## 3 Simulation Results and Discussion  
### Simulation Method  
Previous experimental procedures have demonstrated that micro-hole formation is closely related to the process of femtosecond laser ablation of materials, and thus precise control of micro-hole depth is desired due to the physical nature of femtosecond laser ablation of metals. As a traditional physical model, the two-temperature model can describe the heat transfer process inside the material very well.  
Figure 2: Femtosecond laser-induced surface micro-structure morphology by SEM at 120 \(\upmu\)J (**a**) 5 scans; (**b**) 10 scans; (**c**) 20 scans; (**d**) 30 scans.  
The two-temperature model considers the electron and lattice as two separate systems, with two separate equations describing the process of changing their temperatures as follows [26]:  
\[C_{e}\frac{\partial T_{e}}{\partial t}=k_{e}\bigg{(}\frac{\partial^{2}T_{e}}{ \partial^{2}}\bigg{)}-g(T_{e}-T_{l})+S(z) \tag{2}\]  
\[C_{l}\frac{\partial T_{l}}{\partial t}=-k_{l}\bigg{(}\frac{\partial^{2}T_{l}}{ \partial z^{2}}\bigg{)}+g(T_{e}-T_{l}) \tag{3}\]  
where \(g\) denotes the energy of the electron coupling coefficient [27]. The subscripts \(e\) and \(l\) denote the parameters of the electron and lattice, respectively, \(T\) denotes the temperature, and \(k\) denotes the thermal conductivity. Here, \(k_{e}\) varies with the ratio of the electron temperature to the lattice temperature as shown in Equation (4), while \(k_{l}\) is only related to the lattice temperature [28].  
\[k_{e}=k_{0}\frac{T_{e}}{T_{l}} \tag{4}\]  
where \(k_{0}\) denotes the electronic thermal conductivity [27].  
In Equations (2) and (3), \(C\) denotes the heat capacity, and \(C_{l}\) is considered as a constant [21]. \(C_{e}=\gamma T\epsilon\) denotes the electron heat capacity, and \(\gamma\) denotes the electron-specific heat coefficient [29]. In Equation (2), \(S(z)\) is the energy source of the system. In this model, only the energy change in \(z\)-direction is studied. The equation is as follows [25]:  
\[S(z)=\frac{(1-R)}{\big{(}L_{p}\big{)}}\cdot F\cdot\frac{\omega_{0}{}^{2}}{ \omega^{2}(z)} \tag{5}\]  
where \(R\) denotes the reflectivity of the material [30], \(L_{p}\) denotes the depth of ballistic transport [31], \(\omega_{0}\) denotes the beam radius, and \(\omega(z)\) denotes the variation of the beam radius in the \(z\)-direction, given by Equations (6) and (7) [32]:  
\[\omega(z)=\omega_{0}\bigg{(}1+\frac{z^{2}}{L_{R}^{2}}\bigg{)}^{1/2} \tag{6}\]  
\[L_{R}=\frac{u\cdot\pi\cdot\omega_{0}{}^{2}}{\lambda} \tag{7}\]  
where \(L_{R}\) denotes the Rayleigh length and \(\lambda\) denotes the femtosecond laser wavelength taken as 1030 nm, \(u\) is the refractive index and can be assumed to be 1 [32]. \(F\) in Equation (5) denotes the laser fluence and the relationship between the single pulse energy \(E\) is as follows [33]:  
\[F=\frac{2E}{\pi\cdot\omega_{0}{}^{2}} \tag{8}\]  
The laser device directly controls the single pulse energy in this study. The equations are solved using the finite difference method. The initial temperature of the lattice and electrons is assumed to be the ambient temperature (300 K). The lattice temperature is raised above the evaporation temperature (3003 K), and the material is removed to calculate the laser's ablation depth [34]. The enthalpies absorbed by the solid metal fusion and vaporization are recorded as \(\Delta\)Q. In the calculation, \(\Delta\)Q is converted to temperature to correct for the original temperature of each simulation grid. This method treats the latent heat absorbed when the metal fusion and vaporization as a correction for the temperature of each simulation grid. It should be clarified that, to improve the computational speed and scale, the change of the beam radius with depth on the length scale of the ballistic electron transport is ignored in this simulation. Tables 1 and 2 show the physical parameters of the target material as well as the processing parameters used. All calculations in this study were performed using nickel, the main element in DD15.