## 4 Simulations and Discussion
### The Relationship between the Moving Distance of the MSON and the Oblique Incidence Angle  
Apart from the characteristic analysis, the relationship between the position of the MSON and the angle of incidence should be further stressed. The simplified optical path diagram of the schematic diagram in Figure 1a is shown in Figure 6. In Figure 6,_f1,f2_, and _f3_ represent the object focal lengths of the scan lens, the tube lens, and the high NA lens, respectively. The distance from the reflected beam to the optical axis is \(h\), and the angle between the reflected beam and the incident beam is \(\sigma\). The axis center of the x-axis galvanometer is at the same level as the focus of the scan lens (_f1_), tube lens (_f2_),  
Figure 4: Creation of the movable super-diffraction optical needle (MSON) at different oblique incidence angles. Contours of \(|E|^{2}\) in the _r_-z plane when oblique incident angles are **(a)**\(0^{\circ}\), **(b)**\(10^{\circ}\), **(c)**\(15^{\circ}\), and **(d)**\(20^{\circ}\). **(e)** and (f) are the depth of focus (DOF) and full width at half-maximum (FWHM) of the MSON when the oblique incidence angles are **(a)**\(0^{\circ}\), **(b)**\(10^{\circ}\), **(c)**\(15^{\circ}\), and **(d)**\(20^{\circ}\). The intensity distribution at the longitudinal plane, as marked by the black dashed line in **(a)**, is plotted in **(e)**. The intensity distribution at the transverse plane, as marked by the white dashed line in **(a)**, is plotted in **(f)**. Figure 5: **(a)** The depth of focus (DOF) of the movable super-diffraction optical needle (MSON) before and after phase compensation. The red curve denotes the DOF before compensation, and the purple dotted line denotes the DOF after compensation. **(b)** The full width at half-maximum (FWHM) of the MSON. The black line denotes the FWHM, the blue dashed line denotes the diffraction limit, and the green dashed line denotes the super-oscillation criterion of the lens. and the high \(NA\) lens _(/3)_. According to the geometric optical path shown in Figure 6, a mathematical model can be established. The relationship between the oblique incident angle (\(\theta\)) and the deflection angle of the x-axis galvanometer (\(\gamma\)) can be expressed as Equation (12), and the moving distance (\(h\)') of the MSON and the galvanometer deflection angle (\(\gamma\)) satisfy Equation (13). \[\tan\theta=\frac{f_{1}}{f_{2}\,^{\prime}\tan(\sigma)}=\frac{f_{1}}{f_{2}\tan(2 \gamma)}, \tag{12}\]  
\[h^{\prime}=f_{3}\tan\theta=\frac{f_{1}f_{3}}{f_{2}}\frac{1}{\tan(2\gamma)}. \tag{13}\]  
The radius selected for the high NA lens was 13.2 mm. The moving distances (\(h\)') of the MSON for different oblique incident angles (\(\theta\)) can be calculated by Equation (13). The results are shown in Figure 7. Figure 7a shows that the moving distances (\(h\)') of the MSON and the oblique incident angle (\(\theta\)) were non-linear. In contrast, as shown in Figure 7b, for the case where the oblique incidence angle was less than \(10\lx@math@degree\), the moving distances of the MSON and the oblique incident angle were approximately linear. When the oblique incident angle was \(10\lx@math@degree\), \(h\)' = 765.0176 \(\upmu\)m. The 2D galvanometer system can make the oblique incidence angle change in the range of [\(-10\lx@math@degree\), \(10\lx@math@degree\)]. Therefore, the MSON can be arbitrarily moved in a 2D focusing plane with a size of [\(-765.0176\)\(\upmu\)m, 765.0176 \(\upmu\)m]. This provides a strong basis for constituting small-scale vector scanning. Within the range [\(-765.0176\)\(\upmu\)m, 765.0176 \(\upmu\)m], by controlling the angle of oblique incidence, the MSON can move linearly on the two-dimensional focus plane to achieve vector scanning. When the range exceeds [\(-765.0176\)\(\upmu\)m, 765.0176 \(\upmu\)m], combined with mechanical scanning, the speed of laser processing can also be increased. Figure 6: The schematic of the geometric optical path.