**1. Introduction**

The expression adaptive optics was initially termed for the technology used in telescopes to deform the mirrors for phase correction of the incoming light [1]. Soon, adaptive optics was implemented in microscopes [2], optical communication systems [3], and optical imaging systems [4]. In conventional imaging systems, the lenses are mechanically moved to focus an image, whereas, with the adaptive optics lens, the surface curvature of the lens is changed to focus an image. The tunable focus lens, also known as the adaptive lens, uses different actuation principles to change the curvature of a deformable surface, thereby changing the focus (refractive power) of the lens. One such adaptive lens using a piezoelectric actuation principle to deform a fluid-membrane interface [5] was developed in the Laboratory for Microactuators, IMTEK - Department of Microsystems Engineering, University of Freiburg, Germany.

The developed adaptive lens consists of a piezoelectric actuator, a fluid chamber, and a transparent flexible membrane, as shown in Figure 1a. The flexible membrane bounds the fluid chamber on one side, and hence any change in the fluid chamber pressure will deform the membrane. An electric field applied on the piezoelectric actuator will deform the fluid chamber and change the fluid chamber pressure. By changing the electric field direction and magnitude, the fluid pressure can be varied to positive or negative pressures resulting in a varied refractive power. The refractive power defined as a function of the applied electric field exhibits piezoelectric hysteresis [6]. At higher temperatures, the fluid expansion will also contribute to the membrane deformation [7]. The piezoelectric hysteresis and thermal expansion contribute to a non-linear response of the adaptive lens. As the membrane

deformation is a direct result of the fluid pressure change, the non-linear effects can be addressed by defining the refractive power as a function of the fluid pressure.

To address the non-linear response and hence to compensate for the hysteresis and the temperature effect on the refractive power, it is essential to determine the combined influence of piezoelectric actuation and temperature on the membrane deformation. Hence in this paper, we present a finite-element simulation of the adaptive lens modelled in COMSOL Multiphysics -R (5.3a, COMSOL Inc, Burlington, MA, USA) to define the refractive power linearly as a function of both the fluid pressure and temperature.

COMSOL Multiphysics -R is based on the finite-element method (FEM), which solves engineering problems such as structural mechanics, fluid dynamics, heat transfer by a numerical approach. In FEM, the complex geometry is divided into simpler domains. These domains are defined with the elementary partial differential equations based on the physics. Then the elementary equations are combined to form a system of global equations, which represent the complex geometry [8]. The system of global equations can be solved using FEM-based simulation software such as ANSYS, ABAQUS, ATILA, and COMSOL [9]. To simulate complex geometry with multiple physics domains, COMSOL Multiphysics -R offers a methodological environment to access elementary equations and then couple them with a wide range of available physics modules [10]. Using COMSOL -R , authors in [11–13] simulated adaptive lenses using only the piezoelectric physics module, authors in [14,15] simulated micro-pumps using the fluid-structure interaction physics module and authors in [16,17] simulated thermal actuators using the heat transfer physics module. However, the articles [11–17] did not simulate any kind of solid deformation produced by the fluid forces from both the piezoelectric actuation and the thermal expansion. Furthermore, COMSOL -R does not provide a direct feature to couple the piezoelectric with the fluid-structure and heat transfer physics modules. Hence, in this paper, we present the explicit coupling of multiple physics modules to simulate the membrane deformation due to the fluid forces from both the piezoelectric actuation and the thermal expansion.

We describe the physical design and working principle of the adaptive lens in Section 2. In Section 3, we describe the simulation model of the adaptive lens and describe the explicit coupling of multiple physics modules using a moving mesh physics module. In Section 4, we present the simulation results of the adaptive lens and compare the simulation results with the experimental results in Section 5. We conclude our paper with results in Section 6.

#### **2. The Fluid-Membrane Piezoelectric Lens**

The piezoelectric bi-morph actuator has a circular ring-shaped design with the two piezoelectric ceramic layers glued together in an anti-parallel polarization configuration. The fluid chamber and the flexible membrane are integrated with the actuator using micro-molding techniques to form an active lens chamber [5]. The active lens chamber is glued onto a PCB-based substrate [7] and primed with an optical oil [7] (Figure 1a).

**Figure 1.** (**a**) 2D cut section of the adaptive lens to show the fluid chamber, flexible membrane, and the integrated actuator. (**b**) The adaptive fluid-membrane piezoelectric lens.

The manufactured adaptive lens with the actuator diameter of 20 mm with an aperture of 10 mm is shown in Figure 1b. The adaptive lens has an overall thickness of around 2.2 mm, with substrate thickness of 1 mm, rim thickness of 0.8 mm, membrane thickness of 0.2 mm and bi-morph actuator thickness of 0.2 mm. Depending on the applied electric field/voltage direction, the actuator deforms

the fluid chamber to produce positive or negative fluid chamber pressure. The positive pressure leads to a plano-convex lens (Figure 2a), and the negative pressure leads to a plano-concave lens (Figure 2b).

**Figure 2.** The 2D cross-section of the adaptive lens showing the piezoelectric forces and fluidic forces, which form either (**a**) plano-convex lens or (**b**) plano-concave lens depending on the applied voltage direction.
