# Design of an Internal Focusing Tube Lens for Optical Inspection Systems
## 2 Structure and Design of the Optical System
The chief ray passes through the center of the aperture. Figure 9 shows the distortion aberration of the 10\(\times\) optical system, which shows the distortion for object heights ranging from 0 to 1.7 mm. Figure 9a-c represent the distortion aberration diagram based on the sample movement. The X- and Y-axes represent the distortion aberration and sample height, respectively. We observe that the distortion aberration can be offset by combining the 30 mm focal length objective lens and the tube lens. Table 2 lists the actual data of the distortion aberration based on each sample height in position changes owing to sample movement, as shown in Figure 9a-c. As listed in Table 2, the largest RMS value is 9.73 \(\upmu\)m. When the maximum spot size was compared to that of the airy disk, the maximum spot size was smaller. This implied that an image point could be viewed as a single point. In other words, we observed that the designed 5\(\times\) optical system has a high resolution. As listed in Table 2, the maximum distortion aberration values (\(-\)0.017, 0.012, and 0.0083%) were close to zero. \begin{table}
\begin{tabular}{c c c c} \hline
**Sample Height (mm)** & **(a) RMS (mm)** & **(b) RMS (mm)** & **(c) RMS (mm)** \\ \hline
1.700 & 0.01017 & 0.00684 & 0.00767 \\
1.680 & 0.01017 & 0.00681 & 0.00764 \\
1.350 & 0.01042 & 0.00662 & 0.00717 \\
1.010 & 0.01007 & 0.00676 & 0.00775 \\
0.600 & 0.00912 & 0.00655 & 0.00839 \\
0.000 & 0.00827 & 0.00622 & 0.00896 \\ \hline \end{tabular}
\end{table} 
Table 1: Image RMS data of the 10\(\times\) optical system. Figure 8: Spot diagram of the 10\(\times\) optical system when the sample moves (**a**) \(-\)0.36 mm from the reference point, (**b**) when the sample is located at the reference point, and (**c**) when the sample moves +0.36 mm. As the object moved by \(\pm 50\) um, the top surface moved. Using the internal focusing method, the movement of the lens group inside the optical system was calculated and applied to the image plane correction. Figure 10 shows the optical path diagram of the \(5\times\) optical system. A spot diagram was created to analyze the resolution of the \(5\times\) inspection optical system. Figure 11 shows a spot diagram of the \(5\times\) optical system. Figure 11a-c show spot diagrams based on the sample movement. As in the \(10\times\) optical system, the airy disk of the \(5\times\) optical system is \(13.29\) um and the black circles correspond to the size of the airy disk. In the spot diagram, the RMS of the image point size based on the sample size at a defocus of \(0.00\) mm without image point movement are listed in Table 3.  
The distortion aberration was calculated to confirm that there was no distortion aberration in the \(5\times\) inspection optical system. The results confirm that most of the distortion aberration was canceled at -0.042. Figure 12 shows the distortion aberration diagram of the \(5\times\) inspection optical system, which indicates a distortion in the height of the object in the range of \(0\) to \(3.5\) mm. Figure 12a-c show the distortion aberration diagram based on the sample movement. The X- and Y-axes represent the distortion aberration and height of the sample, respectively. In addition, Table 4 lists the actual data on the distortion aberration based on each sample height in the position changes, as shown in Figure 12a-c. As listed in Table 4, the maximum distortion aberration values (\(-0.019\), \(-0.042\), and \(0.023\)%) were close to zero. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample Height (mm)** & **(a) Distortion (\%)** & **(b) Distortion (\%)** & **(c) Distortion (\%)** \\ \hline
1.700 & 0.0094 & 0.0005 & \(-0.0083\) \\
1.680 & 0.0103 & 0.0016 & \(-0.0070\) \\
1.350 & 0.0170 & 0.0115 & 0.0060 \\
1.010 & 0.0139 & 0.0109 & 0.0078 \\
0.600 & 0.0061 & 0.0051 & 0.0040 \\
0.000 & 0.0000 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Distortion aberration data of the \(10\times\) optical system.