Nonlinear optical waves in liquid crystalline lattices

Liquid crystals (LC) are molecular dielectrics encompassing several properties of both liquids and solids; in particular, they are often characterized by an order parameter which can be employed to distinguish among possible LC phases. In the nematic phase, liquid crystals show a significant degree of orientational order, their elongated organic molecules being aligned in a mean direction in space, as described by a vectorial field n called director. Sincemost nematics are derivative of benzene, they feature "cigar-like" molecules; hence, the macroscopic system can be regarded as an optically uniaxial crystalline fluid. The dielectric tensor , describing the optical polarization of the medium, can be expressed as , with ( is the Kronecker delta) and a rotation tensor. The steady-state director configuration is obtained as an extremal point of the action integral , whose density defines the energy spent by the molecular system to hold a specific director configuration (Frank freeenergy formulation) [1]. The energy density can be further expanded into elastic and electromagnetic terms: . The contribution can be evaluated in the framework of the elastic continuum theory and, in the single constant approximation [2], reads: with K accounting for elastic deformations ([K] = N). The electromagnetic contribution can be calculated by considering that the electric field induces dipoles on the nematic liquid crystal (NLC) molecules; the latter are then subjected to a torque and change their angular orientation towards a minimum energy configuration (e.g., parallel to the applied field). The contribution describing such reorientation process is [2]: being the NLC birefringence and denoting a square time average. The balance between field-induced reorientation and elastic interactions gives rise to the steady state distribution n, found as an extremal of the action integral .