## 2 Methodology
\begin{table}
\begin{tabular}{c c} \hline \hline Kane momentum parameter, \(E_{p}\) & 21 eV [24; 25] \\ \hline Material-independent constant, \(K\) & 0.86 \(\times\) 10\({}^{-8}\)[24; 25] \\ \hline Speed of light, \(c\) & 3.10\({}^{8}\) m/s \\ \hline Planck constant, \(\hbar\) & 6.625 \(\times\) 10\({}^{-34}\) J.s \\ \hline Spectral range, \(\omega^{\prime}\) & 200–400 nm \\ \hline Material’s band-gap energy (eV) & \\ lithium calcium aluminum fluoride (LiCAF) & 12.23 [26] \\ lithium strontium aluminum fluoride (LiSAF) & 11.79 [26] \\ lyttrium lithium fluoride (YLF) & 11.09 [27] \\ lanthanum fluoride (LaF) & 9.00 [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Value of parameters used to solve the nonlinear refractive index \(n_{2}\) using Equation (3). \begin{table}
\begin{tabular}{c c} \hline \hline
2 PA & \(\frac{(x_{1}+x_{2}-1)^{3/2}}{2\,x_{1}x_{2}^{3}}\left(\frac{1}{x_{1}}+\frac{1}{ x_{2}}\right)^{2}\) \\ \hline Raman & \(\frac{(x_{1}-x_{2}-1)^{3/2}}{2\,  
Using the function \(F_{2}(x_{1};x_{2})\) in Table 2, the integration in Equation (4) can be performed analytically to obtain the nondegenerate \(G_{2}(x_{1};x_{2})\) and hence also to calculate the nonlinear refractive index \(n_{2}\) using Equation (3). Table 3 summarizes the nondegenerate nonlinear dispersion functions for the three contributions [29]. For laser generation, only the degenerate dispersion is useful. Therefore, \(n_{2}\) was calculated using \(x_{1}=x_{2}=\hbar\omega/E_{g}\). Previous works have reported that this Kramers-Kronig transformation-based model gives good agreement with experimental nonlinear refractive index data for different materials, including oxides such as sapphire [25] and fluorides such as lithium fluoride [24]. As mentioned previously, the nonlinear refractive index gives rise to nonlinear optical processes such as self-focusing. This occurs if the energy of the laser pulse in the material is greater than the critical energy, which is defined by [30]  
\[E_{cr}(\lambda)=\alpha\frac{\lambda^{2}\tau}{4\pi n_{0}(\lambda)n_{2}(\lambda)} \tag{5}\]  
where \(\lambda\) is the wavelength of the laser (nm), \(\alpha\) is a constant whose value is equal to 1.8962 for a Gaussian beam, \(\tau\) is the duration of the laser pulse (s), and \(n_{0}\) is the linear refractive index of materials. Self-focusing could damage the laser material as well as peripheral optical components if the focal point occurs inside the laser material or optical component. When self-focusing occurs, the material acts as a Kerr lens with a defined focal length \(f\). Therefore, it is necessary to determine the focal length of the Kerr lens and ensure that the focal point is outside the material to avoid the damaging effects of self-focusing. The inverse focal length \(f^{-1}\) is given by [31]  
\[f^{-1}=\frac{8n_{2}(\lambda)d}{\pi r^{4}}P \tag{6}\]  
where \(P\) is the peak power of the laser (W), \(r\) is radius of the laser beam at the surface of the laser material (mm), and \(d\) is the length of the laser material (mm). All calculations were performed using MATLAB software. Calculations were performed in the UV region spanning the wavelength range from 200 nm to 400 nm, with a step size of 1 nm. \begin{table}
\begin{tabular}{c c} \hline
2 PA & H(\(x_{1};x_{2})+\) H(\(-x_{1};x_{2}\)) \\ \hline \multirow{4}{*}{Raman} & H(\(x_{1};-x_{2})+\) H(\(-x_{1};-x_{2}\)) \\  & where for both 2 PA and Raman, H(\(x_{1},\ x_{2}\)) \\  & \(=\frac{1}{2^{\alpha}x_{1}^{\alpha}}\times\left[