## 2 Theory and Methods  
### Modeling of Impurity-Induced Temperature and Thermal Distributions  
The residual impurity defects are generally distributed inside the SSD layers of ground and polished fused silica optics. The laser energy absorbed by these impurity defects is far greater than that by intrinsic thermal absorption of the fused silica itself. On the one hand, the strong absorption of these impurity defects can locally heat the silica materials, inevitably resulting in unsteady and non-uniformtemperature distribution. When the local temperature reaches a certain degree, the optical material may suffer from modification, softening, melting and even boiling, which would certainly incur laser-induced damage [7]. On the other hand, the great temperature gap inside silica materials caused by the absorption of impurity defects could give rise to internal thermal stress, and correspondingly leads to the negative effects of obvious wavefront distortion and weak mechanical strength. These negative effects would directly affect the laser energy capacity and service life of large-aperture fused silica optics [20; 21]. In the present efforts, the heat transfer and thermo-elastic equations governing the laser energy absorbing process are numerically solved by adopting the FEM method to investigate the distributions of temperature and thermal stress induced by impurity defects. The effects of subsurface impurity defects with various species and dimensions on the laser damage resistance are then analyzed, based on which we finally clarify the most dangerous impurity defects in terms of lowering the LIDT of fused silica optics. When the incident laser falls on the silica surface, most of the laser energy is reflected at the interface or transmitted through the bulk, leaving only a fraction of the laser energy absorbed by the materials [34]. As the absorbed energy penetrates, the temperature on fused silica materials increases. The spatially and temporally increasing temperature can be described by the heat transfer equation in the Cartesian coordinate system which is given below [35]:  
\[\frac{\partial T}{\partial\tau}=\alpha(\frac{\partial^{2}T}{\partial x^{2}}+ \frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}})+ \frac{q_{\
u}}{\rho c} \tag{1}\]  
where, \(T\), \(\alpha\), \(c\) and \(\rho\) represent the temperature, thermal diffusivity, specific heat capacity, and density, respectively. \(q_{\
u}\) indicates the heat source determined by the incident laser, which can be expressed as [36]:  
\[q_{\
u}=\alpha\frac{(1-R)P}{\pi a^{2}}\exp(-\frac{x^{2}+y^{2}}{a^{2}})\exp(- \alpha z) \tag{2}\]  
with \(\alpha\) the absorption coefficient, \(R\) the Fresnel reflection coefficient, \(P\) the incident laser power, and \(a\) the radius waist at 1/e. As the laser irradiates the optical surface, part of the heat energy is transported into the bulk, resulting in the rise of surface temperature. Thermal radiation would take place at the hot surface. Besides, the high temperature on the surface may cool down by exchanging heat energy with the ambient air in the form of thermal convection. Hence, the below boundary conditions of heat flux, natural convection cooling and surface-to-ambient radiation are all applied to fully describe the heat transfer process in fused silica under the intense laser irradiation [37]. Under the control of Equation (3), the energy balance would be actually realized by transferring the energy among the sample surface, sample bulk and the surrounding circumstance during the laser heating and natural cooling processes. \[I\Delta\tau=-k\frac{\partial T}{\partial z}+h(T_{s}-T_{0})+\sigma\varepsilon( T_{s}{}^{4}-T_{0}{}^{4}) \tag{3}\]  
where, \(k\), \(h\) and \(\varepsilon\) are the heat conductivity, convection coefficient, and radiation coefficient, respectively. \(T_{s}\) and \(T_{0}\) denote the temperatures of fused silica surface and indoor air temperature. \(I\) represents the incident laser intensity and \(\Delta t\) indicates the laser action period.