Abstract

The dynamics of the speckle pattern from a rotating, diffusely reflecting object under laser illumination are analyzed. These are described by the general spatiotemporal cross-correlation function of the optical intensity. An analytical expression for this function is derived that is interpreted in terms of the observed phenomena of speckle boiling and rotation. Experimental results are presented in quantitative support of this expression.

© 1982 Optical Society of America

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References

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  1. J. Ohtsubo, “Velocity measurement using the space–time cross-correlation of speckle patterns,” Opt. Commun. 34, 147–152 (1980).
    [Crossref]
  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]
  3. J. H. Churnside and H. T. Yura, “Velocity measurement using laser speckle statistics,” Appl. Opt. 20, 3539–3541 (1981).
    [Crossref] [PubMed]
  4. B. E. A. Saleh, “Speckle correlation measurement of the velocity of a small rotating rough object,” Appl. Opt. 14, 2344–2346 (1975).
    [Crossref] [PubMed]
  5. N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
    [Crossref]
  6. T. Sporton, “The scattering of coherent light from a rough surface,” J. Phys. D. 2, 1027–1034 (1969).
    [Crossref]
  7. 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]
  8. N. George, “Speckle from rough, moving objects,” J. Opt. Soc. Am. 66, 1182–1194 (1976).
    [Crossref]
  9. J. C. Erdmann and R. I. Gelbert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser light spot,” J. Opt. Soc. Am. 66, 1194–1204 (1976).
    [Crossref]
  10. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
  11. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [Crossref]
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  13. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [Crossref]
  14. M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt.Soc. Am. 66, 1164–1172 (1976).
    [Crossref]
  15. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  16. R. M. Hardesty, R. J. Keeler, M. J. Post, and R. A. Richter, “Characteristics of coherent lidar returns from calibration targets and aerosols,” Appl. Opt. 20, 3763–3769 (1981).
    [Crossref] [PubMed]
  17. L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

1981 (4)

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. H. Churnside and H. T. Yura, “Velocity measurement using laser speckle statistics,” Appl. Opt. 20, 3539–3541 (1981).
[Crossref] [PubMed]

N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[Crossref]

R. M. Hardesty, R. J. Keeler, M. J. Post, and R. A. Richter, “Characteristics of coherent lidar returns from calibration targets and aerosols,” Appl. Opt. 20, 3763–3769 (1981).
[Crossref] [PubMed]

1980 (1)

J. Ohtsubo, “Velocity measurement using the space–time cross-correlation of speckle patterns,” Opt. Commun. 34, 147–152 (1980).
[Crossref]

1976 (4)

1975 (1)

1971 (1)

1969 (1)

T. Sporton, “The scattering of coherent light from a rough surface,” J. Phys. D. 2, 1027–1034 (1969).
[Crossref]

1966 (1)

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

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]

N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Churnside, J. H.

Drain, L. E.

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

Erdmann, J. C.

Estes, L. E.

Gelbert, R. I.

George, N.

Goodman, J. W.

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[Crossref]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Hardesty, R. M.

Holmes, J. F.

M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt.Soc. Am. 66, 1164–1172 (1976).
[Crossref]

Iwai, T.

N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[Crossref]

Keeler, R. J.

Kerr, J. R.

M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt.Soc. Am. 66, 1164–1172 (1976).
[Crossref]

Kogelnik, H.

Lee, M. H.

M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt.Soc. Am. 66, 1164–1172 (1976).
[Crossref]

Li, T.

Narducci, L. M.

Ohtsubo, J.

J. Ohtsubo, “Velocity measurement using the space–time cross-correlation of speckle patterns,” Opt. Commun. 34, 147–152 (1980).
[Crossref]

Post, M. J.

Richter, R. A.

Saleh, B. E. A.

Sporton, T.

T. Sporton, “The scattering of coherent light from a rough surface,” J. Phys. D. 2, 1027–1034 (1969).
[Crossref]

Takai, N.

N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (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]

Tuft, R. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Yura, H. T.

Appl. Opt. (4)

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]

Appl. Phys. B (1)

N. Takai, T. Iwai, and T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt.Soc. Am. (1)

M. H. Lee, J. F. Holmes, and J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt.Soc. Am. 66, 1164–1172 (1976).
[Crossref]

J. Phys. D. (1)

T. Sporton, “The scattering of coherent light from a rough surface,” J. Phys. D. 2, 1027–1034 (1969).
[Crossref]

Opt. Commun. (1)

J. Ohtsubo, “Velocity measurement using the space–time cross-correlation of speckle patterns,” Opt. Commun. 34, 147–152 (1980).
[Crossref]

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Other (3)

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

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Figures (11)

Fig. 1
Fig. 1

Block diagram of the geometry under consideration.

Fig. 2
Fig. 2

Plot of normalized autocorrelation function CI(P, 0, τ) versus ωτ for several values of the quantity (d2/b2) + (kb/2z)2[P − (z/R) d]2.

Fig. 3
Fig. 3

Experimental configuration used for autocorrelation measurements.

Fig. 4
Fig. 4

Plot of experimental (circles) and theoretical autocorrelation function CI(P, 0, τ) versus τ for several values of transverse detector position P.

Fig. 5
Fig. 5

Plot of experimental (circles) and theoretical autocorrelation function CI(P, 0, τ) versus transverse detector position P for τ = 0.5 msec.

Fig. 6
Fig. 6

Plot of typical correlation function CI(P, p, τ) versus ωτ showing shift of ωτ0 in the location of the peaks.

Fig. 7
Fig. 7

Receiver geometry used for spatiotemporal correlation measurements.

Fig. 8
Fig. 8

Plot of experimental (circles) and theoretical cross-correlation function CI(P, p, τ) versus τ for ϕ = 90°.

Fig. 9
Fig. 9

Plot of experimental (circles) and theoretical cross-correlation function CI(P, p, τ) versus τ for ϕ = 60°.

Fig. 10
Fig. 10

Plot of experimental (circles) and theoretical cross-correlation function CI(P, p, τ) versus τ for ϕ = 30°.

Fig. 11
Fig. 11

Plot of experimental (circles) and theoretical cross-correlation function CI(P, p, τ) versus τ for ϕ = 0°.

Equations (30)

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C I ( p 1 , p 2 , t 1 , t 2 ) = I ( p 1 , t 1 ) I ( p 2 , t 2 ) I ( p 1 , t 1 ) I ( p 2 , t 2 ) [ I 2 ( p 1 , t 1 ) I ( p 1 , t 1 ) 2 ] 1 / 2 [ I 2 ( p 2 , t 2 ) I ( p 2 , t 2 ) 2 ] 1 / 2 ,
I ( p 1 , t ) I ( p 2 , t + τ ) = U ( p 1 , t ) U * ( p 1 , t ) U ( p 2 , t + τ ) U * ( p 2 , t + τ ) = | Γ ( p 1 , p 2 , t ) | 2 + I ( p 1 t ) I ( p 2 , τ + τ ) ,
Γ ( p 1 , p 2 , τ ) = U ( p 1 , t ) U * ( p 2 , t + τ ) .
C I ( p 1 , p 2 , τ ) = | Γ ( p 1 , p 2 , τ ) | 2 Γ ( p 1 , p 1 , 0 ) Γ ( p 2 , p 2 , 0 ) = | γ ( p 1 , p 2 , τ ) | 2 ,
U ( p , t ) = k e i k z 2 π i z U ( ρ , t ) exp [ i k 2 z ( p ρ ) 2 ] d 2 ρ ,
Γ ( p 1 , p 2 , τ ) = ( k 2 π z ) 2 Γ ( ρ 1 , ρ 2 , τ ) × exp { i k 2 z [ ( p 1 ρ 1 ) 2 ( p 2 ρ 2 ) 2 ] } d 2 ρ 1 d 2 ρ 2
Γ ( ρ 1 , ρ 2 , τ ) = U I ( ρ 1 , t ) U I * ( ρ 2 , t + τ ) δ ( ρ 2 ρ 1 ) ,
ρ 1 = ρ 1 cos ω τ + A ρ 1 sin ω τ ,
ρ 1 = ( x 1 y 1 ) .
A = ( 0 1 1 0 )
ρ 1 = ( x 1 y 1 ) cos ω τ + ( 0 1 1 0 ) ( x 1 y 1 ) sin ω τ = ( x 1 cos ω τ + y 1 sin ω τ y 1 cos ω τ x 1 sin ω τ ) .
U I ( ρ , t ) = U 0 exp [ ( 1 b 2 + i k 2 R ) ( ρ d ) 2 ] ,
Γ ( p 1 , p 2 , τ ) = ( k 2 π z ) 2 | U 0 | 2 exp [ ( 1 b 2 + i k 2 R ) ( ρ 1 d ) 2 ( 1 b 2 i k 2 R ) ( ρ 2 d ) 2 + i k 2 z ( p 1 ρ 1 ) 2 i k 2 z ( p 2 ρ 2 ) 2 ] × δ ( ρ 2 ρ 1 cos ω τ A ρ 1 sin ω τ ) d 2 ρ 1 d 2 ρ 2 .
γ ( P , p , τ ) = exp { 1 2 ( k b 2 z ) 2 p 2 ( 1 cos ω τ ) [ d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2 1 4 ( k b 2 z ) 2 p 2 ] sin ω τ ( k b 2 z ) 2 ( P z R d ) A p + i k z [ 1 2 ( 1 + cos ω τ ) d p sin ω τ P A d P p ] } ,
C I ( P , p , τ ) = exp { ( k b 2 z ) 2 p 2 ( 1 cos ω τ ) [ 2 d 2 b 2 + 2 ( k b 2 z ) 2 ( P z R d ) 2 1 2 ( k b 2 z ) 2 p 2 ] 2 sin ω τ ( k b 2 z ) 2 ( P z R d ) A p } .
C I ( P , p , 0 ) = exp [ ( k b 2 z ) 2 p 2 ] ,
C I ( P , 0 , τ ) = exp { 2 [ d 2 b 2 + ( k b 2 z ) 2 × ( P z R d ) 2 ] ( 1 cos ω τ ) } ,
[ d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2 ] .
d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2
C I ( P , 0 , τ ) = exp { [ d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2 ] ω 2 τ 2 } .
τ d 1 b ω d ,
τ d 2 2 z k b ω | ( P z R d ) | 1 .
γ osc ( P , 0 , τ ) = exp ( i k z P A d sin ω τ ) .
γ osc ( P , 0 , τ ) = exp ( i k z P v τ )
γ osc ( P , 0 , τ ) = exp ( i k υ τ cos θ ) = exp ( i ω D τ ) ,
τ 0 = 1 ω tan 1 [ ( k b 2 z ) 2 ( P z R d ) A p d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2 ( k b 4 z ) 2 p 2 ] .
C I ( P , p , τ 0 ) = exp ( 1 2 ( k b 2 z ) 2 p 2 2 d 2 b 2 2 ( k b 2 z ) 2 ( P z R d ) 2 + 2 { ( k b 2 z ) 4 [ ( P z R d ) A p ] 2 + [ d 2 b 2 + ( k b 2 z ) 2 ( P z R d ) 2 ( k b 4 z ) 2 p 2 ] 2 } 1 / 2 ) .
C I ( P , p , τ ) = exp { d 2 b 2 ω 2 τ 2 ( k b 2 z ) 2 [ A p ( P z R d ) ω τ ] 2 + ( k b 4 z ) 2 p 2 ω 2 τ 2 } .
( k b 4 z ) 2 p 2 ω 2 τ 2 ( k b 2 z ) 2 p 2 ,
C I ( P , p , τ ) = exp [ d 2 b 2 ω 2 τ 2 ( k b 2 z ) 2 p 2 + 2 ( k b 2 z ) 2 × p ( P + z R d ) ω τ sin ϕ ( k b 2 z ) 2 ( P + z R d ) 2 ω 2 τ 2 + ( k b 4 z ) 2 p 2 ω 2 τ 2 ] .