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

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References

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  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, 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]

2011 (1)

B. A. Malouin, 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]

2010 (1)

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

2009 (1)

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

2008 (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]

2007 (1)

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]

2005 (2)

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]

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

2004 (1)

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

2000 (1)

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]

1987 (1)

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

Basaran, O. A.

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

Berge, B.

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]

Bostwick, J. B.

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

Cheng, L.

B. A. Malouin, 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]

Hendriks, B. H. W.

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

Hirsa, A. H.

B. A. Malouin, 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]

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

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]

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

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]

Hocking, L. M.

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

Kuiper, S.

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

Laytin, M. A.

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]

Lee, C. C.

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

López, C. A.

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

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]

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

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]

Malouin, B. A.

B. A. Malouin, 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]

Olles, J. D.

B. A. Malouin, 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]

Peseux, J.

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]

Ramalingam, S. K.

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

Steen, P. H.

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

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]

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]

Theisen, E. A.

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]

Vogel, M. J.

B. A. Malouin, 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]

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]

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]

Appl. Phys. Lett. (3)

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

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]

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

Eur. Phys. J. E (1)

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]

J. Fluid Mech. (2)

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

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]

Lab Chip (1)

B. A. Malouin, 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]

Nat. Photonics (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]

Phys. Fluids (2)

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

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

Other (2)

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

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

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Figures (7)

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.

Fig. 2
Fig. 2

Rt , Rb , 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/VS = 0.5 and the system is driven at 69 Hz with a pressure amplitude of 2 Pa. Measurements are represented by blue circles (Rt ) and red squares (Rb ), while solid curves denote computations.

Fig. 3
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.

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/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 f B .

Fig. 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.

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.

Fig. 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 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.

Equations (3)

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ω * = 24 ( 1 y * 2 ) ( y * 2 + 1 ) 3 ( 2 y * + 3 L * ) .
1 f B = ( n ' n 1 ) ( 1 R b + 1 R t + d ( n / n ' 1 ) ) .
1 f = 1 S I + 1 S O ,

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