Abstract

New feasibilities are considered for the optical-correlation diagnostics of rough surfaces with different distributions of irregularities. The influence of deviations of the height surface roughness distribution from a Gaussian probability distribution on the accuracy of optical analysis is discussed. Possibilities for the optical diagnostics of fractal surface structures are shown, and a set of statistical and dimensional parameters of the scattered fields for surface roughness diagnostics is determined. Finally, a multifunctional measuring device for estimating these parameters is proposed.

© 2001 Optical Society of America

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References

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  1. J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  2. J. M. Bennett, “Surface roughness measurement,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997), pp. 341–367.
  3. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  4. O. V. Angelsky, P. P. Maksimyak, S. Hanson, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media (SPIE Press, Bellingham, Wash., 1999), PM71.
  5. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  6. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  7. J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
    [CrossRef]
  8. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [CrossRef] [PubMed]
  9. T. V. Vorburger, E. Marx, T. R. Lettieri, “Regimes of surface roughness measurable with scattering,” Appl. Opt. 32, 3401–3408 (1993).
    [CrossRef] [PubMed]
  10. S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).
  11. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  12. E. L. Church, “Comments on the correlation length,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 102–111 (1986).
    [CrossRef]
  13. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57; Chap. 39, pp. 362–365.
  14. E. Feder, Fractals (Plenum, New York, 1988).
  15. K. Nakagawa, T. Yoshimura, T. Minemoto, “Surface-roughness measurement using Fourier transformation of doubly scattered speckle pattern,” Appl. Opt. 32, 4898–4903 (1993).
    [CrossRef] [PubMed]
  16. A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
    [CrossRef]
  17. K. J. Falconer, Fractal Geometry (Wiley, New York, 1990).
  18. D. A. Zimnyakov, V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. 35, 4325–4333 (1996).
    [CrossRef] [PubMed]
  19. J. F. Nye, Natuaral Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).
  20. O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
    [CrossRef] [PubMed]
  21. Yu. I. Neymark, P. S. Landa, Stochastic and Chaotic Oscillations (Nauka, Moscow, 1987).
  22. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
    [CrossRef]
  23. E. L. Church, P. Z. Takacs, “Effect nonvanishing tip size in mechanical profile measurements,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, P. Grover, ed., Proc. SPIE1332, 504–514 (1991).
  24. K. A. O’Donnell, “Effect of finite stylus width in surface contact profilometry,” Appl. Opt. 32, 4922–4928 (1993).
    [CrossRef]
  25. R. S. Sayles, T. R. Thomas, “Surface topography as a nonstationary random process,” Nature (London) 271, 431–433 (1978).
    [CrossRef]
  26. A. Arneodo, “Wavelet analysis of fractals,” in Wavelets, G. Erlebacher, M. Y. Hussaini, L. M. Jameson, eds. (Oxford University, Oxford, UK, 1996), pp. 352–497.
  27. H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).
  28. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  29. K. S. Clarke, “Computation of the fractal dimention of topographic surfaces using the triangular prism surface area method,” Comput. Geosci. 12, 113–122 (1986).
    [CrossRef]
  30. B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
    [CrossRef]
  31. A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
    [CrossRef]
  32. O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical correlation method for measuring spatial complexity in optical fields,” Opt. Lett. 18, 90–92 (1993).
    [CrossRef] [PubMed]
  33. O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Dimensionality in optical fields and signals,” Appl. Opt. 32, 6066–6071 (1993).
    [CrossRef] [PubMed]
  34. N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).
  35. F. Takens, “Detecting strange attractors in turbulence,” Lect. Notes Math. 898, 366–381 (1981).
    [CrossRef]
  36. O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of slightly rough surfaces,” Appl. Opt. 30, 140–143 (1992).
    [CrossRef]
  37. O. V. Angelsky, P. P. Maksimyak, “Polarization-interference measurement of phase-inhomogeneous objects,” Appl. Opt. 31, 4417–4419 (1992).
    [CrossRef] [PubMed]
  38. O. V. Angelsky, P. P. Maksimyak, “Optical correlation measurements of the structure parameters of random and fractal objects,” Meas. Sci. Technol. 9, 1682–1693 (1998).
    [CrossRef]

1998 (1)

O. V. Angelsky, P. P. Maksimyak, “Optical correlation measurements of the structure parameters of random and fractal objects,” Meas. Sci. Technol. 9, 1682–1693 (1998).
[CrossRef]

1996 (1)

1994 (1)

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

1993 (6)

1992 (2)

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of slightly rough surfaces,” Appl. Opt. 30, 140–143 (1992).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Polarization-interference measurement of phase-inhomogeneous objects,” Appl. Opt. 31, 4417–4419 (1992).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

1988 (1)

1986 (1)

K. S. Clarke, “Computation of the fractal dimention of topographic surfaces using the triangular prism surface area method,” Comput. Geosci. 12, 113–122 (1986).
[CrossRef]

1981 (1)

F. Takens, “Detecting strange attractors in turbulence,” Lect. Notes Math. 898, 366–381 (1981).
[CrossRef]

1980 (1)

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

1979 (2)

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

1978 (1)

R. S. Sayles, T. R. Thomas, “Surface topography as a nonstationary random process,” Nature (London) 271, 431–433 (1978).
[CrossRef]

1970 (1)

1961 (1)

Angelsky, O. V.

O. V. Angelsky, P. P. Maksimyak, “Optical correlation measurements of the structure parameters of random and fractal objects,” Meas. Sci. Technol. 9, 1682–1693 (1998).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical correlation method for measuring spatial complexity in optical fields,” Opt. Lett. 18, 90–92 (1993).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Dimensionality in optical fields and signals,” Appl. Opt. 32, 6066–6071 (1993).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of slightly rough surfaces,” Appl. Opt. 30, 140–143 (1992).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Polarization-interference measurement of phase-inhomogeneous objects,” Appl. Opt. 31, 4417–4419 (1992).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, S. Hanson, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media (SPIE Press, Bellingham, Wash., 1999), PM71.

Arneodo, A.

A. Arneodo, “Wavelet analysis of fractals,” in Wavelets, G. Erlebacher, M. Y. Hussaini, L. M. Jameson, eds. (Oxford University, Oxford, UK, 1996), pp. 352–497.

Asakura, T.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Bennett, H. E.

Bennett, J. M.

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
[CrossRef]

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

J. M. Bennett, “Surface roughness measurement,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997), pp. 341–367.

Church, E. L.

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

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

E. L. Church, P. Z. Takacs, “Effect nonvanishing tip size in mechanical profile measurements,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, P. Grover, ed., Proc. SPIE1332, 504–514 (1991).

E. L. Church, “Comments on the correlation length,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 102–111 (1986).
[CrossRef]

Clarke, K. S.

K. S. Clarke, “Computation of the fractal dimention of topographic surfaces using the triangular prism surface area method,” Comput. Geosci. 12, 113–122 (1986).
[CrossRef]

Dogariu, A.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Dubuc, B.

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

Elson, J. M.

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
[CrossRef]

Falconer, K. J.

K. J. Falconer, Fractal Geometry (Wiley, New York, 1990).

Farmer, J. D.

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

Feder, E.

E. Feder, Fractals (Plenum, New York, 1988).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grutchfield, J. P.

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

Hanson, S.

O. V. Angelsky, P. P. Maksimyak, S. Hanson, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media (SPIE Press, Bellingham, Wash., 1999), PM71.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

Landa, P. S.

Yu. I. Neymark, P. S. Landa, Stochastic and Chaotic Oscillations (Nauka, Moscow, 1987).

Lettieri, T. R.

Maksimyak, P. P.

O. V. Angelsky, P. P. Maksimyak, “Optical correlation measurements of the structure parameters of random and fractal objects,” Meas. Sci. Technol. 9, 1682–1693 (1998).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Dimensionality in optical fields and signals,” Appl. Opt. 32, 6066–6071 (1993).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical correlation method for measuring spatial complexity in optical fields,” Opt. Lett. 18, 90–92 (1993).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, “Polarization-interference measurement of phase-inhomogeneous objects,” Appl. Opt. 31, 4417–4419 (1992).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of slightly rough surfaces,” Appl. Opt. 30, 140–143 (1992).
[CrossRef]

O. V. Angelsky, P. P. Maksimyak, “Optical diagnostics of random phase objects,” Appl. Opt. 29, 2894–2898 (1990).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, S. Hanson, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media (SPIE Press, Bellingham, Wash., 1999), PM71.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57; Chap. 39, pp. 362–365.

Marx, E.

Mattson, L.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Minemoto, T.

Nakagawa, K.

Neymark, Yu. I.

Yu. I. Neymark, P. S. Landa, Stochastic and Chaotic Oscillations (Nauka, Moscow, 1987).

Nicodemus, F. E.

Nye, J. F.

J. F. Nye, Natuaral Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Packard, N. H.

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

Perun, T. O.

Porteus, J. O.

Quiniuo, J. F.

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

Roques-Carmes, C.

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

Sayles, R. S.

R. S. Sayles, T. R. Thomas, “Surface topography as a nonstationary random process,” Nature (London) 271, 431–433 (1978).
[CrossRef]

Shaw, P. S.

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “Effect nonvanishing tip size in mechanical profile measurements,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, P. Grover, ed., Proc. SPIE1332, 504–514 (1991).

Takens, F.

F. Takens, “Detecting strange attractors in turbulence,” Lect. Notes Math. 898, 366–381 (1981).
[CrossRef]

Tatarsky, V. I.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

Thomas, T. R.

R. S. Sayles, T. R. Thomas, “Surface topography as a nonstationary random process,” Nature (London) 271, 431–433 (1978).
[CrossRef]

Tricot, C.

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

Tuchin, V. V.

Uozumi, J.

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Vorburger, T. V.

Yoshimura, T.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Zimnyakov, D. A.

Appl. Opt. (10)

Comput. Geosci. (1)

K. S. Clarke, “Computation of the fractal dimention of topographic surfaces using the triangular prism surface area method,” Comput. Geosci. 12, 113–122 (1986).
[CrossRef]

J. Mod. Opt. (1)

A. Dogariu, J. Uozumi, T. Asakura, “Angular power spectra of fractal structures,” J. Mod. Opt. 41, 729–738 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Lect. Notes Math. (1)

F. Takens, “Detecting strange attractors in turbulence,” Lect. Notes Math. 898, 366–381 (1981).
[CrossRef]

Meas. Sci. Technol. (1)

O. V. Angelsky, P. P. Maksimyak, “Optical correlation measurements of the structure parameters of random and fractal objects,” Meas. Sci. Technol. 9, 1682–1693 (1998).
[CrossRef]

Nature (London) (1)

R. S. Sayles, T. R. Thomas, “Surface topography as a nonstationary random process,” Nature (London) 271, 431–433 (1978).
[CrossRef]

Opt. Eng. (2)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
[CrossRef]

Opt. Lett. (2)

N. H. Packard, J. P. Grutchfield, J. D. Farmer, P. S. Shaw, “Geometry from a time series,” Opt. Lett. 45, 712–716 (1980).

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical correlation method for measuring spatial complexity in optical fields,” Opt. Lett. 18, 90–92 (1993).
[CrossRef] [PubMed]

Phys. Rev. (1)

B. Dubuc, J. F. Quiniuo, C. Roques-Carmes, C. Tricot, “Evaluation the fractal dimensions of profiles,” Phys. Rev. 39, 1500–1512 (1989).
[CrossRef]

Pure Appl. Opt. (1)

A. Dogariu, J. Uozumi, T. Asakura, “Sources of error in optical measurements of fractal dimension,” Pure Appl. Opt. 2, 339–350 (1993).
[CrossRef]

Other (16)

K. J. Falconer, Fractal Geometry (Wiley, New York, 1990).

J. F. Nye, Natuaral Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

E. L. Church, “Comments on the correlation length,” in Surface Characterization and Testing, K. Creath, ed., Proc. SPIE680, 102–111 (1986).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982), Chap. 6, pp. 37–57; Chap. 39, pp. 362–365.

E. Feder, Fractals (Plenum, New York, 1988).

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarsky, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

J. M. Bennett, “Surface roughness measurement,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997), pp. 341–367.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

O. V. Angelsky, P. P. Maksimyak, S. Hanson, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media (SPIE Press, Bellingham, Wash., 1999), PM71.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

E. L. Church, P. Z. Takacs, “Effect nonvanishing tip size in mechanical profile measurements,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, P. Grover, ed., Proc. SPIE1332, 504–514 (1991).

Yu. I. Neymark, P. S. Landa, Stochastic and Chaotic Oscillations (Nauka, Moscow, 1987).

A. Arneodo, “Wavelet analysis of fractals,” in Wavelets, G. Erlebacher, M. Y. Hussaini, L. M. Jameson, eds. (Oxford University, Oxford, UK, 1996), pp. 352–497.

H.-O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, New York, 1988).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (19)

Fig. 1
Fig. 1

Relief maps and height-distribution functions for some modeled surfaces. The histogram shows the real surface height distribution, and the solid curve shows the Gaussian distribution with the same mean value and dispersion: a, F0, nonsmoothed fractal surface; b, F3H1S, three-point smoothed fractal surface obeying the Gaussian law with a quadratic power nonlinearity; c, R3E, three-point smoothed nonfractal random surface obeying exponential distribution; d, R3H2R, three-point smoothed nonfractal random surface obeying the Gaussian law with a power nonlinearity of 0.25.

Fig. 2
Fig. 2

Formation of the field diffracted by a rough surface.

Fig. 3
Fig. 3

Typical behavior of the statistical moments as a function of the observation distance z for surface R3H02 (three-point nonfractal random surface obeying the Gaussian law with h max = 200 nm): Ku f , kurtosis coefficient of the field; Sk, asymmetry coefficient; σφ 2, phase variance of the field; σ A 2, amplitude dispersion of the field; β2, scintillation index of the field.

Fig. 4
Fig. 4

Dependence of the phase variance of the field on the distance of registration z for random surfaces of various heights: curve 1, h max = 100 nm; curve 2, 200 nm; curve 3, 600 nm; curve 4, 1000 nm; curve 5, 2000 nm; curve 6, 4000 nm.

Fig. 5
Fig. 5

Kurtosis coefficient of the field Ku f as a function of z for various height spans of rough surface relief.

Fig. 6
Fig. 6

Dependencies of the statistical parameters of the field as a function of z for random surfaces with h max = 2000 nm and various power nonlinearities: R2, without nonlinearity; R2R2, -k = 0.25, double square root; R2S2, k = 4, double square.

Fig. 7
Fig. 7

Behavior of the statistical parameters of the scattered field for three-point smoothing by applying Gaussian law fractal surfaces: F3H01, h max = 100 nm; F3H04, h max = 400 nm; F3H1, h max = 1000 nm.

Fig. 8
Fig. 8

Transverse coherence function of the field Γ(ρ) for a, nonfractal and, b, fractal objects with various values of h max.

Fig. 9
Fig. 9

Behavior of the fractal dimension of the field D f t as a function of z for, a, nonfractal surfaces: RH02, h max = 200 nm; RH04, h max = 400 nm; RH2, h max = 2000 nm; b, fractal surfaces with h max = 2000 nm: F0H2, nonsmoothed; F2H2, two-point smoothed on Gaussian law; F5H2, five-point smoothed on Gaussian law; c, nonfractal surfaces with h max = 200 nm with applied power nonlinearities: R02, without nonlinearities; R1R02, k = 0.5 (square root), R1S02, k = 2 (square); R2R02, k = 0.25 (double square root); R2S02, k = 4 (double square).

Fig. 10
Fig. 10

Scheme for calculation of the correlation exponent v from the structure function.

Fig. 11
Fig. 11

Behavior of the correlation exponent as a function of z: a, three-point smoothed nonfractal random surface obeying the Gaussian law with the maximal height span, h max = 100, 400, 800, 1600, 3200, and 6400 nm; b, three-point smoothed fractal surface obeying the Gaussian law with the maximal height span, h max = 100, 400, 1000, 2000, and 4000 nm; c, three-point smoothed nonfractal random surface obeying the Gaussian law with the maximal height span, h max = 2000 nm. R0, without nonlinearities; R2R, k = 0.25 (double square root); R1S, k = 2 (square); R3R, k = 0.125 (triple square root); R3S, k = 8 (triple square).

Fig. 12
Fig. 12

Experimental optical arrangement: L, He–Ne laser; T, inverse telescopic system; BS1, BS2, beam splitters; M1, M2, mirrors; S, transmitting object with a rough surface; P1, P2, polarizers; I, interference block; O, objective; CCD, digital camera.

Fig. 13
Fig. 13

Conversion of the amplitude transmittance distribution of a photomask into a transmission phase profile.

Fig. 14
Fig. 14

Surface relief structure found profilometrically.

Fig. 15
Fig. 15

Intensity distribution found experimentally for a field at various distances from the surface: a, z = 20 µm; b, z = 50 µm; c, z = 100 µm.

Fig. 16
Fig. 16

Calculated (index m) and experimentally obtained (index e) statistical moments: scintillation index, β2; kurtosis coefficient, Ku f ; asymmetry coefficient Sk of a field for a nonfractal random surface with h max = 200 nm.

Fig. 17
Fig. 17

a, Measured intensity distribution; b, computed surface relief.

Fig. 18
Fig. 18

a, Height distribution of the modeled object; b, experimentally obtained height distribution. The histogram shows the real height distribution, and the solid curve corresponds to a Gaussian distribution with the same mean value and dispersion.

Fig. 19
Fig. 19

Experimentally found power spectra for fractal surfaces.

Tables (2)

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Table 1 Phase Variance Magnitudes σφ0 2 for Fractal and Nonfractal Surfaces Found from Height Distributions h and Measured Far-Field Coherence Function cf for Various Maximal Height Spans of Surface Irregularities and Power Nonlinearities

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Table 2 Computed and Measured Fractal Dimensions of Surface Relief Found through the Triangular Prism Technique Df t, through Analysis of the Surface Profile D fp, and through Fourier Transformation of the Measured Correlation Function of the Relief Df c

Equations (21)

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Ψφ0ρ=σφ02Kρ,
σφ2=σA2=0.5σφ02,
β2=2σφ02.
Γρ=expσφ02Kφ0ρ-1.
Sk=1σh31Ni=1N hi3
Ku=1σh41Ni=1N hi4
Sfx=Kn/fxn with 1<n<3.
Sf=Γn+1/22Γ1/2Γn/2Knfxn+1.
D=5-n/2,
T3-n=-122πnΓncosnπ/2Kn.
Uξ, ζ=ziλ    Ax, yR2x, y, ξ, ζ, z×exp-ikRx, y, ξ, ζ, z+n-1hx, ydxdy,
Γρ > lφ0=Imax-IminImax+Imin=exp-σφ02.
Iim=Ii, Ii+1,  , Ii+m-1,
Cmε=limN1N2i=1Nj=1N Θε-|Iim-Ijm|,
ν=limε0ln Cmε/ln ε
DIr1, r2=|I˜r1-I˜r22.
DIr1, r2=DIρ=2ΨI0-ΨIρ,
|Iim-Ijm|=k=0m-1Ii+k-Ij+k21/2.
ISξ, ς=A02+A2ξ, ς+2A0Aξ, ςcos φξ, ς,
ISξ, ςI0=σφ2,
hξ, ζ=1kn-1ISξ, ς2I0-1.

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