# Design of an Internal Focusing Tube Lens for Optical Inspection Systems
## 2 Structure and Design of the Optical System
\begin{bmatrix}k_{1},-\frac{d_{1}}{n_{1}},\cdots,-\frac{d_ {i-1}}{n_{i-1}},k_{i}\end{bmatrix} \] \[D\equiv \begin{bmatrix}-\frac{d_{0}}{n_{0}}k_{1},-\frac{d_{1}}{n_{1 }},\cdots,-\frac{d_{i-1}}{n_{i-1}},k_{i}\end{bmatrix} \]  
Figure 4: Paraxial ray-tracing layout. Figure 3: Sag value of the lens. Here, \(A\), \(B\), \(C\), and \(D\), comprising the system matrix, can be obtained using a Gaussian bracket operation. \(A\) Gaussian bracket operation comprising of \(n\) elements is developed, as shown in Equation (5). \[ \begin{array}{c}[]=1\\ \left[a_{1}\right]=a_{1}\\ \left[a_{1},\ a_{2}\right]=1+a_{2}a_{2}\\ \left[a_{1},\ a_{2},\ a_{3}\right]=\left[a_{1},\ a_{2}|a_{3}+a_{1}\right.\\ \left[a_{1},\ a_{2},\ a_{3},a_{4}\right]=\left[a_{1},\ a_{2},\ a_{3}\right]a_{4 }+\left[a_{1},\ a_{2}\right]\\ \vdots\\ \left[a_{1},\ \cdots,\ a_{n-1},\ a_{n}\right]=\left[a_{1},\ \cdots,\ a_{n-1}\right]a_{n}+ \left[a_{1},\ \cdots,a_{n-2}\right]\end{array}  \tag{5}\]  
If the regularity of the Gaussian bracket operation with these rules is well organized, it can be represented by the algorithm shown in Figure 5.  
When the object distance in the optical system changes, the image distance also changes. Equation (6) shows the relationship between the object distance \(l\) and image distance variation in an optical system with focal length \(f\)[51]. \[\Delta f=\frac{f^{2}}{l-f} \tag{6}\]  
To compensate for this variation, the movement of the image plane is restricted to the movement of a specific internal lens group, as per the internal focusing method. To obtain the movement of the lens group for the image plane variance correction, the zoom equation must be solved as follows: An optical system with multiple lens groups is shown in Figure 6. Each lens group is shown in terms of the principal planes (\(H_{1}\), \(H_{2}\)) and refractive power. When the refractive indices of the object and image sides of each lens group were identical, the point of intersection of the extension line of the incident light beam with an angular magnification of 1 and the optical axis was the first principal plane (\(H_{1}\)). In addition, the  
Figure 5: Algorithm of the Gaussian bracket. on its position, whereas the vertical axis refers to the height of the object. An airy disk is a theoretically permissible circle of confusion that an image point can have [12]. When compared to the root mean square (RMS) of the distribution of the image points in the sports diagram, being close to the airy disk, which is the theoretical resolution limit, indicates that it has a high resolution [12]. In other words, a ray departing from one point on an object does not form an image at exactly one point on the image plane, depending on the pupil coordination and wavelength. The spot diagrams magnify these points to show the distribution of the rays. As listed in Table 1, the RMS represents the distribution when a ray departing from one point of an object reaches the image surface. The presented data refer to the spot RMS at the defocus 0.00 mm position without movement of the top surface of the spot diagram based on the sample movement, as shown in Figure 8a-c. The unit of the RMS is millimeters. The area marked with a black circle corresponds to the size of the airy disk; the diffraction-limited airy disk sets the standard for a high resolution [12]. As listed in Table 1, the largest RMS value is 7.67 \(\upmu\)m and the size of the airy disk is 13.29 \(\upmu\)m. The fact that the maximum spot size was smaller than that of the airy disk indicates that the light rays that started at one point converged at one point. In other words, it was observed that the designed 10\(\times\) optical system had a high resolution. Figure 7: Optical path diagram of the 10\(\times\) optical system. Figure 8: _Cont_. The optical inspection system was assumed free of distortion aberrations. Distortion refers to the difference between the height of the chief ray on the top plane and that of the chief ray considering aberrations in paraxial ray tracing [53].