Abstract

When a single complex variable χ is used to specify the ellipse of polarization of light passing through an anisotropic medium, a first-order, second-degree, ordinary differential equation (Riccati’s equation) governs the evolution of χ with distance along the direction of propagation. In this differential equation the properties of the medium are represented by the elements <i>n<sub>ji</sub></i> of its <i>N</i>-matrix, which was first introduced by Jones. Its solution χ(<i>Z</i>,χ<sub>0</sub>) represents a trajectory in the complex plane which is traversed as the distance <i>z</i> is increased, starting from an initial polarization χ0 at <i>z</i> = 0. A stereographic projection onto a tangent sphere produces the corresponding trajectory in the more-familiar Poincaré-sphere representation. The function χ (<i>Z</i>,χ<sub>0</sub>) has been determined for propagation along (i) an arbitrary direction in a homogeneous anisotropic medium, and (ii) the helical axis of a cholesteric liquid crystal. The solution in the first case provides a unified law that leads to all the rules for the use of the Poincaré sphere. For axial propagation in a cholesteric liquid crystal, it is found that two orthogonal polarizations are privileged in that the axes of their ellipses are forced to remain in alignment with the principal axes of birefringence of the molecular planes. The general solution (that satisfies the conditions of propagation) shows that the ellipse of polarization never repeats itself. As to the two parameters of the ellipse, the ellipticity is shown to be periodic with periodicity shorter than the pitch of the helical structure and the azimuth is aperiodic.

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  1. R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948), and the other papers of this series.
  2. G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, XXV, 1, edited by S. Flügge (Springer, Berlin, 1961). This article provides an extensive review of crystal optics.
  3. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972); 62, 336 (1972).
  4. R. M. A. Azzam and N. M. Bashara, Optics Commun. 4, 203 (1971); 5, 5 (1972).
  5. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  6. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, Mass., 1962).
  7. A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971).
  8. A. S. Marathay, Optics Commun. 3, 369 (1971).
  9. D. W. Berreman and J. J. Scheffer, Mol. Cryst. Liquid Cryst. 11, 395 (1970); J. Opt. Soc. Am. 62, 502 (1972).
  10. G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968).
  11. J. L. Fergason, in Liquid Crystals, co-ordinated by G. H. Brown, G. J. Dienes, and M. M. Labes (Gordon and Breach, New York, 1966), p. 89.
  12. R. M. A. Azzam and N. M. Bashara, Optics Commun. 5, 319 (1972).
  13. See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw–Hill, New York, 1961), p. 240.
  14. C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Cryst. 4, 219 (1951).
  15. Note that only forward-traveling waves are being considered. Reflections that may give rise to backward-traveling waves are assumed absent.
  16. The sign of a determines the handedness of the helical structure. Positive and negative values correspond to right- and left-handed structures, respectively.
  17. This result can be determined by the direct application of the definition in Eq. (1). The Jones matrix of a thin slice of thickness Δz located at z is obtained, in the circular complex-plane representation, from its linear eigenpolarizations eiz and -eiz whose eigenvalues are e½gz and egz, respectively.
  18. The dispersion in the two-parameter (α,g0) model of the cholesteric structure results from the dispersion of go which is given by g0=2πΔn/λ, where Δn is the difference between the principal refractive indices of the molecular layer and λ is the vacuum wavelength of light.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972); 62, 336 (1972).

R. M. A. Azzam and N. M. Bashara, Optics Commun. 4, 203 (1971); 5, 5 (1972).

R. M. A. Azzam and N. M. Bashara, Optics Commun. 5, 319 (1972).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Optics Commun. 5, 319 (1972).

R. M. A. Azzam and N. M. Bashara, Optics Commun. 4, 203 (1971); 5, 5 (1972).

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972); 62, 336 (1972).

Berreman, D. W.

D. W. Berreman and J. J. Scheffer, Mol. Cryst. Liquid Cryst. 11, 395 (1970); J. Opt. Soc. Am. 62, 502 (1972).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Conners, G. H.

G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968).

Fergason, J. L.

J. L. Fergason, in Liquid Crystals, co-ordinated by G. H. Brown, G. J. Dienes, and M. M. Labes (Gordon and Breach, New York, 1966), p. 89.

Jones, R. C.

R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948), and the other papers of this series.

Korn, G. A.

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw–Hill, New York, 1961), p. 240.

Korn, T. M.

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw–Hill, New York, 1961), p. 240.

Marathay, A. S.

A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971).

A. S. Marathay, Optics Commun. 3, 369 (1971).

Oseen, C. W.

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Cryst. 4, 219 (1951).

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, XXV, 1, edited by S. Flügge (Springer, Berlin, 1961). This article provides an extensive review of crystal optics.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, XXV, 1, edited by S. Flügge (Springer, Berlin, 1961). This article provides an extensive review of crystal optics.

Scheffer, J. J.

D. W. Berreman and J. J. Scheffer, Mol. Cryst. Liquid Cryst. 11, 395 (1970); J. Opt. Soc. Am. 62, 502 (1972).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, Mass., 1962).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Other (18)

R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948), and the other papers of this series.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik, XXV, 1, edited by S. Flügge (Springer, Berlin, 1961). This article provides an extensive review of crystal optics.

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972); 62, 336 (1972).

R. M. A. Azzam and N. M. Bashara, Optics Commun. 4, 203 (1971); 5, 5 (1972).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, Mass., 1962).

A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971).

A. S. Marathay, Optics Commun. 3, 369 (1971).

D. W. Berreman and J. J. Scheffer, Mol. Cryst. Liquid Cryst. 11, 395 (1970); J. Opt. Soc. Am. 62, 502 (1972).

G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968).

J. L. Fergason, in Liquid Crystals, co-ordinated by G. H. Brown, G. J. Dienes, and M. M. Labes (Gordon and Breach, New York, 1966), p. 89.

R. M. A. Azzam and N. M. Bashara, Optics Commun. 5, 319 (1972).

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw–Hill, New York, 1961), p. 240.

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Cryst. 4, 219 (1951).

Note that only forward-traveling waves are being considered. Reflections that may give rise to backward-traveling waves are assumed absent.

The sign of a determines the handedness of the helical structure. Positive and negative values correspond to right- and left-handed structures, respectively.

This result can be determined by the direct application of the definition in Eq. (1). The Jones matrix of a thin slice of thickness Δz located at z is obtained, in the circular complex-plane representation, from its linear eigenpolarizations eiz and -eiz whose eigenvalues are e½gz and egz, respectively.

The dispersion in the two-parameter (α,g0) model of the cholesteric structure results from the dispersion of go which is given by g0=2πΔn/λ, where Δn is the difference between the principal refractive indices of the molecular layer and λ is the vacuum wavelength of light.

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