## 2 Materials and Methods
\[m_{T}=\frac{u_{0}}{u_{4}}=\frac{u_{0}}{u_{3}}\frac{u_{3}}{u_{4}}=m_{CL}m=\frac{ f}{f_{CL}} \tag{5}\]  
Here, as shown in Figure 2, because \(u_{0}\) and \(u_{4}\) do not significantly change according to the movement of the second lens group, \(m_{T}\) also hardly changes. Therefore, Equation (6) can be obtained by substituting Equations (4) and (5) into Equation (3) and rearranging them. \[[-z_{0},k_{1},-z_{1}-\Delta f,k_{2},-z_{2}+\Delta f,k_{3}]=\frac{1}{m_{CL}}= \frac{u_{3}}{u_{0}}=\frac{m}{m_{T}}=\frac{\Delta l}{m_{T}f}=\frac{f_{CL}\Delta l }{f^{2}} \tag{6}\]  
Therefore, the refracting power of each lens group can be obtained by solving Equations (1), (2), and (6) in series. Here, we have explained that f is the focal length of the test lens, which is determined as the reciprocal of k in Figure 2. \(\Delta l\) is also a determined value. In addition, if the motion distance \(\Delta f\) of the second lens group and the value between each lens group are properly determined, only unknown variables \(k_{1}\), \(k_{2}\), and \(k_{3}\) remain. Distance \(z_{0}\) from the target to group 1 is 5 mm. Distance \(z_{1}\) from the first to the second lens group is 40 mm. Distance \(z_{2}\) from the second to the third lens group is 30 mm. Motion distance \(\Delta f\) of the second lens group is 20 mm. Focal length f of the test lens is 4 mm. If the total focal length \(f_{CL}\) of the collimator optical system is set to 100 mm, \(\Delta l\) becomes 0.1 mm. We substitute this value into Equations (1), (2), and (6). Thereafter, \(f_{1}\)(=1/\(k_{1}\)), \(f_{2}\)(=1/\(k_{2}\)), and \(f_{3}\)(=1/\(k_{3}\)) can be obtained using the MATLAB programming codes. In Figure 3,\(f_{1}\),\(f_{2}\), and \(f_{3}\) represent the focal lengths and the reciprocal of the refracting power of each lens group. Figure 3 shows that three solutions can be obtained. In the first solution, the focal length of the first lens group is very short, and in the third solution, the focal length of the second lens group is very short. Thus, the optically meaningful solution becomes the second solution. The focal length of each lens group can be determined from the obtained solution. Assuming that these lens groups have no aberration, the paraxial layout is shown in Figure 2.0  
Figure 2.1 Result from solving Equations (1), (2), and (6) in a series of solutions using MATLAB codes. the last surface of the first lens group, respectively. The principal plane of each lens group is shown in Figure 5b. To commercialize our designed collimator system for mobile camera phone optics, we need to obtain a relational expression for the displacement of the second lens group and the defocus of the test lens. When the test lens is installed behind the designed collimator, the equation for the height of the axial ray is expressed in Gaussian brackets, as in Equation (7) [27,28]. Figure 2.2  
In Equation (7), \(\Delta f\) represents the motion amount of the second lens group, and \(\Delta l\) denotes the defocus of the test lens. In addition, when the crane design is completed, the refracting power and spacing of each lens group are determined. \(z_{3}\), which has not been previously mentioned, denotes the distance between the last group of the collimator and the test lens. The magnification, \(m_{T}\), calculated using Equation (5), can be expressed as Equation (8) using the Gaussian bracket [29]. Figure 2.3  
Except for \(z_{3}\) in Equation (8), the values of the other variables are already known. Therefore, \(z_{3}\) can be obtained from Equation (8). Hence, the unknown variables in Equation (7) are \(\Delta f\) and \(\Delta l\). In fact, in the optical system designed and shown in Figure 5, \(z_{3}\) is approximately 21.001 mm. The equations for both variables can be found using MATLAB codes.