Abstract

First-order statistics of the speckle phase are studied on an axis of the imaging system under the assumption of Gaussian statistics for the formation of the speckle field. It is found that the speckle phase is most widely distributed at the image plane owing to the noncircularity of the speckle field. Asymmetry of the probability-density distribution of the speckle phase also appears to be due to the inclination of the joint probability density of the complex speckle amplitude. These statistical characteristics of the speckle phase are enhanced for relatively small values of the optical roughness of the object.

© 1985 Optical Society of America

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References

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  1. J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
    [CrossRef]
  2. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [CrossRef]
  3. J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980).
    [CrossRef]
  4. J. Uozumi, T. Asakura, “First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region,” Appl. Opt. 20, 1454–1466 (1981).
    [CrossRef] [PubMed]
  5. E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
    [CrossRef]
  6. K. Ouchi, “Statistics of image plane speckle,” Opt. Quantum Electron. 12, 237–243 (1980).
    [CrossRef]

1981 (1)

1980 (2)

J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980).
[CrossRef]

K. Ouchi, “Statistics of image plane speckle,” Opt. Quantum Electron. 12, 237–243 (1980).
[CrossRef]

1977 (1)

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

1975 (2)

J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Asakura, T.

J. Uozumi, T. Asakura, “First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region,” Appl. Opt. 20, 1454–1466 (1981).
[CrossRef] [PubMed]

J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980).
[CrossRef]

J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Jakeman, E.

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

Ouchi, K.

K. Ouchi, “Statistics of image plane speckle,” Opt. Quantum Electron. 12, 237–243 (1980).
[CrossRef]

Uozumi, J.

J. Uozumi, T. Asakura, “First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region,” Appl. Opt. 20, 1454–1466 (1981).
[CrossRef] [PubMed]

J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980).
[CrossRef]

Welford, W. T.

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

Appl. Opt. (1)

Opt. Commun. (3)

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Opt. Quantum Electron. (2)

J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980).
[CrossRef]

K. Ouchi, “Statistics of image plane speckle,” Opt. Quantum Electron. 12, 237–243 (1980).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Double-diffraction optical imaging system for producing an image speckle field. The speckle field is examined at a distance z away from the image plane.

Fig. 2
Fig. 2

Equiprobability-density ellipse and its relation to the parameters defining the joint probability distribution of the speckle field. The maximum density occurs at the center of the ellipse. The phase angle Δθ between two tangent lines drawn from the origin to the ellipse corresponds roughly to the region of the speckle phase distribution.

Fig. 3
Fig. 3

Variation of the equiprobability-density ellipse on the observation point u. Only the shape and the inclination of the ellipse are shown with the center of the ellipse corresponding to u. The number of contributing scatterers is N = 100, and the variance σϕ2 of ϕ is 0.25.

Fig. 4
Fig. 4

Dependence of the circularity of the speckle field along the optical axis on the variance σϕ2 of ϕ. The number of contributing scatterers is N = 10.

Fig. 5
Fig. 5

Dependence of the probability-density distribution of the speckle phase on the normalized distance u from the image plane to the observation point. The number of contributing scatterers is N = 10, and the variance σϕ2 of ϕ is (a) 0.5, (b) 1.0, (c) 2.0, and (d) 4.0.

Fig. 6
Fig. 6

Asymmetrical probability-density distributions of the speckle phase at (a) u = π and (c) u = −π, and symmetrical distribution at (b) u = 0 with the parameters N and σϕ2 as in Fig. 5(a).

Fig. 7
Fig. 7

Degree of asymmetry of the probability-density distribution of the speckle phase. The number of contributing scatterers is N = 10.

Fig. 8
Fig. 8

Transition of the probability-density distribution of the speckle phase from the asymmetrical and narrow to the symmetrical and uniform distribution at u = π. The number N of contributing scatterers is 10.

Equations (20)

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P r , i ( A r , A i ) = 1 2 π σ r σ i ( 1 ρ 2 ) 1 / 2 × exp [ 1 2 ( 1 ρ 2 ) ( Δ A r 2 σ r 2 2 ρ Δ A r Δ A i σ r σ i + Δ A i 2 σ i 2 ) ] ,
Δ A r = A r A r , Δ A i = A i A i ,
δ = ½ tan 1 [ 2 ρ σ r σ i / ( σ r 2 σ i 2 ) ] .
A r = exp ( σ ϕ 2 / 2 ) ,
A i = 0 ,
σ i 2 = 1 2 N [ T + sinc ( u ) S ] exp ( σ ϕ 2 ) ,
σ i 2 = 1 2 N [ T sinc ( u ) S ] exp ( σ ϕ 2 ) ,
ρ = sinc ( u / 2 ) sin ( u / 2 ) S [ T 2 sinc 2 ( u ) S 2 ] 1 / 2 ,
T = n = 1 ( σ ϕ 2 ) n n ! n , S = n = 1 ( σ ϕ 2 ) n n ! n , N = ( 2 f / k w ξ ) 2 , u = z / ( f 2 / k w 2 ) ,
δ = ½ tan 1 [ tan ( u / 2 ) ] = ¼ ( u 2 n π ) , | u 2 n π | > π , n = 0 , ± 1 , ± 2 , .
δ = u / 4 .
C = σ y / σ x .
P θ ( θ ) = ν 2 π τ { 1 + π ζ exp ( ζ 2 ) [ 1 + erf ( ζ ) ] } exp ( γ / 2 σ x 2 ) ,
μ = θ δ , ν = σ x / σ y , τ = cos 2 μ + ν 2 sin 2 μ , β = A x cos μ + ν 2 A y sin μ , γ = A x 2 + ν 2 A y 2 , ζ = β / 2 τ σ x ,
( 1 ρ 2 ) 1 ( Δ A r 2 σ r 2 2 ρ Δ A r Δ A i σ r σ i + Δ A i 2 σ i 2 ) = 1 .
d Δ A r / d Δ A i = 0 .
d Δ A r d Δ A i = ( 2 ρ Δ A r σ r σ i 2 Δ A i σ i 2 ) / ( 2 Δ A r σ r 2 2 ρ Δ A i σ r σ i ) = 0 .
Δ A i = ρ σ i Δ A r / σ r .
Δ A r 2 = σ r 2 .
Δ A i 2 = σ i 2 .

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