## 4 Results and Discussion
### Tolerance Analysis  
A camera optical system is a relatively large product. In optical systems such as the one devised in this study, many lenses are used, and a driving device for moving the lenses is used too. Therefore, it is expensive to manufacture such an optical system. Consequently, additional costs are incurred from disposal, if a failure occurs. However, defect rates will exist when the lens is assembled at the mass production stage and tested for resolution, such as MTF. The fundamental cause of such defects is that the lenses are decentered or tilted at the assembly stage. Nevertheless, resolution performance can be improved by decentering or tilting a specific lens group. Figure 13 is a histogram showing the sensitivities of spherical aberration and field curvature based on decentering and tilting of each lens group constituting the designed optical system. In Figure 13, SAM is spherical aberration, SAG is the sagittal component of field curvature, and TAN is the tangential component of field curvature. As illustrated in Figure 13, the tangential component of the field curvature changes considerably with respect to tilt, and the spherical aberration changes with respect to the decentering of the third group. Therefore, it is possible to correct the spherical aberration and field curvature caused by the assembly error of the designed optical system. It is also evident that the sagittal component of the field curvature does not change significantly due to the assembly error of the lens group. This can be explained by the third order aberration theory, because the sagittal component is generally about one third smaller than the tangential component [23]. Manufacturing errors for the optical system can be classified as being rotationally symmetrical and asymmetrical with respect to the optical axis. Here, rotationally symmetrical errors include lens shape, thickness, and material refractive index, and rotational asymmetry errors include centering, seat tilt, clearance, group decenter, and group tilt [24]. These errors allow us to calculate spherical aberrations and field curvatures that affect resolution degradation. In this case, the aberration may be corrected by adjusting the decenter and tilt of the specific lens group in order to counteract the aberrations generated during the assembly of the optical system. Figure 14 illustrates the results of spherical aberration and field curvature distribution of the whole optical system by randomly generating the above-mentioned errors of assembly and parts, and also illustrates the change of aberration distribution based on the decentering or tilting of a specific lens group. The blue dot represents the aberration calculated by the random number generated error. The red dot denotes that the spherical aberration is corrected by decentering of the third group. The green dot indicates that the field curvature is corrected by tilting the first group. The purple box here denotes a performance limit, and a resolution box within this range indicates good resolution  
Figure 13: The sensitivities for the decenter and tilt of the lens group. performance. The rest of the spherical aberrations remain after group 1 adjustment, which in turn may lead to defects. Therefore, this should be considered in the optical design stage so that there is no spherical aberration according to the tilt of group 1. To estimate the actual failure rate from the results shown in Figure 12, we need to analyze the process capability index (\(C_{pk}\)). \(C_{pk}\) can be calculated by Equation (13) [25]. \[ \begin{array}{c}C_{pk}=(1-K)C_{p}\\ C_{p}=\frac{USL-LSL}{6\sigma}\end{array} , \tag{13}\]  
where _USL_ represents the upper specification limit, \(LSL\) represents the lower specification limit, \(\sigma\) represents the standard deviation of the distribution, and \(K\) represents kurtosis. Here, _USL_ and _LSL_ are values determined from the inspection standard of the optical system.