## 2 The Working Principle of the Single-Axis Scanning System with Two Galvanometers  
The mirror-based scanning system is responsible for deflecting the beam relative to the working plane. With the advantage of high deflection speed and dynamics compared to the workpiece axis-based lateral movement system, this system significantly improves material  
Figure 1: The simulated image of (**a**) an ideal 2D scanning system and (**b**) a commercial 2D scanning system with an X-scan mirror tilted to the \(Z\) axis at an angle of 15 degrees with the deflection of beam position. processing efficiency [33]. Two-dimensional beam deflection systems typically consist of two galvanometers that can be rotated to deflect the beam for scanning in the \(x\) and \(y\) directions and are known as galvanometer scanners. The major disadvantage of this system is the dependence of the beam deflection of the second mirror on the beam deflection of the first mirror. With the incident beam aligned perpendicular to the axis of rotation of the first mirror, changing the tilt angle of the first mirror will change the tilt angle of incidence to the second mirror, making the trajectory of the second mirror not straight, leading to the scanning field profile not being a square [34; 35]. Figure 2 shows the telecentricity of different locations on the workpiece area of 25 \(\times\) 25 mm\({}^{2}\) using a commercial 2D scanning system. It can be easily seen that the telecentricity is not the same at the same position along the x and y direction from the center point of the workpiece. This is also evident in the distortion of the scan field for a two-galvanometer scanning system and is a major disadvantage. Following the scanning system, a focusing lens with a flat field is responsible for focusing the laser beam onto the flat surface of the workpiece. Most lenses follow a relationship with image height resulting in Y = f. tan(\(\theta\)) with zero distortion. However, it is not reasonable to use such a lens in a scanning system due to the velocity of the scanned spot increasing proportionally to the tangent of the scanning system. Since the motion of the laser scanner is required at a linear speed, conventional mirrors are not acceptable. The selected lens must maintain the relationship, in that changing the linear speed with Y = f. \(\theta\) will produce a linear scan rate suitable for the scanning system [36; 37]. Lenses like these are called F-theta. These lenses are designed with intentional negative or barrel distortion to counteract increased image velocity. The distance of the beam on the workpiece from the center point to the scanning field d is given by the equation  
\[\mathrm{d}(\theta)=\mathrm{f.\theta} \tag{1}\]  
which is, in turn, denoted as the focal length of the F-theta lens f and the beam deflection angle \(\theta\). Combining the scanning field of the scanning system and the f-theta lens can significantly improve the scanning position on the workpiece surface, while still having minor telecentricity. Even though it is less than one percent, it also causes significant positional distortion if the focal length of the F-theta lens is significant and will produce guide beam inclinations that cannot have perpendicular effects on the workpiece, resulting in the beam profile's non-circular shape. With the single-mirror galvanometer system, which orbits in a straight line with the Y-axis and has a scanning field in a broader axis than the two-axis system, telecentricity still exists due to the combination of F-theta lenses. It is because of the disadvantage of these two scanning systems that we propose an approach to eliminate the telecentricity to increase the accuracy of workpiece processing, and at the same time, improve the edge quality of the bore or groove. A scanning system using two  
Figure 2: Telecentricities at different positions on the \(X\) and \(Y\) axis with a scan area of 25 \(\times\) 25 mm\({}^{2}\) of the workpiece. mirrors is considered a solution for minimizing the telecentricity of the system.