## 1 Introduction  
In recent decades, the strong development of the laser material processing industry has significantly contributed to the creation of high-quality micromachining products. Optical scanning devices are seen as an indispensable component in making that remarkable development. A two-axis galvanometer-based scanning system deflects a laser beam in both \(X\) and \(Y\) directions so the laser beam can be guided to any position on the workpiece. They are also used in many other systems along with many other criteria, such as high marking speed, high repeatability, high resolution, and high precision. The most used optical scanners are galvanometer-based, polygon, resonant, holographic, acousto- and electro-optic. Laser scanning technology is divided into two types: pixel-based and vector-based scanning. Polygon and resonant scanning systems represent pixel-based scanning technology with a deflected speed higher than 100 m/s [1,2], but the disadvantage is that the pixelation increase leads to low-precision laser beam positioning. In contrast to pixel-based scanning systems, vector scanning systems such as galvanometers, piezo and MEMs scanners allow the beam to be scanned along a defined trajectory, thereby providing higher spatial resolution and reduced processing time [3]. In addition, in a vector scanning system, optical deflectors such as EOD (electro-optical deflector) and AOD (acousto-optical deflector) provide high scanning speed and accuracy. Still, their disadvantages are small scanning ranges and very high costs. With a cost-effective and large scanning field, a galvanometer scanner is chosen as a competitive commercial solution for the micromaterial processing industry. The galvanometer scanning system was studied and designed in the early 1970s, with the first open-loop scanners, and continuously improved and developed in terms of design and technology to increase the marking speed, as well as the precise positioning [4; 5; 6]. The laser scanner has also been widely used in electronics technology [7; 8; 9; 10], biomedical and ophthalmic imaging [11; 12; 13; 14], and laser material processing [15; 16; 17; 18]. Nowadays, with a state-of-the-art laser material processing system, the two-dimensional galvanometer scanning system is available for commercial markets with high performance, such as positioning speed (75 m/s), low feedback resolution (0.012 mrad), accuracy (50 mrad pk-pk) and marking speed (5 m/s) [19]. However, the scanning system still has inherent disadvantages, such as scanned image distortion and decreased telecentricity. They are considered essential performance criteria which lead to the inhomogeneity of spots in the scan field when the scanning system scans along the two \(X\) and \(Y\) axes in combination with a telecentric f-theta lens. The scanned image distortion and telecentric errors increase with a larger scan field, which is related to the structure of the f-theta lens. The use of a telecentric lens is that the laser beam can move parallel to the optical axis to impact the working plane vertically, after passing through the focusing lens, so that the beam shape can be kept round over the scan field [20]. As a result, when using the 2D galvanometer system, the working scanning area of the laser machining system will be limited by the telecentricity of the f-theta lens. Ideally, the smaller the distance between the two mirrors in a conventional 2D scanning system, the better the performance. However, when increasing the input beam size to obtain a small spot size on the working plane, the \(X\)-axis scanning mirror is usually tilted at an angle to create a short enough distance between the two mirrors to obtain a higher telecentricity in practice. In consequence, the original center point position is shifted, and the telecentricity is thus decreased, as shown in Figure 1.  
Various methods of improving distortion and telecentricity for 2D scanning systems have been proposed. [PERSON] et al. proposed the targeted coordinate transformation to correct the distortion error [21].