## 4 Methodology: Optical Design of the Prime System
### Design and Verification of the Underwater Optical Camera  
The selected wavelengths from the blue-green spectrum are listed in Table 5. We selected green and blue light wavelengths because they have superior penetration ability in lakes. The weight ratio of the three selected wavelengths was 1:1:1, and the center wavelength was in the blue light range. The lens was designed to reduce the chromatic aberration value, particularly that of green and blue light, to a low point. In CODE V, we mainly used wavelength weights to attenuate chromatic aberration. As depicted in Figure 10, the lens consists of a dome window and an air lens assembly, and the optical system has 11 lenses in total. In addition to an infinite object distance, various object distances should be considered in photography. Therefore, we conducted tests with object distances of infinity, 10 m, and 5 m. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Half-Angle of View**} & \multicolumn{1}{c}{**Weight**} \\ \hline
0\({}^{\circ}\) & 3 \\ \hline
2.2\({}^{\circ}\) & 3 \\ \hline
4.5\({}^{\circ}\) & 2 \\ \hline
6.0\({}^{\circ}\) & 2 \\ \hline
8.4\({}^{\circ}\) & 2 \\ \hline
10.8\({}^{\circ}\) & 1 \\ \hline
12.2\({}^{\circ}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fields of view and their weights. \begin{table}
\begin{tabular}{c c} \hline \hline Resolution & 4800 \(\times\) 3600 (1700 K) \\ \hline Sensor size & 36 mm \(\times\) 24 mm (CMOS) \\ \hline Diagonal length & 43.2 mm \\ \hline Actual image height & 21.6 mm \\ \hline Object distance & Infinity \\ \hline Spatial cutoff frequency & 70 lp/mm \\ \hline Half-angle of view & 12.20\({}^{\circ}\) \\ \hline Focal length & 75 mm \\ \hline Aperture & f/2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Specifications of the novel underwater prime lens. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Wavlength**} & \multicolumn{1}{c}{**Weight**} \\ \hline
546.1 nm (\(n_{e}\)) & 1 \\ \hline
486.1 nm (\(n_{F}\)) & 1 \\ \hline
479.9 nm (\(n_{F}\)) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Weight ratios for wavelengths in the blue–green spectrum. The spot size diagrams in Figures 11-13 show how light rays are focused on the ideal imaging surface. The half-angle of view gradually increases from bottom to top, with the bottommost angle on the optical axis being 0\({}^{\circ}\) and the topmost angle being 12.2\({}^{\circ}\). The scales of the diagram are all 0.1 mm. Figures 11-13 present the spot size diagrams with an infinite, 10-m, and 5-m object distance, respectively. Figures 14-16 present the spherical aberration, astigmatism field curvature, and distortion with an object distance of infinity, 10 m, and 5 m, respectively. The scales of the spherical aberration range from \(-\)0.1 to 0.1. The optical axis does not have astigmatism, but the off-axis fields of view do. The distortion of this camera approaches 0 at an infinite object distance. When the object is closer, the negative distortion of the off-axis fields of view increases. The distortion ranges within \(\pm\)2%. Figure 11: Spot size diagram with an infinite object distance (generated by CODE V). Figure 10: Optical design layout of the underwater lenses. Figure 12: Spot size diagram with a 10 m object distance (generated by CODE V). Figure 13: Spot size diagram with a 5 m object distance (generated by CODE V). The modulation transfer function (MTF) is another method for inspecting the quality of an optical system. The horizontal axis is the spatial frequency, and the vertical axis is the modulation. The highest spatial frequency was defined as 70 lp/mm; the highest and lowest modulations were 1 and 0, respectively. Figures 17-19 are the MTF curves with an infinite, 10-m, and 5-m object distance.