# Improved Design of Imaging System for Online Detection of Large-Sized Step-Shaft Runout Errors
## 2 Principle and Simulation Analysis
### Imaging Optical System Design
#### 2.2.2 Improved Imaging Optics Design  
In order to meet the design requirements of the system, it is necessary to optimize the initial structure. First, the focal length of the selected initial structure is scaled, and the operand PMAG is used to control the magnification. Next, the air interval between the devices attached to the lens and the thickness of the lens are set as variables, and the operand MNCA and MNCG are used to control the thickness and air interval of the lens. When optimizing the lens, it is necessary to prioritize the surface where the aberration is prominent, control the aberration within the effective range, and then optimize the other surfaces one by one. If the value still fails to meet the design requirements after several optimization, the existing glass is replaced and the operand MTFT, which controls MTF, is added again for HAMMER optimization. The final structure obtained is shown in Figure 7, and the final parameters of the lens are shown in Table 3.  
Figures 8 and 9 show the MTF function and point diagram of the system, respectively. It can be seen from the figure that the radius of the Avery's spot of the system is 5.363 \(\upmu\)m and the radius of RMS is 4.929 \(\upmu\)m when the spatial frequency is 42 lp/mm. Both these values meet the imaging quality requirements of the imaging optical system. In addition, the imaging energy of PSD position detector is concentrated in an Ailey spot, which meets the requirement of light-energy centroid detection.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Surface** & **Type** & \begin{tabular}{c} **Radius of** \\ **Curvature/mm** \\ \end{tabular} & **Thickness/mm** & **Material** & \
\begin{tabular}{c} **Clear** \\ **Diameter/mm** \\ \end{tabular} &
\begin{tabular}{c} **Argument** \\ **(Tangent)** \\ \end{tabular} \\ \hline
0 & OBJECT & Inclined & Infinity & 10.000 & & 1.237 & 0.840 \\
1 & Standard & 6.665 & 2.962 & H-ZLAF66 & 2.402 & \\
2 & Standard & 5.225 & 1.923 & & 18.706 & \\
3 & Standard & 19.393 & 2.879 & H-ZLAF92 & 25.877 & \\
4 & Standard & \(-\)21.585 & 1.271 & & 34.829 & \\
5 & DIAPHRAGM & Standard & 5.129 & 1.877 & N-LASF9 & 37.846 & \\
6 & Standard & 4.169 & 5.000 & & 46.408 & \\
7 & Standard & \(-\)123.720 & 3.000 & & 47.830 & \\
8 & Standard & \(-\)10.592 & 30.000 & & 50.016 & \\
9 & IMAGE & Inclined & Infinity & & 47.562 & \(-\)2.050 \\ \hline \end{tabular}
\end{table}
Table 2: Imaging optical system lens initial parameters.  
Figure 7: Imaging optical system structure diagram.  
The MTF function and point diagram of the two groups of optical imaging systems were compared using Zemax v.24.1, and the inclination range of the measured object surface was set to be 7\({}^{\circ}\). The comparison of MTF function and point diagram of the two groups of optical imaging systems is shown in Figure 10. Through comparison, it can be seen that the  
Figure 8: MTF function at up to frequency 42 lp/mm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Surface** & **Type** & \
\begin{tabular}{c} **Radius of** \\ **Curvature/mm** \\ \end{tabular} & **Thickness/mm** & **Material** &
\begin{tabular}{c} **Clear** \\ **Diameter/mm** \\ \end{tabular} \\ \hline
0 & OBJECT & Inclined & Infinity & 10.0 & & 1.9 \\
1 & & Standard & 5.8 & 2.2 & N-LASF9 HT & 2.7 \\
2 & & Standard & 4.6 & 4.1 & & 2.3 \\
3 & & Standard & 22.5 & 3.7 & H-ZLAF92 & 2.6 \\
4 & & Standard & \(-\)26.0 & 4.8 & & 2.7 \\
5 & DIAPHRAGM & Standard & 5.4 & 2.0 & SF57 HTULTRA & 2.4 \\
6 & & Standard & 4.3 & 1.3 & & 2.3 \\
7 & & Standard & 142.7 & 3.0 & SF11 & 2.9 \\
8 & & Standard & \(-\)23.3 & 10.0 & & 3.3 \\
9 & IMAGE & Standard & Infinity & & & 4.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Imaging optical system final parameters.  
Figure 9: Imaging optics point list.  
runout error detection imaging optical system designed in this paper can still achieve value greater than 0.3 when the space stop frequency is 42 lp/mm, and the airy spot radius of the system meets the design requirements when the step axis produces a small inclination error.