## 2 Theory and Methods
### Modeling of Impurity-Induced Temperature and Thermal Distributions
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Nomenclature** & **Value (Units)** \\ \hline Molar mass & \(M\) & 60.06 (g/mol) \\ Crystal system & – & Amorphous \\ Density & \(\rho\) & 2.21 (g/cm\({}^{3}\)) \\ Melting point & \(T_{\text{m}}\) & 1900 (\({}^{\circ}\)C) \\ Thermo-optical coefficient & \(\epsilon\) & \(1\times 10^{-5}\) \\ Coefficient of linear expansion & \(\alpha_{t}\) & \(5.5\times 10^{-7}\)\({}^{\circ}\)C\({}^{-1}\) \\ Specific heat capacity & \(c\) & 0.728 J/(g\({}^{\circ}\)C) \\ Heat conductivity coefficient & \(k\) & 1.35 W/(m\({}^{\circ}\)C) \\ Relative dielectric constant & \(\epsilon_{r}\) & 2.25 \\ Young modulus & \(E\) & \(7.36\times 10^{10}\) (Pa) \\ Shear modulus & G & \(3.14\times 10^{10}\) (Pa) \\ Compressive strength & \(P\) & 800–1000 (MPa) \\ Poisson’s ratio & \(\
u\) & 0.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermodynamic parameters of fused silica materials used in calculations [39; 40]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Defect** & **Density** & **Specific Heat** & **Heat Conductivity** & **Coefficient of Linear** & **Young Modulus** & **Poisson’s** \\
**Species** & **(g/cm\({}^{3}\))** & **Capacity** (0/g\({}^{\circ}\)C) & **(W/(cm\({}^{\circ}\)C))** & **Expansion** (\(\times 10^{-8}\) K\({}^{-1}\)) & **(GPa)** & **Ratio** \\ \hline Fe & 7.0 & 0.45 & 0.565 & 11.8 & 152.0 & 0.30 \\ Al & 2.7 & 0.88 & 2.38 & 23.0 & 70.0 & 0.33 \\ CeO\({}_{2}\) & 7.13 & 0.465 & 0.045 & 13.2 & 174 & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermodynamic parameters of various impurity defects used in the calculation [40; 41; 42]. Figure 1 shows the FEM model for simulating the temperature and thermal stress distributions caused by subsurface impurity defect. To improve the computational efficiency, only the domain exposed within the laser spot is calculated and the diameter of fused silica is set to be 400 \(\upmu\)m. The sample thickness is chosen as 10 \(\upmu\)m due to the dramatically attenuating of laser energy in the vertical propagation direction. A two-dimensional axisymmetric finite element model is applied in this work for a cylindrical fused silica optics. The diabatic boundary conditions combined with the surface-to-ambient radiation boundary are applied at the borders of the simulation domain. Since the energy of the applied incident lasers is spatially distributed with a Gaussian profile, the temperature in the area beyond the laser spot would be much lower than those in the central area. Hence, the applied boundary conditions would not affect the simulation accuracy and more attention should be paid on the results of temperature and stress fields near the central area. The laser parameters chosen in the simulations are the same as those applied in the actual laser damage experiments to ensure accurate and reliable simulation results. It is worth noting that the actual laser energy is spatially distributed with an approximate Gaussian distribution. For simplicity, the spatial distribution of incident lasers with Gaussian shape (see Figure 1c) is applied in the simulations with a 390-\(\upmu\)m beam diameter and a 10-ns pulse duration. The laser frequency is 1 Hz and after each laser pulse the fused silica sample cools down naturally. In this work, the evolutions of temperature and stress distributions during the laser heating and free cooling processes are both simulated and the total simulation running time is set to be much longer than the pulse duration to insure the stable temperature and stress fields to be reached finally. For simplicity, the impurity defect inside fused silica in Figure 1b is viewed as a small spherical particle positioned right beneath the laser spot center. The simulation domain is non-uniformly gridded in different regions: the defect region is gridded with triangular mesh, while the other region is gridded with quadrilateral mesh. Further, the refined gridding sizes are applied in the vicinity of fused silica-defect interface to guarantee the simulation accuracy.