## 2 Experimental Methods
### Determination of Focus and Processing Parameters of Nanosecond Fiber Laser System for PCD Material  
Laser processing must be performed within the correct range of the focal length and with appropriate parameters to obtain good machining quality. The processing parameters \begin{table}
\begin{tabular}{c c} Density (g/cm\({}^{3}\)) & 8.35 \\ Young’s modulus (GPa) & 900 \\ Poisson’s ratio & 0.086 \\ Thermal conductivity (W/mK) & 560 \\ Compressive strength (GPa) & 7.55 \\ Transverse rupture strength (GPa) & 1.04 \\ Fracture toughness (MPa \(\times\) m\({}^{1/2}\)) & 6.55 \\ Knoop hardness (GPa) & 50 \\ \end{tabular}
\end{table} 
Table 1: Properties of PCD. of the fiber laser system include the average power, scan rate, pulse width, pulse repetition rate, peak pulse power, and pulse energy. After the laser beam is focused by the lens, the beam will converge to a spot and then diverge. The different focusing positions of the laser affect the distribution of its processing energy. The focus position should be correct to achieve the highest energy density for processing and obtain better results and desired quality. If the laser beam is under-focused or over-focused on the surface of the workpiece, the energy density will be insufficient for effective material removal. The laser energy will accumulate on the surface of the workpiece, which will widen the cutting kerf width [21; 24]. In addition, a laser beam with sufficient energy will produce high-frequency sound when focused on the surface of the material [24]. This study first determined the focal length range based on the audio frequency used in the processing, and then observed the effect of different Z-axis positions within this range on the cutting kerf width and material removal during the processing of the PCD workpiece. The most suitable focal length setting was used in the subsequent processing experiments. The reactive fusion cutting mechanism [16] was used for PCD material processing, and a nozzle provided clean dry air (pressure six bar) to assist in the laser processing. The laser beam was focused by a lens; the focal spot size can be determined from Equation (1), and the depth of focus can be obtained from Equation (2) [24]:  
\[\text{Spot size}:\ 2\omega_{0}=\Big{(}4\ \times\lambda\ \times\ f\times M^{2} \Big{)}/(\pi\times\text{D}), \tag{1}\]  
\[\text{Depth of focus}:\ D\text{OF}=4\sqrt{2}\ \times\omega_{0}. \tag{2}\]  
where, \(\lambda\) denotes the laser wavelength, \(f\) denotes the lens focal length, \(M^{2}\) denotes the coefficient of laser-beam quality, D denotes the input beam diameter, and \(\omega_{0}\) denotes the spot size. It can be seen from the formula that a smaller spot diameter can be obtained by increasing the incident beam diameter or shortening the focal length. However, a smaller spot diameter will reduce the depth of focus. In laser processing, a larger spot diameter and longer depth of focus can improve the cutting efficiency, but the processing accuracy is poor. A smaller spot diameter and shorter depth of focus can improve the processing accuracy, but the processing efficiency is poor. The laser system used in this study was calculated to have a spot diameter of 36 \(\upmu\)m and depth of focus of 102 \(\upmu\)m.  
When the pulse repetition rate is fixed, the scan rate will affect the amount of energy accumulated by the laser in the same processing area. A pulse repetition rate will have a corresponding appropriate range of the scan rate. Much faster scan rate and very low pulse repetition rate will decrease the rate of overlap, which will cause a discontinuous corrugated surface and rough cutting edge; this will affect the processing efficiency and surface quality of the workpiece after processing. The gray area in Figure 1a is the unprocessed area, and the size of this area is the main factor affecting the processing quality.