## 2 Principle of DETL
### Tuning Principle
#### 2.2.2 Displacement-Based  
In contrast to deformation-based lenses, displacement zoom lenses primarily employ DEA to actuate the lens group, creating axial or vertical axial displacement to alter the focal length. 1. Axial displacement  
The researchers employed DE as the primary component to address the drawbacks of the mechanical displacement zoom system, which is considerable in size and weight. They realised flexible drive zoom by mechanically connecting the DEA to either a solid or soft-solid lens. When an electric field is applied to the dielectric elastomer, the elastomer contracts and moves the lens along the optical axis, thereby altering the distance between the lens and the object or imaging plane to achieve the zoom effect. This principle mirrors the focus adjustment method found in traditional optical systems, and its specific expression is illustrated in Equation (2) [29]:  
\[\frac{1}{u}+\frac{1}{v}=\frac{1}{f} \tag{2}\]  
where \(u\) is the object distance, \(v\) is the image distance, and \(f\) is the focal length. [PERSON] et al. [38] fabricated three different structures of DEA-driven micro-optics, where an applied voltage can cause the DEA to produce displacement in the direction of the extended optical axis to achieve zoom. [PERSON] et al. [39] developed a displacement lens via DEA with the structure shown in Figure 4a, which consists of a convex hemispherical PDMS lens and a DEA. By applying a voltage, the deformation of the DE film enables the lens  
Figure 3: Indirectly driven deformable DETL: (**a**) lens zoom schematic, the dotted line is the shape of the DEA and lens without voltage applied, and the realisation is the shape of the DEA and lens after the voltage is applied; (**b**) physical diagram of the lens [33]. to change in the vertical direction and thus change the focal length, with a magnification ratio of up to 118.4%, a displacement attenuation of less than 9% for 250,000 consecutive cycles of operation, and good durability. The DETL, which shifts the zoom along the optical axis, has effectively achieved miniaturisation compared to conventional shift-type lenses. However, compared to the variable-form DETL, it has a larger zoom ratio and a greater axial dimension thickness. 2. Lateral displacement  
[PERSON] et al. [40] first developed a DETL with displacement zoom in the vertical optical axis direction, as shown in Figure 4b, which DE can drive to enable reciprocal motion in the vertical direction along the optical axis. A focal length change can be realised with another solid lens, but continuous zooming is impossible. On the other hand, tuneable lenses based on the principle of Fresnel and Alvarez lenses can achieve the constant zoom function by displacement in the vertical optical axis direction. On the other hand, tuneable lenses based on the principles of Fresnel and [PERSON] lenses achieve a continuous zoom function by displacing in the direction of the perpendicular optical axis while further enhancing the zoom capability. Fresnel lenses satisfy the radius a of the Nth ring band by splitting the continuous surface of a conventional spherical lens into a thin sheet of concentric rings:  
\[R_{N}^{2}=NR_{1}^{2}\quad(N=1,2,3,\ldots) \tag{3}\]  
where \(R_{1}\) is the radius of the first annular band. Each ring is an optical surface with a specific curvature, and these rings are designed so that the lens as a whole still has refractive power, but reduces the thickness and weight of the lens. By varying the spacing of the rings, the Fresnel lens allows for focal length adjustment [41]. The initial focal length F is related to the wavelength \(\lambda\) and the radius of the ring band as:  
\[F=\frac{R_{1}^{2}}{\lambda}=\frac{R_{N}^{2}}{N\lambda} \tag{4}\]  
When the radius of the ring band shrinks due to deformation to S times the original radius \((S=R_{N}^{\prime}/R_{N})\), the new focal length \(F^{\prime}\) is  
\[F^{\prime}=S^{2}F \tag{5}\]  
Based on this principle, [PERSON] et al.