# Design Study of a Large-Angle Optical Scanning System for MEMS LIDAR
## 3 Scanning Optical System Design
For the \(f\)-\(\theta\) lens, it has a specific scanning speed, Equation (6) is derived for scanning time \(t\):  
\[\frac{dh}{dt}=-{f^{\prime}}_{1}\frac{d\gamma}{dt}=-2\
u f^{\prime} \tag{6}\]  
where, \(\
u\) is the rotation angular velocity of the MEMS mirror. As the image height \(h\) and the scanning angle \(\gamma\) of the \(f\)-\(\theta\) lens satisfy the linear relationship, the linear control of the scanning speed can be achieved by controlling the scanning angle \(\gamma\). If the MEMS mirror is deflected at a constant speed, the focused spot of the laser beam can be moved at a constant speed to realize a uniform scanning. Set the tilt angles of the MEMS mirror in ZEMAX in multiple configurations to simulate its actual rotation. Due to the symmetry of the rotation angles, we only set \(0^{\circ}\), \(1^{\circ}\), \(2^{\circ}\), \(3^{\circ}\), \(4^{\circ}\) and \(5^{\circ}\). When the mirror is not deflected (relative to the position \(0^{\circ}\)), the size of the diffuse spot is within \(2\) um, then as the deflection angle increases to \(1^{\circ}\), the diffuse spot is within \(4\) um. The maximum deflection angle is \(5^{\circ}\), and the diffuse spot less than \(20\) um. The dispersion spot distribution at the optical system image plane is shown in Figure 5. Light in green, blue and red referring to the laser wavelength of \(1520\) nm, \(1550\) nm, and \(1570\) nm, respectively. The Airy disk radius is given by \(1.22\) times the wavelength times the F number of the beam, which in general depends upon field position and pupil orientation. If all the rays are well within the Airy disk, then the system is often said to be \"diffraction limited\". From Figure 5, we can see that the light spots are all within the Airy disk, meeting the design requirements. Figure 6 shows the field curvature and distortion diagram of the optimized \(f\)-\(\theta\) lens. It can be seen that the relative distortion is less than \(0.5\%\), which satisfies the design requirement. The linearity of the optimized \(f\)-\(\theta\) lens system is very high. Figure 4: Structure diagram of \(f\)-\(\theta\) lens design by ZEMAX, \(0^{\circ}\) in blue, \(1^{\circ}\) in green, \(2^{\circ}\) in red, \(3^{\circ}\) in gray, \(4^{\circ}\) in purple, and \(5^{\circ}\) in wathet. Figure 5: Light spots with the rotation angles \(0^{\circ}\), \(1^{\circ}\), \(2^{\circ}\), \(3^{\circ}\), \(4^{\circ}\), \(5^{\circ}\), the circles are the Airy disks. The light beam from the collimating lens group should be parallel or approximately parallel light, which must ensure that the object focal plane of the collimating lens group coincides with the image focal plane of the _f_-\(\theta\) lens. To obtain a field of view not less than \(\pm 25^{\circ}\), the focal length of the collimating lens group \(f\)2\({}^{\prime}\) should be:  
\[f_{2}^{\prime}\leq\frac{f_{1}^{\prime}tan\gamma}{tan25^{\circ}}\approx 13.10\ \text{mm} \tag{7}\]  
In order to extend the scan angle to a greater extent, the focal length of the collimating lens group is determined to be 12 mm. The collimating lens group is composed of two approximately symmetrical doublet lenses. Its two-dimensional structure is shown in Figure 7. This structure can not only correct coma and astigmatism, but also has a compact structure and small field curvature. Combining the doublet lens L\({}_{0}\), _f_-\(\theta\) lens L\({}_{1}\), common mirror, MEMS mirror, and collimating lens group L\({}_{2}\) to complete the overall system design. The three-dimensional structure of the large-angle MEMS scanning optical system is shown in Figure 8. The parameters related to the design are listed in Table 2.  
Figure 3:0 Field curvature and distortion of _f_-\(\theta\) lens. Figure 3:1 Two-dimensional structure diagram of symmetrical cemented lens group.