## 2 Methods
### Correction of Longitudinal Chromatic Aberration of a Thin Triplet Lens  
Assuming that the total refractive power \(\mathrm{K_{d}}\) of the fixed three lenses is 0.1 mm\({}^{-1}\), and to calculate the primary color \(\mathrm{\epsilon_{FC}}\) and the second spectrum \(\mathrm{\epsilon_{dC}}\) for eliminating the longitudinal chromatic aberration, we apply the theoretical formulas shown in Equations (5)-(7):  
\[\mathrm{K_{d}}=\frac{1}{\mathrm{f_{d}}}\mathrm{=K_{d_{1}}+K_{d_{2}}+K_{d_{3}} =0.1\leavevmode\
obreak\ mm^{-1}}, \tag{5}\]\[\varepsilon_{\rm FC}=\frac{\rm K_{d_{1}}}{\rm V_{d_{1}}}+\frac{\rm K_{d_{2}}}{\rm V _{d_{2}}}+\frac{\rm K_{d_{3}}}{\rm V_{d_{3}}}=0, \tag{6}\]  
\[\varepsilon_{\rm dC}=\frac{\rm P_{dC_{1}}}{\rm V_{d_{1}}}\rm K_{d_{1}}+\frac{\rm P _{dC_{2}}}{\rm V_{d_{2}}}\rm K_{d_{2}}+\frac{\rm P_{dC_{3}}}{\rm V_{d_{3}}}\rm K _{d_{3}}=0, \tag{7}\]  
where \(\rm f_{d}\) is the focal length of the thin triplet lens and \(\rm f_{d}=10\) mm; \(\rm K_{d_{1}}\), \(\rm K_{d_{2}}\), and \(\rm K_{d_{3}}\) are the refractive powers of the first, second, and third lenses of the thin triplet lens group, respectively; \(\rm V_{d_{1}}\), \(\rm V_{d_{2}}\), and \(\rm V_{d_{3}}\) are the Abbe numbers of the first, second, and third lenses of the thin triplets, respectively; and \(\rm P_{dC_{1}}\), \(\rm P_{dC_{2}}\), and \(\rm P_{dC_{3}}\) are the relative partial dispersion power of the first, second, and third lenses of the thin triplet lens group, respectively.  
By solving Equations (5)-(7) simultaneously, the refractive power formulas for \(\rm K_{d_{1}}\), \(\rm K_{d_{2}}\), and \(\rm K_{d_{3}}\) can be obtained.  
\[\rm K_{d_{1}}=\frac{\rm V_{d_{1}}(\rm P_{dC_{2}}-\rm P_{dC_{3}})\rm K_{d}}{\rm V _{d_{1}}(\rm P_{dC_{2}}-\rm P_{dC_{3}})+\rm V_{d_{2}}(\rm P_{dC_{3}}-\rm P_{dC _{1}})+\rm V_{d_{3}}(\rm P_{dC_{1}}-\rm P_{dC_{2}})}, \tag{8}\]  
\[\rm K_{d_{2}}=\frac{\rm V_{d_{2}}(\rm P_{dC_{3}}-\rm P_{dC_{1}})\rm K_{d}}{\rm V _{d_{1}}(\rm P_{dC_{2}}-\rm P_{dC_{3}})+\rm V_{d_{2}}(\rm P_{dC_{3}}-\rm P_{dC _{1}})+\rm V_{d_{3}}(\rm P_{dC_{1}}-\rm P_{dC_{2}})}, \tag{9}\]  
\[\rm K_{d_{3}}=\frac{\rm V_{d_{3}}(\rm P_{dC_{1}}-\rm P_{dC_{2}})\rm K_{d}}{\rm V _{d_{1}}(\rm P_{dC_{2}}-\rm P_{dC_{3}})+\rm V_{d_{2}}(\rm P_{dC_{3}}-\rm P_{dC _{1}})+\rm V_{d_{3}}(\rm P_{dC_{1}}-\rm P_{dC_{2}})} \tag{10}\]  
Equations (2) and (8)-(10) are solved to obtain the curvature difference of each thin lens as follows.  
\[(\rm C_{1}-C_{2})=\frac{\rm K_{d_{1}}}{(\rm n_{d_{1}}-1)}, \tag{11}\]  
\[(\rm C_{3}-C_{4})=\frac{\rm K_{d_{2}}}{(\rm n_{d_{2}}-1)}, \tag{12}\]  
\[(\rm C_{5}-C_{6})=\frac{\rm K_{d_{3}}}{(\rm n_{d_{3}}-1)} \tag{13}\]  
Here, \(\rm C_{1}\) and \(\rm C_{2}\) are the curvatures of the first and second surfaces of the first thin lens group; \(\rm C_{3}\) and \(\rm C_{4}\) are the curvatures of the first and second surfaces of the second thin lens group; and \(\rm C_{5}\) and \(\rm C_{6}\) are the curvatures of the first and second surfaces of the third thin lens group, respectively.  
After obtaining the curvature of each thin lens, the refractive power of the thin triplet lens at a certain wavelength can be calculated by the following equation.  
\[\rm K_{\lambda}=(n_{\lambda_{1}}-1)(\rm C_{1}-C_{2})+(n_{\lambda_{2}}-1)(\rm C _{3}-C_{4})+(n_{\lambda_{3}}-1)(\rm C_{5}-C_{6}). \tag{14}\]  
The chromatic aberration correction of the three-piece glass lens is related to the glass's \(\rm n_{d}\), \(\rm V_{d}\), and \(\rm P_{dC}\). The 106 kinds of glass produced by the Schott Glass Company are numbered as shown in Table 1.