# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 5 Interaction of Localised Beams with Dielectric Perturbations  
The trajectories of optical solitary waves in NLCs can also be manipulated by localised perturbations--defects--of the refractive index, acting in a manner similar to a lens [7; 89; 90]. There are a number of techniques to generate refractive defects as the orientation of NLC molecules, and so the refractive index can be controlled via numerous mechanisms, including applied electric fields [21; 23; 25; 30; 91; 92], dye-doping [93; 94; 95; 96; 24], polymer dispersion/doping [97; 98], external magnetic fields [99; 100; 101], disinclinations and topological structures [102; 103] and other (incoherent) light beams [22; 24; 89; 104]. However, since a nematic is a wavepacket with an extended profile, it can interact with the localised defect/lens even if its peak is somewhat distant from it--similar to the case of total internal reflection discussed in Section 4. As remarked before, NLCs are nonlocal, nonlinear media for which the reorientation extends well beyond the beam waist forcing it. Let us consider a refractive perturbation of the background director orientation \(\phi_{b}(x,y,z)\) about the uniform pre-tilt \(\theta_{0}\) in a region denoted by \(\Omega\). As the director orientation is linked to the extraordinary refractive index \(n_{e}\), see Equation (3), a light beam can be refracted when passing through, or near, such a localised defect in \(n_{e}\) and modify its trajectory. By adding this index perturbation, the NLC model in Equations (7) and (8) becomes  
\[i\frac{\partial u}{\partial z}+i\gamma\Delta(\theta_{0}+\phi_{b} )\frac{\partial u}{\partial y}+\frac{1}{2}\
abla^{2}u+2\phi_{b}u+2\phi u = 0, \tag{23}\] \[\
u\
abla^{2}\phi-2q\phi = -2|u|^{2}. \tag{24}\]  
Note that the nonlinear term in the electric field equation is of the form \(\Delta nu\), where \(\Delta n\) is the light-induced change in refractive index from its pre-tilt value. As stated above, the dielectric defect acts as a lens to modify the beam trajectory. Nematicon refraction by localised index perturbations induced by an external electric field (of constant value \(u_{0}\) inside the region \(\Omega\) and \(0\) otherwise) was studied by full numerical solutions of Equations (23) and (24) and MT [27] using circular, elliptical and rectangular domains \(\Omega\). The input beam was taken as  
\[u=af(\rho)e^{i\
u+i\Omega(y-\xi)},\quad\rho=\frac{\sqrt{x^{2}+(y-\xi)^{2}}}{w} \tag{25}\]  
with profile \(f(\rho)\) either Gaussian or sech, obtaining comparable results, as expected from previous studies on the role of the beam profile on nematicon propagation [67]. The example of an elliptical refractive index change can illustrate the findings in [27]. The defect can be generated by an applied low-frequency electric field of the form  
\[u_{b}=\left\{ \begin{array}{ll}u_{0},&\Gamma=\frac{(y-Y_{\Omega})^{2}}{R_{y} ^{2}}+\frac{(z-Z_{\Omega})^{2}}{R_{z}^{2}}\leq 1,\\ 0,&\Gamma>1,\end{array} \right. \tag{26}\]  
providing the desired background orientation of the optic axis \(\hat{\mathbf{n}}\). Figure 6 compares the nematicon trajectory \(y=\xi(z)\) as given by full numerical solutions of the NLC model in Equations (23) and (24) with the initial condition defined in Equation (25) and by the modulation theory. It can be seen that MT gives a good prediction of the nematicon trajectory as it is refracted by the index change. These results show that the trajectory of a nematicon can be controlled by adjusting the strength of the dielectric defect, similar to adjusting a lens controls the path of light. The study of nematicon refraction in proximity to dielectric defects was extended to nematicon paths interacting with the perturbation itself [105] based on the assumed additional orientation  
\[\theta_{b}=a_{b}e^{-\left[(y-Y_{b})^{2}+(z-Z_{b})^{2}\right]/w_{b}^{2}} \tag{27}\]  
and the input wavepacket in Equation (25).