Nonlinear Optical Waves in Liquid Crystalline Lattices

Abstract

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 $\overleftrightarrow{\varepsilon \left( r \right)}$, describing the optical polarization of the medium, can be expressed as $\overleftrightarrow{\varepsilon} = \overleftrightarrow{R}^\dag \cdot \overleftrightarrow{\varepsilon} _{{\rm NLC}} \cdot \overleftrightarrow{R}$, with $\overleftrightarrow{\varepsilon}_{{\rm NLC}} = \left[ \varepsilon_\bot , \varepsilon_\bot , \varepsilon_\| \right] I, I_{ij} = \delta_{ij }$ ($\delta_{ij }$ is the Kronecker delta) and $\overleftrightarrow{R}(n)$ a rotation tensor. The steady-state director configuration is obtained as an extremal point of the action integral $\mathcal{I} = \smallint \mathcal{L} d\rm{x}d\rm{y}d\rm{z}$, whose density $\mathcal{L}$ defines the energy spent by the molecular system to hold a specific director configuration (Frank freeenergy formulation) [1]. The energy density $\mathcal{L}$ can be further expanded into elastic $\mathcal{L}_{\rm{el}}$ and electromagnetic $\mathcal{L}_{\rm{em}}$ terms: $\mathcal{L} = \mathcal{L}_{\rm{el}} + \mathcal{L}_{\rm{em}}$. The contribution $\mathcal{L}_{\rm{el}}$ can be evaluated in the framework of the elastic continuum theory and, in the single constant approximation [2], reads:

$$\mathcal{L}_{rm{el}} = \frac{1}{2}K\left[ {\left( {\nabla \cdot n} \right)^2 + \left( {n \cdot \nabla \times n} \right)^2 } \right],$$

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]:

$$\mathcal{L}_{rm{em}} = -\frac{{\Delta \varepsilon }}{2}\left\langle {n \cdot E} \right\rangle ,$$

being $\Delta \varepsilon = \varepsilon_{\|} - \varepsilon_{\bot}$ the NLC birefringence and $\left\langle \ldots \right\rangle$ 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 $\delta \mathcal{L} = 0$.