## 3 Results  
### Effect of Laser Repetition Frequency on the Material Removal Rate  
The nano\(-\)crystalline diamond coating was etched using repetition frequencies of 100 Hz, 500 Hz, 1000 Hz, and 10 kHz when the energy\(-\)flux density of the focused spot was 14.2 J/cm\({}^{2}\) and 56.6 J/cm\({}^{2}\). The morphology of the white\(-\)light interferometer scan is shown in Figure 7.  
As shown in Figure 7, the effect of repetition frequency on the etch profile is minimal. As the repetition frequency increases from 100 Hz to 10 kHz, there is no appreciable change in the etch width, but there is a slight increase in the etch depth, and the center of the ablation pits becomes rougher rather than more uniform. This conclusion is supported by the data in the bar chart, which show that the etch depth slowly increases while the etch breadth maintains a steady value at various repetition frequencies. The etching depth rose from 2.09 \(\upmu\)m to 2.25 \(\upmu\)m at a 14.2 J/cm\({}^{2}\) energy density and from 2.81 \(\upmu\)m to 3.08 \(\upmu\)m at a 56.6 J/cm\({}^{2}\) energy density. Due to the laser beam's Gaussian distribution, as the repetition frequency rises, the ablation in the core region becomes more pronounced and the etching depth increases correspondingly. When the femtosecond laser interacts with the material at high repetition frequencies (>200 kHz), a thermal accumulation effect is created [41], which has a significant impact on the morphology of femtosecond\(-\)laser\(-\)induced micro and nanostructures but is less noticeable at lower repetition frequencies. This is brought on by the lengthy delay between femtosecond laser pulses with low repetition frequencies. The temperature in the laser focus zone has already decreased to room temperature by the time the following pulse reaches the material, and this little thermal accumulation effect also causes the etching depth to significantly deepen with increased repetition frequency.  
The material removal rate at various repetition frequencies is calculated using the data in Figure 8 and the mathematical model for the material removal rate; the resulting curves are displayed in Figure 9.  
The etch depth increases gradually as the femtosecond laser's repetition frequency rises. However, as shown in Figure 9, the repetition frequency's overall impact on the  
In order to study the effect of laser repetition frequency on the material removal rate, the data in Figure 9 were nonlinearly fitted to make the Asymptotic function in Exponential type, where the independent variable is the repetition frequency and the dependent variable is the material removal rate. The corresponding fitting equations are shown in Equation (7) when the energy\(-\)flux density is \(56.6\) J/cm\({}^{2}\) and in Equation (8) when the energy\(-\)flux density is \(14.2\) J/cm\({}^{2}\).  
\[\mathrm{y}=240,099.899-20,449.467\times 0.99^{\mathrm{x}} \tag{7}\]  
\[\mathrm{y}=153,442.415-25,098.606\times 0.999^{\mathrm{x}} \tag{8}\]  
From the fitting curve in Figure 9, it can be seen that when the energy\(-\)flux density of the focused spot and the scanning speed are certain, due to the ablation equilibrium, the changes in the ablation width and ablation depth of the diamond film at different repetition frequencies are not obvious, and the removal rate of the material tends to be stabilized.