## 3 Results and Discussion
### Self-Focusing in the Fluoride Crystals
To achieve a more uniform excitation and emission energy distribution, a smaller material size could be used, but this would also limit the power scalability of the LaF-based laser. As \(n_{2}\) leads to self-focusing, the corresponding focal length\(f\) was calculated for each of the laser host materials and is shown in Figure 5. By calculating the focal length, the optimal length \(d\) of the laser material is determined such that the focal point of self-focusing occurs outside the laser material. Consequently, damage to the laser material resulting from very high intensities at the focal point will be avoided. The dashed line in Figure 5 indicates the condition when the focal length of self-focusing is equal to the length of the laser material (\(f=d\)). The dashed line therefore defines the upper limit of the laser material's length, meaning that a length that is longer than this limit will result in the focal point occurring inside the laser material (\(f<d\)). Therefore, Figure 5 is important in deciding the suitable length of the laser material. The length of the laser material should be shorter than the focal length (\(f>d\)) so that the focal point occurs outside of the laser material. The upper limit on the length of the fluoride materials studied here are 7.5 mm for LaF, 14 mm for YLF, 16 mm for LiSAF, and 18 mm for LiCAF. Notably, the LiCAF laser host material allows for a longer length to be used. This longer length is advantageous for the development of high-power, ultrashort lasers using, for example, multipass laser oscillator and amplifier configurations that result in higher laser emission energies. In order to achieve enough laser fluence to satisfy the laser threshold and enable lasing, the excitation beam radius is usually reduced. As the beam radius is reduced, the intensity increases, triggering self-focusing. Therefore, the focal length was calculated as a function of beam radius from 0.1 mm to 0.6 mm. The calculations assumed a Gaussian beam with 1 mJ energy and 100 fs pulse duration. Figure 6 shows the calculation results. Once again, the dashed line indicates the limit in the laser material's length that defines whether the focal point occurs inside or outside the material. The dashed line also defines the lower limit on the size of the excitation beam. In order for the focal point to occur outside the laser material, the radius of the excitation beam should be larger than the limit (dashed  
Figure 4: Variation of nonlinear refractive index (\(\Delta n_{2}\)) as a function of the intensity distribution of a Gaussian laser pulse. line). The lower limit on the excitation beam radius is 0.55 mm for LaF, 0.42 mm for YLF, 0.39 mm for LiSAF, and 0.37 mm for LiCAF. These results show that a smaller beam radius can be used for LiCAF, effectively achieving a higher laser fluence. Figure 5: Focal length as a function of the length of the laser host material. The dashed line indicates the condition when the focal length of self-focusing is equal to the length of the laser material (\(f=d\)). Figure 6: Focal length as a function of the beam radius. The dashed line indicates the limit in the laser material’s length that defines whether the focal point occurs inside or outside the material. It also defines the lower limit on the size of the excitation beam.