## 2 Principle of DETL
### Tuning Principle
#### 2.2.1 Deformation-Based  
Methods based on deformation zoom mainly utilise DE driven by voltage to produce in-plane expansion. The radius of curvature of the spherical crown of the lens is directly or indirectly modulated to achieve zoom. The principle is shown in Equation (1):  
\[\frac{1}{f}=(n-1)(\frac{1}{R_{1}}-\frac{1}{R_{2}}) \tag{1}\]  
where \(R_{1}\) and \(R_{2}\) are the radii of curvature of the front and back spherical crowns, \(n\) is the refractive index of the lens, and \(f\) is the lens's focal length [29]. 1. Direct drive  
Tuneable lenses with direct actuated deformation are mainly used by DEA to change the shape of liquid or soft-solid lenses directly, enabling the dynamic change in focal length. [PERSON] et al. [30] developed a biconvex lens structure by combining a fluid-filled elastomer lens with a DEA, as shown in Figure 2a. The lens consists of two layers of pre-stretched transparent elastomer film forming a cavity, with the centre filled with transparent fluid and coated with flexible electrodes in an annular region around the lens. When a voltage is applied, the ring actuator generates an electrostatic compressive stress in the thickness direction, which triggers a radial contraction and directly drives the lens to produce a radial dimensional reduction. Since the fluid volume inside the lens is constant, its thickness increases, and the radius of curvature decreases. The lens has a magnification ratio of up to 129.1% and an operating frequency range of 1-10 Hz [PERSON] et al. [31] developed a soft-solid tuneable lens, as shown in Figure 2b, with a magnification ratio of up to 154.6%, by drawing on the regulation mechanism of animal lenses. Its preparation process is more concise and efficient than liquid lenses. The lens designed by [PERSON] et al. [32] achieves the zoom function by deforming the film to create a bump or depression, with a magnification ratio of 124% and a frequency response of up to 20 Hz. Direct-drive anamorphic tuneable lenses feature a compact structure and small size, thanks to the high level of integration between the driver and the lens. This integration provides a significant advantage in portable devices and space-constrained application scenarios. Moreover, direct-drive solutions generally exhibit a faster response time and simpler control, making them ideal for systems with demanding dynamic performance requirements. However, the volume and deformation capacity of the driver limit the direct drive method, resulting in a relatively small focal length adjustment range that makes it challenging to accommodate a wide array of zoom requirements. 2. Indirect drive Compared to direct-drive DETL, indirectly driven zoom lenses are typically configured with a liquid chamber connected to the outside. The DEA induces deformation of the passive film (e.g., polydimethylsiloxane (PDMS) elastomer film) by compressing the liquid medium inside the chamber, thereby achieving the indirect tuning of the lens curvature. [PERSON] et al. [33] constructed a diaphragm-pump-type adjustable liquid lens driven by the DEA, as shown in Figure 3, which can change the focal length through the DEA by adjusting the hydraulic pressure of the diaphragm pump; the lens is filled and discharged to change the focal length, and its magnification ratio can be up to 333%. [PERSON] et al. [34] proposed a hydrogel electrode-coupled tuneable lens structure. They carried out the corresponding theoretical analyses, which utilised the deformation generated by the DEA to change the volume of the other cavity, which provided a new idea for the tuneable lens. [PERSON] et al. [35] developed a concentric annular dielectric elastomer-driven liquid lens; this lens's circular and annular cavities are interconnected and filled with liquid.