# Design of an Imaging Optical System for Large-Sized Stepped Shaft Diameter Detection
## 3 Design and Optimization of Double-Telecentric Optical System  
### Object-Space Telecentric Optical Path Design and Optimization  
In the design process of optical systems, selecting an appropriate initial structure is a critical step. Currently, there are generally two methods for determining the initial structure of a system: The first method is based on the system's relevant design specifications, utilizing the Seidel aberration theory to establish aberration balancing equations for design purposes or employing the PW method directly to design the initial structure of the system [21]. Although the method of using aberration balancing equations has notable advantages for simple system structures, it becomes computationally complex and redundant and requires a considerable workload for complex optical systems. It is difficult to achieve an efficient and high-quality system design using this method. The second method combines patent databases or existing references to determine an initial structure that meets the system's specifications for subsequent design. This approach utilizes patented structures found in optical patent databases or references as the initial structure. Regarding the complex imaging system for large-scale stepped shaft diameter detection, the second method in this article is employed to determine the initial structure of the system [22]. Firstly, a specific initial structure is chosen based on the optical design manual and the design specifications provided in Table 1. Then, scaling is performed according to the size of the image plane. Following that, an optimization function is set up to control the far focus of the object side, using the operand RANG to control the pupil distance and object distance. Using the operand property TTHI allows for controlling the air gap using the operand properties MNCA/MXCA/MNEA, and controlling the glass thickness is achieved using the operand properties MNCG/MXCG/MNEG. Finally, Zemax software is utilized to optimize the initial structure and generate the structure of the object-side far-focus system. The parameters are presented in Table 2, the system layout is illustrated in Figure 3a, and the spot diagram in Figure 3b. \begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameters** & **Value** \\ \hline Magnification ratio & \(-\)0.32 \\ Detection range & 110 mm \\ Working distance & 60 mm \\ F-number & 10 \\ Object-side telecentricity & 0.1 \\ Image-side telecentricity & 0.1 \\ Distortion & 0.1\% \\ Resolution & \textgreater{}30\%@71.4 lp/mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main technical indexes of double-telecentric optical system. From Figure 3b, it can be observed that this is an object-space telecentric system with an Airy disk radius of 9.605 \(\upmu\)m. The optical system is divided into five fields of view, each \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Surface** & **Radius** & **Thickness** & **Material** &
\begin{tabular}{c} **Clear** \\ **Semi-Dia** \\ \end{tabular} \\ \hline
0 & OBJECT & Infinity & Infinity & & Infinity \\
1 & STOP & Infinity & 34.609 & & 2.402 \\
2 & & \(-19.829\) & 9.987 & H-ZLAF71 AGT & 18.706 \\
3 & & \(-27.641\) & 0.196 & & 25.877 \\
4 & & \(-90.856\) & 12.657 & ZF52 & 34.829 \\
5 & & \(-56.337\) & 11.447 & & 37.846 \\
6 & & \(-266.608\) & 10.569 & H-ZLAF75B & 46.408 \\
7 & & \(-123.493\) & 1.083 & & 47.830 \\
8 & & \(524.461\) & 10.920 & H-ZLAF52A & 50.016 \\
9 & & \(-359.472\) & 88.534 & & 50.252 \\
10 & IMAGE & Infinity & & & 47.562 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Structural parameters of the optimized object-space telecentric system. Figure 3: Object-space telecentric system (**a**) layout and (**b**) spot diagram. representing the image height when the rays converge at the image plane after passing through the optical system. The upper-left field corresponds to an image height of 0 mm with an RMS spot radius of 1.814 mm. The upper-right field corresponds to an image height of 16.244 mm with an RMS spot radius of 5.093 mm.