## 2 Methods
Figure 4 shows the optical path diagram of a thick lens where \(n\) is the refractive index, \(h\) is the height of the ray incident on the principal plane, \(H\) is the principal point, \(u\) is the paraxial angle, and its subscript refers to the plane number. Further, \(l\) and \(l^{\prime}\) are the object and image distances, respectively, and \(z\) refers to the position of the principal surface. According to the analytical equivalent lens transformation method, the height of incidence on the principal plane does not change. Using this concept, the heights of the \(h_{1}\) and \(h_{2}\) incidents on each side of Figure 4 can be expressed using Equation (2). \[ \begin{split} h_{1}&=h-zu_{0}=h-\left(\frac{n_{0}u_ {0}d_{1}}{n_{1}K}\right)k_{2}\\ h_{2}&=h-z^{\prime}u_{2}=h+\left(\frac{n_{2}u_{2}d_ {1}}{n_{1}K}\right)k_{1}\end{split}  \tag{2}\]  
Figure 3: The angle of view is according to the focal length. where \(k\) and \(K\) are the refractive powers of the relevant surface and the entire lens, respectively. By substituting Equation (2) into the refraction equation in Equation (3), we obtain Equation (4). \[n_{i+1}u_{i+1}=n_{i}u_{i}-h_{i+1}k_{i+1} \tag{3}\]  
\[ \begin{gathered}\Big{(}h-\frac{d_{1}n_{0}u_{0}k_{2}}{n_{1}K} \Big{)}k_{1}=n_{0}u_{0}-n_{1}u_{1}\\ \Big{(}h+\frac{d_{1}n_{1}+n_{1}k_{1}}{n_{1}k}\Big{)}k_{2}=n_{1}u_{ 1}-n_{2}u_{2}\end{gathered}  \tag{4}\]  
Equation (4) is a nonlinear simultaneous equation for \(k_{1}\) and \(k_{2}\). When the \(k_{1}\) and \(k_{2}\) terms are eliminated, they are expressed as a simultaneous linear equation in Equation (5). \[k_{2}=\frac{n_{1}u_{1}K-n_{2}u_{2}k_{1}}{n_{0}u_{0}} \tag{5}\]  
The simultaneous equations in Equation (5) are converted into quadratic equations for \(k_{1}\), as shown in Equation (6), and the solution to this equation becomes an equivalent lens. \[\frac{n_{2}u_{2}d_{1}}{n_{1}K}k_{1}^{2}+(h-d_{1}u_{1})k_{1}+n_{1}u_{1}-n_{0}u_ {0}=0 \tag{6}\]  
The inspection distance and range of the optical system must be fixed. For a zoom optical system, the position of the image surface must be fixed; therefore, the internal lens or specific lens group has to be moved [37; 38]. An internal focus control method was used to fix the position of the image surface and move the internal focus control group to the target inspection distance such that the overall length of the optical system did not change. Further, Group 1 in the optical system must be arranged to have a positive refractive power. This is because there should be no difference in brightness between the wide-angle and telephoto ends. In this study, the CODE V (Synopsis Inc., East Boothbay, ME, USA) optical design software was used to determine the angle of view corresponding to the three types of imaging devices proposed in the THz wavelength range, and the optical system was constructed by arranging the refractive power of each lens. We aimed to design an optical system wherein the angle of view was fixed, and only a specific lens group was moved such that the image height varied depending on the size of the imaging element. Aberration is not a problem if it is smaller than the array size of the detection module. Therefore, the optical aberration must be corrected at the millimeter level. Further, we aimed to conduct a performance analysis by checking the contrast ratio of the pixels using Light Tools (Synopsis Inc.), which is a lighting optical design software package. As the wavelength in the THz band is very long, the pixel size is considerably larger than that of a visible-light pixel. An imaging device becomes very large if the number of pixels increases; therefore, only imaging devices with a low number of pixels are expected to be developed. A wide-angle end with a short focal length and a telephoto end with a  
Figure 4: Thick single-lens optical layout at finite point. long focal length is required to construct a zoom lens with a variable focal length. This is also referred to as the normal or middle stage between the wide-angle and the telephoto ends.