Abstract

The space–time intensity covariance function for illuminating an object giving rise to fully developed speckle is considered in the case where the object is illuminated with two spatially separated beams, or with a multitude of equidistant but spatially separated spots. Specifically, and to the best of our knowledge for the first time, we obtain the result that the larger speckles will be covered by a fine structure that, in general, translates at a different velocity from that of the larger speckles. In particular, closed-form analytical expressions are found for the space- and time-lagged covariance of irradiance as well as the corresponding power spectrum for each of the two spatially separated, N equidistant separated illuminating beams. The present analysis is valid not only for free-space propagation but also for an arbitrary real ABCD optical system. Finally, the corresponding statistical signal properties, including the power spectrum, are derived and discussed for the (practical) case where the comprehensive speckle field is spatially filtered by a gratinglike structure.

© 2008 Optical Society of America

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References

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  1. T. Asakura and N. Takai, "Dynamics of laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
    [Crossref]
  2. Y. Aizu and T. Asakura, Spatial Filtering Velocimetry: Fundamentals and Applications (Springer-Verlag, 2006).
  3. M. L. Jakobsen and S. G. Hanson, "Micro-lenticular array for spatial-filtering velocimetry on solid surfaces," Meas. Sci. Technol. 15, 1949-1957 (2004).
    [Crossref]
  4. D. V. Semenov, E. Nippolainen, and A. A. Kamshilin, "Accuracy and resolution of a dynamic-speckle profilometer," Appl. Opt. 45, 411-418 (2006).
    [Crossref] [PubMed]
  5. L. E. Drain, The Laser Doppler Technique (Wiley, 1980).
  6. L. H. Tanner, "Aparticle timing laser-velocimeter," (Department of Aeronautical Engineering, Queens University of Belfast, 1972).
  7. S. G. Hanson and B. H. Hansen, "Laser-based measurement scheme for rotational measurement of specularly reflective shafts," Proc. SPIE 2292, 143-153 (1994).
    [Crossref]
  8. J. W. Goodman, Statistical Optics (Wiley, 2000) pp. 84-85.
  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. 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, (2006).
  11. M. L. Jakobsen and S. G. Hanson, "Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces," Appl. Opt. 43, 4643-4651 (2004).
    [Crossref] [PubMed]
  12. S. Wolfram, Mathematica Version 6 (Addison-Wesley, 2007).

2006 (2)

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, (2006).

D. V. Semenov, E. Nippolainen, and A. A. Kamshilin, "Accuracy and resolution of a dynamic-speckle profilometer," Appl. Opt. 45, 411-418 (2006).
[Crossref] [PubMed]

2004 (2)

M. L. Jakobsen and S. G. Hanson, "Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces," Appl. Opt. 43, 4643-4651 (2004).
[Crossref] [PubMed]

M. L. Jakobsen and S. G. Hanson, "Micro-lenticular array for spatial-filtering velocimetry on solid surfaces," Meas. Sci. Technol. 15, 1949-1957 (2004).
[Crossref]

1994 (1)

S. G. Hanson and B. H. Hansen, "Laser-based measurement scheme for rotational measurement of specularly reflective shafts," Proc. SPIE 2292, 143-153 (1994).
[Crossref]

1987 (1)

1981 (1)

T. Asakura and N. Takai, "Dynamics of laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[Crossref]

Appl. Opt. (2)

Appl. Phys. (1)

T. Asakura and N. Takai, "Dynamics of laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[Crossref]

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

Meas. Sci. Technol. (1)

M. L. Jakobsen and S. G. Hanson, "Micro-lenticular array for spatial-filtering velocimetry on solid surfaces," Meas. Sci. Technol. 15, 1949-1957 (2004).
[Crossref]

Proc. SPIE (2)

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, (2006).

S. G. Hanson and B. H. Hansen, "Laser-based measurement scheme for rotational measurement of specularly reflective shafts," Proc. SPIE 2292, 143-153 (1994).
[Crossref]

Other (5)

J. W. Goodman, Statistical Optics (Wiley, 2000) pp. 84-85.

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

L. E. Drain, The Laser Doppler Technique (Wiley, 1980).

L. H. Tanner, "Aparticle timing laser-velocimeter," (Department of Aeronautical Engineering, Queens University of Belfast, 1972).

S. Wolfram, Mathematica Version 6 (Addison-Wesley, 2007).

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Figures (10)

Fig. 1
Fig. 1

Compound speckle field observed in the p plane from a multitude of illuminated spots at the moving object, r plane. An array of facets with pitch Λ followed by a lens and two detectors facilitates the bandpass filtering of the speckle pattern.

Fig. 2
Fig. 2

Normalized intensity space–time correlation function for collimated illumination, where L = 5 mm ; w 0 = 0.1 mm ; λ = 850 nm ; θ = 3.6 ° , and ν τ = 0 μ m , 50 μ m , 100 μ m , and 150 μ m .

Fig. 3
Fig. 3

Normalized power spectrum as a function of frequency for collimated illumination, where λ = 850 nm , θ = 3.6 ° , ν = 1 mm s , and w 0 = 0.1 mm . The black, red (online), and blue (online) curves refer to L = 4 , 6, and 8 mm , respectively.

Fig. 4
Fig. 4

Same as Fig. 3, except w 0 = 0.2 mm .

Fig. 5
Fig. 5

Same as Fig. 3, except w 0 = 0.4 mm .

Fig. 6
Fig. 6

Power at ω = ω 1 relative to that at ω = ω 2 (expressed in decibels) as a function of the detuning parameter α.

Fig. 7
Fig. 7

Power spectrum for collimated illumination (i.e., for ρ = 1 ) as a function of frequency for α = 1 for the same values of the parameters indicated in Fig. 2, except here θ = 1.9 ° .

Fig. 8
Fig. 8

Three-spot (lower curve) and five-spot (upper curve) normalized power spectra as a function of frequency for collimated illumination, where L = 4 mm , w 0 = 0.05 mm , λ = 850 nm , and θ = 1.9 ° .

Fig. 9
Fig. 9

Rectangular array transmission function for Λ = D 10 .

Fig. 10
Fig. 10

Comparison of the periodic rectangular and trigonometric overlap integrals for Λ = D 10 .

Equations (53)

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Γ ( Δ p , 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 r e f ( r ) d 2 r ,
G ( r , p ) = exp [ i k ( r p ) 2 2 L ] ,
U r e f ( r , t ) = η ( r , t ) U i n c ( r ) ,
η ( r , t ) η * ( r , t ) = constant × δ [ r r v ( t t ) ] ,
U i n c ( 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 i k ( r Δ r 2 ) θ ] ,
γ ( Δ p , τ ) = γ 11 ( Δ p , τ ) + γ 12 ( Δ p , τ ) + γ 21 ( Δ p , τ ) + γ 22 ( Δ p , τ ) ,
γ i i ( Δ p , τ ) = G ( r , p ) G * ( r v τ , p Δ p ) U i ( r ) U i * ( r v τ ) d 2 r , i = 1 , 2 ,
γ 12 ( Δ p , τ ) = G ( r , p ) G * ( r v τ , p Δ p ) U 1 ( r ) U 2 * ( r v τ ) d 2 r ,
γ 21 ( Δ p , τ ) = G ( r , p ) G * ( r v τ , p Δ p ) U 2 ( r ) U 1 * ( r v τ ) d 2 r .
Γ ( Δ p , τ ) = γ 11 ( Δ p , τ ) 2 + γ 22 ( Δ p , τ ) 2 + 2 Re [ γ 11 ( Δ p , τ ) γ 22 * ( Δ p , τ ) ] + γ 12 ( Δ p , τ ) 2 + γ 21 ( Δ p , τ ) 2 + 2 Re [ γ 11 ( Δ p , τ ) γ 12 * ( Δ p , τ ) + γ 11 ( Δ p , τ ) γ 21 * ( Δ p , τ ) + γ 12 ( Δ p , τ ) γ 21 * ( Δ p , τ ) + γ 22 ( Δ p , τ ) γ 21 * ( Δ p , τ ) + γ 22 ( Δ p , τ ) γ 12 * ( Δ p , τ ) ] .
Γ ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ν τ ( 1 L R ) ) 2 w S 2 ] cos 2 [ ( Δ p 2 ν τ ) k θ ] ,
w S = 2 L k w 0
Γ ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ν τ ( A B R ) ) 2 w S 2 ] cos 2 [ { Δ p ν τ ( 1 + A ) } k θ ] ,
Γ ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ν τ ) 2 w S 2 ] cos 2 [ ( Δ p 2 ν τ ) k θ ] .
Γ ( Δ p , 0 ) = exp [ Δ p 2 w S 2 ] cos 2 [ Δ p k θ ] .
Γ ( 0 , τ ) = exp [ ( ν τ ) 2 ( 1 w 0 2 + 1 w S 2 ) ] cos 2 [ 2 k θ ν τ ] .
P ( ω ) = Γ ( 0 , τ ) exp [ i ω τ ] d τ .
P ( ω ) 2 exp [ ω N 2 1 + β 2 ( 1 L R ) 2 ] + exp [ ( ω N ω 1 ) 2 1 + β 2 ( 1 L R ) 2 ] + exp [ ( ω N + ω 1 ) 2 1 + β 2 ( 1 L R ) 2 ] ,
ω N = ω k w 0 ν L ,
ω 1 = 4 L θ w 0 ,
β = w S w 0 = 2 L k w 0 2
Δ ω r e l = { 1 + β 2 ω 1 w 0 4 L θ , β 1 2 k θ w 0 , β 1 .
i ( t ) = I ( p , t ) T ( p ) d p ,
T 1 ( p ) = 1 2 ( 1 + cos K p ) exp [ p 2 D 2 ] ,
K = 2 π Λ ,
T 2 ( p ) = 1 2 ( 1 + cos K p ) exp [ p 2 D 2 ] 1 2 ( 1 + cos [ K p + π ] ) exp [ p 2 D 2 ] = exp [ p 2 D 2 ] cos K p .
R ( τ ) = ( i ( t ) i ( t + τ ) i ( t ) i ( t + τ ) ) i ( t ) i ( t + τ ) = [ I ( p 1 , t ) I ( p 2 , t + τ ) I ( p 1 , t ) I ( p 2 , t + τ ) I ( p 1 , t ) I ( p 2 , t + τ ) ] T ( p 1 ) T ( p 2 ) d p 2 d p 1 = Γ ( p 2 p 1 , τ ) T ( p 1 ) T ( p 2 ) d p 2 d p 1 .
R ( τ ) = Γ ( Δ p , τ ) H ( Δ p ) d Δ p ,
H ( Δ p ) = T ( P + Δ p 2 ) T ( P Δ p 2 ) d P .
H 1 ( Δ p ) = 1 2 ( 1 + cos K Δ p ) exp [ Δ p 2 2 D 2 ] ,
H 2 ( Δ p ) = exp [ Δ p 2 2 D 2 ] cos K Δ p ,
R 1 ( τ ) = R 2 ( τ ) + 2 R 2 ( τ ) K = 0 .
R 2 ( τ ) = exp [ ( ν τ w 0 ) 2 ( w S K 2 ) 2 ] cos ( K ν τ ) + 1 2 ( exp [ ( ν τ w 0 ) 2 ( w S ( K + 2 k θ ) 2 ) 2 ] cos [ ( K ρ 2 k θ ( 2 ρ ) ) ν τ ] + exp [ ( ν τ w 0 ) 2 ( w S ( K 2 k θ ) 2 ) 2 ] cos [ ( K ρ + 2 k θ ( 2 ρ ) ) ν τ ] ) ,
ρ = 1 L R
P 2 ( ω ) = exp [ K 2 w s 2 4 ] exp [ ( ω K ρ ν ) 2 ω 0 2 ] + 1 2 exp [ ( K 2 k θ ) 2 w S 2 4 ] exp [ ( ω K ν ρ 2 k ν θ ( 2 ρ ) ) 2 ω 0 2 ] + 1 2 exp [ ( K + 2 k θ ) 2 w S 2 4 ] exp [ ( ω K ν ρ 2 k ν θ ( 2 ρ ) ) 2 ω 0 2 ] + terms ω ω ,
P 2 ( ω ) K = 2 k θ = exp [ ( k θ w S ) 2 ] exp [ ( ω 2 k ρ ν θ ) 2 ω 0 2 ] + 1 2 exp [ ( ω 4 k ν θ ) 2 ω 0 2 ] + 1 2 exp [ ( 2 k θ w S ) 2 ] exp [ ( ω 4 k ν θ ) 2 ω 0 2 ] + terms ω ω .
P 2 ( ω ) K = 2 k θ exp [ ( 2 L θ w 0 ) 2 ] exp [ ( ω 2 k ρ v θ ) 2 ω 0 2 ] + 1 2 exp [ ( ω 4 k ν θ ) 2 ω 0 2 ] , ω 0 .
P 1 ( ω ) = P 2 ( ω ) + 2 P 2 ( ω ) K = 0 .
exp [ ( k θ w S ) 2 ] = exp [ ( 2 L θ w 0 ) 2 ( α 1 ) 2 ] = exp [ ( beam separation spot radius ) 2 ( α 1 ) 2 ] < 1 .
U i n c ( r ) = exp [ p 2 D 0 2 ] m = n m = n U m ( r ) ,
U m ( r ) = U 0 exp [ ( r m L θ ) 2 w 0 2 i k ( r m L θ ) 2 2 R i k ( r m L θ ) m θ ] .
Γ N ( Δ p , τ ) = i = n n γ i i 2 + i = n , n γ i i j = n j i n γ j j * = i = n n γ i i 2 + i = n , n γ i i ( j = n n γ j j * γ i i * ) = i = n n γ i i 2 ,
γ m m = exp [ ( ν τ ) 2 2 w 0 2 ( Δ p ρ ν τ ) 2 2 w S 2 + i k 2 ( 2 p ( ν τ Δ p ) + Δ p ( Δ p + 2 m L θ ) ν τ ( Δ p + 4 m L θ ) ) ] ,
Γ N ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ρ ν τ ) 2 w S 2 ] ( sin [ k θ ( 2 ν τ Δ p ) ( 2 n + 1 ) 2 ] sin [ k θ ( 2 ν τ Δ p ) 2 ] ) 2 .
Γ N ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ( A B R ) ν τ ) 2 w S 2 ] ( sin [ k θ ( ( 1 + A ) ν τ Δ p ) ( 2 n + 1 ) 2 ] sin [ k θ ( ( 1 + A ) ν τ Δ p ) 2 ] ) 2 .
Γ 3 ( Δ p , τ ) = exp [ ( ν τ ) 2 w 0 2 ( Δ p ρ ν τ ) 2 w S 2 ] ( 1 + 2 cos [ k θ ( 2 ν τ Δ p ) ] ) 2 .
P N ( ω ) = R N ( τ ) exp [ i ω τ ] d τ ,
R N ( τ ) = Γ N ( Δ p , τ ) H ( Δ p ) d Δ p ,
P 3 ( ω ) = exp [ ( ω 2 k ν θ ) 2 ω 0 2 ] + 1 2 exp [ ( 3 L θ w 0 ) 2 ( ω ( 4 3 ρ ) k ν θ ) 2 ω 0 2 ] + exp [ ( 2 L θ w 0 ) 2 ( ω 2 ( 1 ρ ) k ν θ ) 2 ω 0 2 ] + 3 2 exp [ ( L θ w 0 ) 2 ( ω k ν ρ θ ) 2 ω 0 2 ] + 1 2 exp [ ( L θ w 0 ) 2 ( ω ( 4 ρ ) k ν θ ) 2 ω 0 2 ] + terms ω ω .
T R = UnitStep [ cos ( 2 π x Λ ) ] exp [ x 2 D 2 ] ,

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