Abstract

Null ellipsometry is analyzed for components with depolarizations for unpolarized incident light. Serious imperfections include sample depolarization D < 0.2, misalignment (δC < 2°) and off-quarter-wave retardance (δτ < 5°) of the compensator. The four-zone null positions are derived analytically to the second order of serious imperfections and are also simulated based on Mueller matrices with depolarization. Errors of all four-zone nulls increase with increasing D. Depolarizations of all components except the analyzer cause errors to the nulls. The errors associated with D always couple with the second order of δC and δτ and are enhanced by csc2 2ψ. These divergent errors limit the applicable region of null ellipsometry where the errors in ψ and Δ are within 0.1°, and the simulation agrees well with the analytic solutions.

© 1999 Optical Society of America

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References

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  1. S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
    [CrossRef]
  2. S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
    [CrossRef]
  3. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
  4. S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
    [CrossRef]
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sections 2.12, 3.8, and Appendix.
  6. R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1971).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  8. S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC, Boca Raton, Fla., 1999), Chap. 60.
    [CrossRef]
  9. S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
    [CrossRef]
  10. G. E. Jellison, D. H. Lowndes, “Time resolved ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
    [CrossRef]
  11. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
    [CrossRef] [PubMed]
  12. R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.
  13. U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
    [CrossRef]
  14. J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
    [CrossRef]
  15. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
    [CrossRef] [PubMed]
  16. S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).
  17. S.-M. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
    [CrossRef]
  18. S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
    [CrossRef]
  19. S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
  20. G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
    [CrossRef]
  21. A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. 9, 1124–1131 (1992).
    [CrossRef]
  22. D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
    [CrossRef]
  23. 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]
  24. M. Kildemo, P. Bulkin, B. Drevillon, O. Hunderi, “Real-time control by multiwavelength ellipsometry of plasma-deposited multilayers on glass by use of an incoherent-reflection model,” Appl. Opt. 36, 6352–6359 (1997).
    [CrossRef]
  25. U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
    [CrossRef]
  26. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
    [CrossRef]
  27. G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
    [CrossRef]
  28. S.-M. F. Nee, “Error reduction for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
    [CrossRef]
  29. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), Chap. 6.
  30. S. F. Nee, J. R. Jokipii, “Interstellar polarization in an irregularly fluctuating medium,” Astrophys. J. 234, 140–153 (1979).
    [CrossRef]
  31. S. F. Nee, “Fluctuation theory of starlight polarization,” Astrophys. J. 237, 471–481 (1980).
    [CrossRef]
  32. S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 31–37 (1990).
  33. S. F. Nee, H. E. Bennett, “Accurate null polarimetry for measuring the refractive index of transparent materials,” J. Opt. Soc. Am. A 10, 2076–2083 (1993).
    [CrossRef]

1998 (6)

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

1997 (2)

1996 (1)

1994 (1)

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

1993 (2)

S. F. Nee, H. E. Bennett, “Accurate null polarimetry for measuring the refractive index of transparent materials,” J. Opt. Soc. Am. A 10, 2076–2083 (1993).
[CrossRef]

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

1992 (3)

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

A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. 9, 1124–1131 (1992).
[CrossRef]

A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
[CrossRef] [PubMed]

1991 (1)

1986 (1)

1985 (1)

1980 (1)

S. F. Nee, “Fluctuation theory of starlight polarization,” Astrophys. J. 237, 471–481 (1980).
[CrossRef]

1979 (1)

S. F. Nee, J. R. Jokipii, “Interstellar polarization in an irregularly fluctuating medium,” Astrophys. J. 234, 140–153 (1979).
[CrossRef]

1971 (1)

Archibald, P. C.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1971).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sections 2.12, 3.8, and Appendix.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1971).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sections 2.12, 3.8, and Appendix.

Bennett, H. E.

S. F. Nee, H. E. Bennett, “Accurate null polarimetry for measuring the refractive index of transparent materials,” J. Opt. Soc. Am. A 10, 2076–2083 (1993).
[CrossRef]

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 31–37 (1990).

Bennett, J. M.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Bulkin, P.

Burge, D.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.

Cole, T.

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).

Drevillon, B.

Forcht, K.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), Chap. 6.

Gombert, A.

Graf, W.

Hu, Y. Z.

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

Hunderi, O.

Jellison, G. E.

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

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

G. E. Jellison, D. H. Lowndes, “Time resolved ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
[CrossRef]

Joerger, R.

Jokipii, J. R.

S. F. Nee, J. R. Jokipii, “Interstellar polarization in an irregularly fluctuating medium,” Astrophys. J. 234, 140–153 (1979).
[CrossRef]

Kildemo, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

M. Kildemo, P. Bulkin, B. Drevillon, O. Hunderi, “Real-time control by multiwavelength ellipsometry of plasma-deposited multilayers on glass by use of an incoherent-reflection model,” Appl. Opt. 36, 6352–6359 (1997).
[CrossRef]

Köhl, M.

Kostinski, A. B.

Lowndes, D. H.

Ludema, K. C.

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

McCamy, J. W.

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

Nee, S. F.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S. F. Nee, H. E. Bennett, “Accurate null polarimetry for measuring the refractive index of transparent materials,” J. Opt. Soc. Am. A 10, 2076–2083 (1993).
[CrossRef]

S. F. Nee, “Fluctuation theory of starlight polarization,” Astrophys. J. 237, 471–481 (1980).
[CrossRef]

S. F. Nee, J. R. Jokipii, “Interstellar polarization in an irregularly fluctuating medium,” Astrophys. J. 234, 140–153 (1979).
[CrossRef]

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 31–37 (1990).

Nee, S.-M. F.

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

S.-M. F. Nee, “Error reduction for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
[CrossRef]

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).

S.-M. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC, Boca Raton, Fla., 1999), Chap. 60.
[CrossRef]

Nee, T. W.

S.-M. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

Ossikovski, R.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

Ramsey, D. A.

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Richter, U.

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

Richter, W.

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

Röseler, A.

A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. 9, 1124–1131 (1992).
[CrossRef]

Rossow, U.

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

Spanos, L.

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

Stchakovsky, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

Trepk, Th.

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Williams, M. W.

Yoo, C.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).

Zettler, J. Th.

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

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

Astrophys. J. (2)

S. F. Nee, J. R. Jokipii, “Interstellar polarization in an irregularly fluctuating medium,” Astrophys. J. 234, 140–153 (1979).
[CrossRef]

S. F. Nee, “Fluctuation theory of starlight polarization,” Astrophys. J. 237, 471–481 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1971).
[CrossRef]

A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. 9, 1124–1131 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Rev. Sci. Instrum. (1)

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Thin Solid Films (6)

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

J. Th. Zettler, Th. Trepk, L. Spanos, Y. Z. Hu, W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films 234, 402–407 (1993).
[CrossRef]

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Other (12)

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S.-M. F. Nee, T. Cole, C. Yoo, D. Burge, “Characterization of infrared polarizers,” in Conference of Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3121, 213–224 (1997).

S.-M. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sections 2.12, 3.8, and Appendix.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC, Boca Raton, Fla., 1999), Chap. 60.
[CrossRef]

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart & Winston, New York, 1975), Chap. 6.

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 31–37 (1990).

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Figures (10)

Fig. 1
Fig. 1

Simulated four-zone analyzer nulls δA versus Δ for sample depolarization D = 0 and 0.1. δA is larger for larger D.

Fig. 2
Fig. 2

Error δψ of the four-zone average versus Δ for D = 0, 0.1, and 0.2. The discrete symbols are from the simulation and the curves are evaluated from Eq. (25). The simulation agrees well with Eq. (25), and δψ is much smaller than δA of Fig. 1.

Fig. 3
Fig. 3

Deviations δP of the simulated four-zone polarizer nulls from the ideal nulls versus Δ for D = 0. Deviations of zones 1 and 3 overlap, as well as those of zones 2 and 4.

Fig. 4
Fig. 4

Simulated four-zone deviations δP versus Δ for D = 0.1. D messes up and increases all deviations. In both Figs. 3 and 4, the four-zone average δΔ is much smaller than δP.

Fig. 5
Fig. 5

Error δΔ of the four-zone average versus Δ for different D. δΔ is much smaller than δP of Figs. 3 and 4. The curves were evaluated from Eq. (24) and agree well with the simulation.

Fig. 6
Fig. 6

Deviations δA of the simulated four-zone analyzer nulls from the ideal nulls versus ψ for D = 0 and 0.1. Δ is set at 30°. δA for D ≠ 0 diverge near ψ = 0° and 90°, while still converging for D = 0.

Fig. 7
Fig. 7

Error δψ (<0.1) from the four-zone average versus ψ for different D. δψ is much smaller than δA of Fig. 6, although it diverges for D ≠ 0 near ψ = 0° and 90°. Curves were evaluated from Eq. (25) and agree well with the simulation in most regions.

Fig. 8
Fig. 8

Same as Fig. 7 except for δψ < 0.5 to show the divergent region where Eq. (25) breaks down.

Fig. 9
Fig. 9

Simulated four-zone polarizer nulls as functions of ψ for D = 0 and 0.1. The polarizer nulls deviate from the ideal nulls divergently as ψ approaches 0° or 90°. The divergence is serious for D = 0.1.

Fig. 10
Fig. 10

Errors δΔ of the four-zone average as functions of ψ for different D. The curves were calculated with Eq. (24) and agree well with the simulation within 20° < ψ < 70° for D ≤ 0.2. Simulated δΔ is still acceptable within 10° < ψ < 80° for D < 0.2.

Tables (1)

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Table 1 Values of C and m That Comply with the Conventional Four-Zone Null Positions

Equations (26)

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M=T1-P cos 2ψ00-P cos 2ψ1-2Dv0000P sin 2ψ cos ΔP sin 2ψ sin Δ00-P sin 2ψ sin ΔP sin 2ψ cos Δ,
M=T1+u+v-cos 2ψ00-cos 2ψ1+u-v0000sin 2ψ cos Δsin 2ψ sin Δ00-sin 2ψ sin Δsin 2ψ cos Δ.
P=1/1+u+v,  D1-P=Du+Dv,  Du=uP, Dv=vP, T=TP.
P0°=Tp1+up+vp10011+up-vp0000000000.
C0°=Tc1+uc+vc00001+uc-vc0000cos τsin τ00-sin τcos τ.
Mϕ=RTϕM0Rϕ
Rϕ=10000cos 2ϕsin 2ϕ00-sin 2ϕcos 2ϕ00001.
So=I01000.
Y0=1+Dp+Dc-P cos 2ψ1+uc-vccos×2C cos 2C-P+cos τ sin 2C sin 2C-P,  Y1=-P cos 2ψ1+Dp+Dc+1-2Dv1+uc-vc×cos 2C cos 2C-P+cos τ sin 2C sin 2C-P,  Y2=P sin 2ψ1+uc-vccos Δ sin 2C cos 2C-P-cos τ cos 2C cos Δ-sin τ sin Δsin 2C-P,  Y3=P sin 2ψ-1+uc-vcsin Δ sin 2C cos 2C-P+cos τ cos 2C sin Δ+sin τ cosΔsin 2C-P.
IP, A=TaTTcTpI0Iˆ,  Iˆ=1+DaY0+Y1 cos 2A+Y2 sin 2A.
Y0=1+Dp+Dc+P cos 2ψ sgn Csin 2δC cos 2C-P+sin δτ sin 2C-P,  Y1=-P cos 2ψ1+Dp+Dc-1-2Dvsgn C×sin 2δC cos 2C-P+sin δτ sin 2C-P,  Y2=P sin 2ψcos 2δC+uc-vccos Δ sgn C cos 2C-P+cos δτ sin Δ-sin δτ sgn C sin 2δC×cos Δsin 2C-P.
IˆP=1+DaY0P+Y1P cos 2A+Y2P sin 2A=0,
IˆA=2-Y1 sin 2A+Y2 cos 2A=0.
cos 2A-P cos 2ψ-sin 2δC sin 2C-P +sin δτ cos 2C-P+P sin 2ψ sin 2A×sin2C-P-Δ sgn C+uc-vc-2 sin2 δC×cos Δ sin 2C-P+2 sgn C sin2δτ/2sin Δ+sin δτ sin 2δC cos Δcos 2C-P=0,
sin 2AP cos 2ψ1+Dp+Dc+1-2Dvsgn C×sin 2δC cos 2C-P-sin δτ sin 2C-P +sgn C P sin 2ψ cos 2Acos2C-P-Δ sgn C +uc-vc-2 sin2 δCcos Δ cos 2C-P -sin δτ sin 2δC cos Δ+2 sgn C sin2δτ/2sin Δsin 2C-P=0.
sin2C-P-Δ sgn C=0,  P=C+mπ/2-0.5 Δ sgn C.
sin2A+2ψ sgn C cos mπ=0,  A=-ψ sgn C cos mπ.
2δP-δCP sin 2ψ+cos mπ cot 2ψDu-Dv×sin δτ sin Δ+sgn C sin 2δC cos Δ+2δAsin δτ cos Δ-sgn C sin 2δC sin Δ=cos mπ cot 2ψDu-Dvsin 2δC sin Δ-sgn C sin δτ cos Δ+sin δτ sin 2δC P sin 2ψ cos2 Δ+sgn C P sin 2ψ sin 2Δ0.5uc-vc+sin2δτ/2-sin2 δC,
2δP-δCsin 2ψsin δτ cos Δ-sgn C sin 2δC sin Δ+2δAP+cos mπ cos 2ψsin δτ sin Δ+sgn C sin 2δC cos Δ=sin 2ψsin 2δC cos Δ+sgn C sin δτ sin Δ+0.25 cos mπ sin δτ sin 2δC P sin 4ψ sin 2Δ+0.5 sgn C cos mπ P sin 4ψDp+Dc-uc-vc-2 sin2 δCcos2 Δ+2 sin2δτ/2sin2 Δ.
δP=δC+0.5 cos mπ P-1 csc 2ψ cot 2ψDu-Dv×sin 2δC sin Δ-sgn C sin δτ cos Δ,  δA=0.5P-1 sin 2ψsin 2δC cos Δ+sgn C sin δτ sin Δ.
δP=δC+0.5 cos mπ P-1 csc 2ψ cot 2ψDu-Dv×sin 2δC sin Δ-sgn C sin δτ cos Δ+0.5 sin δτ sin 2δCcos2 Δ-P-2 cos 2Δ×1+cot2 2ψDu-Dv+0.25 sgn C sin 2Δ×uc-vc+2 sin2δτ/2-2 sin2 δC+P-2sin2 2δC-sin2 δτ1+cot2 2ψDu-Dv,
δA=0.5P-1 sin 2ψsin 2δC cos Δ+sgn C sin δτ sin Δ+0.5 cos mπ sin 2δC sin δτ sin 4ψ sin Δ cos Δ×0.5-P-21+Du-Dvcsc2 2ψ+0.25 sgn C cos mπ sin 4ψDp+Dc+2 sin2δτ/2-P-2 sin2 δτsin2 Δ-uc-vc-2 sin2 δC+P-2 sin2 2δCcos2 Δ+P-2 csc2 2ψDu-Dvsin2 δτ cos2 Δ+sin2 2δC sin2 Δ.
Δ*=-12i=14 sgn Ci Pi*=Δ+δΔ,  ψ*=-14i=14 sgn Ci cos miπ Ai*=ψ+δψ.
δΔ=-0.5 sin 2Δuc-vc+2 sin2δτ/2-2 sin2 δC+P-2sin2 2δC-sin2 δτ1+cot2 2ψDu-Dv,
δψ=0.25 sin 4ψ-Dp+Dc+sin2 ΔP-2 sin2 δτ-2 sin2δτ/2+cos2 Δuc-vc-2 sin2 δC+P-2 sin2 2δC-P-2 csc2 2ψDu-Dv×sin2 δτ cos2 Δ+sin2 2δC sin2 Δ.
sin 2A=-sgn C cos mπ sin 2ψ+2δA cos 2ψ,cos 2A=cos 2ψ+2δA sgn C cos mπ sin 2ψ,sin2C-P-Δ sin 2C=cos mπ2δC-δP,cos2C-P-Δ sin 2C=cos mπ,cos 2C-P=cos mπcos Δ-2δC-δP×sgn C sin Δ,sin 2C-P=cos mπsgn C sin Δ+2δC-δPcos Δ.

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