Abstract

We consider the statistics of the spatially integrated speckle intensity difference obtained from two separated finite collecting apertures. For fully developed speckle, closed-form analytic solutions for both the probability density function and the cumulative distribution function are derived here for both arbitrary values of the mean number of speckles contained within an aperture and the degree of coherence of the optical field. Additionally, closed-form expressions are obtained for the corresponding nth statistical moments.

© 2009 Optical Society of America

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References

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  1. J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367-2368 (1962).
  2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  3. R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91-102 (1987).
    [CrossRef]
  4. P. K. Rastogi, Optical Measurement Techniques (Artech House, 1997).
  5. M. S. Beck and A. Plaskowski, Cross Correlation Flowmeters--Their Design and Application (Hilger, 1987).
  6. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 4.6.2.
  7. S. Wolfram, Mathematica Version 6 (Addison-Wesley, 2007).

1987 (1)

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91-102 (1987).
[CrossRef]

1962 (1)

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367-2368 (1962).

Barakat, R.

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91-102 (1987).
[CrossRef]

Beck, M. S.

M. S. Beck and A. Plaskowski, Cross Correlation Flowmeters--Their Design and Application (Hilger, 1987).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 4.6.2.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Gordon, E. I.

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367-2368 (1962).

Plaskowski, A.

M. S. Beck and A. Plaskowski, Cross Correlation Flowmeters--Their Design and Application (Hilger, 1987).

Rastogi, P. K.

P. K. Rastogi, Optical Measurement Techniques (Artech House, 1997).

Rigden, J. D.

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367-2368 (1962).

Wolfram, S.

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

J. Mod. Opt. (1)

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91-102 (1987).
[CrossRef]

Proc. IRE (1)

J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367-2368 (1962).

Other (5)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

P. K. Rastogi, Optical Measurement Techniques (Artech House, 1997).

M. S. Beck and A. Plaskowski, Cross Correlation Flowmeters--Their Design and Application (Hilger, 1987).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Sec. 4.6.2.

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

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Figures (2)

Fig. 1
Fig. 1

PDF of the integrated speckle intensity difference as a function of the normalized intensity difference for the field correlation coefficient μ and various values of the mean number of speckles over the aperture, m.

Fig. 2
Fig. 2

PDF of the integrated intensity difference as a function of Δ W N for m = 3 and various values of the field correlation coefficient, μ.

Equations (23)

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W ( t ) = 1 A A p E ( r , t ) 2 d r ,
p ( W ) = 1 Γ ( m ) ( m W ) m W m 1 exp ( m W W ) ,
m 1 + A A C ,
p ( W 1 , W 2 ) = ( m W ) 2 1 μ 2 ( m 2 W 1 W 2 μ 2 W 2 ) ( m 1 ) 2 × I m 1 [ 2 μ 1 μ 2 m 2 W 1 W 2 W 2 ] × exp [ m 1 μ 2 ( W 1 + W 2 ) W ] ,
μ ( Δ r ) = E 1 ( r ) E 2 * ( r + Δ r ) E 2 ,
χ ( ω ) = 0 d W 1 0 d W 2 p ( W 1 , W 2 ) exp [ i ω ( W 1 W 2 ) ] .
p ( Δ W ) = 1 2 π d ω χ ( ω ) exp [ i ω Δ W ] .
χ ( ω ) = [ 1 + ( 1 μ 2 ) W 2 ω 2 m 2 ] m .
p ( Δ W ; m , μ ) = 1 π 0 d ω [ 1 + ( 1 μ 2 ) W 2 ω 2 m 2 ] m cos ( ω Δ W ) = 2 1 2 m m 1 2 + m ( 1 μ 2 ) 1 4 m 2 Δ W N m 1 2 K 1 2 m ( m Δ W N 1 μ 2 ) π Γ ( m ) ,
Δ W N = Δ W W ,
p ( Δ W N ; 1 , μ ) = 1 2 1 μ 2 exp [ Δ W N 1 μ 2 ] ,
p ( Δ W N ; 2 , μ ) = 1 2 ( 1 μ 2 ) exp [ 2 Δ W N 1 μ 2 ] ( 2 Δ W N + 1 μ 2 ) ,
p ( Δ W N ; 3 , μ ) = 9 16 ( 1 μ 2 ) 3 2 exp [ 3 Δ W N 1 μ 2 ] ( 3 Δ W N 2 + 3 Δ W N 1 μ 2 + 1 μ 2 ) .
P ( W N ; m , μ ) = W N d W N p ( W N ; m , μ ) .
P ( W N ; m , μ ) = 2 2 m 1 m π Γ ( m ) ( 1 μ 2 ) m + 1 2 { 2 2 m m π Γ ( m ) ( 1 μ 2 ) m + 1 2 + 2 m m 2 W N Γ ( m 1 2 ) ( 1 μ 2 ) m F 2 1 [ 1 2 ; 3 2 , 3 2 m ; m 2 W N 2 4 ( 1 μ 2 ) ] 1 μ 2 ( m W N ) 2 m Γ ( 1 2 m ) F 2 1 [ m ; m + 1 2 , m + 1 ; m 2 W N 2 4 ( 1 μ 2 ) ] } for Δ W N 0
P ( W N ; m , μ ) = 1 2 + 2 2 m 1 m π Γ ( m ) ( 1 μ 2 ) m + 1 2 { 2 2 m m 2 Δ W N Γ ( m 1 2 ) ( 1 μ 2 ) m F 2 1 [ 1 2 ; 3 2 , 3 2 m ; m 2 W N 2 4 ( 1 μ 2 ) ] + ( m Δ W N ) 2 m Γ ( 1 2 m ) 1 μ 2 F 2 1 [ m ; m + 1 2 , m + 1 ; m 2 W N 2 4 ( 1 μ 2 ) ] } for Δ W N 0 ,
P ( Δ W N ; 1 , μ ) = { 1 2 exp ( Δ W N 1 μ 2 ) for Δ W N 0 1 1 2 exp ( Δ W N 1 μ 2 ) for Δ W N 0 } ,
P ( Δ W N ; 2 , μ ) = { 1 2 [ 1 Δ W N 1 μ 2 ] exp ( 2 Δ W N 1 μ 2 ) for Δ W N 0 1 1 2 [ 1 + Δ W N 1 μ 2 ] exp ( 2 Δ W N 1 μ 2 ) for Δ W N 0 } .
M n = ( i ) n d n χ ( ω ) d ω n ω = 0 = ( i ) n d n [ 1 + ( 1 μ 2 ) W 2 ω 2 m 2 ] m d ω n ω = 0 = 2 n 2 ( n 1 ) ! ! ( 1 μ 2 ) n 2 m n 1 j = 1 n 2 1 ( m + j ) ,
for n even , = 0 , for n odd ,
σ Δ W N 2 = M 2 = 2 ( 1 μ 2 ) m ,
S = M 3 M 2 2 = 0 ,
γ = M 4 M 2 2 3 = 3 m .

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