## 2 Materials and Methods  
### Equipment and Helical Drilling Principle  
The experimental platform is a five-axis CNC machine tool equipped with a femtosecond laser, a reflector box, and the helical drilling optical system from Fraunhofer Institute for Laser Technology, as shown in Figure 1. The helical drilling optical system mainly consists of the reflector M\({}_{1}\), half-wave plate, wedge-shaped prism, hollow motor, Dove prism, balance mirror groups, and focusing lens. The internal beam propagation path is shown in Figure 2. The wedge-shaped prisms 1 can adjust the diameter of the micro-hole by deflection, and the mirror M\({}_{2}\) can control the taper of the micro-hole by translation, based on the adjustment of laser incident angle and position. The Dove prism installed in the hollow motor can make the laser beam move in a circular motion around the center of the micro-hole. The optical aperture of the helical drilling system is about 25 mm. The mirror M\({}_{2}\) is fixed on a one-dimensional linear moving platform, which determines the shift distance \(L\) of the laser incoming on the surface of the objective lens. The wedge prism is installed on a rotating motor, named the hollow motor, and its deflection angle \(\varphi\) determines the angle \(\theta\) at which the laser is incident on the surface of the objective lens. The calculation principle is shown in Figure 3a. Figure 1: The experimental equipment. Figure 2: Laser beam propagation path in helical drilling optical system. Assuming the incidence beam before focusing is \(AB\), the angle \(\varphi\) is  
\[\tan f= \frac{\sin\alpha}{n-\cos\alpha}+\Delta f \tag{1}\]  
where \(\alpha\) is the angle between the two faces of a wedge-shaped prism, \(n\) is the refractive index of a wedge prism, \(\Delta\varphi\) is the deflection angle of the wedge prism set by controlling software. The calculation formula for the tilt angle of the beam after focusing is  
\[\tan\theta= \left|\tan\varphi-\frac{L}{f}\right| \tag{2}\]  
where \(L\) is the position of the laser incoming on the surface of the objective lens, \(f\) is the focal length of the objective lens. The formula for calculating the rotation diameter of the spot center on the focal plane is  
\[d=2f\cdot tan\varphi \tag{3}\]  
The theoretical diameter D of the processed micro-hole can be calculated by  
\[D=d+r \tag{4}\]  
where \(r\) is the radius of the focal spot, and \(r\) can be calculated as  
\[r=\frac{\lambda M^{2}f}{\pi D_{f}} \tag{5}\]  
where \(M^{2}\) is the beam quality factor, \(D_{f}\) is the diameter of the unfocused spot. From Equations (3) and (4), it can be found that the diameter of the micro-hole depends on the angle \(\varphi\), the focal length \(f\), and the focal spot radius \(r\). The focal length \(f\) and the focal spot radius \(r\) are decided by the objective lens system, the beam expansion system, and the laser beam quality. So, the diameter of the micro-hole is mainly related to the angle \(\varphi\). Considering the Rayleigh length of the focused spot, there is a certain deviation between the actual diameter and the theoretical diameter \(D\) of the micro-hole. The angle \(\theta\) decides the taper of the micro-hole. From Equation (2), while the angle \(\varphi\) is fixed, the taper of the micro-hole is mainly decided by the position \(L\) of the laser incoming on the surface of the objective lens, which is related to the translation distance of the linear moving platform. When the angle \(\theta\) is \(0^{\circ}\), the cylindrical holes can be drilled without any taper in theory, just as shown in Figure 3b. But, due to the shielding effect of the unremoved material on the incidence beam, the actual diameter of the outlet will be smaller than that of the inlet and  
Figure 3: The drilling principle of helical drilling optical system: (**a**) adjustment mechanism for diameter and taper; (**b**) drilling diagram of a cylindrical hole. thus produce a positive taper error.