## Optical performance of an oscillating, pinned-contact double droplet liquid lens |

Optics Express, Vol. 19, Issue 20, pp. 19399-19406 (2011)

http://dx.doi.org/10.1364/OE.19.019399

Acrobat PDF (1840 KB)

### 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

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]

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]

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

*) or object distance (*

_{B}*S*). 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.

_{O}## 2. Experimental setup

*P*, 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 (

_{e}(t)*R*) and bottom droplet (

_{t}*R*) change for the DDS (

_{b}*V/V*= 0.5) at system resonance (

_{S}*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.

*R*and

_{t}*R*) from the experimental images. To define the experimental space, two parameters must be set. For the case shown in Fig. 2, a volume of 0.5

_{b}*V*and driving pressure amplitude of 2 Pascals (Pa) were chosen. This produces a system with a corresponding resonant frequency of 69 Hz.

_{S}*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]

*R*, and bottom droplet,

_{t}*R*. 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

_{b}*R*and

_{t}*R*within 3%. The asymmetry of the top and bottom drops (

_{b}*R*and

_{t}*R*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

_{b}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]

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]

*), shown in Fig. 3 . From the thick lens equation, f*

_{B}*can be simplified [11*

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

*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,*

_{B}*R*and

_{t}*R*. Using

_{b}*R*and

_{t}*R*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.*

_{B}## 3. Results and discussion

*R*,

_{t}*R*, and f

_{b}*beyond a single volume and driving pressure [1*

_{B}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**(10), 610–613 (2008). [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]

*V/V*< 1 and

_{S}*P*< 7 Pa, good agreement is found for the computed

_{e}(t)*R*,

_{t}*R*, and f

_{b}*to the experimental data (green circles) shown in Fig. 4.*

_{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.*

_{B}*S*is the image distance and

_{I}*S*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,

_{O}*S*and

_{I}*S*, are computed based on the moving apex of

_{O}*R*and

_{t}*R*. Figure 5 shows how the variation in pressure, incremented between 2 and 5 Pa, changes

_{b}*S*of the DDS for

_{O}*V/V*= 0.5. Figure 5a shows the computed object distance and 5b shows the experimentally measured sharpness through the DDS.

_{S}*S*coincides with the computed object distance. In the

_{O}*V/V*= 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

_{S}*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.

*V/V*= 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

_{S}*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*= 0.6, the distortion of the in-focus image was calculated to be no more than 2% and of the barrel type.

_{S}*R*,

_{t}*R*, and f

_{b}*for one period, with*

_{B}*V/V*= 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

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

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

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

_{S}## 4. Conclusion

*V/V*< 1 at low pressures (of order 1 Pa, or 10

_{S}^{−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

**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]

## Acknowledgements

## References and links

1. | C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics |

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 |

3. | B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E |

4. | S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. |

5. | P. G. de Gennes, F. Brochard-Wyart, and D. Quéré, |

6. | L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech. |

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. |

8. | S. K. Ramalingam and O. A. Basaran, “Axisymmetric oscillation modes of a double droplet system,” Phys. Fluids |

9. | R. Kingslake, ``Paraxial rays and first-order optics,” in |

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. |

11. | C. A. López, C. C. Lee, and A. H. Hirsa, “Electrochemically activated adaptive liquid lens,” Appl. Phys. Lett. |

12. | J. B. Bostwick and P. H. Steen, “Capillary oscillations of a constrained liquid drop,” Phys. Fluids |

**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

- C. A. López and A. H. Hirsa, “Fast focusing using a pinned-contact oscillating liquid lens,” Nat. Photonics2(10), 610–613 (2008). [CrossRef]
- 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]
- 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]
- S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett.85(7), 1128 (2004). [CrossRef]
- P. G. de Gennes, F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, (Springer, 2002).
- 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]
- S. K. Ramalingam and O. A. Basaran, “Axisymmetric oscillation modes of a double droplet system,” Phys. Fluids22(11), 112111 (2010). [CrossRef]
- R. Kingslake, ``Paraxial rays and first-order optics,” in Lens Design Fundamentals, (Academic, 1978), pp. 39–71.
- 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]
- J. B. Bostwick and P. H. Steen, “Capillary oscillations of a constrained liquid drop,” Phys. Fluids21(3), 032108 (2009). [CrossRef]

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