## 3 Results and Discussion
### Thermal Stress Distribution Caused by the Subsurface Impurity Defects
#### 3.2.2 Effect of Impurity Defect Parameters on Its Induced Thermal Stress  
In order to discern the most dangerous impurity defect to the laser damage resistance of fused silica optics, we also investigate the von Mises thermal stress caused by impurity defects with various structural parameters (e.g., defect radius \(r\), defect depth \(d\)). Since the CeO\({}_{2}\) impurity defect would induce the highest temperature and thermal stress as discussed above, here we only consider the effect of CeO\({}_{2}\) defect parameters on its induced thermal stress, which is shown in Figure 9. As shown in Figure 9a, with the increase of defect radius, the von Mises stress ascends firstly and then decreases gradually. The maximum thermal stress reaches the summit (7420 MPa) when the defect radius is 0.15 \(\upmu\)m (critical defect radius). One can see from Figure 9b that impurity defect located more closely to the silica surface would induce higher maximum thermal stress. The peak maximum stress of fused silica is 8739 MPa, and when the defect depth is larger than 0.3 \(\upmu\)m, it remains roughly stable. The changing tendency of maximum thermal stress with respect to defect parameters is similar to that of maximum temperature shown in Figure 7. It is worth noting that the thermal stress of fused silica is higher than that of CeO\({}_{2}\) defect as shown in Figure 9b. This is because the defect radius is so small (\(r\) keeps constant at the critical radius of 0.15 \(\upmu\)m) that the heat energy can be promptly transferred to the surrounding fused silica materials. As a result, fused silica with higher heat capacity would induce higher thermal stress. From the simulation results of defect-induced temperature and thermal stress distributions discussed above, one can see that the transient local high temperature can be produced by heat absorbing of impurity defects under the irradiation of intense lasers. The extremely high temperature would trigger the plasma formation inside the impurity defect, result in material expansion, and finally initiate the fracture and breakdown of the surrounding silica materials. In addition, the huge thermal stress caused by the impurity defect far exceeds the intrinsic strength limit of the material, making the fused silica optics susceptible to be crushed down. Furthermore, in the actual process of intense laser irradiation, the high temperature and thermal stress caused by the impurity defect could act together,  
Figure 9: Evolution of the maximum von Mises stress caused by CeO\({}_{2}\) impurity defect with respect to the defect radius (**a**) and depth (**b**) when the fused silica surface is irradiated by incident lasers for 10 ns. The maximum thermal stresses in fused silica and CeO\({}_{2}\) defect are both presented in (**b**). leading to even worse damage of fused silica optics. Based on the results above, we can conclude that the temperature and thermal stress caused by impurity defects strongly depends on the defect species and structural parameters. The CeO\({}_{2}\) defect can induce higher temperature and thermal stress than Al and Fe defects, especially when it is located more closely to the silica surface. The most dangerous subsurface impurity defect is the CeO\({}_{2}\) defect with a diameter of 0.3 \(\upmu\)m, located less than 0.15 \(\upmu\)m beneath the surface, which would induce local temperature up to 7073 K and von Mises equivalent thermal stress up to 8739 MPa. Therefore, we should pay close attention to control and remove this dangerous kind of subsurface defects in the practical manufacturing process of fused silica optics with high precision surfaces. By considering the synthetic effects of temperature and thermal stress caused by impurity defects, it can be concluded from the simulation results shown in Figures 8b-d and 7 that the CeO\({}_{2}\) defect plays the dominant role in decreasing the laser damage resistance of fused silica, followed by Al and Fe defects. It has been also experimentally observed by [PERSON] et al.