## Abstract

One of the limitations in the application of electrowetting-based tunable fluidic lenses is their slow response time. We consider here two approaches for enhancing the response speed of tunable fluidic lenses: optimization of the properties of the fluids employed and modification of the time-dependent actuation voltages. Using a tubular optofluidic configuration, it is shown through simulations how one may take advantage of the interplay between liquid viscosities and surface tension to reduce the actuation time. In addition, by careful designing the actuation pulses, the response speed of both overdamped and underdamped systems may be increased by over an order of magnitude, leading to response times of several ten milliseconds. These performance improvements may significantly enhance the applicability of tunable optofluidic-based components and systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optofluidics has proven to be an attractive technology for a wide range of tunable optical devices [1–4], especially when combined with electrowetting-on-dielectrics (EWOD) as an actuation mechanism, as it combines low power consumption with large tuning range [5,6]. However, in most current state-of-the-art electrowetting devices, the optofluidic elements are actuated quasi-statically and actuation times are usually in the range of several hundred milliseconds [7–9]. Recently, imaging system [10], microscope [11,12], laser scanning system [13] and digital display [14] incorporating EWOD devices have been demonstrated. However, for such electrowetting devices, the focusing speed is often one of the bottlenecks limiting their realistic applications.

Many scientific, industrial, and commercial applications require fast modulation and control to enable high-speed imaging and measurements. For instance, laser processing technology [15] that employs digital mirror devices (DMDs) or deformable mirrors (DMs) has been developed extensively. With such components, high-speed switching at rates up to 40 kHz is achievable. Replacing conventional adaptive optical elements with EWOD devices, provides further scope for high-NA laser fabrication to greater depth ranges and in more complex geometries. Another example of application is three-dimensional optical microscopy which has employed a movable mirror to achieve axial scanning [16]. However, mechanical actuation of the mirror limits the axial scanning rate to 10-100 Hz. To accelerate axial swept light-sheet microscopy by an order of magnitude, electrowetting devices with switching rate in the order of kHz would be thus considered. This allows then an extended depth of field for rapid imaging in three dimensions.

Various approaches have been proposed to reduce the response time of EWOD-based devices. Jung et al. [17] have investigated the effects of viscosity, interfacial tension and substrate roughness on the response time in an electrowetting lens; a critical damping response was achieved by analyzing and applying optimal conditions. Vo et al. [18] have studied systems with both underdamped and overdamped characteristics to understand the transient dynamics of actuating droplets; in their model, the critical viscosity was calculated at which the transition occurs, showing a reduced response time of the device. Kuiper et al. [10] have demonstrated an electrowetting lens for miniature variable-focus camera application. By varying the insulating liquid and the cylinder diameter, the underdamped, critically damped and overdamped meniscus were observed. For different lens diameters, a critical damping time and critical viscosity were calculated. However, both approaches were limited by the liquids employed in the devices and the optimization procedures are cumbersome.

An additional promising approach for increasing actuation speed is the use of time-shaped actuation waveforms. An in-depth study using COMSOL was performed to simulate the meniscus dynamic behavior of the electrowetting lens by Supekar et al. [19]. Based on the results, they have proposed a method using shaped input voltages to enhance the response time of a cylindrical lens. However, they limited their pulses to exponential functions, which only has advantages for underdamped systems, such that a systematic approach to determine the optimal input voltage shape to obtain the highest response speed is still missing.

We thus consider here a more comprehensive study of means to increase EWOD actuation speed. We show which fluid parameters can be optimized to improve time-dependent performance; propose a general mean for defining time-dependent actuation voltages which enhance tuning speed; and verify the validity of these approaches with experimental performance data.

## 2. EWOD lens structure and simulation model

We briefly introduce the mechanical, fluidic and electrical structure of the EWOD-based lens used in the subsequent analysis, simulation and experimental verification.

#### 2.1 EWOD lens structure

The lens structure employed in all the considerations below is shown in Fig. 1(a). It is based on previously published work [9,20] which uses a flexible electrode foil rolled into a glass tube. The flexible polyimide foil consists of an electrode covered by a dielectric stack of 3 µm polyimide and 1 µm of the hydrophobic fluoropolymer Cytop. The foil is rolled and attached inside a 5 mm diameter glass tube, and mounted on a glass substrate; the latter seals the device and provides electrical connections. The electrode thus covers the entire inner circumference of the tube.

The tube, which defines the tubular geometry of the lens, is then selectively filled with two immiscible liquids: a polar aqueous phase surrounded by an oil phase. In its initial (unactuated) state, due to the hydrophobic behavior of the Cytop surface layer, the lens is convex, as shown in Fig. 1(b). By applying a voltage to the electrode embedded in the foil, the contact angle changes and the lens becomes concave, as shown in Fig. 1(c).

#### 2.2 Simulation model

To study the dynamic behavior of the liquid-liquid interface in this EWOD lens, a model based on COMSOL Multiphysics 5.1 was established [21]. As shown in Fig. 2, the 2D axisymmetric model consists of a sealed chamber filled with two immiscible liquids for which we can then vary the interfacial tension and viscosities. The lower fluid is a polar liquid with a kinetic viscosity which may range from 1 cSt to 350 cSt. The upper fluid is non-polar, density-matched to the lower one, and a kinetic viscosity which may be varied in the same range. The surface tension at the interface between the two fluids is a further essential parameter in determining the dynamic response; we define values ranging from 0.01 N/m to 0.07 N/m. The geometry and materials of the model are given in 1. A moving mesh boundary was employed to simulate the dynamics of the liquid-liquid interface. A DC voltage step function was used as the voltage input to the model.

## 3. Effect of contact angle, lens diameter, viscosities and surface tension

Using this numerical model, the effect of contact angle, lens diameter, liquid viscosities and surface tension on the dynamic response of the meniscus could be studied in detail. Firstly, the response time of the lenses was analyzed for different lens diameters. The simulation was then performed for different lens diameters of 1 mm, 3 mm, 5 mm, 7 mm and 9 mm, while holding surface tension constant at 0.02 N/m, polar and non polar liquid viscosities at 10 cSt and voltage constant at 100 V. Then, the response time of the lenses was analyzed for different contact angles. Therefore, voltage steps of 38 V, 56 V, 70 V, 82 V, 92 V and 100V were applied on the electrode, while keeping the surface tension at 0.02 N/m, polar and non polar liquid viscosities at 10 cSt. In the end, for a better comparison, the response time of the lenses was analyzed at the same meniscus height. To accomplish this, voltage steps of 69.7 V, 98.6 V, 120.7 V, 139.4 V, 155.8 V, 170.7 V and 184.3 V for different surface tensions ranging from 0.01 N/m to 0.07 N/m were applied to the electrode and the meniscus height was used as a measure for the dynamic response of the lens. Parametric sweeps were undertaken to examine the effects of surface tension and the viscosities of the non-polar (upper, oil-based) and polar (lower, aqueous) liquids; each of these three parameters was successively varied while holding the other two constant. The results are shown in Fig. 3(c) 3(d), and 3(e).

#### 3.1 Lens response

The lens response to the applied voltage pulse under different conditions is shown in Fig. 3. In Fig. 3(a), when keeping surface tension constant at 0.02 N/m, polar and non-polar liquid viscosities at 10 cSt and voltage constant at 100 V, lens with smaller diameter is faster than lens with larger diameter. Increasing the diameter of the lens, thus extending the response time. In Fig. 3(b), lens behavior changes from overdamped to underdamped for voltage increases from 38 V to 100 V, while keeping the surface tension at 0.02 N/m, polar and non-polar liquid viscosities at 10 cSt. Critical damping would be thus observed at a certain contact angle. In Fig. 3(c), both non-polar and polar liquid kinetic viscosities were set to 10 cSt and the impact of surface tension on lens response is seen. The lens behavior changes from overdamped to underdamped for surface tension increases from 0.01 N/m to 0.07 N/m. The response given in Fig. 3(d) was seen for a constant surface tension of 0.01 N/m and kinetic viscosity of the polar liquid of 10 cSt; we see that the lens response is dominated by non-polar liquid viscosity, resulting in a range of 195 ms to 2.5 s response times. Finally, the influence of the polar liquid viscosity is depicted in Fig. 3(e), keeping the non-polar liquid viscosity at 10 cSt and the surface tension at 0.01 N/m. It is clear that the polar liquid also has a significant effect on the lens response, leading to a response time from 175 ms to 2.1 s.

We see that the both lens diameter and contact angle play a role in lens response. However, for some applications, when lens diameter and change in contact angle are predetermined, it is more useful to characterize the liquids properties on the lens response. We see from Fig. 3, viscosities of both liquids as well as the interfacial tension are factors which also affect the achievable hydrodynamic response and the resulting switching speed. Comparing Fig. 3(d) and 3(e), an increase in polar liquid viscosity results in a sharp fall in meniscus height, indicating that polar liquid viscosity has a greater effect on lens response than non-polar liquid viscosity. However, when comparing the viscosity of polar liquid and the interfacial tension, the question remains: which is more important?

#### 3.2 Parameter optimization

To answer this question and find the optimal condition for the rapid damping of the electrowetting lens based on the multi-parameter variations in this model, design of experiments (DOE) was used to find an optimal set of parameters. DOE is used to obtain maximum information with the minimum number of experiments; it aims at predicting an outcome by introducing a change of the preconditions, which are represented by several input variables. In the current model, there are three relevant factors: interfacial tension, viscosity of the polar liquid and viscosity of the non-polar liquid, and all influence the system response. Through DOE, the relationship between these factors affecting the response time of the device can be determined and a systematic approach to reduce the response time can be then generated.

In the simulation, multi-level factorial design was performed to evaluate multiple parameters (surface tension and both viscosities) set at multi levels. The associated minimal and maximal of the three parameters are: 1 cSt and 350 cSt (a mixture of OHGL & OHZB from Cargille) for the polar liquid, defined based on data sheet values of these commercially available liquids; 1 cSt and 350 cSt for the non-polar liquid (to have the identical processing area as for polar liquid); 0.01 N/m and 0.07 N/m for interfacial tension, determined according to [22]. The output response of the system was taken as the lens settling time, defined as the time required by the response to reach steady-state within 0.5$\%$ of its final value. 2 illustrates the results of 8 runs at the boundaries of the experiment design, with the settling time as output.

By exploiting several working points of the entire experiment space, the overall interactions of the three factors can be detected. Here, in our DOE model, $3^k$ runs were performed, where $k=4$ defines the level of the factors. The relationships between the three factors are represented by the three-dimensional surface plots and a half normal plot [23] as shown in Fig. 4. The 3D surface plots graph the response time in Z direction as a function of two input variables. The input variables are in Fig. 4(a) surface tension and non-polar liquid viscosity; in Fig. 4(b) non-polar liquid viscosity and polar liquid viscosity; and in Fig. 4(c) surface tension and polar liquid viscosity. We see that the interfacial tension between the two liquids plays a crucial role for the actuation speed, since the response time shown in Fig. 4(b) shifts only slightly, from 1.44 s to 2.23 s, when keeping the surface tension at a constant value of 0.02 N/m, it becomes clear that high interfacial tension is a necessary requirement for fast liquid actuation. As shown in 4(a) and 4(c), the response time changes from 0.18 s to 1.76 s when keeping the polar liquid viscosity at a constant value of 20 cSt, the response time changes from 0.07 s to 1.83 s when keeping the non polar liquid viscosity of 20 cSt. Comparing Fig. 4(a) and Fig. 4(c), polar liquid viscosity plays a dominant role in lens response than non polar liquid viscosity. It becomes clear that high interfacial tension is a necessary requirement for fast liquid actuation.

The half normal probability plot (Fig. 3(d)) also helps to identify the effect of response time for all the influencing factors. On the half normal probability plot of the effects, effects that are further from 0 are statistically significant. It is seen that the surface tension has a significant impact on the actuation speed; the second most important factor for fast liquid actuation is the polar viscosity. In contrast to the strong influence of the interfacial tension, the speed of the liquid interface is not significantly affected by the viscosity of the non-polar liquid, as demonstrated in Fig. 3(c) and 3(d).

Once we have qualitatively characterized the response time as a function of surface tension and viscosities, a dynamically-actuated system with different liquid combinations can be analyzed. As an example, suppose we now use the two most common liquids, water as the polar liquid, with a viscosity of 1 cSt, and silicone oil as the non-polar liquid, with a viscosity of 10 cSt [24,25]. As discussed, if fast actuation speed is desired, the highest interfacial tension 0.07 N/m [22] is required to achieve such a high dynamic functionality. Therefore, no surfactant will be added to the polar liquid, since a surfactant reduces the surface tension. However, this high interfacial tension leads to a high actuation voltage device, which has detrimental effect on shelf life of hydrophobic layer. For EWOD devices, therefore, there is a trade off between two fundamental requirements: high actuation speed and low actuation voltage. A design must this consider which of these is to be prioritized.

## 4. Actuation pulse design

Optimal choice of liquids in an EWOD-actuated component is one important tool in enhancing the dynamic response, but this choice may occasionally be limited due to other boundary conditions on, as well as availability of, suitable fluids. We turn now to a second variable parameter which may be tuned to increase actuation speed, namely the shape of the actuation voltage pulse in time.

Key to this analysis is the use of a proportional-integral-derivative (PID) controller for stimulation of the electrodes. The PID controller uses a feedback loop to control the system in such a way as to obtain the expected dynamic response. Figure 5 summarizes the procedure of shaping the input voltage: in the first step, a time based system response was measured by applying a step function using finite element analysis in COMSOL, allowing us to calculate the transfer function of the system, which is referred to the plant. Next, a PID controller was added to regulate the input voltage, setting the optimal gains for P, I and D to obtain an ideal response. This design procedure was performed in MATLAB using the Control System Toolbox. In the following, we discuss each point in this process separately.

#### 4.1 Plant transfer function

The design procedure begins with the determination of the plant transfer function [26]. Since the system is linearly time variant, the plant transfer function is calculated as the ratio of output (system response) to input (step function). The transfer function block models a system by a transfer function using the Laplace-domain variable $s$. The transfer function of the system $Plant\left (s\right )$ can be described by

#### 4.2 PID controller

The transfer function of the PID controller [27] $C$ is defined as

where $K_p$ is the proportional gain; $K_i$ is the integral gain; and $K_d$ is the derivative gain. The goal of the simulation is find the optimal value of $K_p$, $K_i$ and $K_d$ such that a fast rise time and minimal overshoot with zero steady-state error can be achieved. The closed-loop transfer function of the entire given system $T$ with a PID controller is thenSubsequently, still using the Control System Toolbox in MATLAB, a PID controller is used to control and tune the input step voltage by defining the response time and robustness of the system as the design criteria. The tuner computes PID parameters that robustly stabilize the system and these parameters are exported back to the PID controller block to allow verification of the controller performance. After performing several tuning iterations, the values of $K_p$, $K_i$ and $K_d$ can be determined.

#### 4.3 Enhanced system performance

To better demonstrate the potential of this approach to reduce the response time, two systems, one underdamped and one overdamped, were thus optimized. The temporally-shaped voltage pulse obtained using the PID controller model was imported into COMSOL Multiphysics 5.1, and the dynamic behavior of the meniscus was analyzed, as above. A table containing values of the voltage at discrete times is used as input to describe the voltage shape. The new system response was thus acquired by performing a finite element analysis, while keeping other system properties identical as before.

### 4.3.1 Underdamped system

For the underdamped system, a combination of water (1 cSt viscosity) with silicone oil (10 cSt) was employed. No surfactant was added to the water so that the resulting surface tension value is approximately 0.07 N/m. The dielectric surface layer is a stack of 3 µm polyimide and a thin layer of 1 µm Cytop, as before.

At the initial stage, the contact angle is 130$\circ$ due to the hydrophobic behavior of Cytop; the contact angle changes to 60$\circ$ as 176 V applied to the electrode. As depicted in Fig. 6(a), the red curve represents the 176 V DC step function. From the COMSOL simulation using the step function, the lens time response is obtained and shown in Fig. 6(b) (red curve); we see considerable ringing. The calculated transfer function of the system is plotted as Fig. 6(b) (green curve). Since these two curves show excellent agreement, we see that the calculated transfer function is adequate, and can be integrated into the PID controller design.

The PID controller predicts that a shaped input voltage of the form shown in Fig. 6(a) (blue curve) would provide an optimal response. The system response for this input stimulus is shown in Fig. 6(b) (blue curve) from which we see that the response time decreases from 76 ms for the step function to 58 ms for the shaped input voltage. This corresponds to an increased speed of 23$\%$ without compromising optical performance.

### 4.3.2 Overdamped system

For the overdamped system, the two liquids employed are a mixture of OHGL&OHZB from Cargille as the polar liquid, resulting in a viscosity of 337.3 cSt, and a density of 1.9 g/cm$^3$. The non-polar liquid used in the model is a density matched fluid named laser liquid 433 from Cargille with a viscosity of 5 cSt. The dielectric layer remains the same as above. Initially the contact angle is 160°, and the contact angle changes to 90° as 142 V is applied to the electrode. As depicted in Fig. 7(a), the red curve represents the step function of 142 V DC. Through the COMSOL simulation, the lens time response is shown in Fig. 7(b) (red curve). The transfer function of the system is then plotted as Fig. 7(b) (green curve) which agrees well and confirms that a fourth order Fourier transform is appropriate for representing the transfer function of the system.

A voltage with an overshoot was then applied to the input voltage, where the overshoot voltage is limited by the breakdown strength of the dielectric. The shaped input voltage after executing the PID controller is shown in the Fig. 7(a) (blue curve) and the improved system response is shown in Fig. 7(b) (blue curve). For the overdamped system the response time for the overshoot input voltage is then 17 ms, compared with 239 ms for the stepped DC input. A dramatically enhanced actuation speed, better than a factor ten, is thus achieved.

## 5. Experimental verification

Experimental verification of these results was undertaken using the setup shown schematically in Fig. 8. A 2 mm collimated laser source (CPS635R, $\lambda$=635 nm) generates a signal on a photodetector after passing through the EWOD-tunable lens and an aperture. The detector signal shifts with a change in the optical power of the lens, enabling measurement of the response time. An arbitrarily shaped waveform is generated by a programmable function generator; this signal is amplified by a 67X piezo amplifier and applied to the electrode of the lens.

To quantitatively compare the system response with the simulation result, the simulated lens profiles at different time are converted to the total power on the detector by importing the surfaces as Zernike profiles. As shown in Fig. 9, this is achieved by positioning a 0.8 mm micro structured aperture on top of the lens; the aperture is used to restrict the total power on the detector, such that a change in focal length leads to a change in transmitted optical power. The total power on the detector as a function of lens profile is modeled using Opticstudio v15.

#### 5.1 Underdamped system

Using this system, an underdamped EWOD lens, filled with water and 10 cSt silicone oil was first characterized for a DC step function. In the initial state, the lens is convex and is actuated from an initial contact angle of 130° at 0 V to 60° at 176 V; Fig. 10(a) (black solid line) shows the response of the lens. The red dashed line shows the corresponding simulation result, with the lens profile converted to the normalized optical power on the detector.

Subsequently a shaped input voltage, as depicted in Fig. 7, was applied to the electrode, and the corresponding signal change is shown in Fig. 10(b) (black solid line); the blue dashed line shows the simulation result. As seen in the figure, the measurement curve agrees well with the simulation results and the improvement achieved using the shaped input voltage is noticeable. The measured response time for the step function is 90 ms and for the shaped input voltage is 65 ms, which corresponds to an increased speed of 27$\%$.

#### 5.2 Overdamped system

For the overdamped case, a lens using laser liquid 433 and a combination of OHGL&OHZB was used. The first measurement was carried out with a step voltage of 142 V with which the lens was actuated from an initial contact angle of 160° at 0 V to 90°; the measured response behavior is shown in Fig. 11(a) (black solid line). The simulation result is shown as the red dashed line.

Figure 10(b) black solid line illustrates the recorded signal on the detector when an overshoot voltage, as depicted in Fig. 7, was applied. As is clearly seen in the figure, the improvement in response speed by using shaped input voltage is considerable. The measured response time for the shaped input function is around 22 ms, compared with 219 ms for the step input voltage. For the overdamped EWOD lens, then, using shaped input voltages results in an order-of-magnitude response speed improved when compared to conventional stepped input voltage.

## 6. Conclusion

We have delved into the response behavior of electrowetting structures and seen that three fluid parameters, namely interfacial tension and liquid viscosities, have a strong influence on the response speed of EWOD lenses. It was shown that response time is strongly dependent upon the interfacial surface tension and less dominated by the viscosities, yielding a model which facilitates selecting correct liquids for fast electrowetting lenses. In addition, shaping the driving voltage is also a useful tool for the realization of fast dynamic lenses actuation, allowing order-of-magnitude faster response for overdamped lenses. To the best of our knowledge, the tuning speed demonstrated here significantly exceed those of comparable devices found in the literature.

## Funding

Deutsche Forschungsgemeinschaft (411766042).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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