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 nij of its N-matrix, which was first introduced by Jones. Its solution χ(z,χ0) represents a trajectory in the complex plane which is traversed as the distance z is increased, starting from an initial polarization χ0 at z = 0. A stereographic projection onto a tangent sphere produces the corresponding trajectory in the more-familiar Poincaré-sphere representation. The function χ(z,χ0) 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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