# Design of an Internal Focusing Tube Lens for Optical Inspection Systems
## 1 Introduction
In addition, the tube lens forms an image of parallel light projected from the objective lens onto an imaging device [24]. Figure 1: Airy disk [12]. For magnification, the objective lens was placed close to the object, aberrations were corrected, and an image formed in the image plane using the tube lens [25]. Thus, the sample can be imaged on an imaging device that excessively widens the focus-adjustment range by moving a specific lens group within the tube lens. This focus adjustment is referred to as the internal focusing method [26]. Considering these characteristics, the magnification in a zoomed system with a Gaussian bracket remains constant, requiring the use of a telecentric optical system [27]. Therefore, we intend to design and evaluate the performance of a combined optical system in which the magnification does not change and the focus adjustment range is increased using optical design software [28]. In the design software, the zoom locus is derived using equations governing Gaussian brackets [29]. In the currently developed optical inspection system, the entire lens group must be focused, thereby compensating for the resolution based on the sample movement [30]. Therefore, current optical inspection systems are typically heavy because of the use of multiple optical lenses for aberration correction [31]. Previous studies have used telecentric optical systems for inspection applications. The structural optical design equation can aid in the design and analysis of telecentric optical systems [32]. In a double-sided telecentric optical system, a mathematical method was used to determine the design parameters using the third-order aberration method [33]. Using a Gaussian bracket and a lens module, a telecentric optical system with focus-tunable optical lenses and a moving aperture stop was analyzed [34]. Mechanical movement was reduced by combining an adaptive liquid lens with a double-sided telecentric optical system [35]. A telecentric lens module was designed for vision inspection systems with a field of view greater than 100 mm [36]. An optical design layout with three telecentric objectives for dimensional inspection systems was developed using the shadow projection method in the object space [37]. A two-dimensional telecentric system was designed by splitting the pupils and moving the principal planes using the transformed focal power [38]. An image-side telecentric anastigmatic optical system with a coaxial parent mirror was designed to obtain an ultrawide field of view [39]. A prism dispersion imaging spectrometer system was designed using imaging theory from a telecentric off-axis three-mirror imaging system [40]. A telecentric three-mirror solar simulator was proposed for large irradiation surfaces requiring high uniformity [41]. The sample arm for an optical coherent tomography system was designed [42]. The AF speed increased in the telephoto interchangeable lens [43]. To achieve high brightness in a wide-angle lens, the camera system uses two or more lens groups to accurately calculate the amount of lens movement [44]. To the best of our knowledge, this is the first optical design study for an internal focusing tube lens used in an optical inspection system. The main contributions of our study are as follows:  
1. The designed telecentric optical system provides a wide focus adjustment range using an internal focusing function;
2. The height of the image was fixed, whereas the magnification did not change in the proposed telecentric optical system;  
Figure 2: Schematic of a general microscope optical system. As focusing is performed by moving only a specific lens group without moving the entire optical system, the inspection device can be miniaturized. Therefore, inertia control by weight is convenient, and inspection speed and accuracy are enhanced. Section 2 describes the structure and design of the telecentric optical system. Section 3 describes the performance analysis of the designed internal focusing-type tube lens integrated into a telecentric optical system.