## 2 Theory and Methods
### Sample Preparation, SSD Removal, and Etching Process Characterization  
In the practical chemical-mechanical polishing process of fused silica optics, mechanical and chemical actions are both involved in generating the polished surfaces by mutual contact, friction, squeezing and deformation between polishing agent and optical surface. Under the combined work of mechanical and chemical actions, the optical materials are removed, leaving the subsurface damage (SSD) distributed beneath the polished surface. The SSD distribution profile is presented in Figure 2, which includes re-deposition, defect and deformed layers [17; 26; 31; 43]. The re-deposition layer consists of impurity defects and silica compounds, resulting from the hydrolysis reaction of fused silica. Figure 1: The finite element method (FEM) model for simulating the temperature and thermal stress distributions initiated by impurity defects under intense laser irradiation: (**a**) the FEM model; (**b**) the designed gridding; (**c**) the spatial energy distribution of applied Gaussian laser pulse. The defect layer is mainly formed by cracks and scratches introduced in the initial rough grinding process. Besides, in the final polishing process, the impurity defects are easily imbedded and hidden inside the defect layer as well. The deformed layer is generated by the deformation of optical materials, which ascribes to the pressure stress of grinding wheel and buffing pad in the processing processes. To promote the laser damage resistance of fused silica optics, the HF acid etching technique is employed in this work to remove the subsurface impurity defects. In the etching process, the topside re-deposition layer including most of the impurity defects is firstly dissolved by chemical reaction between the HF acid and hydrolysis products. Then, the crack and scratch defects are blunted and the embedded impurity can be also easily wiped off. In this way, the detrimental impact of these SSD defects on the laser damage resistance can be greatly mitigated for the silica optics. In the sample preparing process, W14-sized SiC abrasive particles are adopted to grind the ten pieces of initial fused silica samples (30-mm diameter and 5-mm thickness) for half an hour. The ground samples are then polished for 2 h with 1-um-diameter CeO\({}_{2}\) polishing slurry to ensure that large brittle scratches and cracks are totally removed or buried under the re-deposition layer. The grinding and polishing spindle speeds are 50 r/min and 40 r/min. The grinding and polishing solution concentrations are 10 wt % and 8 wt %, respectively, with a 22-kPa load pressure. Figure 3 shows the HF acid etching schematic for removing the subsurface impurity defects. The HF-based etchant with 2% HF (mass fraction) and 5% NH\({}_{4}\)F (mass fraction) is applied with solvent of deionized water. The NH\({}_{4}\)F solvent, regarded as buffered oxide etch (BOE), is added in the etching process to promote the F\({}^{-}\) population and eventually guarantee the stable etching rate and low evaporation pressure. It should be noted that one half side of the topmost sample surface is set as reference with the paraffin applied over it to prevent the acid etching. Meanwhile, the other half side of the polished silica surface is etched when submerged into the HF-based etchant. After different periods of acid etching, the samples are cleaned with deionized water and the coated paraffin layers are removed with acetone reagent. The effectiveness of acid etching on removing the subsurface defects of polished silica optics is presented by the etching depths at various etching times, which can be directly calculated from a key parameter of etching rate. The etching rate in this work is measured with a stylus profilometer and the principle diagram of the measuring process is shown in Figure 3b. For silica samples etched for various periods, the profilometer stylus scans perpendicularly to the dividing line of the un-etched and etched surfaces to obtain the height profiles before and after HF acid etching.