Abstract

Three-dimensional (3D) speckle dynamics are investigated within the paraxial approximation as represented by ABCD-matrix theory. Within the paraxial approximation, exact expressions are derived for the space–time-lagged intensity covariance that results from an in-plane translation, an out-of-plane rotation, or an in-plane rotation of a diffuse scattering object that is illuminated by a Gaussian-shaped laser beam. As illustrative examples we consider the 3D dynamical nature of speckles that are formed in free space and in Fourier transform and imaging systems. The spatiotemporal characteristics of the observed 3D speckle patterns are interpreted in terms of boiling, decorrelation, rotation, translation, and tilting. Experimental results, which support the quantitative theory, are presented and discussed.

© 1999 Optical Society of America

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References

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  1. See, for example, J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIV, Chap. 1; S. M. Kozel, G. R. Lokshin, “Longitudinal correlation properties of coherent radiation scattered from a rough surface,” Opt. Spectrosc. 33, 89–90 (1972); T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981); T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986); B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998); T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. XXXIV, Chap. 3; H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [CrossRef]
  2. T. Yoshimura, S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 324–328 (1993).
    [CrossRef]
  3. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  4. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.
  5. A. S. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.
  6. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  7. The generalization of the results derived here to systems that are not cylindrically symmetric is straightforward.
  8. H. T. Yura, B. Rose, S. G. Hanson, “Dynamic laser speckle in complex ABCD systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
    [CrossRef]
  9. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).
  10. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [CrossRef]
  11. J. H. Churnside, “Speckle from a rotating diffuse object,” J. Opt. Soc. Am. 72, 1464–1469 (1982).
    [CrossRef]
  12. Because each scatterer appears in the same position in the illuminated region once with each revolution of the object, the speckle pattern exhibits a periodicity with a frequency equal to the frequency of revolution.
  13. Substituting Eqs. (4.6a) and (4.6b) into Eq. (4.1) yields that the maximum value of the space–time cross covariance equals unity.
  14. This because the orientation of the (static) speckles is independent of the motion of the object.
  15. H. T. Yura, B. Rose, S. G. Hanson, “Speckle dynamics from in-plane rotating diffuse objects in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1167–1171 (1998).
    [CrossRef]
  16. Note that the complex cross covariance is proportional to exp[ikJ-ik(J-vZτ)]=exp[iωDτ], where J is the path length along the optic axis and ωD=kvZ is the Doppler shift caused by the longitudinal motion of the object.

1998

1993

1987

1982

Churnside, J. H.

Dainty, J. C.

See, for example, J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIV, Chap. 1; S. M. Kozel, G. R. Lokshin, “Longitudinal correlation properties of coherent radiation scattered from a rough surface,” Opt. Spectrosc. 33, 89–90 (1972); T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981); T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986); B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998); T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. XXXIV, Chap. 3; H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

Grum, T. P.

Hanson, S. G.

Iwamoto, S.

Rose, B.

Siegman, A. S.

A. S. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Wolfram, S.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

Yoshimura, T.

Yura, H. T.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

See, for example, J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIV, Chap. 1; S. M. Kozel, G. R. Lokshin, “Longitudinal correlation properties of coherent radiation scattered from a rough surface,” Opt. Spectrosc. 33, 89–90 (1972); T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981); T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986); B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998); T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995), Vol. XXXIV, Chap. 3; H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

A. S. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

The generalization of the results derived here to systems that are not cylindrically symmetric is straightforward.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

Because each scatterer appears in the same position in the illuminated region once with each revolution of the object, the speckle pattern exhibits a periodicity with a frequency equal to the frequency of revolution.

Substituting Eqs. (4.6a) and (4.6b) into Eq. (4.1) yields that the maximum value of the space–time cross covariance equals unity.

This because the orientation of the (static) speckles is independent of the motion of the object.

Note that the complex cross covariance is proportional to exp[ikJ-ik(J-vZτ)]=exp[iωDτ], where J is the path length along the optic axis and ωD=kvZ is the Doppler shift caused by the longitudinal motion of the object.

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Figures (7)

Fig. 1
Fig. 1

Propagation of reflected light from the object plane through an arbitrary ABCD optical system to an observation plane. The vector ρ is a 2D vector in a plane perpendicular to the optic axis at z.

Fig. 2
Fig. 2

Optical diagram for the measurement setup. The first image is acquired at the position z2, whereafter additional measurements are taken for displacements of Δz. For free-space measurements, the lens is omitted and the distance z2+Δz denotes the distance between the target and the image sensor.

Fig. 3
Fig. 3

Division of the captured image (speckle pattern) into subimages.

Fig. 4
Fig. 4

Lorentzian fit to the measured peak values of the cross covariance for free-space propagation. The reference image is captured at z2=70 mm.

Fig. 5
Fig. 5

(a) Speckle slope in the x direction for various transverse positions for a free-space configuration. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction for various transverse positions for a free-space configuration. Note that for positions 1, 2, and 3, the measurements are overlapping.

Fig. 6
Fig. 6

(a) Speckle slope in the x direction at various transverse positions for an imagelike configuration. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction at various transverse positions for an imagelike configuration. Note that for positions 1, 2, and 3, the measurements are overlapping.

Fig. 7
Fig. 7

(a) Speckle slope in the x direction at various transverse positions for a Fourier transform system. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction at various transverse positions for a Fourier transform system. Note that for positions 1, 2, and 3, the measurements are overlapping.

Equations (51)

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CI(p1, p2; t1, t2)=I(p1, t1)I(p2, t2)-I(p1, t1)I(p2, t2){[I(p1, t1)2-I(p1, t1)2][I(p2, t2)2-I(p2, t2)2]}1/2,
CI(p1, p2;τ)=|Γ(p1, p2; τ)|2Γ(p1, p1; 0)Γ(p2, p2; 0),
Γ(p1, p2; τ)=U(p1, t1)U*(p2, t2)
P(I1, I2)=exp-I1+I2I(1-CI)I2(1-CI)I02I1I2CII(1-CI),
U(p, t)=d2rU0(r, t)G(r, p),
G(r, p)=-ik2πBexp-ik2B(Ar2-2r·ρ+Dρ2),
U0(r, t)=Ui(r, t)ψ(r, t),
ψ(r1, t)ψ*(r2, t)=const.×δ(r1-r2),
Γ0(r1, r2; τ)=U0(r1, t)U0*(r2, t+τ)=Ui(r1)Ui*(r2)δ(r2-r1),
Γ(p1, p2; τ)=d2rUi(r)Ui*(r)G(r, p1)G*(r, p2).
r=r+vτ.
xy=x cos ωτ-y sin ωτx sin ωτ+y cos ωτ.
AdBdCcDd=1Δz01ABCD.
Ui(r)=exp-r21w2+ik2R,
CI(ρ, Δρ, Δz; τ)=1[1+(Δz/lz)2]exp-1ρ02|Δρ-A˜vτ-DΔz(ρ+A˜vτ)/B|2×exp[-(vτ/w)2],
A˜=A+B/R,
lz=4B(B+DΔz)kw24B2kw2,
ρ0=2(B+DΔz)kw[1+(Δz/lz)2]1/22Bkw[1+(Δz/lz)2]1/2.
CI(ρ, Δρ;τ)=exp[-(vτ/w)2]exp-|Δρ-A˜vτ|2(2B/kw)2,
CI(ρ, Δρ, Δz; τ)
=1[1+(Δz/lz)2]exp[-(vτ/w)2]
×exp-1ρ02|Δρ-A0vτ-Δz(ρ+A0vτ)/z|2,
A0=1+z/R,lz=4z2/kw2,
ρ0=2z[1+(Δz/lz)2]1/2/kw.
Δρ=A˜vτ,
Δz=0.
CI(ρ, Δρ, Δz; 0)=1[1+(Δz/lz)2]exp-1ρ02|Δρ-DΔzρ/B|2.
Δρ=DBρΔz.
CI(Δz, 0)=1[1+(Δz/lz)2].
LZ=4B2kw2.
L=4Bkw2B2+D2ρ2.
θ(τ)Δρ/Δz=DB(ρ+A˜vτ).
θ(τ)=ρ+A˜vτz.
Δθ(τ)=DBA˜vτ.
VS(Δz)=A˜v+DBΔzA˜v.
VS(LZ)-VS(-LZ)=2DLZA˜|v|/B=8A˜BD|v|/kw2.
CI(Δρ, Δz;τ)=exp[-(vτ/w)2][1+(Δz/LZI)2]×exp-|Δρ-mvτ|2(2 f2/kσ)2[1+(Δz/LZI)2]
LZI=4f22kσ2
CI(ρ, Δρ, Δz; τ)=1[1+(Δz/lz)2]exp-1ρ02Δρ-DBΔzρ2+21+DBΔz2ρ·(ρ+Δρ)sin2ωτ2+Δρ·R·ρ sin ωτ
R=01-10
CI(ρ, 0; τ)=exp[-(τ/τR)2],
τR=ρ0/ρω.
x(τ)=(x cos ωτ-y sin ωτ),
y(τ)=(y cos ωτ+x sin ωτ),
Δz=0,
x(τ)y(τ)=x cos ωτ-(y+a)sin ωτx sin ωτ+(y+a)cos ωτ-a,
AdBdCcDd=1Δz01ABCD1-vZτ01.
C1(ρ, Δρ, Δz; τ)
=exp-1ρ02Δρ-DBρΔz+vZτρB(A˜+CΔz)21+kw24B22Δz-A˜2vZτ1+(D/B)Δz-(A˜/B)vZτ2,
Δz(τ)=A˜2vZτ
Δρ(τ)=DρΔzB-vZτρB(A˜+CΔz)=vZτρA˜B(A˜D-1-A˜CvZτ)vZτρA˜C.

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