Abstract

Calculations are performed to relate the stylus profile of a one-dimensionally rough surface to the angular distribution of the light scattered by such a surface. In the direct problem, the angular distribution of the scattered light calculated from the profile is shown to agree with the measured one. In the inverse problem, the rms roughness and the autocorrelation function are found by a least-squares fit to the measured angular distribution. For the smoother surfaces, the rms roughness is mostly determined by the ratio between the power of the specular beam and the total power of the scattered light; the computed values are proportional to those calculated directly from the stylus profiles. The values of the parameters obtained by the least-squares fit are affected by a variety of errors and agree only partially with those obtained from the stylus profile.

© 1990 Optical Society of America

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References

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  1. R. E. Reason, “Progress in the Appraisal of Surface Topography During the First Half-Century of Instrument Development,” Wear 57, 1–16 (1979).
    [CrossRef]
  2. T. R. Thomas, Ed., Rough Surfaces (Longman, London, 1982), Chap. 2.
  3. D. J. Whitehouse, “Stylus Techniques,” in Characterization of Solid Surfaces, P. E. Kane, G. R. Larrabee, Eds. (Plenum, New York, 1975), Chap. 3.
  4. J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785–1802 (1981).
    [CrossRef] [PubMed]
  5. T. V. Vorburger, “Measurement of Roughness of Very Smooth Surfaces,” CIRP Annals 36/2, 503–509 (1987).
    [CrossRef]
  6. E. C. Teague, “Evaluation, Revision, and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” Nat. Bur. Stand. (U.S.), Tech. Note 902, Gaithersburg, MD (1976).
  7. J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering Vol. 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York1979), Chap. 7.
  8. ASTM Standard F1084-87, Standard Test Method for Measuring the Effect of Surface Roughness of Optical Components by Total Integrated Scattering,” in Annual Book of ASTM Standards (ASTM, Philadelphia, 1987).
  9. C. S. Gardner, W. E. Streight, “Surface Roughness Gauge and Method,” U.S. Patent No. 4,364,663, 12/82.
  10. R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
    [CrossRef]
  11. P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
    [CrossRef]
  12. P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
    [CrossRef]
  13. D. H. Hensler, “Light Scattering from Fused Polycrystalline Aluminum Oxide Surfaces,” Appl. Opt. 11, 2522–2528 (1972).
    [CrossRef] [PubMed]
  14. 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]
  15. T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).
  16. P. Beckmann, in The Scattering of Electromagnetic Waves from Rough Surfaces, P. Beckmann, A. Spizzichino (Pergamon, London, 1963).
  17. Certain commercial equipment is identified in this report to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the equipment identified are necessarily the best available for the purpose.
  18. ANSI/ASME B46.1-1985, “Surface Texture” (American Society of Mechanical Engineers, New York, 1985).
  19. R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), pp. 5–7.
  20. T. V. Vorburger, “fastmenu: A Set of fortran Programs for Analyzing Surface Texture,” Nat. Bur. Stand. (U.S.) NBSIR 83-2703, Gaithersburg, MD (1983).
  21. G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), pp. 171 ff.
  22. T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.
  23. J. C. Dainty, Ed., Laser-Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9 (Springer-Verlag, Berlin, 1975).
  24. R. Brodmann, M. Allgäuer, “Comparison of Light Scattering from Rough Surfaces with Optical and Mechanical Profilometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1009, 111–118 (1988).
  25. J. H. Rakels, “Recognizable Surface Finish Parameters Obtained from Diffraction Patterns of Rough Surfaces,” in Surface Measurement and Characterization, Proc. Soc. Photo-Opt. Instrum. Eng. 1009, p. 119–125 (1988).
  26. D. G. Chetwynd, “The Digitization of Surface Profiles,” Wear 57, 137–145 (1979).
    [CrossRef]
  27. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, 1989), pp. 39–41.
  28. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 121.
  29. J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116–124 (1979).
    [CrossRef]
  30. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125–136 (1979).
    [CrossRef]

1988 (2)

R. Brodmann, M. Allgäuer, “Comparison of Light Scattering from Rough Surfaces with Optical and Mechanical Profilometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1009, 111–118 (1988).

J. H. Rakels, “Recognizable Surface Finish Parameters Obtained from Diffraction Patterns of Rough Surfaces,” in Surface Measurement and Characterization, Proc. Soc. Photo-Opt. Instrum. Eng. 1009, p. 119–125 (1988).

1987 (1)

T. V. Vorburger, “Measurement of Roughness of Very Smooth Surfaces,” CIRP Annals 36/2, 503–509 (1987).
[CrossRef]

1984 (2)

R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
[CrossRef]

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

1981 (1)

1979 (4)

R. E. Reason, “Progress in the Appraisal of Surface Topography During the First Half-Century of Instrument Development,” Wear 57, 1–16 (1979).
[CrossRef]

D. G. Chetwynd, “The Digitization of Surface Profiles,” Wear 57, 137–145 (1979).
[CrossRef]

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]

1976 (2)

P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
[CrossRef]

P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

1972 (1)

1961 (1)

Allgäuer, M.

R. Brodmann, M. Allgäuer, “Comparison of Light Scattering from Rough Surfaces with Optical and Mechanical Profilometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1009, 111–118 (1988).

Beckmann, P.

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

Bennett, H. E.

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]

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering Vol. 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York1979), Chap. 7.

Bennett, J. M.

J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785–1802 (1981).
[CrossRef] [PubMed]

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

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering Vol. 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York1979), Chap. 7.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, 1989), pp. 39–41.

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 121.

Blackman, R. B.

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), pp. 5–7.

Brodmann, R.

R. Brodmann, M. Allgäuer, “Comparison of Light Scattering from Rough Surfaces with Optical and Mechanical Profilometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1009, 111–118 (1988).

R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
[CrossRef]

Cao, L. X.

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Chandley, P. J.

P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
[CrossRef]

P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

Chetwynd, D. G.

D. G. Chetwynd, “The Digitization of Surface Profiles,” Wear 57, 137–145 (1979).
[CrossRef]

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Dancy, J. H.

Elson, J. M.

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

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering Vol. 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York1979), Chap. 7.

Fullana, L.

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Gardner, C. S.

C. S. Gardner, W. E. Streight, “Surface Roughness Gauge and Method,” U.S. Patent No. 4,364,663, 12/82.

Gast, T.

R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
[CrossRef]

Giauque, C. H. W.

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Gilsinn, D. E.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Hensler, D. H.

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), pp. 171 ff.

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]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, 1989), pp. 39–41.

McLay, M. J.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

Porteus, J. O.

Raja, J.

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Rakels, J. H.

J. H. Rakels, “Recognizable Surface Finish Parameters Obtained from Diffraction Patterns of Rough Surfaces,” in Surface Measurement and Characterization, Proc. Soc. Photo-Opt. Instrum. Eng. 1009, p. 119–125 (1988).

Reason, R. E.

R. E. Reason, “Progress in the Appraisal of Surface Topography During the First Half-Century of Instrument Development,” Wear 57, 1–16 (1979).
[CrossRef]

Scire, F. E.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

Streight, W. E.

C. S. Gardner, W. E. Streight, “Surface Roughness Gauge and Method,” U.S. Patent No. 4,364,663, 12/82.

Teague, E. C.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

E. C. Teague, “Evaluation, Revision, and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” Nat. Bur. Stand. (U.S.), Tech. Note 902, Gaithersburg, MD (1976).

Thurn, G.

R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
[CrossRef]

Tukey, J. W.

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), pp. 5–7.

Vorburger, T. V.

T. V. Vorburger, “Measurement of Roughness of Very Smooth Surfaces,” CIRP Annals 36/2, 503–509 (1987).
[CrossRef]

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

T. V. Vorburger, “fastmenu: A Set of fortran Programs for Analyzing Surface Texture,” Nat. Bur. Stand. (U.S.) NBSIR 83-2703, Gaithersburg, MD (1983).

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), pp. 171 ff.

Whitehouse, D. J.

D. J. Whitehouse, “Stylus Techniques,” in Characterization of Solid Surfaces, P. E. Kane, G. R. Larrabee, Eds. (Plenum, New York, 1975), Chap. 3.

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]

Appl. Opt. (2)

CIRP Annals (2)

T. V. Vorburger, “Measurement of Roughness of Very Smooth Surfaces,” CIRP Annals 36/2, 503–509 (1987).
[CrossRef]

R. Brodmann, T. Gast, G. Thurn, “An Optical Instrument for Measuring the Surface Roughness in Production Control,” CIRP Annals 33/1, 403–406 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Res. NBS (1)

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface Roughness Studies with DALLAS-Detector Array for Laser Light Angular Scattering,” J. Res. NBS 89, 3–16 (1984).

Opt. Eng. (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]

Opt. Quantum Electron. (2)

P. J. Chandley, “Surface Roughness Measurements from Coherent Light Scattering,” Opt. Quantum Electron. 8, 323–327 (1976).
[CrossRef]

P. J. Chandley, “Determination of the Autocorrelation Function of Height on a Rough Surface from Coherent Light Scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. Brodmann, M. Allgäuer, “Comparison of Light Scattering from Rough Surfaces with Optical and Mechanical Profilometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1009, 111–118 (1988).

Surface Measurement and Characterization (1)

J. H. Rakels, “Recognizable Surface Finish Parameters Obtained from Diffraction Patterns of Rough Surfaces,” in Surface Measurement and Characterization, Proc. Soc. Photo-Opt. Instrum. Eng. 1009, p. 119–125 (1988).

Wear (2)

D. G. Chetwynd, “The Digitization of Surface Profiles,” Wear 57, 137–145 (1979).
[CrossRef]

R. E. Reason, “Progress in the Appraisal of Surface Topography During the First Half-Century of Instrument Development,” Wear 57, 1–16 (1979).
[CrossRef]

Other (16)

T. R. Thomas, Ed., Rough Surfaces (Longman, London, 1982), Chap. 2.

D. J. Whitehouse, “Stylus Techniques,” in Characterization of Solid Surfaces, P. E. Kane, G. R. Larrabee, Eds. (Plenum, New York, 1975), Chap. 3.

E. C. Teague, “Evaluation, Revision, and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” Nat. Bur. Stand. (U.S.), Tech. Note 902, Gaithersburg, MD (1976).

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering Vol. 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York1979), Chap. 7.

ASTM Standard F1084-87, Standard Test Method for Measuring the Effect of Surface Roughness of Optical Components by Total Integrated Scattering,” in Annual Book of ASTM Standards (ASTM, Philadelphia, 1987).

C. S. Gardner, W. E. Streight, “Surface Roughness Gauge and Method,” U.S. Patent No. 4,364,663, 12/82.

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

Certain commercial equipment is identified in this report to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the equipment identified are necessarily the best available for the purpose.

ANSI/ASME B46.1-1985, “Surface Texture” (American Society of Mechanical Engineers, New York, 1985).

R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), pp. 5–7.

T. V. Vorburger, “fastmenu: A Set of fortran Programs for Analyzing Surface Texture,” Nat. Bur. Stand. (U.S.) NBSIR 83-2703, Gaithersburg, MD (1983).

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), pp. 171 ff.

T. V. Vorburger, L. X. Cao, C. H. W. Giauque, J. Raja, D. E. Gilsinn, L. Fullana, “Optical Scattering from Rough Surfaces: Experiment and Theory,” Seventh International Colloquium on Surfaces, H. Trumpold, Ed. (Technische Universität, Karl-Marx-Stadt, DDR, 1988), pp. 308–316.

J. C. Dainty, Ed., Laser-Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9 (Springer-Verlag, Berlin, 1975).

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, 1989), pp. 39–41.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 121.

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Figures (13)

Fig. 1
Fig. 1

Autocorrelation functions calculated from the profile data for specimens #6 and #14, respectively the roughest and the smoothest in the set.

Fig. 2
Fig. 2

Photograph of DALLAS.

Fig. 3
Fig. 3

Schematic diagram of detector yoke, showing five of the eighty-seven detectors. The yoke is positioned for the central scan in the plane of incidence (j = 3). The detector positions for the other scans (j = 1,2,4,5) are also shown. These positions are reached by rotating the yoke through angles ϕ(j,θ).

Fig. 4
Fig. 4

Detector positions in a typical run showing the five positions for θ = 0° (zenith) and θ = 86°. The spacing between different scans is calculated to be the same distance for the various polar angles θ, resulting in different angles ϕ.

Fig. 5
Fig. 5

Angular scattering distributions for specimen #15 showing the central scan (j = 3) and the sum of all five scans. The dots show single-detector measurements taken at higher resolution in selected positions.

Fig. 6
Fig. 6

Measured and computed angular distributions of scattered light intensity for specimens #6 and #14 (the direct problem).

Fig. 7
Fig. 7

Surface roughness σ measured by optical scattering vs σ measured by a stylus instrument.

Fig. 8
Fig. 8

ACL determined from optical scattering by a least-squares fit vs that obtained from stylus measurements.

Fig. 9
Fig. 9

Measured and computed angular distributions for specimens #6 and #14, using a four-parameter fit of the angular distribution. The roughness σ is derived from the specular beam for #14 and from the stylus data for #6. The specular beam is added to the diffuse scattering distribution for #14.

Fig. 10
Fig. 10

Measured and computed angular distributions for specimens #6 and #14. The parameter σ was computed directly from the stylus data and the parameters T, α1, α2, and α3 were determined by a least-squares fit of the ACF obtained from the stylus data.

Fig. 11
Fig. 11

Measured and computed angular distributions for specimens #6 and #14 (five-parameter fit).

Fig. 12
Fig. 12

Logarithmic autocorrelation functions computed from stylus data and from the parameters obtained in the least-squares fit of the angular distributions shown in Fig. 11.

Fig. 13
Fig. 13

Measured and computed angular distributions for specimens #6 and #14 (five-parameter fit). Measured intensities for large angles are included in the calculations.

Tables (2)

Tables Icon

Table I Roughness σ and Autocorrelation length T

Tables Icon

Table II Results of Numerical Experiments

Equations (28)

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

σ 2 = 1 2 L - L L ζ ( x ) 2 d x .
σ [ 1 N k = 1 N z k 2 ] 1 / 2 .
C ( τ ) = lim L 1 2 L σ 2 - L L ζ ( x ) ζ ( x + τ ) d x .
C j = 1 σ 2 ( N - j ) k = 1 N - j z k z j + k ,             j = 0 , 1 , .
C ( τ ) = exp ( - τ / T α ) ,
ln ( - ln C ) = α ( ln τ - ln T )
ϕ ( θ , j ) = 2 ( j - 3 ) arcsin [ 1 / 4 π d / ( 2 r cos θ ) ] = 2 ( j - 3 ) arcsin ( 0.01042 / cos θ ) ,
ρ ( θ ) = F ( θ ) 2 L - L L exp ( i v · r ) d x ,
F ( θ ) = [ 1 + cos ( θ inc - θ ) ] / [ cos θ inc ( cos θ inc + cos θ ) ] ,
r ( x ) = x e ^ 1 + ζ ( x ) e ^ 3 ,             v ( θ ) = k inc - k ( θ ) ,
v x ( θ ) = - ( 2 π / λ ) ( sin θ inc + sin θ ) , v z ( θ ) = - ( 2 π / λ ) ( cos θ inc + cos θ ) .
w ( z ) = 1 σ ( 2 π ) exp ( - z 2 / 2 σ 2 ) ,
ρ ( θ ) = F ( θ ) exp { - ½ [ σ v z ( θ ) ] 2 } sin [ v x ( θ ) L ] / v x ( θ ) L ,
ρ ρ * = [ F ( θ ) ] 2 4 L 2 - L L - L L exp [ i v x ( x 1 - x 2 ) ] × exp [ i v z ( ζ 1 - ζ 2 ) ] d x 1 d x 2 ,
i ( θ ) = [ F ( θ ) ] 2 2 T - exp [ i v x ( θ ) τ - g ( θ ) ] { exp [ g ( θ ) C ( τ ) ] - 1 } d τ ,
g ( θ ) = [ σ v z ( θ ) ] 2 = [ ( 2 π σ / λ ) ( cos θ inc + cos θ ) ] 2 .
i k = F k 2 J k ,
J k = 0 cos ( ω k τ ) [ exp { g k [ exp ( - τ α ) - 1 ] } - exp ( - g k ) ] d τ ,
F k = F ( θ k ) ,             g k = g ( θ k ) ,             ω k = v x ( θ k ) T ,             τ = τ / T .
Q = k = 1 K ( ln I k - ln I k c ) 2 ,
Q ( σ , T , α , A ) = [ ln I k - ln i k ( σ , T , α ) - A ] 2 ,
A = ( 1 / K ) ( ln I k - ln i k ) .
v x ( θ ) - ( 2 π / λ ) ( θ inc + θ ) cos θ inc ,             v z ( θ ) - ( 4 π / λ ) cos θ inc ,
i spec = exp [ - ( 4 π σ / λ ) 2 cos 2 θ inc ,
I tot = f I k - I spec ( f - 1 ) ,
α ( τ ) = α 1 / [ 1 + ( α 3 τ ) ν ] + α 2 / [ 1 + ( α 3 τ ) - ν ] ,
Q = Q + w spec ( I spec / I tot - i spec ) 2 ˙ ,
sin θ + sin θ inc = ± n λ / d , n = 1 ,

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