# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 4 Refraction and Reflection of Self-Guided Beams at Interfaces
However, since the nematicon transverse profile can exist on both sides of the interface, the beam can penetrate the less optically dense region, turn around and re-enter the denser NLC: this is denoted as Goos-Hanchen TIR [76] in Figure 3. Reflection type 4 illustrates TIR for which the nematicon bends towards the incidence region without its centre \"touching\" the interface, whilst its tail enters the less dense region. Figure 4 shows experimental results for the refraction from a less to a more optically dense medium and total internal reflection from a less to a more optically dense medium of a nematicon at an interface to illustrate these refraction and reflection regimes. The refraction of a nematicon going from a less to a more optically dense NLC region resembles that of linear waves [28; 75]. Modulation theory was developed to model nematicon refraction at interfaces, as governed by the NLC Equations (17) and (18) [75; 77]. This model essentially treats the nematicon as an equivalent particle in a varying medium (mechanical potential) [12; 78], the essential concept behind the term \"soliton\". This work [75; 77] found excellent agreement between these MT solutions and full numerical solutions of Equations (17) and (18), including the incidence angles separating the various refraction and reflection types illustrated in Figure 3. The pertinent initial conditions for this MT model are  
\[u=a\,\text{sech}\,\frac{\sqrt{\chi^{2}+(y-\xi)^{2}}}{w}e^{i\sigma+iU(y-\xi)}, \quad\phi=\alpha\,\text{sech}^{2}\,\frac{\sqrt{\chi^{2}+(y-\xi)^{2}}}{\beta}. \tag{21}\]  
As for the initial condition in Equation (14) for spiralling nematicons, \(U\) is related to the input angle of the beam in the \((y,z)\) plane and can be considered the input \(y\) \"velocity\" of the beam, which is inputted at \(x=0\) and \(y=\xi\) at \(z=0\). The analysis of NLCs' self-confined beams refracting at an interface was further extended to optical vortex beams (i.e., propagating vortices rather than those discussed in [79]) based on the input wavepacket [80]  
\[u=are^{-r/w}e^{i\sigma+iU(y-\xi)+i\varphi}, \tag{22}\]  
which is a vortex of (topological) charge one [6], with \((r,\varphi)\) being plane polar coordinates and \(U\) and \(\xi\) having the same meaning as for the nematicon initial condition in  
Figure 4: Observation of a near-infrared 4.5 mW nematicon interacting with a voltage-controlled dielectric interface in NLCs as it travels from the left to the right. (**Left**): Refraction of the nematicon going towards a denser region; (**Right**): total internal reflection of the nematicon propagating towards a less dense region at an incidence angle exceeding the critical value. The opposite values of the applied voltage difference \(\Delta V\) marked above the panels are consistent with the opposite refractive index contrast imposed between the two dielectric regions in the two cases. Reproduced with permission from Springer Nature. All rights reserved [28], doi: [[https://doi.org/10.1038/nphys427](https://doi.org/10.1038/nphys427)]([https://doi.org/10.1038/nphys427](https://doi.org/10.1038/nphys427)). Equation (21). In local media, optical vortices are unstable to a mode 2 azimuthal instability, but they stabilise in sufficiently nonlocal media, such as nematic liquid crystals [81; 82; 83] and thermo-optic media [73], including dye-doped NLCs [84]. Optical vortices share similar refraction/reflection at a NLC dielectric interface as nematicons, including Snell's Law type and TIR. The major difference is that a vortex is less stable than a nematic; hence, its deformation upon refraction can destabilise it, so that it easily breaks up into stable nematicons [80], as illustrated in Figure 5. This figure shows snapshot cross-sections in the \((x,y)\) plane of the evolving vortex at various downcell distances \(z\) as it crosses the refractive index interface given by Equations (19) and (20). The deformation of a vortex hitting the interface is clearly visible in Figure 5b and increases as more of the vortex interacts with it, as in Figure 5c.