# New Optical Design Method of Floating Type Collimator for Microscopic Camera Inspection
## 3 Results and Discussion  
The distortion of the designed optical system, as shown in Figure 4, is illustrated in Figure 3. The lens group is changed by the cam, which is the component that changes the rotational motion into linear motion [44; 45]. The angle at which the cam is rotated can be changed according to the mechanical design [46; 47; 48]. Therefore, in this study, the value obtained by normalizing the maximum rotation cam angle to 1 is referred to as the normalized cam angle. In Figure 4, a normalized zero cam angle indicates that the virtual image is created at an infinite distance by the collimator, and one cam angle means that the virtual image is made at a distance of 200 mm from the front of the test lens. Figure 4 shows the appearance of the rectangular grid in an optical system combined with a collimator and a test lens. Figure 3: Optical layout for the final design. In the optical design stage, the focal length of the test lens was 4 mm, and the image height was assumed to be 5.8 mm. Because the optical system distortion changes, the entire optical system magnification changes from \(-\)0.0293 to \(-\)0.0308. Therefore, although the target size must be changed according to the inspection distance, the target size must be fixed [49]. Depending on the magnification and distortion of the optical system, the target size can be calculated using Equation (5) [50,51]. As shown in Figure 3, the distortion \(D\) can be calculated as Equation (5), \(y_{\text{p}}\) is the height calculated by finite ray tracing, and \(y_{0}\) is the height by paraxial ray tracing. \[D=\frac{y_{p}-y_{0}}{y_{0}} \tag{5}\]  
\[D=\frac{m\cdot y-y_{0}}{y_{0}} \tag{6}\]  
\[y=\frac{1+D}{m}y_{0}\]  
Here, if the target size is \(y\) and the magnification is \(m,y_{0}\), the height of the chief ray at the image surface, becomes \(y_{p}=m\cdot y\). Therefore, Equation (5) can be converted to be Equation (6). In addition, assuming that the magnification of the optical system is \(-\)0.03, and the distortion is approximately +10%, the target size is 174 mm. Figure 5 shows the field MTF and the longitudinal aberration. Figure 5a,b describe the field MTF and longitudinal aberration when the inspection distance of the mobile phone lens is infinite, Figure 5c,d describe the field MTF and longitudinal aberration when the inspection distance is 396 mm, and Figure 5e,f when the inspection distance is 200 mm. The MTF decreases from the center of the screen to the edge, however, there is no problem when inspecting the phone lens. As shown in the longitudinal aberration in Figure 5, the field curvature is within \(\pm\)5 \(\upmu\)m, so the optical system aberration has been sufficiently corrected. Figure 4: Distortion graphs for the final collimator design. Figure 5: (**a**) The field MTF and (**b**) longitudinal aberration when the inspection distance of the mobile phone lens is infinite. (**c**) The field MTF and (**d**) longitudinal aberration when the inspection distance is 396 mm. (**e**) The field MTF and (**f**) longitudinal aberration when the inspection distance is 200 mm. To manufacture the optical system, as shown in Figure 3, we need to know the locus of all lens groups. Equations (1) and (3) must be used to determine the locus. All lens group locus can be obtained by combining Equations (1) and (3). Here, the unknowns are \(z_{1},z_{2}\), \(z_{3},z_{4}\), \(\Delta z\), and \(m\). However, it was previously assumed that the overall magnification \(m\) of the optical system was fixed, but \(m\) was not fixed when the actual design was applied. This is because the entire optical system distortion remains as the lens group moves [21]. It is extremely difficult to correct the optical system distortion that is not asymmetric around the aperture, as in the collimator system proposed in [20]. However, it is not necessary to determine the overall magnification of the optical system. Therefore, only Equations (1) and (3) need to be solved in parallel. Because there is a total of four unknowns, additional conditions are required [52; 53].