Abstract

The doubly and singly clipped correlation functions of aperture integrated laser speckle are evaluated and compared.

© 1986 Optical Society of America

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References

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  1. R. Barakat, J. Blake, “Second-Order Statistics of Speckle Patterns Observed through Finite-Size Scanning Apertures,” J. Opt. Soc. Am. 68, 614 (1978).
    [CrossRef]
  2. H. Pedersen, “Theory of Speckle-Correlation Measurements Using Nonlinear Detectors,” J. Opt. Soc. Am. A 1, 850 (1985).
    [CrossRef]
  3. J. Ohtsubo, “Intensity Clipping Correlation of Speckle Patterns,” Appl. Opt. 24, 746 (1985).
    [CrossRef] [PubMed]
  4. J. Churnside, “Speckle Correlation Measurements using Clipped Intensity Signals,” Appl. Opt. 24, 2488 (1985).
    [CrossRef] [PubMed]
  5. J. Marron, A. Martino, G. M. Morris, “Generation of Random Arrays using Clipped Laser Speckle,” Appl. Opt. 25, 26 (1986).
    [CrossRef] [PubMed]
  6. J. Marron, G. M. Morris, “Correlation Properties of Clipped Laser Speckle,” J. Opt. Soc. Am A 2, 1403 (1985).
    [CrossRef]
  7. J. Marron, G. M. Morris, “Correlation Measurements Using Clipped Laser Speckle,” Appl. Opt. 25, 789 (1986).
    [CrossRef] [PubMed]
  8. R. Barakat, “Second Order Statistics of Integrated Intensities and of Detected Photoelectrons,” Opt. Acta (accepted for publication).
  9. M. Lawrentjew, B. Schabat, Methoden der komplexen funktionen theorie (Deutscher Verlag, Berlin, 1967).
  10. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, New York, 1966), pp. 337–342.
  11. R. Barakat, J. Blake, “Theory of Photoelectron Counting Statistics: an Essay,” Phys. Rep. 60, 225 (1980).
    [CrossRef]
  12. G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, RI, 1939). Chapt. 3.

1986

1985

1980

R. Barakat, J. Blake, “Theory of Photoelectron Counting Statistics: an Essay,” Phys. Rep. 60, 225 (1980).
[CrossRef]

1978

Barakat, R.

R. Barakat, J. Blake, “Theory of Photoelectron Counting Statistics: an Essay,” Phys. Rep. 60, 225 (1980).
[CrossRef]

R. Barakat, J. Blake, “Second-Order Statistics of Speckle Patterns Observed through Finite-Size Scanning Apertures,” J. Opt. Soc. Am. 68, 614 (1978).
[CrossRef]

R. Barakat, “Second Order Statistics of Integrated Intensities and of Detected Photoelectrons,” Opt. Acta (accepted for publication).

Blake, J.

Churnside, J.

Lawrentjew, M.

M. Lawrentjew, B. Schabat, Methoden der komplexen funktionen theorie (Deutscher Verlag, Berlin, 1967).

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, New York, 1966), pp. 337–342.

Marron, J.

Martino, A.

Morris, G. M.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, New York, 1966), pp. 337–342.

Ohtsubo, J.

Pedersen, H.

Schabat, B.

M. Lawrentjew, B. Schabat, Methoden der komplexen funktionen theorie (Deutscher Verlag, Berlin, 1967).

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, New York, 1966), pp. 337–342.

Szego, G.

G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, RI, 1939). Chapt. 3.

Appl. Opt.

J. Opt. Soc. Am A

J. Marron, G. M. Morris, “Correlation Properties of Clipped Laser Speckle,” J. Opt. Soc. Am A 2, 1403 (1985).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rep.

R. Barakat, J. Blake, “Theory of Photoelectron Counting Statistics: an Essay,” Phys. Rep. 60, 225 (1980).
[CrossRef]

Other

G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, RI, 1939). Chapt. 3.

R. Barakat, “Second Order Statistics of Integrated Intensities and of Detected Photoelectrons,” Opt. Acta (accepted for publication).

M. Lawrentjew, B. Schabat, Methoden der komplexen funktionen theorie (Deutscher Verlag, Berlin, 1967).

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, New York, 1966), pp. 337–342.

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Figures (4)

Fig. 1
Fig. 1

Probability density function of the aperture integrated intensity: – – –, α = 1; ——, α = 3; …, α = 5; — · —, α = 7.

Fig. 2
Fig. 2

Doubly clipped correlation function for b2 = b1 = 1: – – –, α = 1; ——, α = 3; …, α = 5; — · —, α = 7.

Fig. 3
Fig. 3

Doubly clipped correlation function for α = 5 as a function of the clipping levels: …, b2 = b1 = 0.5; —— (curve B) b2 = b1 = 1; — · —, b2 = b1 = 1.5. Solid curve A, b2 = 1.05; b1 = 1; solid curve B, b2 = 0.95, b1 = 1.

Fig. 4
Fig. 4

Singly clipped correlation function for b2 = b1 = 1: – – –, α = 1; ——, α = 3; …, α = 5; — · —, α = 7.

Equations (33)

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Ω j = A j E ( x ) 2 d x             j = 1 , 2.
Q ( λ 2 , λ 1 ) = exp ( - λ 2 Ω 2 - λ 1 Ω 1 ) = 0 W ( Ω 2 , Ω 1 ) exp ( - λ 2 Ω 2 - λ 1 Ω 1 ) d Ω 2 d Ω 1 .
Q ( λ 2 , λ 1 ) = [ ( 1 + Ω λ 2 ) ( 1 + Ω λ 1 ) - Ω 2 r 2 λ 2 λ 1 ] - 1 ,
W ( Ω 2 , Ω 1 ) = 1 ( 1 - r ) μ 2 I 0 [ 2 ( 1 - r ) ( r Ω 2 Ω 1 μ 2 ) 2 ] × exp [ - ( 1 - r ) - 1 ( Ω 2 μ + Ω 1 μ ) ] ,
r E 2 2 E 1 2 - E 2 2 E 4 - E 2 2
W ( Ω 2 , Ω 1 ) = W ( Ω 2 ) δ ( Ω 2 - Ω 1 ) .
W ( Ω j ) = 1 μ exp ( - Ω j / μ ) .
W g ( Ω j ) = 1 Γ ( α ) ( α μ ) α Ω j α - 1 exp ( - α Ω j / μ ) ,
α = Ω 2 var ( Ω ) .
W ( Ω 2 , Ω 1 ) = W g ( Ω 2 ) W g ( Ω 1 ) × n = 0 r Ω n ( n + α - 1 n ) L n ( α - 1 ) ( α Ω 2 μ ) L n ( α - 1 ) ( α Ω 1 μ ) ,
r Ω ( l ) = Ω 2 Ω 1 - Ω 2 Ω 2 - Ω 2 .
L n ( α - 1 ) ( x ) n = 0 N ( n + α - 1 n - l ) 1 l ! ( - x ) l
0 W g ( Ω j ) L m ( α - 1 ) ( α Ω j μ ) L n ( α - 1 ) ( α Ω j μ ) d Ω j = ( n + α - 1 n ) δ m n .
g 2 ( Ω 2 ) g 1 ( Ω 1 ) = 0 g 2 ( Ω 2 ) g 1 ( Ω 1 ) W ( Ω 2 , Ω 1 ) d Ω 2 d Ω 1 .
g 2 ( Ω 2 ) g 1 ( Ω 1 ) = n = 0 r n ( n + α - 1 n ) g 2 ( Ω 2 ) L n - 1 ( α ) ( α Ω 2 μ ) × g 1 ( Ω 1 ) L n - 1 ( α ) ( α Ω 1 μ ) ,
g j ( Ω j ) L n - 1 ( α ) ( α Ω j μ ) = 0 g j ( Ω j L n - 1 ( α ) ( α Ω j μ ) W g ( Ω j ) d Ω j .
q ( Ω j ) = 1 , if Ω j b j Ω = 0 , if Ω j < b j Ω ,
R c ( 2 ) ( r Ω ) = q ( Ω 2 ) q ( Ω 1 ) = b 2 u b 1 u W ( Ω 2 , Ω 1 ) d Ω 2 d Ω 1 ,
R c ( 2 ) = n = 0 r Ω n ( n + α - 1 n ) b 2 u W g ( Ω 2 ) L n ( α - 1 ) ( α Ω 2 μ ) d Ω 2 × b 1 u W g ( Ω 1 ) L n ( α - 1 ) ( α Ω 1 μ ) d Ω 1 = 1 Γ 2 ( α ) n = 0 r Ω n ( n + α - 1 n ) Ψ n ( b 2 , α ) Ψ n ( b 1 , α ) ,
Ψ n ( b j , α ) b j α x α - 1 L n ( α - 1 ) ( x ) exp ( - x ) d x .
x α - 1 exp ( - x ) L n ( α - 1 ) ( x ) = - 1 n d d x [ x a exp ( - x ) d d x L n ( α - 1 ) ( x ) ]
Ψ n ( b j , α ) = 1 n β j α exp ( - β j ) d d β j L n α - 1 ( β j ) = - 1 n ( b j α ) α exp ( - b j α ) L n - 1 ( α ) ( b j α ) .
Ψ 0 ( b j , α ) = 1 Γ ( α ) Γ ( α , b j α ) ,
Γ ( α , y ) = y x α - 1 exp ( - x ) d x .
R c ( 2 ) ( r Ω ) = 1 Γ 2 ( α ) Γ ( α , b 2 α ) Γ ( α , b 1 α ) + ( b 2 α ) α ( b 1 α ) α Γ 2 ( α ) exp ( - b 2 α ) exp ( - b 1 α ) n = 1 × r Ω n ( n + α - 1 n ) n 2 L n - 1 ( α ) ( b 2 - α ) L ( n - 1 ) ( α ) ( b 1 α ) ,
R c ( 2 ) ( 1 ) = b 1 u W g ( Ω 1 ) d Ω 1 b 2 u δ ( Ω 2 - Ω 1 ) d Ω 2 = b 1 u W g ( Ω 1 ) H ( Ω 1 - b 2 u ) d Ω 1 .
R c ( 2 ) ( 1 ) = 1 - 1 Γ ( α ) δ ( α , b ˜ j α ) ,
r c ( 2 ) ( r Ω ) R c ( 2 ) ( r Ω ) R c ( 2 ) ( 1 ) ,
R c ( 1 ) ( r Ω ) = 1 Γ ( α ) n = 0 r Ω n ( n + α - 1 n ) Ψ n ( α , b α ) × 0 W g ( Ω 1 ) L n ( α - 1 ) ( α Ω 1 μ ) d Ω 1 .
0 W g ( Ω ) L 0 ( α - 1 ) ( α Ω μ ) Ω d Ω = Ω , 0 W g ( Ω ) L 1 ( α - 1 ) ( α Ω μ ) Ω d Ω = - Ω ,
R c 1 ( r n ) = Ω Γ ( α ) [ Γ ( α , b α ) ] + b α α α - 1 exp ( - b α ) r Ω ]
r 0 ( 1 ) ( r Ω ) R c ( 1 ) ( r Ω ) R c ( 1 ) ( 1 ) .
r c ( 1 ) ( r ) = ( 1 + b r 1 + b ) .

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