## 5 Femtosecond Laser Ablation Threshold of CVD Diamond Coatings  
The laser ablation threshold of materials is the minimum laser energy density required to induce irreversible surface damage during laser-target interaction. Obtaining the ablation threshold of a specific material is of great significance to the study of its machinability. Since the single pulse energy of an fs laser obeys the Gauss distribution (see Figure 14), the laser threshold of material \(\varphi_{th}\) is equivalent to laser fluence at the border of the effective ablation region, which can be presented by Equation (3), as follows:  
\[\varphi_{th}=\varphi_{0}\cdot e^{-D^{2}/2\omega_{0}^{2}}, \tag{3}\]  
where \(\varphi_{0}\) is the central energy density of fs laser and \(D\) is the diameter of the effective ablation region.  
Thus,  
\[D^{2}=2\omega_{0}^{2}(\ln\varphi_{0}-\ln\varphi_{th}). \tag{4}\]  
Given the threshold fluence \(\varphi_{th}\) holds constant under a specific effective number of pulses, it can be found that the function represents a curve of the first degree with respect to independent variable  
Figure 14: Schematic diagram of femtosecond laser energy distribution. The threshold of the target is equivalent to the laser fluence distributed at the border of the effective ablation region.  
Figure 13: Responses for the S/N of Ra and MRR measurements. The significant rank in the MRR response is \(f>E>v\), the Ra response ratio follows \(f>v>E\).  
In\(\varphi_{0}\) and dependent variable \(D^{2}\) with a slope of 2\(\omega_{0}\)\({}^{2}\) and an \(x\)-intercept of \(\varphi_{th}\). Since the relationship between laser fluence \(\varphi\) and single pulse energy \(E\) follows  
\[\varphi=\frac{2E}{\pi\omega_{0}^{2}}, \tag{5}\]  
Equation (4) can be expressed accordingly as the following equation:  
\[D^{2}=2\omega_{0}^{2}(\ln E-\ln E_{th}), \tag{6}\]  
where \(E_{th}\) is the threshold of single pulse energy under given conditions. In the case of continuous linear ablation of the diamond coating by an fs laser, multiple pulses arrive at the same spot. Thanks to the extreme narrow pulse width of the fs laser, which is much shorter than the thermal relaxation time, multiphoton absorption occurs and makes incubation effect severe. Some studies have suggested that the occurrence of the incubation effect enables the ablation threshold to be decreased, which is inversely proportional to the effective number of pulses [32]. It can be discovered from Figure 12a that when EUV equals 0.7873 (\(v=1.2\) mm/s), the ablation mechanism begins to transform from weak ablation to intense ablation, and when EUV reaches 0.8464 (\(v=1\) mm/s), the ablation of diamond coating is saturated. At this time, owing to the extremely high absorption coefficient of graphite, the material removal is hard to increase. Therefore, according to Equation (6), the saturated ablation threshold of CVD diamond coating can be calculated by obtaining the In\(E\)-\(D^{2}\) curve at the scanning speed of 1 mm/s.  
The beam waist \(\omega_{0}\) and the threshold of pulse energy \(E_{th}\) can be obtained from In\(E\)-\(D^{2}\) fitting (see Figure 15), and the laser ablation threshold of material \(\varphi_{th}\) can finally be calculated by Equation (5).  
The obtained beam waist \(\omega_{0}\) equals 45.05 \(\upmu\)m and the threshold of pulse energy can be found at 4.41 \(\upmu\)J. Thus, the fs laser ablation threshold of NCD diamond coating is 0.138 J/cm\({}^{2}\) under the effective pulses of 90, calculated by Equation (1). The calculated threshold is much smaller than that of SCD (8.80 J/cm\({}^{2}\)) [12] or PCD (1-1.6 J/cm\({}^{2}\)) [33].  
Figure 15: Scatter plot of the square of ablation width against the logarithm of laser power. The red line is the linear fitting of the plot, the fitting function is shown in the upper left.