## 5 Optimization Program and Design Results  
When the thickness of the thin lens is added and the distance between the lenses is adjusted, the height of the marginal ray on the surface of the lens will change. The curvature of the surface will also change, the longitudinal chromatic aberration will increase, and the lateral chromatic aberration in the thin lens system will be zero. When the thickness of the lens increases, the lateral chromatic aberration also increases. In order to maintain the desired chromatic aberration result for the thin lens design, we use an optimization program to eliminate both the increase in the longitudinal and lateral chromatic aberrations due to the influence of the lens thickness. A flow chart of the optimization program is shown in Figure 6.  
The eight triplet lens combinations selected by the illustration method (as shown in Table 2) are used as the starting values for optimization. During the optimization process, the thickness of the lens and the distance between the triplet lens are appropriately increased. The selected variables \(x_{0}\) are the respectively refractive powers of the triplet lenses k\({}_{\rm d_{1}}\), k\({}_{\rm d_{2}}\), and k\({}_{\rm d_{3}}\). The optimization functions include the focal length \(f_{d}\) of the triplet lens, the longitudinal chromatic aberration curve area (LCA), and the lateral chromatic aberration curve area (TCA) of the triplet lens group. The designed target value sets are \(t_{fd}\) = 10 mm, \(t_{LCA}\) = 0, and \(t_{TCA}\) = 0. The optimization function is  
\[\phi=\sum_{i=1}^{3}f_{i}^{2}=w_{1}^{2}(f_{d}-10)^{2}+w_{2}^{2}(LCA-t_{LCA})^{2 }+w_{3}^{2}(TCA-t_{TCA})^{2}, \tag{40}\]  
where \(\phi\) stands for the merit function; \(f_{i}\) is the design target; and \(w_{i}\) is the corresponding weight for the design target.  
The optimization results for the correction of the real longitudinal aberration and lateral chromatic aberration for the eight triplet lens groups selected by the illustration method are shown in Table 3, Figures 7 and 8. Figures 7 and 8 show graphs of the real longitudinal and lateral chromatic aberrations, respectively. The optimized focal length is 10 mm, and the real longitudinal chromatic aberration area value is up to \(10^{-7}\), which is similar to the chromatic aberration area obtained with the thin-lens illustration method, and the real lateral chromatic aberration area ranges from \(10^{-5}\) to \(10^{-6}\).  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(f_{d}\) (mm) & **LCA (mm\(\times\)\(\mu\)m)** & **TCA (mm\(\times\)\(\mu\)m)** \\ \hline A & 10 & \(5.6328\times 10^{-7}\) & \(8.2488\times 10^{-6}\) \\ \hline B & 10 & \(4.8447\times 10^{-7}\) & \(9.2612\times 10^{-6}\) \\ \hline C & 10 & \(5.8346\times 10^{-7}\) & \(8.1079\times 10^{-5}\) \\ \hline D & 10 & \(4.5206\times 10^{-7}\) & \(3.7359\times 10^{-5}\) \\ \hline E & 10 & \(9.7842\times 10^{-7}\) & \(4.4330\times 10^{-6}\) \\ \hline F & 10 & \(4.3850\times 10^{-7}\) & \(1.8181\times 10^{-5}\) \\ \hline G & 10 & \(3.3362\times 10^{-7}\) & \(2.1194\times 10^{-5}\) \\ \hline H & 10 & \(5.4256\times 10^{-7}\) & \(1.4754\times 10^{-5}\) \\ \hline \end{tabular}
\end{table}
Table 3: Optimization results for the eight triplet lens groups selected by the illustration method.