## 3 Results and Discussion  
### Linear and Nonlinear Refractive Indices of the Fluoride Laser Host Materials  
The linear dispersion of the fluoride materials in the UV region were calculated using their individual Sellmeier equations [32; 33; 34; 35] and are shown in Figure 1. As expected, the linear refractive index of the materials decreased for longer wavelengths. Among the four fluoride materials, LiSAF exhibited the smallest variation in its \(n_{0}\), changing from 1.44 at 200 nm to 1.41 at 400 nm, corresponding to a \(\Delta n_{0}\) of 0.03. LiCAF also exhibited a small variation in its \(n_{0}\) which changed from 1.43 at 200 nm to 1.39 at 400 nm and a \(\Delta n_{0}\) of 0.04. The value of \(n_{0}\) at the 290 nm peak laser emission wavelength is ~1.4, which is in good agreement with the previously reported experimental values [32]. The refractive index of the doped and undoped materials are similar, differing by only <1% of each other. For example, the refractive index of undoped and Ce\({}^{3+}\)-doped LiCAF at the 290 nm UV laser emission wavelength is 1.40 [32] and 1.41 [36], respectively. Similarly, the difference in the refractive index between a chromium-doped LiCAF (Cr\({}^{3+}\):LiCAF) and undoped LiCAF is about \(5\times 10^{-3}\) throughout the spectral range from 200-400 nm [37]. YLF had a \(\Delta n_{0}\) of 0.05, with its \(n_{0}\) changing from 1.52 to 1.47 at the same wavelength range as the others. Notably, LaF had the largest \(\Delta n_{0}\) of 0.08 as its \(n_{0}\) ranged from 1.70 at 200 nm to 1.62 at 400 nm. In general, LaF also had higher \(n_{0}\) values compared to the other fluoride materials.  
Figure 2 shows the nonlinear refractive index of the fluoride host materials in the UV wavelength region. Compared to the behavior of \(n_{0}\) in Figure 1, a larger variation in \(n_{2}\) is observed from LiCAF, LiSAF, and YLF with a steep decrease from 200 nm to about 250 nm before settling at a stable value from around 300 nm. LiCAF has the smallest variation in \(n_{2}\), which changed from \(2\times 10^{-16}\) cm\({}^{2}\)/W to \(0.7\times 10^{-16}\) cm\({}^{2}\)/W giving a \(\Delta n_{2}\) of \(1.3\times 10^{-16}\) cm\({}^{2}\)/W, followed by LiSAF with a \(\Delta n_{2}\) of \(1.7\times 10^{-16}\) cm\({}^{2}\)/W and then YLF with a \(\Delta n_{2}\) of \(2.0\times 10^{-16}\) cm\({}^{2}\)/W. Interestingly, LaF exhibited a different pattern wherein its \(n_{2}\) initially increases, peaking at 262 nm, before decreasing as the wavelength becomes longer. LaF has a maximum \(n_{2}\) value of 5.6 \(\times\) 10\({}^{-16}\) cm\({}^{2}\)/W, which occurred at 262 nm. The larger \(n_{2}\) values from LiSAF, YLF, and LaF indicate that nonlinear processes will be more evident in these materials compared to LiCAF. For the case of LaF, exciting this laser host material around 260 nm wavelength could lead to self-focusing effects even at lower  
Figure 1: Linear refractive index (\(n_{0}\)) of the fluoride laser materials.  
excitation energies. Self-focusing effects in these laser host materials are further analyzed in the succeeding paragraphs.  
The spectral profile of laser emission from the Ce\({}^{3+}\)-doped fluoride materials is broad, allowing the wavelength of the laser emission to be tuned within a range of 10 nm to 35 nm [38]. Calculations were performed over this broad spectral tuning range, but detailed analysis of the nonlinear effects are considered at the peak laser emission wavelength, which is also expected to express the highest laser output energy. The peak laser emission wavelength of Ce\({}^{3+}\)-doped LiCAF, LiSAF, and LaF occurs at around 290 nm, while it is at around 308 nm for Ce:YLF. Investigation of self-focusing is particularly important when scaling up the power of the laser emission towards the terawatt level, as self-focusing could easily damage the laser material at this high intensity level. The peak laser emission wavelengths and \(n_{2}\) values of the fluoride laser materials at their respective peak laser emission wavelengths are summarized in Table 4.