## 2 Methods
As presented in Table 1, we aimed to construct an optical system that supported three types of imaging devices: wide-angle, telephoto, and intermediate ends. The reciprocal of the focal length of a lens is known as the refractive power or the power of the lens. If the refractive power is positive, a positive refractive power is obtained and vice versa. When indicating the type, the previous abbreviations were used and written as P (positive) and N (negative). In this optical system, the first group was determined to have a positive refractive power, such that the brightness of the wide-angle and telephoto ends was maintained. A Group 4 lens with the refractive power of the PNPP (a lens arrangement of positive-negative-positive-positive) was designed as the basis. Our optical system utilized the Gaussian bracket method to express the total refractive power and that overall length using Equation (7). Here, \(z\) is the distance between the principal surfaces of each lens group, \(k\) is the refractive power of each lens group, \(K\) is the refractive power of the entire lens, and \(T\) is the total length of the optical system. Further, the subscripts \(k\) and \(z\) represent the lens group numbers. The focal length of the optical system was determined by the sensor size of the wide-angle and telephoto ends of the imaging device, as expressed in Equation (7). Because the reciprocal of the focal length is the refractive power of the optical system, \(K_{1}\) and \(K_{2}\) are the refractive powers at the wide-angle and telephoto ends, respectively. \[ \begin{array}{c}[k_{1},-z_{1,1},k_{2},-z_{1,2},k_{3},-z_{1,3},k_{4}]=K_{1}\\ [k_{1},-z_{1,1},k_{2},-z_{1,2},k_{3},-z_{1,3},k_{4},-z_{1,4}]=0\\ [k_{1},-z_{2,1},k_{2},-z_{2,2},k_{3},-z_{2,3},k_{4}]=K_{2}\\ [k_{1},-z_{2,1},k_{2},-z_{2,2},k_{3},-z_{2,3},k_{4},-z_{2,4}]=0\\ z_{1,1}+z_{1,2}+z_{1,3}+z_{1,4}=T\\ z_{2,1}+z_{2,2}+z_{2,3}+z_{2,4}=T\end{array}  \tag{7}\]  
By solving the above Gaussian bracketing method and arranging the refractive power, the optical system was composed of four groups. Figure 5 shows an optical path diagram of a zoom lens wherein the image height was fixed in the form of a general zoom lens, and the angle of view changed such that the lens group moved and the focal length changed. It had a fixed whole length that utilized an internal focus movement method to move between Groups 2 and 4. For the fundamental optical system design with a refractive power arrangement, a spot diagram was checked to confirm the optical system performance. Figure 6a-c show the spot diagrams at the wide-angle, middle, and telephoto ends, respectively, in the case of an infinite object distance. The \(x\)-axis of the spot diagram represents the defocusing range, and the \(y\)-axis represents the maximum image height standardized to 1, indicating the number of fields. Because the pixel size of the sensor exceeded 1 mm, a 1 mm \(\times\) 1 mm square border was placed and checked to determine whether it was larger than the pixel size. As shown in Figure 6, the boundary line was exceeded in all zoomed cases. Further, lens shape improvement and aberration correction were required; therefore, performance improvement is required to obtain a focus smaller than the pixel size. \begin{table}
\begin{tabular}{c c c} \hline
**Detector** & **Number of Pixels** & **Pixel Size (mm)** \\ \hline Wide & 64 \(\times\) 64 & 1.5 \\ Middle & 128 \(\times\) 128 & 1 \\ Tele & 200 \(\times\) 100 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: The image detector proposed in this study. Figure 5: The fundamental layout of the zoom optical system is based on the placement of refractive power. Figure 6: \(Cont\).