# Design of an Internal Focusing Tube Lens for Optical Inspection Systems
## 2 Structure and Design of the Optical System  
Distortion correction using software is difficult owing to the characteristics of optical inspection systems. Therefore, the distortion control theory has been used [45; 46]. Before designing the inspection optical system, we used an optical system that operated in the visible light region. Therefore, the design wavelengths were 656.27, 587.56, 546.07, 486.13, and 435.83 nm, and the center wavelength was set at 546.07 nm. The optical system was comprised of an objective and a tube lens. When the two optical systems were designed and combined, a 10\(\times\) and 5\(\times\) inspection optical system was configured. The specifications of the optical system were as follows: an objective lens with an NA of 0.4, a focal length of 30 mm, and an image height of 1.7 of mm; another objective lens with an NA of 0.2, a focal length of 60 mm, and an image height of 3.4 mm; and a tube lens with F/8, a focal length of 300 mm, and an image height of 17 mm. Subsequently, the objective and tube lenses were combined. The design of each objective lens begins at an infinite object point, and the optical system is configured to form an image on the image plane. By inputting the object distance at which the point moves by 50 \(\upmu\)m and 0.36 mm based on the infinite object point at the focal lengths of 30 and 60 mm objective lenses, respectively, the design was optimized for performance close to the diffraction limit at the input object distance. The tube lens was designed to have a performance close to the diffraction limit while being able to adjust the focus for the input object's distance based on the objective lens's focal length of 30 mm. The objective and tube lenses were designed such that they acted in combination as a telecentric optical system, where the magnification did not change with the movement of the sample. Paraxial ray tracing can be used in optical system designs [47]. Paraxial ray tracing is an ideal ray-tracing method without aberrations, where the sag value owing to the curvature of the lens is approximated as zero [48]. Figure 3 shows the sag values of the lens. If these sag values are minimized, paraxial ray tracing can be performed, as explained by refraction and transfer equations. Equation (2) is the refraction equation for the refraction plot of light rays at the interface between the media and the optical system. Equation (3) is the transfer equation, which describes the height of the arrival point when a ray departing from a height at a certain point enters the next boundary at a specific angle [49]. \[n_{i+1}u_{i+1}=n_{i}u_{i}-h_{i+1}k_{i+1} \tag{2}\]  
\[h_{i+1}=h_{i}+d_{i}u_{i} \tag{3}\]  
where \(h\) refers to the height of the light rays on each lens surface. A ray of light departing from one point passes at a certain angle through one lens surface and then passes through the next lens plane. Therefore, the height of the ray on the \(i_{\text{th}}\) side is denoted as \(h_{i}\). \(k\) denotes the refractive power of the lens surface. The light rays are refracted based on the lens's curvature and refractive index. The physical quantity that describes this refraction is known as the refractive power. The refractive power in the \(i_{\text{th}}\) plane is denoted as \(k_{i}\). Ray tracing on multiple lens surfaces is represented by the schematic shown in Figure 4.  
If paraxial ray tracing is repeated on multiple lens surfaces, the series of processes can be represented as system matrices. Equation (4) describes ray tracing from the object to the \(i_{\text{th}}\) plane as a matrix operation [50]. \[ \begin{bmatrix}h_{i}\\ n_{i}u_{i}\end{bmatrix} = \begin{bmatrix}A&-B\\ -C&D\end{bmatrix}\begin{bmatrix}h_{0}\\ n_{0}u_{0}\end{bmatrix}  \tag{4}\] \[A\equiv \begin{bmatrix}k_{1},-\frac{d_{1}}{n_{1}},\cdots,k_{i-1}, -\frac{d_{i-1}}{n_{i-1}}\end{bmatrix} \] \[B\equiv \begin{bmatrix}-\frac{d_{0}}{n_{0}}k_{1},-\frac{d_{1}}{n_{1 }},\cdots,k_{i-1},-\frac{d_{i-1}}{n_{i-1}}\end{bmatrix} \] \[C\equiv