Abstract

Utilizing transmission ellipsometry at small angles of incidence, it is shown that c-cut uniaxial samples can be used to determine both the miscut of the optic axis with respect to the plane of incidence as well as very accurate values of the spectroscopic birefringence. For example, wafers of ZnO, LiNbO3, and 6H-SiC single-crystals are examined and the miscut direction and the spectroscopic birefringence are determined. While all materials show strong dispersion in birefringence, ZnO exhibits a distinct isotropic point at 396.8 nm.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge1997).
  2. W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
    [CrossRef]
  3. F. A. Modine, R. W. Major, E. Sonder, “High frequency polarization modulation method for measuring birefringence,” Appl. Opt. 14, 757–760 (1975).
    [CrossRef] [PubMed]
  4. C. F. Wong, “Birefringence measurement using a photoelastic modulator,” Appl. Opt. 18, 3996–3999 (1979).
    [CrossRef] [PubMed]
  5. T. Fukazawa, Y. Fujita, “Polarization modulated transmission spectro-ellipsometry,” Rev. Sci. Instrum. 67, 1951–1955 (1996).
    [CrossRef]
  6. B. L. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
    [CrossRef]
  7. B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators” Opt. Eng. 41, 981–987 (2002).
    [CrossRef]
  8. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.
  9. G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Transmission 2- modulator generalized ellipsometry (2-MGE) measurements,” Appl. Opt. 41, 6555–6566 (2002).
    [CrossRef] [PubMed]
  10. G. E. Jellison, F. A. Modine, “Two modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
    [CrossRef]
  11. G. E. Jellison, F. A. Modine, “Two modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  12. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  13. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  14. C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
    [CrossRef]
  15. J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
    [CrossRef]
  16. C. M. Herzinger, “System and mathematical regression-based method utilizing optical data, for identifying optical axis orientation in material systems such as optical compensators and retarders,” U. S. Patent5,835,222 (10November1998).
  17. G. E. Jellison, L. A. Boatner, “Optical functions of uniaxial ZnO determined by generalized ellipsometry,” Phys. Rev. B 58, 3586–3589 (1998).
    [CrossRef]
  18. E. D. Palik “Lithium Niobate (LiNbO3)” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 695–702.
    [CrossRef]
  19. D. F. Nelson, R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688–3689 (1974).
    [CrossRef]
  20. J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
    [CrossRef]
  21. G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
    [CrossRef]
  22. N. W. Tibault, “Morphological and structural crystallography and optical properties of silicon carbide. Part II: Structural crystallography and optical properties,” Am. Mineral. 29, 327–362 (1944).
  23. P. T. B. Schaffer, “Refractive index, dispersion, and birefringence of silicon carbide polytypes,” Appl. Opt. 10, 1034–1036 (1971).
    [CrossRef]
  24. W. J. Choyke, E. D. Palik “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed., (Academic Press, New York, 1985), pp. 587–595.
    [CrossRef]
  25. M. Kildemo, “Optical properties of silicon carbide polytypes below and around bandgap,” Thin Solid Films, 455–456, 187–195 (2004).
    [CrossRef]

2004 (2)

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

M. Kildemo, “Optical properties of silicon carbide polytypes below and around bandgap,” Thin Solid Films, 455–456, 187–195 (2004).
[CrossRef]

2002 (2)

1999 (1)

B. L. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

1998 (2)

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

G. E. Jellison, L. A. Boatner, “Optical functions of uniaxial ZnO determined by generalized ellipsometry,” Phys. Rev. B 58, 3586–3589 (1998).
[CrossRef]

1997 (2)

1996 (1)

T. Fukazawa, Y. Fujita, “Polarization modulated transmission spectro-ellipsometry,” Rev. Sci. Instrum. 67, 1951–1955 (1996).
[CrossRef]

1992 (1)

1979 (1)

1978 (1)

1975 (1)

1974 (1)

D. F. Nelson, R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688–3689 (1974).
[CrossRef]

1971 (1)

1968 (1)

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

1965 (1)

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[CrossRef]

1944 (1)

N. W. Tibault, “Morphological and structural crystallography and optical properties of silicon carbide. Part II: Structural crystallography and optical properties,” Am. Mineral. 29, 327–362 (1944).

An, I.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Ashkin, A.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Ballman, A. A.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Berman, J. G.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Boatner, L. A.

G. E. Jellison, L. A. Boatner, “Optical functions of uniaxial ZnO determined by generalized ellipsometry,” Phys. Rev. B 58, 3586–3589 (1998).
[CrossRef]

Bond, W. L.

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge1997).

Chen, C.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Chipman, R. A.

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

Choyke, W. J.

W. J. Choyke, E. D. Palik “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed., (Academic Press, New York, 1985), pp. 587–595.
[CrossRef]

Collins, R. W.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Dziedzic, J. M.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Elman, J. F.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

Ferreira, G. M.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Fujita, Y.

T. Fukazawa, Y. Fujita, “Polarization modulated transmission spectro-ellipsometry,” Rev. Sci. Instrum. 67, 1951–1955 (1996).
[CrossRef]

Fukazawa, T.

T. Fukazawa, Y. Fujita, “Polarization modulated transmission spectro-ellipsometry,” Rev. Sci. Instrum. 67, 1951–1955 (1996).
[CrossRef]

Goldstein, D. H.

Greener, J.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

Griffiths, C. O.

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Transmission 2- modulator generalized ellipsometry (2-MGE) measurements,” Appl. Opt. 41, 6555–6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
[CrossRef]

Hauge, P. S.

Herzinger, C. M.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

C. M. Herzinger, “System and mathematical regression-based method utilizing optical data, for identifying optical axis orientation in material systems such as optical compensators and retarders,” U. S. Patent5,835,222 (10November1998).

Holcomb, D. E.

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Transmission 2- modulator generalized ellipsometry (2-MGE) measurements,” Appl. Opt. 41, 6555–6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
[CrossRef]

Jellison, G. E.

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Transmission 2- modulator generalized ellipsometry (2-MGE) measurements,” Appl. Opt. 41, 6555–6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, L. A. Boatner, “Optical functions of uniaxial ZnO determined by generalized ellipsometry,” Phys. Rev. B 58, 3586–3589 (1998).
[CrossRef]

G. E. Jellison, F. A. Modine, “Two modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
[CrossRef]

G. E. Jellison, F. A. Modine, “Two modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
[CrossRef]

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
[CrossRef]

Johs, B.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

Kildemo, M.

M. Kildemo, “Optical properties of silicon carbide polytypes below and around bandgap,” Thin Solid Films, 455–456, 187–195 (2004).
[CrossRef]

Levinstein, H. J.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Major, R. W.

Mikulyak, R. M.

D. F. Nelson, R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688–3689 (1974).
[CrossRef]

Modine, F. A.

Nelson, D. F.

D. F. Nelson, R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688–3689 (1974).
[CrossRef]

Oakberg, T. C.

B. L. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Palik, E. D.

W. J. Choyke, E. D. Palik “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed., (Academic Press, New York, 1985), pp. 587–595.
[CrossRef]

E. D. Palik “Lithium Niobate (LiNbO3)” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 695–702.
[CrossRef]

Podraza, N. J.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Rouleau, C. M.

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Transmission 2- modulator generalized ellipsometry (2-MGE) measurements,” Appl. Opt. 41, 6555–6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
[CrossRef]

Schaffer, P. T. B.

Smith, R. G.

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

Sonder, E.

Tibault, N. W.

N. W. Tibault, “Morphological and structural crystallography and optical properties of silicon carbide. Part II: Structural crystallography and optical properties,” Am. Mineral. 29, 327–362 (1944).

Wang, B. L.

B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators” Opt. Eng. 41, 981–987 (2002).
[CrossRef]

B. L. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge1997).

Wong, C. F.

Zapien, J. A.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

Am. Mineral. (1)

N. W. Tibault, “Morphological and structural crystallography and optical properties of silicon carbide. Part II: Structural crystallography and optical properties,” Am. Mineral. 29, 327–362 (1944).

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. G. Berman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92–94 (1968).
[CrossRef]

J. Appl. Phys. (2)

D. F. Nelson, R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688–3689 (1974).
[CrossRef]

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators” Opt. Eng. 41, 981–987 (2002).
[CrossRef]

Phys. Rev. B (1)

G. E. Jellison, L. A. Boatner, “Optical functions of uniaxial ZnO determined by generalized ellipsometry,” Phys. Rev. B 58, 3586–3589 (1998).
[CrossRef]

Rev. Sci. Instrum. (2)

T. Fukazawa, Y. Fujita, “Polarization modulated transmission spectro-ellipsometry,” Rev. Sci. Instrum. 67, 1951–1955 (1996).
[CrossRef]

B. L. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Thin Solid Films (3)

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films, 455–456, 14–23 (2004).
[CrossRef]

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially- stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
[CrossRef]

M. Kildemo, “Optical properties of silicon carbide polytypes below and around bandgap,” Thin Solid Films, 455–456, 187–195 (2004).
[CrossRef]

Other (6)

W. J. Choyke, E. D. Palik “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed., (Academic Press, New York, 1985), pp. 587–595.
[CrossRef]

C. M. Herzinger, “System and mathematical regression-based method utilizing optical data, for identifying optical axis orientation in material systems such as optical compensators and retarders,” U. S. Patent5,835,222 (10November1998).

E. D. Palik “Lithium Niobate (LiNbO3)” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 695–702.
[CrossRef]

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge1997).

G. E. Jellison, C. O. Griffiths, D. E. Holcomb, C. M. Rouleau, “Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer (2-MGE)” in Polarization Measurement, Analysis, and Application V, D. H. Goldstein, D. B. Chenault; eds., Proc. SPIE4819, 9–19 (2002).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the off-normal incidence measurement of birefringence. The incident light is at an angle of incidence of θ, while the c axis is at an angle γ with respect to the sample normal represented by the dashed line.

Fig. 2
Fig. 2

Retardation δ, fast axis angle ϕ, and the magnitude of the diattenuation |N| as a function of the angle of incidence for a ZnO crystal with the c axis nominally perpendicular to the surface. The sample was 0.558 mm thick, and the measurements were done at 633 nm. The solid curve in the top panel was a fit to the data, with the fit parameters as shown in the figure. The solid curve in the bottom panel is calculated from Eq. (7c) with using the refractive indices of Ref. 17.

Fig. 3
Fig. 3

Observed retardation and direction of the fast axis (given in degrees) of a c-cut ZnO crystal at various angles of incidence.

Fig. 4
Fig. 4

Birefringence of ZnO as a function of wavelength. The inset shows an expanded view near the isotropic point. The square data points are taken from Ref. 2.

Fig. 5
Fig. 5

Birefringence of LiNbO3 determined in this work and the data presented by Nelson and Mikulyak.19

Fig. 6
Fig. 6

Birefringence of 6H-SiC obtain in this work (solid curve), as well as from the work of Tibault,22 Schaffer,23 and Kildemo.25

Equations (11)

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

M = [ 1 C ϕ N S ϕ N 0 C ϕ N C ϕ 2 + S ϕ 2 C C ϕ S ϕ ( 1 C ) S ϕ S S ϕ N C ϕ S ϕ ( 1 C ) S ϕ 2 + C ϕ 2 C C ϕ S 0 S ϕ S C ϕ S C ] ,
δ = 2 π d Δ n / λ
S = ( 1 N 2 ) sin δ ,
C = ( 1 N 2 ) cos δ .
N 2 + S 2 + C 2 = 1 ,
Δ n obs = n e n o
1 n e 2 = cos 2 γ r n o 2 + sin 2 γ r n e 2 ,
δ = 2 π d n o sin 2 ( γ r ) λ cos ( θ 1 ) x 1 + x / 2 ( 1 + x ) 2 ,
R p = r p r p * = [ n 1 cos ( θ 0 ) n 0 cos ( θ 1 ) n 1 cos ( θ 0 ) + n 0 cos ( θ 1 ) ] 2 ,
R s = r s r s * = [ n 0 cos ( θ 0 ) n 1 cos ( θ 1 ) n 0 cos ( θ 0 ) + n 1 cos ( θ 1 ) ] 2 ,
| N | = | ( 1 R s ) 2 ( 1 R p ) 2 ( 1 R s ) 2 + ( 1 R p ) 2 | .

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