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We show that the optical structure of the helical phase of a chiral nematicis naturally associated with the Bianchi VII(0) group manifold, of which wegive a full account. The Joets-Ribotta metric governing propagation of theextraordinary rays is invariant under the simply transitive action of theuniversal cover of the three dimensional Euclidean group of two dimensions.Thus extraordinary light rays are geodesics of a left-invariant metric on thisBianchi type VII(0) group. We are able to solve by separation of variables boththe wave equation and the Hamilton-Jacobi equation for this metric. The formerreduces to Mathieu's equation and the later to the quadrantal pendulumequation. We discuss Maxwell's equations for uniaxial optical materials wherethe configuration is invariant under a group action and develop a formalism totake advantage of these symmetries. The material is not assumed to be impedancematched, thus going beyond the usual scope of transformation optics. We showthat for a chiral nematic in its helical phase Maxwell's equations reduce to ageneralised Mathieu equation. Our results may also be relevant to helicalphases of some magnetic materials and to light propagation in certaincosmological models.

The helical phase of chiral nematic liquid crystals as the Bianchi VII(0) group manifold