# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 2 NLC Equations
Here, the \(\theta\)-dependent extraordinary refractive index of the NLC is  
\[n_{e}=\left[\frac{n_{\perp}^{2}n_{\parallel}^{2}}{n_{\parallel}^{2}\cos^{2} \theta+n_{\perp}^{2}\sin^{2}\theta}\right]^{1/2}. \tag{3}\]  
An extraordinary wave undergoes walkoff of the Poynting vector so that the energy flux propagates in the \((Y,Z)\) plane at an angle \(\delta=\tan^{-1}\Delta\) to the wavevector (i.e., the \(Z\) direction in Figure 1), where \(\Delta\) is given by  
\[\Delta(\theta)=\frac{\Delta(\theta)\epsilon\sin 2\theta}{\Delta(\theta) \epsilon+2n_{\perp}^{2}+\Delta(\theta)\epsilon\cos 2\theta}. \tag{4}\]  
In the above equations, \(\Delta\epsilon=n_{\parallel}^{2}-n_{\perp}^{2}\) is the optical anisotropy, \(\Delta\epsilon_{LF}\) is the low-frequency dielectric anisotropy and \(\epsilon_{0}\) is the electrical permittivity of free space. In addition, \(K\) is the elastic (Frank) constant in the scalar approximation for which the strengths of bend, twist and splay deformations are taken to be equal [7; 13]. Finally, the Laplacians \(\
abla^{2}E\) and \(\
abla^{2}\phi\) are in the transverse coordinates \((X,Y)\). Figure 1: Sketch of planar NLC glass cells in the two main configurations. (**a**) Biased cell: The orientation angle \(\theta_{0}\) is the pre-tilt induced by the external voltage \(V\) across the cell thickness, and \(\phi\) is the all-optical rotation of the induced dipoles (ellipses) in the plane of propagation \((Y,Z)\). \(\hat{\textbf{n}}\) is the molecular director (optic axis) parallel to the long axis of the elongated NLC molecules. (**b**) Bias-free cell: the background orientation \(\theta_{0}\) is obtained by anchoring the molecular director at the planar boundaries parallel to \((Y,Z)\). In both arrangements (**a**,**b**), the input light beam undergoing nonlinear reorientation is an extraordinary-wave launched from the left with the electric field (double red arrow) polarised in the principal plane \((Y,Z)\). The NLC Equations (1) and (2) are highly nonlinear and thus difficult to analyse. However, for continuous-wave milliwatt power beams, the light-induced response \(\phi\) is small, \(|\phi|\ll|\theta_{0}|\), so the model can be expanded in Taylor series around \(\theta_{0}\). In addition, these equations can be recast in a dimensionless form to reduce the number of parameters involved by using typical length scales \(L_{Z}\) and \(W\) down and across the cell, respectively, as well as an amplitude scale \(A_{b}\) for the electric field of the beam. Then,  
\[Z=L_{Z}z,\quad X=Wx,\quad Y=Wy,\quad E=A_{b}u. \tag{5}\]  
Here, \((x,y,z)\) is the non-dimensional coordinate system, and \(u\) is the non-dimensional electric field of the beam. Suitable scales are [8; 34]  
\[L_{Z}=\frac{4n_{e}}{k_{0}\Delta e\sin 2\theta_{0}},\quad W=\frac{2}{k_{0} \sqrt{\Delta e\sin 2\theta_{0}}},\quad A_{b}^{2}=\frac{2P_{b}}{\pi\Gamma W_{b}^{2}}, \quad\Gamma=\frac{1}{2}e_{0}cn_{e} \tag{6}\]  
based on a Gaussian input wavepacket of power \(P_{b}\), amplitude \(A_{b}\) and width \(W_{b}\). The simplified non-dimensional equations governing beam propagation and the NLC response become  
\[i\frac{\partial u}{\partial z}+i\gamma\Delta(\theta_{0}+\phi) \frac{\partial u}{\partial y}+\frac{1}{2}\
abla^{2}u+2\phi u = 0, \tag{7}\] \[\
u\
abla^{2}\phi-2\phi q = -2|u|^{2}. \tag{8}\]  
Here, the dimensionless elasticity \(\
u\) and pre-tilt parameter \(q\) (when present) are given by  
\[+k_{0}^{2}\Big{[}n_{\perp}^{2}\cos^{2}\theta+n_{\parallel}^{2} \sin^{2}\theta-n_{\perp}^{2}\cos^{2}\theta_{0}-n_{\parallel}^{2}\sin^{2} \theta_{0}\Big{]}E=0, \tag{1}\]0  
Finally, the non-dimensional walkoff factor \(\gamma\) is  
\[+k_{0}^{2}\Big{[}n_{\perp}^{2}\cos^{2}\theta+n_{\parallel}^{2} \sin^{2}\theta-n_{\perp}^{2}\cos^{2}\theta_{0}-n_{\parallel}^{2}\sin^{2} \theta_{0}\Big{]}E=0, \tag{1}\]1  
Note that the bias-free case (Figure 1b) corresponds to \(q=0\). The model in Equations (7) and (8) is a focusing nonlocal, nonlinear Schrodinger (NLS) equation-type system with a refractive index increasing with the intensity \(|u|^{2}\).