Abstract

The evolution of the Stokes parameters in optically anisotropic media is characterized by a set of coupled nonlinear first-order differential equations. The incident quasi-monochromatic plane-wave field is assumed to be statistically stationary and of arbitrary state of polarization. The optical medium is assumed to be linear, passive, and characterized by a differential Mueller matrix. It is shown that the set of these equations provides an efficient tool for the analysis of the propagation of partially polarized light in anisotropic media. As an example, we analyze the evolution of a beam of light propagating in a cholesteric liquid crystal. We also investigate how an additive temporal randomness on the differential Mueller matrix can modify the evolution of Stokes parameters in this medium.

© 1995 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. See, for example,A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  3. R. M. A. Azzam, J. Opt. Soc. Am. 68, 1756 (1978).
    [CrossRef]
  4. R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948).
    [CrossRef]
  5. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 1252 (1972); B. E. Merrill, R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 64, 731 (1974).
    [CrossRef]
  6. C. R. Menyuk, P. K. A. Wai, J. Opt. Soc. Am. B 11, 1288 (1994).
    [CrossRef]
  7. See, for example,I. C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
  8. C. Brosseau, R. Barakat, Opt. Commun. 84, 127 (1991).
    [CrossRef]
  9. K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).
  10. C. Brosseau, D. Bicout, Phys. Rev. E 50, 4997 (1994).
    [CrossRef]
  11. C. Brosseau, “Statistics of the normalized Stokes parameters for a Gaussian stochastic plane-wave field,” Appl. Phys. (to be published).
  12. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).
  13. See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

1994 (2)

1991 (1)

C. Brosseau, R. Barakat, Opt. Commun. 84, 127 (1991).
[CrossRef]

1987 (1)

K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).

1978 (1)

1972 (1)

1948 (1)

Azzam, R. M. A.

Barakat, R.

C. Brosseau, R. Barakat, Opt. Commun. 84, 127 (1991).
[CrossRef]

Bashara, N. M.

Bicout, D.

C. Brosseau, D. Bicout, Phys. Rev. E 50, 4997 (1994).
[CrossRef]

Brosseau, C.

C. Brosseau, D. Bicout, Phys. Rev. E 50, 4997 (1994).
[CrossRef]

C. Brosseau, R. Barakat, Opt. Commun. 84, 127 (1991).
[CrossRef]

C. Brosseau, “Statistics of the normalized Stokes parameters for a Gaussian stochastic plane-wave field,” Appl. Phys. (to be published).

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

Flannery, B. P.

See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Jones, R. C.

Khoo, I. C.

See, for example,I. C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).

Kim, K.

K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).

Mandel, L.

K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).

Menyuk, C. R.

Press, W. H.

See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Teukolshy, S. A.

See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Wai, P. K. A.

Wolf, E.

K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).

Wu, S. T.

See, for example,I. C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).

Yariv, A.

See, for example,A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

See, for example,A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

J. Opt. Soc. Am. (3)

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

K. Kim, L. Mandel, E. Wolf, J. Opt. Soc. Am. A 4, 43 (1987).

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

Opt. Commun. (1)

C. Brosseau, R. Barakat, Opt. Commun. 84, 127 (1991).
[CrossRef]

Phys. Rev. E (1)

C. Brosseau, D. Bicout, Phys. Rev. E 50, 4997 (1994).
[CrossRef]

Other (6)

C. Brosseau, “Statistics of the normalized Stokes parameters for a Gaussian stochastic plane-wave field,” Appl. Phys. (to be published).

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

See, for example,W. H. Press, B. P. Flannery, S. A. Teukolshy, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

See, for example,I. C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).

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

See, for example,A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (2)

Fig. 1
Fig. 1

(a) Trajectory, represented as a parametric plot of the 〈σ1〉 and 〈σ3〉 coordinates, that describes the propagation of the state of polarization in a uniform-pitch cholesteric liquid crystal [Eq. (7)], with g = 1 and 0 < αx3 < 12, starting from an incident circular polarization state (〈σ1i = 0, 〈σ2i = 0, 〈σ3i = 1) indicated by the arrow. The circle 〈σ12 + 〈σ32 = 1 is the circular boundary separating the physical (interior) and the nonphysical (exterior) states of polarization. (b) Same as (a) but with a random perturbation [Eq. (8)] added to Eq. (7). The matrix elements in Eq. (8) are a = 0, b = 0.04, c = 0.09, d = 0.09, and e = 0.02.

Fig. 2
Fig. 2

(a) Trajectory, represented as a parametric plot of the 〈σ1〉 and 〈σ2〉 coordinates, that describes the propagation of the state of polarization in a uniform-pitch cholesteric liquid crystal [Eq. (7)], with g = 1 and 0 < αx3 < 12, starting from an incident circular polarization state (〈σ1i = 0, 〈σ2i = 0, 〈σ3i = 1) indicated by the arrow. The circle 〈σ12 + 〈σ22 = 1 is the circular boundary separating the physical (interior) and the nonphysical (exterior) states of polarization. (b) Same as (a) but with a random perturbation [Eq. (8)] added to Eq. (7). The matrix elements in Eq. (8) are a = 0, b = 0.05, c = 0.09, d = 0.07, and e = 0.09.

Equations (13)

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d S ( x 3 ) d x 3 = mS ( x 3 ) .
d σ j d x 3 1 S 0 d S j d x 3 - σ j d S 0 S 0 d x 3 .
1 S 0 d S 0 d x 3 = m 00 + m 01 σ 1 + m 02 σ 2 + m 03 σ 3 ,
1 S 0 d S 1 d x 3 = m 10 + m 11 σ 1 + m 12 σ 2 + m 13 σ 3 ,
1 S 0 d S 2 d x 3 = m 20 + m 21 σ 1 + m 22 σ 2 + m 23 σ 3 ,
1 S 0 d S 3 d x 3 = m 30 + m 31 σ 1 + m 32 σ 2 + m 33 σ 3 .
d σ 1 d x 3 = m 10 + ( m 11 - m 00 ) σ 1 - m 01 σ 1 2 + m 12 σ 2 + m 13 σ 3 - m 02 σ 1 σ 2 - m 03 σ 1 σ 3 ,
d σ 2 d x 3 = m 20 + ( m 22 - m 00 ) σ 2 - m 02 σ 2 2 + m 21 σ 1 + m 23 σ 3 - m 01 σ 1 σ 2 - m 03 σ 2 σ 3 ,
d σ 3 d x 3 = m 30 + ( m 33 - m 00 ) σ 3 - m 03 σ 3 2 + m 31 σ 1 + m 32 σ 2 - m 01 σ 1 σ 3 - m 02 σ 2 σ 3 .
d P d x 3 = 1 P ( σ 1 d σ 1 d x 3 + σ 2 d σ 2 d x 3 + σ 3 d σ 3 d x 3 ) .
m ¯ = m + m R ( t ) ,
m = [ 0 0 0 0 0 0 0 g sin ( 2 α x 3 ) 0 0 0 g cos ( 2 α x 3 ) 0 - g sin ( 2 α x 3 ) - g cos ( 2 α x 3 ) 0 ] ,
m R ( t ) = [ a 0 0 0 0 b 0 e 0 0 c e 0 e e d ] ,

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