Abstract

A new wavelength-scanning two-channel polarization modulation ellipsometer is described, where a photo-elastic modulator is used and the analyzed light is separated into orthogonally polarized beams using a Wollaston prism. Both beams are detected using phototubes whose bias voltage is dynamically controlled for constant dc. The dc from each phototube is measured with a digital voltmeter, and the fundamental and second harmonic of the phototube current are measured using individual lock-in amplifiers. All three of the associated ellipsometric parameters (N = cos2ψ, S = sin2ψ sinΔ, and C = sin2ψ cosΔ) can be determined simultaneously in a single scan. The versatility of the instrument is demonstrated by the determination of the optical functions of Si from 238 to 652 nm (5.3–1.9 eV).

© 1990 Optical Society of America

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. S. N. Jasperson, S. E. Schnatterly, “An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
    [CrossRef]
  3. G. E. Jellison, F. A. Modine, “Optical Constants for Silicon at 300 and 10 K Determined from 1.64 to 4.73 eV by Ellipsometry,” J. Appl. Phys. 53, 3745–3753 (1982).
    [CrossRef]
  4. V. M. Bermudez, V. H. Ritz, “Wavelength-Scanning Polarization-Modulation Ellipsometry: Some Practical Considerations,” Appl. Opt. 17, 542–552 (1978).
    [CrossRef] [PubMed]
  5. B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
    [CrossRef]
  6. D. E. Aspnes, A. A. Studna, “High Precision Scanning Ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [PubMed]
  7. F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
    [CrossRef]
  8. G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
    [CrossRef]
  9. J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
    [CrossRef]
  10. N. V. Smith, “Optical Constants of Sodium and Potassium from 0.5 to 4.0 eV by Split-Beam Ellipsometry,” Phys. Rev. B 183, 634–644 (1969).
    [CrossRef]
  11. R. M. A. Azzam, “Division-of Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982); “Beam-Splitters for the Division-of-Amplitude Photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
    [CrossRef]
  12. G. E. Jellison, D. H. Lowndes, “Time-Resolved Ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
    [CrossRef] [PubMed]
  13. A. Moritani, J. Nakai, “High-Speed Retardation Modulation Ellipsometry,” Appl. Opt. 21, 3231–3232 (1982); A. Moritani, C. Harmaguchi, “High-Speed Ellipsometry of Arsenic-Implanted Si During cw Laser Annealing,” Appl. Phys. Lett. 46, 746–748 (1985).
    [CrossRef] [PubMed]
  14. D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interface Between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic Analysis of the Interface Between Si and its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
    [CrossRef]
  15. D. E. Aspnes, A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  16. G. E. Jellison, F. A. Modine, “Optical Absorption of Silicon Between 1.6 and 4.7 at Elevated Temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
    [CrossRef]
  17. G. E. Jellison, F. A. Modine, “Optical Functions of Silicon Between 1.7 and 4.7 eV at Elevated Temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
    [CrossRef]
  18. G. E. Jellison, H. H. Burke, “The Temperature Dependence of the Refractive Index of Silicon at Elevated Temperatures at Several Laser Wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
    [CrossRef]
  19. P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
    [CrossRef]
  20. P. S. Hauge, “Recent Developments in Instrumentation in Ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  21. G. E. Jellison, F. A. Modine, “Accurate Calibration of a Photoelastic Modulator in a Polarization Modulation Ellipsometry Experiment,” Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 231–241 (1990).
  22. O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
    [CrossRef]
  23. G. E. Jellison, F. A. Modine, “A Simple Implementation of a Power Supply for Constant Phototube Current in Light Modulation Spectroscopy,” Rev. Sci. Instrum. 60, 3345 (1989).
    [CrossRef]
  24. R. C. O’Handley, “Modified Jones Calculus for the Analysis of Errors in Polarization-Modulation Ellipsometry,” J. Opt. Soc. Am. 63, 523–528 (1973).
    [CrossRef]
  25. R. M. A. Azzam, “Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder,” Optik 45, 209–218 (1976).
  26. F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in Ellipsometry Measurements Made with a Photoelastic Modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
    [CrossRef]
  27. D. E. Aspnes, A. A. Studna, “Chemical Etching and Cleaning Procedures for Si, Ge, and Some III-V Compound Semiconductors,” Appl. Phys. Lett. 39, 316–318 (1981).
    [CrossRef]
  28. W. C. Dash, R. Newman, “Intrinsic Optical Absorption in Single-Crystal Germanium and Silicon at 77°K and 300°K,” Phys. Rev. 99, 1151–1155 (1955).
    [CrossRef]
  29. I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [CrossRef]
  30. G. E. Jellison, F. A. Modine, “Optical Nature on Interface Layers: a Comparative Study of the Si–SiO2 Interface,” J. Opt. Soc. Am. 72, 1253–1257 (1982).
    [CrossRef]
  31. E. A. Taft, L. Cordes, “Optical Evidence for a Silicon–Silicon Oxide Interlayer,” J. Electrochem. Soc. 126, 131–134 (1979).
    [CrossRef]
  32. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten vor heterogenen substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
    [CrossRef]
  33. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969); P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
    [CrossRef]
  34. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Applied Mathematics Series Vol. 55, 1964), p. 361.

1990 (1)

G. E. Jellison, F. A. Modine, “Accurate Calibration of a Photoelastic Modulator in a Polarization Modulation Ellipsometry Experiment,” Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 231–241 (1990).

1989 (2)

O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

G. E. Jellison, F. A. Modine, “A Simple Implementation of a Power Supply for Constant Phototube Current in Light Modulation Spectroscopy,” Rev. Sci. Instrum. 60, 3345 (1989).
[CrossRef]

1988 (1)

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

1987 (1)

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

1986 (2)

G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

G. E. Jellison, H. H. Burke, “The Temperature Dependence of the Refractive Index of Silicon at Elevated Temperatures at Several Laser Wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[CrossRef]

1985 (1)

1983 (3)

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in Ellipsometry Measurements Made with a Photoelastic Modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Functions of Silicon Between 1.7 and 4.7 eV at Elevated Temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

D. E. Aspnes, A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1982 (6)

G. E. Jellison, F. A. Modine, “Optical Absorption of Silicon Between 1.6 and 4.7 at Elevated Temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Constants for Silicon at 300 and 10 K Determined from 1.64 to 4.73 eV by Ellipsometry,” J. Appl. Phys. 53, 3745–3753 (1982).
[CrossRef]

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

R. M. A. Azzam, “Division-of Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982); “Beam-Splitters for the Division-of-Amplitude Photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Nature on Interface Layers: a Comparative Study of the Si–SiO2 Interface,” J. Opt. Soc. Am. 72, 1253–1257 (1982).
[CrossRef]

A. Moritani, J. Nakai, “High-Speed Retardation Modulation Ellipsometry,” Appl. Opt. 21, 3231–3232 (1982); A. Moritani, C. Harmaguchi, “High-Speed Ellipsometry of Arsenic-Implanted Si During cw Laser Annealing,” Appl. Phys. Lett. 46, 746–748 (1985).
[CrossRef] [PubMed]

1981 (1)

D. E. Aspnes, A. A. Studna, “Chemical Etching and Cleaning Procedures for Si, Ge, and Some III-V Compound Semiconductors,” Appl. Phys. Lett. 39, 316–318 (1981).
[CrossRef]

1980 (1)

P. S. Hauge, “Recent Developments in Instrumentation in Ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

1979 (2)

E. A. Taft, L. Cordes, “Optical Evidence for a Silicon–Silicon Oxide Interlayer,” J. Electrochem. Soc. 126, 131–134 (1979).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interface Between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic Analysis of the Interface Between Si and its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

1978 (1)

1976 (1)

R. M. A. Azzam, “Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder,” Optik 45, 209–218 (1976).

1975 (1)

1973 (1)

1969 (3)

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969); P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

S. N. Jasperson, S. E. Schnatterly, “An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

N. V. Smith, “Optical Constants of Sodium and Potassium from 0.5 to 4.0 eV by Split-Beam Ellipsometry,” Phys. Rev. B 183, 634–644 (1969).
[CrossRef]

1965 (1)

1955 (1)

W. C. Dash, R. Newman, “Intrinsic Optical Absorption in Single-Crystal Germanium and Silicon at 77°K and 300°K,” Phys. Rev. 99, 1151–1155 (1955).
[CrossRef]

1952 (1)

J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten vor heterogenen substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Applied Mathematics Series Vol. 55, 1964), p. 361.

Acher, O.

O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Archard, J. F.

J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

D. E. Aspnes, A. A. Studna, “Chemical Etching and Cleaning Procedures for Si, Ge, and Some III-V Compound Semiconductors,” Appl. Phys. Lett. 39, 316–318 (1981).
[CrossRef]

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interface Between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic Analysis of the Interface Between Si and its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

D. E. Aspnes, A. A. Studna, “High Precision Scanning Ellipsometer,” Appl. Opt. 14, 220–228 (1975).
[PubMed]

Azzam, R. M. A.

R. M. A. Azzam, “Division-of Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982); “Beam-Splitters for the Division-of-Amplitude Photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

R. M. A. Azzam, “Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder,” Optik 45, 209–218 (1976).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bermudez, V. M.

Bernoux, F.

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

Bigan, E.

O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten vor heterogenen substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Bu-Abbud, G. H.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

Burke, H. H.

G. E. Jellison, H. H. Burke, “The Temperature Dependence of the Refractive Index of Silicon at Elevated Temperatures at Several Laser Wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[CrossRef]

Cardona, M.

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

Clegg, P. L.

J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
[CrossRef]

Cordes, L.

E. A. Taft, L. Cordes, “Optical Evidence for a Silicon–Silicon Oxide Interlayer,” J. Electrochem. Soc. 126, 131–134 (1979).
[CrossRef]

Dalby, J. L.

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

Dash, W. C.

W. C. Dash, R. Newman, “Intrinsic Optical Absorption in Single-Crystal Germanium and Silicon at 77°K and 300°K,” Phys. Rev. 99, 1151–1155 (1955).
[CrossRef]

Drevillon, B.

O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

Ferrieu, F.

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

Garriga, M.

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

Gruzalski, G. R.

Hauge, P. S.

P. S. Hauge, “Recent Developments in Instrumentation in Ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, “An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Jellison, G. E.

G. E. Jellison, F. A. Modine, “Accurate Calibration of a Photoelastic Modulator in a Polarization Modulation Ellipsometry Experiment,” Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 231–241 (1990).

G. E. Jellison, F. A. Modine, “A Simple Implementation of a Power Supply for Constant Phototube Current in Light Modulation Spectroscopy,” Rev. Sci. Instrum. 60, 3345 (1989).
[CrossRef]

G. E. Jellison, H. H. Burke, “The Temperature Dependence of the Refractive Index of Silicon at Elevated Temperatures at Several Laser Wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[CrossRef]

G. E. Jellison, D. H. Lowndes, “Time-Resolved Ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
[CrossRef] [PubMed]

G. E. Jellison, F. A. Modine, “Optical Functions of Silicon Between 1.7 and 4.7 eV at Elevated Temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in Ellipsometry Measurements Made with a Photoelastic Modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Absorption of Silicon Between 1.6 and 4.7 at Elevated Temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Constants for Silicon at 300 and 10 K Determined from 1.64 to 4.73 eV by Ellipsometry,” J. Appl. Phys. 53, 3745–3753 (1982).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Nature on Interface Layers: a Comparative Study of the Si–SiO2 Interface,” J. Opt. Soc. Am. 72, 1253–1257 (1982).
[CrossRef]

Lautenschlager, P.

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

Lowndes, D. H.

Malitson, I. H.

Marbot, R.

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

Modine, F. A.

G. E. Jellison, F. A. Modine, “Accurate Calibration of a Photoelastic Modulator in a Polarization Modulation Ellipsometry Experiment,” Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 231–241 (1990).

G. E. Jellison, F. A. Modine, “A Simple Implementation of a Power Supply for Constant Phototube Current in Light Modulation Spectroscopy,” Rev. Sci. Instrum. 60, 3345 (1989).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Functions of Silicon Between 1.7 and 4.7 eV at Elevated Temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in Ellipsometry Measurements Made with a Photoelastic Modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Absorption of Silicon Between 1.6 and 4.7 at Elevated Temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Constants for Silicon at 300 and 10 K Determined from 1.64 to 4.73 eV by Ellipsometry,” J. Appl. Phys. 53, 3745–3753 (1982).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Nature on Interface Layers: a Comparative Study of the Si–SiO2 Interface,” J. Opt. Soc. Am. 72, 1253–1257 (1982).
[CrossRef]

Moritani, A.

Muller, R. H.

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969); P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

Nakai, J.

Newman, R.

W. C. Dash, R. Newman, “Intrinsic Optical Absorption in Single-Crystal Germanium and Silicon at 77°K and 300°K,” Phys. Rev. 99, 1151–1155 (1955).
[CrossRef]

O’Handley, R. C.

Perrin, J.

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

Ritz, V. H.

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, “An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Smith, N. V.

N. V. Smith, “Optical Constants of Sodium and Potassium from 0.5 to 4.0 eV by Split-Beam Ellipsometry,” Phys. Rev. B 183, 634–644 (1969).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Applied Mathematics Series Vol. 55, 1964), p. 361.

Stehle, J. L.

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

Studna, A.

D. E. Aspnes, A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Chemical Etching and Cleaning Procedures for Si, Ge, and Some III-V Compound Semiconductors,” Appl. Phys. Lett. 39, 316–318 (1981).
[CrossRef]

D. E. Aspnes, A. A. Studna, “High Precision Scanning Ellipsometer,” Appl. Opt. 14, 220–228 (1975).
[PubMed]

Taft, E. A.

E. A. Taft, L. Cordes, “Optical Evidence for a Silicon–Silicon Oxide Interlayer,” J. Electrochem. Soc. 126, 131–134 (1979).
[CrossRef]

Taylor, A. M.

J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
[CrossRef]

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interface Between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic Analysis of the Interface Between Si and its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

Thomas, O.

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

Vina, L.

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

Violet, A.

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

Woollam, J. A.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

Ann. Phys. Leipzig (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten vor heterogenen substanzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

D. E. Aspnes, A. A. Studna, “Chemical Etching and Cleaning Procedures for Si, Ge, and Some III-V Compound Semiconductors,” Appl. Phys. Lett. 39, 316–318 (1981).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Absorption of Silicon Between 1.6 and 4.7 at Elevated Temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

J. Appl. Phys. (2)

G. E. Jellison, H. H. Burke, “The Temperature Dependence of the Refractive Index of Silicon at Elevated Temperatures at Several Laser Wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Constants for Silicon at 300 and 10 K Determined from 1.64 to 4.73 eV by Ellipsometry,” J. Appl. Phys. 53, 3745–3753 (1982).
[CrossRef]

J. Electrochem. Soc. (1)

E. A. Taft, L. Cordes, “Optical Evidence for a Silicon–Silicon Oxide Interlayer,” J. Electrochem. Soc. 126, 131–134 (1979).
[CrossRef]

J. Opt. Soc. Am. (4)

Mat. Res. Symp. Proc. (1)

F. Ferrieu, J. L. Stehle, F. Bernoux, O. Thomas, Mat. Res. Symp. Proc. 101, 403 (1988).
[CrossRef]

Opt. Acta (1)

R. M. A. Azzam, “Division-of Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light,” Opt. Acta 29, 685–689 (1982); “Beam-Splitters for the Division-of-Amplitude Photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

Optik (1)

R. M. A. Azzam, “Alternate Arrangement and Analysis of Systematic Errors for Dynamic Photometric Ellipsometers Employing an Oscillating-Phase Retarder,” Optik 45, 209–218 (1976).

Phys. Rev. (1)

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[CrossRef]

Phys. Rev. B (4)

D. E. Aspnes, A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

N. V. Smith, “Optical Constants of Sodium and Potassium from 0.5 to 4.0 eV by Split-Beam Ellipsometry,” Phys. Rev. B 183, 634–644 (1969).
[CrossRef]

P. Lautenschlager, M. Garriga, L. Vina, M. Cardona, “Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon,” Phys. Rev. B 36, 4821–4830 (1987).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical Functions of Silicon Between 1.7 and 4.7 eV at Elevated Temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

D. E. Aspnes, J. B. Theeten, “Optical Properties of the Interface Between Si and its Thermally Grown Oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic Analysis of the Interface Between Si and its Thermally Grown Oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

Proc. Phys. Soc. London Ser. B (1)

J. F. Archard, P. L. Clegg, A. M. Taylor, “Photoelectric Analysis of Elliptically Polarized Light,” Proc. Phys. Soc. London Ser. B 65, 758 (1952).
[CrossRef]

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

G. E. Jellison, F. A. Modine, “Accurate Calibration of a Photoelastic Modulator in a Polarization Modulation Ellipsometry Experiment,” Proc. Soc. Photo-Opt. Instrum. Eng. 1166, 231–241 (1990).

Rev. Sci. Instrum. (4)

O. Acher, E. Bigan, B. Drevillon, “Improvements of Phase-Modulated Ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[CrossRef]

G. E. Jellison, F. A. Modine, “A Simple Implementation of a Power Supply for Constant Phototube Current in Light Modulation Spectroscopy,” Rev. Sci. Instrum. 60, 3345 (1989).
[CrossRef]

B. Drevillon, J. Perrin, R. Marbot, A. Violet, J. L. Dalby, “Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis,” Rev. Sci. Instrum. 53, 969–977 (1982).
[CrossRef]

S. N. Jasperson, S. E. Schnatterly, “An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Surf. Sci. (2)

P. S. Hauge, “Recent Developments in Instrumentation in Ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surf. Sci. 16, 14–33 (1969); P. S. Hauge, R. H. Muller, C. G. Smith, “Conventions and Formulas for Using the Mueller-Stokes Calculus in Ellipsometry,” Surf. Sci. 96, 81 (1980).
[CrossRef]

Thin Solid Films (1)

G. H. Bu-Abbud, N. M. Bashara, J. A. Woollam, “Variable Wavelength, Variable Angle Ellipsometry Including a Sensitivities Test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

Other (2)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Applied Mathematics Series Vol. 55, 1964), p. 361.

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of several typical ellipsometers: (a) general ellipsometer (appropriate for nulling, RAE, or RPE instruments); (b) PME in the measurement configuration; and (c) PME in the calibration configuration. P, P′ = polarizer, C = compensator, A = analyzer, M = modulator, W = Wollaston prism analyzer, D, D1, D2 are the detectors, and W1 and W2 are the entrance and exit windows, respectively.

Fig. 2
Fig. 2

Schematic diagram of the photoelastic modulator used in these experiments. The solid black bars represent the light aperture through the device, which is set at a distance e from the intersection of the crystalline quartz (c-SiO2) and the fused quartz (a-SiO2) pieces.

Fig. 3
Fig. 3

Wollaston prism and phototube assembly. The heavy line represents the light beam as it enters the denvice. The dynode resistor chain which distributes the bias voltage across the dynodes of the phototubes is enclosed in the space behind the phototubes.

Fig. 4
Fig. 4

General equipment connections used in the PME experiment. The numbers below the boxes in the form 7xx represent the current IEEE-488 addresses used in our laboratory. DAC and ADC refer to the built-in digital–analog converters and the analog–digital converters found on the rear panel of the Stanford Research SR-530 lock-in amplifiers (referenced as LIAxy, where x is the channel number and y is the harmonic number). The Kepco 1000Bs (OPS1 and OPS2) bias the photomultiplier tubes PMT1 and PMT2, and the dc from each PMT is measured using DMM1 and DMM2.

Fig. 5
Fig. 5

Modulator drive voltage required to yield a phase retardation of 2.4048 (137.8°) vs the wavelength of the light.

Fig. 6
Fig. 6

Measured static strain retardation of the modulator in radians vs the photon energy of the light. The nonlinearity observable at higher photon energies is due to the dispersion of the refractive index and the stress-optic coefficients of fused silica.

Fig. 7
Fig. 7

Wavelength dependence of the channel gain factors ξij for channels 1 and 2 and for the fundamental and the second harmonic.

Fig. 8
Fig. 8

Effective dielectric functions (〈〉 = 〈1〉 + i2〉) of Si measured using the two-channel PME. Results of the three data sets (BEFORE, before chemical treatment; BRM, after BRM treatment; HF5, after HF5 treatment; see text) are shown without oxide overlayer correction, while the data labeled Corrected have been corrected for the oxide overlayer. The calculated data shown were taken from the HF5 data, although there is little difference between these spectra and the corrected spectra from the BEFORE or BRM data.

Fig. 9
Fig. 9

Optical absorption coefficient of Si determined from three data sets (see Fig. 8 and test) without an oxide overlayer correction and from the data for HF5 corrected for the appropriate overlayer. There is little difference between the HF5 corrected data and the corrected data of BEFORE or BRM.

Tables (5)

Tables Icon

Table I Values of N, S, C, ψ, and Δ for the Straight through Configuration at Various Photon Energies and Azimuthal Angles of the Modulator and Analyzer; Polarizer–Modulator Azimuthal Angle Set to θb = 45°

Tables Icon

Table II Values of N, S, C, β, ψ, and Δ for Silicon Without Surface Treatment (BEFORE), Taken In the Measurement Configuration at Various Photon Energies and Azimuthal Angles of the Modulator and Analyzer; Polarizer–Modulator Azimuthal Angle set to θb = 45°

Tables Icon

Table III Resultant Angle of Incidence φ and Oxide Thickness (doxide) Determined from the Ellipsometric Angles ψ and Δ at λ = 652.4 nm; Also Shown are the Associated Ellipsometric Parameters N S, and C Defined in Eqs. (2).

Tables Icon

Table IV Fractions of a-SIO2 and the Thicknesses of the Layers for the Various Calculations Discussed In the Test; the Real and Imaginary Parts of the Effective Dielectric Functions of the Oxide at λ = 652 nm are Shown, Resulting from the Effective Medium Calculation

Tables Icon

Table V Resultant Dielectric Functions at Several Photon Energies of Interest Calculated from the Data Shown In Fig. 8, Corrected for the Oxide Overlayer in Various Ways (see Text)

Equations (89)

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r p / r s = tan ψ exp ( i Δ ) ,
N = cos 2 ψ ,
S = sin 2 ψ sin Δ ,
C = sin 2 ψ cos Δ ,
N 2 + S 2 + C 2 = 1.
β 2 = N m 2 + S m 2 + C m 2 .
n i ( λ , V m , t , l ) = n i 0 ( λ ) + V m c i ( λ ) sin ( ω t ) sin ( π l / l 0 ) ,
n j ( λ , V m , t , l ) = n j 0 ( λ ) + V m c j ( λ ) sin ( ω t ) sin ( π l / l 0 ) ,
δ ( λ , t ) = 2 π d ( n i - n j ) / λ ,
δ ( V m , λ , t ) = ( 2 π d / λ ) { ( n i 0 - n j 0 ) + V m [ c i ( λ ) - c j ( λ ) ] sin ( ω t ) } ,
δ ( V m , λ , t ) = δ 0 ( λ ) + A sin ( ω t ) ,
A = 2 π d V m [ c i ( λ ) - c j ( λ ) ] / λ
I i ( j ω ) a j 0 + a j N N + a j s S + a j C C ,
I i ( dc ) 1 ,
I i ( ω ) ± b ± i 2 J 1 ( A ) S ,
I i ( 2 ω ) ± b m 2 J 2 ( A ) N ,
I i ( dc ) 1 ,
I i ( ω ) ± b ± i 2 J 1 ( A ) S ,
I i ( 2 ω ) ± b ± i m 2 J 2 ( A ) C .
I i ( dc ) 1 ,
I i ( ω ) ± b ± i 2 J 1 ( A ) S ,
I i ( 2 ω ) ± b 2 J 2 ( A ) ( x C ± y N ) / 2 ,
R i j ( λ ) = k i j ξ i j ( λ ) I i ( j ω , λ ) / I i ( d c , λ ) ,
ψ = 0.5 tan - 1 [ ( C 2 + S 2 ) 1 / 2 / N ] ,
Δ = tan - 1 ( S / C ) .
1 = sin 2 φ [ 1 + tan 2 φ ( N 2 - S 2 ) / ( 1 + C ) 2 ] ,
2 = 2 sin 2 φ tan 2 φ N S / ( 1 + C ) 2 .
n ˜ = n - i k = ,
α = 4 π k / λ ,
I i = I 0 ( 1 - N Z ± i cos ( 2 θ a ) ( Z - N ) ± i sin ( 2 θ a ) ( C Y + S X ) ± i S w { S Z sin ( 2 θ a ) + X [ ± i N - cos ( 2 θ a ) ] } ± i ( S Y - C X ) [ S w 2 cos ( 2 θ a ) - ( C w 1 + C w 2 ) sin ( 2 θ a ) ] ) ,
X = sin ( 2 θ b ) sin δ ,
Y = sin ( 2 θ m ) cos ( 2 θ b ) - cos ( 2 θ m ) sin ( 2 θ b ) cos δ ,
Z = cos ( 2 θ m ) cos ( 2 θ b ) + sin ( 2 θ m ) sin ( 2 θ b ) cos δ .
C w 1 = cos ( 2 θ w 1 ) δ w 1 ,
S w 1 = sin ( 2 θ w 1 ) δ w 1 ,
C w 2 = cos ( 2 θ w 2 ) δ w 2 ,
S w 2 = sin ( 2 θ w 2 ) δ w 2 .
I i , c = I 0 Q i { 1 + cos ( 2 θ b ) Z - sin ( 2 θ b ) [ cos δ Y - sin δ X ] } ,
Q i = 1 - N cos ( 2 θ p ) ± i { cos ( 2 θ a ) [ cos ( 2 θ p ) - N ] + C sin ( 2 θ a ) sin ( 2 θ p ) } ± i S [ sin ( 2 θ a ) cos ( 2 θ p ) S w 1 + cos ( 2 θ a ) sin ( 2 θ p ) S w 2 ] + S sin ( 2 θ p ) ( C w 1 + C w 2 ) ,
X = sin ( 2 θ e ) sin δ ,
Y = sin ( 2 θ d ) cos ( 2 θ e ) + cos ( 2 θ d ) sin ( 2 θ e ) cos δ ,
Z = cos ( 2 θ d cos ( 2 θ e ) - sin ( 2 θ d ) sin ( 2 θ e ) cos δ .
sin ( δ ) = J 0 ( A ) δ 0 + 2 [ J 1 ( A ) sin ω t + J 2 ( A ) δ 0 cos 2 ω t + ] ,
cos ( δ ) = J 0 ( A ) + 2 [ - J 1 ( A ) δ 0 sin ω t + J 2 ( A ) cos 2 ω t - + ] .
I i ( dc ) = I 0 ( a 00 + a 0 N N + a 0 C C + a 0 s S ) ,
I i ( ω ) = 2 I 0 J 1 ( A ) sin 2 θ b sin ω t ( a 10 + a 1 N N + a 1 C C + a 1 s S ) ,
I i ( 2 ω ) = 2 I 0 J 2 ( A ) sin 2 θ b cos 2 ω t ( a 20 + a 2 N N + a 2 C C + a 2 S S ) .
a 00 = 1 ± i cos 2 θ a [ cos 2 θ b cos 2 θ m + sin 2 θ b sin 2 θ m J 0 ( A ) ] ,
a 0 N = - [ ± i cos 2 θ a + cos 2 θ b cos 2 θ m + sin 2 θ b sin2 θ m J 0 ( A ) ] ,
a 0 C = ± i sin 2 θ a [ cos 2 θ b sin 2 θ m - sin 2 θ b cos 2 θ m J 0 ( A ) ] ,
a 0 S = ± i cos 2 θ b { cos 2 θ m sin 2 θ a S w 1 + sin 2 θ m [ cos 2 θ a S w 2 - sin 2 θ a ( C w 1 + C w 2 ) ] } ,
a 10 = i cos 2 θ a ( sin 2 θ m δ 0 + S w 1 ) ,
a 1 N = sin 2 θ m δ 0 + S w 1 ,
a 1 C = ± i [ sin 2 θ a ( cos 2 θ m δ 0 + C w 1 + C w 2 ) - cos 2 θ a S w 2 ] ,
a 1 S = ± i sin 2 θ a ,
a 20 = ± i cos 2 θ a sin 2 θ m ,
a 2 N = - sin 2 θ m ,
a 2 C = i sin 2 θ a cos 2 θ m ,
a 2 S = ± i [ sin 2 θ a [ δ 0 + sin 2 θ m S w 1 + cos 2 θ m ( C w 1 + C w 2 ) ] - cos 2 θ a cos 2 θ m S w 2 } .
I i , c ( dc ) = I 0 Q i [ 1 + cos 2 θ b Z - sin 2 θ b Y J 0 ( A ) ] ,
I i , c ( ω ) = 2 I 0 Q i J 1 ( A ) sin 2 θ b ( X + Y δ 0 ) ,
I i , c ( 2 ω ) = 2 I 0 Q i J 2 ( A ) sin 2 θ b ( Y - X δ 0 ) ,
I i ( dc ) = ( 1 i N ) ( 1 ± b ± i m 2 b ) ,
I i ( ω ) = ± b ± i 2 J 1 ( A ) [ 2 S a - C S w 2 - ( 1 i N ) S w 1 ] ,
I i ( 2 ω ) = ± b ± i m 2 J 2 ( A ) [ 2 C a - 2 ( 1 i N ) m + S S w 2 ] ,
I i ( dc ) = ( 1 i N ) [ 1 ± b ± i ± m J 0 ( A ) ] ,
I i ( ω ) = ± b ± i 2 J 1 ( A ) [ 2 S a - ( 1 i N ) ( S w 1 ± m δ 0 ) - C S w 2 ] ,
I i ( 2 ω ) = ± b ± i ± m 2 J 2 ( A ) ( 1 i N ) ,
I i ( dc ) = 1 + 2 N ( ± i a ± b ± m b ) ± b ± i ± m C J 0 ( A ) ,
I i ( ω ) = ± b ± i 2 J 1 ( A ) [ S + C ( W 0 ± m δ 0 ) ± i N S w 1 ] ,
I i ( 2 ω ) = ± b ± i m 2 J 2 ( A ) [ C ± i 2 N m - S ( W 0 ± m δ 0 ) ] ,
I i ( dc ) = 1 ± i 2 N a - ± b ± m [ N J 0 ( A ) ± i 2 C b ] ,
I i ( ω ) = ± b ± i 2 J 1 ( A ) [ S ± i N ( S w 1 ± m δ 0 ) + C W 0 ] ,
I i ( 2 ω ) = ± b m 2 J 2 ( A ) { N ± i [ 2 ( a - C m ) - S ( S w 1 ± m δ 0 ) ] } ,
I i ( dc ) = 1 ± i 2 N a ± b [ 2 b χ - J 0 ( A ) ζ ] ,
I i ( ω ) = ± b ± i 2 J 1 ( A ) ( S ± i δ 0 ζ ± i N S w 1 + C W 0 ) ,
I i ( 2 ω ) = b 2 J 2 ( A ) [ ζ ± i ( 2 α x - S δ 0 + 2 m χ ) i S ( x S w 1 + y W 0 ) ] ,
I i ( dc ) = Q i ( 1 ± b ± d ± e 2 b ) ,
I i ( ω ) = ± b ± e 4 J 1 ( A ) Q i e sin δ ,
I i ( 2 ω ) = b ± d ± e 4 J 2 ( A ) Q i ( d + e cos δ ) ,
I i ( dc ) = Q i [ 1 b ± d ± e J 0 ( A ) ] ,
I i ( ω ) = ± b ± e 2 J 1 ( A ) Q i ( sin δ ± d δ 0 cos δ ) ,
I i ( 2 ω ) = ± b ± e 2 J 2 ( A ) Q i ( δ 0 sin δ d cos δ ) ,
I i ( dc ) = Q i [ 1 b ± d ± e J 0 ( A ) ] ,
I i ( ω ) = ± b ± e 2 J 1 ( A ) Q i ( 2 e sin δ ± d C δ 0 ) ,
I i ( 2 ω ) = b ± d ± e 2 J 2 ( A ) Q i ,
I i ( dc ) = Q i ( b ± d ± e 2 b cos δ ) ,
I i ( ω ) = ± b ± e 2 J 1 ( A ) Q i sin δ ,
I i ( 2 ω ) = ± b ± e 2 J 2 ( A ) Q i [ δ 0 sin δ + 2 ( e + d cos δ ) ] .

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