## 3 The Combination of a Thin Triplet Lens Using the Illustration Method
### Illustration Method  
According to Equation (22), the achromatic formulation of a thin triplet lens is \(\varepsilon_{\rm AB}=(\rm E/G)K_{d}\). If we want to obtain a smaller chromatic aberration \(\varepsilon_{\rm AB}\) between the A-line and the B-line, then the larger the value of the denominator G and the smaller the value of the numerator E. Coordinate diagrams for the 106 kinds of glass material (V\({}_{d}\), P\({}_{\rm dC}\), P\({}_{\rm AB}\)) being selected from the Schott Optical Glass Catalog and numbered as listed in Table 1 can be drawn. As shown in Figure 4, P\({}_{\rm AB}\) is on right vertical axis, V\({}_{\rm d}\) is on the left vertical axis, and P\({}_{\rm dC}\) is on the horizontal axis. Blue indicates data for the V\({}_{\rm d}-\)P\({}_{\rm dC}\) coordinate group and orange indicates data for the P\({}_{\rm AB}-\)P\({}_{\rm dC}\) group. For the blue data group in the V\({}_{\rm d}-\)P\({}_{\rm dC}\) diagram, the larger the area enclosed by the three glass options the better. Among them, blue No. 26 (N-FK58), No. 25 (N-FK51A), and No. 59 (N-PK52A) are farthest away from the blue data group, so these three types of glass material can be chosen as the benchmark, with the other two kinds farthest away from these three selected to obtain a larger area. However, the distribution of the orange data points in the P\({}_{\rm AB}-\)P\({}_{\rm dC}\) diagram is denser, and the area surrounding the three types of glass should be the smallest, which is difficult to observe by viewing the diagram. Thus, we  
Figure 4: Diagram of V\({}_{\rm d}-\)P\({}_{\rm dC}\) and \(\Delta\)P\({}_{\rm AB}-\)P\({}_{\rm dC}\). must expand the relative coordinates of the orange data points. A line shows the trend of the orange data points for linear adjustment. The function for calculating the trend is PAB = 0.5278 PdC + 0.3021. Let PAB be the adjusted coordinate value and \(\Delta\)PAB = PAB - (0.5278 PdC + 0.3021). The triangular area E\({}^{\prime}\) of the adjusted \(\Delta\)PAB - PdC is expressed as follows:  
\[\mathrm{E}^{\prime}=\Delta\mathrm{P}_{\mathrm{AB_{1}}}(\mathrm{P}_{\mathrm{dC_ {2}}}-\mathrm{P}_{\mathrm{dC_{3}}})+\Delta\mathrm{P}_{\mathrm{AB_{2}}}( \mathrm{P}_{\mathrm{dC_{3}}}-\mathrm{P}_{\mathrm{dC_{1}}})+\Delta\mathrm{P}_{ \mathrm{AB_{3}}}(\mathrm{P}_{\mathrm{dC_{1}}}-\mathrm{P}_{\mathrm{dC_{2}}}) \tag{27}\]  
\[\mathrm{E}^{\prime}=\mathrm{P}_{\mathrm{AB_{1}}}(\mathrm{P}_{\mathrm{dC_{2}}}- \mathrm{P}_{\mathrm{dC_{3}}})+\mathrm{P}_{\mathrm{AB_{2}}}(\mathrm{P}_{\mathrm{dC _{3}}}-\mathrm{P}_{\mathrm{dC_{1}}})+\mathrm{P}_{\mathrm{AB_{3}}}(\mathrm{P}_{ \mathrm{dC_{1}}}-\mathrm{P}_{\mathrm{dC_{2}}})=\mathrm{E}, \tag{28}\]  
Figure 4 can be obtained by re-plotting the coordinate values of \(\Delta\)PAB - PdC. We use the illustration method to design an example of a triplet lens combination. Looking at the section with the blue data point G in the figure, we first select glass No. 26 (N-FK58), and then choose the other two glasses in the data group which are the farthest apart PdC. The greater the base of the triangle, the larger the G value which can be obtained. At the same time, the three glass numbers selected also need to be close to the orange data point line, to obtain a smaller E value. Thus, we selected number 16 (N-KZFS11) and number 105 (P-SF6). The focal length shift \(\Delta\)f\({}_{\mathrm{S}}\) is defined as the difference between the focal length of any wavelength (\(\bar{f}_{\lambda}\)) and the focal length of the center wavelength (\(\bar{f}_{d}\)), as shown in the following equation:  
\[\Delta f_{\mathrm{S}}=f_{\lambda}-f_{d} \tag{29}\]  
The illustration method was used to select eight triplet lens combinations. Figure 5 shows the chromatic aberration curve, in which the vertical axis shows the focal length shift \(\Delta\)f\({}_{\mathrm{S}}\) and the horizontal axis shows the wavelength. Figure 6 shows the chromatic aberration curves for the eight groups comprised of three thin lenses selected by the illustration method.