Abstract

The surface-roughness dependence of the intensity correlation function of the speckle pattern, produced in the Fresnel region with fully developed speckle-pattern illumination, has been theoretically investigated and has been discussed as follows. This correlation function is represented by two correlation functions of scattered and unscattered components. As the diffuse object becomes rough, the speckle size varies from the speckle size of the illumination light to that obtained with the condition, so that the object is a deep phase screen. The speckle contrast, however, is always one.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 1.
  3. T. Asakura, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), p. 11.
    [Crossref]
  4. J. Ohtubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
    [Crossref]
  5. P. J. Chandley, H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
    [Crossref]
  6. M. H. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
    [Crossref]
  7. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 716–772 (1970).
    [Crossref]
  8. H. Fujii, T. Asakura, “A constant variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
    [Crossref]
  9. H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle pattern,” Opt. Commun. 12, 156–159 (1974).
    [Crossref]
  10. H. Fujii, T. Asakura, “Statistical properties of image speckle pattern in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
    [Crossref]
  11. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [Crossref]
  12. G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
    [Crossref]
  13. T. Yoshimura, K. Nakagawa, “Properties of light scattered by a diffuse object under dynamic speckle illumination and their application to the roughness distribution measurement,” Opt. Commun. 60, 139–144 (1986).
    [Crossref]
  14. K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [Crossref]
  15. D. Newman, “Kdistributions from doubly scattered light,” J. Opt. Soc. Am. A 2, 22–26 (1985).
    [Crossref]
  16. R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
    [Crossref]
  17. E. Menzel, B. Stoffregen, “Autocorrelation function of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).
  18. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  19. A. Starkov, E. Wolf, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 923–928 (1982).
  20. N. Takai, T. Asakura, “Statistical properties of laser speckles produced under illumination from a multimode optical fiber,” J. Opt. Soc. Am. A 2, 1282–1290 (1985).
    [Crossref]

1986 (2)

T. Yoshimura, K. Nakagawa, “Properties of light scattered by a diffuse object under dynamic speckle illumination and their application to the roughness distribution measurement,” Opt. Commun. 60, 139–144 (1986).
[Crossref]

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

1985 (2)

1983 (2)

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[Crossref]

M. H. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[Crossref]

1982 (3)

1979 (1)

P. J. Chandley, H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[Crossref]

1978 (1)

J. Ohtubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[Crossref]

1976 (2)

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

E. Menzel, B. Stoffregen, “Autocorrelation function of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

1975 (1)

H. Fujii, T. Asakura, “Statistical properties of image speckle pattern in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

1974 (2)

H. Fujii, T. Asakura, “A constant variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle pattern,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 716–772 (1970).
[Crossref]

Asakura, T.

N. Takai, T. Asakura, “Statistical properties of laser speckles produced under illumination from a multimode optical fiber,” J. Opt. Soc. Am. A 2, 1282–1290 (1985).
[Crossref]

J. Ohtubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[Crossref]

H. Fujii, T. Asakura, “Statistical properties of image speckle pattern in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

H. Fujii, T. Asakura, “A constant variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

T. Asakura, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), p. 11.
[Crossref]

Barakat, R.

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

Chandley, P. J.

P. J. Chandley, H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[Crossref]

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 716–772 (1970).
[Crossref]

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 1.

Escamilla, H. M.

P. J. Chandley, H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[Crossref]

Escamilla, M. H.

M. H. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[Crossref]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Fujii, H.

H. Fujii, T. Asakura, “Statistical properties of image speckle pattern in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

H. Fujii, T. Asakura, “A constant variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

Goodman, J. W.

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]

Menzel, E.

E. Menzel, B. Stoffregen, “Autocorrelation function of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Nakagawa, K.

T. Yoshimura, K. Nakagawa, “Properties of light scattered by a diffuse object under dynamic speckle illumination and their application to the roughness distribution measurement,” Opt. Commun. 60, 139–144 (1986).
[Crossref]

Newman, D.

O’Donnell, K. A.

Ohtubo, J.

J. Ohtubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[Crossref]

Pedersen, H. M.

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

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle pattern,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

Rao, V. V.

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[Crossref]

Reddy, G. R. C.

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[Crossref]

Starkov, A.

Stoffregen, B.

E. Menzel, B. Stoffregen, “Autocorrelation function of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Takai, N.

Wolf, E.

Yoshimura, T.

T. Yoshimura, K. Nakagawa, “Properties of light scattered by a diffuse object under dynamic speckle illumination and their application to the roughness distribution measurement,” Opt. Commun. 60, 139–144 (1986).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Nouv. Rev. Opt. (1)

H. Fujii, T. Asakura, “Statistical properties of image speckle pattern in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975).
[Crossref]

Opt. Acta (4)

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[Crossref]

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[Crossref]

M. H. Escamilla, “Speckle contrast in the diffraction field of a weak random-phase screen when the illuminated region contains a few correlation areas,” Opt. Acta 30, 1655–1664 (1983).
[Crossref]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 716–772 (1970).
[Crossref]

Opt. Commun. (6)

H. Fujii, T. Asakura, “A constant variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974).
[Crossref]

H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle pattern,” Opt. Commun. 12, 156–159 (1974).
[Crossref]

J. Ohtubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[Crossref]

P. J. Chandley, H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151–154 (1979).
[Crossref]

T. Yoshimura, K. Nakagawa, “Properties of light scattered by a diffuse object under dynamic speckle illumination and their application to the roughness distribution measurement,” Opt. Commun. 60, 139–144 (1986).
[Crossref]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Optik (1)

E. Menzel, B. Stoffregen, “Autocorrelation function of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Other (3)

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]

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics XIV, E. Wolf, ed. (North-Holland, Amsterdam, 1976), p. 1.

T. Asakura, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), p. 11.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Diagram of the optical system for analyzing the intensity correlation function.

Fig. 2
Fig. 2

Dependences of the function γ on the roughness parameter 〈ϕ21/2 under the condition of rsi = 10rss.

Fig. 3
Fig. 3

Intensity correlation functions at L = 200 mm for the three objects. The solid and the dashed curves show the experimental and the theoretical results, respectively.

Fig. 4
Fig. 4

Dependences of the speckle size on the roughness parameters.

Fig. 5
Fig. 5

Photographic results of the speckle pattern with the fully developed speckle-pattern illumination. The diffuse object placed at the object plane in Fig. 1 is (a) nothing, (b) #3000, (c) #1500, (d) #600.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E ( X ) = E I ( x ) exp [ i ϕ ( x ) ] K 0 ( x , X ) d x ,
E I ( x ) = E L ( ξ ) exp [ i Φ ( ξ ) ] K I ( ξ , x ) d ξ .
E ( X 1 ) E * ( X 2 ) = E I ( x I ) E I * ( x 2 ) × exp { i [ ϕ ( x 2 ) ϕ ( x 1 ) ] } K 0 ( x 1 , X 1 ) K 0 * ( x 2 , X 2 ) d x 1 d x 2 ,
K 0 ( x , X ) = [ k / 2 π i L ] exp ( i k L ) exp [ i k / ( 2 L ) | x X | 2 ] .
E I ( x 1 ) E I * ( x 2 ) = exp [ | x 1 + x 2 | 2 / ( 2 d 2 ) ] × exp [ | x 2 x 1 | 2 / ( 2 r s i 2 ) ] ,
exp { i [ ϕ ( x 2 ) ϕ ( x 1 ) ] } = exp { ϕ 2 [ 1 exp ( | x 2 x 1 | 2 / R c 2 ) ] } .
exp { i [ ϕ ( x 2 ) ϕ ( x 1 ) ] } = exp ( ϕ 2 ) + [ 1 exp ( ϕ 2 ) ] × exp ( η 2 | x 2 x 1 | 2 / R c 2 )
η = { 1 ( ϕ 2 < 1 ) ϕ 2 1 / 2 ( ϕ 2 1 ) .
g ( 2 ) ( r ) 1 = I ( 0 ) I ( r ) / I ( 0 ) 2 1 = ( γ exp { | r | 2 / [ 2 ( r s s 2 + r s i 2 ) ] } + ( 1 γ ) exp { | r | 2 / [ 2 ( r s s 2 + r s s 2 ) ] } ) 2 ,
γ = [ r s i 2 / ( r s s 2 + r s i 2 ) ] exp ( ϕ 2 ) [ r s i 2 / ( r s s 2 + r s i 2 ) ] exp ( ϕ 2 ) + [ r s r 2 / ( r s s 2 + r s r 2 ) ] [ 1 exp ( ϕ 2 ) ] ,
r s s = 2 L / k d ,
r s r = r s i / ( 1 + 2 η 2 N ) 1 / 2 ,
N = r s i 2 / R c 2 ,
D u = C / ( 1 + C ) = r s i 2 ( r s s 2 + r s i 2 )
r s r 2 / ( r s s 2 + r s r 2 ) [ 1 exp ( ϕ 2 ) ] = ( α r s i ) 2 / [ r s s 2 + ( α r s i ) 2 ] [ 1 exp ( ϕ 2 ) ] ,
α 2 = [ R c 2 / ( 2 η 2 ) ] / [ R c 2 / ( 2 η 2 ) + r s i 2 ] .

Metrics