Abstract

Shearing interferometry is employed for the analysis of basic statistical properties of rough curved surfaces. Using this method the root-mean-square values (standard deviations) of heights and slopes of the surface irregularities are determined. Moreover, autocorrelation lengths and functions of the surfaces mentioned are evaluated. A statistical analysis of randomly rough parabolic mirror collimators is performed by the method presented.

© 1985 Optical Society of America

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  1. H. E. Bennett, “Specular Reflectance of Aluminized Ground Glass and the Height Distribution of Surface Irregularities,” J. Opt. Soc. Am. 53, 1389 (1963).
    [CrossRef]
  2. I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
    [CrossRef]
  3. I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
    [CrossRef]
  4. P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329 (1976).
    [CrossRef]
  5. A. Abdulkadir, R. Birkebak, “Optical Surface Roughness and Slopes Measurements with a Double Beam Spectrophotometer,” Rev. Sci. Instrum. 45, 1356 (1974).
    [CrossRef]
  6. D. L. Hornauer, H. Raether, “Determination of Roughness of LiF Films Using Guided Light Modes,” Opt. Commun. 40, 105 (1981).
    [CrossRef]
  7. I. J. Hodgkinson, “A Simple Scatter Method for Optical Surface Roughness and Slope Measurements. Roughness of Polished Fused Silica,” J. Phys. E 3, 341 (1970).
    [CrossRef]
  8. I. J. Hodgkinson, “The Application of Fringes of Equal Chromatic Order to Assessment of the Surface Roughness of Polished Fused Silica,” J. Phys. E. 3, 300 (1970).
    [CrossRef]
  9. P. J. Chandley, “Determination of the Standard Deviation of Height on a Rough Surface Using Interference Microscopy,” Opt. Quantum Electron. 11, 407 (1979).
    [CrossRef]
  10. R. P. Edwin, “Light Scattering as a Technique for Measuring the Roughness of Optical Surfaces,” J. Phys. E 6, 55 (1973).
    [CrossRef]
  11. W. B. Ribbens, “Interferometric Surface Roughness Measurement,” Appl. Opt. 8, 2173 (1969).
    [CrossRef] [PubMed]
  12. W. B. Ribbens, “Surface Roughness Measurement by Holographic Interferometry,” Appl. Opt. 11, 807 (1972).
    [CrossRef] [PubMed]
  13. W. B. Ribbens, “Surface Roughness Measurement by Two Wavelength Holographic Interferometry,” Appl. Opt. 13, 1085 (1974).
    [CrossRef] [PubMed]
  14. J. Eastman, P. Baumeister, “The Microstructure of Polished Optical Surfaces,” Opt. Commun. 12, 418 (1974).
    [CrossRef]
  15. 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 (1976).
    [CrossRef] [PubMed]
  16. K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
    [CrossRef]
  17. S. Hard, O. Nilsson, “Laser Heterodyne Apparatus for Roughness Measurements of Polished Surfaces,” Appl. Opt. 17, 3827 (1978).
    [CrossRef] [PubMed]
  18. B. P. Hildebrand, R. L. Gordon, E. V. Allen, “Instrument for Measuring the Roughness of Supersmooth Surfaces,” Appl. Opt. 13, 177 (1974).
    [CrossRef] [PubMed]
  19. R. J. Whitefield, “Noncontact Optical Profilometer,” Appl. Opt. 14, 2480 (1975).
    [CrossRef] [PubMed]
  20. M. Henzler, “The Roughness of Cleaved Semiconductor Surfaces,” Surf. Sci. 36, 109 (1973).
    [CrossRef]
  21. J. Motyčka, “Proposed Interferometric Method of Measurement of Roughness and Autocorrelation Function in Smooth-Finished Surfaces,” Appl. Opt. 8, 1435 (1969).
    [CrossRef]
  22. H. F. Velzel, “Measurement of Surface Roughness by Interferential Contrast—an Application of Shearing Interferometry to the Study of Phase Objects,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970).
  23. S. Mallick, “Interference Arrangement for Measuring the Autocorrelation Function of a Smoothly Finished Surface,” Opt. Commun. 2, 396 (1971).
    [CrossRef]
  24. H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974).
    [CrossRef]
  25. R. A. Sprague, “Surface Roughness Measurement Using White Light Speckle,” Appl. Opt. 11, 2811 (1972).
    [CrossRef] [PubMed]
  26. H. Fujii, T. Asakura, “A Contrast Variation of Image Speckle Intensity under Illumination of Partially Coherent Light,” Opt. Commun. 12, 32 (1974).
    [CrossRef]
  27. D. Légér, J. C. Perrin, “Real-Time Measurement of Surface Roughness by Correlation of Speckle Patterns,” J. Opt. Soc. Am. 66, 1210 (1976).
    [CrossRef]
  28. H. Fujii, J. W. Y. Lit, “Surface Roughness Measurement Using Dichromatic Speckle Pattern: an Experimental Study,” Appl. Opt. 17, 2690 (1978).
    [CrossRef] [PubMed]
  29. J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785 (1981).
    [CrossRef] [PubMed]
  30. N. Schwartz, R. Brown, “A Stylus Method for Evaluating the Thickness of Thin Films and Substrate Roughness,” in Transections of the Eighth Vacuum Symposium and Second International Congress (Pergamon, New York, 1961).
  31. M. Kubo, “Instrument for the Measurement of Slope and Height Distribution of Surface Roughness,” Rev. Sci. Instrum. 36, 236 (1965).
    [CrossRef]
  32. G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
    [CrossRef]
  33. G. Rasigni, F. Varnier, M. Rasigni, J.-P. Palmari, “Spectral-Density Function of the Surface Roughness for Polished Optical Surfaces,” J. Opt. Soc. Am. 73, 1235 (1983).
    [CrossRef]
  34. G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
    [CrossRef]
  35. M. P. Rimmer, C. M. King, D. G. Fox, “Computer Program for the Analysis of Interferometric Test Data,” Appl. Opt. 11, 2790 (1972).
    [CrossRef] [PubMed]
  36. O. Y. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
    [CrossRef] [PubMed]
  37. Interferenzmikroskop Epival Interphako, Gebrauchsanleitung (VEB Carl Zeiss, Jena, 1977), in German.
  38. B. R. Levin, The Theory of Random Processes and Their Application to Radio Engineering (Sovyetskoe Radio, Moscow, 1960), in Russian.
  39. H. Cramér, Mathematical Methods of Statistics (Princeton U.P., Princeton, 1946).
  40. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

1983 (2)

G. Rasigni, F. Varnier, M. Rasigni, J.-P. Palmari, “Spectral-Density Function of the Surface Roughness for Polished Optical Surfaces,” J. Opt. Soc. Am. 73, 1235 (1983).
[CrossRef]

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

1982 (1)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

1981 (2)

J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785 (1981).
[CrossRef] [PubMed]

D. L. Hornauer, H. Raether, “Determination of Roughness of LiF Films Using Guided Light Modes,” Opt. Commun. 40, 105 (1981).
[CrossRef]

1980 (1)

1979 (1)

P. J. Chandley, “Determination of the Standard Deviation of Height on a Rough Surface Using Interference Microscopy,” Opt. Quantum Electron. 11, 407 (1979).
[CrossRef]

1978 (2)

1977 (1)

I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
[CrossRef]

1976 (3)

1975 (1)

1974 (7)

B. P. Hildebrand, R. L. Gordon, E. V. Allen, “Instrument for Measuring the Roughness of Supersmooth Surfaces,” Appl. Opt. 13, 177 (1974).
[CrossRef] [PubMed]

W. B. Ribbens, “Surface Roughness Measurement by Two Wavelength Holographic Interferometry,” Appl. Opt. 13, 1085 (1974).
[CrossRef] [PubMed]

J. Eastman, P. Baumeister, “The Microstructure of Polished Optical Surfaces,” Opt. Commun. 12, 418 (1974).
[CrossRef]

A. Abdulkadir, R. Birkebak, “Optical Surface Roughness and Slopes Measurements with a Double Beam Spectrophotometer,” Rev. Sci. Instrum. 45, 1356 (1974).
[CrossRef]

I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
[CrossRef]

H. Fujii, T. Asakura, “A Contrast Variation of Image Speckle Intensity under Illumination of Partially Coherent Light,” Opt. Commun. 12, 32 (1974).
[CrossRef]

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974).
[CrossRef]

1973 (3)

R. P. Edwin, “Light Scattering as a Technique for Measuring the Roughness of Optical Surfaces,” J. Phys. E 6, 55 (1973).
[CrossRef]

K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
[CrossRef]

M. Henzler, “The Roughness of Cleaved Semiconductor Surfaces,” Surf. Sci. 36, 109 (1973).
[CrossRef]

1972 (3)

1971 (1)

S. Mallick, “Interference Arrangement for Measuring the Autocorrelation Function of a Smoothly Finished Surface,” Opt. Commun. 2, 396 (1971).
[CrossRef]

1970 (2)

I. J. Hodgkinson, “A Simple Scatter Method for Optical Surface Roughness and Slope Measurements. Roughness of Polished Fused Silica,” J. Phys. E 3, 341 (1970).
[CrossRef]

I. J. Hodgkinson, “The Application of Fringes of Equal Chromatic Order to Assessment of the Surface Roughness of Polished Fused Silica,” J. Phys. E. 3, 300 (1970).
[CrossRef]

1969 (2)

1965 (1)

M. Kubo, “Instrument for the Measurement of Slope and Height Distribution of Surface Roughness,” Rev. Sci. Instrum. 36, 236 (1965).
[CrossRef]

1963 (1)

Abdulkadir, A.

A. Abdulkadir, R. Birkebak, “Optical Surface Roughness and Slopes Measurements with a Double Beam Spectrophotometer,” Rev. Sci. Instrum. 45, 1356 (1974).
[CrossRef]

Allen, E. V.

Asakura, T.

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974).
[CrossRef]

H. Fujii, T. Asakura, “A Contrast Variation of Image Speckle Intensity under Illumination of Partially Coherent Light,” Opt. Commun. 12, 32 (1974).
[CrossRef]

Baumeister, P.

J. Eastman, P. Baumeister, “The Microstructure of Polished Optical Surfaces,” Opt. Commun. 12, 418 (1974).
[CrossRef]

Beckmann, P.

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

Bennett, H. E.

Bennett, J. M.

Birkebak, R.

A. Abdulkadir, R. Birkebak, “Optical Surface Roughness and Slopes Measurements with a Double Beam Spectrophotometer,” Rev. Sci. Instrum. 45, 1356 (1974).
[CrossRef]

Brown, R.

N. Schwartz, R. Brown, “A Stylus Method for Evaluating the Thickness of Thin Films and Substrate Roughness,” in Transections of the Eighth Vacuum Symposium and Second International Congress (Pergamon, New York, 1961).

Chandley, P. J.

P. J. Chandley, “Determination of the Standard Deviation of Height on a Rough Surface Using Interference Microscopy,” Opt. Quantum Electron. 11, 407 (1979).
[CrossRef]

P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329 (1976).
[CrossRef]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U.P., Princeton, 1946).

Dancy, J. H.

Eastman, J.

J. Eastman, P. Baumeister, “The Microstructure of Polished Optical Surfaces,” Opt. Commun. 12, 418 (1974).
[CrossRef]

Edwin, R. P.

R. P. Edwin, “Light Scattering as a Technique for Measuring the Roughness of Optical Surfaces,” J. Phys. E 6, 55 (1973).
[CrossRef]

Fox, D. G.

Fujii, H.

H. Fujii, J. W. Y. Lit, “Surface Roughness Measurement Using Dichromatic Speckle Pattern: an Experimental Study,” Appl. Opt. 17, 2690 (1978).
[CrossRef] [PubMed]

H. Fujii, T. Asakura, “A Contrast Variation of Image Speckle Intensity under Illumination of Partially Coherent Light,” Opt. Commun. 12, 32 (1974).
[CrossRef]

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974).
[CrossRef]

Gordon, R. L.

Hard, S.

Hayslett, C. R.

Henzler, M.

M. Henzler, “The Roughness of Cleaved Semiconductor Surfaces,” Surf. Sci. 36, 109 (1973).
[CrossRef]

Hildebrand, B. P.

Hodgkinson, I. J.

I. J. Hodgkinson, “A Simple Scatter Method for Optical Surface Roughness and Slope Measurements. Roughness of Polished Fused Silica,” J. Phys. E 3, 341 (1970).
[CrossRef]

I. J. Hodgkinson, “The Application of Fringes of Equal Chromatic Order to Assessment of the Surface Roughness of Polished Fused Silica,” J. Phys. E. 3, 300 (1970).
[CrossRef]

Hornauer, D. L.

D. L. Hornauer, H. Raether, “Determination of Roughness of LiF Films Using Guided Light Modes,” Opt. Commun. 40, 105 (1981).
[CrossRef]

King, C. M.

Kubo, M.

M. Kubo, “Instrument for the Measurement of Slope and Height Distribution of Surface Roughness,” Rev. Sci. Instrum. 36, 236 (1965).
[CrossRef]

Kwon, O. Y.

Légér, D.

Levin, B. R.

B. R. Levin, The Theory of Random Processes and Their Application to Radio Engineering (Sovyetskoe Radio, Moscow, 1960), in Russian.

Lit, J. W. Y.

Llebaria, A.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Lukeš, F.

I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
[CrossRef]

I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
[CrossRef]

Mallick, S.

S. Mallick, “Interference Arrangement for Measuring the Autocorrelation Function of a Smoothly Finished Surface,” Opt. Commun. 2, 396 (1971).
[CrossRef]

Mayani, N.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Motycka, J.

Nagata, K.

K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
[CrossRef]

Navrátil, K.

I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
[CrossRef]

I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
[CrossRef]

Nilsson, O.

Nishiwaki, J.

K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
[CrossRef]

Ohlidal, I.

I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
[CrossRef]

Ohlidel, I.

I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
[CrossRef]

Palmari, J. P.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

Palmari, J.-P.

Perrin, J. C.

Raether, H.

D. L. Hornauer, H. Raether, “Determination of Roughness of LiF Films Using Guided Light Modes,” Opt. Commun. 40, 105 (1981).
[CrossRef]

Rasigni, G.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J.-P. Palmari, “Spectral-Density Function of the Surface Roughness for Polished Optical Surfaces,” J. Opt. Soc. Am. 73, 1235 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

Rasigni, M.

G. Rasigni, F. Varnier, M. Rasigni, J.-P. Palmari, “Spectral-Density Function of the Surface Roughness for Polished Optical Surfaces,” J. Opt. Soc. Am. 73, 1235 (1983).
[CrossRef]

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

Ribbens, W. B.

Rimmer, M. P.

Schwartz, N.

N. Schwartz, R. Brown, “A Stylus Method for Evaluating the Thickness of Thin Films and Substrate Roughness,” in Transections of the Eighth Vacuum Symposium and Second International Congress (Pergamon, New York, 1961).

Spizzichino, A.

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

Sprague, R. A.

Umehara, T.

K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
[CrossRef]

Varnier, F.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J.-P. Palmari, “Spectral-Density Function of the Surface Roughness for Polished Optical Surfaces,” J. Opt. Soc. Am. 73, 1235 (1983).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

Velzel, H. F.

H. F. Velzel, “Measurement of Surface Roughness by Interferential Contrast—an Application of Shearing Interferometry to the Study of Phase Objects,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970).

Whitefield, R. J.

Wyant, J. C.

Appl. Opt. (13)

W. B. Ribbens, “Interferometric Surface Roughness Measurement,” Appl. Opt. 8, 2173 (1969).
[CrossRef] [PubMed]

W. B. Ribbens, “Surface Roughness Measurement by Holographic Interferometry,” Appl. Opt. 11, 807 (1972).
[CrossRef] [PubMed]

W. B. Ribbens, “Surface Roughness Measurement by Two Wavelength Holographic Interferometry,” Appl. Opt. 13, 1085 (1974).
[CrossRef] [PubMed]

S. Hard, O. Nilsson, “Laser Heterodyne Apparatus for Roughness Measurements of Polished Surfaces,” Appl. Opt. 17, 3827 (1978).
[CrossRef] [PubMed]

B. P. Hildebrand, R. L. Gordon, E. V. Allen, “Instrument for Measuring the Roughness of Supersmooth Surfaces,” Appl. Opt. 13, 177 (1974).
[CrossRef] [PubMed]

R. J. Whitefield, “Noncontact Optical Profilometer,” Appl. Opt. 14, 2480 (1975).
[CrossRef] [PubMed]

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 (1976).
[CrossRef] [PubMed]

J. Motyčka, “Proposed Interferometric Method of Measurement of Roughness and Autocorrelation Function in Smooth-Finished Surfaces,” Appl. Opt. 8, 1435 (1969).
[CrossRef]

R. A. Sprague, “Surface Roughness Measurement Using White Light Speckle,” Appl. Opt. 11, 2811 (1972).
[CrossRef] [PubMed]

H. Fujii, J. W. Y. Lit, “Surface Roughness Measurement Using Dichromatic Speckle Pattern: an Experimental Study,” Appl. Opt. 17, 2690 (1978).
[CrossRef] [PubMed]

J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785 (1981).
[CrossRef] [PubMed]

M. P. Rimmer, C. M. King, D. G. Fox, “Computer Program for the Analysis of Interferometric Test Data,” Appl. Opt. 11, 2790 (1972).
[CrossRef] [PubMed]

O. Y. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Phys. E (2)

I. J. Hodgkinson, “A Simple Scatter Method for Optical Surface Roughness and Slope Measurements. Roughness of Polished Fused Silica,” J. Phys. E 3, 341 (1970).
[CrossRef]

R. P. Edwin, “Light Scattering as a Technique for Measuring the Roughness of Optical Surfaces,” J. Phys. E 6, 55 (1973).
[CrossRef]

J. Phys. E. (1)

I. J. Hodgkinson, “The Application of Fringes of Equal Chromatic Order to Assessment of the Surface Roughness of Polished Fused Silica,” J. Phys. E. 3, 300 (1970).
[CrossRef]

J. Phys. Paris (1)

I. Ohlidal, F. Lukeš, K. Navrátil, “The Problem of Surface Roughness in Ellipsometry and Reflectometry,” J. Phys. Paris 38, C5-77 (1977) (Colloque C-5, supplèment au No 11).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Nagata, T. Umehara, J. Nishiwaki, “The Determination of rms Roughness and Correlation Length of Rough Surface by Measuring Spatial Coherence Function,” Jpn. J. Appl. Phys. 12, 1693 (1973).
[CrossRef]

Opt. Commun. (6)

H. Fujii, T. Asakura, “A Contrast Variation of Image Speckle Intensity under Illumination of Partially Coherent Light,” Opt. Commun. 12, 32 (1974).
[CrossRef]

S. Mallick, “Interference Arrangement for Measuring the Autocorrelation Function of a Smoothly Finished Surface,” Opt. Commun. 2, 396 (1971).
[CrossRef]

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974).
[CrossRef]

D. L. Hornauer, H. Raether, “Determination of Roughness of LiF Films Using Guided Light Modes,” Opt. Commun. 40, 105 (1981).
[CrossRef]

J. Eastman, P. Baumeister, “The Microstructure of Polished Optical Surfaces,” Opt. Commun. 12, 418 (1974).
[CrossRef]

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Opt. Quantum Electron. (2)

P. J. Chandley, “Determination of the Standard Deviation of Height on a Rough Surface Using Interference Microscopy,” Opt. Quantum Electron. 11, 407 (1979).
[CrossRef]

P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329 (1976).
[CrossRef]

Phys. Rev. B (1)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance Functions, Root-Mean-Square-Roughness Height, and Autocovariance Length for Rough Deposits of Copper, Silver and Gold,” Phys. Rev. B 25, 2315 (1982).
[CrossRef]

Rev. Sci. Instrum. (2)

A. Abdulkadir, R. Birkebak, “Optical Surface Roughness and Slopes Measurements with a Double Beam Spectrophotometer,” Rev. Sci. Instrum. 45, 1356 (1974).
[CrossRef]

M. Kubo, “Instrument for the Measurement of Slope and Height Distribution of Surface Roughness,” Rev. Sci. Instrum. 36, 236 (1965).
[CrossRef]

Surf. Sci. (2)

I. Ohlidel, F. Lukeš, K. Navrátil, “Rough Silicon Surface Studied by Optical Method,” Surf. Sci. 45, 91 (1974).
[CrossRef]

M. Henzler, “The Roughness of Cleaved Semiconductor Surfaces,” Surf. Sci. 36, 109 (1973).
[CrossRef]

Other (6)

N. Schwartz, R. Brown, “A Stylus Method for Evaluating the Thickness of Thin Films and Substrate Roughness,” in Transections of the Eighth Vacuum Symposium and Second International Congress (Pergamon, New York, 1961).

Interferenzmikroskop Epival Interphako, Gebrauchsanleitung (VEB Carl Zeiss, Jena, 1977), in German.

B. R. Levin, The Theory of Random Processes and Their Application to Radio Engineering (Sovyetskoe Radio, Moscow, 1960), in Russian.

H. Cramér, Mathematical Methods of Statistics (Princeton U.P., Princeton, 1946).

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

H. F. Velzel, “Measurement of Surface Roughness by Interferential Contrast—an Application of Shearing Interferometry to the Study of Phase Objects,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970).

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Figures (6)

Fig. 1
Fig. 1

Schematic arrangement of the Zeiss Epival Interphako interference microscope: 1, light source; 2, sample; 3, objective; 4, beam splitter; 5 and 6, beam splitter prisms; 7, wedge with variable wedge angle; 8 and 9, wedges of the interferometer; 10, eyepiece.

Fig. 2
Fig. 2

Schematic diagram of fringe contours of the rough parabolic collimators studied.

Fig. 3
Fig. 3

Interferogram of a particular sample of the rough parabolic collimator investigated.

Fig. 4
Fig. 4

Dependence of the parameter σh on the shear parameter τ for the same sample.

Fig. 5
Fig. 5

Dependence of the function C(τ) on the shear parameter τ implied by the dependence σh(τ) plotted in Fig. 4.

Fig. 6
Fig. 6

Height distribution function of the quantity h for the sample under investigation corresponding to τ > τ*.

Equations (26)

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Z ( x , y ) = x 2 2 R 1 y 2 2 R 2 ,
u 1 = A 1 exp i ( ω t k 1 r + φ 1 ) ,
u 2 = A 2 exp i ( ω t k 2 r + φ 2 ) ,
φ j = k j { s j Z ( x j , y j ) + n j ξ [ x j , y j , Z ( x j , y j ) ] } + φ 0 j ,
k j = k ( sin α j cos β j i x + sin α j sin β j i y + cos α j i z ) ,
n j = [ N x ( x j , y j ) cos α j cos β j N y ( x j , y j ) cos α j sin β j + N z ( x j , y j ) sin α j cos ( φ j + β j ) ] i x + [ N x ( x j , y j ) cos α j sin β j + N y ( x j , y j ) cos α j cos β j + N z ( x j y j ) sin α j sin ( φ j + β j ) ] i y + [ N z ( x j , y j ) cos α j sin ϑ j sin α j ] i z ,
N ( x , y ) = Z x ( x , y ) i x + Z y ( x , y ) i y + i z .
Δ = k ( ( α 1 cos β 1 α 2 cos β 2 ) x + ( α 1 sin β 1 α 2 sin β 2 ) y + 2 [ Z ( x 1 , y 1 ) Z ( x 2 , y 2 ) ] + 2 { ξ [ x 1 , y 1 , Z ( x 1 , y 1 ) ] ξ [ x 2 , y 2 , Z ( x 2 , y 2 ) ] } ] φ 0 .
L | R 1 | ( 1 + 1 | R 2 | R 1 2 + R 2 2 ) + | α j | 1 / | α j | ,
Δ = ( 2 m 1 ) π ,
x 1 x 2 = τ cos Ө and y 1 y 2 = τ sin Ө,
x ( α 1 2 cos β 1 α 2 2 cos β 2 τ cos Ө R 1 ) + y ( α 1 2 sin β 1 α 2 2 sin β 2 τ sin Ө R 2 ) + ξ [ x 1 , y 1 , Z ( x 1 , y 1 ) ] ξ [ x 2 , y 2 , Z ( x 2 , y 2 ) ] = ( 2 m 1 ) λ 4 + φ 0 λ 4 π .
x ( α 1 2 cos β 1 α 2 2 cos β 2 τ cos Ө R 1 ) + y ( α 1 2 sin β 1 α 2 2 sin β 2 τ sin Ө R 2 ) ( 2 m 1 + φ 0 π ) λ 4 = 0 .
q = λ 2 1 A 2 + B 2 ,
p = h / A 2 + B 2 ;
h = λ 2 p q .
σ h 2 ( τ ) = 2 σ 2 [ 1 C ( τ ) ] ,
B ( τ ) = σ 2 C ( τ ) ,
lim τ C ( τ ) = 0 ,
0 C ( τ ) 1 , C ( 0 ) = 1 .
lim τ σ h 2 ( τ ) = 2 σ 2 ,
0 σ h ( τ ) σ 2 , σ h ( 0 ) = 0 .
σ h ( τ ) = σ 2 ( τ > τ * ) .
C ( τ ) = 1 σ h 2 ( τ ) 2 σ 2 .
σ h = ( i = 1 N h i 2 N ) 1 / 2
tan β 0 = 2 σ / T .

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