## 2 Experiments  
The samples used in this paper are experimental soda-lime glass sheets with a thickness of 1 mm and a size of 50 \(\times\) 50 mm, which are pre-processed to 15 \(\times\) 15 mm with an error range of \(\pm\)0.2 mm before the experiment. Mechanical parameters related to soda-lime glass are shown in Table 1. The experimental device is shown in Figure 1. The positioning accuracy of the 3D mobile platform (CE4550-I-PG, HUAYI LASER, Dongguan, China) is \(\pm\)5 \(\upmu\)m, and the repetition accuracy is \(\pm\)2 \(\upmu\)m. The X-Y working table is responsible for the plane movement, and the Z direction is realized by the movement of the camera controlled by the machine tool. The relevant parameters of the selected picosecond laser (YP-IR-30, HUAKUAI, Jieyang, China) are shown in Table 2. The position synchronized output (PSO) mode can be used in all stages of the trajectory, including acceleration, deceleration, and curve motion, to achieve uniform pulse energy irradiation on the processed object. Burst mode is used to select multiple high-frequency sub-pulses with the same repetition rate as the seed source through a controlled optical switch to form a pulse string output. Figure 1: Schematic diagram of picosecond laser with Bessel beam cutting soda–lime glass experimental system. (**a**) The structure diagram of the cutting system. (**b**) The physical diagram of the system. Picosecond laser outputs laser through the laser beam expanding mirror to obtain a radius of 3 mm Gaussian beam and Gaussian beam through the Bessel lens to obtain the Bessel beam. In this paper, the Bessel lens internal selection of the base angle \(\alpha\) was \(5^{\circ}\), refractive index was \(n=1.45\) for the axial cones, and the main Bessel beam cone angle \(\theta\) was calculated using Equation (1) as follows:  
\[\theta=\arcsin(n\times\sin\alpha)-\alpha \tag{1}\]  
The primary Bessel beam produced the secondary Bessel beam by using a telescope system with a scaling factor \(M=8\) (4f system). The cone angle \(\varphi\), the diameter \(D\), and the length \(l\) of the secondary Bessel beam were calculated by the following equations:  
\[\varphi=\arctan(\tan\,(\theta)\times M) \tag{2}\]  
\[D=4.8096/k\sin\,(\theta) \tag{3}\]  
\[l=\omega_{0}/M\tan(\theta) \tag{4}\]  
where \(\omega_{0}=3\) mm is the radius of the beam waist of the incident Gaussian beam, and \(k\) is the wave number of the medium. The intensity distribution of the secondary beam can be calculated by Equation (5):  
\[I\ =I_{0}\frac{z}{l}J_{0}^{2}(4.8096r/D)\exp\,(-2z^{2}/l^{2}) \tag{5}\]  
where \(J_{0}\) is the 0 th-order Bessel function, and \(I_{0}\) is the incident laser energy. The Bessel spot diagram is shown in Figure 2a, where the center spot and the surrounding interference ring together form the Bessel spot. The Bessel spot energy score was filamentary, its distribution in air was simulated using COMSOL Multiphysics 6.0, and the results are shown in Figure 2b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mechanical Properties** & **Standard Values** \\ \hline Density & 2.2 g/cm\({}^{3}\) \\ Poisson’s ratio & 0.14–0.17 \\ Thermal conductivity (20 \({}^{\circ}\)C) & 1.4 W/m \({}^{\circ}\)C \\ Mohs hardness & 7 \\ Refractive index & 1.45845 \\ Coefficient of thermal expansion & 5.4 \(\times\) 10\({}^{-7}\) \\ Young’s modulus & 72,000 MPa \\ Specific heat capacity (20–350 \({}^{\circ}\)C) & 670 J/Kg \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of mechanical parameters related to soda–line glass. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters Related to Processing Systems** & **Standard Values** \\ \hline Laser operating frequency & 50 kHz–1000 kHz \\ Pulse width & 15 ps \\ Maximum output power & 30 W \\ Wavelength & 1064 nm \\ Positioning accuracy of the platform & \(\pm\)5 μm \\ Repetition accuracy & \(\pm\)2 μm \\ Scaling factor of telescope system & 8 \\ Refractive index of the axial cones & 1.45 \\ \hline \hline \end{tabular}
\end{table}