## 3 Results and Discussion
### Temperature Distribution Caused by the Subsurface Impurity Defects
#### 3.1.2 Effect of Impurity Defect Parameters on Temperature Distribution  
In order to figure out the most dangerous impurity defect to the laser damage resistance of fused silica, we investigate the temperature distributions caused by impurity defects with various structural parameters (e.g., defect radius \(r\), defect depth \(d\)). The effect of defect structural parameters on its induced temperature rise is demonstrated in Figure 7. As shown in Figure 7a, after being exposed in lasers for 10 ns, the maximum temperatures caused by impurity defects ascend sharply and then decrease gradually with the increase of defect radius. The critical defect radius corresponding to the highest temperatures for Al and Fe impurity defects is 0.15 \(\upmu\)m. The highest temperatures are 3037 K and 2965 K, respectively, for the Al and Fe defects. Meanwhile, for the CeO\({}_{2}\) defect, the critical radius is 0.15 \(\upmu\)m with the highest temperature of 4531 K. Temperature with this order of magnitude is high enough to incur matter evaporation, or even plasma formation, which would consequently induce local volume expansion, surrounding material fracture and finally laser damage. The phenomenon of maximum temperature evolution with respect to the defect radius can be explained as follows: as the defect radius increases, the defect area irradiated by incident lasers will enlarge, resulting in more laser energy to be absorbed by the defect. However, the enlarged irradiation area of impurity defect will increase the energy loss by thermal diffusion as well. Besides, with the increase of defect radius, the increase rate of absorbed energy would gradually decrease owing to the Gaussian distribution of the incident laser energy. Based on this, there must be a critical defect radius existing between the heat absorbing and diffusing processes, at which the temperature would reach the summit. This is consistent with the results shown in Figure 7a. Figure 7b shows the dynamic evolution of maximum temperature versus the defect radius. One can see that during the laser loading period (\(t<\) 10 ns), the maximum temperature increases firstly and then decreases gradually similar to the results as shown in Figure 7a. When the laser pulse ends (\(t>\) 10 ns), the temperature keeps rising all the time with the increase of defect radius. This is because under the same heat diffusing circumstance, an impurity defect with large size will hold more thermal energy and correspondingly induce a higher temperature. This agrees with the explanation discussed above. The simulation results at the loading times before (\(t=\) 1 ns, 3 ns, 5 ns, 7 ns, 10 ns) and after (\(t=\) 30 ns, 50 ns, 70 ns, 100 ns, 300 ns) the incident laser  
Figure 6: The temperature distributions on the longitudinal cross-section (\(z\)-\(r_{\mathrm{p}}\) plane) caused by CeO\({}_{2}\) defect for laser loading times of (**a**) 10 ns; (**b**) 100 ns; (**c**) 1000 ns and (**d**) the temperature distributions along the central axis (\(r_{\mathrm{p}}=\) 0 \(\upmu\)m) caused by CeO\({}_{2}\) and Al (**e**) defects for various laser loading times. heating process as shown in Figure 7b validate that the maximum temperatures caused by impurity defects with various radii generally take place at the ending time (\(t=10\) ns) of the laser pulse. Hence, the variations of maximum temperatures at \(t=10\) ns with respect to the defect radius and depth are summarized in Figure 7a,c to obtain the most dangerous impurity defects in the polished fused silica optics. Figure 7c exhibits the dependence of maximum temperature on the defect depth. It is shown that the maximum temperature caused by impurity defect is higher when it is located more closely to the fused silica surface. When the defect depth is deeper than 0.3 \(\upmu\)m, the maximum temperature will keep constant as the defect depth increases. This is because the heat conductivity coefficient of ambient air (0.023 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)) is lower than that of fused silica materials.