## 4 Methodology: Optical Design of the Prime System  
### Specifications for Underwater Optical Systems  
The design was based on the specification requirements of Polaris. We first designed the prime lenses followed by zoom lenses. In terms of chromatic aberration, spherical-chromatic aberration, and different spectra, the executive software CODE V was powerful enough to obtain relatively small chromatic aberration when appropriate settings were used. This suggested that we could achieve satisfactory results for prime lens design without using the genetic algorithm.  
Underwater lenses are composed of a dome window and an air lens assembly, and the two components should be designed together. The initial architecture of the underwater lenses designed in this study was a dome window and a double-Gauss lens, as illustrated in Figure 9. We used quartz glass as the material for the dome window in consideration of water pressure and Schott lenses for the air lens assembly.  
Figure 8: (**a**) Polaris, (**b**) cabin configuration, and (**c**) data-processing center.  
Theoretically, in an optical system where the object is at infinity, the refractive indices of the object space and the image space are different. The refractive index of the object space is \(n\), and that of the image space is \(n^{\prime}\). The focal length is \(f\), and the image height is \(y^{\prime}\). The first and second principle points are \(P\) and \(P^{\prime}\), respectively; the first and second nodal points are \(N\) and \(N^{\prime}\), respectively. The incident and emergent angles of the chief ray are \(\overline{u}\). The included angle between the marginal ray and the optical axis is \(u\) in the object space and \(u^{\prime}\) in the image space. The heights of the incident point and emergent point of the marginal ray are \(h\) and \(h^{\prime}\), respectively. By using the optical invariant [12], we could infer the relationship among refractive index, focal length, angle of view, and image height, as expressed in Equation (8) [13].  
On the image surface  
\[H=n\left(\overline{h}u-h\overline{u}\right)=-nh\overline{u} \tag{2}\]  
On the object surface  
\[H=n\left(\overline{h}u-h\overline{u}\right)=n^{\prime}y^{\prime}u^{\prime} \tag{3}\]  
Equivalent image and object surface according to the optical invariant  
\[-nh\overline{u}=n^{\prime}y^{\prime}u^{\prime} \tag{4}\]  
Image height  
\[y^{\prime}=\frac{n}{n^{\prime}}\frac{h}{-u^{\prime}}\overline{u} \tag{5}\]  
Focal length  
\[f=\frac{h}{-u^{\prime}} \tag{6}\]  
Image height obtained by substituting Equation (6) into Equation (5)  
\[y^{\prime}=\frac{n}{n^{\prime}}f\overline{u} \tag{7}\]  
Image height in a nonparaxial optical system  
\[y^{\prime}=\frac{n}{n^{\prime}}fan\ \overline{u} \tag{8}\]  
The specifications of the underwater lenses designed for the full and blue-green spectra are listed in Table 3. The actual image height detected by the sensor was 21.6 mm, the pixel size was 7.13 \(\upmu\)m, and the spatial frequency was 70 lp/mm. The aperture and focal  
Figure 9: Optical system with the object at infinity.  
length were f/2 and 75 mm, respectively, and a half-angle of view of 12.20\({}^{\circ}\) was obtained. The half-angles of view and their weight ratios are listed in Table 4. We selected seven fields of view for the design of the prime lens. The weight of the on-axis field of view was the largest, and that of the outermost field of view was the smallest.