Design and wavefront characterization of an electrically tunable aspherical optofluidic lens

We present a novel design of an exclusively electrically controlled adaptive optofluidic lens that allows for manipulating both focal length and asphericity. The device is totally encapsulated and contains an aqueous lens with a clear aperture of 2mm immersed in ambient oil. The design is based on the combination of an electrowetting-driven pressure regulation to control the average curvature of the lens and a Maxwell stress-based correction of the local curvature to control spherical aberration. The performance of the lens is evaluated by a dedicated setup for the characterization of optical wavefronts using a Shack Hartmann Wavefront Sensor. The focal length of the device can be varied between 10 and 27mm. At the same time, the Zernike coefficient Z40, characterising spherical aberration, can be tuned reversibly between 0.059waves and 0.003waves at a wavelength of λ=532nm. Several possible extensions and applications of the device are discussed.


Introduction
For a long time, the development of lenses with variable focal length has been a central focus of adaptive microoptics [1][2][3]. Various approaches have been demonstrated, including deformable polymeric lenses, liquid lenses with free surfaces and liquid lenses covered by an elastomeric membrane. Most of the proposed approaches aimed at spherical lenses of variable curvature. In particular in optofluidics, various actuation mechanisms were explored to tune liquid lenses, including variations of the pressure or the volume of the lens fluid and the wettability of the substrate. Some approaches even involved dynamic excitation of the fluid in combination with synchronized image acquisition using high speed cameras to achieve ultrafast actuation [4]. Electrowetting (EW) proved to be a particularly versatile approach in this respect because it allows for very fast actuation and a wide range allowing to achieve both positive and negative focal lengths with the same device, simply by a applying more or less voltage [5][6][7]. If operated with density matched ambient liquid media, such lenses proved to be very reliable, fast, and resistant against mechanical vibrations.
A key problem in microoptics is often the presence of strong aberrations, in particular if the full aperture of a lens is used to collect a sufficient amount of light. Because of constrained space, microoptical systems often don't allow for standard combinations of lenses to compensate for aberrations. This typically leads to rather poor image quality. To respond to this challenge, several approaches have been proposed in recent years in order to generate non-spherical microlenses with tunable shape to compensate for various forms of geometric aberrations. In case of elastomeric lenses, mechanical actuators were used to distort surface profiles and suppress or deliberately induce astigmatism [8]. In case of membrane-covered liquid lenses, membranes with custom-engineered thickness profiles were used to achieve lenses of minimum spherical aberration within certain ranges of focal length [9]. More recently, sophisticated tubular lenses actuated by EW with segmented electrodes on the inside of the tube were demonstrated to efficiently compensate astigmatism [10]. Numerical simulations using a genetic algorithm indicate that this approach allows for very efficient improvements of the point spread function and Strehl ratio of an imaging system [10,11].
However, using this approach it will always be difficult to compensate for spherical aberration because any free liquid surface, no matter how complex the boundary condition, is always a surface of constant mean curvature by the laws of capillarity, unless additional external forces are applied. This is in contrast to the fundamental origin of spherical aberration, which arises from the fact that perfect imaging can only be obtained if the curvature of the refracting surface is not constant but decreases with increasing distance from the optical axis. A convincing solution to overcome this problem was first demonstrated by Zhan et al. [12]. These authors demonstrated that electric fields could be used to distort liquid surfaces in a manner that approaches an ideal aspherical lens shape. This was achieved by placing a homogeneous flat electrode at a fixed voltage above electrically insulated drops of photo-curable polymers. This lead to aspherical microlenses that were subsequently crosslinked in their deformed state under voltage. As a consequence of solidification, the drops obviously lost their tunability. Moreover, the suffered from surface roughness.
A few years later, inspired by EW-experiments on the Cassie-to-Wenzel transition on superhydrophobic surfaces [13] and numerical calculations of the equilibrium liquid surface profiles in electric fields [14], Mishra et al. [15] implemented a liquid lens design that allowed for reversible tuning of both longitudinal spherical aberration (LSA) and focal length over a substantial range. While the equilibrium shape of the lens is determined by the local balance of Laplace pressure and electrical Maxwell stress as in the approach of [12], the possibility of simultaneous independent variation of the hydrostatic pressure difference between the aqueous lens fluid and an ambient oil in combination with the asphericitycontrolling voltage enabled independent variation of LSA and focal length. This allowed for instance, to vary the focal length from 2 to 20mm while keeping the LSA -as inferred from side view images of the lens -at zero. In subsequent extensions of the approach, the flat electrode used to suppress spherical aberration was replaced in numerical simulations first by a stripe-shaped electrode to induce controlled astigmatism [16] and eventually by an array of individually addressable electrodes [17]. Ray-tracing analysis of these numerically calculated surface profiles as well as experimental ones [18] and their analysis in terms of Zernike coefficients and modulation transfer function suggested that various types of geometric aberrations could indeed be controlled by this approach.
The purpose of the present contribution is twofold: First, we combine the approaches of Murade et al. [19] with an EW-controlled pressure regulation to vary the overall lens curvature and the one of Mishra et al. [15] with a Maxwell stress-controlled local variation of the local curvature to suppress spherical aberration into a single, all-encapsulated device. This lens is actuated exclusively by electrical control signals and allows for a wide tuning range of both focal length and spherical aberration. Second, we quantify experimentally the wavefront distortions generated by our device. To this end, we set up a testing platform using a Shack-Hartman wavefront sensor (SHWS) that allows us to measure wavefronts following previous reports in the literature [8,20,21].

Device design and operating principle
The design of the device is shown in Fig. 1. The core of the lens consists of a top plate (number (1) in Fig. 1a Variations of the EW voltage P U change the contact angle on the bottom plate and thereby also the curvature of the annular part of the drop surface. As a consequence, the radius of the lens surface and hence the focal length of the device change. For the present device, varying P U between 0 and 70V leads to a perfectly reversible variation of the focal length between approximately 10mm and 27mm, as shown in Fig. 3a. In these experiments, the focal length is measured using the SHWS. After applying a voltage to the device, the microscope objective in the measurement arm is displaced until the Zernike coefficient corresponding to defocussing ( 0 2 Z or Z4) is minimized. (We consider the wavefront on the SHWS as flat if its radius of curvature exceeds 20m.) The corresponding displacement is noted as the variation of the focal length. At the same time, all other Zernike coefficients are measured using the SHWS. Figure 3b shows the corresponding variation of the primary spherical aberration 0 4 Z . (We use here the indexing based on the radial (subscript) and azimuthal (superscript) degree of the Zernike function. According the OSA standard, this coefficient would be denoted as Z12; according to Noll Z11; and according to the popular ray-tracing software package Zemax as Z13.) As expected given the spherical shape and the fixed aperture diameter, the ice of the ation that refractive he tuning range of the focal length could be increased by choosing a different ratio between the aperture diameter and the spacing between the middle and bottom plate. We note, however, that the presently achieved tuning range already exceeds the one demonstrated e.g. for elastomeric lenses with tunable geometric aberrations [8]. Clearly, combining the present approach with an array of structured electrodes on the top plate instead of the homogeneous one described here would enable systematic addressing of other geometric aberrations such as coma and astigmatism in arbitrary directions [16,17]. Moreover, it is conceivable to include a feedback mechanism to the electrical actuation. In this manner, the system could be used to actively control the shape of the wavefront and to further optimize imaging properties e.g. within a confocal microscope. The response speed of our lens is determined by the hydrodynamic response time of the drop, which scales with the aperture diameter 3 2 D − . Although not tested explicitly in the work, we expect for the dimensions of the present device that this time will be of the order of a few tens of ms [5]. It can be increased substantially, if smaller aperture diameters are acceptable [19].

Conclusion
The integration of an electrowetting based pressure control in a liquid lens and an additional Maxwell stress controlled deformation of the refracting liquid-liquid interface leads to an all electrically controlled tunable optofluidic lens with a wide range of reversibly tunable focal length and asphericity. This all electrical control is expected to enable the implementation of feedback mechanisms for adaptive wavefront shaping, which is particularly attractive in combination with segmented electrodes that allow to address specific primary aberrations in a targeted manner.

Funding
Dutch Science Foundation NWO Foundation for Technical science STW, VICI program 11380.