## 2 Materials and Methods  
Figure 2 shows a layout of the paraxial optical path of our proposed focus-adjustable collimator. Here, k is the refracting power of each lens group, which is reciprocal to the focal length of each lens group [18; 19]. z is the distance between the lens groups, and u is the angle of the axial ray [20]. The subscripts of these variables indicate the number of lens group, and \"0\" (zero) indicates the object plane. \(k_{1}\)-\(k_{3}\) indicate the refracting power of the lens group that constitutes the collimator system, and k is the refracting power of the optical system to be tested. Figure 1: Test layout of the mobile phone camera optics. The axial ray from the target passes through the collimator and enters the test lens as a collimated beam. At this time, when some lens groups in the collimator system are moved, a diverged beam (not a collimated beam) is incident to the test lens. Therefore, the point where the axial ray incident to the test lens meets the optical axis becomes the object viewed by the test lens. In Figure 2, \(l\) denotes the object distance from the virtual target to the test lens. Here, when some lens groups of the collimator system are moved, the object distance can be varied from infinity to object distance 1. However, the target position is fixed, although the target size does not change. \(\Delta f\) in Figure 2 denotes the amount of movement of the lens group that moves in the collimator system, and \(\Delta l\) denotes the amount of change in the image plane of the test lens. In Figure 2, \(k\) represents the refracting power of each lens group. Thus, it is a value provided by the optical design of the collimator system. In addition, \(l\) can be considered as the minimum inspection distance. When \(k\) and \(l\) are given, \(\Delta l\) represents the value that can be calculated from the imaging equation [21]. Before the lens group moves in the collimator optical system, the collimated beam is incident to the test lens, and the magnification of the collimator optical system becomes infinite. If the magnification is calculated using the Gaussian bracket at the refracting power and interval of each lens group, the result is expressed in Equation (1) [22; 23]. In Equation (1), \(m_{\text{inf}}\) denotes the magnification when the collimated beam is incident to the test lens, and this value is infinite. z and k are the same as the values shown in Figure 2.  
\[[-z_{0},k_{1},-z_{1},k_{2},-z_{2},k_{3}]=\frac{1}{m_{\text{inf}}}=0 \tag{1}\]  
In addition, the refractive power KCL of the collimator system can be expressed by Equation (2), using the Gaussian bracket. \[[k_{1},-z_{1},k_{2},-z_{2},k_{3}]=K_{CL} \tag{2}\]  
Figure 2 shows that the first, second, and third lens groups are fixed. Therefore, the collimator system can be conveniently used. On the other hand, when the second lens group moves by \(\Delta f\), the magnification is calculated in the same manner as that in Equation (1); thus, it is expressed by Equation (3). \[[-z_{0},k_{1},-z_{1}-\Delta f,k_{2},-z_{2}+\Delta f,k_{3}]=\frac{1}{m_{CL}}= \frac{u_{3}}{u_{0}} \tag{3}\]  
Here, \(m_{CL}\) may be expressed as angle \(u_{0}\) of an axial ray starting from the target and angle \(u_{3}\) of an axial ray incident to the test lens when the second lens group moves. In addition, the virtual target created by the collimator system is imaged again by the test lens  
Figure 2: Paraxial layout of our proposed focus-adjustable collimator. into an image sensor, and the magnification of the test lens is calculated from the imaging equation. The result is calculated using Equation (4). \[m=\frac{u_{3}}{u_{4}}=\frac{\Delta l}{f} \tag{4}\]  
Finally, the magnification of the optical system, including the collimator optical system and the entire test lens, is denoted as \(m_{T}\), which is calculated using Equation (5).