## 2 Materials and Methods
The transmission characteristics as a function of the transmitted beam power show that the threshold beam power for the thinnest cell (12 \(\upmu\)m) is greater than 3 W (approx. \(>\) 190 W/cm\({}^{2}\) for experimental beamwidth). The threshold powers for cells with a 50 \(\upmu\)m and 75 \(\upmu\)m thickness are approximately \(\text{P}_{\text{th}}\) = 1.4 W and \(\text{P}_{\text{th}}\) = 0.95 W, respectively. These values are not suitable for safeguarding CCD cameras from harm or glare; nonetheless, they illustrate the overall working principle of the proposed liquid crystal laser beam power limiter. To overcome this limitation, we exploit the simultaneous impact of thermal nonlinearity, which is the other significant nonlinearity of liquid crystals that substantially amplifies the rate of change of the transmission drop above the threshold power. The thermal effects associated with molecular reorientation are related to the absorption of light by the NLC medium, resulting in an increase in temperature. This leads, among other things, to a decrease in the order parameter and a more significant disruption of the spatial arrangement of the molecules within the NLC cell as well as changes in refractive indices of the NLC [7; 35; 38]. These thermal-induced modifications directly reduce the birefringence experienced by the polarized beam. This process is similar to the untwisting of the director axis caused by the reorientation effect, further diminishing the NLC's ability to effectively modulate the light polarization. In this way, high enough power laser exposure causesmultiple refractive index changes (birefringence changes) in the NLC layer through mechanisms such as director axis reorientation and modifications to the order parameter due to temperature increases. As a consequence, the exiting light's polarization significantly differs from its initial state, and the overall transmission decreases. At sufficiently high powers, these effects can result in a state where the birefringence is negligible, causing the sample to appear nearly isotropic. The relationship between the refractive indices and the light intensity reveals a direct correlation between the two. Since the refractive indices are temperature-sensitive, and the temperature depends on laser intensity, the refractive indices are also intensity-dependent: \(n_{e}=n_{e}(I)\), \(n_{o}=n_{o}(I)\), in this way the birefringence is dependent on the optical intensity I: \(\Delta n=\Delta n(I)\). We can write \(\Delta n(I)\) as \(\Delta n(I)=\Delta n-n_{2}I\) in terms of the nonlinear coefficient \(n_{2}=\Delta n/I_{ISO}\), where \(I_{ISO}\) is the light intensity needed to drive the NLC into the isotropic state when the birefringence vanishes. Our research has shown that adjusting the order parameter via thermal buildup is a more effective strategy for reducing birefringence than merely relying on the optical (reorientational) untwisting of the director axis. In order to enhance these thermal effects, we have incorporated dye-doped nematics into the device design. These dyes increase light absorption, which accelerates changes in the order parameter due to thermal effects, resulting in a more effective reduction in birefringence across a range of laser intensities. The magnitude of \(n_{2}\) is dependent on the dye concentration: higher concentration gives lower \(I_{\text{ISO}}\) and, therefore, higher \(n_{2}\). The benefit of this solution is that, for sufficiently absorbent NLC media, the twisted nematic structure is perturbed by illumination with a sufficiently dense light beam at a significantly lower threshold power (compared to pure NLC). Although there is an inevitable loss of transmission in the NLC system, this cost can be easily offset at the data-processing level using devices such as CCD cameras and photodetectors. To enhance the thermal nonlinear response, we deployed a Sudan Blue II (SBII) absorption dye, which purely enhances the thermal nonlinear response [52; 53; 54; 55].