## 2 Experiments and Measurement Methods  
The grinding experiment was conducted on the UPL-450 ultra-precision three-axis grinder developed by National University of Defense Technology [26], as shown in Figure 1. The ultra-precision grinder draws on the layout and structure of the ultra-precision lathe bed. The X-axis is used to adjust the lateral movement of the workpiece relative to the grinding wheel and to control the workpiece feed during the grinding process. The Z-axis is used to adjust the axial position of the grinding wheel relative to the workpiece and to control the feed in the cutting depth direction during the grinding process. The grinder adopts an open horizontal structure with air static pressure spindles for both the grinding wheel spindle and the workpiece rotary spindle. The grinding wheel spindle rotation accuracy is less than 50 nm, and the workpiece spindle rotation accuracy is less than 25 nm. The X-axis and Z-axis of the grinder's translation axis both use liquid static pressure guides and nano-level drive feed control systems with linear motion accuracy better than 0.4 \(\upmu\)m/400 mm, and the grinder takes measures of temperature control and vibration isolation. The grinding mode adopted is cross grinding. During the grinding process, the workpiece spindle and the grinding wheel spindle rotate simultaneously, and the Z-axis feeds slowly along the cutting depth direction. The process parameters of the grinding experiment are shown in Table 1. The grinding experimental parameters that affect the subsurface damage mainly include grain size of wheel, cutting speed, depth of cut, workpiece rotation speed and workpiece feed speed, etc. In order to explore the correlation relationship between subsurface depth, subsurface damage and laser damage characteristics after grinding, it was necessary to obtain different results of subsurface damage depth. Therefore, the cutting speed and depth of cut in grinding experiments were changed to obtain different results of subsurface damage. Meanwhile, a fine-grained grinding wheel, low workpiece rotation speed and low workpiece feed speed were chosen to achieve ultra-precision grinding and ensure good surface integrity of fused silica samples. The grinding wheels were diamond flake wheels with a grain size of 5 \(\upmu\)m from [PERSON] [PERSON] DIAMANMTETALL(r) AG, Weilheim, Germany. Moreover, the surface integrity and subsurface crack distribution of fused silica have been investigated under different depths of cut and cutting speeds in ultra-precision grinding in a previous study [26]. The published materials could provide references in terms of surface roughness and subsurface crack depth, but they did not involve the study of laser damage characteristics of fused silica. Based on the previous study, this work delves deeper into assessing subsurface laser damage characteristics by employing damage precursor multimodal analysis techniques, in order to obtain the correlation relationship between subsurface depth, subsurface damage and laser damage characteristics of fused silica after ultra-precision grinding. The fused silica samples in the experiment were sourced from Qinhuangdao Aouaqurtz High-Technology Development Co., Ltd, Qinhuangdao, China. with a specimen size of \(\mathcal{O}\)50 mm \(\times\) 10 mm. The samples were processed by chemical mechanical polishing, and the surface roughness Ra was about 1 nm under a ZYGO NV700S white light interferometer. Cracks were not detected on the subsurface of the samples. Measurement of subsurface damage depth was carried out using the damage precursor multimodal measuring instrument combined with the MRF spot method. First, an MRF machine was used to throw two polishing wedges on the surface of the sample to be measured, as shown in Figure 2a. Two wedges of each sample were measured with the same test parameters. By averaging the test data of two MRF wedges, the SSD results of each sample were obtained. After polishing, the surface was covered with a hydrolytic layer to cover up the subsurface cracks.