Abstract

The statistical properties of the speckle phase in the diffraction region are theoretically and experimentally investigated. The statistical parameters describing the equiprobability density ellipse, by which the speckle field is characterized, are first derived for the general case. By using the equiprobability density ellipse, the statistical properties of the speckle phase are next investigated theoretically as a function of the observation point in space and the illumination condition. The speckle phases were experimentally measured by means of a heterodyne interferometer to verify the theoretical results. In particular, when the diffuse object is illuminated by a Gaussian beam around its waist region, the statistical properties of the speckle phase are revealed to be similar to those observed already in the optical imaging system with respect to the defocusing.

© 1986 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
    [Crossref]
  2. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [Crossref]
  3. 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]
  4. J. Ohtsubo, T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).
  5. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [Crossref]
  6. E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
    [Crossref]
  7. K. Ouchi, “Statistics of image plane speckle,” Opt. Quantum Electron. 12, 237–243 (1980).
    [Crossref]
  8. J. Uozumi, T. Asakura, “First-order probability density function of the laser speckle phase,” Opt. Quantum Electron. 12, 447–494 (1980),
    [Crossref]
  9. 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]
  10. H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
    [Crossref]
  11. H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).
  12. N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng.25 (to be published, 1986).
    [Crossref]

1985 (2)

H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).

H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
[Crossref]

1981 (1)

1980 (2)

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

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

1977 (1)

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

1976 (2)

J. Ohtsubo, T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
[Crossref]

1975 (2)

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

Asakura, T.

H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).

H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
[Crossref]

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 patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

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]

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng.25 (to be published, 1986).
[Crossref]

Goodman, J. W.

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
[Crossref]

Jakeman, E.

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

Kadono, H.

H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
[Crossref]

H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng.25 (to be published, 1986).
[Crossref]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

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]

Pedersen, H. M.

Takai, N.

H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng.25 (to be published, 1986).
[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)

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).

Opt. Commun. (3)

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

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

Opt. Quantum Electron. (2)

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

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

Optik (1)

J. Ohtsubo, T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

Other (2)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
[Crossref]

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng.25 (to be published, 1986).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1

Equiprobability density ellipse and its felation to the statistical parameters in the ArAi and AxAy coordinate systems.

Fig. 2
Fig. 2

Optical system under consideration.

Fig. 3
Fig. 3

Behavior of the equiprobability density ellipse for the normalized off-axis distance | x ˆ | = 0, 0.5, 1.0, and 1.5 for the three cases of N0 = 17 and z ˆ = 72, N0 = 50 and z ˆ = 25, and N0 = 135 and z ˆ = 9 with σϕ = 1.14.

Fig. 4
Fig. 4

Dependence of the phase extent Δθ on the normalized off-axis distance | x ˆ | for the threecases of N0 = 17 and z ˆ = 72, N0 = 50 and z ˆ = 25, and N0 = 135 and z ˆ = 9 with σϕ = 1.14. The dashed lines represent the standard deviations σθ of the speckle phase, which are numerically evaluated from the probability density function of the speckle phase.

Fig. 5
Fig. 5

Behavior of the equiprobability density ellipses for the normalized distance r ˆ = 0, 1, 2, 3, and 4 in the case of N0 = 17, z ˆ = 72, and σϕ = 1.14.

Fig. 6
Fig. 6

Dependence of the phase extent Δθ on r ˆ. The dashed line represents the approximated values calculated from Eqs. (24a)(24d) and (25) for N0 = 17 and σϕ = 1.14. The solid lines represent the phase extent calculated from Eqs. (17h) and (19a)(19d) for z ˆ = 50, 100, and 500.

Fig. 7
Fig. 7

Experimental arrangement using the heterodyne interferometer.

Fig. 8
Fig. 8

Measured probability density distributions of the speckle phase for | x ˆ | = 0, 0.54, 1.0, and 1.21.

Fig. 9
Fig. 9

Dependence of the measured standard deviations σθ of the speckle phase on | x ˆ | and N0 for three cases of N0 = 17 and z ˆ = 72, N0 = 50 and z ˆ = 25, and N0 = 135 and z ˆ = 9 with σϕ = 1.14. The solid lines are fitted to the experimental data.

Fig. 10
Fig. 10

Dependence of the standard deviations σθ of the measured speckle phase on the optical roughness σϕ of the diffuse object for | x ˆ | = 0, 0.49, 0.97, 1.21, and 1.45 for case of N0 = 135 and z ˆ = 9. The solid lines are fitted to the experimental data.

Fig. 11
Fig. 11

Dependence of the measured standard deviations σθ on the normalized distance r ˆ on the optical axis | x ˆ | = 0 for the values of N0 = 17, z ˆ = 72, and σϕ = 1.14. The solid line represents the theoretical values of the phase extent Δθ evaluated from Eq. (26).

Fig. 12
Fig. 12

Dependence of the measured speckle contrast on the normalized distance r ˆ on the optical axis | x ˆ | = 0 for the values of N0 = 17, z ˆ = 72, and σϕ = 1.14. The solid lines are fitted to the experimental data.

Equations (72)

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A = A r + i A i = I exp ( i θ ) ,
p ( 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 , σ r 2 = Δ A r 2 , σ i 2 = Δ A i 2 , ρ = Δ A r Δ A i / σ r σ i .
δ = 1 2 tan 1 [ 2 ρ σ r σ i / ( σ r 2 σ i 2 ) ] .
tan ( 2 Δ θ ) = 2 [ A x 2 σ y 2 + A y 2 σ x 2 σ x 2 σ y 2 ] 1 / 2 / [ A x 2 + A y 2 ( σ x 2 + σ y 2 ) ] .
A x = A r cos δ + A i sin δ ,
A y = A i cos δ A r sin δ ,
σ x 2 = ( σ r 2 cos 2 δ σ i 2 sin 2 δ ) / cos ( 2 δ ) ,
σ y 2 = ( σ i 2 cos 2 δ σ r 2 sin 2 δ ) / cos ( 2 δ ) .
A ( x ) = E ( ξ ) exp [ i ϕ ( ξ ) ] K ( ξ , x ) d ξ .
T ( ξ , x ) = E ( ξ ) K ( ξ , x ) .
A r = exp ( σ ϕ 2 / 2 ) Re T ( ξ , x ) d ξ ,
A i = exp ( σ ϕ 2 / 2 ) Im T ( ξ , x ) d ξ ,
σ r 2 = Δ S ϕ 2 [ 1 exp ( σ ϕ 2 ) ] B ( x ) [ 1 exp ( σ ϕ 2 ) C ( x ) ] ,
σ i 2 = Δ S ϕ 2 [ 1 exp ( σ ϕ 2 ) ] B ( x ) [ 1 + exp ( σ ϕ 2 ) C ( x ) ] ,
σ r σ i ρ = Δ S ϕ 2 exp ( σ ϕ 2 ) [ 1 exp ( σ ϕ 2 ) ] B ( x ) S ( x ) ,
B ( x ) = | T ( ξ , x ) | 2 d ξ ,
C ( x ) = Re T 2 ( ξ , x ) d ξ / B ( x ) ,
S ( x ) = Im T 2 ( ξ , x ) d ξ / B ( x ) ,
E ( ξ ) = I 0 ( w 0 / w i ) exp ( | ξ | 2 / w i 2 ) exp ( i π | ξ | 2 / λ μ i ) ,
K ( ξ , x ) = 1 λ z exp [ i π | ξ x | 2 / λ z ] ,
w i 2 = w 2 ( r ) = w 0 2 [ 1 + ( r / z 0 ) 2 ] ,
μ i = μ ( r ) = r [ 1 + ( z 0 / r ) 2 ] .
A r = I 0 1 + q ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) cos ( q ˆ | x ˆ | 2 + ϕ e 0 ) ,
A i = I 0 1 + q ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) sin ( q ˆ | x ˆ | 2 + ϕ e 0 ) ,
σ r 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 exp ( σ ϕ 2 ) exp ( 2 | x ˆ | 2 ) × cos ( 2 q ˆ | x ˆ | 2 + ϕ e 0 ) / 1 + γ e 2 ] ,
σ i 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 + exp ( σ ϕ 2 ) exp ( 2 | x ˆ | 2 ) × cos ( 2 q ˆ | x ˆ | 2 + ϕ e 0 ) / 1 + γ e 2 ] ,
σ r σ i ρ = I 0 4 N 0 z ˆ 2 exp ( σ ϕ 2 ) [ 1 exp ( σ ϕ 2 ) ] exp ( 2 | x ˆ | 2 ) × sin ( 2 q ˆ | x ˆ | 2 + ϕ e 0 ) / 1 + γ e 2 ,
N 0 = π w 0 2 / Δ S ϕ ,
x ˆ = x / w ( z + r ) ,
r ˆ = r / z 0 ,
z ˆ = z / z 0 ,
γ e = z e 0 / z e ,
z e 0 = π w i 2 / λ ,
1 / z e = 1 / z + 1 / μ i ,
ϕ e 0 = tan 1 ( γ e ) ,
q ˆ = r ˆ + z ˆ ,
δ = q ˆ | x ˆ | 2 + ϕ e 0 / 2 .
A x = I 0 1 + q ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) cos ( ϕ e 0 / 2 ) ,
A y = I 0 1 + q ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) sin ( ϕ e 0 / 2 ) ,
σ x 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 exp ( σ ϕ 2 ) × exp ( 2 | x ˆ | 2 ) / 1 + γ e 2 ] ,
σ y 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 + exp ( σ ϕ 2 ) × exp ( 2 | x ˆ | 2 ) / 1 + γ e 2 ] .
A x = I 0 1 + z ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) cos ( ϕ e 0 / 2 ) ,
A y = I 0 1 + z ˆ 2 exp ( σ ϕ 2 / 2 ) exp ( | x ˆ | 2 ) sin ( ϕ e 0 / 2 ) ,
σ x 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 exp ( σ ϕ 2 ) × exp ( 2 | x ˆ | 2 ) z ˆ / 1 + z ˆ 2 ] ,
σ y 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 + exp ( σ ϕ 2 ) × exp ( 2 | x ˆ | 2 ) z ˆ / 1 + z ˆ 2 ] ,
ϕ e 0 = tan 1 ( 1 / z ˆ ) .
tan ( 2 Δ θ ) = 1 N 0 { [ 1 exp ( σ ϕ 2 ) ] × [ 1 + exp ( σ ϕ 2 ) / exp ( 2 | x ˆ | 2 ) ] } 1 / 2 ,
Δ θ = 1 2 N 0 { [ 1 exp ( σ ϕ 2 ) ] [ 1 + exp ( σ ϕ 2 ) / exp ( 2 | x ˆ | 2 ) ] } 1 / 2 .
A x = I 0 z ˆ exp ( σ ϕ 2 / 2 ) cos ( ϕ e 0 / 2 ) ,
A y = I 0 z ˆ exp ( σ ϕ 2 / 2 ) sin ( ϕ e 0 / 2 ) ,
σ x 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 exp ( σ ϕ 2 ) / 1 + r ˆ 2 ,
σ y 2 = I 0 4 N 0 z ˆ 2 [ 1 exp ( σ ϕ 2 ) ] [ 1 + exp ( σ ϕ 2 ) / 1 + r ˆ 2 ,
ϕ e 0 = tan 1 ( r ˆ ) ,
Δ θ = 1 2 N 0 { [ 1 exp ( σ ϕ 2 ) ] × [ 1 + exp ( σ ϕ 2 ) / ( 1 + r ˆ 2 ) ] / exp ( σ ϕ 2 ) } 1 / 2 ,
σ ϕ = 2 π λ ( n 1 n 2 ) σ h ,
T ( ξ , x ) = I 0 w 0 λ z w i exp ( | ξ | 2 / w i 2 ) exp ( i π | ξ | 2 / λ μ i ) × exp ( i π | ξ x | 2 / λ z ) ,
w i 2 = w 2 ( r ) = w 0 2 [ 1 + ( r / z 0 ) 2 ] , μ i = μ ( r ) = r [ 1 + ( z 0 / r ) 2 ] .
exp ( A x 2 ) exp ( i B x 2 ) exp ( i C x ) d x = π ( A 2 + B 2 ) 1 / 4 exp [ A C 2 4 ( A 2 + B 2 ) ] × exp { i [ B C 2 4 ( A 2 + B 2 ) ( 1 2 ) tan 1 ( B / A ) ] } .
T ( ξ , x ) d ξ = I 0 w 0 λ z w i I ξ ( x ) I η ( y ) ,
I ξ ( x ) = exp ( i π x 2 / λ z ) exp ( ξ 2 / w i 2 ) × exp [ i π ξ 2 λ ( 1 μ i + 1 z ) ] exp ( i 2 π λ z x ξ ) d ξ ,
I η ( y ) = exp ( i π y 2 / λ z ) exp ( η 2 / w i 2 ) × exp [ i π η 2 λ ( 1 μ i + 1 z ) ] exp ( i 2 π λ z y η ) d η .
I ξ ( x ) = π w i ( γ e 2 + 1 ) 1 / 4 exp ( x ˆ 2 ) exp ( i q ˆ x ˆ 2 ) exp ( i 2 tan 1 γ e ) ,
x ˆ = x / w ( z + r ) , r ˆ = r / z 0 , z ˆ = z / z 0 , γ e = z e 0 / z e , z e 0 = π w i 2 / λ , 1 / z e = 1 / z + 1 / μ i , q ˆ = r ˆ + z ˆ .
T ( ξ , x ) d ξ = [ I 0 / ( q ˆ + 1 ) ] 1 / 2 exp ( | x ˆ | 2 ) exp ( i q ˆ | x ˆ | 2 ) × exp ( i tan 1 γ e ) .
w 0 z e 0 w i z 1 ( γ e 2 + 1 ) 1 / 2 = ( q ˆ + 1 ) 1 / 2
B ( x ) = | T ( ξ , x ) | 2 d ξ = I 0 z e 0 2 λ z 2 ( w 0 w i ) 2 .
C ( x ) + i S ( x ) = T 2 ( ξ , x ) d ξ / B ( x ) = 1 ( γ e 2 + 1 ) 1 / 2 I ξ ( 2 x ) I η ( 2 y ) = 1 ( γ e 2 + 1 ) 1 / 2 exp ( 2 | x ˆ | 2 ) exp ( i 2 q ˆ | x ˆ | 2 ) × exp ( i tan 1 γ e ) ,
γ e = r ˆ + ( r ˆ 2 + 1 ) / z ˆ .
γ e = 1 / z ˆ .
γ e = r ˆ ( μ i / z + 1 ) .
γ e r ˆ .

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