Abstract

Virtual reality-head mounted displays require a display with high resolution over 2000 ppi, super-fast response time and high contrast ratio for realizing super image quality at near-eyes. Several liquid crystal devices utilizing fringe-field switching (FFS) mode, having response times less than of half of conventional FFS mode, were proposed for this purpose. However, its contrast ratio is still less than 2000:1 because of intrinsic electro-optic characteristics of homogenous alignment mode and also realizing high resolution like 2000 ppi has some difficulty because twist deformation of liquid crystals can easily affect liquid crystal orientation near pixels. In this paper, we propose a vertically aligned liquid crystal device in which bend deformation occurs in a confined area by an oblique electric field, exhibiting 4 times faster decay response time than that of conventional FFS mode, higher contrast ratio over 5000:1, and pixel pitch less than 4 μm. The proposed liquid crystal device has a strong potential to be the main display for high-resolution virtual reality over 2000 ppi.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

2017 (2)

2016 (4)

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

H. Chen, F. Peng, F. Gou, Y.-H. Lee, M. Wand, and S.-T. Wu, “Nematic LCD with motion picture response time comparable to organic LEDs,” Optica 3(9), 1033–1034 (2016).
[Crossref]

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

1998 (1)

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73(20), 2881–2883 (1998).
[Crossref]

1997 (1)

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

1974 (1)

R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45(12), 5466–5468 (1974).
[Crossref]

Bos, P. J.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

M. Kim, H. G. Ham, H.-S. Choi, P. J. Bos, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric in-plane switching (IPS) mode with ultra-high-transmittance, low-voltage, low-frequency, and a flicker-free image,” Opt. Express 25(6), 5962–5971 (2017).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

Chen, H.

Choi, H.-S.

Choi, T.-H.

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

Choi, Y.

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

Gou, F.

Ham, H. G.

Hanaoka, K.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

Higashida, S.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

Iwakabe, Y.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Jeong, K.-U.

Jin, H. S.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

Kanashima, A.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

Katayama, T.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

Keum, C.-M.

Kim, D.-W.

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

Kim, H. Y.

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73(20), 2881–2883 (1998).
[Crossref]

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Kim, M.

Kim, M. S.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

Kimura, S.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Komura, S.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Lee, C. H.

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Lee, J. H.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

M. Kim, H. G. Ham, H.-S. Choi, P. J. Bos, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric in-plane switching (IPS) mode with ultra-high-transmittance, low-voltage, low-frequency, and a flicker-free image,” Opt. Express 25(6), 5962–5971 (2017).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

Lee, S. H.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

M. Kim, H. G. Ham, H.-S. Choi, P. J. Bos, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric in-plane switching (IPS) mode with ultra-high-transmittance, low-voltage, low-frequency, and a flicker-free image,” Opt. Express 25(6), 5962–5971 (2017).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73(20), 2881–2883 (1998).
[Crossref]

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Lee, S. J.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

Lee, S. L.

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73(20), 2881–2883 (1998).
[Crossref]

Lee, S.-L.

Lee, Y.-H.

Li, M.-C.

Mastushima, T.

T. Mastushima, K. Okazaki, Y. Yang, and K. Takizawa, “New fast response time in-plane switching liquid crystal mode,” SID Symposium Digest of Technical Papers46, 648–651 (2015).

Matsushima, T.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Nakanura, T.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Oh, S.-W.

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

Okazaki, K.

T. Mastushima, K. Okazaki, Y. Yang, and K. Takizawa, “New fast response time in-plane switching liquid crystal mode,” SID Symposium Digest of Technical Papers46, 648–651 (2015).

Park, H. S.

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Park, I. C.

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Park, J. S.

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Park, Y.-J.

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

Peng, F.

Rho, B. G.

S. H. Lee, H. Y. Kim, I. C. Park, B. G. Rho, J. S. Park, H. S. Park, and C. H. Lee, “Rubbing-free, vertically aligned nematic liquid crystal display controlled by in-plane field,” Appl. Phys. Lett. 71(19), 2851–2853 (1997).
[Crossref]

Seki, K.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Shimada, S.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

Shin, Y.-H.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

Soref, R. A.

R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45(12), 5466–5468 (1974).
[Crossref]

Takizawa, K.

T. Mastushima, K. Okazaki, Y. Yang, and K. Takizawa, “New fast response time in-plane switching liquid crystal mode,” SID Symposium Digest of Technical Papers46, 648–651 (2015).

Uchida, M.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Wand, M.

Watanabe, Y.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Wu, S.-T.

Yang, D.-K.

M. S. Kim, H. S. Jin, S. J. Lee, Y.-H. Shin, H. G. Ham, D.-K. Yang, P. J. Bos, J. H. Lee, and S. H. Lee, “Liquid crystals for superior electro-optic performance display device with power-saving mode,” Adv. Opt. Mater. 6(11), 1800022 (2018).
[Crossref]

M. Kim, H. G. Ham, H.-S. Choi, P. J. Bos, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric in-plane switching (IPS) mode with ultra-high-transmittance, low-voltage, low-frequency, and a flicker-free image,” Opt. Express 25(6), 5962–5971 (2017).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, D.-K. Yang, J. H. Lee, and S. H. Lee, “Flexoelectric effect in an in-plane switching (IPS) liquid crystal cell for low-power consumption display devices,” Sci. Rep. 6(1), 35254 (2016).
[Crossref] [PubMed]

M. S. Kim, P. J. Bos, D.-W. Kim, C.-M. Keum, D.-K. Yang, H. G. Ham, K.-U. Jeong, J. H. Lee, and S. H. Lee, “Field-symmetrization to solve luminance deviation between frames in a low-frequency-driven fringe-field switching liquid crystal cell,” Opt. Express 24(26), 29568–29576 (2016).
[Crossref] [PubMed]

Yang, Y.

T. Mastushima, K. Okazaki, Y. Yang, and K. Takizawa, “New fast response time in-plane switching liquid crystal mode,” SID Symposium Digest of Technical Papers46, 648–651 (2015).

Yata, T.

T. Matsushima, K. Seki, S. Kimura, Y. Iwakabe, T. Yata, Y. Watanabe, S. Komura, M. Uchida, and T. Nakanura, “Optimal fast-response LCD for high-definition virtual reality head mounted display,” SID Symposium Digest of Technical Papers49, 667–670 (2018).
[Crossref]

Yoon, T.-H.

T.-H. Choi, S.-W. Oh, Y.-J. Park, Y. Choi, and T.-H. Yoon, “Fast fringe-field switching of a liquid crystal cell by two-dimensional confinement with virtual walls,” Sci. Rep. 6(1), 27936 (2016).
[Crossref] [PubMed]

Yoshida, H.

T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
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T. Katayama, S. Higashida, A. Kanashima, K. Hanaoka, H. Yoshida, and S. Shimada, “Development of in-plane super fast response (ip-SFR) LCD for VR-HMD,”SID Symposium Digest of Technical Papers49, 671–673 (2018).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic cross-sectional view of the UFS device with LC director and electric field lines in (a) dark and (b) bright state. The red arrows indicate electric field direction between pixel and common electrodes. Even in voltage-on state, the distance l in which LC does not reorient by the field exists as indicated in Fig. 1(b).
Fig. 2
Fig. 2 Calculated electrode-position dependent transmittances at three different voltages when g is (a) 1 μm, (b) 2 μm, and (c) 3 μm in the UFS device. (d) Calculated voltage-dependent transmittances at three different gs and (e) time-dependent transmittance curves in the FFS and UFS devices. Here, the electrode width and gap between patterned electrodes in the FFS device are 3 μm and 4.5 μm, respectively. The transmittance is normalized to two parallel polarizers.
Fig. 3
Fig. 3 Measured (a) voltage-dependent and (b) time-dependent transmittance (averaged one over the entire pixel) curves between FFS and UFS devices. The inset POM images represent dark and white states of the UFS device.
Fig. 4
Fig. 4 Calculated electrode-position dependent transmittances at three different voltages when the pixel pitch is 4 μm with gap of 2 μm in the normal (a) and the optimized (b) UFS device. (c) Calculated voltage-dependent transmittances and (d) Time-dependent transmittance curves at two different device conditions. Here, the transmittance is normalized to two parallel polarizers. When the pixel pitch reduces from 8 μm to 4 μm, both l and leff. become reduced, resulting in much faster response times in the reduced pitch.
Fig. 5
Fig. 5 Evaluation of optical crosstalk considering 3 x 3 pixels using three-dimensional simulator: (a) Top-view of transmittance profile when three pixels in second row are on state, and (b) Transmittance profile along horizontal direction in which it is defined well within 4 μm.

Equations (2)

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T T 0 = sin 2 ( 2ψ( V ) ) sin 2 ( πdΔn( V ) λ ),
τ d = γ π 2 ( K 2 d 2 + K 1 l 2 ) ,

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