Abstract

There is a phase difference between s and p polarizations when a circularly polarized heterodyne light beam is reflected from a birefringent crystal. It can be measured accurately with a common-path heterodyne interferometric technique. We have derived an equation that describes the relationship between the phase differences and n e, n o, and α. Two groups of solutions for (n e, n o) can be obtained from this equation by the phase measurements performed at three incident angles under moderate conditions. Each group consists of three pairs of solutions for (n e, n o). Finally, by justifying with physical conditions, we obtained the correct solution for (n e, n o). Azimuth angle α of the birefringent crystal optical axis can also be determined. And the feasibility of this method is demonstrated.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. X. Wang, J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505–4508 (1992).
    [CrossRef] [PubMed]
  2. J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
    [CrossRef]
  3. I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
    [CrossRef]
  4. R. S. Weis, T. K. Gaylord, “Magnetooptic multilayered memory structure with a birefringent superstrate: a rigorous analysis,” Appl. Opt. 28, 1926–1930 (1989).
    [CrossRef] [PubMed]
  5. R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
    [CrossRef]
  6. M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1929 (1996).
    [CrossRef]
  7. Y. C. Huang, C. Chou, M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11–16 (1997).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam1989), pp. 269–363.
  9. M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
    [CrossRef]
  10. J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
    [CrossRef]
  11. G. E. Jellison, F. A. Modine, L. A. Boatner, “Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2),” Opt. Lett. 22, 1808–1810 (1997).
    [CrossRef]
  12. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  13. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
    [CrossRef]
  14. J. Y. Lee, D. C. Su, “A method for measuring Brewster’s angle by circularly polarized heterodyne interferometry,” J. Opt. 29, 349–353 (1998).
    [CrossRef]
  15. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1991), pp. 232–239.
  16. R. M. A. Azzam, N. M. Bashara, “Application of generalized ellipsometry to anisotropic crystals,” J. Opt. Soc. Am. 64, 128–133 (1974).
    [CrossRef]
  17. R. L. Burden, J. D. Faires, Numerical Analysis, 5th ed. (PWS-Kent, Boston, Mass., 1993), pp. 553–560.
  18. E. D. Palik, ed., Handbook of Optical Constants of Solids III (Academic, New York, 1998), p. 708.
  19. Ref. 18, p. 729.
  20. M. H. Chiu, J. Y. Lee, D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt. 38, 4047–4052 (1999).
    [CrossRef]

1999

1998

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

J. Y. Lee, D. C. Su, “A method for measuring Brewster’s angle by circularly polarized heterodyne interferometry,” J. Opt. 29, 349–353 (1998).
[CrossRef]

1997

1996

1992

1990

J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
[CrossRef]

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

1989

1974

Allison, D. B.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Application of generalized ellipsometry to anisotropic crystals,” J. Opt. Soc. Am. 64, 128–133 (1974).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam1989), pp. 269–363.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Application of generalized ellipsometry to anisotropic crystals,” J. Opt. Soc. Am. 64, 128–133 (1974).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam1989), pp. 269–363.

Boatner, L. A.

Burden, R. L.

R. L. Burden, J. D. Faires, Numerical Analysis, 5th ed. (PWS-Kent, Boston, Mass., 1993), pp. 553–560.

Chang, M.

Y. C. Huang, C. Chou, M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11–16 (1997).
[CrossRef]

Chen, C. D.

Chiu, M. H.

Chou, C.

Y. C. Huang, C. Chou, M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11–16 (1997).
[CrossRef]

D’Nelly, K. G.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Davis, J. A.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Eifler, A.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Faires, J. D.

R. L. Burden, J. D. Faires, Numerical Analysis, 5th ed. (PWS-Kent, Boston, Mass., 1993), pp. 553–560.

Gaylord, T. K.

George, M. C.

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

Hecht, J. D.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Herzinger, C. M.

Huang, Y. C.

Y. C. Huang, C. Chou, M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11–16 (1997).
[CrossRef]

Jellison, G. E.

Johs, B.

Kramer, V.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Krauss, G.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Lee, J. Y.

M. H. Chiu, J. Y. Lee, D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt. 38, 4047–4052 (1999).
[CrossRef]

J. Y. Lee, D. C. Su, “A method for measuring Brewster’s angle by circularly polarized heterodyne interferometry,” J. Opt. 29, 349–353 (1998).
[CrossRef]

Modine, F. A.

Moreno, I.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Perera, G. M.

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

Rheinlander, B.

Riede, V.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Schubert, M.

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[CrossRef]

Shukla, R. P.

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

Su, D. C.

Valencia, C. L.

J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
[CrossRef]

Valley, J. F.

J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
[CrossRef]

Venkateswarlu, P.

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

Wang, X.

Weis, R. S.

Woollam, J. A.

Wu, J. W.

J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
[CrossRef]

Yao, J.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1991), pp. 232–239.

Appl. Opt.

Appl. Phys. Lett.

J. F. Valley, J. W. Wu, C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084–1086 (1990).
[CrossRef]

J. Opt.

J. Y. Lee, D. C. Su, “A method for measuring Brewster’s angle by circularly polarized heterodyne interferometry,” J. Opt. 29, 349–353 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

R. P. Shukla, G. M. Perera, M. C. George, P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7–12 (1990).
[CrossRef]

Y. C. Huang, C. Chou, M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11–16 (1997).
[CrossRef]

Opt. Eng.

I. Moreno, J. A. Davis, K. G. D’Nelly, D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. B

J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037–7042 (1998).
[CrossRef]

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam1989), pp. 269–363.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1991), pp. 232–239.

R. L. Burden, J. D. Faires, Numerical Analysis, 5th ed. (PWS-Kent, Boston, Mass., 1993), pp. 553–560.

E. D. Palik, ed., Handbook of Optical Constants of Solids III (Academic, New York, 1998), p. 708.

Ref. 18, p. 729.

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 (2)

Fig. 1
Fig. 1

Schematic structure for the measurement of phase differences owing to reflection at a birefringent crystal: EO, electro-optic modulator; Q, quarter-wave plate; BC, birefringent crystal; AN t , analyzer; D t , photodetector; FG, function generator; PM, phasemeter.

Fig. 2
Fig. 2

Reflection at the surface of a birefringent crystal.

Tables (4)

Tables Icon

Table 1 Experimental Conditions and Measurement Results

Tables Icon

Table 2 Calculated Solutions and Results

Tables Icon

Table 3 Estimated Results and Their Average for Calcitea

Tables Icon

Table 4 Estimated Results and Their Average for Quartza

Equations (32)

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

Ei=Q0°EOωtE0=cosωt2-sinωt2=121iexpi ωt2+121-iexp-i ωt2.
Et=ANβSEi=ANβrpprpsrsprssEi =rpp cos β+rsp sin βcosωt2-rps cos β+rss sin βsinωt2cos βsin β,
rpp=A1A6+A2A5A1+A2,
rps=A1A2A4-A3A1+A2,
rsp=A6-A5A1+A2,
rss=A1A3+A2A4A1+A2,
A1=Csin2 θ+C cos θtan α,
A2=no tan αB+no cos θBno cos θ+C2,
A3=cos θ-Ccos θ+C,
A4=no cos θ-Bno cos θ+B,
A5=no2 cos θ-Cno2 cos θ+C,
A6=Bno cos θ-C2Bno cos θ+C2,
B2=no2ne2-sin2 θno2 sin2 α+no2 cos2 α,
C2=no2-sin2 θ.
It=|Er|2=I01+cosωt+ϕ,
I0=rpp cos β+rsp sin β2+rps cos β+rss sin β22,
ϕ=tan-12rpp cos β+rsp sin βrps cos β+rss sin βrpp cos β+rsp sin β2-rps cos β+rss sin β2.
ϕ=ϕne, no, α.
ϕ=tan-12rpprpsrpp2-rps2.
ϕ=tan-1sin 2β rpprssrpp2 cos2 β-rss2 sin2 β.
rpp=none cos θ-no2-sin2 θ1/2none cos θ+no2-sin2 θ1/2,
rss=cos θ-no2-sin2 θ1/2cos θ+no2-sin2 θ1/2.
rpp=no2 cos θ-no2-sin2 θ1/2no2 cos θ+no2-sin2 θ1/2,
rss=cos θ-ne2-sin2 θ1/2cos θ+ne2-sin2 θ1/2.
ϕ1=ϕ1ne, no, α,
ϕ2=ϕ2ne, no, α,
ϕ3=ϕ3ne, no, α.
|Δα|=1|dϕ/dα| |Δϕ|,
|Δϕ1|=ϕ1ne |Δne|+ϕ1no |Δno|,
|Δϕ2|=ϕ2ne |Δne|+ϕ2no |Δno|.
|Δne|=ϕ2no|Δϕ1|+ϕ1no|Δϕ2|ϕ1neϕ2no-ϕ2neϕ1no,
|Δno|=ϕ1ne|Δϕ1|+ϕ2ne|Δϕ2|ϕ1neϕ2no-ϕ2neϕ1no,

Metrics