# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal
## 3 Mathematical Modeling for Photo-Isomerization in Dye-Doped Liquid Crystals
### Light-Induced Ring Pattern  
Figure 5 summarizes the different behaviors presented by the model in Equation (15). From this chart, we conclude that the system has a coexistence region between the nematic state and the pattern. Note that this pattern alternates between areas of higher and lower orientation order [21; 22]. This affects the sample's refractive index; therefore, if a light beam passes through the sample in a patterned state, one expects to observe interference fringes. As the light intensity increases, the bifurcation parameter \(\tilde{A}\) grows. Then, if one considers a light intensity with a Gaussian profile, the parameter \(\tilde{A}(I)\) is characterized by being inhomogeneous, with a bell-like shape. Thus, if the cell is in a nematic phase when the sample is illuminated, the central part of the light beam can induce the cell to leave the coexistence region, and only the pattern will be stable. For this type of region, we would expect to find that the illuminated area shows patterns because the central area of the light beam is circular, and for a small-waist beam, one would expect to see ring-like patterns. Figure 6 shows the typical equilibrium ring pattern observed numerically for the model Equation (9). All numerical simulations presented are obtained by considering finite differences coded with the Runge-Kutta order-4 algorithm. If the system is not illuminated, \(I_{0}=0\), the uniform nematic phase is the equilibrium of the system. By illuminating the system with a low intensity, we found numerically that there was a slight decrease in the reorientation order and an increase in the _cis_ concentration in the central part of the Gaussian (cf. Figure 7). As the intensity \(I_{0}\) increased, the size of the central spot showed greater orientation disorder; that is, the order parameter \(S\) decreased in the central zone as \(I_{0}\) increased. Note that the spot of the orientational disorder was smaller than the waist of the Gaussian forcing. As \(I_{0}\) increased in the central zone, the parameter of order \(S\) approached zero (isotropic liquid). When it hit zero, it generated a new dynamical behavior; the central point expanded, creating a ring. The origin of the clearing out of the central zone was due to the fact that the order parameter \(S\) became negative. This could be interpreted as two ordered regions separated by a circular interface of the disordered state. A dark ring in Figure 7 represents this region. As \(I_{0}\) increased further, the ring continued to expand. By further increasing the intensity of the Gaussian forcing, we observed the emergence of a new central spot surrounded by a ring. Note that for this parameter region, when beginning with a uniform nematic state and applying Gaussian forcing, a central spot of disorder state initially emerged that expanded, forming a ring that continued to propagate; later, another central spot of disorder state emerged, and finally the ring pattern stopped and remained in a stationary state. Experimentally, we observed a similar behavior to that observed numerically (see Figure 3). As the intensity of the Gaussian forcing \(I_{0}\) increased further, new central spots emerged, which became new rings of disordered states (see Figure 7). The waist of the light beam limits the above process. This process is similar to the experimental process (see Figure 3). In brief, the effective model in Equation (9) described the dynamics of the light-induced ring pattern in a dye-doped nematic liquid crystal well in qualitative terms. Figure 6: Numerical stationary ring pattern in a dye-doped nematic liquid crystal using the effective model in Equation (9) for \(A=-0.5\), \(\beta=0.3\), \(\
u=1.05\), \(b=0.1\), \(I_{0}=1.45\), and \(w=4\). (**a**) Contour plot of the squared order parameter \(S\). (**b**) Profile of the cut of the order parameter \(S\) in the diameter of the ring pattern. (**c**) Surface plot of the squared order parameter \(S\).