Abstract

We present a microlens array characterized by the electroclinic effect of chiral smectic A (SmA*) liquid crystals, which show the very fast dynamic switching characteristics required in high-speed optical devices. In order to easily control the intensity at the focal length of the proposed dynamic microlens structure, we adopt a solid-type liquid crystal polymer with optical anisotropy, which can split the beam intensity into two directions, depending on the vectorial portion of the polarization state of the light. The proposed microlens shows a focal intensity tunable by controlling the polarization of light at the SmA* liquid crystal. The lens has a very fast switching time of about 24μs, which is several times faster than conventional microlens arrays with surface-stabilized ferroelectric liquid crystals.

© 2009 Optical Society of America

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  1. K. Rastani, C. Lin, and J. S. Patel, “Active-fiber star coupler that uses arrays of microlenses and liquid-crystal modulators,” Appl. Opt. 31, 3046-3050 (1992).
    [CrossRef] [PubMed]
  2. B. Löfving and S. Hård, “Beam steering with two ferroelectric liquid-crystal spatial light modulators,” Opt. Lett. 23, 1541-1543 (1998).
    [CrossRef]
  3. B. Lee, S. Jung, and J.-H. Park, “Viewing-angle-enhanced integral imaging by lens switching,” Opt. Lett. 27, 818-820 (2002).
    [CrossRef]
  4. M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
    [CrossRef]
  5. A. F. Naumov, G. Love, M. Yu. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express 4, 344-352 (1999).
    [CrossRef] [PubMed]
  6. H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
    [CrossRef]
  7. H.-S. Ji, S. Kumar, and J.-H. Kim, “Electrically controllable microlens array fabricated by anisotropic phase separation from liquid-crystal and polymer composite materials,” Opt. Lett. 28, 1147-1149 (2003).
    [CrossRef] [PubMed]
  8. H. Ren, Y.-H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515-1517 (2003).
    [CrossRef]
  9. H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
    [CrossRef]
  10. J.-H. Kim and S. Kumar, “Fast switchable and bistable microlens array using ferroelectric liquid crystals,” Jpn. J. Appl. Phys. Part 1 43, 7050-7053 (2004).
    [CrossRef]
  11. Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
    [CrossRef]
  12. N. A. Clark and S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899-901 (1980).
    [CrossRef]
  13. S. Garoff and R. B. Meyer, “Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal,” Phys. Rev. Lett. 38, 848-851 (1977).
    [CrossRef]
  14. G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
    [CrossRef]
  15. G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
    [CrossRef]
  16. Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
    [CrossRef]
  17. S. D. Lee and J. S. Patel, “Temperature and field dependence of the switching behavior of induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 55, 122-124 (1989).
    [CrossRef]
  18. S. D. Lee and J. S. Patel, “Nonlinear behavior of the field-induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 54, 1653-1655 (1989).
    [CrossRef]

2005 (1)

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

2004 (4)

J.-H. Kim and S. Kumar, “Fast switchable and bistable microlens array using ferroelectric liquid crystals,” Jpn. J. Appl. Phys. Part 1 43, 7050-7053 (2004).
[CrossRef]

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
[CrossRef]

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

2003 (2)

2002 (1)

1999 (1)

1998 (1)

1994 (1)

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

1992 (1)

1989 (3)

S. D. Lee and J. S. Patel, “Temperature and field dependence of the switching behavior of induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 55, 122-124 (1989).
[CrossRef]

S. D. Lee and J. S. Patel, “Nonlinear behavior of the field-induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 54, 1653-1655 (1989).
[CrossRef]

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

1987 (1)

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

1980 (1)

N. A. Clark and S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

1977 (1)

S. Garoff and R. B. Meyer, “Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal,” Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

Andersson, G.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Choi, Y.

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

Clark, N. A.

N. A. Clark and S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

Dahl, I.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Fan, Y. H.

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

Fan, Y.-H.

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

H. Ren, Y.-H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515-1517 (2003).
[CrossRef]

Garoff, S.

S. Garoff and R. B. Meyer, “Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal,” Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

Gauza, S.

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

Hård, S.

Ji, H.-S.

Jung, S.

Kalmykov, Y. P.

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Keller, P.

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Kim, J.-H.

J.-H. Kim and S. Kumar, “Fast switchable and bistable microlens array using ferroelectric liquid crystals,” Jpn. J. Appl. Phys. Part 1 43, 7050-7053 (2004).
[CrossRef]

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

H.-S. Ji, S. Kumar, and J.-H. Kim, “Electrically controllable microlens array fabricated by anisotropic phase separation from liquid-crystal and polymer composite materials,” Opt. Lett. 28, 1147-1149 (2003).
[CrossRef] [PubMed]

Komitov, L.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

Kuczynsky, W.

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Kumar, S.

J.-H. Kim and S. Kumar, “Fast switchable and bistable microlens array using ferroelectric liquid crystals,” Jpn. J. Appl. Phys. Part 1 43, 7050-7053 (2004).
[CrossRef]

H.-S. Ji, S. Kumar, and J.-H. Kim, “Electrically controllable microlens array fabricated by anisotropic phase separation from liquid-crystal and polymer composite materials,” Opt. Lett. 28, 1147-1149 (2003).
[CrossRef] [PubMed]

Lagerwall, S. T.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

N. A. Clark and S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

Lee, B.

Lee, S. D.

S. D. Lee and J. S. Patel, “Temperature and field dependence of the switching behavior of induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 55, 122-124 (1989).
[CrossRef]

S. D. Lee and J. S. Patel, “Nonlinear behavior of the field-induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 54, 1653-1655 (1989).
[CrossRef]

Lee, S.-D.

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

Lin, C.

Lin, Y.-H.

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

Löfving, B.

Loktev, M. Yu.

Love, G.

Meyer, R. B.

S. Garoff and R. B. Meyer, “Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal,” Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

Naumov, A. F.

Park, J.-H.

Patel, J. S.

K. Rastani, C. Lin, and J. S. Patel, “Active-fiber star coupler that uses arrays of microlenses and liquid-crystal modulators,” Appl. Opt. 31, 3046-3050 (1992).
[CrossRef] [PubMed]

S. D. Lee and J. S. Patel, “Nonlinear behavior of the field-induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 54, 1653-1655 (1989).
[CrossRef]

S. D. Lee and J. S. Patel, “Temperature and field dependence of the switching behavior of induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 55, 122-124 (1989).
[CrossRef]

Rappaport, A.

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Rastani, K.

Ren, H.

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

H. Ren, Y.-H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515-1517 (2003).
[CrossRef]

Sato, S.

M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
[CrossRef]

Skarp, K.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Stebler, B.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

Vij, J. K.

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Vladimirov, F. L.

Wand, M. D.

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Wu, S. T.

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

H. Ren, Y.-H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515-1517 (2003).
[CrossRef]

Xu, Huan

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Ye, M.

M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
[CrossRef]

Yokoyama, Y.

M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
[CrossRef]

Yu, C.-J.

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (6)

H. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84, 4789-4791 (2004).
[CrossRef]

H. Ren, Y.-H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515-1517 (2003).
[CrossRef]

N. A. Clark and S. T. Lagerwall, “Submicrosecond bistable electro-optic switching in liquid crystals,” Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

G. Andersson, I. Dahl, P. Keller, W. Kuczynsky, S. T. Lagerwall, K. Skarp, and B. Stebler, “Submicrosecond electro-optic switching in the liquid-crystal smectic A phase: the soft-mode ferroelectric effect,” Appl. Phys. Lett. 51, 640-642 (1987).
[CrossRef]

S. D. Lee and J. S. Patel, “Temperature and field dependence of the switching behavior of induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 55, 122-124 (1989).
[CrossRef]

S. D. Lee and J. S. Patel, “Nonlinear behavior of the field-induced molecular tilt near the smectic A−C* transition,” Appl. Phys. Lett. 54, 1653-1655 (1989).
[CrossRef]

Ferroelectrics (1)

Y. Choi, C.-J. Yu, J.-H. Kim, and S.-D. Lee, “Fast switching characteristics of surface-relief microlens array based on a ferroelectric liquid crystal,” Ferroelectrics 312, 25-30(2004).
[CrossRef]

J. Appl. Phys. (1)

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, “Device physics of the soft-mode electro-optic effect,” J. Appl. Phys. 66, 4983-4985 (1989).
[CrossRef]

Jpn. J. Appl. Phys. Part 1 (1)

J.-H. Kim and S. Kumar, “Fast switchable and bistable microlens array using ferroelectric liquid crystals,” Jpn. J. Appl. Phys. Part 1 43, 7050-7053 (2004).
[CrossRef]

Opt. Commun. (1)

H. Ren, Y.-H. Fan, Y.-H. Lin, and S. T. Wu, “Tunable-focus microlens array using nanosized polymer-dispersed liquid crystal droplets,” Opt. Commun. 247, 101-106 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys Rev. E. (1)

Y. P. Kalmykov, J. K. Vij, Huan Xu, A. Rappaport, and M. D. Wand, “Dielectric study of the electroclinic effect in the smectic-A phase,” Phys Rev. E. 50, 2109-2114 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

S. Garoff and R. B. Meyer, “Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal,” Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

Proc. SPIE (1)

M. Ye, Y. Yokoyama, and S. Sato, “Liquid crystal lens with voltage and azimuth-dependent focus,” Proc. SPIE 5639, 124-128 (2004).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagrams of the proposed microlens array using electroclinic effect: (a) defocused state at the applied field of 20 V , (b) focused state at the applied field of + 20 V .

Fig. 2
Fig. 2

Texture images of (a)  focusing section without an LC tuning section, (b) microlens array. R, A, and P represent the rubbing direction, the optic axis of the analyzer, and the optic axis of the polarizer, respectively.

Fig. 3
Fig. 3

Induced tilt angle θ as a function of applied voltage.

Fig. 4
Fig. 4

(a) Images of the focused spot patterns. (b) Measured beam intensity profiles at the applied voltages of 20 , 0, and 20 V , respectively.

Fig. 5
Fig. 5

Switching speed measured at the LC microlens array. The solid curve and the circles with the dotted curve represent applied voltage and light transmittance, respectively.

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