## 2 Materials and Methods  
### Theory  
As mentioned in Section 1, the nodal slide or magnification methods are generally used to measure the EFL of an optical system [9]. Figure 1 shows the layout of the focal length measurement system using a nodal bench. As shown in Figure 1, the measurement system is composed of a collimator, which converts light from a light source into a collimated beam, a test lens to be measured, and an image analyzer, which magnifies the image formed by the test lens [10]. In other words, light starts as a pattern from an object plane illuminated by a light source and is converted into a collimated beam by the collimator lens. The collimated beam then passes through the test lens to form an image and, finally, an image analyzer magnifies the image to ascertain the exact image position. Here, if the sizes of the object plane pattern and the image formed by the test lens are known, we can determine the image magnification of the system equipped with a collimator and the test lens. In addition, if the EFL of the collimator is correctly known, we can find the EFL of the test lens [11]. Next, we move the image analyzer laterally to find the image formed by the test lens after an off-axis beam is incident on the test lens as the collimator rotates. Using the EFL of the test lens obtained  
Figure 1: Layout of the focal length and distortion measurement system. earlier and the rotation angle of the collimator, the paraxial image height can be calculated, unless there is distortion by the test lens [12]. The deviation between the calculated paraxial image height and the measured moving distance of the image analyzer becomes a distortion and is described by the following equation [13]:  
\[D\var{IY_{\textit{real}}}=\frac{y_{T_{i}}-f\cdot\tan\theta_{i}}{f\cdot\tan \theta_{i}} \tag{1}\]  
where \(f\) and \(\theta\) are the EFL of the test lens and the rotation angle of the collimator, respectively. And \(y_{T_{i}}\) is the chief ray height measured as the moving distance of the image analyzer. And the subscript \(i\) stands for the number of the data measurement. Due to the obstruction of the central area in an omni-directional system, the image magnification of systems with a collimator and the test lens cannot be measured using a designed measurement system, such as the one depicted in Figure 1. In addition to this image magnification measurement, an alternative method of measurement entails rotating the test lens imperceptibly about its nodal point. However, this method can be used only if the image is formed near the center of the screen. For this reason, it would be impossible to measure the EFL of a center-obstructed test lens using the layout shown in Figure 1.  
If the EFL for the layout shown in Figure 1 is not given, it is worth  
move the image analyzer, which are extremely expensive. The purpose of this study, therefore, is to use an affordable method to measure the focal length of an omni-directional system. Figure 2 shows the layout for the method used in this study; the illustration in the upper right of Figure 2 depicts the optical layout of the system to be measured in this study. The pattern in Figure 2 is designed to be equally spaced. \(Y_{T}\), \(Y_{B}\), \(Y_{p}\), and \(X\) refer to the overall height of the pattern, the height from the floor surface to the reference plane, the height from the reference plane to the top of the pattern, and the distance between the optical axis and the pattern, respectively. Here, the reference plane is perpendicular to the FOV, which includes the chief ray. A ray reflected from the pattern is captured on the image sensor by the omni-directional system. We can obtain the image height for each position of the pattern as well as the angle \(\theta\) incident on the optical system from the configuration in Figure 2. The angle of the ray, \(\theta\), can be evaluated using the position of each pattern and the distance between the pattern and the optical system.