1. Introduction

Adaptive optics has emerged through the years as a promising technology to improve the performance of imaging systems. In contrast to conventional optical devices that are usually based on a fixed design, adaptive optics systems have the capability to improve both sensing and image quality by dynamically correcting the aberrations related to the object. Devices like liquid crystal spatial light modulators [1] and deformable mirrors [2] control these optical aberrations by measuring and correcting the wavefronts. Adjustable lenses, on the other hand, can compensate the aberrations by modulating the geometry of elastomeric and/or liquid lenses using different techniques [3]. Their advantages are given by the simplicity and the small spatial scales, as well as relatively low fabrication costs [3–5].

The principle of adjustable lenses is mainly to introduce a conjugate aberration that cancels out possible distortions of the input beam. Most tunable lenses only allow for tuning the focal length while retaining a spherical shape. Recent developments, however, have shown the potential of some optofluidic approaches to obtain non-spherical lens shapes that allow to correct the corresponding aberrations. (For a recent review, see [6]). For instance, the use of pre-molded membranes [7] or non-circular apertures [8] enables tuning of spherical aberration, astigmatism, coma, etc. Among these methods, electric actuation proved to be particularly versatile because it provides a large degree of tunability in focal length and in aberrations without the need of mechanical actuation, membranes or other predefined (e.g. non-circular) apertures [9–11]. We recently demonstrated experimentally [12] and numerically [13] the efficiency of electrostatically actuated lenses for tuning focal length, asphericity and stigmatism.

In this work, we extend our previous numerical study [13] of lenses with variable astigmatism controlled by a stripe-shaped electrode to the most general case of an array of individually addressable electrodes. With independent control of each electrode, we are able to generate complex electric field distributions and, hence, complex lens shapes allowing us to tune several types of aberrations, namely spherical aberration, 0° astigmatism, 45° astigmatism (oblique astigmatism) and coma in terms of the corresponding orthonormal Zernike coefficients. Modulation transfer function (MTF), spot diagrams and wavefront maps are calculated to provide a complete overview of the lens performance. The details of the numerical simulation (equations, physical properties, etc) are included in the appendix.

2. Geometry configuration and methods

2.1 Geometric setup and operating principle

The setup used in our simulations is based on our previous works [6, 13]. It consists of two immiscible fluids, an insulating (yellow in Fig. 1(c)) and a conductive one (blue in Fig. 1(c)), located between parallel plates separated by a middle plate with an aperture of radius a = 0.5 mm. The insulating liquid is located between the top and the middle plate and the conductive aqueous phase is placed between the middle and the bottom plate. The lens is formed by the interface of the two fluids and is pinned along the edge of the aperture.

 

Fig. 1 (a) Image of the simulation domain with a sliced computational mesh. (b) Sketch of the top view showing the position of the aperture (dashed line) with respect to the top plate. (c) Sketch of the side view. The electric field distribution is represented by the red lines.

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As in refs [6]. and [13], the lens shape is controlled by two independent parameters: a hydrostatic pressure$\Delta P_h$, applied between the two fluid phases; and the potential difference U between the aqueous phase and the top plate. In contrast to the previous works, we employ here an array of one hundred square electrodes (small red squares in Fig. 1) on the top plate rather than a single one as before. The array is centered on top of the aperture, covering the entire surface of the lens as depicted in Fig. 1(b). Each electrode is 80 × 80 ${\text{μm}}^2$ in size with a gap of 20 $\text{μm}$ in between. Geometry and applied voltages in the simulations are chosen such that the maximum electric field between adjacent electrodes does not exceed$10V/\mu m$, a threshold that was found to be safe to prevent dielectric breakdown in the presence of insulating silicone oil in experiments.

At zero voltage, the hydrostatic pressure $\Delta P_h$ is simply balanced by the Laplace pressure $\Delta P_L=\gamma \kappa$; where $\gamma$ is the surface tension and $\kappa$ is the mean curvature of the interface. In the absence of other forces, the free liquid surface displays a spherical shape with a constant mean curvature $\kappa =2/R$ ($R$: radius of the sphere.) Yet, when a potential difference is applied between the top plate and the aperture plate, an electric field is generated across the insulating phase according to $\tilde{E}=-\nabla \text{Φ}$ (red lines in Fig. 1(c); $\text{Φ}$: electrostatic potential). The electric field induces two different effects: free charges migrate to the surface of the conducting phase and electric dipoles are induced in the dielectric medium. This phenomenon gives rise to electric stresses (Maxwell stresses). In mechanical equilibrium, the applied hydrostatic pressure$\Delta P_h$, the Laplace pressure $\Delta P_L$ and the Maxwell stress ${\Pi }_{el}(\overrightarrow{r})$ are balanced:

2.2 Numerical calculation of field distribution and equilibrium shape

We make use of the widely used open-source CFD (computational fluid dynamics) package OpenFOAM to calculate numerically the equilibrium shape of the lens. The mathematical model used is based on the leaky dielectric model [16,17] that couples Maxwell’s equations with fluid mechanics equations. A full three-dimensional mesh was built due to the non-symmetric nature of the problem, see Fig. 1a. We used a dynamic meshing tool to smoothen the interface between both fluids and to reduce the computational costs. (See appendix for details of the governing equations and the boundary conditions and refs [13, 18, 19]. for general aspects of the numerical implementation.)

The shape of the lens is affected by the applied electrical potential on each electrode and by the applied hydrostatic pressure $\Delta P_h$. For the present array of 10 × 10 electrodes, this gives rise to a (100 + 1)-dimensional parameter space. For the present study, we limit ourselves to a series of patterns to address the electrodes, as represented in the left column of Fig. 2. The patterns are chosen to (primarily) excite specific aberrations, namely spherical aberration, astigmatism and coma. Red color of the electrodes indicates the location of the maximum applied voltage U_max. Abrupt changes around U_max were avoided by reducing the voltage of neighboring electrodes in steps of 100 volts. In other words, we have U_max on the well-defined excitation pattern (dark red), U_max – 100 V on the adjacent electrodes (light red) followed by U_max – 200 V on the surrounding electrodes. The middle plate and the aqueous phase are kept grounded. For the gaps between the electrodes and its surroundings we use a Neumann boundary condition to generate a diffusive distribution of the voltage on the top plate that smoothly decreases with the distance of the electrodes (see the left column of Fig. 2).

 

Fig. 2 Left column: simulated actuation patterns aiming at (a) spherical aberration Z11, (b) astigmatism Z6, (c) 45° astigmatism Z5 and (d) coma Z7. Middle column: corresponding equilibrated surface profiles in perpendicular planes. The inset exhibit the details of the curves; Right column: corresponding spot diagrams. The black circle represents the airy disc.

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The numerically calculated equilibrium surface profiles are extracted ($\approx {10}^4$ points for each surface) from the OpenFOAM simulation and fitted in Matlab using a conic asphere surface base equation [20] with a polynomial expansion:

2.2 Optical analysis

For each calculated lens profile, a merit function is set to optimize the simulations under zero defocus condition by translating a point object along the optical axis until the Zernike coefficient for defocus (Z4) is minimum. The raytracing analysis is done with wavelength of $\lambda =550$ nm, the default setting in Zemax.

The optical response of our liquid lenses was subsequently evaluated by decomposing the aberrated wavefront in terms of the Zernike coefficients related to 45° astigmatism (Z5), 0° astigmatism (Z6), coma (Z7) and spherical aberration (Z11). The values are given in units of waves, signifying the phase difference between the reference planar wavefront and the aberrated one. The image quality was characterized by two different types of analysis: the spot diagrams; and the MTF curves (modulus of the optical transfer function OTF), that quantify the resolution of the lens.

3. Results

Figure 2 depicts the response of the lens for four different electrodes excitation patterns (left column) for U_max = 400 V and $\Delta P_h$ = 30 Pa. The middle column shows the perpendicular lens profiles normalized by the radius of the aperture and the right column shows the corresponding spot diagrams. In Fig. 2(a) all electrodes were activated with the same voltage. Simultaneous actuation of all electrodes allows us to deform the lens into an aspherical shape keeping its cylindrical symmetry as presented in the profiles plot. This configuration already showed to be an effective method to tune spherical aberration [6, 10, 22]. The difference from other works, however, is in the finite electrode size used here. Due to the square geometry of the electrodes grid, a non-rotationally symmetric spot diagram is obtained (right column of Fig. 2(a)). This suggests that the system generates, even if negligible, a small amount of astigmatism.

By setting the maximum voltage on a stripe-shaped pattern, as considered in Fig. 2(b) and 2(c), one can break the symmetry of the lens generating two different profiles in orthogonal directions: one parallel and the other perpendicular to the stripe [13]. demonstrated, but with a fixed single stripe electrode, that tunable astigmatism can be achieved using this concept. Evidently the rays start spanning in one direction outside the airy disc once an astigmatic aberrated wavefront is generated.

Tunable coma, to our knowledge, still hasn’t been studied in liquid lenses due to the difficulty in dislocating the apex of the lens with respect to the optical axis. In our design, tunable coma can be achieved by exciting the electrodes with a non-centrosymmetric pattern. To do so, we apply the maximum voltage in a rectangular array of electrodes located in an off-center position, as shown in Fig. 2(d). The induced comatic aberration is depicted by the shifted profile of the lens (where the dashed line on the inset represents the optical axis). Similarly, to the astigmatic result, the image spot lies outside the airy disc, but in this case it is concentrated in one side of the image plane as expected for a comatic lens.

The subsequent results depict the performance and the tuning range of each excitation pattern presented in Fig. 2 considering the combinations of three maximum voltages U_max (200 V, 300 V and 400 V) with three initial hydrostatic pressures (10 Pa, 30 Pa and 50 Pa).

At zero voltage, the lens assumes a spherical cap shape with a radius of curvature R. The latter decreases with increasing the pressure accordingly to 7.6 mm, 2.5 mm and 1.5 mm for 10 Pa, 30 Pa and 50 Pa, respectively (for a = 0.0005 m and $\gamma$ = 0.038 N/m). Consequently, the focal length decreases, which in turn enhances the spherical aberration at fixed aperture [10]. As the voltage is applied, the spherical aberration is compensated by the aspherical shape generated on the lens. Figure 3 shows the results for spherical aberration tuning using the electrode excitation pattern presented in Fig. 2(a).

 

Fig. 3 Results of spherical aberration tuning: (a) MTF plots. The black line represents the diffraction-limited curve. (b) Wavefront map for 50 Pa and U_max = 200 V. (c) Zernike coefficients.

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The predominantly spherically aberrated wavefront generated by a hydrostatic pressure of 50 Pa and 200 V can be observed in Fig. 3(b), while the MTF plots from Fig. 3(a) illustrate the variation in the optical performance as the voltage is applied. Initially, due to a large curvature of the liquid meniscus, the lens has considerable spherical aberration, which characterizes a larger deviation from the diffraction-limited black curve. For 30 Pa and 400 V the spherical aberration is almost entirely suppressed. The Zernike coefficients plotted in Fig. 3(c) reinforce this observation. For 10 Pa the electric field distribution on the surface of the lens is almost uniform due to the small initial curvature. Therefore, for 400 V the lens becomes more curved, but the electric stresses still aren’t strong enough to generate aspherical shapes. At fixed aperture, this means more spherical aberration and the value of Z11 slightly increases.

When subjecting the liquid meniscus to an electric field considering the maximum applied voltage on a stripe-shaped pattern, as shown in Fig. 2(b) and 2(c), stronger electrostatic forces are induced parallel to the stripe. This anisotropic distribution of the electric field generates two different surface profiles giving rise to two different focal lengths in perpendicular directions and, consequently, to astigmatism [13].

Figure 4 shows the optical response of the lens when the horizontal stripe pattern of Fig. 2(b) is considered. The latter produces a closer paraxial focal length on the tangential plane (z-y plane) generating negative values of astigmatism (0° astigmatism). The difference between the profiles of the lens is also emphasized on the MTF plots of Fig. 4(a). The tangential (dashed lines) and sagittal (straight lines) curves deviate from each other confirming the presence of astigmatism on the system. This is also clearly manifested in the wavefront map as shown in Fig. 4(b). The latter resembles to a characteristic Zernike mode corresponding to an astigmatic wavefront shape (see [23]). Although in most cases the wavefront is dominated by spherical aberration, Fig. 4(c) clearly shows that the targeted coefficient Z6 related to astigmatism increases with increasing the voltage for each applied pressure. These trends are consistent with the results reported by [13] using a single stripe electrode.

 

Fig. 4 Results of 0° astigmatism tuning: (a) MTF plots. The black line represents the diffraction-limited curve. (b) Wavefront map for 30 Pa and U_max = 400 V. (c) Zernike coefficients.

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Similar observations can be made when we rotate the stripe pattern in 45° as depicted in Fig. 2(c). However, in this case, the MTF plots of Fig. 5(a) show that symmetry is retained in the tangential and sagittal planes, demonstrating that both types of astigmatism can be tuned independently from each other. The differences between the wavefronts from Fig. 4(b) and Fig. 5(b), which should ideally be identical up to a rotation of 45°, are caused by the discrete patterning of the electrodes. This suggests that an array of 10x10 electrodes is insufficient to compose effective stripe electrodes of equal width in the two different orientations. Besides 45° astigmatism, spherical aberration is always present in our system as visualized in Fig. 5(c). Moreover, the Z11 coefficient behaves in a non-monotonically manner with the voltage showing that other configurations, rather than a plane electrode, have different effects on the overall asphericity of the lens.

 

Fig. 5 Results of 45° astigmatism tuning: (a) MTF plots. The black line represents the diffraction-limited curve. (b) Wavefront map for 30 Pa and U_max = 400 V. (c) Zernike coefficients.

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In order to tune coma in liquid lenses, the apex of the lens needs to be decentered. Like for all the other aberrations, there are many ways of combining the electrodes to achieve this goal. Due to the lack of works regarding decentered liquid lenses, in Fig. 6 we show the effect of five different configurations with decentered patterns intended to generate positive vertical coma (Z7). To produce highly aberrated wavefronts, we disregard the hydrostatic pressure and use 700 V as the maximum applied voltage. Considering our setup, where H = 1, this voltage is close to the threshold between stable and unstable regions. So, the aberrations are also expected to be strong in this limit.

 

Fig. 6 (a) Decentered configurations used to produce vertical positive coma. (b) Effect of each configuration on the Zernike coefficients considering U_max = 700 V and zero applied pressure. The inset shows a zoom on the Z6 astigmatism coefficient.

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We observe from Fig. 6(b) that mainly three aberrations are affected by the applied voltage: Z6, Z7 and Z11. Negative spherical aberration is the predominant aberration, because these lenses are highly aspherical in the absence of a hydrostatic pressure. Moreover, due to its non-symmetric nature, decentered electrode patterns always generate astigmatism as well. The first pattern was the one that produced less astigmatism when compared to the other ones, but keeping the coma in a relatively large tuning range (around 0.25 waves). The effect of variable values of U_max and $\Delta P_L$ using this pattern (number 1 in Fig. 6(a)) is shown in Fig. 7.

 

Fig. 7 Results of vertical coma Z7 tuning using the excitation pattern 1 from Fig. 6(a): (a) MTF plots. The black line represents the diffraction-limited curve. (b) Wavefront map for 30 Pa and U_max = 400 V. (c) Zernike coefficients.

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Indeed, astigmatism is inherent to comatic lenses as depicted by the divergence between the tangential and sagittal curves in the MTF plots of Fig. 7(a). The wavefront map, for the case of 30 Pa and 400 V presented in Fig. 7(b), also shows the asymmetric characteristic of the lens including the presence of spherical aberration. Interestingly is that, for 30 Pa and 300 V, Fig. 7(c) shows that we could reach a value of 0.058 waves for coma while keeping the spherical aberration and the astigmatism lower than 0.01 waves. This approaches the characteristics of a pure comatic lens.

An increase in the tuning ranges of specific aberrations with increasing the voltage was observed for all cases. However, the results presented in this work show only a limited sample of the capabilities of the device. Given the number of degrees of freedom it is clear that further investigation of the effects of the excitation patterns and parameters is required to fully elucidate the potential of the approach and ultimately to optimize the decoupling of the different types of aberration.

4. Conclusion

Self-consistent numerical simulations were performed to calculate the equilibrium shape of optofluidic lenses actuated by a 10x10 grid of individually addressable electrodes and to demonstrate the concept of an optofluidics lens with flexible aberration tuning. For reasons of limited computational resources, a number of selected electrode excitation patterns addressing specifically the Z5, Z6, Z7, and Z11 Zernike coefficients, related to horizontal astigmatism, oblique astigmatism, coma, and spherical aberration, were analyzed. All coefficients could be addressed and varied by up to 0.5waves. While coma and astigmatism in different orientations could be addressed independently from each other, all excitation patterns studied also induced variations of the spherical aberration leading to a non-linear and in some cases non-monotic coupling of Z11 to the other aberrations. The latter is particularly pronounced in case of high numerical apertures, i.e. at high average lens curvatures. These limitations could presumably be overcome by using either even larger arrays of electrodes (e.g. 1000x1000) or by using different electrode geometries that match the symmetry of specific Zernike functions. In the latter case, specific types of aberrations might be addressed more directly with fewer electrodes, which would be desirable both from a computational and from a practical perspective.

The technology required to manufacture such a device is available. For practical applications, the high voltages necessary to control the lens shape might be a concern. Suitable choices of combinations of liquids with reduces interfacial tension as well as different schemes of placing the electrodes may mitigate this concern. In any case, the simulation platform presented here provides all the needs to optimize excitation patterns and geometry geometries for any experimental implementation.

Appendix

To account for the electrohydrodynamic (EHD) effects, we use a model based on the leaky dielectric model, which was developed by [24] and consolidated (reviewed) by [16]. It considers basically the coupling of Maxwell’s equations with fluid mechanics equations. Based on the boundary condition imposed for the voltage, the electric potential distribution can be calculated using Gauss’s law (Eq. (3)), which initially, due to the lack of electric charges, is given by the Laplace equation (Eq. (3) for${\rho }_e=0$):

(3)$$\nabla \cdot \left(\epsilon \nabla U\right)=-{\rho }_e$$

where ${\rho }_e$ is the electric charge density. The latter is advected with the fluid flow and conducted according to the fluid conductivity using the following transport equation:

(4)$$\frac{\partial {\rho }_e}{\partial t}+\nabla \cdot \left(\sigma \overrightarrow{E}+{\rho }_e\overrightarrow{u}\right)=0$$

where $\overrightarrow{u}$ is the velocity vector. Based on the electric charge and the electric field distributions, the electrostatic force can be calculated taking the divergence of the Maxwell stress tensor:

(5)$$\overrightarrow{F_e}=\nabla \cdot {\Pi }_{el}$$

$\overrightarrow{F_e}$ enters as a new contribution in the momentum equation, which, for an incompressible an isothermal problem, is given by:

(6)$$\rho \left(\frac{\partial \overrightarrow{u}}{\partial t}+\overrightarrow{u}\cdot \nabla \overrightarrow{u}\right)-\overrightarrow{F_{\mu }}=-\nabla p+\rho \overrightarrow{g}+\overrightarrow{F_{\gamma }}+\overrightarrow{F_e}$$

where $\rho$ is the density and $\overrightarrow{g}$ is the acceleration of gravity vector. Besides the electric force, Eq. (6) takes into consideration other two main contributions: forces related to the flow (viscous stresses) $\overrightarrow{F_{\mu }}=\nabla \cdot \left[\mu \left(\nabla \overrightarrow{u}+\nabla {\overrightarrow{u}}^T\right)\right]$ and due to the surface tension $\overrightarrow{F_{\gamma }}=\gamma \kappa \nabla {\alpha }_1$, where $\mu$ is the viscosity and ${\alpha }_1$ is the volume fraction, which is a parameter provided by the volume of fluid (VOF) method to deal with two-phase problems [25]. We observe that when the problem reaches a steady state ($\overrightarrow{u}=0$), the left side of Eq. (6) is canceled and the momentum equation becomes the force balance related to the stresses from Eq. (1).

Finally, we need to account for a transport equation for ${\alpha }_1$:

(7)$$\frac{\partial {\alpha }_1}{\partial t}+\nabla \cdot \left({\alpha }_1\overrightarrow{u}\right)+\nabla \cdot \left[{\alpha }_1\left(1-{\alpha }_1\right){\overrightarrow{u}}_r\right]=0$$

where ${\overrightarrow{u}}_r$ can be set as ${\overrightarrow{u}}_1-{\overrightarrow{u}}_2$, i.e. the velocity between phase 1 and phase 2. The volume fraction ${\alpha }_1$ is an indicator function that defines one phase ${\alpha }_1=0$ and the other one as ${\alpha }_1=1$. In the transition region ($0<{\alpha }_1<1$), the physical properties are obtained via weighted arithmetic averaging with the volume fraction as:

(8)$$\theta ={\alpha }_1{\theta }_{phase1}+\left(1-{\alpha }_1\right){\theta }_{phase2}$$

$\theta$ represents the physical properties $\rho$, $\mu$ and $\epsilon$. For the conductivity, a harmonic weighted averaging was used to smooth the electric charge density on the interface [26]. Table 1 summarizes the properties used for both fluids, which are based on the ones considered in [12].

Table 1. Summary of the physical properties of both fluids: conductivity σ; permitivitty ε; density ρ; viscosity μ; and refractive index n.

View Table

Funding

Dutch Science Foundation (NWO) and the Foundation for Technical science (STW) within the VICI program grant (11380).

Acknowledgments

The present work was performed with the support of CNPq, Conselho Nacional de Desenvolvimento Cientifico e Tecnológico - Brasil. We gratefully acknowledge the Dutch Science Foundation NWO and the Foundation for Technical science STW for financial support within the VICI program.

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