Abstract

An out-of-plane rotating object is illuminated with two spatially separated coherent beams, giving rise to fully developed speckles, which will translate and gradually decorrelate in the observation plane, located in the far field. The speckle pattern is a compound structure, consisting of random speckles modulated by a smaller and repetitive structure. Generally, these two components of the compound speckle structure will move as rigid structures with individual velocities determined by the characteristics of the two illuminating beams. Closed-form analytical expressions are found for the space- and time-lagged covariance of irradiance and the corresponding power spectrum for the two spatially separated illuminating beams. The present analysis is valid for propagation through an arbitrary ABCD system, though the focus for the experimental evaluation is far-field observations using an optical Fourier transform system. It is shown that the compound speckle structures move as two individual structures with the same decorrelation length. The velocity of the random speckles is a combination of angular and peripheral velocity, where the peripheral velocity is inversely proportional to the radius of the wavefront curvature of the incident beams. The velocity of the repetitive structure is a combination of angular and peripheral velocity, where the peripheral velocity is proportional to the ratio of the angle to the distance between the beams in the object plane. Experimental data demonstrate good agreement between theory and measurements for selected combinations of beam separation, angle between beams, and radius of wavefront curvature at the object.

© 2009 Optical Society of America

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References

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  1. H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of opticaly rough surface accuracy,” Opt. Commun. 5, 271-274 (1972).
    [CrossRef]
  2. D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201-213 (1976).
    [CrossRef]
  3. 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]
  4. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, and R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119-2129 (1998).
    [CrossRef]
  5. H. T. Yura, S. G. Hanson, and M. L. Jakobsen, “Speckle dynamics resulting from multiple interfering beams,” J. Opt. Soc. Am. A 25, 318-326 (2008).
    [CrossRef]
  6. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomenon, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap 2.
  7. J. A. Leendertz, “Interferometric displacement measurements on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214-218 (1970).
    [CrossRef]
  8. Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Sciences (Springer, 2005).
  9. S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
    [CrossRef]
  10. N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), pp. 84-85.
  12. A. E. Siegman, Lasers (University Science Books, 1986).

2008 (1)

2006 (1)

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[CrossRef]

2005 (1)

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Sciences (Springer, 2005).

1998 (1)

1986 (1)

A. E. Siegman, Lasers (University Science Books, 1986).

1985 (1)

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), pp. 84-85.

1984 (2)

N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
[CrossRef]

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

1981 (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]

1976 (1)

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201-213 (1976).
[CrossRef]

1972 (1)

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of opticaly rough surface accuracy,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

1970 (1)

J. A. Leendertz, “Interferometric displacement measurements on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214-218 (1970).
[CrossRef]

Aizu, Y.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Sciences (Springer, 2005).

Asakura, T.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Sciences (Springer, 2005).

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]

Dam-Hansen, C.

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[CrossRef]

Eastwood, P. G.

N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), pp. 84-85.

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

Gregory, D. A.

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201-213 (1976).
[CrossRef]

Halliwell, N. A.

N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
[CrossRef]

Hansen, R. S.

Hanson, S. G.

Imam, H.

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]

Jakobsen, M. L.

H. T. Yura, S. G. Hanson, and M. L. Jakobsen, “Speckle dynamics resulting from multiple interfering beams,” J. Opt. Soc. Am. A 25, 318-326 (2008).
[CrossRef]

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[CrossRef]

Leendertz, J. A.

J. A. Leendertz, “Interferometric displacement measurements on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214-218 (1970).
[CrossRef]

Pedersen, H. C.

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[CrossRef]

Pickering, C. J. D.

N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
[CrossRef]

Rose, B.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Stubager, J.

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[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]

Tiziani, H. J.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of opticaly rough surface accuracy,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

Yura, H. T.

Appl. Opt. (1)

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. A (1)

J. Phys. E (1)

J. A. Leendertz, “Interferometric displacement measurements on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214-218 (1970).
[CrossRef]

J. Sound Vib. (1)

N. A. Halliwell, C. J. D. Pickering, and P. G. Eastwood, “The laser torsional vibrometer: a new instrument,” J. Sound Vib. 93, 588-592 (1984).
[CrossRef]

Opt. Commun. (1)

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of opticaly rough surface accuracy,” Opt. Commun. 5, 271-274 (1972).
[CrossRef]

Opt. Laser Technol. (1)

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201-213 (1976).
[CrossRef]

Proc. SPIE (1)

S. G. Hanson, M. L. Jakobsen, H. C. Pedersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006).
[CrossRef]

Other (4)

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Series in Optical Sciences (Springer, 2005).

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

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), pp. 84-85.

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the rotating object, illuminated by two beams, while the reflected light is collected by an optical Fourier transforming system, giving rise to compound speckles in the observation plane.

Fig. 2
Fig. 2

The field reflected off the object’s surface is propagated to the observation plane through an arbitrary optical A B C D system.

Fig. 3
Fig. 3

Schematics of the experimental set up. The components are a beam splitter (BS1) for dividing the beam into two beams, a polarizing beam splitter (PBS2), a quarter-wave plate (QW) for guiding the beams, a Fourier transforming lens (FTL), and a CCD camera. The lens ( L C ) is inserted for experiment III.

Fig. 4
Fig. 4

Speckle displacement measured for an angular displacement of 0.27 mrad as a function of object radius for experiment I.

Fig. 5
Fig. 5

Speckle displacement measured for an angular displacement of 0.13 mrad as a function of object radius for experiment II.

Fig. 6
Fig. 6

Speckle displacement measured for an angular displacement of 0.13 mrad as a function of object radius for experiment III.

Tables (3)

Tables Icon

Table 1 Experimental Parameters for Experiment I

Tables Icon

Table 2 Experimental Parameters for Experiment II

Tables Icon

Table 3 Experimental Parameters for Experiment III

Equations (29)

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Γ N ( Δ p , τ ) Γ ( Δ p , τ ) Γ ( 0 , τ ) = | γ ( Δ p , τ ) | 2 ,
γ ( Δ p , τ ) = U ( p , t ) U * ( p Δ p , t τ ) U ( p , t ) U * ( p , t ) .
U ( p , t ) = G ( r , p ) U ref ( r , t ) d 2 r .
G ( r , p ) exp [ i k 2 B ( A r 2 2 r · p + D p 2 ) ] ,
U ref ( r , t ) = η ( r , t ) U inc ( r ) ,
η ( r , t ) η * ( r , t ) = constant × δ [ r r v ( t t ) ] ,
U inc ( r ) = U 1 ( r ) + U 2 ( r ) .
U 1 , 2 ( r ) = U 0 exp [ ( r ± Δ r / 2 ) 2 w 0 2 i k ( r ± Δ r / 2 ) 2 2 R C i k ( r ± Δ r / 2 ) · θ ] ,
Γ ( Δ p , τ ) = exp [ ( v τ ) 2 w 0 2 ( Δ p v τ [ A B / R C ] ) 2 w S 2 ] cos 2 [ k Δ x 2 B ( Δ p v τ ( 2 B θ Δ x + A ) ) ] ,
w S = 2 B / k w 0 ,
A = 2 f / R ,
B = f .
v SS = ( A B R C ) v = 2 f ω R ( 1 R 2 R C ) ,
v FS = ( 2 f θ Δ x + 2 f R ) v = 2 f ω R ( 1 + R θ Δ x ) .
θ = Δ x / 2 R C
v SS = v FS = 2 f ω R .
v SS = 2 f ω R f R C v ,
v FS = 2 f ω R + v .
P ( ω ) = Γ ( 0 , τ ) exp [ i ω τ ] d τ .
P ( ω ) 2 exp [ ω N 2 ( A B R C ) 2 + β 2 ] + exp [ ( ω N ω 1 ) 2 ( A B R C ) 2 + β 2 ] + exp [ ( ω N + ω 1 ) 2 ( A B R C ) 2 + β 2 ] ,
ω N = B k w 0 v ω ,
ω 1 = 2 B w 0 ( θ + A Δ x 2 B )
β = w S w 0 = 2 B k w 0 2 .
ω offset = ± k Δ x B v FS .
Δ ω offset = 2 ( v SS w S ) 2 + ( v w 0 ) 2 ,
P ( κ ) = Γ ( Δ p , 0 ) exp [ i κ Δ p ] d Δ p .
P ( κ ) = 1 2 ( 2 exp [ ( κ w S ) 2 4 ] + exp [ ( κ k Δ x / B ) 2 w S 2 4 ] + exp [ ( κ + k Δ x / B ) 2 w S 2 4 ] ) ,
( α R 2 α + θ ) = ( 1 0 1 F 1 ) ( α R θ ) .
F = R / 2 ,

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