Abstract

The characteristics of a fully developed speckle pattern that results from in-plane rotation of a diffuse object observed in an arbitrary observation plane, as described by the spatiotemporal cross-correlation function of the optical intensity in complex ABCD optical systems, are derived and discussed. Here we consider off-axis illumination, which is in contrast to previous work in which the illuminating beam was assumed to be parallel to the axis of rotation. The spatiotemporal characteristics of the observed pattern are interpreted in terms of speckle boiling, rotation, and translation. For off-axis illumination it is shown theoretically and experimentally that, for Fourier transform optical systems, in-plane rotation causes the speckles to translate in a direction perpendicular to the direction of surface motion, whereas for an imaging system, the translation is parallel to the direction of surface motion. On this basis, we discuss a novel method, which is independent of both the optical wavelength and the position of the laser spot on the object, for determining either the angular velocity or the corresponding in-plane displacement of the target object.

© 1998 Optical Society of America

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References

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  1. B. E. A. Saleh, “Speckle correlation measurements of the velocity of a small rotating object,” Appl. Opt. 14, 2344–2246 (1975).
    [CrossRef] [PubMed]
  2. N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B: Photophys. Laser Chem. 26, 185–192 (1981).
    [CrossRef]
  3. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular displacement sensor: a theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998).
    [CrossRef]
  4. J. H. Churnside, “Speckle from a rotating diffuse object,” J. Opt. Soc. Am. 72, 1464–1469 (1982).
    [CrossRef]
  5. As discussed in Ref. 4, “This is the only qualitative difference between the motion of a speckle pattern produced by a flat, constant-velocity object and that produced by a curved, constantly rotating object as long as the axis of rotation is perpendicular to the illuminating beam,” p. 1464.
  6. Included in Ref. 4 is a thorough review of previous work regarding speckle resulting from in-plane rotation.
  7. T. Yoshimura, K. Nakagawa, N. Wakabayashi, “Rotational and boiling motion of speckles in a two-lens imaging system,” J. Opt. Soc. Am. A 3, 1018–1022 (1986).
    [CrossRef]
  8. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  9. T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. B: Photophys. Laser Chem. 25, 179–194 (1981).
    [CrossRef]
  10. 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.
  11. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.
  12. We omit in the following the unimportant multiplicative factor exp[iω0t], where ω0 is the angular frequency of the incident light.
  13. 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]
  14. In all the measurements indicated in Figs. 4–7 we applied negative angular displacements θ, but to simplify the graphs, we inserted the magnitude of θ.
  15. S. G. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics, Proceedings of the Fifth International Conference at Kiel-Damp, Germany, May 23–26, 1982, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 212–220.

1998 (2)

1987 (1)

1986 (1)

1982 (1)

1981 (2)

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

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

1975 (1)

Asakura, T.

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

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

Churnside, J. H.

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.

Hansen, R. S.

Hanson, S. G.

Imam, H.

Iwai, T.

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

Nakagawa, K.

Rose, B.

Saleh, B. E. A.

Siegman, A. E.

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

Takai, N.

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

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

Wakabayashi, N.

Yoshimura, T.

Yura, H. T.

Appl. Opt. (2)

Appl. Phys. B: Photophys. Laser Chem. (2)

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

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

J. Opt. Soc. Am. (1)

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

Other (7)

As discussed in Ref. 4, “This is the only qualitative difference between the motion of a speckle pattern produced by a flat, constant-velocity object and that produced by a curved, constantly rotating object as long as the axis of rotation is perpendicular to the illuminating beam,” p. 1464.

Included in Ref. 4 is a thorough review of previous work regarding speckle resulting from in-plane rotation.

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

We omit in the following the unimportant multiplicative factor exp[iω0t], where ω0 is the angular frequency of the incident light.

In all the measurements indicated in Figs. 4–7 we applied negative angular displacements θ, but to simplify the graphs, we inserted the magnitude of θ.

S. G. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics, Proceedings of the Fifth International Conference at Kiel-Damp, Germany, May 23–26, 1982, E. O. Schulz-DuBois, ed. (Springer-Verlag, Berlin, 1983), pp. 212–220.

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Figures (8)

Fig. 1
Fig. 1

Off-axis illumination at position a along the y axis of a flat diffuse target rotating about the Z axis.

Fig. 2
Fig. 2

Propagation of reflected light from the object plane through an arbitrary ABCD system to the detector plane.

Fig. 3
Fig. 3

Optical diagram for the measurement setup.

Fig. 4
Fig. 4

Measured linear displacement in the x direction for an angle of incidence ϕ=0° (open symbols) and ϕ=45° (solid symbols). The image sensor was placed in the Fourier plane, and the target distance was fixed throughout the measurements but was chosen arbitrarily.

Fig. 5
Fig. 5

Measured linear displacement in the y direction for an angle of incidence ϕ=0° (open symbols) and ϕ=45° (solid symbols). The image sensor was placed in the Fourier plane, and the target distance was fixed throughout the measurements but was chosen arbitrarily.

Fig. 6
Fig. 6

Measured linear displacement in the y direction for varying angles of incidence at position a=0 cm. The image sensor was placed in the Fourier plane, and the target distance was fixed throughout the measurements but was chosen arbitrarily.

Fig. 7
Fig. 7

Measured linear displacement in the x direction (open symbols) and y direction (solid symbols) versus position of the image sensor (z1=2 f, a=2 cm, and θ=-100 mdeg).

Fig. 8
Fig. 8

Angular direction of speckle motion with respect to the px axis versus the position of the image sensor.

Equations (26)

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CI(p1, p2; t, t+τ)=I(p1, t)I(p2, t+τ)-I(p1, t)I(p2, t+τ){[I(p1, t)2-I(p1, t)2][I(p2, t+τ)2-I(p2, t+τ)2]}1/2,
I(p1, t)I(p2, t+τ)
=U(p1, t)U*(p1, t)U(p2, t+τ)U*(p2, t+τ)=|Γ(p1, p2; τ)|2+I(p1, t)I(p2, t+τ),
Γ(p1, p2; τ)=U(p1, t)U*(p2, t+τ).
CI(p1, p2; τ)=|Γ(p1, p2; τ)|2Γ(p1, p1; 0)Γ(p2, p2; 0)=|γ(p1, p2; τ)|2,
U(p, t)=drUr(r, t)G(r, p),
G(r, p)=-ik2πBexp-ik2B(Ar2-2r·p+Dp2),
Γ(p1, p2; τ)
=dr1dr2Γ0(r1, r2; τ)G(r1, p1)G*(r2, p2),
Γ0(r1, r2; τ)=Ur(r1, t)Ur*(r2, t+τ)=Ui(r1)Ui*(r2)δ(r2-r1),
Γ(p1, p2; τ)=drUi(r)Ui*(r)×G(r, p1)G*(r, p2).
Ui(r)=U0 expikx sin ϕ-x2rs2(sec ϕ)2+y2rs2,
XY=cos ωτ-sin ωτsin ωτcos ωτXY,
r=xy=x cos ωτ-(y+a)sin ωτx sin ωτ+(y+a)cos ωτ-a.
CI(P, p; τ)=exp-p2ρ021-1-cos ωτ2×exp-2 sin ωτρ02p·R(P+aA¯)×exp-2(1-cos ωτ)×a2rs2+(P+aA¯)2ρ02,
ρ0=4|B|2k2rs2+2kIm(BA*)1/2,
A¯=Ar+2Bikrs2;
R=01-10.
CI(P, p; τ)=exp-{[p+R(P+aA¯)]xωτ}2ρ0x2+{[p+R(P+aA¯)]yωτ}2ρ02×exp-aωτrsβ2-(kαrsωτ)24+(ωτ)2β2px24ρ0x2+py24β2ρ02×exp-2Brα(ωτ)2Pxρ0x2+ωτpyρ02,
ρ0x=4|B|2k2β2rs2+2kIm(BA*)1/2.
CI(P, 0; τ)=exp-(ωτ)2(Px+αBr)2ρ0x2+(Py+aA¯)2ρ02+a2(rsβ)2,
CI(p; τ)=exp-(ωτ)2a2rs2β2-β2px24ρ0x2-py24β2ρ02×exp-(px+aAω¯τ)2ρ0x2×exp-(py+αBrωτ)2ρ02.
A=1-z2/f,
B=z1+z2(1-z1/f),
px0=-aA¯θ=-a(1-z2/f)θ,
py0=-αBrθ=-sin ϕ(z1+z2-z1z2/f)θ,

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