Abstract

A tunable-focus large aperture liquid lens is constructed using shape memory alloy (SMA) springs as actuators. The lens mainly consists of a shallow liquid-filled cylindrical cavity bound by a thin compressible annular rim and encapsulated by a flexible circular membrane on the top of the rim and a rigid circular plate at the rim bottom. The lens optical power is adjusted by a controlled compression of the annular rim via actuation of the three shape-memory alloy (SMA) springs. Since the volume of the cavity liquid is constant, the rim compression bulges the flexible membrane outward thus reducing its radius of curvature and the lens focal length. The fabricated tunable lens demonstrated an optical power range of 0-4 diopters utilizing a driving voltage less than 3V. Lens optical wavefront profiling was done using a Shack-Hartmann sensor displaying a RMS wave front error of 0.77 µm and 1.68 µm at 0 D and + 4 D. The aperture diameter and thickness of the fabricated lens are 34 mm and 9 mm, respectively, while weighing 16.7 g.

© 2016 Optical Society of America

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References

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    [Crossref]
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  6. H. Ren, D. W. Fox, B. Wu, and S.-T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express 15(18), 11328–11335 (2007).
    [Crossref] [PubMed]
  7. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S.-T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14(18), 8031–8036 (2006).
    [Crossref] [PubMed]
  8. C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).
  9. M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
  13. M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators,” Sens. Actuators A Phys. 134(2), 494–504 (2007).
    [Crossref]
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  15. M. Kohl, Shape Memory Microactuators (Springer, 2004).
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    [Crossref]
  17. Y. Liu and Z. Xie, “Detwinning in shape memory alloy,” Prog. In Smart Materials and Structures P. L. Reece etd, 29–65 (2007).
  18. E. Khan and S. M. Srinivasan, “A new approach to the design of helical shape memory alloy spring actuators,” Smart Mater. Res. 2011, 5 (2011).
    [Crossref]
  19. T. C. Waram, Actuator Design Using Shape Memory Alloys (Tom Waram, 1993).
  20. R. C. Juvinall and K. M. Marshek, Machine Component Design (Willey, 2012).
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    [Crossref]
  22. C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
    [Crossref] [PubMed]
  23. H. Song, E. Kubica, and R. Gorbet, “Resistance modeling of SMA wire actuators,” in International workshop on Smart Materials, Structures & NDT in Aerospace Conference (2011).
  24. M. D. Giovanni, Flat and Corrugated Diaphragm Design Handbook (CRC Press, 1982).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  27. F. Schneider, J. Draheim, R. Kamberger, P. Waibel, and U. Wallrabe, “Optical characterization of adaptive fluidic silicone-membrane lenses,” Opt. Express 17(14), 11813–11821 (2009).
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2015 (1)

M. Hossain, D. K. Vu, and P. Steinmann, “A comprehensive characterization of the electromechanically coupled properties of VHB 4910 polymer,” Arch. Appl. Mech. 85(4), 523–537 (2015).
[Crossref]

2013 (1)

L. Wang, H. Oku, and M. Ishikawa, “Variable-focus lens with 30 mm optical aperture based on liquid-membrane-liquid structure,” Appl. Phys. Lett. 102(13), 131111 (2013).
[Crossref]

2012 (2)

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

2011 (3)

E. Khan and S. M. Srinivasan, “A new approach to the design of helical shape memory alloy spring actuators,” Smart Mater. Res. 2011, 5 (2011).
[Crossref]

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13(12), 125705 (2011).
[Crossref]

2009 (1)

2008 (1)

2007 (2)

H. Ren, D. W. Fox, B. Wu, and S.-T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express 15(18), 11328–11335 (2007).
[Crossref] [PubMed]

M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators,” Sens. Actuators A Phys. 134(2), 494–504 (2007).
[Crossref]

2006 (2)

H. Ren, D. Fox, P. A. Anderson, B. Wu, and S.-T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14(18), 8031–8036 (2006).
[Crossref] [PubMed]

M. Ye, B. Wang, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 78–81 (2006).

2004 (2)

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

E. Hornbogen, “Thermo-mechanical fatigue of shape memory alloys,” J. Mater. Sci. 39(2), 385–399 (2004).
[Crossref]

1997 (1)

J. E. Huber, N. A. Fleck, and M. F. Ashby, “The selection of mechanical actuators based on performance indices,” Proc. R. Soc. Lond. 453(1965), 2185–2205 (1997).

1993 (1)

Anderson, P. A.

Aschwanden, M.

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Ashby, M. F.

J. E. Huber, N. A. Fleck, and M. F. Ashby, “The selection of mechanical actuators based on performance indices,” Proc. R. Soc. Lond. 453(1965), 2185–2205 (1997).

Barbero, S.

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13(12), 125705 (2011).
[Crossref]

Blum, M.

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Büeler, M.

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Chang, F.-C.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Chen, J.-K.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Chiang, T.-J.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Chiu, C.-P.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Christian, W.

Chu, C.-W.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Copland, J.

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

Draheim, J.

Eliceiri, K.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Fan, S.-K.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Fleck, N. A.

J. E. Huber, N. A. Fleck, and M. F. Ashby, “The selection of mechanical actuators based on performance indices,” Proc. R. Soc. Lond. 453(1965), 2185–2205 (1997).

Fox, D.

Fox, D. W.

Giger, J.

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Gorbet, R.

H. Song, E. Kubica, and R. Gorbet, “Resistance modeling of SMA wire actuators,” in International workshop on Smart Materials, Structures & NDT in Aerospace Conference (2011).

Grätzel, C.

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Hall, G.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Hornbogen, E.

E. Hornbogen, “Thermo-mechanical fatigue of shape memory alloys,” J. Mater. Sci. 39(2), 385–399 (2004).
[Crossref]

Hossain, M.

M. Hossain, D. K. Vu, and P. Steinmann, “A comprehensive characterization of the electromechanically coupled properties of VHB 4910 polymer,” Arch. Appl. Mech. 85(4), 523–537 (2015).
[Crossref]

Huber, J. E.

J. E. Huber, N. A. Fleck, and M. F. Ashby, “The selection of mechanical actuators based on performance indices,” Proc. R. Soc. Lond. 453(1965), 2185–2205 (1997).

Ishikawa, M.

L. Wang, H. Oku, and M. Ishikawa, “Variable-focus lens with 30 mm optical aperture based on liquid-membrane-liquid structure,” Appl. Phys. Lett. 102(13), 131111 (2013).
[Crossref]

Jiang, H.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Kamberger, R.

Khan, E.

E. Khan and S. M. Srinivasan, “A new approach to the design of helical shape memory alloy spring actuators,” Smart Mater. Res. 2011, 5 (2011).
[Crossref]

Ko, F.-H.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Kobrin, P.

Kubica, E.

H. Song, E. Kubica, and R. Gorbet, “Resistance modeling of SMA wire actuators,” in International workshop on Smart Materials, Structures & NDT in Aerospace Conference (2011).

Kuo, S.-W.

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

Li, C.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Mazza, E.

M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators,” Sens. Actuators A Phys. 134(2), 494–504 (2007).
[Crossref]

Morita, S.

Narayanaswamy, S.

Neal, D. A.

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

Neal, D. R.

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

Oku, H.

L. Wang, H. Oku, and M. Ishikawa, “Variable-focus lens with 30 mm optical aperture based on liquid-membrane-liquid structure,” Appl. Phys. Lett. 102(13), 131111 (2013).
[Crossref]

Ren, H.

Riera, P.

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

Rubinstein, J.

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13(12), 125705 (2011).
[Crossref]

Sato, S.

M. Ye, B. Wang, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 78–81 (2006).

Schneider, F.

Seabury, C.

Song, H.

H. Song, E. Kubica, and R. Gorbet, “Resistance modeling of SMA wire actuators,” in International workshop on Smart Materials, Structures & NDT in Aerospace Conference (2011).

Srinivasan, S. M.

E. Khan and S. M. Srinivasan, “A new approach to the design of helical shape memory alloy spring actuators,” Smart Mater. Res. 2011, 5 (2011).
[Crossref]

Steinmann, P.

M. Hossain, D. K. Vu, and P. Steinmann, “A comprehensive characterization of the electromechanically coupled properties of VHB 4910 polymer,” Arch. Appl. Mech. 85(4), 523–537 (2015).
[Crossref]

Sugiura, N.

Topa, D. M.

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

Vu, D. K.

M. Hossain, D. K. Vu, and P. Steinmann, “A comprehensive characterization of the electromechanically coupled properties of VHB 4910 polymer,” Arch. Appl. Mech. 85(4), 523–537 (2015).
[Crossref]

Waibel, P.

Wallrabe, U.

Wang, B.

M. Ye, B. Wang, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 78–81 (2006).

Wang, L.

L. Wang, H. Oku, and M. Ishikawa, “Variable-focus lens with 30 mm optical aperture based on liquid-membrane-liquid structure,” Appl. Phys. Lett. 102(13), 131111 (2013).
[Crossref]

Wissler, M.

M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators,” Sens. Actuators A Phys. 134(2), 494–504 (2007).
[Crossref]

Wu, B.

Wu, S.-T.

Yang, Q.

Ye, M.

M. Ye, B. Wang, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 78–81 (2006).

Zeng, X.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Zhu, D.

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

C. Li, G. Hall, X. Zeng, D. Zhu, K. Eliceiri, and H. Jiang, “Three-dimensional surface profiling and optical characterization of liquid microlens using a Shack-Hartmann wave front sensor,” Appl. Phys. Lett. 98(17), 171104 (2011).
[Crossref] [PubMed]

L. Wang, H. Oku, and M. Ishikawa, “Variable-focus lens with 30 mm optical aperture based on liquid-membrane-liquid structure,” Appl. Phys. Lett. 102(13), 131111 (2013).
[Crossref]

Arch. Appl. Mech. (1)

M. Hossain, D. K. Vu, and P. Steinmann, “A comprehensive characterization of the electromechanically coupled properties of VHB 4910 polymer,” Arch. Appl. Mech. 85(4), 523–537 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Ye, B. Wang, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 78–81 (2006).

J. Adhes. Sci. Technol. (1)

C.-P. Chiu, T.-J. Chiang, J.-K. Chen, F.-C. Chang, F.-H. Ko, C.-W. Chu, S.-W. Kuo, and S.-K. Fan, “Liquid lenses and driving mechanisms: a review,” J. Adhes. Sci. Technol. 26:12–17, 1773–1788 (2012).

J. Mater. Sci. (1)

E. Hornbogen, “Thermo-mechanical fatigue of shape memory alloys,” J. Mater. Sci. 39(2), 385–399 (2004).
[Crossref]

J. Opt. (1)

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13(12), 125705 (2011).
[Crossref]

Opt. Express (3)

Proc. R. Soc. Lond. (1)

J. E. Huber, N. A. Fleck, and M. F. Ashby, “The selection of mechanical actuators based on performance indices,” Proc. R. Soc. Lond. 453(1965), 2185–2205 (1997).

Proc. SPIE (2)

D. R. Neal, J. Copland, D. A. Neal, D. M. Topa, and P. Riera, “Measurement of lens focal length using multi-curvature analysis of Shack-Hartmann wavefront data,” Proc. SPIE 5523, 243–255 (2004).
[Crossref]

M. Blum, M. Büeler, C. Grätzel, J. Giger, and M. Aschwanden, “Optotune focus tunable lenses and laser speckle reduction based on electroactive polymers,” Proc. SPIE 8252, 825207 (2012).
[Crossref]

Sens. Actuators A Phys. (1)

M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators,” Sens. Actuators A Phys. 134(2), 494–504 (2007).
[Crossref]

Smart Mater. Res. (1)

E. Khan and S. M. Srinivasan, “A new approach to the design of helical shape memory alloy spring actuators,” Smart Mater. Res. 2011, 5 (2011).
[Crossref]

Other (10)

T. C. Waram, Actuator Design Using Shape Memory Alloys (Tom Waram, 1993).

R. C. Juvinall and K. M. Marshek, Machine Component Design (Willey, 2012).

M. Kohl, Shape Memory Microactuators (Springer, 2004).

Y. Liu and Z. Xie, “Detwinning in shape memory alloy,” Prog. In Smart Materials and Structures P. L. Reece etd, 29–65 (2007).

R. G. Batchko and A. Szilagyi, “Fluidic lens with manually-adjustable focus,” US patent application 20140313590 (2014).

H. Ren and S.-T. Wu, Introduction to Adaptive Lenses (Wiley, 2012).

H. Jiang and X. Zeng, Microlenses: Properties, Fabrication and Liquid Lenses (CRC, 2013).

H. Zappe and C. Duppe, Tunable Micro-Optics (Cambridge, 2016).

H. Song, E. Kubica, and R. Gorbet, “Resistance modeling of SMA wire actuators,” in International workshop on Smart Materials, Structures & NDT in Aerospace Conference (2011).

M. D. Giovanni, Flat and Corrugated Diaphragm Design Handbook (CRC Press, 1982).

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Figures (5)

Fig. 1
Fig. 1 Schematic showing the fluid lens construction. The lens consists of a top elastic membrane that produces the lens curved surface, a rigid bottom plate and a compressible notched annular rim of height ho . The interior of the lens cavity is filled with an optical fluid and the cavity is compressed at the rim with three vertical spring actuators spaced 120 degrees apart. (a) Lens in planar state, (b) Lens in convex state with compressed rim height h < ho, and (c) exploded view showing the lens components (excluding the spring actuators).
Fig. 2
Fig. 2 (a) Photograph of the complete lens and SMA actuator assembly. (b) Photograph of a single NiTiCu SMA spring actuator spot welded to flat copper tabs used to mount the springs on the rim top washer and bottom plate. The copper tabs serve as electrical contacts for the SMA springs.
Fig. 3
Fig. 3 (a) optical set up for proximity technique, (b) 4f optical set up using Shack-Hartmann sensor, and (c) wave front from SH sensor at lens power + 4 D.
Fig. 4
Fig. 4 (a) Lens power versus SMA spring voltage and (b) Lens power versus rim height. The standard deviation of the lens power were less than 3% .
Fig. 5
Fig. 5 (a) Image of text recorded through liquid lens at its default state and (b) at a lens power + 3 D

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

P= 1 f =(n1)[ 1 R 1 1 R 2 + (n1)d n R 1 R 2 ]= n1 R .
V l = π 3 (R R 2 r r 2 ) 2 (2R R 2 r r 2 )+π r r 2 hπ r r 2 (h+ r r 2 4R ),
V l (P)π r r 2 (h+ r r 2 P 4(n1) ).
ΔP= 4(n1)Δh r r 2 .
F act =π( ( r r + w r ) 2 r r 2 ) E r Δh h o = k r Δh,
k= G d 4 8N D 3 .
ε s π W ( D d ) 2 θ π 2W ( D d ) 2 ε D .
h C = ( k r h s +3 k C ( h o + h a )) (3 k C + k r ) ,
Δh= 3 k r ( k H ( T av ) k C )( h o + h a h s ) (3 k H ( T av )+ k r )(3 k c + k r ) .

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