Abstract

Autocovariance functions (ACF's) G(x) for polished optical surfaces of CaF<sub>2</sub>, MgF<sub>2</sub>, and LiF are deduced from surface profiles determined by using microdensitometer analysis of micrographs of surface-shadowed carbon replicas. The different estimators allowing the ACF computation from its standard definition are reviewed, and their values are compared. ACF's are also computed by using the fast-Fourier-transform algorithm. Results of both computations are in good agreement. It is shown that initial portions of ACF's have a reasonable Gaussian form. The rms roughness height δ and the autocovariance length σ are deduced for each surface. The ACF's of the surface slopes <i>G</i><sub>ś</sub>, (<i>x</i>) are also computed, and it is shown that results obtained are consistent with results deduced from ACF's for surface profiles. In particular, the standard relation between the second derivative of <i>G</i> (<i>x</i>) and <i>G</i><sub>ś</sub>, (<i>x</i>) is reasonably verified. Finally, the exponential ACF model is discussed, and it is shown that this model would not be suitable to escribe the initial portions of the ACF's for the polished optical surfaces that we have studied.

© 1983 Optical Society of America

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  2. M. Rasigni, G. Rasigni, J. P. Palmari, and A. Llebaria, "Study of surface roughness using a microdensitometer analysis of electron micrographs of surface replicas: II. Autocovariance functions," J. Opt. Soc. Am. 71, 1230–1237 (1981).
  3. M. Rasigni, F. Varnier, G. Rasigni, J. P. Palmari, and A. Llebaria, "Validity of surface-roughness study using microdensitometer analysis of electron micrographs of surface replicas," J. Opt. Soc. Am. 71, 1549–1550 (1981).
  4. G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, "Autocovariance functions, root-mean-square roughness height, and autocovariance length for rough deposits of copper, silver and gold," Phys. Rev. B 25, 2315–2323 (1982).
  5. J. Eastman and P. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12, 418–420 (1974).
  6. J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2720 (1976).
  7. J. M. Elson and J. M. Bennett, "Relation between the angular dependance of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979).
  8. J. M. Bennett, S. M. Wong, and G. Krauss, "Relation between the optical and metallurgical properties of polished molybdenum mirrors," Appl. Opt. 19, 3562–3584 (1980).
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  11. F. Varnier, M. Rasigni, G. Rasigni, J. P. Palmari, and A. Llebaria, "Height and slope distributions for surfaces of rough metallic deposits," Appl. Opt. 21, 3681–3684 (1982).
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  13. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon. New York, 1963).
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  15. C. K. Carniglia, "Scalar scattering theory for multilayer optical coatings," Opt. Eng. 18, 104–115 (1979).
  16. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 174.
  17. A. V. Oppenhein and R. W. Shafer, Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), p. 539.
  18. F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).
  19. S. A. Armstrong and P. T. Gough, "Investigations of errors in sample autocovariance functions and their corresponding power spectra," J. Opt. Soc. Am. 68, 568–572 (1978).
  20. L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 105.
  21. Ref. 17, P. 556.
  22. R. K. Otnes and L. Enochson, Digital Time Series Analysis (Wiley, New York, 1972), p. 225.
  23. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 110.
  24. H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithm (Springer-Verlag, Berlin, 1981), p. 32.
  25. Ref. 20, p. 403.
  26. Ref. 20, p. 404.
  27. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 72.
  28. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 189.
  29. J. L. Doob, Stochastic Processes (Wiley, London, 1953).
  30. H. Reather, "Surface plasma oscillations and their applications," in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1977), Vol. 9, p. 145.
  31. E. Kretschmann, "Die Bestimmung der Oberflächenrauhigkeit dunner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung Von oberflächen Plasmaschwingungen," Opt. Commun. 10, 353–356 (1974).
  32. E. Kretschmann and E. Kröger, "Reflection and transmission of light by a rough surface, including results for surfaces plasmon effects," J. Opt. Soc. Am. 65, 150–154 (1975).
  33. E. Kröger and E. Kretschmann, "Scattering of light by slightly rough surfaces on thin films including plasma resonance emission," Z. Phys. 237, 1–15 (1970).
  34. J. M. Elson, "Light scattering from surfaces with a single dielectric overlayer," J. Opt. Soc. Am. 66, 682–694 (1976); "Diffraction and diffuse scattering from dielectrics multilayers," J. Opt. Soc. Am. 69, 48–54 (1979).
  35. J. M. Elson, J. P. Rahn, and J. M. Bennett, "Light scattering from multilayers optics: comparison of theory and experiments," Appl. Opt. 19, 669–679 (1980).
  36. P. Bousquet, F. Flory, and P. Roche, "Scattering from multilayer thin films: theory and experiment," J. Opt. Soc. Am. 71, 1115–1123 (1981).
  37. J. C. Dainty and R. Shaw, Image Science, Principles, Analysis and Evaluation of Photographic-Type Imaging Processes (Academic, London, 1974), p. 197.

1982 (2)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, "Autocovariance functions, root-mean-square roughness height, and autocovariance length for rough deposits of copper, silver and gold," Phys. Rev. B 25, 2315–2323 (1982).

F. Varnier, M. Rasigni, G. Rasigni, J. P. Palmari, and A. Llebaria, "Height and slope distributions for surfaces of rough metallic deposits," Appl. Opt. 21, 3681–3684 (1982).

1981 (5)

1980 (2)

1979 (3)

J. M. Elson and J. M. Bennett, "Relation between the angular dependance of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979).

C. K. Carniglia, "Scalar scattering theory for multilayer optical coatings," Opt. Eng. 18, 104–115 (1979).

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

1978 (1)

1976 (1)

1975 (2)

G. Rasigni, J. P. Palmari, and M. Rasigni, "Plasma resonance in granular deposits and rough surfaces of magnesium," Phys. Rev. B 12, 1121–1131 (1975).

E. Kretschmann and E. Kröger, "Reflection and transmission of light by a rough surface, including results for surfaces plasmon effects," J. Opt. Soc. Am. 65, 150–154 (1975).

1974 (2)

J. Eastman and P. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12, 418–420 (1974).

E. Kretschmann, "Die Bestimmung der Oberflächenrauhigkeit dunner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung Von oberflächen Plasmaschwingungen," Opt. Commun. 10, 353–356 (1974).

1970 (1)

E. Kröger and E. Kretschmann, "Scattering of light by slightly rough surfaces on thin films including plasma resonance emission," Z. Phys. 237, 1–15 (1970).

Aimé, C.

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

Armstrong, S. A.

Baumeister, P.

J. Eastman and P. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12, 418–420 (1974).

Beckmann, P.

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

Bendat, J. S.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 189.

Bennett, J. M.

Borgnino, J.

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

Bousquet, P.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 110.

Carniglia, C. K.

C. K. Carniglia, "Scalar scattering theory for multilayer optical coatings," Opt. Eng. 18, 104–115 (1979).

Dainty, J. C.

J. C. Dainty and R. Shaw, Image Science, Principles, Analysis and Evaluation of Photographic-Type Imaging Processes (Academic, London, 1974), p. 197.

Dancy, J. H.

Doob, J. L.

J. L. Doob, Stochastic Processes (Wiley, London, 1953).

Eastman, J.

J. Eastman and P. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12, 418–420 (1974).

Eastman, J. M.

J. M. Eastman, "Scattering by all-dielectric multilayers, bandpass filters and mirror for lasers," in Physics of Thin Films, G. Hass and M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10, p. 167.

Elson, J. M.

J. M. Elson, J. P. Rahn, and J. M. Bennett, "Light scattering from multilayers optics: comparison of theory and experiments," Appl. Opt. 19, 669–679 (1980).

J. M. Elson and J. M. Bennett, "Relation between the angular dependance of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979).

J. M. Elson, "Light scattering from surfaces with a single dielectric overlayer," J. Opt. Soc. Am. 66, 682–694 (1976); "Diffraction and diffuse scattering from dielectrics multilayers," J. Opt. Soc. Am. 69, 48–54 (1979).

Enochson, L.

R. K. Otnes and L. Enochson, Digital Time Series Analysis (Wiley, New York, 1972), p. 225.

Flory, F.

Gold, B.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 105.

Gough, P. T.

Jenkins, G. M.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 174.

Krauss, G.

Kretschmann, E.

E. Kretschmann and E. Kröger, "Reflection and transmission of light by a rough surface, including results for surfaces plasmon effects," J. Opt. Soc. Am. 65, 150–154 (1975).

E. Kretschmann, "Die Bestimmung der Oberflächenrauhigkeit dunner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung Von oberflächen Plasmaschwingungen," Opt. Commun. 10, 353–356 (1974).

E. Kröger and E. Kretschmann, "Scattering of light by slightly rough surfaces on thin films including plasma resonance emission," Z. Phys. 237, 1–15 (1970).

Kröger, E.

E. Kretschmann and E. Kröger, "Reflection and transmission of light by a rough surface, including results for surfaces plasmon effects," J. Opt. Soc. Am. 65, 150–154 (1975).

E. Kröger and E. Kretschmann, "Scattering of light by slightly rough surfaces on thin films including plasma resonance emission," Z. Phys. 237, 1–15 (1970).

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 72.

Llebaria, A.

Martin, F.

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

Nussbaumer, H. J.

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithm (Springer-Verlag, Berlin, 1981), p. 32.

Oppenhein, A. V.

A. V. Oppenhein and R. W. Shafer, Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), p. 539.

Otnes, R. K.

R. K. Otnes and L. Enochson, Digital Time Series Analysis (Wiley, New York, 1972), p. 225.

Palmari, J. P.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 437 and 503.

Rabiner, L. R.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 105.

Rahn, J. P.

Rasigni, G.

Rasigni, M.

Reather, H.

H. Reather, "Surface plasma oscillations and their applications," in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1977), Vol. 9, p. 145.

Ricort, G.

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

Roche, P.

Roddier, F.

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

Shafer, R. W.

A. V. Oppenhein and R. W. Shafer, Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), p. 539.

Shaw, R.

J. C. Dainty and R. Shaw, Image Science, Principles, Analysis and Evaluation of Photographic-Type Imaging Processes (Academic, London, 1974), p. 197.

Spizzichino, A.

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

Varnier, F.

Watts, D. G.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 174.

Wong, S. M.

Appl. Opt. (5)

J. Opt. Paris (1)

F. Martin, J. Borgnino, C. Aimé, G. Ricort, and F. Roddier, "Erreurs d'estimation sur la détermination de la covariance de processus aléatoires, stationnaires, ergodiques, à valeur moyenne non nulle," .J. Opt. (Paris) 10, 99–108 (1979).

J. Opt. Soc. Am. (7)

Opt. Commun. (2)

J. Eastman and P. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12, 418–420 (1974).

E. Kretschmann, "Die Bestimmung der Oberflächenrauhigkeit dunner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung Von oberflächen Plasmaschwingungen," Opt. Commun. 10, 353–356 (1974).

Opt. Eng. (1)

C. K. Carniglia, "Scalar scattering theory for multilayer optical coatings," Opt. Eng. 18, 104–115 (1979).

Phys. Rev. (2)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, "Autocovariance functions, root-mean-square roughness height, and autocovariance length for rough deposits of copper, silver and gold," Phys. Rev. B 25, 2315–2323 (1982).

G. Rasigni, J. P. Palmari, and M. Rasigni, "Plasma resonance in granular deposits and rough surfaces of magnesium," Phys. Rev. B 12, 1121–1131 (1975).

Z. Phys. (1)

E. Kröger and E. Kretschmann, "Scattering of light by slightly rough surfaces on thin films including plasma resonance emission," Z. Phys. 237, 1–15 (1970).

Other (18)

J. M. Elson, "Light scattering from surfaces with a single dielectric overlayer," J. Opt. Soc. Am. 66, 682–694 (1976); "Diffraction and diffuse scattering from dielectrics multilayers," J. Opt. Soc. Am. 69, 48–54 (1979).

J. C. Dainty and R. Shaw, Image Science, Principles, Analysis and Evaluation of Photographic-Type Imaging Processes (Academic, London, 1974), p. 197.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 174.

A. V. Oppenhein and R. W. Shafer, Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), p. 539.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 437 and 503.

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

J. M. Eastman, "Scattering by all-dielectric multilayers, bandpass filters and mirror for lasers," in Physics of Thin Films, G. Hass and M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10, p. 167.

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 105.

Ref. 17, P. 556.

R. K. Otnes and L. Enochson, Digital Time Series Analysis (Wiley, New York, 1972), p. 225.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 110.

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithm (Springer-Verlag, Berlin, 1981), p. 32.

Ref. 20, p. 403.

Ref. 20, p. 404.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), p. 72.

J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), p. 189.

J. L. Doob, Stochastic Processes (Wiley, London, 1953).

H. Reather, "Surface plasma oscillations and their applications," in Physics of Thin Films, G. Hass, ed. (Academic, New York, 1977), Vol. 9, p. 145.

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