Abstract

An anisotropic incoherent reflection model for the Mueller matrix elements of an optically thick uniaxial anisotropic semitransparent substrate with its anisotropy axis along its surface normal is developed. The Mueller matrix elements are measured by phase-modulated spectroscopic ellipsometry (SE) and compared with incoherent reflection model simulations. In the case of a sapphire substrate the oscillations observed are accurately modeled, and, in addition, the oscillating degree of polarization is correctly predicted. A straightforward generalization of the optical model, in the case of an arbitrary stack of layers containing a thick anisotropic semitransparent substrate, is also proposed and experimentally validated. The model is further applied to study the anisotropic dielectric function of a semi-insulating 4H-SiC wafer. An approximation based on a simple variation in the optical transition element is proposed to model the SiC birefringence. In conclusion, SE is shown to be a powerful alternative for investigating and predicting the behavior of optically thick birefringent materials.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Rossow, W. Richter, in Optical Characterisation of Epitaxial Semiconductor Layers, G. Bauer, W. Richter, eds. (Springer, Berlin, 1995).
  2. 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), and references therein.
    [CrossRef]
  3. S.-M. F. Nee, T. Cole, “Effects of depolarization of polarimetric components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
    [CrossRef]
  4. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).
  5. M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
    [CrossRef]
  6. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977), pp. 491, 358.
  7. M. Schubert, “Generalized ellipsometry and complex systems,” Thin Solid Films 313–314, 323–332 (1998).
    [CrossRef]
  8. 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]
  9. Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
    [CrossRef]
  10. B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer structures,” Appl. Phys. B 39, 165–170 (1986).
    [CrossRef]
  11. B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
    [CrossRef]
  12. B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 552.
  14. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  15. K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
    [CrossRef]
  16. W. J. Tropf, M. E. Thomas, T. J. Harris, “Optical constants of crystals,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, New York, 1995), Vol. II, p. 33.61.
  17. See, e.g., W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull.March, 25–29 (1997).
  18. D. L. Greenaway, G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, Oxford, 1968), p. 87.
    [CrossRef]
  19. S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
    [CrossRef]
  20. A. B. Djurisic, E. H. Li, “Modelling of the optical constants of 6H-SiC in the energy region 1–30 eV,” Opt. Commun. 157, 67–71 (1998).
    [CrossRef]
  21. A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
    [CrossRef]
  22. M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).
  23. W. J. Choyke, E. D. Palik, in Handbook of Optical constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
    [CrossRef]

1998 (4)

M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
[CrossRef]

M. Schubert, “Generalized ellipsometry and complex systems,” Thin Solid Films 313–314, 323–332 (1998).
[CrossRef]

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

A. B. Djurisic, E. H. Li, “Modelling of the optical constants of 6H-SiC in the energy region 1–30 eV,” Opt. Commun. 157, 67–71 (1998).
[CrossRef]

1997 (5)

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

See, e.g., W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull.March, 25–29 (1997).

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

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]

1995 (1)

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

1993 (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

1987 (2)

K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

1986 (2)

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer structures,” Appl. Phys. B 39, 165–170 (1986).
[CrossRef]

Abelson, J. R.

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

Adachi, S.

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

Adolph, B.

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977), pp. 491, 358.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977), pp. 491, 358.

Bechstedt, F.

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Bloomer, I.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 552.

Choyke, W. J.

See, e.g., W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull.March, 25–29 (1997).

W. J. Choyke, E. D. Palik, in Handbook of Optical constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

Cole, T.

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

Crean, G.

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

Djurisic, A. B.

A. B. Djurisic, E. H. Li, “Modelling of the optical constants of 6H-SiC in the energy region 1–30 eV,” Opt. Commun. 157, 67–71 (1998).
[CrossRef]

Drévillon, B.

M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
[CrossRef]

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

Forcht, K.

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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]

Forouhi, A. R.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

Gavrilenko, V. I.

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Gombert, A.

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]

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

Graf, W.

Greenaway, D. L.

D. L. Greenaway, G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, Oxford, 1968), p. 87.
[CrossRef]

Harbecke, B.

B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer structures,” Appl. Phys. B 39, 165–170 (1986).
[CrossRef]

Harbeke, G.

D. L. Greenaway, G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, Oxford, 1968), p. 87.
[CrossRef]

Harris, T. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Optical constants of crystals,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, New York, 1995), Vol. II, p. 33.61.

Hunderi, O.

M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
[CrossRef]

Joerger, R.

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]

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

Kelly, P.

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

Kildemo, M.

M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

Kim, K.

Köhl, M.

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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]

Li, E. H.

A. B. Djurisic, E. H. Li, “Modelling of the optical constants of 6H-SiC in the energy region 1–30 eV,” Opt. Commun. 157, 67–71 (1998).
[CrossRef]

Mandel, L.

Mooney, M.

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

Nee, S.-M. F.

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

Ossikovski, R.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

Palik, E. D.

W. J. Choyke, E. D. Palik, in Handbook of Optical constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

W. J. Choyke, E. D. Palik, in Handbook of Optical constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

Pensl, G.

See, e.g., W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull.March, 25–29 (1997).

Richter, W.

U. Rossow, W. Richter, in Optical Characterisation of Epitaxial Semiconductor Layers, G. Bauer, W. Richter, eds. (Springer, Berlin, 1995).

Rossow, U.

U. Rossow, W. Richter, in Optical Characterisation of Epitaxial Semiconductor Layers, G. Bauer, W. Richter, eds. (Springer, Berlin, 1995).

Schubert, M.

M. Schubert, “Generalized ellipsometry and complex systems,” Thin Solid Films 313–314, 323–332 (1998).
[CrossRef]

Stchakovsky, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

Sudre, C.

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

Tenelsen, K.

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Thomas, M. E.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Optical constants of crystals,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, New York, 1995), Vol. II, p. 33.61.

Tropf, W. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Optical constants of crystals,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, New York, 1995), Vol. II, p. 33.61.

Wolf, E.

Yang, Y. H.

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer structures,” Appl. Phys. B 39, 165–170 (1986).
[CrossRef]

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

J. Vac. Sci. Technol. A (1)

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

MRS Bull. (1)

See, e.g., W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull.March, 25–29 (1997).

Opt. Commun. (1)

A. B. Djurisic, E. H. Li, “Modelling of the optical constants of 6H-SiC in the energy region 1–30 eV,” Opt. Commun. 157, 67–71 (1998).
[CrossRef]

Phys. Rev. B (3)

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

S. Adachi, “Model dielectric constants of GaP, GaAs, GaSb, InP, and InSb,” Phys. Rev. B 35, 7454–7463 (1987).
[CrossRef]

B. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Prog. Cryst. Growth Charact. (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

Thin Solid Films (5)

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of absorption edge of thick transparent substrates using incoherent reflection model and spectroscopic UV-visible–near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

M. Kildemo, B. Drévillon, O. Hunderi, “A direct robust feedback method for growth control of optical coatings by multiwavelength ellipsometry,” Thin Solid Films 313–314, 484–489 (1998).
[CrossRef]

M. Schubert, “Generalized ellipsometry and complex systems,” Thin Solid Films 313–314, 323–332 (1998).
[CrossRef]

Other (8)

M. Kildemo, M. Mooney, P. Kelly, C. Sudre, G. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” in Proceedings of the International Conference on Silicon Carbide and Related Materials (to be published).

W. J. Choyke, E. D. Palik, in Handbook of Optical constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977), pp. 491, 358.

U. Rossow, W. Richter, in Optical Characterisation of Epitaxial Semiconductor Layers, G. Bauer, W. Richter, eds. (Springer, Berlin, 1995).

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), and references therein.
[CrossRef]

W. J. Tropf, M. E. Thomas, T. J. Harris, “Optical constants of crystals,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, New York, 1995), Vol. II, p. 33.61.

D. L. Greenaway, G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon, Oxford, 1968), p. 87.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 552.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Multiple-beam model of light reflection from an anisotropic layer on a semi-infinite medium. Numbers 0, 1, and 2 refer to ambient, layer, and semi-infinite medium, respectively.

Fig. 2
Fig. 2

PMSE measured and model fitted intensities I s and I c of a 1.0-mm-thick double-polished sapphire substrate. The refractive-index difference n o - n c is found to be approximately 8.0 × 10-3.

Fig. 3
Fig. 3

Simulated degree of polarization of the 1.0-mm-thick double-polished sapphire substrate from Fig. 2.

Fig. 4
Fig. 4

PMSE measured and model fitted intensities I s (dotted curves) and I c (solid curves) of a 245-nm vanadium oxide layer on a 1-mm-thick, double-polished sapphire substrate.

Fig. 5
Fig. 5

PMSE measured and model fitted intensities I s and I c of a 382-µm-thick double-polished semi-insulating 4H-SiC substrate.

Fig. 6
Fig. 6

Fitted ordinary (lower curve) and extraordinary (upper curve) refractive indices of 4H-SiC of Fig. 5, with the modified Sellmeier model in Eq. (8).

Equations (9)

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

M=rprp*+rsrs*21rprp*-rsrs*rprp*+rsrs*00rprp*-rsrs*rprp*+rsrs*10000rprs*+rsrp*rprp*+rsrs*-irprs*-rsrps*rprp*+rsrs*00irprs*-rsrp*rprp*+rsrs*rprs*-rsrp*rprp*+rsrs*,
Isγ=2 Imrprs*rsrs*+rprp*,Ic2β=2 Rerprs*rsrs*+rprp*,Ic3α=rsrs*-rprp*rsrs*+rprp*,
rs=r01s+r12s exp2iβs1+r01sr12s exp2iβs,rp=r01p+r12p exp2iβp1+r01pr12p exp2iβp,
βs=2π dλno2-n2 sin2 ϕo1/2,βp=2π dλnonene2-n2 sin2 ϕo1/2,
rx=r01x+t01xt10xr12x exp2iβx+t01xt10xr10xr12x2exp4iβx+=r01x+t01xt10xr12x exp2iβx×k=0r10xr12xkexp2ikβx,
rxry*=r01xr01y*+t01xt01y*t10xt10y*r12xr12y*exp2iβx-βy*×k=0r10xr10y*kr12xr12y*k exp2ikβx-βy*+t01xt01y*t10xt10y*r12xr12y*exp2iβx-βy*×k=0l=0lkr10xr12xkr10y*r12y*l exp2ikβx-1βy*+r01xt01y*t10y*r12y* exp-2iβy*×k=0r10y*r12y*k exp-2ikβy*+r01y*t01xt10xr12xexp2iβx×k=0r10xr12xk exp2ikβx.
rxry*=r01xr01y*+t01xt01y*t10xt10y*r12xr12y*exp2iβx-βy*1-r10xr10y*r12xr12y*exp2iβx-βy*.
qEnq2E=+fqEq2Eq2-E2,
q=ordinary direction, perpendicular to c axisextraordinary direction, parallel to c axis.

Metrics