## 2 Theory and Methods
### Modeling of Impurity-Induced Temperature and Thermal Distributions
\(\sigma\) is the Boltzmann constant. The historical temperature evolution on fused silica optics can be obtained by numerically solving the heat transfer equation of Equation (1) with the consideration of boundary conditions of Equation (3). It is worth noting that in this work we only focus on the temperature evolution inside the laser spot for the reason that the amount of heat energy transferred outside the light spot is much smaller. Under the irradiation of the Gaussian laser pulse, the thermal distortion caused by absorptive impurity defects could be resolved and analyzed on the basis of the temperature distribution obtainedfrom Equation (1). Ignoring the volume and inertia forces inside fused silica, the thermal distortion arising from internal temperature gradient would be described as follows [38]:  
\[\left\{ \begin{array}{l}\
abla^{2}u_{r}-\frac{u_{z}}{r^{2}}+\frac{1}{1-2v}\frac {\partial\epsilon}{\partial r}-\frac{2(1+v)}{1-2v}\alpha_{t}\frac{\partial T}{ \partial r}=0\\ \
abla^{2}u_{z}+\frac{1}{1-2v}\frac{\partial\epsilon}{\partial z}-\frac{2(1+ v)}{1-2v}\alpha_{t}\frac{\partial T}{\partial z}=0\end{array} \right. \tag{4}\]  
where \(u_{r}\) and \(u_{z}\) are the displacement components in \(r\) and \(z\) directions of cylindrical coordinate system. \(e\) represents the volumetric strain of fused silica. \(\
u\) and \(\alpha_{t}\) are the Poisson's ratio and coefficient of thermal expansion, respectively. Using the relations between stress and strain, the expression of potential energy principle can be gained based on the principle of virtual work. Then, by dispersing the potential energy expression and solving the nodal displacement matrix with the FEM method, we would obtain the corresponding thermal stress distribution initiated by the absorptive impurity defects. With the application of advanced testing techniques like synchrotron radiation X-ray fluorescence (SXRF) and secondary ion mass spectrometry (SIMS), it has been previously reported [17; 18; 20; 22; 25] that the precision grinding and polishing of optical materials can probably introduce impurity defects with sub-wavelength size on the processed surface or subsurface. The impurity defects mainly contain cerium oxide (CeO\({}_{2}\)), zirconium dioxide (ZrO\({}_{2}\)), iron (Fe), aluminum (Al), chromium (Cr) and so on [17; 20; 25]. In this work, we choose CeO\({}_{2}\), Fe and Al as the three representative defects coming from the oxide polishing slurries, metal polishing tools and cleaning solutions, respectively. The effects of defect species, size and spatial location are all investigated and compared to reveal the most dangerous impurity to laser damage resistance. The optical and thermodynamic parameters of fused silica and impurity defects are exhibited in Tables 1 and 2, respectively. It should be noted that the optical damage of fused silica materials initiated by impurity defects under intense laser irradiations is really a complex process, which involves in the material modification, softening, melting, boiling, material fracture and even ejection. All of these processes are closely associated with the rise of temperature and stress during the energy absorption of incident lasers. Therefore, in the present work, the material definition models for both fused silica and impurity defects are simplified with constant thermodynamic parameters to model the temperature and stress distributions caused by representative impurity defects for evaluating their effects on the laser damage resistance.