## 2 Theory and Methods
### Laser Damage Test  
The laser-induced damage thresholds (LIDTs) of fused silica optics before and after HF acid etching are measured following R-on-1 test protocol to validate the effect of impurity on laser damage resistance. In this test protocol, the laser pulse with fluence far below the damage threshold is initially applied. Then, incident lasers with pulse fluence increasingly ramped up are adopted until damage is observed. The LIDT is defined as the lowest fluence at which the damage occurs [44]. The laser damage experiment is carried out using a Q-switched Nd: YAG pump lasers (SGR-Exra-10, provided by the Beamtech Company (Beijing, China). It is capable of steadily providing 1064 nm and 532 nm wavelength lasers with a 1-Hz repetition rate and 10-ns pulse width. The detailed parameters of the laser damage test are listed in Table 3.  
The setup used to test the LIDT of fused silica is sketched in Figure 4. The pump laser is effectively delivered and focused on the surface of tested fused silica sample by propagating through focusing lens, wedged splitter and reflection mirrors. The attenuator consisting of polarizer and half-wave plate is used to adjust the energy output of the laser system. The pulse energy of each laser shot is calculated and recorded by monitoring the partial light energy, split from the main pump lasers using a wedged splitter. The sample is firmly mounted on the 3D translation stage to assure its position is accurately adjusted during the laser damage test. The judgment of the laser damage initiation is crucial to the laser damage test in that the measured LIDT is directly determined by the laser energy at which the laser damage is initiated. The Normaski microscope camera with magnification ranging from 100 to 200 times is employed to image any permanent changes (i.e., laser-induced damage) on the tested sample surface. Furthermore, the laser damage morphology is also real-time monitored with a He-Ne laser scattering system as shown in Figure 4b. By travelling through the focus lens, the He-Ne probe lasers scattered from the tested surface can be focused on the photoelectric detector. The detector  
Figure 3: Sketch of the HF acid etching experiment (**a**) and the principle diagram; (**b**) for determining the etching depth using stylus profilometer in the etching process of fused silica optics.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pulse Width** & **Wavelength** & **Repetition** & **Beam Diameter** & **Incident Angle** & **Laser** & **Divergence** \\ \(\tau_{p}\)**/ns** & **\(\lambda\)/nm** & **Rate \(\
u\)/Hz** & **\(D\)/im** & **\(\theta\)/deg** & **Modal** & **Angle \(\theta_{d}\)/mrad** \\ \hline
10 & 355 & 1 & 390 & 0 & TEM\({}_{00}\) & \(\leq\)2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The experimental parameters for laser damage test on fused silica.  
linked with a phase-locked amplifier is capable of recognizing the weak scattering light signal. When the laser damage is initiated on the tested surface, the signal becomes stronger. As a result, the laser damage can be precisely judged by real-time monitoring the change of the scattering light signal.