## 4 Discussion  
Considering that the energy gap of quartz is 9 eV, electron excitation from the valence band to the conduction band by a 248 nm KrF laser (5 eV) under an extremely low absorption rate is impossible, such that no ablation trace can be found after direct ablation without the metal target, as shown in Figure 3a. However, with the metal sacrificial plate below, LIPA can effectively etch the quartz sample (Figure 3b). The plasma formation directly depends on laser power, such that the etching depth increases linearly with the irradiation influence, as shown in Figure 4.  
From the morphology result displayed in Figures 7 and 8, there appears to be another ablation mechanism when the metal target is absent. After reviewing the literature [22; 23], we propose a two-step ablation hypothesis during LIPA to describe the processing dynamics. A schematic of the process in our experiment is depicted in Figure 9.  
Figure 8: The ablation depth of the first and second ablation series. Figure 7: Photographs of the quartz sample without the stainless-steel target: (**a**) prior to secondary direct ablation; (**b**) after secondary direct ablation (in the red circle). The inset is the SEM image of the second ablation pattern. The excimer irradiation pulse train heats the stainless-steel target, and the martial temperature continuously rises, as shown in Figure 9a. When the fluence is high enough, the upper layer of the metal target is completely melted and vaporized, and a plasma plume including a metal ion and radical is formed, as shown in Figure 9b[19]. The metal vaporization threshold \(E_{th}\) can be expressed as follows[24]:  
\[\frac{E_{th}}{d\cdot\tau}=\frac{\pi^{3/2}(T_{v}-T_{m})}{\sqrt{2}A_{t}\tan^{-1} \sqrt{\frac{32\lambda_{t}}{\rho_{\text{s}}C_{\text{s}}\omega d}}}+\frac{\pi^{3 /2}(T_{m}-T_{0})\lambda_{s}}{\sqrt{2}A_{s}\tan^{-1}\sqrt{\frac{32\lambda_{t}}{ \rho_{\text{s}}C_{\text{s}}\omega d}}}, \tag{1}\]  
where \(T_{v}\) and \(T_{m}\) are the vaporizing and melting temperature values of the material, and \(T_{0}\) is room temperature; \(d\) is the laser spot diameter, and \(\tau\) is the pulse duration; \(\lambda_{t}\), \(A_{t}\), \(C_{t}\), and \(\rho_{l}\) are the mean values of thermal conductivity, absorptivity, heat capacity, and density in the liquid state. Likewise, \(\lambda_{s}\), \(A_{s}\), \(C_{s}\), and \(\rho_{s}\) are the mean values corresponding to the solid state. The kinetic energy of the ions and free radicals in the plasma vaporization is very high, in the range of a few to several hundred eV. Afterward, the kinetic energy is transferred to the rear surface of the quartz sample, the plasma plume heats the quartz substrate, and the temperature rapidly rises to the 2000 K melting point, at which the quartz sample is melted and vaporized. We assume the etching depth during each pulse as the same, and the etching duration as shorter than the pulse duration, such that the etching depth \(h\) after a single laser pulse of energy \(E\) can be expressed simply as follows:  
\[h=\alpha(E-E_{th})=\alpha(I\cdot S-E_{th}), \tag{2}\]  
where \(\alpha\) is the etching coefficient. The etching depth is expected to be a linear function of the influence \(I\) in a given laser spot area \(S\), which as validated by the result in Figure 4. While considering that the influence distribution is not the uniform, the influence in the central part is higher than at the edge, such that a slightly arched etched top instead of flat top on the quartz rear surface should be formed after the ablation (see Figure 9e). On the other hand, because of the thermal expansion and contraction strain after the rapid melting and cooling of quartz, some micro-cracks appear under the recast layer in Figure 3c,d[19]. Afterward, the metal vapor cools down and freezes, and the black microscale molten metal residuum remains on the rear surface of the quartz.