1. Introduction

Liquid lenses have been demonstrated with features unparalleled by most solid lens counterparts. In one liquid lens strategy, a spherical lens (droplet) is formed by capillarity, pinned at its base through a combination of surface chemistry and geometry. By coupling two droplets at their base through a conduit, a lens with a straight, continuous optical axis is created. In this double droplet system (DDS), one can manipulate the surface curvatures and, therefore, focal distance by controlling the differential pressure across the two droplets [1]. The system has shown good performance in a gas environment (liquid droplets in air) [1] or submerged in an immiscible liquid (liquid droplets immersed in another liquid) [2]. In either configuration, smooth spherical interfaces can be produced. Making the droplets fixed or “pinned” at their contact line eliminates the hysteresis associated with advancing and receding contact lines [3–6].

Here, the gas/liquid DDS was studied in order to characterize its optical performance as a practical adaptive liquid lens. Using a sinusoidal external pressure on the DDS, the capillary surfaces are forced to oscillate (shown in Fig. 1 ).

 

Fig. 1 (a) is a cut-away of the experimental setup (air surrounds the droplet caps) and (b) is a time (t) series of images during one period of oscillation (T) for V/V_S = 0.5 at resonance ω = 69 Hz (time between images is approximately 3.6 ms). Vertical axis shows effect of increasing sinusoidal external pressure amplitude (2.5, 5, and 10 Pa) on droplet motion.

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The dynamic effects of liquid inertia and surface tension on the DDS can be understood by considering the model by Theisen et al. [7]. Assuming spherical cap droplets, Theisen et al. model the DDS through a force balance with simple components: a mass (liquid volume), springs (capillary pressure on each droplet), and a damper (liquid viscosity). Since the introduction of the one-dimensional model representing the DDS, there has been additional work in understanding and predicting the oscillating motion in the presence of external forces (gravity and pressure) [1,8]. Even with this simplified linear model, however, a resonant frequency of the system can be extracted. The resonant frequency provides the greatest change in curvature for a given applied pressure. The theoretical resonant frequency found (on the order of 100 Hz) has been corroborated with experiments [1] and multi-dimensional computational models based on the Navier-Stokes equations [8]. In the case where V/V_S < 1, (the protruding droplet volumes V nondimensionalized by the volume of a sphere V_S, with a radius of the orifice) Theisen et al. report the nondimensionalized resonant frequency, ω ^*, as

In this equation y^* = y/r is the height of a droplet and 2L^* = 2L/r is the length of the conduit connecting the DDS, where r is the radius of the orifice (see Fig. 2 ). In the context of lenses, the length of the conduit is the edge thickness. The frequency has been nondimensionalized by the mode of a spherical drop with the lowest non-zero frequency, 8σ/(ρ r ³), where σ is surface tension, and ρ is density. Here the volume of the droplet used to quantify the liquid lens is related to y through the equation, V=π y/6(3r ² +y ² ).

 

Fig. 2 R_t, R_b, and other DDS definitions, are depicted along with experimental and computational data for both the top and bottom droplets in one period of oscillation. In this figure and in Fig. 3, V/V_S = 0.5 and the system is driven at 69 Hz with a pressure amplitude of 2 Pa. Measurements are represented by blue circles (R_t) and red squares (R_b), while solid curves denote computations.

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In this paper general fluid and optical characteristics of the system are elaborated upon through both experiments and computations. The dynamic (varying) focal length of the DDS is reported as either the back focal distance [9] (f_B) or object distance (S_O). We show how changing the liquid volume and external driving pressure affects the time-varying focal length through direct measurements of radii of curvature, as well as image sharpness measurements. Lastly, we show the limits of the spherical cap assumption inherent in the model and how it breaks down as the forcing pressure increases.

2. Experimental setup

To create the DDS, a liquid (water) is injected through a port in a plate, filling an orifice which extends through the substrate, so that a desired volume protrudes. The droplet caps are constrained at the base by a pinned contact line, which is a result of the orifice edge sharpness as well as the interfacial properties of the liquid and substrate material. For these experiments a Teflon substrate machined to a thickness of 1.92 mm and an orifice diameter of 1.68 mm is used. Enclosing one side of the DDS in a chamber and providing a time-varying (t) external pressure source, P_e(t), from an audio speaker induces motion of the DDS. The setup is shown in a cut-away in Fig. 1a. Figure 1b presents side views throughout one period of oscillation (T), illustrating how the radii of curvature for the top droplet (R_t) and bottom droplet (R_b) change for the DDS (V/V_S = 0.5) at system resonance (ω = 69 Hz) for three different pressure amplitudes. Besides the amplitude of pressure, these time-varying changes in curvature depend on the total liquid volume. With an understanding of the dynamics of the DDS, it is straightforward to exploit this system as a dynamic liquid lens.

Due to image distortion near the edge of the liquid lens, the droplet caps in the DDS have a maximum usable diameter of about $2r/\sqrt{2}$, i.e., half the orifice area. This region of the droplet caps can be used in order to extract the radii of curvature (R_t and R_b) from the experimental images. To define the experimental space, two parameters must be set. For the case shown in Fig. 2, a volume of 0.5V_S and driving pressure amplitude of 2 Pascals (Pa) were chosen. This produces a system with a corresponding resonant frequency of 69 Hz.

Optical characteristics of the lens are predicted using a computational model (based on the work of Theisen et al. [7]) and validated through experiments. In Fig. 2 the model shows smooth oscillatory motion for both the top droplet, R_t, and bottom droplet, R_b. This is expected since the forcing pressure on the fluid is set to be sinusoidal. The model and the experiments show agreement of the expected oscillatory motion, with the model predicting the droplet R_t and R_b within 3%. The asymmetry of the top and bottom drops (R_t and R_b in Fig. 2) is predominantly due to gravity producing unequal droplet volumes. In practice, the asymmetry can be compensated by applying a pressure bias [10], however in this paper all experiments and predictions were conducted without compensating for gravity. With smaller scale gas/liquid systems [10], or by using liquid/liquid systems, the effect of gravity can be made negligibly small.

The shape of the optical surfaces can be utilized to determine back focal distance (f_B), shown in Fig. 3 . From the thick lens equation, f_B can be simplified [11] to

 

Fig. 3 f_B definition is depicted with experimental and computational data for f_B throughout one period of oscillation. In the plot where circular symbols are found through measurements and the curve is the computational prediction.

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To determine f_B experimentally, a time sequence of side profiles is taken of the DDS during one oscillation. Each image is analyzed (using only half of the orifice area) and a circle is fit to the top and bottom droplet caps to extract the radii of curvature, R_t and R_b. Using R_t and R_b from Fig. 2, f_B is plotted for both experimental and computational data in Fig. 3. Figure 3 shows good agreement between the predicted and measured optical performance of the DDS throughout a period of oscillation.

To capture time sequence events, a high-speed camera (Redlake HG-100K) is used with a 1.0X TV tube, a high magnification zoom lens (Thales-Optem 70XL), and a 1.5X objective. For the pressure excitation, a function generator (Agilent 33210A) is used as an input to a subwoofer speaker (Yamaha YSTII), with pressure measurements from a ½ inch microphone (Brüel & Kjær 4189). To gather data from these devices, LabView was used in conjunction with a data acquisition card (National Instruments USB-6009). When gathering image sharpness data, a standard resolution target was imaged throughout a focal scan of the DDS and analyzed using a variance filter algorithm.

3. Results and discussion

In order to extend the investigation of R_t, R_b, and f_B beyond a single volume and driving pressure [1], a range of volumes and pressure amplitudes were examined and the results are presented in Fig. 4 . Each experimental data set shows the variation in focal distance throughout a period of oscillation. Using the spring-mass model [1,7] and computationally analyzing (solid curve) a finite parameter space for V/V_S < 1 and P_e(t) < 7 Pa, good agreement is found for the computed R_t, R_b, and f_B to the experimental data (green circles) shown in Fig. 4.

 

Fig. 4 Plot (a) shows the effect of volume on f_B at a pressure of 2 Pa. (b) shows how the operational f_B range varies with pressure amplitude (for V/V_S = 0.5). All dotted lines depict a full period of motion for the DDS found through measurement, while solid curves show the computational maximum and minimum of the f_B.

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During the oscillation of the driving pressure, the liquid lens geometry changes. Thus, the lens shape at pressure peaks and pressure troughs are different, and lead to a minimum and a maximum f_B during each period of oscillation, respectively. An increase in volume tends to decrease both the maximum (upper) and minimum (lower) limit values of f_B, whereas the variation of f_B throughout a period of oscillation (determined by the difference of the upper and lower limits) increases with volume (see Fig. 4a). Similarly, increasing the pressure amplitude increases the variation of f_B, as depicted in Fig. 4b. The range of f_B that is scanned per oscillation increases in a roughly linear fashion, from no range at zero pressure, up to just over 1 mm of range at 5.5 Pa. Although these focal lengths are on the order of a millimeter, incorporating this liquid lens into an appropriate optical train would greatly expand the useful range.

Placing the DDS as the objective lens for an imaging system, the basic lens equation can be used

 

Fig. 5 To show the effects of driving pressure on the system, a variety of driving pressures were tested. Plot (a) shows curves of the computational object distances in focus for several different driving pressure amplitudes (hereV/V_S = 0.5). Plot (b) shows experimental measurements of relative sharpness of the target in (a) for each driving pressure. Insets show images of the resolution target taken through the DDS at different instances during its oscillation while driven at 2 Pa (green curve), in-focus (left inset) and out of focus (right inset). The system was driven at a resonance of 69 Hz.

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A standard resolution target is placed at 10.2 mm above the Teflon substrate (denoted by a dashed pink line in Fig. 5a). Images of that target were captured through the DDS and analyzed for sharpness. The relative sharpness (for each driving pressure) is then plotted in Fig. 5b. Increases in sharpness are shown to occur as expected (throughout the focal distance scan of the lens) wherever S_O coincides with the computed object distance. In the V/V_S = 0.5 case, the resonant oscillation occurs at a period of 14.5 ms (69 Hz). For most cases of pressure variation (3-5 Pa) in Fig. 5, the sharpest images for this DDS are at t/T=0.3 and 0.7. At the lowest pressure, 2 Pa, the reduced movement of the DDS causes the sharpness to be greater for the entire cycle and also changes the sharpest image times to occur at t/T=0.36 and 0.64, consistent with computations of Fig. 5a.

This analysis of the experimentally obtained image sharpness was extended for volume ratios of V/V_S = 0.4, 0.5, and 0.6 at a pressure of 3 Pa, with the period of oscillation varying according to the resonance of each particular volume, results are presented in Fig. 6 . For these cases the in-focus images are at t/T=0.25 and 0.75, t/T=0.3 and 0.7, and t/T=0.3 and 0.7 respectively. For the case of V/V_S = 0.6, the distortion of the in-focus image was calculated to be no more than 2% and of the barrel type.

 

Fig. 6 To show the effect of volume on system performance, a number of volumes were tested. Plot (a) shows curves of the computational object distances (V/V_S = 0.4, 0.5, and 0.6) for an external pressure amplitude of 3 Pa. Plot (b) shows the relative sharpness of the images taken through the DDS of each volume during its oscillation at the corresponding volumes and pressure, which were driven at resonance of; 89, 69, and 56 Hz respectively.

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Note the subtle, but important difference between changing droplet volume (Fig. 5) and driving pressure amplitude (Fig. 6). For the cases of varying pressure magnitude, the equilibrium (zero pressure) state of the system is identical for all cases, which likely explains why all the cases have identical focal distances as they pass through that equilibrium state. On the other hand, changing the total volume of droplets changes the base state (equilibrium) optics, resulting in more variation in the curves of Fig. 6a. Pressure variations of 3-5 Pa in Fig. 5a, and all volumes in Fig. 6a, show computational results tending towards infinity. These computed object distances are a result of the overall equivalent lens (liquid lens used in conjunction with additional lenses) making a transition from a positive focal length lens to a negative lens.

A key issue in the dynamics of the DDS is how the pressure amplitude affects the sphericity. Since only side view images are captured, radii of curvature of the droplet caps are analyzed assuming axisymmetry. Using a 1% deviation threshold from circularity on the radii of curvature of the droplet caps, two volumes are experimentally measured and plotted against the computational results. Figure 7 shows R_t, R_b, and f_B for one period, with V/V_S = 0.5 and 0.7, when a 1% deviation from circularity occurs (at pressures of 5.5 and 4 Pa respectively). These experimentally obtained limits of the pressure provide an upper bound for the model predictions. More recent theoretical and computational fluid dynamics studies predict the deviation from spherical caps [8,12]. Ideally a threshold plot would be constructed through experiments to show pressure amplitudes for all volumes where a deviation of 1% from circularity occurs. However, volumes under V/V_S = 0.5 cause the DDS to have a concave meniscus (and no longer visible from the side) before reaching a measurable deviation of 1%.

 

Fig. 7 1% deviation from sphericity occurs at an external pressure amplitude of 5.5 Pa and 4 Pa for V/V_S = 0.5 and V/V_S = 0.7, respectively. Plots in (a) show R_t and R_b for computational results, solid curves, and experimental data, circular and square symbols. Plots in (b) depict f_B for one period, with solid curves for computations and circular symbols for experimental data. These were driven at 69 and 49 Hz, respectively.

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4. Conclusion

The variable focal length double droplet liquid lens has been shown to perform as a viable lens at frequencies on the order of 100 Hz. Here we have shown that for the parameter space V/V_S < 1 at low pressures (of order 1 Pa, or 10⁻⁵ atm.), the one-dimensional computational model predicts the dynamics of the DDS well. However, at larger amplitudes, for example, the model’s assumption of spherical cap droplets can break down, limiting the utility of the model in such cases. Some data has been presented to illustrate representative experimental conditions under which departures from spherical caps inhibit the full use of the model. In practice, driving with larger pressure amplitudes may still be of practical use, by producing some desired effects including increased focal distance of the liquid lens. However, this increased focal distance may come at the expense of decreasing the undistorted image area. The fluid motion within the DDS in a gas environment [1], and also around the DDS while immersed in another liquid [2], still needs to be studied further and may lead to a better understanding and prediction of the actual shape of the DDS throughout a period of oscillation.