## 2 Basic Theory of Optical Design  
The proposed design consists of a liquid lens used to achieve a 9\(\times\) zoom ratio based on the structure described in [13]. The intermediate optics concept was applied to design the layout of the zoom lens; two subsystem groups were cascaded with a 3\(\times\) zoom lens to achieve a 9\(\times\) zoom ratio. In the proposed optical system, the compensating liquid lens is placed at the final position in the second subsystem, a 3\(\times\) zoom lens group. A real image (e.g., intermediate image) is observed between two zoom lens groups. The effect of the intermediate image enables the power of the two groups of lenses to function independently, wherein a lower power in the front lens group can be assigned to alleviate the diameter of the lens elements. The traditional zoom ratio is defined by \(\mathrm{Mtotal}=\mathrm{flele/fwid}\), where fwid is the EFL with wide-angle and flele is the EFL for the telescope. Figure 1 illustrates the total zoom ratio for an intermediate image. The method of cascading two lens systems has been used in developing various applications. Several applications, such as afocal optics, can be used to realize an entire lens system. Two well-known afocal lens designs are the telescope and microscope, two lens systems that cannot be used to create an image without cascading an additional image system, similar to the function of the human eye. A telescope is used to view an object at long distances, in contrast to a microscope, which is employed when an object must be viewed in a larger size to observe details. These two lens systems produce a virtual image, which is the object of the additional image system. The additional image system must catch the virtual image to form the image at a focal plane. However, ensuring that the pupils match is a challenge; that is, the exit pupil of the front afocal lens must be the same as the entrance pupil of the second image system. When the two systems are combined, the pupil match is crucial because it is related to the F-number. The size and diagonal of the CCD are 1/2.7\" and 6.592 mm, respectively. All design specifications are show in Table 1. The maximum field angle of the proposed zoom lens is 50\({}^{\circ}\); accordingly, the semifield angle is 25\({}^{\circ}\). When the image high and field angle have been determined, the relationships among the EFL, image high, and field angle can be defined as  
\[E=\frac{H}{\mathit{Tan}(\theta)}. \tag{1}\]  
where \(E\) is the EFL, \(H\) is the image high, and \(\theta\) is the semifield angle. Equation (1) can be used to determine that the EFL is 7.068 mm at the maximum field angle. Because the zoom ratio of the proposed  
Figure 1: The illustration of zoom ratio with relay image (M1 is first group of zoom lens and M2 is second group of zoom lens). zoom lens is 9\(\times\), the EFL of a telescope in which the proposed lens is applied is 63.612 mm. The specifications of the 9\(\times\) zoom lens are provided in Table 1.  
The proposed intermediate optics structure consists of a front zoom lens subsystem and back zoom lens subsystem, as shown in Figure 2. The front zoom lens involves applying Zoom 1-Zoom 3 to realize a zoom ratio of 3\(\times\); its f number ranges from 3.2 to 4.4. The back zoom lens involves using Zoom 4-Zoom 6 to implement a zoom ratio of 3\(\times\); its f number ranges from 4.4 to 5.6. The proposed zoom lens system was designed using a 9\(\times\) zoom lens by cascading the front zoom lens at a 3\(\times\) zoom ratio, and the back zoom lens at a 3\(\times\) zoom ratio. Before the two subsystems were combined, the front zoom lens was designed as the infinity conjugate and the back zoom lens was designed as the finite conjugate. The intermediate optics structure in the infinite conjugate system has the object located at infinity; its specification is shown in Table 2. Under the same previous specification of image height and wavelength spectrum, the full-field angle is set at 50\({}^{\circ}\).