2 / ( w 2 / k 4 z 0 ) .
γ = V 0 x τ + ɛ [ ( x 0 + V 0 x τ ) / z 0 υ x τ / ( 2 z 0 ) ] = V 0 x τ + ɛ m x ( τ ) , δ = V 0 y τ + ɛ [ ( y 0 + V 0 y τ ) / z 0 υ y τ / ( 2 z 0 ) ] = V 0 y τ + ɛ m y ( τ ) .
g ( R 0 ; r , τ ) = [ 1 + ( ɛ / l z ) 2 ] 1 exp ( τ / τ T ) 2 .
V 1 = V 0 ( z 0 + ɛ ) / z 0 v ɛ / ( 2 z 0 ) , V 0 = v ( 1 + z 0 ) / ρ ) , V 1 = V 0 ( z 0 ɛ ) / z 0 + υ ɛ / ( 2 z 0 ) .

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