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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19399–19406
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Optical performance of an oscillating, pinned-contact double droplet liquid lens

Joseph D. Olles, Michael J. Vogel, Bernard A. Malouin, Jr., and Amir H. Hirsa  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19399-19406 (2011)
http://dx.doi.org/10.1364/OE.19.019399


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Abstract

Liquid droplets can produce spherical interfaces that are smooth down to the molecular scale due to surface tension. For typical gas/liquid systems, spherical droplets occur on the millimeter and smaller scales. By coupling two droplets, with contact lines pinned at each edge of a cylindrical hole through a plate, a biconvex lens is created. Using a sinusoidal external pressure, this double droplet system (DDS) can be readily forced to oscillate at resonance. The resulting change in the curvatures of the droplets produces a time-varying focal length. Such an oscillating DDS was introduced in 2008 [Nat. Photonics 2, 610 (2008)]. Here we provide a more comprehensive description of the system’s optical performance, showing the effects of liquid volume and driving pressure amplitude on the back focal distance, radii of curvature, object distance, and image sharpness.

© 2011 OSA

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

1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

]. The system has shown good performance in a gas environment (liquid droplets in air) [1

1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

] or submerged in an immiscible liquid (liquid droplets immersed in another liquid) [2

2. B. A. Malouin Jr, M. J. Vogel, J. D. Olles, L. Cheng, and A. H. Hirsa, “Electromagnetic liquid pistons for capillarity-based pumping,” Lab Chip 11(3), 393–397 (2011). [CrossRef] [PubMed]

]. 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

3. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000). [CrossRef]

6

6. L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech. 179(-1), 253–266 (1987). [CrossRef]

].

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/VS = 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.
).

ω*=24(1y*2)(y*2+1)3(2y*+3L*).
(1)

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

9. R. Kingslake, ``Paraxial rays and first-order optics,” in Lens Design Fundamentals, (Academic, 1978), pp. 39–71.

] (fB) or object distance (SO). 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, Pe(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 (Rt) and bottom droplet (Rb) change for the DDS (V/VS = 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/2, i.e., half the orifice area. This region of the droplet caps can be used in order to extract the radii of curvature (Rt and Rb) from the experimental images. To define the experimental space, two parameters must be set. For the case shown in Fig. 2, a volume of 0.5VS 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

7. E. A. Theisen, M. J. Vogel, C. A. López, A. H. Hirsa, and P. H. Steen, “Capillary dynamics of coupled spherical-cap droplets,” J. Fluid Mech. 580, 495–505 (2007). [CrossRef]

]) and validated through experiments. In Fig. 2 the model shows smooth oscillatory motion for both the top droplet, Rt, and bottom droplet, Rb. 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 Rt and Rb within 3%. The asymmetry of the top and bottom drops (Rt and Rb 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

10. A. H. Hirsa, C. A. López, M. A. Laytin, M. J. Vogel, and P. H. Steen, “Low-dissipation capillary switches at small scales,” Appl. Phys. Lett. 86(1), 014106 (2005). [CrossRef]

], however in this paper all experiments and predictions were conducted without compensating for gravity. With smaller scale gas/liquid systems [10

10. A. H. Hirsa, C. A. López, M. A. Laytin, M. J. Vogel, and P. H. Steen, “Low-dissipation capillary switches at small scales,” Appl. Phys. Lett. 86(1), 014106 (2005). [CrossRef]

], 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 (fB), shown in Fig. 3
Fig. 3 fB definition is depicted with experimental and computational data for fB throughout one period of oscillation. In the plot where circular symbols are found through measurements and the curve is the computational prediction.
. From the thick lens equation, fB can be simplified [11

11. C. A. López, C. C. Lee, and A. H. Hirsa, “Electrochemically activated adaptive liquid lens,” Appl. Phys. Lett. 87(13), 134102 (2005). [CrossRef]

] to

1fB=(n'n1)(1Rb+1Rt+d(n/n'1)).
(2)

To determine fB 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, Rt and Rb. Using Rt and Rb from Fig. 2, fB 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 Rt, Rb, and fB beyond a single volume and driving pressure [1

1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

], a range of volumes and pressure amplitudes were examined and the results are presented in Fig. 4
Fig. 4 Plot (a) shows the effect of volume on fB at a pressure of 2 Pa. (b) shows how the operational fB range varies with pressure amplitude (for V/VS = 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 fB.
. Each experimental data set shows the variation in focal distance throughout a period of oscillation. Using the spring-mass model [1

1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

,7

7. E. A. Theisen, M. J. Vogel, C. A. López, A. H. Hirsa, and P. H. Steen, “Capillary dynamics of coupled spherical-cap droplets,” J. Fluid Mech. 580, 495–505 (2007). [CrossRef]

] and computationally analyzing (solid curve) a finite parameter space for V/VS < 1 and Pe(t) < 7 Pa, good agreement is found for the computed Rt, Rb, and fB to the experimental data (green circles) shown in Fig. 4.

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 fB during each period of oscillation, respectively. An increase in volume tends to decrease both the maximum (upper) and minimum (lower) limit values of fB, whereas the variation of fB 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 fB, as depicted in Fig. 4b. The range of fB 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
1f=1SI+1SO,
(3)
where f is the focal length, SI is the image distance and SO is the object distance. Here, the image distance is set as the standoff of the high-speed camera (with optical train), 52.5 mm to the camera side of the Teflon substrate. The distances, SI and SO, are computed based on the moving apex of Rt and Rb. Figure 5
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/VS = 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.
shows how the variation in pressure, incremented between 2 and 5 Pa, changes SO of the DDS for V/VS = 0.5. Figure 5a shows the computed object distance and 5b shows the experimentally measured sharpness through the DDS.

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 SO coincides with the computed object distance. In the V/VS = 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/VS = 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
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/VS = 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.
. 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/VS = 0.6, the distortion of the in-focus image was calculated to be no more than 2% and of the barrel type.

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

8. S. K. Ramalingam and O. A. Basaran, “Axisymmetric oscillation modes of a double droplet system,” Phys. Fluids 22(11), 112111 (2010). [CrossRef]

,12

12. J. B. Bostwick and P. H. Steen, “Capillary oscillations of a constrained liquid drop,” Phys. Fluids 21(3), 032108 (2009). [CrossRef]

]. 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/VS = 0.5 cause the DDS to have a concave meniscus (and no longer visible from the side) before reaching a measurable deviation of 1%.

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/VS < 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

1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

], and also around the DDS while immersed in another liquid [2

2. B. A. Malouin Jr, M. J. Vogel, J. D. Olles, L. Cheng, and A. H. Hirsa, “Electromagnetic liquid pistons for capillarity-based pumping,” Lab Chip 11(3), 393–397 (2011). [CrossRef] [PubMed]

], 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.

Acknowledgements

This work was supported by National Science Foundation grant DMII-0500408 and DARPA grant HR0011-09-1-0052. The authors would like to thank David Posada for many useful discussions.

References and links

1.

C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics 2(10), 610–613 (2008). [CrossRef]

2.

B. A. Malouin Jr, M. J. Vogel, J. D. Olles, L. Cheng, and A. H. Hirsa, “Electromagnetic liquid pistons for capillarity-based pumping,” Lab Chip 11(3), 393–397 (2011). [CrossRef] [PubMed]

3.

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000). [CrossRef]

4.

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128 (2004). [CrossRef]

5.

P. G. de Gennes, F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, (Springer, 2002).

6.

L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech. 179(-1), 253–266 (1987). [CrossRef]

7.

E. A. Theisen, M. J. Vogel, C. A. López, A. H. Hirsa, and P. H. Steen, “Capillary dynamics of coupled spherical-cap droplets,” J. Fluid Mech. 580, 495–505 (2007). [CrossRef]

8.

S. K. Ramalingam and O. A. Basaran, “Axisymmetric oscillation modes of a double droplet system,” Phys. Fluids 22(11), 112111 (2010). [CrossRef]

9.

R. Kingslake, ``Paraxial rays and first-order optics,” in Lens Design Fundamentals, (Academic, 1978), pp. 39–71.

10.

A. H. Hirsa, C. A. López, M. A. Laytin, M. J. Vogel, and P. H. Steen, “Low-dissipation capillary switches at small scales,” Appl. Phys. Lett. 86(1), 014106 (2005). [CrossRef]

11.

C. A. López, C. C. Lee, and A. H. Hirsa, “Electrochemically activated adaptive liquid lens,” Appl. Phys. Lett. 87(13), 134102 (2005). [CrossRef]

12.

J. B. Bostwick and P. H. Steen, “Capillary oscillations of a constrained liquid drop,” Phys. Fluids 21(3), 032108 (2009). [CrossRef]

OCIS Codes
(080.3630) Geometric optics : Lenses
(110.1080) Imaging systems : Active or adaptive optics

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: June 8, 2011
Revised Manuscript: August 16, 2011
Manuscript Accepted: August 24, 2011
Published: September 22, 2011

Citation
Joseph D. Olles, Michael J. Vogel, Bernard A. Malouin, and Amir H. Hirsa, "Optical performance of an oscillating, pinned-contact double droplet liquid lens," Opt. Express 19, 19399-19406 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19399


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References

  1. C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics2(10), 610–613 (2008). [CrossRef]
  2. B. A. Malouin, M. J. Vogel, J. D. Olles, L. Cheng, and A. H. Hirsa, “Electromagnetic liquid pistons for capillarity-based pumping,” Lab Chip11(3), 393–397 (2011). [CrossRef] [PubMed]
  3. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E3(2), 159–163 (2000). [CrossRef]
  4. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett.85(7), 1128 (2004). [CrossRef]
  5. P. G. de Gennes, F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, (Springer, 2002).
  6. L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech.179(-1), 253–266 (1987). [CrossRef]
  7. E. A. Theisen, M. J. Vogel, C. A. López, A. H. Hirsa, and P. H. Steen, “Capillary dynamics of coupled spherical-cap droplets,” J. Fluid Mech.580, 495–505 (2007). [CrossRef]
  8. S. K. Ramalingam and O. A. Basaran, “Axisymmetric oscillation modes of a double droplet system,” Phys. Fluids22(11), 112111 (2010). [CrossRef]
  9. R. Kingslake, ``Paraxial rays and first-order optics,” in Lens Design Fundamentals, (Academic, 1978), pp. 39–71.
  10. A. H. Hirsa, C. A. López, M. A. Laytin, M. J. Vogel, and P. H. Steen, “Low-dissipation capillary switches at small scales,” Appl. Phys. Lett.86(1), 014106 (2005). [CrossRef]
  11. C. A. López, C. C. Lee, and A. H. Hirsa, “Electrochemically activated adaptive liquid lens,” Appl. Phys. Lett.87(13), 134102 (2005). [CrossRef]
  12. J. B. Bostwick and P. H. Steen, “Capillary oscillations of a constrained liquid drop,” Phys. Fluids21(3), 032108 (2009). [CrossRef]

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