## 2 Methodology  
The optical Kerr effect is a nonlinear effect whereby the refractive index of a material changes in response to light propagating in the material. For a particular wavelength, the refractive index is proportional to the local irradiance of the light, being widely defined as [23]:  
\[n(\omega)=n_{0}(\omega)+n_{2}(\omega)I, \tag{1}\]  
where \(n_{0}(\omega)\) and \(n_{2}(\omega)\) are the linear and nonlinear refractive indices at the frequency \(\omega\), respectively, and \(I\) is the laser light intensity. It follows that \(\Delta n_{2}(\omega)=n_{2}(\omega)I\) is the change in the nonlinear refractive index. The Kramers-Kronig relation model proposed by [PERSON] and [PERSON] predicts the dispersion and the magnitude of the nonlinear refractive index (\(n_{2}\)) of wide-bandgap materials. Calculation of \(n_{2}\) for solid-state oxide materials using the Kramers-Kronig relation model has excellent agreement with experimental values [24; 25]. Hence, we use the same model to calculate \(n_{2}\) for the solid-state fluoride laser host materials LiCAF, LiSAF, LaF, and YLF. According to the Kramers-Kronig dispersion relation, the change in the refractive index is given by  
\[\Delta n(\omega)=\frac{c}{\eta}\int_{0}^{\infty}\frac{\Delta\alpha(\omega^{ \prime};\omega)}{\omega^{\prime 2}-\omega^{2}}d\omega^{\prime}, \tag{2}\]where \(\Delta n(\omega)\) is the change in the refractive index due to the presence of pump or excitation light of frequency \(\omega\). The change in the index is associated with changes in the absorption coefficient \(\Delta\alpha(\omega^{\prime};\omega)\) at the frequency \(\omega\) throughout the spectrum \(\omega\prime\)[24]. The equation for calculating the nonlinear refractive index \(n_{2}\) can be derived from the Kramers-Kronig dispersion relation in Equation (2), as discussed in detail by [PERSON] et al. [24] and given in Equation (3), as follows:  
\[n_{2}(\omega)=\frac{4\times 10^{5}\pi K\sqrt{E_{p}}}{cn_{0}^{2}(\omega)E_{B}^{4}} G_{2}\left(\frac{\hbar\omega}{E_{g}}\right), \tag{3}\]  
where \(E_{g}\) is the bandgap of the material (eV), \(n_{0}\) is the linear refractive index of the material which can be calculated using the material's Sellmeier equation, \(c\) is the speed of light (cm/s), \(E_{p}\) is a Kane momentum parameter that has a value of 21 eV, \(K\) is a material-independent constant equal to 0.86 \(\times\) 10\({}^{-8}\) when \(E_{g}\) and \(E_{p}\) are in electron volts (eV), and \(n_{2}\) is in square centimeters per watt (cm\({}^{2}\)/W) [24]. The values of these parameters, including the band-gap energies of the laser host materials studied in this paper, are summarized in Table 1.  
In Equation (3), \(G_{2}\left(\hbar\omega/E_{g}\right)\) is a dispersion function given by  
\[G_{2}\left(\frac{\hbar\omega}{E_{g}}\right)=\frac{2}{\pi}\int_{0}^{\infty} \frac{F_{2}(x_{1};x_{2})}{x_{1}^{2}-x_{2}{}^{2}}dx, \tag{4}\]  
where \(x_{1}=\hbar\omega_{1}/E_{g}\), \(x_{2}=\hbar\omega_{2}/E_{g}\), and \(F_{2}(x_{1};x_{2})\) is a nonlinear absorption spectral function that considers the contribution of two-photon absorption (2 PA), Raman, and optical Stark effects [24; 29]. The function \(F_{2}(x_{1};x_{2})\) reflects the band structure of the material and involves only the parameters \(x_{1}\) and \(x_{2}\); meaning it is a function only of the ratio of the photon energy \(\hbar\omega\) to the band gap \(E_{g}\) of the material [25; 29]. Therefore, calculation of the dispersion of the nonlinear refractive index \(n_{2}\) requires the band-gap energy of the fluoride host material and the energy of the exciting photon. The expressions of \(F_{2}(x_{1};x_{2})\) for the three contributions are listed in Table 2[29].