# Design of an Internal Focusing Tube Lens for Optical Inspection Systems
## 1 Introduction  
Achieving a high resolution in optical inspection systems is a topic of significant research interest. To improve the resolution, the diffraction limit must be increased [1; 2; 3], which is proportional to the wavelength and inversely proportional to the numerical aperture (NA) [3; 4; 5]. Therefore, a high resolution can be achieved with shorter wavelengths such as ultraviolet rays rather than in the visible range [6; 7; 8]. However, generating short-wavelength light is expensive, and the samples may be damaged by high-energy light at high frequencies during manufacturing/fabrication [9; 10]. According to the diffraction limit, the smallest spot in an optical system is referred to as an airy disk, as shown in Figure 1. Generally, an objective lens is not used alone in an optical inspection system but in conjunction with a tube lens to form an image on a sensor surface [11]. An airy disk is used as an indicator of resolution. Therefore, the size of the airy disk is proportional to the wavelength and inversely proportional to the NA [12]. The incident energy of the zeroth-order diffraction was 95.21% when observing the intensity distribution of light passing through a circular aperture. Light passing through a circular hole diffracts and forms an interference pattern [12]. The interference pattern appeared as a repeating pattern of bright and dark shapes, with the first dark shape appearing after the first bright shape at approximately 1.22 [12]. In addition, the first point at which the zeroth-order diffraction becomes zero is approximately 1.22. Equation (1)represents the size of an airy disk, where \(\sigma\) indicates the airy disk, \(\rho_{max}\) denotes the radius of the entrance pupil, \(R\) denotes the focal length, and \(\lambda\) represents the wavelength [12]. \[\text{Airy Disk}=(2.44)\cdot\frac{\lambda R}{2\rho_{max}}=2.44\cdot\lambda\cdot (F/\#)=1.22\frac{\lambda}{NA} \tag{1}\]  
It can be observed that the resolution increased as the number of airy disks decreased. An optical system's resolution is directly proportional to its \(NA\) and inversely proportional to its wavelength [13]. Studies have focused on optical systems with high NA to increase the resolution [14]. However, as the NA increases, the depth of focus of the optical system decreases, and the resolution may deteriorate as the distance between the sample and optical system changes [15]. An NA-increasing lens, also known as a plano-convex optical system, was used to increase the amount of light on the object [16; 17]; however, the NA constraint on the resolution increased for the microscopy system. To achieve a high resolution, a plastic objective lens with an NA of 0.85 was proposed for use at three laser diode wavelengths [18]. A significant amount of power is required to conduct fast inspections using such systems [19]. Therefore, a fast and accurate inspection can be achieved if the sample is imaged by moving a specific lens group rather than the entire optical system. The instrument size can be minimized by reducing its motor size and weight. Additionally, heavier moving mechanisms generate more inertial forces, which can affect the results. Moreover, the equipment's durability can also be a problem, making it difficult to control. The amount of lens group movement must be calculated for internal autofocusing by a specific lens group at a finite object distance, including an infinite object distance. Gaussian brackets can be used for this purpose [20]. For example, an analytical calculation technique has been used for zoom lens systems [21]. The Pade approximation method was used to compute the analytical zoom locus [22]. Generally, an objective lens is not used alone in an optical inspection system but in conjunction with a tube lens to form an image on a sensor surface. Figure 2 shows a schematic of a general microscope optical system with an objective and a tube lens. In Figure 2, the objective lens magnifies the object and produces parallel light [23].