Abstract

We control the pretilt angle of liquid crystals (LCs) by simultaneously doping silica nanoparticles (SNs) and reactive monomers into the LC cell. Application of AC high voltage (ACHV) to the cell compels the lifting force and the facilitation of polar groups to move the SNs and monomers toward the substrate surface. Polymer networks and SNs are stabilized at the substrate surface after UV exposure, sustaining the LCs at high pretilt angles. The deposited SNs on the substrate surface increases the anchoring energy of the substrate; the dispersed SNs in the cell decrease the relaxation constant of LCs. Therefore, the response time of the high-pretilted-polymer-stabilized LC cell is decreased. The method enables the control of the LC pretilt angle over a broad range. The slow response time of the polymer-stabilized LC cell from high monomer dose can also be prevented following the addition of SNs.

© 2014 Optical Society of America

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  1. Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
    [Crossref]
  2. F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
    [Crossref]
  3. W. Y. Wu, C. C. Wang, and A. Y. G. Fuh, “Controlling pre-tilt angles of liquid crystal using mixed polyimide alignment layer,” Opt. Express 16(21), 17131–17137 (2008).
    [Crossref] [PubMed]
  4. T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
    [Crossref]
  5. J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
    [Crossref]
  6. F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
    [Crossref]
  7. J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
    [Crossref]
  8. H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
    [Crossref]
  9. V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
    [Crossref]
  10. H. Ren, S. Xu, Y. Liu, and S. T. Wu, “Switchable focus using a polymeric lenticular microlens array and a polarization rotator,” Opt. Express 21(7), 7916–7925 (2013).
    [Crossref] [PubMed]
  11. O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
    [Crossref] [PubMed]
  12. C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
    [Crossref]
  13. S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
    [Crossref]
  14. C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
    [Crossref]
  15. W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
    [Crossref]
  16. X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
    [Crossref]
  17. L. Lu, V. Sergan, T. Van Heugten, D. Duston, A. Bhowmik, and P. J. Bos, “Surface localized polymer aligned liquid crystal lens,” Opt. Express 21(6), 7133–7138 (2013).
    [Crossref] [PubMed]
  18. L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
    [Crossref] [PubMed]
  19. X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
    [Crossref]

2013 (2)

2012 (2)

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
[Crossref] [PubMed]

2011 (1)

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

2010 (1)

V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
[Crossref]

2009 (2)

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

2008 (2)

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

W. Y. Wu, C. C. Wang, and A. Y. G. Fuh, “Controlling pre-tilt angles of liquid crystal using mixed polyimide alignment layer,” Opt. Express 16(21), 17131–17137 (2008).
[Crossref] [PubMed]

2007 (6)

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

2006 (1)

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

2004 (1)

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

1980 (1)

T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
[Crossref]

Ahn, H. J.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

Baik, H. K.

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Bhowmik, A.

Bos, P. J.

L. Lu, V. Sergan, T. Van Heugten, D. Duston, A. Bhowmik, and P. J. Bos, “Surface localized polymer aligned liquid crystal lens,” Opt. Express 21(6), 7133–7138 (2013).
[Crossref] [PubMed]

L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
[Crossref] [PubMed]

V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
[Crossref]

Cao, H.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Chen, J. H.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Choi, C. J.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Ding, X.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Duston, D.

Fuh, A. Y. G.

Gwag, J. S.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Ho, J. Y.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Horng, L.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Hsieh, C. T.

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Huang, C. Y.

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Hung, W.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Hwang, B. H.

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

Hwang, S. J.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Hyun, D. C.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

Jo, M. K.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Jo, S. I.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Jo, S. J.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Kang, D.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

Kang, S. G.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Kelly, J. R.

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

Kim, C. S.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Kim, J. B.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

Kim, J. H.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Kim, J. T.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

Kim, K. C.

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

Kim, Y. K.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Kim, Y. S.

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Kuo, C. C.

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

Kwok, H. S.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

Lai, C. C.

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Lavrentovich, O. D.

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

Lee, F. K.

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

Lee, Y. J.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Li, B.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Li, W.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Li, Y. W.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Liao, S. W.

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

Liu, Y.

Lo, K. Y.

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Lu, L.

L. Lu, V. Sergan, T. Van Heugten, D. Duston, A. Bhowmik, and P. J. Bos, “Surface localized polymer aligned liquid crystal lens,” Opt. Express 21(6), 7133–7138 (2013).
[Crossref] [PubMed]

L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
[Crossref] [PubMed]

Lu, R.

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

Nie, X.

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

Ohgawara, M.

T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
[Crossref]

Park, J. S.

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

Park, Y. R.

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

Pishnyak, O. P.

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

Ren, H.

Sergan, T. A.

V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
[Crossref]

Sergan, V.

L. Lu, V. Sergan, T. Van Heugten, D. Duston, A. Bhowmik, and P. J. Bos, “Surface localized polymer aligned liquid crystal lens,” Opt. Express 21(6), 7133–7138 (2013).
[Crossref] [PubMed]

L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
[Crossref] [PubMed]

Sergan, V. V.

V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
[Crossref]

Sheng, P.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

Shih, Y. T.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Shiyanovskii, S. V.

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

Song, H. C.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Tang, S.

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

Tien, C. J.

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Tseng, Y. H.

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Tsui, O. K.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Tsui, O. K. C.

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

Uchida, T.

T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
[Crossref]

Van Heugten, T.

Wada, M.

T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
[Crossref]

Wang, C. C.

Wang, Y. W.

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Wu, S.

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

Wu, S. T.

H. Ren, S. Xu, Y. Liu, and S. T. Wu, “Switchable focus using a polymeric lenticular microlens array and a polarization rotator,” Opt. Express 21(7), 7916–7925 (2013).
[Crossref] [PubMed]

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

Wu, T. X.

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

Wu, W. Y.

Xianyu, H.

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

Xie, F. C.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Xu, S.

Yang, H.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Yang, Y. T.

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Yang, Z.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Yeung, F. S.

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Zhang, B.

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

Zhu, M.

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

Appl. Phys. Lett. (8)

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, J. T. Kim, S. J. Jo, C. S. Kim, H. K. Baik, C. J. Choi, M. K. Jo, Y. S. Kim, J. S. Park, and D. Kang, “No bias pi cell using a dual alignment layer with an intermediate pretilt angle,” Appl. Phys. Lett. 91(2), 023507 (2007).
[Crossref]

F. K. Lee, B. Zhang, P. Sheng, H. S. Kwok, and O. K. C. Tsui, “Continuous liquid crystal pretilt control through textured substrates,” Appl. Phys. Lett. 85(23), 5556–5558 (2004).
[Crossref]

J. B. Kim, K. C. Kim, H. J. Ahn, B. H. Hwang, D. C. Hyun, and H. K. Baik, “Variable liquid crystal pretilt angles on various compositions of alignment layers,” Appl. Phys. Lett. 90(4), 043515 (2007).
[Crossref]

H. J. Ahn, J. B. Kim, K. C. Kim, B. H. Hwang, J. T. Kim, H. K. Baik, J. S. Park, and D. Kang, “Liquid crystal pretilt angle control using adjustable wetting properties of alignment layers,” Appl. Phys. Lett. 90(25), 253505 (2007).
[Crossref]

S. W. Liao, C. T. Hsieh, C. C. Kuo, and C. Y. Huang, “Voltage-assisted ion reduction in liquid crystal-silica nanoparticle dispersions,” Appl. Phys. Lett. 101(16), 161906 (2012).
[Crossref]

C. Y. Huang, C. C. Lai, Y. H. Tseng, Y. T. Yang, C. J. Tien, and K. Y. Lo, “Silica-nanoparticle-doped nematic display with multistable and dynamic modes,” Appl. Phys. Lett. 92(22), 221908 (2008).
[Crossref]

Y. J. Lee, J. S. Gwag, Y. K. Kim, S. I. Jo, S. G. Kang, Y. R. Park, and J. H. Kim, “Control of liquid crystal pretilt angle by anchoring competition of the stacked alignment layers,” Appl. Phys. Lett. 94(4), 041113 (2009).
[Crossref]

F. S. Yeung, J. Y. Ho, Y. W. Li, F. C. Xie, O. K. Tsui, P. Sheng, and H. S. Kwok, “Variable liquid crystal pretilt angles by nanostructured surfaces,” Appl. Phys. Lett. 88(5), 051910 (2006).
[Crossref]

Chem. Phys. Lett. (1)

V. V. Sergan, T. A. Sergan, and P. J. Bos, “Control of the molecular pretilt angle in liquid crystal devices by using a low-density localized polymer network,” Chem. Phys. Lett. 486(4-6), 123–125 (2010).
[Crossref]

J. Appl. Phys. (1)

X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101(10), 103110 (2007).
[Crossref]

J. Appl. Polym. Sci. (1)

W. Li, M. Zhu, X. Ding, B. Li, W. Hung, H. Cao, Z. Yang, and H. Yang, “Studies on electro-optical properties of polymer matrix/LC/SiO2 nanoparticles composites,” J. Appl. Polym. Sci. 111(3), 1449–1453 (2009).
[Crossref]

J. Disp. Tech. (1)

X. Nie, H. Xianyu, R. Lu, T. X. Wu, and S. T. Wu, “Pretilt angle effects on liquid crystal response time,” J. Disp. Tech. 3(3), 280–283 (2007).
[Crossref]

Jpn. J. Appl. Phys. (2)

T. Uchida, M. Ohgawara, and M. Wada, “SiO2 liquid crystal orientation on the surface of obliquely-evaporated silicon monoxide with homeotropic surface treatment,” Jpn. J. Appl. Phys. 19(11), 2127–2136 (1980).
[Crossref]

C. Y. Huang, J. H. Chen, C. T. Hsieh, H. C. Song, Y. W. Wang, L. Horng, Y. T. Shih, and S. J. Hwang, “Effect of the polyimide concentration on the memory stability of the silica-nanoparticle-doped hybrid aligned nematic Cell,” Jpn. J. Appl. Phys. 50(2R), 021702 (2011).
[Crossref]

Opt. Express (3)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. Lu, V. Sergan, and P. J. Bos, “Mechanism of electric-field-induced segregation of additives in a liquid-crystal host,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5), 051706 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii, and O. D. Lavrentovich, “Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal,” Phys. Rev. Lett. 99(12), 127802 (2007).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Substrate surface image of the pristine SN-doped LC cell after ACHV treatment. (b) Pretilt angles and fall times of the cells at various SN concentrations before and after ACHV treatment. The arrows in (b) indicate the reference axes of the obtained data.

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Fig. 2
Fig. 2 (a) Substrate surface image of the pristine RM-doped LC cell after ACHV–UV treatment. (b) Pretilt angles and fall times of the cells at various RM concentrations before and after ACHV–UV treatment. The arrows in (b) indicate the reference axes of the obtained data.

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Fig. 3
Fig. 3 Schematic of ACHV–UV-treated SN–RM-doped LC cell.

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Fig. 4
Fig. 4 (a) Substrate surface images of the SN–RM-doped LC cells after ACHV–UV treatment. (b) Pretilt angles of the SN–RM-doped LC cells at various SN concentrations after ACHV–UV treatment.

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Tables (2)

Tables Icon

Table 1 Measured Pretilt Angles of the LC Cells after ACHV–UV Treatment at Various SN and RM Concentrations

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Table 2 Measured Fall Times of the LC Cells after ACHV–UV Treatment at Various SN and RM Concentrations

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Equations (2)

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

τ d = γ 1 β 2 K ,
β = 2 d cos − 1 ( θ p θ m ) ,

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