# Interactions of Self-Localised Optical Wavepackets in Reorientational Soft Matter
## 3 Interacting Beams
The spiralling of two skew nenactions about each other due to nonlocal, nonlinear attraction is illustrated in Figure 2. This figure shows experimental results for two identical solitary waves launched and evolving out-of-plane and spiralling about one another for various input powers. The interacting beams are imaged in the observation plane \((Y,Z)\), their output spots in the \((X,Y)\) plane (Figure 2c,d,g,h) at the end of the sample after they evolved in the down cell \(Z\) direction. The output spots appear to rotate as the beam power goes up and the effective angular momentum of the \"two nematicon molecule\" cluster increases, with an augmented angular velocity [64]. Nematicon spiralling was theoretically investigated using MT based on the input beams (14) [66]. Both beam profiles were either Gaussian, \(f(\rho)=\exp(-\rho^{2})\), or \(sech\), \(f(\rho)=\text{sech}\,\rho\). Good to excellent agreement was obtained between full numerical solutions of the NLC Equations (11)-(13) and MT for the trajectories, with the Gaussian and sech profiles providing comparable results in the highly nonlocal limit, the agreement improving for increasing angular momentum of the input cluster [66]. Figure 2: Spiralling nematicons. The NLC planar cell is realised as in Figure 1b. Top left: Numerical solution of the nematic model in Equations (11)–(13), illustrating the mutual spiralling of two identical skew nematicons launched out-of-plane \((Y,Z)\). Top right: Artist’s rendering of the beam spots rotating about one another at the end of the sample for increasing excitation. (**a–h**): Experimental results; (**a**,**b**,**e**,**f**) acquired images of beam evolution in \((Y,Z)\); (**c**,**d**,**g**,**h**) of the solitary wave cluster in the transverse plane \((X,Y)\) at the sample output for input powers 2.1, 2.7, 3.3 and 3.9 mW, respectively. These findings confirmed that, in the highly nonlocal limit, nematicon trajectories are weakly dependent on the initial wavepacket profile [51; 67]. It is important to underline that besides those based on the full nonlocal response of NLC, some studies rely on simplified models as the NLC Equation (13) can be solved in terms of a Green's function \(G\) as  
\[\phi=-2\int\int_{\text{cell}}G(x-x^{\prime},y-y^{\prime})\Big{[}A_{u}|u(x^{ \prime},y^{\prime},z)|^{2}+A_{v}|v(x^{\prime},y^{\prime},z)|^{2}\Big{]}\,dx^{ \prime}dy^{\prime}. \tag{16}\]  
Since in \((2+1)\) dimensions, the Green's function \(G\) is the modified Bessel function of order \(0\), \(K_{0}\), in general, the solution (16) is not useful for analytical studies. For this reason, much work on light beam propagation in nonlocal, nonlinear media has adopted simplified responses for \(G\)--the most common being Gaussian and exponential. Of particular relevance to the present review, the in-plane interaction of two solitary waves was investigated by means of a Gaussian [68; 69] and an exponential response [70] and their spiralling with a Gaussian [71]; such studies were in qualitative agreement with those based on the actual NLC response. At the end of this section, we deem it appropriate to mention that a range of experimental and numerical studies were undertaken on self-guided beams interacting in other nonlocal media [72], particularly those with a self-focusing thermo-optic response [39; 73]. The understanding and intuitive perception of nonlinear phenomena in reorientational soft matter, in fact, may benefit from the insight afforded by solitary waves in thermal media. For the latter scenario, the governing equations are the Schrodinger-Poisson model shown in Equations (7) and (8), where \(\phi\) denotes the temperature, with the walk-off factor and the pre-tilt set to zero, \(\gamma=0\) and \(q=0\). Employing lead-glass, it was reported in [39] that in \((1+1)\)D, two propagating solitary waves attract each other; in \((2+1)\)D, spiralling can occur, with orbits dependent on the nonlocality.