Abstract

A new ellipsometer is described that uses two photoelastic modulator–polarizer pairs, where the photoelastic modulators are operating at differing resonant frequencies. The time-dependent intensity of the light beam is extremely complicated but can be analyzed so that all elements of the sample Mueller matrix are obtained. For a given configuration, nine of the Mueller matrix elements can be measured at any one time; the other seven elements are accessible when the azimuthal angles of the photoelastic modulators are changed. The single-configuration measurement is often sufficient to characterize a number of real situations completely, such as film growth in a vacuum environment, anisotropic samples, and simple depolarization.

© 1997 Optical Society of America

Full Article  |  PDF Article

Errata

Gerald E. Jellison and Frank A. Modine, "Two-modulator generalized ellipsometry: theory—erratum," Appl. Opt. 42, 3765-3765 (2003)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-42-19-3765

References

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  1. M. M. Decker, H. Mueller, “Transmitting data by light modulation,” Control Eng. 4, 63–67 (1957).
  2. M. Billardon, J. Badoz, “Birefringence modulator,” C. R. Acad. Sci. (Paris) 262B, 1672–1675 (1966).
  3. L. F. Mollenauer, D. Downie, H. Engstrom, W. B. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. Opt. 8, 661–664 (1969).
    [CrossRef] [PubMed]
  4. 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); Errata 41, 152 (1970).
    [CrossRef]
  5. J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).
  6. G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).
  7. J. P. Badoz, M. P. Silverman, J. C. Canit, “New model of a photoelastic modulator with distributed birefringence,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 478–488 (1989); “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
  8. F. A. Modine, G. E. Jellison, “Errors in polarization measurements due to static retardation in photoelastic modulators,” Appl. Phys. Commun. 12, 121–139 (1993).
  9. V. M. Bermudez, V. H. Ritz, “Wavelength-scanning polarization-modulation ellipsometry: some practical considerations,” Appl. Opt. 17, 542–552 (1978).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. 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]
  13. O. Acher, E. Bigan, B. Drevillon, “Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
    [CrossRef]
  14. G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
    [CrossRef] [PubMed]
  15. D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [CrossRef] [PubMed]
  16. G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “Variable wavelength, variable angle ellipsometry including a sensitivities test,” Thin Solid Films 137, 27–41 (1986).
    [CrossRef]
  17. P. Chindaudom, K. Vedam, “Determination of the optical function n(λ) of vitreous silica by spectroscopic ellipsometry with an achromatic compensator,” Appl. Opt. 32, 6391–6397 (1993).
    [CrossRef] [PubMed]
  18. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  19. R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.
  20. R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 27.
  21. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  22. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  23. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  24. D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. 7, 693–700 (1990).
    [CrossRef]
  25. R. C. Thompson, J. R. Bottinger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1978).
    [CrossRef]
  26. E. Compain, B. Drevillon, “Complete Mueller Matrix measurement with a high frequency coupled-phase modulator,” to be published in Thin Solid Films.
  27. B. Drevillon, Ecole Polytechnique, 91128 Palaiseau, France (personal communication, 1997).
  28. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 767–777 (1985).
    [CrossRef]
  29. R. Anderson, “Measurement of Mueller matrices,” Appl. Opt. 31, 11–13 (1992).
    [CrossRef] [PubMed]
  30. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt.8184–8189 (1997).
  31. D. G. M. Anderson, R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305–2319 (1994).
    [CrossRef]
  32. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  33. D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).
  34. G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–29140–45 (1996).
    [CrossRef]
  35. G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS of GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
    [CrossRef]
  36. R. Joerger, K. Forcht, A. Gombert, M. Köhl, W. Graf, “Influence of incoherent superposition of light on ellipsometric coefficients,” Appl. Opt. 36, 319–327 (1997).
    [CrossRef] [PubMed]

1997 (1)

1996 (1)

G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–29140–45 (1996).
[CrossRef]

1994 (1)

1993 (2)

F. A. Modine, G. E. Jellison, “Errors in polarization measurements due to static retardation in photoelastic modulators,” Appl. Phys. Commun. 12, 121–139 (1993).

P. Chindaudom, K. Vedam, “Determination of the optical function n(λ) of vitreous silica by spectroscopic ellipsometry with an achromatic compensator,” Appl. Opt. 32, 6391–6397 (1993).
[CrossRef] [PubMed]

1992 (3)

R. Anderson, “Measurement of Mueller matrices,” Appl. Opt. 31, 11–13 (1992).
[CrossRef] [PubMed]

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS of GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[CrossRef] [PubMed]

1990 (2)

1989 (1)

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

1986 (1)

G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “Variable wavelength, variable angle ellipsometry including a sensitivities test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

1985 (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 767–777 (1985).
[CrossRef]

1983 (1)

1982 (2)

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]

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]

1980 (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

1978 (4)

1975 (1)

1969 (3)

1966 (1)

M. Billardon, J. Badoz, “Birefringence modulator,” C. R. Acad. Sci. (Paris) 262B, 1672–1675 (1966).

1957 (1)

M. M. Decker, H. Mueller, “Transmitting data by light modulation,” Control Eng. 4, 63–67 (1957).

Acher, O.

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

Anderson, D. G. M.

Anderson, R.

Aspnes, D. E.

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, 767–777 (1985).
[CrossRef]

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 27.

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

Badoz, J.

M. Billardon, J. Badoz, “Birefringence modulator,” C. R. Acad. Sci. (Paris) 262B, 1672–1675 (1966).

Badoz, J. P.

J. P. Badoz, M. P. Silverman, J. C. Canit, “New model of a photoelastic modulator with distributed birefringence,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 478–488 (1989); “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).

Barakat, R.

Bashara, N. M.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “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.

Bigan, E.

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

Billardon, M.

M. Billardon, J. Badoz, “Birefringence modulator,” C. R. Acad. Sci. (Paris) 262B, 1672–1675 (1966).

Bottinger, J. R.

Bu-Abbud, G. H.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “Variable wavelength, variable angle ellipsometry including a sensitivities test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

Canit, J. C.

J. P. Badoz, M. P. Silverman, J. C. Canit, “New model of a photoelastic modulator with distributed birefringence,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 478–488 (1989); “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).

Chindaudom, P.

Chipman, R. A.

D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. 7, 693–700 (1990).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.

Compain, E.

E. Compain, B. Drevillon, “Complete Mueller Matrix measurement with a high frequency coupled-phase modulator,” to be published in Thin Solid Films.

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]

Decker, M. M.

M. M. Decker, H. Mueller, “Transmitting data by light modulation,” Control Eng. 4, 63–67 (1957).

Downie, D.

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]

E. Compain, B. Drevillon, “Complete Mueller Matrix measurement with a high frequency coupled-phase modulator,” to be published in Thin Solid Films.

B. Drevillon, Ecole Polytechnique, 91128 Palaiseau, France (personal communication, 1997).

Engstrom, H.

Forcht, K.

Fry, E. S.

Goldstein, D. H.

Gombert, A.

Graf, W.

Grant, W. B.

Gruzalski, G. R.

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
[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); Errata 41, 152 (1970).
[CrossRef]

Jellison, G. E.

G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–29140–45 (1996).
[CrossRef]

F. A. Modine, G. E. Jellison, “Errors in polarization measurements due to static retardation in photoelastic modulators,” Appl. Phys. Commun. 12, 121–139 (1993).

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS of GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

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 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, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt.8184–8189 (1997).

Joerger, R.

Kemp, J. C.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).

Köhl, M.

Lewis, J. W.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).

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]

McCamy, J. W.

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS of GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

Modine, F. A.

F. A. Modine, G. E. Jellison, “Errors in polarization measurements due to static retardation in photoelastic modulators,” Appl. Phys. Commun. 12, 121–139 (1993).

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

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 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, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt.8184–8189 (1997).

Mollenauer, L. F.

Mueller, H.

M. M. Decker, H. Mueller, “Transmitting data by light modulation,” Control Eng. 4, 63–67 (1957).

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]

Randall, C. E.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).

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); Errata 41, 152 (1970).
[CrossRef]

Silverman, M. P.

J. P. Badoz, M. P. Silverman, J. C. Canit, “New model of a photoelastic modulator with distributed birefringence,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 478–488 (1989); “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).

Studna, A. A.

Thompson, R. C.

Vedam, K.

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]

Woolam, J. A.

G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “Variable wavelength, variable angle ellipsometry including a sensitivities test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

Appl. Opt. (9)

Appl. Phys. Commun. (1)

F. A. Modine, G. E. Jellison, “Errors in polarization measurements due to static retardation in photoelastic modulators,” Appl. Phys. Commun. 12, 121–139 (1993).

Appl. Phys. Lett. (1)

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS of GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

C. R. Acad. Sci. (Paris) (1)

M. Billardon, J. Badoz, “Birefringence modulator,” C. R. Acad. Sci. (Paris) 262B, 1672–1675 (1966).

Control Eng. (1)

M. M. Decker, H. Mueller, “Transmitting data by light modulation,” Control Eng. 4, 63–67 (1957).

J. Appl. Phys. (1)

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. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

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, 767–777 (1985).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (3)

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]

O. Acher, E. Bigan, B. Drevillon, “Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[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); Errata 41, 152 (1970).
[CrossRef]

Surf. Sci. (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Thin Solid Films (2)

G. H. Bu-Abbud, N. M. Bashara, J. A. Woolam, “Variable wavelength, variable angle ellipsometry including a sensitivities test,” Thin Solid Films 137, 27–41 (1986).
[CrossRef]

G. E. Jellison, “The calculation of thin film parameters from spectroscopic ellipsometry data,” Thin Solid Films 290–29140–45 (1996).
[CrossRef]

Other (9)

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

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).

G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt.8184–8189 (1997).

E. Compain, B. Drevillon, “Complete Mueller Matrix measurement with a high frequency coupled-phase modulator,” to be published in Thin Solid Films.

B. Drevillon, Ecole Polytechnique, 91128 Palaiseau, France (personal communication, 1997).

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

J. P. Badoz, M. P. Silverman, J. C. Canit, “New model of a photoelastic modulator with distributed birefringence,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 478–488 (1989); “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.

R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, Vol. 2, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 27.

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Figures (3)

Fig. 1
Fig. 1

General schematic for a polarization-sensitive reflection or transmission measurement. The initial light beam is conditioned by the PSG, and the reflected or transmitted beam is further conditioned by the PSD. The light intensity is measured at the detector.

Fig. 2
Fig. 2

First 100 channels (at 1 µs/channel) of the simulated wave form from a 2-MGE operating in the straight-through configuration. The modulators are assumed to be operating at ω0 = 50 kHz, δ0 = 0.01 and ω1 = 60 kHz, δ1 = 0.03 with A0 = A1 = 2.4048 and ϕ0 = ϕ1 = 0. The wave form repeats every 100 channels.

Fig. 3
Fig. 3

Fast Fourier transform power spectrum for the waveform shown in Fig. 2, assuming that the waveform extends for 4096 channels.

Tables (1)

Tables Icon

Table 1 Some Frequency Components for the Fourier Integral Analysis of the Time-Dependent Intensity Vector [Eq. (12)]

Equations (61)

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

PPSG=I021cos2θm0cos2θb0+sin2θm0sin2θb0Y0δsin2θm0cos2θb0-cos2θm0sin2θb0Y0δsin2θb0X0δ,
X0δ=sinA0 sinω0t+ϕ0+δ0,
Y0δ=cosA0 sinω0t+ϕ0+δ0.
M=m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33.
M=1-N00-N10000CS00-SC,
I=PPSDMPPSG.
It=Idc+IX0X0δ+IY0Y0δ+IX1X1δ+IY1Y1δ+IX0X1X0δX1δ+IX0Y1X0δY1δ+IY0X1Y0δX1δ+IY0Y1Y0δY1δ.
Xiδ=sinAi sinωit+ϕi+δiXi+δiYi,
Yiδ=cosAi sinωit+ϕi+δiYi-δiXi,
Xi=sinAi sinωit+ϕi,
Yi=cosAi sinωit+ϕi.
It=BtTδI,
BtT=1 X0 Y0 X1 Y1 X0X1 X0Y1 Y0X1 Y0Y1,
IT=Idc IX0 IY0 IX1 IY1 IX0X1 IX0Y1 IY0X1 IY0Y1,
δ=10000000001δ00000000-δ010000000001δ10000000-δ110000000001δ1δ0000000-δ110δ000000-δ001δ1000000-δ0-δ11.
Xi=sinAi sinωit=2k=1J2k-1Aisin2k-1ωit,
Yi=cosAi sinωit=J0Ai+2k=1J2kAicos2kωit.
It=Idc+2k=1Rkαk cosΩkt+βk sinΩkt.
signIX0=signsinϕm+ϕ0  Ω=ω0,
signIY0=signcosϕm+ϕ0  Ω=2ω0,
signIX1=signsinϕm+ϕ1  Ω=ω1,
signIY1=signcosϕm+ϕ1  Ω=2ω1,
signIX0X1=signcosϕm+ϕ0+ϕ1  Ω=ω0+ω1,
signIX0Y1=signsinϕm+ϕ0+2ϕ1  Ω=ω0+2ω1,
signIY0X1=signsinϕm+2ϕ0+ϕ1  Ω=2ω0+ω1,
signIY0Y1=signcosϕm+2ϕ0+2ϕ1  Ω=2ω0+2ω1.
0TBtItdt=0TBtBtTdtδI=KδI.
I=δ-1K-10TBtItdt.
M=Idcσ0sIY0IX0σ1sIY1σ0sσ1sIY0Y1σ1sIX0Y1-IX1σ0sIY0X1-IX0X1,
M=Idcσ0sIY0IX0-σ1cIY1-σ0sσ1cIY0Y1-σ1cIX0Y1-IX1σ0sIY0X1-IX0X1,
M=Idc-σ0cIY0IX0σ1sIY1-σ0cσ1sIY0Y1σ1sIX0Y1-IX1-σ0cIY0X1-IX0X1,
M=Idc-σ0cIY0IX0-σ1cIY1σ0cσ1cIY0Y1-σ1cIX0Y1-IX1-σ0cIY0X1-IX0X1,
σ0s=signsin2θm0,  σ1s=signsin2θm1,  σ0c=signcos2θm0,  σ1c=signcos2θm1.
M=1-N0NδSw0-N1SδSw1-δSw0-CδSw10SδSw0C-SδWS+CδW-NδSw1δSw1+CδSw0-S-CδWC-SδW,
δSw0=δw0 sin2θw0,
δSw1=δw1 sin2θw1,
δW=δw0 cos2θw0+δw1 cos2θw1.
J=rpprpsrsprss=rssγpp expiΔppγps expiΔpsγsp expiΔsp1,
M=A·JJ*·A-1,
A=1001100-101100-ii0.
M=1-N-αpsCsp+ζ1Ssp+ζ2-N-αsp1-αsp-αps-Csp+ζ1-Ssp+ζ2Cps+ξ1-Cps+ξ1C+β1S+β2-Sps+ξ2Sps+ξ2-S+β2C-β1,
N=1-γpp2-γsp2-γps2/D,  D=1+γpp2+γsp2+γps2=2/1+N,
S=2γpp sinΔpp/D,  C=2γpp cosΔpp/D,
Ssp=2γsp sinΔsp/D,  Csp=2γsp cosΔsp/D,
Sps=2γps sinΔps/D,  Cps=2γps cosΔps/D,
αsp=2γsp2/D,  αps=2γps2/D,
β1=D/2CspCps+SspSps,  β2=D/2CspSps-SspCps,
ζ1=D/2CCps+SSps,  ζ2=D/2CSps-SCps,
ξ1=D/2CCsp+SSsp  ξ2=D/2CSsp-SCsp.
N2+S2+C2+Ssp2+Csp2+Sps2+Cps2=1.
Sps=KscSm02+m12-Cm03+m13,
Cps=KscCm02+m12+Sm03+m13,
Ssp=m03-m13/2,
Csp=m02-m12/2,
Ksc=1+N/2S2+C2.
Sps=-m30+m31/2,
Cps=m20-m21/2,
Ssp=KscSm20+m21+Cm30+m31,
Csp=KscCm20+m21-Sm30+m31.
M=p1m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33+1-p1000000000000000,
N2+S2+C2+Ssp2+Csp2+Sps2+Cps2=p2.

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