Helical phase of chiral nematic liquid crystals as the Bianchi VII0group manifold

Abstract

We show that the optical structure of the helical phase of a chiral nematic is naturally associated with the Bianchi VII(0) group manifold, of which we give a full account. The Joets-Ribotta metric governing propagation of the extraordinary rays is invariant under the simply transitive action of the universal cover E(2) of the three-dimensional Euclidean group of two dimensions. Thus extraordinary light rays are geodesics of a left-invariant metric on this Bianchi type VII(0) group. We are able to solve, by separation of variables, both the wave equation and the Hamilton-Jacobi equation for this metric. The former reduces to Mathieu's equation, and the latter to the quadrantal pendulum equation. We discuss Maxwell's equations for uniaxial optical materials where the configuration is invariant under a group action and develop a formalism to take advantage of these symmetries. The material is not assumed to be impedance matched, thus going beyond the usual scope of transformation optics. We show that for a chiral nematic in its helical phase Maxwell's equations reduce to a generalized Mathieu equation. Our results may also be relevant to helical phases of some magnetic materials and to light propagation in certain cosmological models.

Helical Superconducting Black Holes

“…It is natural to next use the AdS/CFT correspondence to analyse transport. Based on the rich optical properties of other helical orders (such as the chiral nematic phase of liquid crystals, recently discussed in [14]) we expect interesting results.…”

Eisenhart lifts and symmetries of time-dependent systems

“…A further generalization would be to let the fixed point move, in which case we would replace SO(3) by the Euclidean group E(3). The Bianchi group V II 0 has been applied to the optical geometry of the ground state of chiral nematics [38]. The time-dependent theory might well have relevance in that case as well.…”

Nontrivial causal structures engendered by knotted solitons

“…Waves are described by pencils of null geodesics scattered by the corresponding geometry and it is possible that they remain trapped for some regions of the effective spacetime. In order to understand better these aspects, it would be interesting to analyse in more details the geometrical and topological properties of the above metrics, including their symmetries (see [30] for a Bianchi classification in the helical phase of chiral nematic liquid crystals).…”