Abstract

The modeling procedure for analyzing the rotating analyzer spectroellipsometric data using the linear regression analysis technique is examined. The best set of optical parameters which should be employed in the modeling procedure is searched out with the help of variance–covariance matrices and the correlation properties of optical parameters. It was found that α and β, which are the Fourier coefficients of sinusoidal components of the intensity, constitute the best set for such an analysis for the case of single-crystal silicon and gold film evaporated on glass substrate. Also suitable expressions of α and β in terms of the ellipsometric parameters Δ and ψ have been derived to enable a linear regression analysis using α and β.

© 1986 Optical Society of America

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  1. D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interfaces between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046 (1979).
    [CrossRef]
  2. J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
    [CrossRef]
  3. H. J. Mattausch, D. E. Aspnes, “Optical Properties of InSb and its Electrochemically Grown Anodic Oxide,” Phys. Rev. B 23, 1896 (1981).
    [CrossRef]
  4. H. Arwin, D. E. Aspnes, “Nondestructive Analysis of Hg1−xCdxTe (x = 0.00, 0.20, 0.29, and 1.00) by Spectroscopic Ellipsometry. II. Substrate, Oxide, and Interface Properties,” J. Vac. Sci. Technol. A2, 1316 (1984); J. Vac. Sci. Technol. A2, 1309 (1984).
  5. D. E. Aspnes, J. B. Theeten, “Spectroscopic Analysis of the Interface Between Si and Its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359 (1980).
    [CrossRef]
  6. J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.
  7. J. B. Theeten, M. Erman, “Depth Profiling and Interface Analysis Using Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 20(3), 471 (1982).
    [CrossRef]
  8. J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
    [CrossRef]
  9. J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
    [CrossRef]
  10. P. J. McMarr, “Characterization of Surfaces, Thin Films and Ion-Implanted Silicon by Spectroscopic Ellipsometry,” Ph.D. Thesis, Pennsylvania State U. (1984)
  11. J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).
  12. D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
    [CrossRef]
  13. D. E. Aspnes, “Dielectric Function and Surface Roughness Measurement of InSb by Spectroscopic Ellipsometry,” J. Opt. Soc. Am. 17, 1057 (1980).
  14. D. E. Aspnes, “Effects of Component Optical Activity in Data Reduction and Calibration of Rotating-Analyzer Ellipsometers,” J. Opt. Soc. Am. 64, 812 (1974).
    [CrossRef]
  15. D. E. Aspnes, A. A. Studna, “High-Precision Scanning Ellipsometer,” Appl. Opt. 14, 220 (1975).
    [PubMed]
  16. D. E. Aspnes, “Microstructural Information from Optical Properties in Semiconductor Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188 (1981).
  17. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977), p. 319.
  18. W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chap. 4.
  19. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1955).
  20. C. G. Granqvist, D. Hunderi, “Optical Properties of Ultra-fine Gold Particles,” Phys. Rev. B 16, 3513 (1977); “Optical Properties of Ag-SiO2 Cermet Films: A Comparison of Effective Medium Theories,” Phys. Rev. B 18, 2897 (1978).
    [CrossRef]
  21. R. M. A. Azzam, N. M. Bashara, “Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers,” J. Opt. Soc. Am. 64, 1459 (1974).
    [CrossRef]
  22. P. J. McMarr, G. R. Mariner, K. Vedam, to be published.

1985

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

1984

H. Arwin, D. E. Aspnes, “Nondestructive Analysis of Hg1−xCdxTe (x = 0.00, 0.20, 0.29, and 1.00) by Spectroscopic Ellipsometry. II. Substrate, Oxide, and Interface Properties,” J. Vac. Sci. Technol. A2, 1316 (1984); J. Vac. Sci. Technol. A2, 1309 (1984).

1982

J. B. Theeten, M. Erman, “Depth Profiling and Interface Analysis Using Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 20(3), 471 (1982).
[CrossRef]

1981

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

H. J. Mattausch, D. E. Aspnes, “Optical Properties of InSb and its Electrochemically Grown Anodic Oxide,” Phys. Rev. B 23, 1896 (1981).
[CrossRef]

D. E. Aspnes, “Microstructural Information from Optical Properties in Semiconductor Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188 (1981).

1980

D. E. Aspnes, “Dielectric Function and Surface Roughness Measurement of InSb by Spectroscopic Ellipsometry,” J. Opt. Soc. Am. 17, 1057 (1980).

D. E. Aspnes, J. B. Theeten, “Spectroscopic Analysis of the Interface Between Si and Its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359 (1980).
[CrossRef]

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

1979

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interfaces between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046 (1979).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
[CrossRef]

1977

C. G. Granqvist, D. Hunderi, “Optical Properties of Ultra-fine Gold Particles,” Phys. Rev. B 16, 3513 (1977); “Optical Properties of Ag-SiO2 Cermet Films: A Comparison of Effective Medium Theories,” Phys. Rev. B 18, 2897 (1978).
[CrossRef]

1975

1974

Adams, T. E.

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

Arwin, H.

H. Arwin, D. E. Aspnes, “Nondestructive Analysis of Hg1−xCdxTe (x = 0.00, 0.20, 0.29, and 1.00) by Spectroscopic Ellipsometry. II. Substrate, Oxide, and Interface Properties,” J. Vac. Sci. Technol. A2, 1316 (1984); J. Vac. Sci. Technol. A2, 1309 (1984).

Aspnes, D. E.

H. Arwin, D. E. Aspnes, “Nondestructive Analysis of Hg1−xCdxTe (x = 0.00, 0.20, 0.29, and 1.00) by Spectroscopic Ellipsometry. II. Substrate, Oxide, and Interface Properties,” J. Vac. Sci. Technol. A2, 1316 (1984); J. Vac. Sci. Technol. A2, 1309 (1984).

H. J. Mattausch, D. E. Aspnes, “Optical Properties of InSb and its Electrochemically Grown Anodic Oxide,” Phys. Rev. B 23, 1896 (1981).
[CrossRef]

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

D. E. Aspnes, “Microstructural Information from Optical Properties in Semiconductor Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188 (1981).

D. E. Aspnes, “Dielectric Function and Surface Roughness Measurement of InSb by Spectroscopic Ellipsometry,” J. Opt. Soc. Am. 17, 1057 (1980).

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Spectroscopic Analysis of the Interface Between Si and Its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interfaces between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046 (1979).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
[CrossRef]

D. E. Aspnes, A. A. Studna, “High-Precision Scanning Ellipsometer,” Appl. Opt. 14, 220 (1975).
[PubMed]

D. E. Aspnes, “Effects of Component Optical Activity in Data Reduction and Calibration of Rotating-Analyzer Ellipsometers,” J. Opt. Soc. Am. 64, 812 (1974).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers,” J. Opt. Soc. Am. 64, 1459 (1974).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977), p. 319.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Analysis of Systematic Errors in Rotating-Analyzer Ellipsometers,” J. Opt. Soc. Am. 64, 1459 (1974).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977), p. 319.

Bennett, J. M.

J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).

Blanco, J. R.

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).

Chang, R. P. H.

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

Erman, M.

J. B. Theeten, M. Erman, “Depth Profiling and Interface Analysis Using Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 20(3), 471 (1982).
[CrossRef]

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.

Granqvist, C. G.

C. G. Granqvist, D. Hunderi, “Optical Properties of Ultra-fine Gold Particles,” Phys. Rev. B 16, 3513 (1977); “Optical Properties of Ag-SiO2 Cermet Films: A Comparison of Effective Medium Theories,” Phys. Rev. B 18, 2897 (1978).
[CrossRef]

Hamilton, W. C.

W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chap. 4.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1955).

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
[CrossRef]

Hunderi, D.

C. G. Granqvist, D. Hunderi, “Optical Properties of Ultra-fine Gold Particles,” Phys. Rev. B 16, 3513 (1977); “Optical Properties of Ag-SiO2 Cermet Films: A Comparison of Effective Medium Theories,” Phys. Rev. B 18, 2897 (1978).
[CrossRef]

Mariner, G. R.

P. J. McMarr, G. R. Mariner, K. Vedam, to be published.

Mattausch, H. J.

H. J. Mattausch, D. E. Aspnes, “Optical Properties of InSb and its Electrochemically Grown Anodic Oxide,” Phys. Rev. B 23, 1896 (1981).
[CrossRef]

McMarr, P. J.

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

P. J. McMarr, “Characterization of Surfaces, Thin Films and Ion-Implanted Silicon by Spectroscopic Ellipsometry,” Ph.D. Thesis, Pennsylvania State U. (1984)

J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).

P. J. McMarr, G. R. Mariner, K. Vedam, to be published.

Messier, R.

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

Murau, P. C.

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

Pernas, J.

J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.

Simondet, F.

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.

Studna, A. A.

Theeten, J. B.

J. B. Theeten, M. Erman, “Depth Profiling and Interface Analysis Using Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 20(3), 471 (1982).
[CrossRef]

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Spectroscopic Analysis of the Interface Between Si and Its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359 (1980).
[CrossRef]

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interfaces between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046 (1979).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
[CrossRef]

J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.

Vedam, K.

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).

P. J. McMarr, G. R. Mariner, K. Vedam, to be published.

Appl. Opt.

J. Appl. Phys.

J. B. Theeten, D. E. Aspnes, F. Simondet, M. Erman, P. C. Murau, “Nondestructive Analysis of Si3N4/SiO2/Si Structures Using Spectroscopic Ellipsometry,” J. Appl. Phys. 52, 6788 (1981).
[CrossRef]

J. Electrochem. Soc.

J. B. Theeten, R. P. H. Chang, D. E. Aspnes, T. E. Adams, “In-Situ Measurement and Analysis of Plasma-Grown GaAs Oxides with Spectroscopic Ellipsometry,” J. Electrochem. Soc. 127, 378 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Spectroscopic Analysis of the Interface Between Si and Its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Vac. Sci. Technol.

J. B. Theeten, M. Erman, “Depth Profiling and Interface Analysis Using Spectroscopic Ellipsometry,” J. Vac. Sci. Technol. 20(3), 471 (1982).
[CrossRef]

H. Arwin, D. E. Aspnes, “Nondestructive Analysis of Hg1−xCdxTe (x = 0.00, 0.20, 0.29, and 1.00) by Spectroscopic Ellipsometry. II. Substrate, Oxide, and Interface Properties,” J. Vac. Sci. Technol. A2, 1316 (1984); J. Vac. Sci. Technol. A2, 1309 (1984).

Mat. Res. Soc. Sym. Proc.

J. R. Blanco, R. Messier, K. Vedam, P. J. McMarr, “Spectroscopic Ellipsometric Study of rf-Sputtered a-Ge Films,” Mat. Res. Soc. Sym. Proc. 38, 301 (1985).
[CrossRef]

Phys. Rev. B

H. J. Mattausch, D. E. Aspnes, “Optical Properties of InSb and its Electrochemically Grown Anodic Oxide,” Phys. Rev. B 23, 1896 (1981).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of Effective-Medium Models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292 (1979).
[CrossRef]

C. G. Granqvist, D. Hunderi, “Optical Properties of Ultra-fine Gold Particles,” Phys. Rev. B 16, 3513 (1977); “Optical Properties of Ag-SiO2 Cermet Films: A Comparison of Effective Medium Theories,” Phys. Rev. B 18, 2897 (1978).
[CrossRef]

Phys. Rev. Lett.

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interfaces between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046 (1979).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

D. E. Aspnes, “Microstructural Information from Optical Properties in Semiconductor Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 188 (1981).

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977), p. 319.

W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chap. 4.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1955).

J. B. Theeten, F. Simondet, M. Erman, J. Pernas, “Thin Films and Interface Analysis Using Spectroscopic Ellipsometry,” in Proceedings, Fourth International Conference on Solid Surfaces, Cannes, France (1980), Vol. 201, p. 1071.

P. J. McMarr, “Characterization of Surfaces, Thin Films and Ion-Implanted Silicon by Spectroscopic Ellipsometry,” Ph.D. Thesis, Pennsylvania State U. (1984)

J. R. Blanco, P. J. McMarr, K. Vedam, J. M. Bennett, “Study of Surface Roughness of Al and Si by Spectroscopic Ellipsometry,” in Multiple Scattering of Waves, V. V. Varadan, V. K. Varadan, Eds. (Technomic Publ. Co., Lancaster, PA, 1986).

P. J. McMarr, G. R. Mariner, K. Vedam, to be published.

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Tables (10)

Tables Icon

Table I Expressions of fj and Associated S for Some Frequently Used Sets

Tables Icon

Table II Variance–Covariance Matrices of a Few Optical Parameter Sets for Gold Film; Au Film: Variance–Covariance Matrix Element (×108) Table of α, β

Tables Icon

Table III Variance–Covariance Matrices of a Few Optical Parameter Sets for Gold Film; Au Film: Variance–Covariance Matrix Element (×108) Table of Δ, ψ

Tables Icon

Table IV Variance–Covariance Matrices of a Few Optical Parameter Sets for Gold Film; Au Film: Variance–Covariance Matrix Element (×108) Table of tanψ, cosΔ

Tables Icon

Table V Variance–Covariance Matrices of a Few Optical Parameter Sets for Gold Film; Au Film: Variance–Covariance Matrix Element (×108) Table of Re(ρ), Im(ρ)

Tables Icon

Table VI Variance–Covariance Matrices of a Few Optical Parameter Sets for Crystalline Silicon; c-Si: Variance–Covariance Matrix Element (×108) Table of α, β

Tables Icon

Table VII Variance–Covariance Matrices of a Few Optical Parameter Sets for Crystalline Silicon; c-Si: Variance–Covariance Matrix Element (×108) Table of Δ, ψ

Tables Icon

Table VIII Variance–Covariance Matrices of a Few Optical Parameter Sets for Crystalline Silicon; c-Si: Variance–Covariance Matrix Element (×108) Table of tanψ, cosΔ

Tables Icon

Table IX Variance–Covariance Matrices of a Few Optical Parameter Sets for Crystalline Silicon; c-Si: Variance–Covariance Matrix Element (×108) Table of Re(ρ), Im(ρ)

Tables Icon

Table X Summary of the Variance–Covariance Matrices

Equations (35)

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

I = I 0 ( 1 + α cos 2 A + β sin 2 A ) ,
ρ = tan ψ exp ( i Δ ) = cot ( Q - A s ) - i a 1 + i a cot ( Q - A s ) · tan ( P - P s ) + i γ P 1 - i γ P tan ( P - P s ) ,
Q = 1 / 2 tan - 1 ( β / α ) + π / 2 u ( - α ) sgn ( β ) ,
a = - 2 γ A ζ ± ( 1 - γ A 2 ) ( 1 - ζ 2 ) 1 / 2 ( 1 + ζ ) - γ A 2 ( 1 - ζ ) ,
G = 1 N - p - 1 ( i N ρ calc - ρ exp 2 ) E i ,
x 1 , x 2 , , x p .
f 1 = a 11 x 1 + a 12 x 2 + + a 1 p x p + e 1 , f 2 = a 21 x 1 + a 22 x 2 + + a 2 p x p + e 2 , f N = a N 1 x 1 + a N 2 x 2 + + a N p x p + e N .
F = AX + e or f i = A i k X k + e i ,             k = 1 , 2 , , p ,
S = V M f - 1 V ,
V = F - F ^ = F - AX ^ ,
M f , i i = e i e i = var ( f i f i ) = f i f i - f i f i = 1 n l = 1 n f i , l f i , l - ( 1 n l = 1 n f i , l ) 2 ,
M f , i j = e i e j = cov ( f i f j ) = f i f j - f i f j = 1 n l = 1 n f i , l f j , l - ( 1 n l = 1 n f i , l ) ( 1 n l = 1 n f j , l )             ( i j ) ,
X ^ = B - 1 A M f - 1 F ,
B = A M f - 1 A ,
M f = σ 2 I ,
X ^ = B - 1 A F / σ 2 ,             B = A A / σ 2 ,
σ 2 = V V / ( N - p - 1 )
= S / ( N - p - 1 ) .
I / I 0 = 1 + α cos ( 2 A ) + β sin ( 2 A )
= 1 + R a γ [ cos ( 2 A ) cos ( 2 Q ) + sin ( 2 A ) sin ( 2 Q ) ] ,
R a γ = ( 1 - a 2 ) ( 1 - γ A 2 ) ( 1 + a γ A ) 2 + ( a + γ A ) 2 .
α = R a γ cos ( 2 Q ) ;
β = R a γ sin ( 2 Q ) .
tan ( x + i y ) = ρ [ 1 - i γ P tan ( P - P s ) ] [ tan ( P - P s ) + i γ P ]
tan ( x + i y ) = tan x + i tanh y 1 - i tan x tanh y = cot ( Q - A s - i a ) 1 + i a cot ( Q - A s ) .
Q = A s ± π / 2 - x ,
a = - tanhy .
tan ( x + i y ) = sin 2 x + i sinh 2 y cos 2 x + cosh 2 y = ξ exp ( i η ) .
ξ 2 = sin 2 2 x + sinh 2 2 y ( cos 2 x + cosh 2 y ) 2 ,
tan η = sinh 2 y sin 2 x .
ξ 2 = sin 2 2 x ( 1 + tan 2 η ) [ cos 2 x + ( 1 + tan 2 η sin 2 2 x ) 1 / 2 ] 2 .
x = 1 / 2 tan - 1 [ 2 ξ cos η / ( 1 - ξ 2 ) ] + π / 2 u ( ξ 2 - 1 ) sgn [ cos ( η ) ] .
a = - ξ sin η cos 2 x + ξ cos η sin 2 x + ξ 2 sin 2 x .
C i j = M i j / ( M i i · M j j ) 1 / 2 ,
Δ F = F Δ X + e or Δ f i = ( f i / X k ) Δ X k + e i , k = 1 , 2 , , p .

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