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Abstract
In this study, a focus-tunable liquid cylindrical lens based on electrowetting was designed and fabricated. The cylindrical cavity usually used in common electrowetting zoom spherical lenses was replaced by a 20 mm × 10 mm × 8 mm cuboid cavity, in which the interface of two liquids formed a toroid owing to the electrowetting effect. The proposed liquid cylindrical lens can serve as either a converging or diverging lens with the response time under 110 ms by changing the supplied voltage. The zoom lens we fabricated worked stably under 0–110 V voltage for a long time, guaranteeing that the focal length of the liquid cylindrical lens can range within (–∞, –148.36 mm) ∪ (697.21 mm, +∞). By combining the liquid lens that we designed with a simple fixed cylindrical lens, a cylindrical lens system with an arbitrary focal length suitable for various tasks in beam manipulation can be realized.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A zoom lens is a basic optical element without which many optical devices, such as telescopes [1], microscopes [2], and optical measuring instruments [3] cannot operate. However, the traditional zoom lens contains many mechanical structures [4,5], and the focal variation process is complicated. Alvarez zoom lens [6–8] varies its focal length with only a small relative lateral shift between its two constituent elements, but it still needs mechanical structure and its surface profile is complex. Therefore, achieving a micro automatic zoom system is a major challenge. The focal lengths of liquid lenses, including polymer-based liquid lenses [9–11], liquid crystal zoom lenses [12,13], and electrowetting zoom lenses [14–16], can be changed rapidly without any optical element moving, which results in a more compact structure. This has recently attracted the attention of many researchers. Polymer-based liquid lenses [17] wear easily and may even be damaged irreversibly by high liquid pressure, and the light transmittance of liquid crystal lenses is low, resulting in large light energy loss. Compared to these two methods, the electrowetting lens can be reused and has high light transmittance and a large zoom range [14]. The zoom principle of the electrowetting lens is based on changing the liquid wetting condition between two liquids in contact with each other by applying a variable external voltage to the conducting solution, so that the curvature of the liquid interface changes, causing variation of the focus [15]. The electrowetting lens has excellent performance, but most research has focused on spherical lenses [18,19]. Cylindrical lenses with no circular symmetry around the optical axis, which are widely used in beam shaping [20,21], scanning devices [22–24], and holographic displays [25–27], are relatively unexplored in relation to zoom capability for higher adaptability. Reference [28] proposed a method to realize an electrowetting zoom cylindrical lens by controlling the voltages in four directions; however, only simulation results were presented. We aimed to achieve an electrowetting-based zoom liquid cylindrical lens using a single narrow cuboid cavity filled with immiscible oil and ionic liquid by manipulating only two electrodes pasted inside the long sidewalls, assisted by a rectangular aperture to filter the irregular edges.
In the following sections, the principle of the electrowetting zoom systems and the lens structure we designed are presented in detail, and then the theoretical derivation, numerical analyses, and experimental confirmation are successively introduced to demonstrate the zoom capability of the proposed liquid cylindrical lens system.
2. Design principle and liquid cylindrical lens structure
The theoretical basis of the electrowetting effect is Young’s equation, which can be applied to explain the principle of a double-liquid lens. The voltage U applied to the conductive liquid alters the contact angle θ between the conductive liquid and another immiscible and non-conductive liquid according to the Young–Lippmann equation [15]:
To achieve a zoom liquid cylindrical lens, we considered a narrow cuboid chamber with two independent electrodes, as shown in Fig. 1. The four sidewalls of the chamber were composed of smooth quartz glass with a thickness of 2 mm, and the inner dimensions of the chamber were 20 mm × 11 mm × 8 mm. Two pieces of 20 mm × 20 mm × 0.5 mm ITO glass, coated with 15 µm thick Perylene N film as the dielectric layer, stuck against the two long sidewalls of the chamber were used as the electrodes. The bottom of the cuboid chamber was a piece of ITO glass with a thickness of 0.58 mm, and the roof was a piece of 20 mm × 10 mm × 0.5 mm quartz glass with a liquid injection hole in the middle. The final inner dimensions of the chamber were 20 mm × 10 mm × 8 mm. Then, two equal volumes of immiscible liquids, one conductive and the other nonconductive, were injected into the chamber to form a liquid lens. In consideration of the cost effectiveness of the materials, the usual n-dodecane and 5% NaCl solutions with different refractive indices were chosen as the experimental liquids injected into the chamber. The density of n-dodecane is 753 kg/m3 and that of 5% NaCl solution is 1034 kg/m3. To overcome the effect of gravity, the lighter liquid occupied the upper part of the cuboid cavity.
Fig. 1. (a) Schematic of the liquid cylindrical lens based on electrowetting. (b) Method of applied voltage.
As shown in Fig. 1, two controllable electrodes were attached to the lateral ITO glass, and the bottom ITO glass was grounded. Ignoring the edge effect, the interface between the two liquids is always straight along the long-side direction, while the curvature varies along the short side when the applied voltage changes. In other words, the double-liquid interface turns into a cylindrical shape, and a focus-tunable liquid cylindrical lens based on electrowetting is achievable.
3. Analysis of the zoom ability
COMSOL Multiphysics 5.4 was employed to simulate the shape of the double-liquid interface of the liquid cylindrical lens with different applied voltages based on Eq. (1). According to the lens structure and selected liquid types described in Section 2, the relevant parameters were set as follows: θ0 = 155°, εr = 2.65, d = 15 µm, and γ12 = 0.01 N/m. The simulated figures of the double-liquid interface at different applied direct current voltages shown in Fig. 2 directly visualize the interface shape and the changes in curvature of the surface, which demonstrates that the interface can serve as a varying cylinder if ignoring the edges of the long side.
Fig. 2. Simulated figures of the double-liquid interface at different voltages: (a) U = 0 V, (b) U = 80 V, and (c) U = 100 V.
The interfaces of the central region shown in Fig. 2 are curved in the Y-Z plane and straight in the X-Z plane. The curve in the Y-Z plane [29] can be defined as
where c is the curvature (reciprocal of the radius), and k is the conic constant. The surface data in the Y-Z plane at different voltages were extracted and fitted using MATLAB, as shown in Fig. 3.The cylindrical lens bends the light only along the radial direction. The front view (Y-Z plane) of the focus-tunable liquid cylindrical lens is shown in Fig. 4. The refractive indices of ITO glass, 5% NaCl solution, n-dodecane, quartz glass, and air are 1.5200 (n1), 1.3401 (n2), 1.4185 (n3), 1.4585 (n4), and 1.0003 (n5), respectively. Special to note is that, the ITO glass refers to a combination of substrate glass with 0.58 mm thick and an ITO film coated on it with a thickness of 200 nm, which is thin enough that the optical path distance is negligible, so that n1 closes to the refractive index of substrate glass. The center thickness of n-dodecane is h2. The central curvature radius of the double-liquid interface is r, and the incident angle and refraction angle at this surface are defined as i and i′, respectively. The aperture angles of the liquid surface and the last surface of the lens are defined as β and β′, respectively. To obtain the location of the focal plane, we can trace a paraxial ray that is parallel to the optical axis at an incident height h.
According to geometrical optics theory [30], the following relationships exist at the liquid interface in the paraxial area:
After two refractions by the quartz glass roof, the aperture angle β′ can be expressed as
Then, the effective focal length f, which is the distance between the image principal plane H′ and focal plane of the liquid cylindrical lens, can be expressed as
Substituting the curvature of the double-liquid interface shown in Fig. 3 into Eq. (5), the effective focal length values corresponding to the voltage were obtained and are plotted in Fig. 5. The theoretically calculated effective focal length range is (–∞, –148.36 mm) ∪ (120.37 mm, +∞) when the applied voltage ranges from 0 V to 140 V.
4. Experiment
The proposed focus-tunable liquid cylindrical lens based on electrowetting is composed of an outer shell and inner liquid, as described in Section 2. In the experiment, all parts were bonded by UV light–curable adhesive. After curing, 0.8 mL of 5% NaCl solution (bought from Quanzhou Yida Technology Co., Ltd.) and 0.8 mL of n-dodecane solution (purity ≥ 99%, bought from Tianjin Kermel Chemical Reagent Co., Ltd.) were injected into the chamber. Images of the liquid lens without applied voltage are shown in Fig. 6. The front view (Y-Z plane) of the double-liquid cylindrical lens shown in Fig. 6(a) indicates that the interface of the two liquids is a curve along the short side, and the side view (X-Z plane) shown in Fig. 6(b) demonstrates that the middle area of the interface is straight along the long side, which is consistent with the analysis and simulation in Section 3. Therefore, for incident light passing through the liquid lens across a rectangular aperture parallel to the bottom surface of the chamber, the liquid lens acts as a cylindrical lens.
The focal length of the cylindrical lens is determined by the curvature in the Y-Z plane. The front view (Y-Z plane) simulated and experimental surfaces at different direct current voltages were compared, as shown in Fig. 7. To obtain a clear interface of the two liquids, we added a small amount of red pigment to the 5% NaCl solution. For convenience in comparing the experimental and simulation results, solid red lines representing the liquid surface curves simulated by MATLAB, as introduced in Section 3, are marked in the experimental drawings, showing good fits and pointing to the good feasibility of the experiment.
Fig. 7. Front view of the double-liquid interface: (a)–(c) are the simulation shape and (d)–(f) are the experimental shape. (a), (d) U = 0 V; (b), (e) U = 80 V; (c), (f) U = 110 V.
To experimentally investigate the zoom performance of the designed and fabricated liquid cylindrical lens, an experimental optical system was set up as shown in Fig. 8. A light-emitting diode (LED, wavelength: 589 nm) point light source was expanded to parallel light with a diameter of 30 mm by setting it at the focal point of a convex lens (f = 180 mm). After reflection by a 45° turning mirror, the parallel light passing through a rectangular aperture with 6.0 mm wide and 8.8 mm long irradiated the bottom of the electrowetting cylinder lens system at a right angle. The light refracted by the liquid lens reached a CCD with 5496 × 3672 pixels and 2.4 × 2.4 µm2 per pixel size that was connected to a PC via a USB for observation.
Experiments to verify that the proposed electrowetting liquid lens worked as a cylindrical lens were carried out first. We placed an extremely thin circular obscuration with a diameter of 1.2 mm attaching to the top (the last optical surface) of the electrowetting cylindrical lens to observe the images with a distance of 69.0 mm between CCD and the obscuration when the applied voltage of the liquid lens was varied. The images received by the CCD are shown in Figs. 9(a)–(c), which show extremely small variations in the vertical direction that can nearly be ignored. Subsequently, we obtained images of the rectangular aperture placed in front of the liquid lens at different applied voltages, as shown in Figs. 9(d)–9(f). Figure 9 demonstrates that the double-liquid electrowetting lens we designed can serve as a liquid cylindrical lens because it varies the beam profile in only one dimension. The CCD was set to save images every 10 ms to roughly measure the response time of the electrowetting liquid cylindrical lens, which was defined as the time from the moment of turning on power to the image stabilization. By comparing the recorded images, we get that the cylindrical lens can always zoom in 110 ms.
Fig. 9. Images received by the CCD at different voltages: (a)–(c) are the images of a circular obscuration and (d)–(f) are the images of a rectangular aperture. (a), (d) U = 0 V; (b), (e) U = 80 V; (c), (f) U = 110 V.
Because the focal length of the electrowetting liquid lens is relatively long and even negative in some cases, it is inconvenient to experimentally find the focal plane directly. Therefore, the change in the focal length of the lens can be indirectly determined by observing the change in the images with variable applied voltages when the CCD is fixed. The widths of parallel beams passing through the electrowetting liquid cylindrical lens were observed and measured at different voltages when the width of the rectangular aperture was 6.0 mm and the distance between the rear surface of the lens and the CCD was 69.0 mm. The experimental results shown in Fig. 10 indicate that the width of the spot diagram gradually decreased as the applied voltage increased. The images in Fig. 10 are cropped short because we focused on the width variation of the graphs to verify the zoom ability of the liquid cylindrical lens.
Fig. 10. Images of the parallel light passing through the lens system at different voltages when the width of the rectangular aperture was 6.0 mm and the distance between the rear surface of the lens and the CCD was 69.0 mm.
To confirm our experimental data, we performed simultaneous simulations. The curve parameters simulated by MATLAB were imported into the optical design software Zemax, and the toroidal surface type was selected to fit the liquid interface data. The simulation results when the width of the rectangular aperture was 6.0 mm, the length was 8.8 mm, and the image plane was placed 69.0 mm behind the liquid cylindrical lens are shown in Fig. 11.
Fig. 11. (a)–(g) Ray tracing images of the liquid lens based on Zemax at different voltages. (h)–(j) Simulation results about the light intensity distribution of the receiving plane (69.0 mm behind the liquid cylindrical lens) when the width of the rectangular aperture was 6.0 mm and the length was 8.8 mm at different voltages: (h) U = 0 V; (i) U = 80 V; (j) U = 110 V.
It is obvious that the beams gradually converged with increasing voltage according to the ray-tracing images of the double-liquid cylindrical lens based on Zemax shown in Figs. 11(a)–11(g). The simulation results about the light intensity distribution of the receiving plane, 69.0 mm behind the liquid cylindrical lens, at U = 0 V, 80 V, and 110 V are shown in Figs. 11(h)–11(j). The double-liquid lens is initially a concave lens, so the parallel light diverges after passing through the lens system, and the width of the beam shown on the plane is larger than the aperture width. With changing voltage, the curvature of the double-liquid interface also changed significantly. When the voltage was just over 100 V, the parallel light passing through the double-liquid lens began to converge.
The experimental width data measured based on Fig. 10 and the simulation width data obtained based on Zemax are plotted together in Fig. 12. High goodness of fit is evident. Figure 12 demonstrates that the proposed double-liquid electrowetting lens can attain the zoom function by changing the applied voltage, as the analysis and simulation indicated. However, the experimental setup became extremely unstable when the voltage reached 120 V, because electrolysis often occurred in the cylindrical lens. Therefore, all experimental data included in this study correspond to voltages less than or equal to 110 V.
Fig. 12. Width of the images when the width of the rectangular aperture was 6.0 mm and the image plane was placed 69.0 mm behind the liquid cylindrical lens according to applied voltage.
5. Conclusion
In summary, a focus-tunable electrowetting-based liquid cylindrical lens created using a 20 mm × 10 mm × 8 mm cuboid chamber filled with two immiscible liquids that can operate stably for a long time when the applied voltage is below 120 V was demonstrated. The cylindrical lens gradually changes from a concave lens to a convex lens with the response time under 110 ms, and the focal length ranges within (–∞, –148.36 mm) ∪ (697.21 mm, +∞), as the applied voltage increased from 0 V to 110 V. In future theoretical and experimental studies, improving the imaging quality and decreasing the applied voltage should be considered.
Funding
National Natural Science Foundation of China (61705192, 62065019); Reserve Talents Project for Young and Middle-aged Academic and Technical Leaders of Yunnan Province (202205AC160029).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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