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Applied Optics

Applied Optics


  • Vol. 12, Iss. 4 — Apr. 1, 1973
  • pp: 764–771

Trajectories Describing the Evolution of Polarized Light in Homogeneous Anisotropic Media and Liquid Crystals

R. M. A. Azzam, B. E. Merrill, and N. M. Bashara  »View Author Affiliations

Applied Optics, Vol. 12, Issue 4, pp. 764-771 (1973)

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Trajectories are given that describe the evolution of the ellipse of polarization in the complex plane for light propagating in a homogeneous anisotropic medium and along the helical axis of a cholesteric liquid crystal. For the general homogeneous anisotropic medium that exhibits combined birefringence and dichroism the trajectory is a spiral that converges to the low-absorption eigenpolarization. For pure birefringence the trajectory becomes a complete circle that encloses one eigenpolarization, whereas for pure dichroism the trajectory becomes an arc of a circle that ends at the low-absorption eigenstate. The case of a cholesteric (or twisted nematic) liquid crystal leads to an interesting family of trajectories that can be considered as distorted hypo- or epicycloids. These trajectories are nonrepetitive (open) and may show multilobes or branches depending upon the initial polarization and the properties of the liquid crystal. Graphs are also presented where the ellipticity and azimuth are plotted separately as functions of distance along the helical axis.

© 1973 Optical Society of America

Original Manuscript: August 14, 1972
Published: April 1, 1973

R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, "Trajectories Describing the Evolution of Polarized Light in Homogeneous Anisotropic Media and Liquid Crystals," Appl. Opt. 12, 764-771 (1973)

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  1. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 1252 (1972). [CrossRef]
  2. R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948). [CrossRef]
  3. The cholesteric liquid crystal problem has also been treated by C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Crystallogr. 4, 219 (1951); G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968); D. W. Berreman, T. J. Scheffer, Mol. Crystallogr. Liquid Crystallogr. 11, 395 (1970), J. Opt. Soc. Am. 62, 502 (1972); A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971), Optics Commun. 3, 369 (1971). This is not an exhaustive list. [CrossRef]
  4. The proof follows from finding the ratios A/C and B/D using Eq. (8) and noticing that the condition of their independence of χ0 leads to the definition of β in Eq. (3). That these ratios give the eigenpolarizations can be checked by substitution in Eq. (7).
  5. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972). [CrossRef]
  6. R. M. A. Azzam, N. M. Bashara, Optics Commun. 5, 319 (1972). [CrossRef]
  7. A conformal transformation preserves the shape of any curve in the immediate vicinity of a point in the complex plane. The spiraling behavior of exp(i2βz) around the origin (or the point at infinity) is therefore reproduced by χ(z,χ0) around the low-absorption eigenpolarization.
  8. P. K. Rees, Analytic Geometry (Prentice-Hall, Englewood Cliffs, N. J., 1970), p. 177.

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