## Abstract

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

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### Equations (19)

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(1)
$$d\chi /dz={n}_{12}{\chi}^{2}+({n}_{22}-{n}_{11})\chi +{n}_{21},$$
(2)
$$\chi (z,{\chi}_{0})=\{\left[\beta -\frac{1}{2}({n}_{11}-{n}_{22})\hspace{0.17em}\text{tan}\beta z\right]{\chi}_{0}+({n}_{21}\hspace{0.17em}\text{tan}\beta z)\}/\{({n}_{12}\hspace{0.17em}\text{tan}\beta z){\chi}_{0}+\left[\beta +\frac{1}{2}({n}_{11}-{n}_{22})\hspace{0.17em}\text{tan}\beta z\right]\},$$
(3)
$$\beta ={\left[-\frac{1}{4}{({n}_{11}-{n}_{22})}^{2}-{n}_{12}{n}_{21}\right]}^{1/2}$$
(4)
$$\overline{\chi}(z,{\chi}_{0})=\frac{[\beta -i\alpha \hspace{0.17em}\text{tan}\beta z){\chi}_{0}+(-i{g}_{0}\hspace{0.17em}\text{tan}\beta z)]}{[(-i{g}_{0}\hspace{0.17em}\text{tan}\beta z){\chi}_{0}+(\beta +i\alpha \hspace{0.17em}\text{tan}\beta z)]\hspace{0.17em}\text{exp}(i2\alpha z)},$$
(6)
$$\beta ={g}_{0}/{[{(\alpha /{g}_{0})}^{2}+1]}^{1/2}.$$
(7)
$$\text{tan}\beta z=i[1-\text{exp}(i2\beta z)]/[1+\text{exp}(i2\beta z)],$$
(8)
$$\chi (z,{\chi}_{0})=[A+B\hspace{0.17em}\text{exp}(i2\beta z)]/[C+D\hspace{0.17em}\text{exp}(i2\beta z)],$$
(9)
$$\begin{array}{l}A=i{n}_{21}+\left[\beta -\frac{1}{2}i({n}_{11}-{n}_{22})\right]\hspace{0.17em}{\chi}_{0},\\ B=-i{n}_{21}+\left[\beta +\frac{1}{2}i({n}_{11}-{n}_{22})\right]\hspace{0.17em}{\chi}_{0},\\ C=\left[\beta +\frac{1}{2}i({n}_{11}-{n}_{22})\right]\hspace{0.17em}+i{n}_{12}{\chi}_{0},\\ D=\left[\beta -\frac{1}{2}i({n}_{11}-{n}_{22})\right]\hspace{0.17em}-i{n}_{12}{\chi}_{0}.\end{array}$$
(10)
$${\chi}_{e1}=A/C,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\chi}_{e2}=B/D.$$
(11)
$${\chi}_{e1,2}=(1/2{n}_{12})\{({n}_{22}-{n}_{11})\pm {[{({n}_{22}-{n}_{11})}^{2}+4{n}_{12}{n}_{21}]}^{1/2}\}.$$
(13)
$${\chi}_{f}=A/C={\chi}_{e1},$$
(14)
$${\chi}_{f}=B/D={\chi}_{e2}.$$
(15)
$$\text{Im}\hspace{0.17em}\left[\frac{1}{4}{(\text{Tr}N)}^{2}-(\text{Det}N)\right]=0,$$
(16)
$${N}_{\text{eb}}=\left[\begin{array}{ll}A\hfill & B\hfill \\ -{B}^{*}\hfill & {A}^{*}\hfill \end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{N}_{\text{ed}}=\left[\begin{array}{ll}p\hfill & R\hfill \\ {R}^{*}\hfill & q\hfill \end{array}\right],$$
(17)
$$u(z,{u}_{0})=[{A}^{\prime}+{B}^{\prime}\hspace{0.17em}\text{exp}(i2\beta z)]/[{C}^{\prime}+{D}^{\prime}\hspace{0.17em}\text{exp}(i2\beta z)],$$
(18)
$$\begin{array}{l}{A}^{\prime}={g}_{0}+(\alpha +\beta ){\chi}_{0},\\ {B}^{\prime}=-{g}_{0}-(\alpha -\beta ){\chi}_{0},\\ {C}^{\prime}=-(\alpha -\beta )+{g}_{0}{\chi}_{0},\\ {D}^{\prime}=(\alpha +\beta )-{g}_{0}{\chi}_{0}.\end{array}$$
(19)
$$\begin{array}{l}\text{Re}{\chi}_{0}=\text{Re}{u}_{0}=\frac{1}{2}({k}_{1}+{k}_{2})\\ =\alpha /{g}_{0}\end{array}$$