# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 5 Interaction of Localised Beams with Dielectric Perturbations
If the beam waist is small compared with the defect width, then self-localisation is preserved and the solitary wave refracts similarly to the case of propagation around the defect [27]. However, if the beam and defect have comparable sizes, a large enough refractive index contrast can pull apart the nematicon, as is apparent in Figure 7, where the latter breaks up into two beams. These two beams are the counterparts of the caustics forming in the linear regime [105]. The work summarised above was extended to experimental and theoretical studies in unbiased planar samples with a background director orientation \(\theta_{0}\) varying across the transverse \(Y\) direction but uniform in the down cell \(Z\) direction [106; 107]. The NLC  
Figure 6: Comparison of nematicon trajectories \(y=\xi(z)\) nearing an elliptical region \(\Omega\) of differing refractive indexes in an NLC cell prepared as in Figure 1b. The initial condition is Equation (25). Numerical solutions of Equations (23) and (24): red line; solution of MT equations [27]: dashed green line. The initial condition is \(f(\rho)=\mathrm{sech}\,\rho\), \(a=2.5\), \(w=2.0\), \(U=-0.1\), \(\xi=-10.0\) with \(\
u=200\) and \(q=2\). The defect parameters are \(u_{0}=1\), \(Y_{\Omega}=0\), \(Z_{\Omega}=20\) and \(R_{y}=5\), \(R_{z}=9\). The boundary of \(\Omega\) is given by the pink line. Reproduced with permission from the American Physical Society. All rights reserved [27], doi: [[http://dx.doi.org/10.1103/PhysRevA.82.053843](http://dx.doi.org/10.1103/PhysRevA.82.053843)]([http://dx.doi.org/10.1103/PhysRevA.82.053843](http://dx.doi.org/10.1103/PhysRevA.82.053843)). Figure 7: Evolution of \(|u|\) in the plane \(x=0\), as given by numerical solution of Equations (23) and (24) in a sample as in Figure 1b. The initial condition is Equation (25) with \(f(\rho)=\mathrm{sech}\,\rho\), \(w=3.0\), \(U=-0.05\), \(\xi=2.0\) with \(\
u=400\) and \(q=2\). The defect parameters are \(a_{b}=0.5\), \(w_{b}=3.0\), \(Y_{b}=0\) and \(Z_{b}=30\). Reproduced with permission from the American Physical Society. All rights reserved [105], doi: [[http://dx.doi.org/10.1103/PhysRevA.85.013804](http://dx.doi.org/10.1103/PhysRevA.85.013804)]([http://dx.doi.org/10.1103/PhysRevA.85.013804](http://dx.doi.org/10.1103/PhysRevA.85.013804)). Equations (23) and (24) govern such propagation with \(q=0\) since in the experiments [106], the NLC molecules were pre-tilted by the physical treatment of the glass plates (Figure 1b). The modulation theory used to model the experiments was based on the beam and director distributions  
\[u = af(\rho)e^{i\sigma+iU(z)(y-\xi(z))},\quad\rho=\frac{\sqrt{x^{2}+(y- \xi(z))^{2}}}{w}\] \[\theta = \pi f^{2}(\mu),\quad\mu=\frac{\sqrt{x^{2}+(y-\xi(z))^{2}}}{\beta}. \tag{28}\]  
Here, \(a\) and \(w\) are the amplitude and width of the beam and \(\alpha\) and \(\beta\) are the amplitude and width of the director response to the optical forcing. The variable \(\xi(z)\) is the \(y\) position of the nematicon peak as it travels down the cell with \(z\) increasing, \(y=\xi(z)\) as it evolves in the \((y,z)\) plane, with fixed \(x=0\), and \(U(z)\) is the nematicon \(y\) \"velocity\"--physically, the angle it makes with the \(z\) direction in the \((y,z)\) plane. We underline that the beam profile is not specified above. It was found that if the length scale of the refractive index change is larger than the beam width, the MT results are independent of the profile and the MT equations reduce to momentum conservation equations for the wavepacket trajectory \(\xi(z)\), as for interacting nematicons [51]. This is important since, as stated in Section 4, there are no known exact nematicon solutions on which to base the modulation theory. Figure 8 compares experimental and MT results (momentum conservation) for the nematicon trajectory, with an excellent match.