## 3 Results and Discussion
### Thermal Stress Distribution Caused by the Subsurface Impurity Defects  
#### 3.2.1 Comparison of Thermal Stress Distributions with and without Impurity Defect  
By numerically solving Equations (1) and (4) with finite element method (FEM), the physics of heat conduction and solid mechanics involved in the intense laser irradiation process can be well coupled for obtaining the thermal stress distribution caused by impurity defects. The thermal stress could be applied to describe the negative effect of impurity defects on the mechanical property and wavefront distortion of fused silica optics with high precision surfaces. In this section, the von Mises equivalent stress is employed to evaluating the yielding behavior of fused silica material with the presence of subsurface impurity defects. Figure 8 presents the comparison of von Mises equivalent stress on fused silica with and without impurity defects. The parameters of incident lasers are the same to those applied in Section 3.1 (10 J/cm\({}^{2}\), 355 nm, 10 ns) and the defects of CeO\({}_{2}\), Fe and Al are all investigated with a 1 \(\upmu\)m defect radius and a 1 \(\upmu\)m defect depth. The negative effect of impurity defects on laser damage resistance of fused silica optics is primarily related to the maximum temperature and stress during the laser irradiating process. Hence, in this work, we only concentrate on the transient simulation results of the maximum temperature and stress, which generally take place at roughly 10 ns when the laser pulse ends. Figure 7: Evolutions of the maximum temperatures caused by various impurity defects with respect to the defect radius (**a**) and depth (**c**) during the laser loading period of 10 ns. (**b**) The dependence of maximum temperature on defect radius for CeO\({}_{2}\) defects under various laser loading times. As shown in Figure 8a, for fused silica optics without a subsurface defect, the induced von Mises stress appears in standard Gaussian distribution under the laser irradiation. The maximum thermal stress is 52 MPa, which is far lower than the strength of fused silica (800-1000 MPa). It means that the mechanical property and surface deformation of fused silica would be not strongly affected under the laser irradiation with this level of power. For the thermal stress in fused silica optics with impurity defects, it is demonstrated in Figure 8b-d that different types of impurity defects can induce different thermal stress distribution under the intense laser irradiation. The highest thermal stress is induced by the CeO\({}_{2}\) impurity defect, followed by Al and Fe defects. The maximum von Mises equivalent thermal stress induced by CeO\({}_{2}\) defect is 7432 MPa located on the topmost surface of defect and the maximum stress on the fused silica material is 3617 MPa. For Al and Fe defects, the induced maximum Mises equivalent thermal stresses are 4476 MPa and 2236 MPa, respectively. By comparing the simulated thermal stress distributions and thermodynamic parameters of Al and Fe defects, one can see that the coefficient of thermal expansion plays a dominant role in producing the higher thermal stress. It is also shown in Figure 8c,d that sharp changes exist at the intersections of fused silica material and impurity defect in the evolutions of von Mises stress caused by Al and Fe defects (the thermal stress curves have discontinuous points at \(z\)-positions of 8.5 \(\upmu\)m and 9.5 \(\upmu\)m). This phenomenon can be explained below: (1) under the intense laser irradiation, the temperature rise of the impurity defect is greater than that of surrounding fused silica material and the local concentrated thermal load may be formed; (2) the thermal expansion coefficient of fused silica is much smaller than that of impurity defects that the impurity defect undergoing thermal expansion would be spatially constrained by the local surrounding silica materials; and (3) a big difference exists in the rigidity property of neighboring FEM elements due to the natural mechanical properties of the involved materials.