Abstract

We analyze the dynamics of laser speckle patterns, designed for sensing with a receiver, based on spatial filtering. The speckle translation arises after free-space propagation of light scattered from nonspecular surfaces of a solid object in motion. The speckle pattern is manipulated by modulating the intensity of the coherent light, illuminating the target. The space–time normalized cross covariance of speckle patterns incident on the spatial sensor is calculated for the field distribution of three Gaussian beams having arbitrary directions and separations when incident on the target. The modulation of the intensity distribution at the target introduce a higher spatial frequency component in the speckle pattern. The theoretical analysis provides the statistical parameters for both the speckles and the higher spatial frequency component. The analysis reveals that the speckles and the higher spatial frequency component do not necessarily translate as a rigid structure. The theoretical findings are supported by measurements.

© 2008 Optical Society of America

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References

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  1. Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).
  2. U. Schnell, J. Piot, and R. Dändliker, “Detection of motion with laser speckle patterns: statistical properties,” J. Opt. Soc. Am. A 15, 207-216 (1998).
    [CrossRef]
  3. M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering of laser speckle from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
    [CrossRef] [PubMed]
  4. M. L. Jakobsen and S. G. Hanson, “Miniaturized lenticular array for laser speckle from solid surfaces,” Meas. Sci. Technol. 15, 1949-1957 (2004).
    [CrossRef]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), Chap. 2.
  6. S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
    [CrossRef]
  7. H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1160-1166 (1998).
    [CrossRef]
  8. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16, 1402-1412 (1999).
    [CrossRef]
  9. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931-1948 (1987).
    [CrossRef]
  10. A. E. Siegman, Lasers (University Science, 1986), Chap. 20.
  11. A. E. Siegman, Lasers (University Science, 1986), Chap. 17.

2006

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

2004

M. L. Jakobsen and S. G. Hanson, “Miniaturized lenticular array for laser speckle from solid surfaces,” Meas. Sci. Technol. 15, 1949-1957 (2004).
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering of laser speckle from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
[CrossRef] [PubMed]

1999

1998

1987

Aizu, Y.

Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).

Asakura, T.

Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).

Dam-Hansen, C.

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

Dändliker, R.

Goodman, J. W.

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

Hansen, R. S.

Hanson, S. G.

Jakobsen, M. L.

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering of laser speckle from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
[CrossRef] [PubMed]

M. L. Jakobsen and S. G. Hanson, “Miniaturized lenticular array for laser speckle from solid surfaces,” Meas. Sci. Technol. 15, 1949-1957 (2004).
[CrossRef]

Petersen, H. C.

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

Piot, J.

Rose, B.

Schnell, U.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), Chap. 17.

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

Stubager, J.

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

Yura, H. T.

Appl. Opt.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

M. L. Jakobsen and S. G. Hanson, “Miniaturized lenticular array for laser speckle from solid surfaces,” Meas. Sci. Technol. 15, 1949-1957 (2004).
[CrossRef]

Proc. SPIE

S. G. Hanson, M. L. Jakobsen, H. C. Petersen, C. Dam-Hansen, and J. Stubager, “Miniaturized optical speckle-based sensor for cursor control,” Proc. SPIE 6341, 63411U (2006) doi:10.1117/12.695469.
[CrossRef]

Other

Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).

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

A. E. Siegman, Lasers (University Science, 1986), Chap. 17.

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

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for measuring the simultaneous translation of speckle and fine structure.

Fig. 2
Fig. 2

(Left) Speckle pattern with fine structure observed with the CMOS camera and (right) the corresponding autocorrelation function.

Fig. 3
Fig. 3

Ratio of the gain factor for the speckle pattern and the gain factor for the fine structure (circles and squares, left axis) as a function of Δ z between the back focal plane of L and the VCSEL. The circles represent data points obtained with the SM VCSEL, while the squares represent data points obtained with the MM VCSEL. The gain factor of for the fine structure is normalized to the expected gain (triangles, right axis) as a function of Δ z .

Fig. 4
Fig. 4

Modeled gain ratios as functions of the decollimation ( Δ z ) of the laser beam for various beam divergences of a SM VCSEL.

Fig. 5
Fig. 5

R at the target as a function of decollimation ( Δ z ) of the laser beam at various beam divergences of a SM VCSEL.

Equations (18)

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C ( p 1 , p 2 ; τ ) = I ( p 1 , t ) I ( p 2 , t + τ ) I ( p 1 , t ) I ( p 2 , t + τ ) { [ I ( p 1 , t ) 2 I ( p 1 , t ) 2 ] [ I ( p 2 , t + τ ) 2 I ( p 2 , t + τ ) 2 ] } 1 / 2 ,
C I ( p 1 , p 2 ; τ ) = | Γ ( p 1 , p 2 ; τ ) | 2 Γ ( p 1 , p 2 ; 0 ) Γ ( p 1 , p 2 ; 0 ) ,
Γ ( p 1 , p 2 , τ ) = U ( p 1 , t ) U * ( p 2 , t + τ )
U ( p , t ) = S d 2 r U 0 ( r , t ) G ( r , p ) ,
G ( r , p ) = i k 2 π z exp ( i k 2 z ( r 2 2 r · p + p 2 ) )
U 0 ( r , t ) = U i ( r ) Ψ ( r , t ) ,
Ψ ( r 1 , t ) Ψ * ( r 2 , t ) = const . × δ ( r 1 r 2 ) ,
Γ 0 ( r 1 , r 2 ; τ ) = U 0 ( r 1 , t ) U 0 * ( r 2 , t + τ ) = U i ( r 1 ) U i * ( r 2 ) δ ( r 2 r 1 ) ,
Γ ( p 1 , p 2 ; τ ) = S d 2 r U i ( r ) U i * ( r ) G ( r , p 1 ) G * ( r , p 2 ) ,
r = r + v τ .
U i ( r ) = n = M 1 M 1 m = M 2 M 2 ( 2 P n , m π r s 2 ) 1 / 2 exp ( ( r Λ n , m ) 2 ( 1 r s 2 + i k 2 R ) + i k ( r Λ n , m ) · θ n , m + i φ n , m ) .
C I ( p 1 , p 1 + Δ p ; τ ) = C 0 ( n = M 1 M 1 n = M 1 M 1 ( exp ( 1 ρ 2 [ ( 2 z k r s 2 ) 2 ( v τ ( n + n ) Λ 1 , 0 ) 2 + [ Δ p ( 1 + z R ) v τ ( n + n ) ( z θ 1 , 0 z R Λ 1 , 0 ) ] 2 ] ) cos ( k ( n n ) 2 z [ Λ 1 , 0 · ( Δ p ( 1 + z α Λ ) v τ + ( n + n ) z θ 1 , 0 ) ] ) ) ) 2 ,
C 0 = ( n = M 1 M 1 n = M 1 M 1 [ exp ( ( n + n ) 2 2 ( Λ 1 , 0 r s ) 2 ( n + n ) 2 ρ 2 ( z θ 1 , 0 z R Λ 1 , 0 ) 2 ) × cos ( k ( n n ) ( n + n ) 2 Λ 1 , 0 · θ 1 , 0 ) ] ) 2 .
1 ρ 2 ( 2 z k r s 2 ) 2 d dec 2 = 1 d dec = 2 1 / 2 r s .
lim Λ 0 [ C I ( p 1 , p 1 + Δ p ; τ ) ] ( n = M 1 M 1 n = M 1 M 1 ( exp ( 1 ρ 2 [ ( 2 z k r s 2 ) 2 ( v τ ) 2 + [ Δ p ( 1 + z R ) v τ ( n + n ) z θ 1 , 0 ] 2 ] ) cos ( k α 2 ( n n ) v τ ) ) ) 2 .
C I ( p 1 , p 1 + Δ p ; τ ) ( exp ( 1 ρ 2 [ ( 2 z k r s 2 ) 2 ( v τ ) 2 + [ Δ p ( 1 + z R ) v τ ] 2 ] ) ( 1 + 2 cos ( k z [ Λ 1 , 0 · ( Δ p ( 1 + z α Λ ) v τ ) ] ) ) ) 2 .
g sp = 1 + z R , g fs = 1 + z α Λ .
g sp g fs = 1 7 ( 1 + 60 mm R ) .

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