# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal
## 2 Experimental Observations of the Ring Patterns
### Light-Induced Ring Patterns  
When the dye-doped liquid crystal cell was illuminated with a probing light, the monitoring CCD camera showed a homogeneous dark greenish color throughout the cell, as illustrated in the snapshot of Figure 3b at \(t_{0}\). When applying the blue light beam, we observed that the illuminated area immediately began to transmit more light. Figure 3a illustrates how the total transmitted intensity measured in the green channel of the CCD camera (\(\Delta I_{S}\)) evolved over time with respect to the transmitted light without the blue light beam (\(I_{g,0}\)). The temporal evolution of the detected light intensity was characterized by growth and subsequent saturation. Figure 3a shows two regions in which the growth (region \(I\)) and saturation region (\(II\)) could be distinguished. In order to describe the growth and saturation process, we modeled it using the following expression: \(\Delta I_{S}(t)/I_{g,0}=A(1-e^{t/\tau})\), where \(A=2.79\) and \(\tau=55.71\) s. Namely, the establishment of the stationary ring pattern required a time period on the order of one minute.  
When the blue light was applied, a region of light green with a darker center emerged; as time elapsed, this dark spot became a propagative ring. Note that the lightened region continued to grow, becoming even larger than the waist of the blue laser. Figure 1b compares the light green region to the waist size of the blue laser. As time continued to elapse, the dark ring continued to move away from the center, and a new dark spot emerged in the center, which then became a new dark ring. Figure 3b summarizes the temporal sequence of snapshots in the ring pattern formation process. In order to determine the process of spot emergence and ring propagation, we consider the spatiotemporal evolution of a diameter cut section. Figure 3c illustrates the observed spatiotemporal diagram evolution. From this chart, we see how the dark rings emerge, spread, and stop.  
When we applied low powers of the blue laser (few mW, cf. Figure 4), the system did not show the formation of ring patterns, and we only observed the emergence of a light green spot. As the power was increased, this light green spot increased in size. With powers close to 40 mW, we began to observe the emergence of a dark spot in the center of the illuminated region (see Figure 4). Physically, we interpreted this region as a region of greater orientational disorder due to the consideration of the dye-doped liquid crystal sample between crossed polarizers. When we further increased the power of the blue laser, we observed the emergence of the first ring. Figure 4 shows the observed equilibrium ring. As the power increased, the diameter of the equilibrium ring grew. For powers close to 70 mW, we observed the emergence of a ring with a dark spot in the center as a state of equilibrium. As the power of the blue laser continued to increase, we observed that the central dark spot grew, and at a higher power, it became unstable, generating a new ring. Figure 4 summarizes the equilibria found for different blue laser powers.  
Figure 3: Experimental ring pattern emergence induced by a blue light (with a 445 nm wavelength) applied to a dye-doped liquid crystal cell, E7 NLC with azo-dye methyl red methyl ester at a concentration of 1 wt%. (**a**) Temporal evolution of transmitted total intensity, measured in the green channel of the CCD camera (\(\Delta I_{g}\)) with respect to the transmitted light without the blue light beam (\(I_{g,0}\)). The points were obtained experimentally, and the continuous curve was acquired using the expression \(\Delta I_{g}(t)/I_{g,0}=A(1-e^{t/\tau})\), where \(A=2.79\) and \(\tau=55.71\) s. Painted areas I and II account for the growth and saturation regions, respectively. (**b**) Temporal sequence of snapshots in the ring pattern formation process (\(t_{0}=0\) s, \(t_{1}=1\) s, \(t_{2}=5\) s, \(t_{3}=84\) s, \(t_{4}=360\) s, and \(t_{5}=570\) s). (**c**) Spatiotemporal diagram evolution of a diameter cut section.