Abstract

We present a method of image-speckle contrast for the nonprecalibration measurement of the root-mean-square roughness and the lateral-correlation length of random surfaces with Gaussian correlation. We use the simplified model of the speckle fields produced by the weak scattering object in the theoretical analysis. The explicit mathematical relation shows that the saturation value of the image-speckle contrast at a large aperture radius determines the roughness, while the variation of the contrast with the aperture radius determines the lateral-correlation length. In the experimental performance, we specially fabricate the random surface samples with Gaussian correlation. The square of the image-speckle contrast is measured versus the radius of the aperture in the 4f system, and the roughness and the lateral-correlation length are extracted by fitting the theoretical result to the experimental data. Comparison of the measurement with that by an atomic force microscope shows our method has a satisfying accuracy.

© 2002 Optical Society of America

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References

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  1. Y.-P. Zhao, G. C. Wang, T.-H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic Press, New York, 2001).
  2. T. R. Thomas, Rough Surfaces, 2nd edition (Imperial College Press, London, 1999).
  3. J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
    [CrossRef]
  4. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  5. J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
    [CrossRef] [PubMed]
  6. J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Adam Hilger, New York, 1991).
  7. Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
    [CrossRef]
  8. T. R. Michel, K. A. O’Dennell, “Angular correlation functions of amplitudes scattered from one-dimensional, perfectly conduction rough surfaces,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
    [CrossRef]
  9. B. J. Pernick, “Surface roughness measurements with an optical Fourier spectrum analyzer,” Appl. Opt. 18, 796–801 (1979).
    [CrossRef] [PubMed]
  10. C. Gorecki, “Optical classification of machined metal surfaces by Fourier spectrum sampling,” Wear 137, 187–298 (1990).
    [CrossRef]
  11. E. Fontana, R. H. Pantell, “Characterization of multilayer rough surfaces by use of surface-plason spectroscopy,” Phys. Rev. B 37, 3164–3182 (1988).
    [CrossRef]
  12. R.-J. M. van der Bijl, O. M. Fähnle, H. van Brug, J. J. M. Braat, “In-process monitoring of grinding and polishing of optical surfaces,” Appl. Opt. 39, 3300–3303 (2000).
    [CrossRef]
  13. R. K. Erf, Speckle Metrology (Academic Press, New York, 1978).
  14. R. B. Crane, “Use of a laser-produced speckle pattern to determine surface roughness,” J. Opt. Soc. Am. 60, 1658–1663 (1970).
    [CrossRef]
  15. T. Yashimura, K. Kazuo, K. Nakagawa, “Surface roughness dependence of the intensity correlation function under speckle pattern illumination,” J. Opt. Soc. Am. A 7, 2254–2259 (1990).
    [CrossRef]
  16. L. Basano, S. Leporatti, V. Palestini, R. Rolandi, “Measurements of surface roughness: use of the CCD camera to correlate doubly scattered speckle patterns,” Appl. Opt. 34, 7286–7290 (1995).
    [CrossRef] [PubMed]
  17. P. Lehmann, “Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination,” Appl. Opt. 38, 1144–1152 (1999).
    [CrossRef]
  18. D. Léger, E. Matieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation technique,” Appl. Opt. 14, 872–877 (1975).
    [CrossRef] [PubMed]
  19. B. Ruffing, J. Anschutz, “Surface roughness measurement by 2-D digital correlation of speckle images,” in Photomechanics and Speckle Metrology, C. Dainty, ed., Proc. SPIE814, 105–112 (1987).
    [CrossRef]
  20. C. J. Tay, S. L. Toh, H. M. Shang, J. Zhang, “Whole-field determination of surface roughness by speckle correlation,” Appl. Opt. 34, 2324–2334 (1995).
    [CrossRef] [PubMed]
  21. R. A. Sprague, “Surface roughness measurement using white light speckle,” Appl. Opt. 11, 2811–2816 (1972).
    [CrossRef] [PubMed]
  22. H. Fujii, I. Asakura, Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66, 1217–1221 (1976).
    [CrossRef]
  23. H. Fujii, T. Asakura, “Roughness measurement of metal surfaces using laser speckle,” J. Opt. Soc. Am. 67, 1171–1176 (1977).
    [CrossRef]
  24. U. Persson, “Roughness measurement of machined surfaces by means of the speckle technique in the visible and infrared regions,” Opt. Eng. 32, 3327–3332 (1993).
    [CrossRef]
  25. U. Persson, “Real-time measurement of surface-roughness on ground surfaces using speckle-contrast technique,” Opt. Lasers Eng. 17, 61–67 (1992).
    [CrossRef]
  26. L. C. Leonard, V. Toal, “Roughness measurement of metallic surfaces based on the laser speckle contrast method,” Opt. Laser Eng. 30, 433–440 (1998).
    [CrossRef]
  27. H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
    [CrossRef]
  28. H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).
  29. H. M. Pedersen, “The roughness dependence of partially developed monochromatic speckle patterns,” Opt. Commun. 14, 30–34 (1975).
  30. J. C. Dainty, “The Statistics of speckle patterns,” in Progress in Optics: Volume XIV, E. Wolf, ed., (Elsevier, New York, 1976), pp. 1–46.
  31. J. Ohtsubo, T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
    [CrossRef] [PubMed]
  32. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [CrossRef]
  33. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [CrossRef]
  34. J. Ohtsubo, T. Asakura, “Statistical properties of speckle intensity variations in the diffraction filed under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
    [CrossRef]
  35. D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).
  36. N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

2001 (1)

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

2000 (3)

J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
[CrossRef]

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

R.-J. M. van der Bijl, O. M. Fähnle, H. van Brug, J. J. M. Braat, “In-process monitoring of grinding and polishing of optical surfaces,” Appl. Opt. 39, 3300–3303 (2000).
[CrossRef]

1999 (1)

1998 (2)

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

L. C. Leonard, V. Toal, “Roughness measurement of metallic surfaces based on the laser speckle contrast method,” Opt. Laser Eng. 30, 433–440 (1998).
[CrossRef]

1995 (2)

1993 (2)

U. Persson, “Roughness measurement of machined surfaces by means of the speckle technique in the visible and infrared regions,” Opt. Eng. 32, 3327–3332 (1993).
[CrossRef]

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

1992 (2)

U. Persson, “Real-time measurement of surface-roughness on ground surfaces using speckle-contrast technique,” Opt. Lasers Eng. 17, 61–67 (1992).
[CrossRef]

T. R. Michel, K. A. O’Dennell, “Angular correlation functions of amplitudes scattered from one-dimensional, perfectly conduction rough surfaces,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
[CrossRef]

1990 (2)

1988 (3)

E. Fontana, R. H. Pantell, “Characterization of multilayer rough surfaces by use of surface-plason spectroscopy,” Phys. Rev. B 37, 3164–3182 (1988).
[CrossRef]

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

1987 (1)

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

1979 (1)

1977 (2)

1976 (2)

1975 (4)

D. Léger, E. Matieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation technique,” Appl. Opt. 14, 872–877 (1975).
[CrossRef] [PubMed]

H. M. Pedersen, “The roughness dependence of partially developed monochromatic speckle patterns,” Opt. Commun. 14, 30–34 (1975).

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 filed under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

1972 (1)

1970 (1)

Anschutz, J.

B. Ruffing, J. Anschutz, “Surface roughness measurement by 2-D digital correlation of speckle images,” in Photomechanics and Speckle Metrology, C. Dainty, ed., Proc. SPIE814, 105–112 (1987).
[CrossRef]

Asakura, I.

Asakura, T.

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

H. Fujii, T. Asakura, “Roughness measurement of metal surfaces using laser speckle,” J. Opt. Soc. Am. 67, 1171–1176 (1977).
[CrossRef]

J. Ohtsubo, T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

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

Basano, L.

Block, U.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Braat, J. J. M.

Bruynseraede, Y.

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

Cheng, C. F.

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Cheng, C.-F.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Church, E. L.

Crane, R. B.

Dainty, J. C.

J. C. Dainty, “The Statistics of speckle patterns,” in Progress in Optics: Volume XIV, E. Wolf, ed., (Elsevier, New York, 1976), pp. 1–46.

Erf, R. K.

R. K. Erf, Speckle Metrology (Academic Press, New York, 1978).

Fähnle, O. M.

Fontana, E.

E. Fontana, R. H. Pantell, “Characterization of multilayer rough surfaces by use of surface-plason spectroscopy,” Phys. Rev. B 37, 3164–3182 (1988).
[CrossRef]

Fujii, H.

Goodman, J. W.

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

Gorecki, C.

C. Gorecki, “Optical classification of machined metal surfaces by Fourier spectrum sampling,” Wear 137, 187–298 (1990).
[CrossRef]

Haesendock, C.

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

Henley, C. L.

J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
[CrossRef]

Hevaet, I.

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

Kadono, H.

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Kazuo, K.

Kondev, J.

J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
[CrossRef]

Krim, J.

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

Léger, D.

Lehmann, P.

Leonard, L. C.

L. C. Leonard, V. Toal, “Roughness measurement of metallic surfaces based on the laser speckle contrast method,” Opt. Laser Eng. 30, 433–440 (1998).
[CrossRef]

Leporatti, S.

Liu, D. L.

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Lu, T.-H.

Y.-P. Zhao, G. C. Wang, T.-H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic Press, New York, 2001).

Lu, T.-M.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Matieu, E.

Michel, T. R.

Nakagawa, K.

O’Dennell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Adam Hilger, New York, 1991).

Ohtsubo, J.

J. Ohtsubo, T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

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

Palestini, V.

Pantell, R. H.

E. Fontana, R. H. Pantell, “Characterization of multilayer rough surfaces by use of surface-plason spectroscopy,” Phys. Rev. B 37, 3164–3182 (1988).
[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 patterns,” Opt. Commun. 14, 30–34 (1975).

Pernick, B. J.

Perrin, J. C.

Persson, U.

U. Persson, “Roughness measurement of machined surfaces by means of the speckle technique in the visible and infrared regions,” Opt. Eng. 32, 3327–3332 (1993).
[CrossRef]

U. Persson, “Real-time measurement of surface-roughness on ground surfaces using speckle-contrast technique,” Opt. Lasers Eng. 17, 61–67 (1992).
[CrossRef]

Qi, D. P.

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Rolandi, R.

Ruffing, B.

B. Ruffing, J. Anschutz, “Surface roughness measurement by 2-D digital correlation of speckle images,” in Photomechanics and Speckle Metrology, C. Dainty, ed., Proc. SPIE814, 105–112 (1987).
[CrossRef]

Salinas, D. G.

J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
[CrossRef]

Shang, H. M.

Shindo, Y.

Sprague, R. A.

Takai, N.

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

Tay, C. J.

Teng, S. Y.

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Thomas, T. R.

T. R. Thomas, Rough Surfaces, 2nd edition (Imperial College Press, London, 1999).

Toal, V.

L. C. Leonard, V. Toal, “Roughness measurement of metallic surfaces based on the laser speckle contrast method,” Opt. Laser Eng. 30, 433–440 (1998).
[CrossRef]

Toh, S. L.

van Brug, H.

van der Bijl, R.-J. M.

Wang, G. C.

Y.-P. Zhao, G. C. Wang, T.-H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic Press, New York, 2001).

Wang, G.-C.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Wu, I.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Yashimura, T.

Zhang, J.

Zhang, N. Y.

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Zhao, Y.-P.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

Y.-P. Zhao, G. C. Wang, T.-H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic Press, New York, 2001).

Acta Optica Sinica (1)

N. Y. Zhang, D. L. Liu, S. Y. Teng, D. P. Qi, C. F. Cheng, “Experimental study on the properties of autocorrelation function of image speckles produced by weak random screens,” Acta Optica Sinica 21, 696–701 (2001) (in Chinese).

Acta Physica Sinica (1)

D. P. Qi, D. L. Liu, S. Y. Teng, N. Y. Zhang, C. F. Cheng, “Morphological analysis by atomic force microscope and light scattering study for random scattering screens,” Acta Physica Sinica 49, 1260–1266 (2000) (in Chinese).

Appl. Opt. (9)

Appl. Phys. B (1)

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B 44, 167–173 (1987).
[CrossRef]

J. Appl. Phys. (1)

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–1582 (1998).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Commun. (3)

H. M. Pedersen, “The roughness dependence of partially developed monochromatic speckle patterns,” Opt. Commun. 14, 30–34 (1975).

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 filed under illumination of coherent light,” Opt. Commun. 14, 30–34 (1975).
[CrossRef]

Opt. Eng. (1)

U. Persson, “Roughness measurement of machined surfaces by means of the speckle technique in the visible and infrared regions,” Opt. Eng. 32, 3327–3332 (1993).
[CrossRef]

Opt. Laser Eng. (1)

L. C. Leonard, V. Toal, “Roughness measurement of metallic surfaces based on the laser speckle contrast method,” Opt. Laser Eng. 30, 433–440 (1998).
[CrossRef]

Opt. Lasers Eng. (1)

U. Persson, “Real-time measurement of surface-roughness on ground surfaces using speckle-contrast technique,” Opt. Lasers Eng. 17, 61–67 (1992).
[CrossRef]

Optik (1)

H. Kadono, T. Asakura, N. Takai, “Roughness and correlation-length measurements of rough-surface objects using the speckle contrast in the diffraction field,” Optik 80, 115–120 (1988).

Phys. Rev. B (1)

E. Fontana, R. H. Pantell, “Characterization of multilayer rough surfaces by use of surface-plason spectroscopy,” Phys. Rev. B 37, 3164–3182 (1988).
[CrossRef]

Phys. Rev. E (1)

J. Kondev, C. L. Henley, D. G. Salinas, “Nonlinear measures for characterizing rough surface morphologies,” Phys. Rev. E 61, 104–125 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

J. Krim, I. Hevaet, C. Haesendock, Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in a ionbombarded film surface,” Phys. Rev. Lett. 70, 57–60 (1993).
[CrossRef] [PubMed]

Wear (1)

C. Gorecki, “Optical classification of machined metal surfaces by Fourier spectrum sampling,” Wear 137, 187–298 (1990).
[CrossRef]

Other (6)

R. K. Erf, Speckle Metrology (Academic Press, New York, 1978).

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Adam Hilger, New York, 1991).

Y.-P. Zhao, G. C. Wang, T.-H. Lu, Characterization of Amorphous and Crystalline Rough Surfaces: Principles and Applications (Academic Press, New York, 2001).

T. R. Thomas, Rough Surfaces, 2nd edition (Imperial College Press, London, 1999).

J. C. Dainty, “The Statistics of speckle patterns,” in Progress in Optics: Volume XIV, E. Wolf, ed., (Elsevier, New York, 1976), pp. 1–46.

B. Ruffing, J. Anschutz, “Surface roughness measurement by 2-D digital correlation of speckle images,” in Photomechanics and Speckle Metrology, C. Dainty, ed., Proc. SPIE814, 105–112 (1987).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the 4f system.

Fig. 2
Fig. 2

AFM images of samples: (a) No. 1, (b) No. 2, and (c) No. 3.

Fig. 3
Fig. 3

Numerical distributions of the height correlation of samples and their Gaussian fits.: (a) No. 1, (b) No. 2, and (c) No. 3.

Fig. 4
Fig. 4

Height probability density of the samples calculated from the AFM image data: (a) No. 1, (b) No. 2, and (c) No. 3.

Fig. 5
Fig. 5

Experimental data of C 2 versus R, and the fit of Eq. (21) to these data, from which the a and b are obtained and then w and ξ are extracted.

Equations (24)

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Ur=U0r+Usr,
Ir=UrU*r=I0r+Isr+U0rUs*r+U0*rUsr,
Ir=I0r+Isr,
C2=I2-I2I2=Is2+2I0IsI02+Is2+2I0Is.
Ufrf=-expi 2πλn-1hr0×exp-i 2πλfr0·rfd2r0.
Ur=-- Prfexpi 2πλn-1hr0×exp-i 2πλfr0-r·rfd2r0d2rf.
Ir=I0=U0U*0 =----expi 2πλn-1×hr0-hr0PrfP*rf×exp-i 2πλfr0·rf-r0rfd2r0d2r0d2rfd2rf.
expi 2πλn-1hr0-hr0 =- Phr0, hr0expi 2πλn-1×hr0-hr0dhr0dhr0 =exp2πλn-12w2-Rhr0, r0,
Rhr0, r0=w2 exp-|r0-r0|2/ξ2,
I=Ir =---- PrfP*rf×exp-2πλn-1w21-exp-|r0-r0|2/ξ2exp-i 2πλfr0 · rf-r · rfd2r0d2r0d2rfd2rf.
I=exp-2πλn-1w2×---- PrfP*rf×m=02πn-1w/λ2mm!exp-m|Δr0|2/ξ2]×exp-i 2πλfr0·rf-rf+Δr0 · rf×d2r0d2Δr0d2rfd2rf.
2πλn-1w21.
I=exp-2πλn-1w2×-- |Prf|21+2πλn-1w2×exp-|Δr0|2/ξ2exp-i 2πλf Δr0·rfd2Δr0d2rf.
I=λ2f2 exp-2πλn-1w2×-  |Prf|2δrfλf+πξ22πλn-1w2×exp-πξ2λ2f2 |rf|2d2rf.
I=I0+Is,
I0=λ2f2 exp-2πλn-1w2×- |Prf|2δrfλfd2rf,
Is=λ2f2exp-2πλn-1w2×- |Prf|2πξ22πλn-1w2×exp-πξ2λ2f2 |rf|2d2rf.
Prf=1,|rf|=ρ=xf2+yf2R0,ρ>R.
I0=λ4f4exp-2πλn-1w2.
Is=2π2λ2f22πλn-1w2exp-2πλn-1w20R ξ2exp-πξλf2ρ2dρ =λ4f42πλn-1w2 exp-2πλn-1w2×1-exp-πξλf2R2.
C2=1-11+2a1-exp-R2/b2,
a=2πλn-1w2,
b=λf/πξ.
C2=2a1-exp-R2/b2.

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