Abstract

A relation from vector scattering theory has been used to predict the angular distribution of scattered light from optical surfaces as a function of the wavelength, optical constants of the material, and spectral density function. For calculations of one-dimensional (two-dimensional) scattering, the spectral density function of the surface roughness is obtained from the Fourier transform (Hankel transform) of the autocovariance function, which in turn is determined from surface-profile data. Measured statistics presented for various types of optical surfaces indicate that there are three basic components of surface structure: long-range waviness, short-range random roughness, and periodicity; one or more of which may be present on a given surface. Averaged and unaveraged surface-profile data for the same surface are shown to be consistent. Experimental data are presented that yield an exponential autocovariance function, and give a reasonably good fit to a Poisson distribution of zero crossings. Finally, angular scattering values calculated using measured surface statistics with vector scattering theory are compared to scattering values measured on the same surface. The shapes of the measured and calculated curves are similar, but the magnitudes are not. However, the rms surface roughnesses calculated from total integrated scattering measurements are in excellent agreement with values measured directly on these same surfaces.

© 1979 Optical Society of America

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  1. H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961);H. E. Bennett, ibid.,  53, 1389 (1963);J. O. Porteus, ibid. 53, 1395 (1963).
    [Crossref]
  2. J. M. Bennett, Appl. Opt. 15, 2705 (1976).
    [Crossref] [PubMed]
  3. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963);J. A. Holzer and C. C. Sung, J. Appl. Phys. 47, 3363 (1976).
    [Crossref]
  4. J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974);V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975);A. A. Maradudin and D. L. Mills, ibid. 11, 1392 (1975);J. M. Elson, ibid. 12, 2541 (1975);J. M. Elson, Appl. Opt. 16, 2872 (1977).
    [Crossref] [PubMed]
  5. J. Eastman and P. W. Baumeister, Opt. Commun. 12, 418 (1974);P. J. Chandley, Opt. Quantum Electron. 8, 329 (1976);J. C. Leader, McDonnell Aircraft Company, Report No. MCAIR 71–013, 1971 (unpublished).
    [Crossref]
  6. P. J. Chandley, Opt. Quantum Electron. 8, 323 (1976).
    [Crossref]
  7. J. C. Stover, Appl. Opt. 14, 1796 (1975).
    [Crossref] [PubMed]
  8. I. J. Hodgkinson, J. Phys. E,  3, 300 (1970);E. C. Teague, “Evaluation, Revision and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” NBS Technical Note 902 (U. S. Department of Commerce, Washington, DC, April1976);M. Kubo, Acta Imeko 21, 115 (1964);M. Kubo, Rev. Sci. Instrum. 36, 236 (1965);J. A. Greenwood and J. B. P. Williamson, Proc. Roy. Soc. Lond. A 295, 300 (1966);J. B. P. Williamson, Proc. Inst. Mech. Eng. 182, 21 (1967);T. R. Thomas and S. D. Probert, J. Phys. D. 3, 277 (1970);D. J. Whitehouse and J. F. Archard, Proc. Roy. Soc. Lond. A 316, 97 (1970);C. J. Pellerin, J. Christensen, R. C. Jerner, and J. H. Peavey, J. Vac. Sci. Technol. 12, 496 (1975);J. Renau and J. A. Collinson, Bell System Tech. J. 44, 2203 (1965);J. Peklenik, Proc. Inst. Mech. Eng. 182, 108 (1967).
    [Crossref]
  9. D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
    [Crossref] [PubMed]
  10. E. L. Church and J. M. Zavada, Appl. Opt. 14, 1788 (1975).
    [Crossref] [PubMed]
  11. E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 16, 360 (1977).
    [Crossref]
  12. Equation (5.1) in the paper by Elson and Ritchie in Ref. 4. In this reference the surface factor is given as |ζk¯|2/L2.
  13. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1966), p. 272.
  14. The complex dielectric constant is related to the complex index of refraction n¯=n+ik where n is the real part of the refractive index and k the extinction coefficient by the relation ∊1 = n2 − k2 and ∊2 = 2nk. The extinction coefficient is not to be confused with the surface wave vector k which is used throughout the paper.
  15. To a good approximation the second and third terms in the denominator can be neglected relative to the first term, and sin2θ in the A term can be neglected relative to ∊1. If the log of the optical factor is plotted versus θ for a given wavelength, the factor multiplying cos2θ, |1−∊|2/(∊22−∊12)1/2, merely shifts the curve vertically and does not affect the angular dependence of the function.
  16. J. M. Elson, J. Opt. Soc. Am. 66, 682 (1976).
    [Crossref]
  17. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 73–74.
  18. R. B. Blackman and J. W. Tukey, Bell System Tech. J. 37, 485 (1958), see especially, p. 499.
    [Crossref]
  19. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 314–321.
  20. See Ref. 19, pp. 311–312. Equation (13) is equivalent to Eq. (9.98) in Ref. 19. This equation and Eq. (9.95) in Ref. 19 are equally good definitions of the auto-covariance function, especially when used with lag windows.
  21. See Ref. 18, pp. 228–231.
  22. See Ref. 19, pp. 83, 173.
  23. See Ref. 19, pp. 376–377.
  24. See Ref. 19, p. 22.
  25. See Ref. 17, p. 177.
  26. J. L. Doob, Stochastic Processes (Wiley, New York, 1953), p. 233.
  27. J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
    [Crossref]
  28. Manufactured by Rank Precision Industries, Ltd., Leicester, England.
  29. Manufactured by Ernst Fr. Weinz WEKA-OHG, 658 Idar-Oberstein 2, Germany.
  30. J. H. Dancy and J. M. Bennett, “Study of Diamond Styluses for a Talystep Profilometer,” in High Energy Laser Mirrors and Windows, Annual Report No. 9, NWC TP5988, NWC TP5988, (Naval Weapons Center, China Lake, California1977), pp. 133–144.
  31. H. E. Bennett and J. L. Stanford, J. Res. Natl. Bur. Stand. (U.S.) 80A, 643 (1976);P. C. Archibald and H. E. Bennett, “Scattering from Infrared Missile Domes,” in Proceedings of Seminar on Optics in Missile Engineering, January 16–18, 1978, Los Angeles, California, Vol. 133, (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1978);also, Opt. Eng. 17, 647 (Nov/Dec 1978).
    [Crossref]
  32. R. W. Dietz and J. M. Bennett, Appl. Opt. 5, 881 (1966).
    [PubMed]
  33. R. Anderson and J. M. Bennett, “Properties of KCl Optical Elements Forged Between Optically Polished Dies,” in Proceedings of the High Power Laser Optical Components and Component Materials Meeting, (Defense Advanced Research Projects Agency, Arlington, Virginia, 1977), p. 177.
  34. T. T. Saito, Appl. Opt. 14, 1773 (1975);J. B. Arnold, P. J. Steger, and T. T. Saito, ibid. 14, 1777 (1975).
    [Crossref] [PubMed]
  35. J. M. Blakely and D. L. Olson, J. Appl. Phys. 39, 3476 (1968).
    [Crossref]
  36. W. J. Choyke, W. D. Partlow, E. P. Supertzi, F. J. Venskytis, and G. B. Brandt, Appl. Opt. 16, 2013 (1977).
    [Crossref] [PubMed]
  37. Frederick Scmid, U. S. PatentNo. 3,898,051, Crystal Growing, 5Aug.1975.

1977 (2)

1976 (4)

J. M. Elson, J. Opt. Soc. Am. 66, 682 (1976).
[Crossref]

J. M. Bennett, Appl. Opt. 15, 2705 (1976).
[Crossref] [PubMed]

P. J. Chandley, Opt. Quantum Electron. 8, 323 (1976).
[Crossref]

H. E. Bennett and J. L. Stanford, J. Res. Natl. Bur. Stand. (U.S.) 80A, 643 (1976);P. C. Archibald and H. E. Bennett, “Scattering from Infrared Missile Domes,” in Proceedings of Seminar on Optics in Missile Engineering, January 16–18, 1978, Los Angeles, California, Vol. 133, (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1978);also, Opt. Eng. 17, 647 (Nov/Dec 1978).
[Crossref]

1975 (3)

1974 (2)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974);V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975);A. A. Maradudin and D. L. Mills, ibid. 11, 1392 (1975);J. M. Elson, ibid. 12, 2541 (1975);J. M. Elson, Appl. Opt. 16, 2872 (1977).
[Crossref] [PubMed]

J. Eastman and P. W. Baumeister, Opt. Commun. 12, 418 (1974);P. J. Chandley, Opt. Quantum Electron. 8, 329 (1976);J. C. Leader, McDonnell Aircraft Company, Report No. MCAIR 71–013, 1971 (unpublished).
[Crossref]

1971 (1)

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[Crossref]

1970 (1)

I. J. Hodgkinson, J. Phys. E,  3, 300 (1970);E. C. Teague, “Evaluation, Revision and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” NBS Technical Note 902 (U. S. Department of Commerce, Washington, DC, April1976);M. Kubo, Acta Imeko 21, 115 (1964);M. Kubo, Rev. Sci. Instrum. 36, 236 (1965);J. A. Greenwood and J. B. P. Williamson, Proc. Roy. Soc. Lond. A 295, 300 (1966);J. B. P. Williamson, Proc. Inst. Mech. Eng. 182, 21 (1967);T. R. Thomas and S. D. Probert, J. Phys. D. 3, 277 (1970);D. J. Whitehouse and J. F. Archard, Proc. Roy. Soc. Lond. A 316, 97 (1970);C. J. Pellerin, J. Christensen, R. C. Jerner, and J. H. Peavey, J. Vac. Sci. Technol. 12, 496 (1975);J. Renau and J. A. Collinson, Bell System Tech. J. 44, 2203 (1965);J. Peklenik, Proc. Inst. Mech. Eng. 182, 108 (1967).
[Crossref]

1969 (1)

D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
[Crossref] [PubMed]

1968 (1)

J. M. Blakely and D. L. Olson, J. Appl. Phys. 39, 3476 (1968).
[Crossref]

1966 (1)

1961 (1)

1958 (1)

R. B. Blackman and J. W. Tukey, Bell System Tech. J. 37, 485 (1958), see especially, p. 499.
[Crossref]

Anderson, R.

R. Anderson and J. M. Bennett, “Properties of KCl Optical Elements Forged Between Optically Polished Dies,” in Proceedings of the High Power Laser Optical Components and Component Materials Meeting, (Defense Advanced Research Projects Agency, Arlington, Virginia, 1977), p. 177.

Baumeister, P. W.

J. Eastman and P. W. Baumeister, Opt. Commun. 12, 418 (1974);P. J. Chandley, Opt. Quantum Electron. 8, 329 (1976);J. C. Leader, McDonnell Aircraft Company, Report No. MCAIR 71–013, 1971 (unpublished).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963);J. A. Holzer and C. C. Sung, J. Appl. Phys. 47, 3363 (1976).
[Crossref]

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 314–321.

Bennett, H. E.

H. E. Bennett and J. L. Stanford, J. Res. Natl. Bur. Stand. (U.S.) 80A, 643 (1976);P. C. Archibald and H. E. Bennett, “Scattering from Infrared Missile Domes,” in Proceedings of Seminar on Optics in Missile Engineering, January 16–18, 1978, Los Angeles, California, Vol. 133, (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1978);also, Opt. Eng. 17, 647 (Nov/Dec 1978).
[Crossref]

H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961);H. E. Bennett, ibid.,  53, 1389 (1963);J. O. Porteus, ibid. 53, 1395 (1963).
[Crossref]

Bennett, J. M.

J. M. Bennett, Appl. Opt. 15, 2705 (1976).
[Crossref] [PubMed]

R. W. Dietz and J. M. Bennett, Appl. Opt. 5, 881 (1966).
[PubMed]

J. H. Dancy and J. M. Bennett, “Study of Diamond Styluses for a Talystep Profilometer,” in High Energy Laser Mirrors and Windows, Annual Report No. 9, NWC TP5988, NWC TP5988, (Naval Weapons Center, China Lake, California1977), pp. 133–144.

R. Anderson and J. M. Bennett, “Properties of KCl Optical Elements Forged Between Optically Polished Dies,” in Proceedings of the High Power Laser Optical Components and Component Materials Meeting, (Defense Advanced Research Projects Agency, Arlington, Virginia, 1977), p. 177.

Blackman, R. B.

R. B. Blackman and J. W. Tukey, Bell System Tech. J. 37, 485 (1958), see especially, p. 499.
[Crossref]

Blakely, J. M.

J. M. Blakely and D. L. Olson, J. Appl. Phys. 39, 3476 (1968).
[Crossref]

Brandt, G. B.

Chandley, P. J.

P. J. Chandley, Opt. Quantum Electron. 8, 323 (1976).
[Crossref]

Chinnock, E. L.

D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
[Crossref] [PubMed]

Choyke, W. J.

Church, E. L.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 16, 360 (1977).
[Crossref]

E. L. Church and J. M. Zavada, Appl. Opt. 14, 1788 (1975).
[Crossref] [PubMed]

Dancy, J. H.

J. H. Dancy and J. M. Bennett, “Study of Diamond Styluses for a Talystep Profilometer,” in High Energy Laser Mirrors and Windows, Annual Report No. 9, NWC TP5988, NWC TP5988, (Naval Weapons Center, China Lake, California1977), pp. 133–144.

Dietz, R. W.

Doob, J. L.

J. L. Doob, Stochastic Processes (Wiley, New York, 1953), p. 233.

Earl, H. E.

D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
[Crossref] [PubMed]

Eastman, J.

J. Eastman and P. W. Baumeister, Opt. Commun. 12, 418 (1974);P. J. Chandley, Opt. Quantum Electron. 8, 329 (1976);J. C. Leader, McDonnell Aircraft Company, Report No. MCAIR 71–013, 1971 (unpublished).
[Crossref]

Elson, J. M.

J. M. Elson, J. Opt. Soc. Am. 66, 682 (1976).
[Crossref]

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974);V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975);A. A. Maradudin and D. L. Mills, ibid. 11, 1392 (1975);J. M. Elson, ibid. 12, 2541 (1975);J. M. Elson, Appl. Opt. 16, 2872 (1977).
[Crossref] [PubMed]

Gloge, D.

D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
[Crossref] [PubMed]

Hodgkinson, I. J.

I. J. Hodgkinson, J. Phys. E,  3, 300 (1970);E. C. Teague, “Evaluation, Revision and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” NBS Technical Note 902 (U. S. Department of Commerce, Washington, DC, April1976);M. Kubo, Acta Imeko 21, 115 (1964);M. Kubo, Rev. Sci. Instrum. 36, 236 (1965);J. A. Greenwood and J. B. P. Williamson, Proc. Roy. Soc. Lond. A 295, 300 (1966);J. B. P. Williamson, Proc. Inst. Mech. Eng. 182, 21 (1967);T. R. Thomas and S. D. Probert, J. Phys. D. 3, 277 (1970);D. J. Whitehouse and J. F. Archard, Proc. Roy. Soc. Lond. A 316, 97 (1970);C. J. Pellerin, J. Christensen, R. C. Jerner, and J. H. Peavey, J. Vac. Sci. Technol. 12, 496 (1975);J. Renau and J. A. Collinson, Bell System Tech. J. 44, 2203 (1965);J. Peklenik, Proc. Inst. Mech. Eng. 182, 108 (1967).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1966), p. 272.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 16, 360 (1977).
[Crossref]

Leader, J. C.

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[Crossref]

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 73–74.

Olson, D. L.

J. M. Blakely and D. L. Olson, J. Appl. Phys. 39, 3476 (1968).
[Crossref]

Partlow, W. D.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 314–321.

Porteus, J. O.

Ritchie, R. H.

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974);V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975);A. A. Maradudin and D. L. Mills, ibid. 11, 1392 (1975);J. M. Elson, ibid. 12, 2541 (1975);J. M. Elson, Appl. Opt. 16, 2872 (1977).
[Crossref] [PubMed]

Saito, T. T.

Scmid, Frederick

Frederick Scmid, U. S. PatentNo. 3,898,051, Crystal Growing, 5Aug.1975.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963);J. A. Holzer and C. C. Sung, J. Appl. Phys. 47, 3363 (1976).
[Crossref]

Stanford, J. L.

H. E. Bennett and J. L. Stanford, J. Res. Natl. Bur. Stand. (U.S.) 80A, 643 (1976);P. C. Archibald and H. E. Bennett, “Scattering from Infrared Missile Domes,” in Proceedings of Seminar on Optics in Missile Engineering, January 16–18, 1978, Los Angeles, California, Vol. 133, (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1978);also, Opt. Eng. 17, 647 (Nov/Dec 1978).
[Crossref]

Stover, J. C.

Supertzi, E. P.

Tukey, J. W.

R. B. Blackman and J. W. Tukey, Bell System Tech. J. 37, 485 (1958), see especially, p. 499.
[Crossref]

Venskytis, F. J.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 16, 360 (1977).
[Crossref]

E. L. Church and J. M. Zavada, Appl. Opt. 14, 1788 (1975).
[Crossref] [PubMed]

Appl. Opt. (6)

Bell System Tech. J. (2)

D. Gloge, E. L. Chinnock, and H. E. Earl, Bell System Tech. J. 48, 511 (1969);I. J. Hodgkinson, J. Phys. E,  3, 341 (1970);R. P. Edwin, ibid.,  6, 55 (1973);D. Heitman and V. Permien, Opt. Commun. 23, 131 (1977);B. P. Hildebrand, R. L. Gordon, and E. V. Allen, Appl. Opt. 13, 177 (1974).
[Crossref] [PubMed]

R. B. Blackman and J. W. Tukey, Bell System Tech. J. 37, 485 (1958), see especially, p. 499.
[Crossref]

J. Appl. Phys. (2)

J. M. Blakely and D. L. Olson, J. Appl. Phys. 39, 3476 (1968).
[Crossref]

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. E (1)

I. J. Hodgkinson, J. Phys. E,  3, 300 (1970);E. C. Teague, “Evaluation, Revision and Application of the NBS Stylus/Computer System for the Measurement of Surface Roughness,” NBS Technical Note 902 (U. S. Department of Commerce, Washington, DC, April1976);M. Kubo, Acta Imeko 21, 115 (1964);M. Kubo, Rev. Sci. Instrum. 36, 236 (1965);J. A. Greenwood and J. B. P. Williamson, Proc. Roy. Soc. Lond. A 295, 300 (1966);J. B. P. Williamson, Proc. Inst. Mech. Eng. 182, 21 (1967);T. R. Thomas and S. D. Probert, J. Phys. D. 3, 277 (1970);D. J. Whitehouse and J. F. Archard, Proc. Roy. Soc. Lond. A 316, 97 (1970);C. J. Pellerin, J. Christensen, R. C. Jerner, and J. H. Peavey, J. Vac. Sci. Technol. 12, 496 (1975);J. Renau and J. A. Collinson, Bell System Tech. J. 44, 2203 (1965);J. Peklenik, Proc. Inst. Mech. Eng. 182, 108 (1967).
[Crossref]

J. Res. Natl. Bur. Stand. (U.S.) (1)

H. E. Bennett and J. L. Stanford, J. Res. Natl. Bur. Stand. (U.S.) 80A, 643 (1976);P. C. Archibald and H. E. Bennett, “Scattering from Infrared Missile Domes,” in Proceedings of Seminar on Optics in Missile Engineering, January 16–18, 1978, Los Angeles, California, Vol. 133, (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1978);also, Opt. Eng. 17, 647 (Nov/Dec 1978).
[Crossref]

Opt. Commun. (1)

J. Eastman and P. W. Baumeister, Opt. Commun. 12, 418 (1974);P. J. Chandley, Opt. Quantum Electron. 8, 329 (1976);J. C. Leader, McDonnell Aircraft Company, Report No. MCAIR 71–013, 1971 (unpublished).
[Crossref]

Opt. Eng. (1)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 16, 360 (1977).
[Crossref]

Opt. Quantum Electron. (1)

P. J. Chandley, Opt. Quantum Electron. 8, 323 (1976).
[Crossref]

Phys. Status Solidi B (1)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974);V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975);A. A. Maradudin and D. L. Mills, ibid. 11, 1392 (1975);J. M. Elson, ibid. 12, 2541 (1975);J. M. Elson, Appl. Opt. 16, 2872 (1977).
[Crossref] [PubMed]

Other (19)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963);J. A. Holzer and C. C. Sung, J. Appl. Phys. 47, 3363 (1976).
[Crossref]

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), pp. 314–321.

See Ref. 19, pp. 311–312. Equation (13) is equivalent to Eq. (9.98) in Ref. 19. This equation and Eq. (9.95) in Ref. 19 are equally good definitions of the auto-covariance function, especially when used with lag windows.

See Ref. 18, pp. 228–231.

See Ref. 19, pp. 83, 173.

See Ref. 19, pp. 376–377.

See Ref. 19, p. 22.

See Ref. 17, p. 177.

J. L. Doob, Stochastic Processes (Wiley, New York, 1953), p. 233.

Equation (5.1) in the paper by Elson and Ritchie in Ref. 4. In this reference the surface factor is given as |ζk¯|2/L2.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1966), p. 272.

The complex dielectric constant is related to the complex index of refraction n¯=n+ik where n is the real part of the refractive index and k the extinction coefficient by the relation ∊1 = n2 − k2 and ∊2 = 2nk. The extinction coefficient is not to be confused with the surface wave vector k which is used throughout the paper.

To a good approximation the second and third terms in the denominator can be neglected relative to the first term, and sin2θ in the A term can be neglected relative to ∊1. If the log of the optical factor is plotted versus θ for a given wavelength, the factor multiplying cos2θ, |1−∊|2/(∊22−∊12)1/2, merely shifts the curve vertically and does not affect the angular dependence of the function.

Manufactured by Rank Precision Industries, Ltd., Leicester, England.

Manufactured by Ernst Fr. Weinz WEKA-OHG, 658 Idar-Oberstein 2, Germany.

J. H. Dancy and J. M. Bennett, “Study of Diamond Styluses for a Talystep Profilometer,” in High Energy Laser Mirrors and Windows, Annual Report No. 9, NWC TP5988, NWC TP5988, (Naval Weapons Center, China Lake, California1977), pp. 133–144.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 73–74.

R. Anderson and J. M. Bennett, “Properties of KCl Optical Elements Forged Between Optically Polished Dies,” in Proceedings of the High Power Laser Optical Components and Component Materials Meeting, (Defense Advanced Research Projects Agency, Arlington, Virginia, 1977), p. 177.

Frederick Scmid, U. S. PatentNo. 3,898,051, Crystal Growing, 5Aug.1975.

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Figures (14)

FIG. 1
FIG. 1

(a) Graph of the optical factor for silver from Eq. (2b) plotted vs scattering angle for wavelengths of 0.44, 1.15, and 10.6 μm. The units are μm−3. The graphs are identical for gold, aluminum, and molybdenum. (b) Graph of the surface factor (spectral density function, in units of μm3) for polished fused quartz plotted vs the surface wave vector k. Only k values smaller than the value marked “FECO limit” (π/τ0, where τ0 = 1.95 μm) are obtainable from interferometric data. (c) Calculated angular scattering curves for a silver-coated, polished fused quartz mirror using the data in (a) and (b). Only the near-angle portions of the 0.44- and 1.15-μm curves are obtainable from interferometric data (sin θmax = kmaxλ/2π = λ/2τ0).

FIG. 2
FIG. 2

(a) Interferometric profile for a polished fused quartz surface. Height data are averaged in segments of length τ0. (b) Profilometer height-profile data for the same surface using a stylus with a radius of approximately 1 μm and a sampling distance 32 times smaller than in (a). (c) Same data as in (b) but adjacent points are connected by straight line segments for clarity. The vertical scales are identical in (a), (b), and (c), but the total profile length in (b) and (c) is one-tenth that in (a). To have the horizontal and vertical scales equal, the profile length in (a) should be multiplied by 500, and by 5000 in (b) and (c).

FIG. 3
FIG. 3

Schematic representation of a sinusoidal surface component with amplitude h and period d, and the stylus of radius r (not to scale).

FIG. 4
FIG. 4

(a) Initial portion of the autocovariance function for fresh feed polished fused quartz plotted vs the lag length τ with τ0 = 0.152 μm to simulate random data. Probability curves for (b) zero, (c) one, and (d) two zero crossings in an interval τ. Solid curves are measured and dashed curves are calculated using Eqs. (23) and (24), assuming γ = 0.617 μm.

FIG. 5
FIG. 5

Three basic types of surface profiles (top line), their autocovariance functions (center line), and calculated angular scattering curves (bottom line).

FIG. 6
FIG. 6

Surface profile and autocovariance function for a fresh feed polished fused quartz surface having short-range random roughness and some waviness. The initial portion of the G(τ) curve is replotted on an expanded τ scale.

FIG. 7
FIG. 7

Surface profile and autocovariance function for a bowl feed polished fused quartz surface having long-range waviness but no short-range random roughness. The initial portion of the G(τ) curve is replotted on an expanded τ scale.

FIG. 8
FIG. 8

Autocovariance function for a commercial diamond-turned copper sample showing periodicities of 6.9, 13.8, and 290 μm, with the initial portion of the G(τ) curve replotted on an expanded τ scale. The calculated angles at which scatter peaks would occur (for normally incident light) are listed for a visible and an infrared wavelength.

FIG. 9
FIG. 9

Surface profile and autocovariance function for a polished silicon carbide sample having a smooth but wavy surface. The initial portion of the G(τ) curve is replotted on an expanded τ scale (the spike in this curve is instrumental noise).

FIG. 10
FIG. 10

Surface profile and autocovariance function for a sapphire sample with a chemical-mechanical polish, showing waviness but no short-range random roughness. The initial portion of the G(τ) curve is replotted on an expanded τ scale and the dashed curve is a Gaussian.

FIG. 11
FIG. 11

Nomogram giving spatial wavelengths of surface features that produce scattering at a particular angle and wavelength. The dashed lines are an illustrative example, as explained in the text.

FIG. 12
FIG. 12

(a) Initial portion of autocovariance function for a polished fused quartz surface with τ0 = 0.061 μm. (b) Same as (a) except that surface data have been averaged in blocks of 32 to simulate interferometric data; τ0 = 1.95 μm. (c) Autocovariance function for the same surface obtained from interferometric data.

FIG. 13
FIG. 13

Calculated angular scattering curves for a polished fused quartz sample whose roughness is 18.6 Å rms. The solid curve is calculated from the measured autocovariance function and the dashed and dash-dot curves from an analytical model of the autocovariance function (see text). The solid and dashed curves are for one-dimensional scattering and the dash-dot curve is for two-dimensional scattering.

FIG. 14
FIG. 14

Calculated (solid) and measured (dashed) curves showing the angular scattering for polished fused quartz samples whose roughnesses are (a) 18.6 Å rms and (b) 5.2 Å rms, respectively. The wavelength is 0.6471 μm.

Equations (35)

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d P d Ω = ( ω c ) 4 { | 1 | 2 cos 2 θ π 2 × [ | ν | 2 cos 2 ϕ | ν i q | 2 + ( ω c ) 2 sin 2 ϕ | ν i q | 2 ] } g ( k ) .
d P d Ω = 16 π 2 λ 4 [ | 1 | 2 cos 2 θ A + cos 2 θ 2 A sin ( α / 2 ) cos θ ] g ( 2 ) ( k ) ,
d P d θ = 16 π 2 λ 3 [ | 1 | 2 cos 2 θ A + cos 2 θ 2 A sin ( α / 2 ) cos θ ] g ( 1 ) ( k ) ,
z ( ρ ) av = 1 L 0 L z ( ρ ) d ρ ,
G L ( τ ) = 1 L τ ( L τ ) / 2 ( L τ ) / 2 z ( ρ τ / 2 ) z ( ρ + τ / 2 ) d ρ ,
G ( τ ) = z ( ρ ) z + ( ρ + τ ) .
Z ( k ) = z ( ρ ) e i k ρ d ρ ,
| Z ( k ) | 2 L = G ( τ ) e i k τ d τ ,
g L ( k ) = | Z ( k ) | 2 / L ,
g ( k ) = G ( τ ) e i k τ d τ .
G M ( τ ) = 1 L τ 0 L τ z ( ρ ) z ( ρ + τ ) d ρ
g d ( k ) = D ( τ ) G M ( τ ) e i k τ d τ .
g d ( k ) = D ( τ ) G ( τ ) e i k τ d τ .
g d ( k ) = Q ( k η ) g ( η ) d η ,
D ( τ ) = ( ½ ) [ 1 + cos ( π τ / τ M ) ] .
Q ( k ) = τ M ( 1 4 F ( k π / τ M ) + 1 2 F ( k ) + 1 4 F ( k + π / τ M ) )
1 N τ 0 n = 1 N z ( n ) = 0 .
G M ( m ) = 1 N n = 1 N m z ( n ) z ( n + m ) .
g d ( 1 ) ( k ) = τ 0 ( G M ( 0 ) + 2 m = 1 M 1 D ( m ) G M ( m ) cos k m τ 0 ) ,
k = K k max / M , K = 0 , ± 1 , ± 2 , , ± M .
g d ( 1 ) ( k ) = 1 2 ( g d ( k ) + 1 2 g d ( k + k max / M ) + 1 2 g d ( k k max / M ) ) ,
g d ( k ) = τ 0 ( G M ( 0 ) + 2 m = 1 M 1 G M ( m ) cos k m τ 0 + ( 1 ) K G M ( M ) )
z ( ρ ) av = 1 τ 0 τ 0 / 2 τ 0 / 2 z ( ρ + η ) d η .
G M ( m ) = 1 N n = 1 N m z ( n ) av z ( n + m ) av ,
G M ( m ) = 1 N τ 0 2 n = 1 N m τ 0 / 2 τ 0 / 2 d η τ 0 / 2 τ 0 / 2 Z ( n τ 0 + η ) × z [ ( n + m ) τ 0 + μ ] d μ ,
z ( n τ 0 + η ) z [ ( n + m ) τ 0 + μ ] = G ( m τ 0 + μ η ) ,
δ N 0 / δ 2 = 2 τ 0 0 τ 0 ( 1 σ τ 0 ) Ĝ ( σ ) d σ ,
k < 1 / h r ,
P ( n , τ ) = [ ( τ / γ ) n / n ! ] e τ / γ , n = 0 , 1 , 2 , ,
G ( τ ) = δ 2 e 2 | τ | / γ ,
G ( τ ) = A e τ 2 / α 2 + B e | τ | / β .
g ( 1 ) ( k ) = A π α e k 2 α 2 / 4 + 2 B β / ( 1 + k 2 β 2 ) .
g ( 2 ) ( k ) = G ( τ ) e i k τ d 2 τ ,
g ( 2 ) ( k ) = 2 π 0 τ G ( τ ) J 0 ( k τ ) d τ .
g ( 2 ) ( k ) = π ( A α 2 e k 2 α 2 / 4 + 2 B β 2 ( 1 + k 2 β 2 ) 3 / 2 ) .