Interaction of liquid crystals with electromagnetic fields: Mauguin theorem, angular momentum conservation, and optical Fréedericksz transitions in twisted nematic liquid crystals

We study the propagation of an elliptically polarized light beam normally incident onto an arbitrarily oriented liquid crystal in steady-state conditions. The Euler-Lagrange equations for the molecular director and the equations describing the evolution of the beam polarization in the birefringent medium are derived from a unique variational principle, which is proved to be consistent with the geometric-optics approximation. The Hamiltonian formulation of the theory is studied in detail. The conservation of total angular momentum and total free energy in the process is derived from Noether's theorem, and the theory of the adiabatic invariants is used to obtain a new proof of Mauguin's theorem of crystal optics. The general analytical solution of the propagation problem is presented for the important case of pure twisted structures. It is proved that two particular solutions exist (called Mauguin's solutions) obeying Mauguin's theorem rigorously. Only these solutions may exhibit the occurrence of the optical Fr\'eedericksz transition. In general, multiple optical thresholds are found. An analytical formula to obtain the thresholds is also derived.