Abstract

We develop the theory of the speckle velocimeter that is based on use of a photorefractive real-time hologram in four-wave mixing as a time-integrative correlator. The theory of the speckle velocimeter has been developed for the time correlation between the far-field spectrum of light scattered from the diffuser and the reference wave that is Doppler shifted. Our theoretical derivation shows that it is possible to extract the velocity with minor processing of the output correlation.

© 2001 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. P. G. Simkins, T. D. Dudderar, “Laser speckle measurements of transient Benard convection,” J. Fluid Mech. 89, 665–671 (1978).
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  3. S. H. Collicott, L. Hesselink, “Anamorphic optical processing of multiple-exposure speckle photographs,” Opt. Lett. 11, 410–412 (1986).
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  9. S. H. Collicott, L. Hesselink, “Real-time photorefractive recording and optical processing for speckle velocimetry,” Opt. Lett. 13, 348–350 (1988).
    [CrossRef] [PubMed]
  10. K. Nakagawa, T. Minemoto, “Readout properties of the specklegram recorded in photorefractive Bi12SiO20 crystal,” Appl. Opt. 30, 2386–2392 (1991).
    [CrossRef] [PubMed]
  11. H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
    [CrossRef]
  12. L. Hesselink, Handbook of Flow Visualization (Hemisphere, New York, 1988).
  13. J. Khoury, V. Ryan, C. L. Woods, M. Cronin-Golomb, “Photorefractive optical lock-in detector,” Opt. Lett. 16, 1442–1444 (1991).
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  14. J. Khoury, V. Ryan, M. Cronin-Golomb, C. L. Woods, “Photorefractive frequency converter and phase-sensitive detector,” J. Opt. Soc. Am. B 10, 72–82 (1993).
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  15. J. Khoury, V. Ryan, C. L. Woods, M. Cronin-Golomb, “Photorefractive time correlation motion detection,” Opt. Commun. 85, 5–9 (1991).
    [CrossRef]
  16. T. C. Hale, K. L. Telshow, V. A. Deason, “Photorefractive optical lock-in vibration measurement,” Appl. Opt. 36, 8248–8285 (1997).
    [CrossRef]
  17. B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
    [CrossRef]
  18. B. Cairns, E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922–1928 (1991).
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  19. 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).
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  20. L. E. Estes, L. M. Narducci, R. A. Tuft, “Scattering of light from a rotating ground glass,” J. Opt. Soc. Am. 61, 1301–1306 (1971).
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  21. J. H. Churnside, “Speckle from a rotating diffuse object,” J. Opt. Soc. Am. 72, 1464–1469 (1982).
    [CrossRef]

1997 (1)

1995 (1)

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (4)

1989 (1)

1988 (1)

1987 (1)

1986 (1)

1983 (1)

1982 (1)

1980 (2)

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]

H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
[CrossRef]

1978 (1)

P. G. Simkins, T. D. Dudderar, “Laser speckle measurements of transient Benard convection,” J. Fluid Mech. 89, 665–671 (1978).
[CrossRef]

1975 (1)

1971 (1)

Asakura, T.

Barakat, R.

Cairns, B.

Churnside, J. H.

Collicott, S. H.

Craig, P. N.

Cronin-Golomb, M.

Deason, V. A.

Dudderar, T. D.

P. G. Simkins, T. D. Dudderar, “Laser speckle measurements of transient Benard convection,” J. Fluid Mech. 89, 665–671 (1978).
[CrossRef]

Estes, L. E.

Goldberg, W. M.

Hale, T. C.

Hesselink, L.

Khoury, J.

Kippelen, B.

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Klenk, J.

H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
[CrossRef]

Kukhtarev, N. V.

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Law, R. L.

Leonhard, K.

H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
[CrossRef]

Lugannani, R.

Meerholz, S. K.

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Meynart, R.

Minemoto, T.

Moran, S. E.

Nakagawa, K.

Narducci, L. M.

Peyghambarian, N.

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Ryan, V.

Saleh, B. E. A.

Sandalphon,

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Simkins, P. G.

P. G. Simkins, T. D. Dudderar, “Laser speckle measurements of transient Benard convection,” J. Fluid Mech. 89, 665–671 (1978).
[CrossRef]

Sutanto,

Takai, N.

Telshow, K. L.

Tiziani, H. J.

H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
[CrossRef]

Tuft, R. A.

Volodin, B. L.

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Wolf, E.

Woods, C. L.

Appl. Opt. (5)

J. Fluid Mech. (1)

P. G. Simkins, T. D. Dudderar, “Laser speckle measurements of transient Benard convection,” J. Fluid Mech. 89, 665–671 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

H. J. Tiziani, K. Leonhard, J. Klenk, “Real-time displacement and tilt analysis by a speckle technique using Bi12SiO20 crystal,” Opt. Commun. 34, 327–331 (1980).
[CrossRef]

J. Khoury, V. Ryan, C. L. Woods, M. Cronin-Golomb, “Photorefractive time correlation motion detection,” Opt. Commun. 85, 5–9 (1991).
[CrossRef]

Opt. Eng. (1)

B. L. Volodin, Sandalphon, S. K. Meerholz, B. Kippelen, N. V. Kukhtarev, N. Peyghambarian, “Highly efficient photorefractive polymers for dynamic holography,” Opt. Eng. 34, 2213–2223 (1995).
[CrossRef]

Opt. Lett. (3)

Other (2)

L. Hesselink, Handbook of Flow Visualization (Hemisphere, New York, 1988).

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

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Figures (3)

Fig. 1
Fig. 1

(a) Diagram of the proposed speckle velocimeter by use of an acousto-optic modulator (AO) that splits the incident beam into a Doppler-shifted beam A1 and a transmitted non-Doppler-shifted beam A4. PR, photorefractive medium; BS, beam splitter; D, detector; PZM2, piezoelectric mirror; and RF, radio frequency (driving AO). Beam A2 is the grating readout beam, and beam A3 is the phase conjugate of A4. (b) A cross-sectional view of the rotating diffuser. The black spot near the edge indicates where the beam is hitting the diffuser.

Fig. 2
Fig. 2

Geometry of the diffuser aperture area and the far-field observation point P. The origin of coordinates is at the center of the diffuser aperture. rû′ is the vector from the origin to point P. The unit vector û′ projects onto a unit vector û′ in the x, y plane. θ is the angle that rû′ makes with the z axis, which is normal to the aperture plane. β represents the normalized velocity vector of the diffuser in the x, y plane.

Fig. 3
Fig. 3

Schematic diagram to illustrate the time-integrative correlator. PR, photorefractive medium; BS, beam splitter; D, detector; and PZM1 and PZM2, piezoelectric mirrors. The incident beam is split into the two beams A1 and A4. Beam A2 is the grating readout beam and beam A3 is the phase conjugate of A4.

Equations (18)

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SΩ, ruˆ=AlD2 cos2 θnΩ-12πrσh1-β·uˆ2 SiΩ×exp-lc sin2 θ2σhnΩ-11-β·uˆ2.
Ω=Ω1-β·uˆ=Ω1-β cos Φ,
A3=γA1A2A4/I01/τ1+iωg×--S1ωS4ω-iω-ω+ωg+1τ1×exp-iω-ω+ωgtτdωdω,
A1t=A1 expiω1t,
S1ω=A1δω-ω1.
A4ω=δω-Ω.
Esc=A1A4AlD2 cos2 θn0-122πrσh1-β·uˆ2 ×exp-lD sin θ22σhn0-11-β·uˆ2 ×γI01τ1+iωgexp iω1-Ω-iω1-Ω+ωg+1τ1.
ad=lDσhn0-1.
A3=γA1A2A4I0AL2πr2 ad2cos θ1-β cos ϕ2×exp-ad22sin2 θ1-β cos ϕ2×1τ1+iωgexpiω1-Ω-iω1-Ω+ωg+1τ1,
Iout=02π-ππ |A4|2L2dθdϕ.
|ω1-Ω+ωgτ|1.
Had-ππ ad2 cos4 θ exp-ad2sin2 θdθ.
02π1τ1+iωgexpiω1-Ω-iω1-Ω+ωg+1τ12dϕ=1+ωgτ1202πdϕ1+|ω1-Ω+ωg|2τ12.
1+ωgτ1202π1-|ω1-Ω+ωgτ1|2dϕ+02π |ω1-Ω+ωgτ1|4dϕ.
ω1=ω01-β1=ω01-v1c,
ω1-Ω+ωg=ωg-ω0β1+ω0βcos ϕ.
1+ωgτ22π-πω0β2+2πω0β1-ωg2τ12.
Iout=ΓI1I4I2I0A2πr Had1+ωgτ122π-πω0β2+2πω0β1-ωg2τ12,

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