## 3 Simulation Results and Discussion
### Simulation Process  
Figure 3a depicts the relationship established by the two-temperature model between the depth of femtosecond laser ablation of the target material and the laser fluence. The results show that the ablation depth increases rapidly when the laser fluence reaches a critical value for the ablation of the material and then gradually increases. It indicates that even at higher energy, the deeper the ablation depth appears unattainable, and the ablation rate is a matter of the efficiency of the industrial application of femtosecond laser, the efficiency of which has been questioned. The energy at the bottom of the micro-structure is difficult to process efficiently due to the defocusing effect and reflection loss, and a similar phenomenon is observed in laser drilling. Because the pulse energy in this experiment is roughly Gaussian, Equation (5) shows that energy in the \(z\)-direction decays from the focal plane (\(z=0\)) to both sides. Figure 3a it can be seen shows that the same pulse has different ablation depths for the material at different \(z\)-directional positions. As shown in Figure 1, the focal plane coincides with the upper surface of the specimen at the start of the experiment, and the part that interacts with the material is all below the focal plane. As a result, the \(z\)-directional distance involved in the following model is below the focal plane, and the distance above the focal plane is ignored. Based on our findings, the surface of the femtosecond laser-induced material primarily exists in the form of a neatly formed micro-hole. Figure 3b depicts the spatial position relationship between the simplified femtosecond pulse and the micro-hole to investigate the evolution law of the micro-hole depth direction. The diagram depicts the ablation of the material by pulses with a predetermined overlap rate, with the position of the pulse in the \(z\)-direction remaining constant as the number of scans increases. The depth of the micro-hole is calculated as the difference between the bottom position \(Z_{2}\) and the orifice position \(Z_{1}\). In addition to the above calculation results, the initial positions of the orifice and bottom of the micro-hole must be entered into the model for calculation. Three different single pulse energies of femtosecond laser processing experiments were performed at 80, 100, and 120 \(\upmu\)J, as shown in Figure 4, to validate the model's applicability. As previously stated, the coating properties are positively influenced when the material's surface undulations stabilise after micro-hole generation, and based on previous experimental results, \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Physical Parameters** & **Symbol** & **Value** & **Reference** \\ \hline Reflectance of the target material & \(R\) & 0.65 & [30] \\ Depth of ballistic transportation & \(L_{P}\) & 13.5 nm & [31] \\ Coefficient of the thermal conductivity of electron & \(k_{0}\) & 91 J/(m\(\cdot\)K-s) & [27] \\ Heat capacity of lattice & \(C_{l}\) & \(4.1\times 10^{6}\) J (m\({}^{3}\cdot\)K) & [21] \\ Thermal conductivity of lattice & \(k_{l}\) & \(3.496+0.026733\times T_{l}-1.11803\times 10^{-5}\,T_{l}^{2}+\) & [28] \\  & & J/(m\(\cdot\)K-s) & \\ Electron-specific heat coefficient & \(\gamma\) & \(188.86\) J/(m\({}^{3}\cdot\)K\({}^{2}\)) & [29] \\ Electron-phonon coupling strength & \(g\) & \(3.6\times 10^{17}\) W/(m\({}^{3}\cdot\)K) & [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the target material. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Physical Parameters** & **Symbol** & **Value** \\ \hline Pulse energy & \(E\) & 80/100/120 \(\upmu\)J \\ Laser fluence & \(F\) & 203.8/254.8/305.7 J/cm\({}^{2}\) \\ Repetition rate & \(f\) & 100 kHz \\ Scanning speed & \(\upsilon\) & 100/200/300/400/500/600/700 mm/s \\ Number of pulses at \(Z=0\) & \(n\) & 10/5/3.3/2.5/2/1.7/1.4 \\ Wavelength & \(\lambda\) & 1030 nm \\ Beam radius at \(Z=0\) & \(\omega_{0}\) & 5 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table}