Abstract

Statistical properties of triple-random-modulated dynamic speckles (TRMDS) are studied theoretically and experimentally. The spatiotemporal correlation function of the intensity fluctuation of the speckles is obtained; it is a Gaussian distribution. The correlation time is inversely proportional to the velocity of the moving diffuser. The average radius of TRMDS is smaller than that of single-random-modulated dynamic speckles (SRMDS). The correlation time of SRMDS is approximately 70–80 times longer than that of TRMDS. The theoretical results are consistent with the experimental results.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. N. Takai, Sutanto, 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]
  4. K. A. O’Donnell, “Correlations of time-varying speckle near the focal plane,” J. Opt. Soc. Am. 72, 191–197 (1982).
    [CrossRef]
  5. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986).
    [CrossRef]
  6. R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
    [CrossRef] [PubMed]
  7. Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
    [CrossRef] [PubMed]
  8. M. Araie, K. Muta, “Effect of long-term topical betaxolol on tissue circulation in the iris and optic nerve head,” Exp. Eye Res. 64, 167–172 (1997).
    [CrossRef] [PubMed]
  9. N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
    [CrossRef]
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    [CrossRef]
  11. S. N. Ma, Q. Lin, “Laser speckle velocimetry: using modulated dynamic speckle to measure the velocity of moving diffusers,” Appl. Opt. 25, 22–25 (1986).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. T. Yoshimura, H. Doi, N. Wakabayashi, “Correlation properties of heterodyned dynamic speckle in a two-lens imaging system and its application to two-dimensional velocity measurements,” J. Opt. Soc. Am. A 1, 1078–1084 (1984).
    [CrossRef]
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    [CrossRef]
  16. K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [CrossRef]
  17. H. Fujii, T. Asakura, K. Nohira, Y. Shintomi, T. Ohura, “Blood flow observed by time-varying laser speckle,” Opt. Lett. 10, 104–106 (1985).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. T. Okamoto, T. Asakura, “Velocity measurements of two diffusers using a temporal correlation length of doubly scattered speckle,” J. Mod. Opt. 37, 389–408 (1990).
    [CrossRef]
  22. T. Okamoto, T. Asakura, “Velocity dependence of image speckles produced by a moving diffuser under dynamic speckle illumination,” Opt. Commun. 77, 113–120 (1990).
    [CrossRef]
  23. Y. Liu, S. H. Ma, H. Y. Lin, S. H. Ye, “Speckle from cascaded diffusers in an imaging system,” J. Opt. Soc. Am. A 10, 951–956 (1993).
    [CrossRef]
  24. T. Iwai, T. Asakura, “Dynamic properties of speckled speckles with relation to velocity measurements of a diffuse object,” Opt. Laser Technol. 21, 31–35 (1989).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.
  26. J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomana, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).
  27. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5.

1997

M. Araie, K. Muta, “Effect of long-term topical betaxolol on tissue circulation in the iris and optic nerve head,” Exp. Eye Res. 64, 167–172 (1997).
[CrossRef] [PubMed]

1996

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

1993

1990

T. Okamoto, T. Asakura, “Velocity measurements of two diffusers using a temporal correlation length of doubly scattered speckle,” J. Mod. Opt. 37, 389–408 (1990).
[CrossRef]

T. Okamoto, T. Asakura, “Velocity dependence of image speckles produced by a moving diffuser under dynamic speckle illumination,” Opt. Commun. 77, 113–120 (1990).
[CrossRef]

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

1989

T. Iwai, T. Asakura, “Dynamic properties of speckled speckles with relation to velocity measurements of a diffuse object,” Opt. Laser Technol. 21, 31–35 (1989).
[CrossRef]

L. G. Shirley, N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6, 765–781 (1989).
[CrossRef]

1988

1986

1985

1984

1982

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 72, 1213–1218 (1982).
[CrossRef]

K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
[CrossRef]

K. A. O’Donnell, “Correlations of time-varying speckle near the focal plane,” J. Opt. Soc. Am. 72, 191–197 (1982).
[CrossRef]

1981

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

1980

1976

1971

Araie, M.

M. Araie, K. Muta, “Effect of long-term topical betaxolol on tissue circulation in the iris and optic nerve head,” Exp. Eye Res. 64, 167–172 (1997).
[CrossRef] [PubMed]

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

Asakura, T.

T. Okamoto, T. Asakura, “Velocity measurements of two diffusers using a temporal correlation length of doubly scattered speckle,” J. Mod. Opt. 37, 389–408 (1990).
[CrossRef]

T. Okamoto, T. Asakura, “Velocity dependence of image speckles produced by a moving diffuser under dynamic speckle illumination,” Opt. Commun. 77, 113–120 (1990).
[CrossRef]

T. Iwai, T. Asakura, “Dynamic properties of speckled speckles with relation to velocity measurements of a diffuse object,” Opt. Laser Technol. 21, 31–35 (1989).
[CrossRef]

N. Takai, T. Asakura, “Laser speckles produced by a diffuse object under illumination from a multimode optical fiber: an experimental study,” Appl. Opt. 27, 557–562 (1988).
[CrossRef] [PubMed]

N. Takai, T. Asakura, “Statistical properties of laser speckles produced under illumination from a multimode optical fiber,” J. Opt. Soc. Am. A 2, 1282–1290 (1985).
[CrossRef]

H. Fujii, T. Asakura, K. Nohira, Y. Shintomi, T. Ohura, “Blood flow observed by time-varying laser speckle,” Opt. Lett. 10, 104–106 (1985).
[CrossRef] [PubMed]

T. Asakura, 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, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, Sutanto, 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]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5.

Berkovits, R.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

Doi, H.

Estes, L. E.

Feng, S.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

Fujii, H.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

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

Holmes, J. F.

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 72, 1213–1218 (1982).
[CrossRef]

Iwai, T.

T. Iwai, T. Asakura, “Dynamic properties of speckled speckles with relation to velocity measurements of a diffuse object,” Opt. Laser Technol. 21, 31–35 (1989).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

Jakeman, E.

Lin, H. Y.

Lin, Q.

Liu, Y.

Ma, S. H.

Ma, S. N.

McWhirter, J. G.

Muta, K.

M. Araie, K. Muta, “Effect of long-term topical betaxolol on tissue circulation in the iris and optic nerve head,” Exp. Eye Res. 64, 167–172 (1997).
[CrossRef] [PubMed]

Narducci, L. M.

Nohira, K.

O’Donnell, K. A.

Ohtsubo, J.

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

Ohura, T.

Okamoto, T.

T. Okamoto, T. Asakura, “Velocity measurements of two diffusers using a temporal correlation length of doubly scattered speckle,” J. Mod. Opt. 37, 389–408 (1990).
[CrossRef]

T. Okamoto, T. Asakura, “Velocity dependence of image speckles produced by a moving diffuser under dynamic speckle illumination,” Opt. Commun. 77, 113–120 (1990).
[CrossRef]

Pusey, P. N.

Rao Gudimetla, V. S.

V. S. Rao Gudimetla, J. F. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. A 72, 1213–1218 (1982).
[CrossRef]

Schmidt-Harms, C. A.

Shintomi, Y.

Shirley, L. G.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5.

Sutanto,

Takai, N.

Tamaki, Y.

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

Tomidokoro, A.

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

Tomita, K.

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

Tuft, R. A.

Wakabayashi, N.

Ye, S. H.

Yoshimura, T.

Appl. Opt.

Appl. Phys.

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

Curr. Eye Res.

Y. Tamaki, M. Araie, K. Tomita, A. Tomidokoro, “Time change of nicardipine effect in choroidal circulation in rabbit eyes,” Curr. Eye Res. 15, 543–548 (1996).
[CrossRef] [PubMed]

Exp. Eye Res.

M. Araie, K. Muta, “Effect of long-term topical betaxolol on tissue circulation in the iris and optic nerve head,” Exp. Eye Res. 64, 167–172 (1997).
[CrossRef] [PubMed]

J. Mod. Opt.

T. Okamoto, T. Asakura, “Velocity measurements of two diffusers using a temporal correlation length of doubly scattered speckle,” J. Mod. Opt. 37, 389–408 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

T. Okamoto, T. Asakura, “Velocity dependence of image speckles produced by a moving diffuser under dynamic speckle illumination,” Opt. Commun. 77, 113–120 (1990).
[CrossRef]

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

Opt. Laser Technol.

T. Iwai, T. Asakura, “Dynamic properties of speckled speckles with relation to velocity measurements of a diffuse object,” Opt. Laser Technol. 21, 31–35 (1989).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

R. Berkovits, S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123 (1990).
[CrossRef] [PubMed]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

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

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5.

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Figures (6)

Fig. 1
Fig. 1

Optical arrangement for the formation of TRMDS at the far-field diffraction plane by two diffusers under illumination by a Gaussian beam.

Fig. 2
Fig. 2

Block diagram for detection of the correlation function of the TRMDS intensity fluctuation. A/D, analog to digital; PMT, photomultiplier.

Fig. 3
Fig. 3

Correlation curves (from top to bottom) corresponding to v1=21 µm/s, v2=38 µm/s, v3=60 µm/s, v4=81 µm/s, and v5=100 µm/s.

Fig. 4
Fig. 4

Experimental results of the reciprocal of the correlation time as a function of the diffuser velocity.

Fig. 5
Fig. 5

Calculated spatiotemporal correlation function of the TRMDS intensity fluctuation.

Fig. 6
Fig. 6

Experimental results for the temporal correlation function of the TRMDS intensity fluctuation when H=10 mm (solid curve) and H=12 mm (dotted–dashed curve).

Equations (61)

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E1(ξ)=-E(α)tD1(α)h1(ξ, α)d2α,
E(α)=exp[-(α2/ω2)]
h1(ξ, α)=exp(ikH)iλHexpik2H(ξ-α)2.
E2(α)=-E1(ξ)t2(ξ, t)h2(α, ξ)d2ξ,
h2(α, ξ)=exp(ikH)iλHexpik2H(α-ξ)2.
h3(x, α)=exp(ikz)iλzexpik2zx2×exp-i2πλzα·x.
E(x, t)=-E2(α)tD1(α)h3(x, α)d2α.
E(x, t)=-E(α)tD1(α)tD2(ξ, t)tD1(α)×h1(ξ, α)h2(α, ξ)h3(x, α)d2αd2ξd2α.
Γ(x1, x2; t1, t2)=E(x1, t1)E*(x2, t2),
Γ(x1, x2; t1, t2)
=-E(α1)E0*(α2)×tD1(α1)tD2(ξ1, t1)tD1(α1)tD1*(α2)tD2*(ξ2, t2)×tD1*(α2)h1(ξ1, α1)h2(α1, ξ1)h3(x1, α1)h1*(ξ2, α2)×h2*(α2, ξ2)h3*(x2, α2)d2α1d2ξ1d2×α1d2α2d2ξ2d2α2.
u1*  uk* uk+1  u2k=πu1*upu2*uq  uk*ur,
Γ(x1, x2; t1, t2)=-E(α1)E0*(α2)×tD1(α1)tD2(ξ1, t1)tD1(α)tD1*(α2)×tD2*(ξ2, t2)tD1*(α2)h1(ξ1, α1)×h2(α1, ξ1)h3(x1, α1)h1*(ξ2, α2)×h2*(α2, ξ2)h3*(x2, α2)d2α1d2ξ1×d2α1d2α2d2ξ2d2α2.
tD1(α)tD2*(ξ)=tD2(ξ)tD1*(α)=0.
tD1(α1)tD2(ξ1, t1)tD1(α1)tD1*(α2)tD2*(ξ2, t2)tD1*(α2)
=tD1(α1)tD1*(α2)tD2(ξ1, t1)tD2*(ξ2, t2)×tD1(α1)tD1*(α2)+tD1(α)tD1*(α2)×tD2(ξ1, t1)tD2*(ξ2, t2)tD1(α1)tD1*(α2)2tD1(α1)tD1*(α2)tD2(ξ1, t1)tD2*(ξ2, t2)×tD1(α1)tD1*(α2).
tD1(α)=exp[-iϕ1(α)],
tD2(ξ, t)=exp[-iϕ2(ξ, t)].
RD1(Δα)=tD1(α1)tD1*(α2)=exp[-π(Δα2/a2)],
RD1(Δα)=tD1(α1)tD1*(α2)=exp[-π(Δα2/a2)],
RD2(Δξ-vτ)=tD2(ξ1, t1)tD2*(ξ2, t2)=exp[-π(Δξ-vτ)2/b2],
Γ(x1, Δx; τ)
=2λ6H4z2expiπλz(2x1-Δx)Δx×- exp-2α12-2α1Δα+Δα2ω2×RD1(Δα)RD2(Δξ1-vτ)RD1(Δα)×exp-iπλH(Δα2+Δξ2-2ΔαΔξ+2Δαξ1-2Δξξ1+2Δξα1-2Δαα1)exp-iπλH×(Δα2+Δξ2-2ΔαΔξ+2Δαξ1-2Δξξ1+2Δξα1-2Δαα1)exp-i2πλz×(α1Δx+Δαx-ΔαΔx)d2α1d2ξ1×d2α1d2Δαd2Δξd2Δα.
δ(x)=12π- exp(-ikx)dk=12π- exp(ikx)dk,
δ(at)=1|a|δ(t),
g(x, y)=A exp[-a(x2+y2)],
F(g(x, y))=-g(x, y)exp[-j2π(fxx+fyy)]dxdy=Aπa exp-π2(fx2+fy2)a,
Γ(x1, Δx; τ)
=2/m3πωλ4H3z2exp-iπλz(2x1Δx-Δx2)2Hz-1+i2πλzx1-Δx21m32πHa2zΔx+πHb2zΔx-πb2vτ×exp-π2ω2Δx22λ2z2-π2(x1-Δx/2)2m3λ2z2-4πH2a2z2Δx2-πb2HzΔx-vτ2+1m32πHa2zΔx+πHb2zΔx-πb2vτ2,
m3=12ω2+2πa2+πb2.
ΓI(x1, x2; t1, t2)=I(x1, t1)I(x2, t2)+|ΓE(x1, x2; t1, t2)|2.
ΔI(x, t)=I(x, t)-I(x, t),
ΓΔI(x1, x2; t1, t2)=|ΓE(x1, x2; t1, t2)|2.
γΔI(x1, x2; t1, t2)=|ΓΔI(x1, x2; t1, t2)|I1I2=|ΓE(x1, x2; t1, t2)|2I1I2,
I1(x1, t1)=|E(x1, t1)|2,
I1=E(x1, t1)E*(x1, t1),
I2(x2, t2)=|E(x2, t2)|2,
I2=E(x2, t2)E*(x2, t2).
γΔI(Δx, τ)=exp-π2ω2Δx2λ2z2+π2Δx22m3λ2z2-8πH2a2z2Δx2-2πb2HzΔx-vτ2+2m32πHa2zΔx+πHb2zΔx-πb2vτ2.
γΔI(0, τ)=exp(-τ2/τc32),
τc3=2πb21-πm3b2-1/2|v|-1.
γΔI(Δx, 0)=exp(-Δx2/rs2),
rs=rs01+10H2λ2πω2a2-12m3ω2-18H2λ2m3ω2a4-1/2
1τc1=2πa21-πm1a21/2v,m1=12ω2+πa2
1τc3=2πb21-πm3b21/2v,
m3=12ω2+2πa2+πb2.
α1α3=τc1τc3.
τc1τc3=m1(m3a2-π)m3(m1a2-π)1/2=1+4πω2(a4/2ω2)+3πa21/2.
α1α3=τc1τc32ω3a.
rsrs01+4H2λ2πω2a2-1/2.
E(x, t)=-E(α)tD(α, t)h(α, x)d2α,
E(α)=exp[-(α2/ω2)],
h(x, α)=exp(ikz)iλzexpik2zx2exp-i2πλzα·x
RD(Δα-vτ)=tD(α1, t1)tD*(α2, t2)=exp[-π(Δα-vτ)2/a2].
Γ(x1, Δx; τ)=2m1πω2λ2z2expiπλz(2x1Δx-Δx2)-i2πλzx1-Δx21m1πa2vτ×exp-π2ω2Δx22λ2z2-π2(x1-Δx/2)2m1λ2z2-πa21-πm1a2v2τ2,
m1=12ω2+πa2.
γΔI(Δx, τ)=exp-π2ω2Δx2λ2z2+π2Δx22m1λ2z2×exp-2πa21-πm1a2v2τ2.
γΔI(0, τ)=exp(-τ2/τc12),
τc1=2πa21-πm1a2-1/2|v|-1.
γΔI(Δx, 0)=exp(-Δx2/rs02),
rs0=λzπω1-12ω2m1-1/2λzπω,(aω)

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