# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 3 Interacting Beams  
A simple manner in which to control a nematicon and its trajectory is to use a second one as a control beam, the interaction between the two being mediated by the nonlinear NLC response. Owing to the high nonlocality characterising the medium, the wavepackets can \"sense\" one another without an apparent collision [51; 52]. Let us first consider the interaction of two incoherent nematicons, with two non-interfering beams of electric fields \(u\) and \(v\) in relative proximity; neglecting walkoff, the NLC Equations (7) and (8) can be extended to [53]  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}D_{u}\
abla^{2}u+2A_{u }\phi u = 0, \tag{11}\] \[i\frac{\partial v}{\partial z}+\frac{1}{2}D_{v}\
abla^{2}v+2A_{ v}\phi v = 0,\] (12) \[\
u\
abla^{2}\phi-2\phi\phi = -2A_{u}|u|^{2}-2A_{v}|v|^{2}. \tag{13}\]  
If the two beams share the same wavelength, then the diffraction coefficients can be scaled to \(D_{u}=D_{v}=1\), as can the coupling coefficients \(A_{u}=A_{v}=1\)[51; 53]. The simplest collisional case is in-plane interactions [51; 52; 54; 55; 56; 57; 58; 59; 60], with solitary waves attracting and periodically interleaving as they propagate in the principal plane. While stable, steady multipole vector solitary waves can exist if the nonlocality \(\
u\) is large enough [55; 56]; in most other cases, the long range attraction mediated by the medium results in the beams oscillating about each other in the plane, irrespective of their relative phase, at variance with NLS solitary waves [52; 54; 61]. In the local limit \(\
u\to 0\), the coupled NLC Equations (11)-(13) reduce to coupled NLS equations. In-phase NLS solitary waves attract and out-of-phase NLS solitary waves repel [6], so the high nonlocality of NLC results in a significantly modified behaviour. The in-plane interaction of nematicons can be modelled using modulation theory, with good agreement with full numerical solutions of the coupled nematic Equations (11)-(13), a notable exception being the period of oscillation of the beams about each other [51; 57]. This period difference greatly affects the beam phase for increasing \(z\). A major result of this modelling is that nematicon trajectories are essentially determined by momentum conservation. Moreover, in the highly nonlocal limit, nematicons shed minimal radiation as they evolve [62], contributing minimal momentum to diffractive radiation. Although standard MT approximates solitary waves as point particles interacting under a potential [12], this was also extended to account for the non-point form of solitary waves [63], resulting in improved agreement with numerical solutions, albeit at the cost of increased complexity in the derivation of the MT equations and the equations themselves. As nematicons propagate in the bulk of a thick planar cell, typically with 100 \(\upmu\)m between parallel glass plates, two interacting nematicons of equal power and size can spiral about each other and form a rotating cluster in three dimensions, exhibiting an angular momentum as a whole due to their nonlocal interaction via the NLCs [54; 64; 65; 66]. Spiralling nematicons can be generated by the initial conditions  
\[u=a_{u}f(\rho_{u})\;e^{i\phi_{u}},\quad v=a_{v}f(\rho_{v})\;e^{i\phi_{v}}, \tag{14}\]  
where\[\rho_{u}=\frac{\sqrt{\left(x-\xi_{u}\right)^{2}+\left(y-\eta_{u} \right)^{2}}}{w_{u}}, \rho_{v}=\frac{\sqrt{\left(x-\xi_{v}\right)^{2}+\left(y-\eta_{v} \right)^{2}}}{w_{v}},\] \[\psi_{u}=\sigma_{u}+U_{u}(x-\xi_{u})+V_{u}(y-\eta_{u}), \psi_{v}=\sigma_{v}+U_{v}(x-\xi_{v})+V_{v}(x-\eta_{v}) \tag{15}\]  
at \(z=0\) in the two colour NLC Equations (11)-(13), where \(f\) is the electric field profile of the beams, noting that the input wavepackets are usually Gaussian. The variables \((U_{u},V_{u})\) and \((U_{v},V_{v})\) are related to the angles at which the beams are input into the nematic cell and can be considered the input \((x,y)\) \"velocities\" of the beams, with \((\xi_{u},\eta_{u})\) and \((\xi_{v},\eta_{v})\) the input positions of the beams.