# New Optical Design Method of Floating Type Collimator for Microscopic Camera Inspection
## 2 Materials and Methods  
The floating optical system refers to an optical system in which two or more lens groups move to adjust focus. The floating-type optical system is extremely useful for correcting a field curvature [27]. Floating focus control is generally employed in the macro lenses of interchangeable lenses [28]. In the floating system applied to an interchangeable lens, the image plane must be fixed. However, in the collimator mentioned in this paper, it should be used in combination with the optical system for the phone camera. In the optical system for the phone camera, the entire optical system is moved to adjust the focus. Therefore, the optical system is fixed, and the image plane moves to adjust the focus when the optical system for the phone camera forms the virtual object created by the collimator [29; 30; 31]. In the case of the first lens group closest to the target, it is inappropriate to use the optical system for focusing because the diameter of the lens is extremely large [32; 33; 34]. Because the optical system for the phone camera is continuously replaced for an inspection of the resolution, it is advantageous that the last lens group also be fixed. Thus, the moving lens group becomes the second and third lens groups, and is located between the fixed lens groups. Therefore, the developed collimator should consist of four lens groups. In Figure 2, the refracting power of each lens group is denoted as \(k\), and the spacing of each lens group is indicated as \(z\). In addition, \(m\) is the magnification, and the subscript \(i\) is the lens group number. The refracting power and spacing of this optical system satisfy the Gaussian bracket, as shown in Equation (1) [35; 36]. Gaussian bracket is a kind of arithmetic rule [37; 38]. Paraxial ray-tracing can be expressed with a simple formula by expressing the refraction angle and height on each surface of the optical system with Gaussian brackets [24]. The first equation in Equation (1) refers to the height of the axial light beam from the image plane, and the second formula refers to the angular ratio of the axial rays from the object plane to the image plane. In addition, \(k_{T}\) is the refracting power of the test lens, coupled to the collimator, and \(\Delta z\) is the amount of movement of the image plane from the virtual image. If the distance from the virtual image to the test lens is \(l\) and the focal length of the test lens is \(f_{T}\), Equation (2) is satisfied [39; 40]. \[ \begin{split}&[-z_{0},k_{1},-z_{1},k_{2},-z_{2},k_{3},-z_{3},k_{4},- z_{4},k_{T},-z_{\text{inf}}-\Delta z]=0\\ &[-z_{0},k_{1},-z_{1},k_{2},-z_{2},k_{3},-z_{3},k_{4},-z_{4},k_{T }]=\frac{1}{m}\\ \end{split}  \tag{1}\]  
\[\Delta z=\frac{f_{T}^{2}}{I-f_{T}} \tag{2}\]  
The collimator used in this study should create a virtual image at infinity and at a near distance. Therefore, the distance between the lenses has two values: infinity and near. Therefore, Equation (3) is satisfied:  
\[ \begin{split}&[-z_{0},k_{1},-z_{1,\text{inf}},k_{2},-z_{2,\text{ inf}},k_{3},-z_{3,\text{inf}},k_{4},-z_{4},k_{T},-z_{\text{inf}}]=0\\ &[-z_{0},k_{1},-z_{1,\text{inf}},k_{2},-z_{2,\text{inf}},k_{3},- z_{3,\text{inf}},k_{4},-z_{4},k_{T}]=\frac{1}{m_{\text{inf}}}\\ &[-z_{0},k_{1},-z_{1,near},k_{2},-z_{2,near},k_{3},-z_{3,near},k_{4},-z_{4},k_{T},-z_{\text{inf}}-\Delta z]=0\\ &[-z_{0},k_{1},-z_{1,near},k_{2},-z_{2,near},k_{3},-z_{3,near},k_{4},-z_{4},k_{T}]=\frac{1}{m_{\text{max}}}\\ \end{split}  \tag{3}\]  
Figure 2: Paraxial optical layout for the third solution. \[z_{1,\text{inf}}+z_{2,\text{inf}}+z_{3,\text{inf}}=z_{1,near}+z_{2,near}+z_{3,near}=T \tag{4}\]  
In Equation (3), \(z_{1,\text{inf}}\), \(z_{2,\text{inf}}\), and \(z_{3,\text{inf}}\) are the cases in which the virtual image becomes infinite. Likewise, \(z_{1,\text{n}ar}\), \(z_{2,narr}\), and \(z_{3,narr}\) are the cases in which a virtual image is created at a close distance. As the reason why \(z_{0}\) and \(z_{4}\) do not have subscripts, the first and fourth lens groups are fixed.