# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 2 NLC Equations  
Let us consider the propagation of a linearly, extraordinarily polarised, coherent light beam of wavenumber \(k_{0}\), wavelength \(\lambda_{0}=2\pi/k_{0}\), through a planar cell filled with fully oriented nematic liquid crystals. The beam is assumed to propagate down the cell in the \(Z\) direction, with its electric field \(E\) initially oscillating in the \(Y\) direction. The coordinate \(X\) then completes the coordinate triad. The refractive indices of the medium are \(n_{\parallel}\) for light polarised along the molecular director \(\mathbf{\hat{n}}\) (long axis of the molecules) and \(n_{\perp}\) for fields polarised orthogonal to it. The molecular director \(\mathbf{\hat{n}}\) corresponds to the optic axis of the positive uniaxial medium with \(n_{\parallel}>n_{\perp}\). A fundamental property of oriented NLC is the Freedericksz transition, whereby a threshold optical power (or electric voltage) is needed to rotate the NLC molecules and thus increase the refractive index when the initial \(\mathbf{\hat{n}}\) is orthogonally aligned to the electric field [13]. Since large optical powers are never desirable as they could lead to heating, two main approaches can be adopted to overcome the optical Freedericksz threshold and adjust (maximise) the nonlinear reorientation. One is to pre-tilt the NLC molecules in the \((Y,Z)\) plane at an angle \(\theta_{0}\) with respect to the \(Z\) direction by the application of an external low-frequency electric field \(E_{LF}\) so that milliwatt power beams can rotate the molecular director \(\mathbf{\hat{n}}\) and induce self-focusing [14]. A typical planar glass cell for the study of NLC solitary waves in the presence of an external voltage bias is sketched in Figure 1a; in this case, a small pre-orientation \(\delta\theta_{0}\) of the order of 1-2\({}^{\circ}\) is ensured when the planar surfaces are treated for molecular anchoring in order to overcome the electric Freedericksz transition. The alternative method is to chemically treat or mechanically rub the planar walls of the cell (parallel to the \((Y,Z)\) plane in Figure 1) so that the NLC molecules are anchored at a given orientation \(\theta_{0}\) with respect to \(Z\). Elastic forces then transfer the rotation into the bulk of the fluid dielectric, providing the homogeneous orientation of the sample. A beam propagating through the NLC and polarised with electric field in the principal \((Y,Z)\) plane can then reorient the molecules by an additional angle \(\phi\) from the pre-tilt so that the total orientation of the molecular director to \(Z\) becomes \(\theta=\theta_{0}+\phi\). With these assumptions and in the paraxial, slowly varying envelope approximation, the -dimensional equations governing the propagation of an extraordinarily polarised light beam (electric field in the \((Y,Z)\) plane) in a biased NLC sample (Figure 1a) can be cast as [7; 8; 33]  
\[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+2 ik_{0}n_{e}\Delta( \theta)\frac{\partial E}{\partial Y}+\
abla^{2}E\] \[+k_{0}^{2}\Big{[}n_{\perp}^{2}\cos^{2}\theta+n_{\parallel}^{2} \sin^{2}\theta-n_{\perp}^{2}\cos^{2}\theta_{0}-n_{\parallel}^{2}\sin^{2} \theta_{0}\Big{]}E=0, \tag{1}\]  
for the electric field of the beam and  
\[K\
abla^{2}\phi+\left[\frac{1}{4}\epsilon_{0}\Delta\epsilon|E|^{2}+\frac{1}{ 2}\Delta\epsilon_{LF}E_{LF}^{2}\right]\sin 2(\theta_{0}+\phi)=0, \tag{2}\]  
for the reorientational nonlinearity [7; 8; 16; 33].