## 1 Introduction  
Because of technological advancement and product innovation, people increasingly demand a high quality of life, especially regarding optomechanical system products. Therefore, people desire products featuring adjustable sizes and high image quality, such as telescopes [1,2], mobile lenses [3,4], pinhole cameras [5,6], and medical endoscopes [7].  
Based on specification requirements, the optical designer typically builds a lens structure with a constant effective focal length (EFL) initially, to facilitate predicting the highest-quality image during optimization simulations. When an object must be enlarged or shrunk according to different distances, the user can only approximate this by physically moving a lens with a constant EFL close to, or far away from, the object involved. Therefore, a zoom lens is extremely effective for taking photos of objects at different distances.  
Zoom lenses with multiple EFLs are built, enabling lens systems to acquire objects at various distances [8,9]. Zoom systems are highly complex optical systems [10,11]. A traditional zoom lens [12] is composed of four parts, namely a front lens group, a zoom lens group, a back lens group, and a compensating lens group. The front lens group and back lens group are fixed, but the location of the elements of the zoom lens group and compensating lens group can be adjusted to vary the EFL and refine the image quality on an image plane. Only one zoom lens group is applied in a traditional zoom lens to enable the use of multiple EFLs. In [13], the intermediate optics concept was used to design a zoom lens layout and a zoom lens with a 9\(\times\) ratio, which was achieved by cascading two subsystem groups with a 3\(\times\) zoom lens.  
In a traditional optical system, field lenses are used to collect edge rays and transmit them through another lens; thus, the optical power at the edge of an optical beam is not easily diverted when edge rays leave the lens [14,15]. The droplet shape and location of liquid lenses can be adjusted reversibly, enabling the focal length and position of the lens to be tuned by adjusting the voltage applied to a set of electrodes [16,17]. Hence, liquid lenses can replace complex lens combinations to reduce the total length of optomechanical systems. But the liquid thickness varies non-linearly with the radius of the interface; it's very difficult to optimize a real liquid lens using commercial optical design software directly [18].  
There is also a point for attention while using liquid lens as compensating lens. First, the negative power of zoom optics limitations make field curvature becomes inherent, which seriously degrades the system modulation transfer function (MTF). Second, the abbe numbers of materials of liquid optics are not similar to those of extra dispersion optical materials and consequently, axial chromatic aberration becomes severe. Third, the chief rays may be vastly bent to fit the limited bore of liquid optics; but this may complicate the optical design, due to aberrations [19].  
The research [13] a zoom lens comprising lens elements of a small diameter was designed by applying intermediate optics to distribute optical power into two zoom subsystems. Therefore, the diameter of the lens element for the zoom lens with intermediate optics is smaller than that of the traditional zoom lens. Moreover, the proposed zoom subsystems can respectively compensate for their aberrations because of the nonlinear compensation of the zoom lens group in the zoom lens system. Cascading these zoom subsystems increases the zoom ratio for the overall zoom lens system.  
This study proposes using a liquid lens as a compensating lens and combining it with a 9\(\times\) zoom lens. The CODE V built-in optimization method was used to determine the process parameters. The proposed technology can be used to improve the total MTF and reduce the Petzval surface curvature (PTZ) of zoom lenses.