Abstract

Optical and electro-optical methods of liquid crystal (LC) director pretilt angle measurement are described for LC cells with homogeneous and inhomogeneous LC director distribution. The LC pretilt on both LC substrates can have the same or opposite direction. The phase retardation difference of both extraordinary and ordinary polarized rays passing through an LC cell with homogeneous and inhomogeneous LC director distribution has been calculated versus the LC pretilt angle θ0 on the cell’s substrates in the range 0θ090°. The experimental procedure for phase retardation difference determination by measurement of the LC cell transmission between crossed polarizers for cells with LC tilted alignment is described. The method developed can also be used in optical compensator design.

© 2013 Optical Society of America

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2013 (1)

2011 (1)

2010 (1)

2009 (2)

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

2005 (3)

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

2004 (1)

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

2002 (1)

M. Warenghem and S. Peralta, “Nematic film under electric field: total internal reflection, surface tilt angle and anchoring energy,” Mol. Cryst. Liq. Cryst. 375, 553–562 (2002).
[CrossRef]

2001 (2)

1999 (1)

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

1996 (1)

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

1994 (1)

K. Y. Han, T. Miyashita, and T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

1993 (1)

1988 (1)

T. Opara, J. W. Baran, and J. Zmija, “Interferential method for determining the inclination angle of molecules in plane-parallel liquid crystalline layers,” Cryst. Res. Technol. 23, 1073–1083 (1988).
[CrossRef]

1985 (1)

V. V. Belyaev, S. A. Ivanov, and M. F. Grebenkin, “Temperature dependence of rotational viscosity of nematic liquid crystals,” Sov. Phys. Crystallogr. 30, 674–682 (1985).

1984 (1)

V. V. Belyaev and M. F. Grebenkin, “Rotational viscosity of mixtures of nematic liquid crystals,” Sov. Phys. Crystallogr. 29, 483–484 (1984).

1983 (1)

M. F. Grebyonkin, G. A. Beresnev, and V. V. Belyaev, “Visco-elastic properties of liquid crystalline mixtures,” Mol. Cryst. Liq. Cryst. 103, 1–18 (1983).
[CrossRef]

1982 (2)

V. G. Chigrinov, “Orientation effects in nematic liquid crystals in electric and magnetic fields,” Sov. Phys. Crystallogr. 27, 245–264 (1982).

G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K33/K11 ratio in nematic liquid crystals,” Sov. Phys. Crystallogr. 27, 1019–1021 (1982).

1980 (1)

H. P. Hinov and S. Sainov, “Total internal reflection from nematic liquid crystals,” Rev. Phys. Appl. 15, 1307–1321 (1980).
[CrossRef]

1978 (1)

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

Baran, J. W.

T. Opara, J. W. Baran, and J. Zmija, “Interferential method for determining the inclination angle of molecules in plane-parallel liquid crystalline layers,” Cryst. Res. Technol. 23, 1073–1083 (1988).
[CrossRef]

Barnik, M. I.

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Belyaev, S. V.

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

Belyaev, V.

V. Belyaev, A. Solomatin, and D. Chausov, “Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration,” Opt. Express 21, 4244–4249 (2013).
[CrossRef]

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

Belyaev, V. V.

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

V. V. Belyaev and V. G. Chigrinov, “Figure of merit of liquid-crystalline materials for optically addressed spatial modulators,” Appl. Opt. 32, 141–146 (1993).
[CrossRef]

V. V. Belyaev, S. A. Ivanov, and M. F. Grebenkin, “Temperature dependence of rotational viscosity of nematic liquid crystals,” Sov. Phys. Crystallogr. 30, 674–682 (1985).

V. V. Belyaev and M. F. Grebenkin, “Rotational viscosity of mixtures of nematic liquid crystals,” Sov. Phys. Crystallogr. 29, 483–484 (1984).

M. F. Grebyonkin, G. A. Beresnev, and V. V. Belyaev, “Visco-elastic properties of liquid crystalline mixtures,” Mol. Cryst. Liq. Cryst. 103, 1–18 (1983).
[CrossRef]

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

V. V. Belyaev and V. G. Mazaeva, “Green technologies of LC alignment on the base of organosilicon compunds,” in SID’11 Digest (Society for Information Display, 2011), 1412–1415.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, and A. A. Gorbunov, “Measurement of the LC pretilt angle and polar anchoring in cells with homogeneous and inhomogeneous LC director configuration and weak anchoring on organosilicon aligning films,” in SID’12 Digest (Society for Information Display, 2012), pp. 1422–1425.

V. V. Belyaev, V. G. Mazaeva, S. N. Timofeyev, and A. A. Min’ko, “Liquid crystal alignment by linear and cyclic organosiloxanes with molecular microrelief,” in Proceedings of 28th International Display Res. Conf. EuroDisplay’09, Rome, Italy, 14–17 September (2009), pp. 139–142.

Beresnev, G. A.

M. F. Grebyonkin, G. A. Beresnev, and V. V. Belyaev, “Visco-elastic properties of liquid crystalline mixtures,” Mol. Cryst. Liq. Cryst. 103, 1–18 (1983).
[CrossRef]

G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K33/K11 ratio in nematic liquid crystals,” Sov. Phys. Crystallogr. 27, 1019–1021 (1982).

Blinov, L. M.

L. M. Blinov, Structure and Properties of Liquid Crystals (Springer, 2011).

Bobylev, Y. P.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

Bos, P. J.

P. J. Bos, “Passive optical phase retarders for liquid crystal displays,” in Proceedings of 14th International Display Research Conference (IDRC, 1994), p. 118.

X.-D. Mi, M. Xu, D.-K. Yang, and P. J. Bos, “Effects of pretilt angle on electro-optical properties of Pi-cell LCDs,” in SID’99 Digest (1999), pp. 24–27.

Cerrolaza, B.

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystals (Cambridge University, 1977).

Chausov, D.

Chausov, D. N.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, and A. A. Gorbunov, “Measurement of the LC pretilt angle and polar anchoring in cells with homogeneous and inhomogeneous LC director configuration and weak anchoring on organosilicon aligning films,” in SID’12 Digest (Society for Information Display, 2012), pp. 1422–1425.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

Chernova, N. I.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

Chigrinov, V.

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Chigrinov, V. G.

V. V. Belyaev and V. G. Chigrinov, “Figure of merit of liquid-crystalline materials for optically addressed spatial modulators,” Appl. Opt. 32, 141–146 (1993).
[CrossRef]

G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K33/K11 ratio in nematic liquid crystals,” Sov. Phys. Crystallogr. 27, 1019–1021 (1982).

V. G. Chigrinov, “Orientation effects in nematic liquid crystals in electric and magnetic fields,” Sov. Phys. Crystallogr. 27, 245–264 (1982).

Cognard, J.

J. Cognard, Alignment of Nematic Liquid Crystals and Their Mixtures, Molecular Crystals and Liquid Crystals (Gordon and Breach, 1982).

Dascalu, C.

C. Dascalu, “Asymmetric electrooptic response in a nematic liquid crystal,” Revista Mexicana de Fisica. 47, 281–285 (2001).

Dobrovolskyy, A.

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

Fercher, A.

Geday, M. A.

Germroth, T. C.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Goetzinger, E.

Gorbunov, A. A.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, and A. A. Gorbunov, “Measurement of the LC pretilt angle and polar anchoring in cells with homogeneous and inhomogeneous LC director configuration and weak anchoring on organosilicon aligning films,” in SID’12 Digest (Society for Information Display, 2012), pp. 1422–1425.

Grebenkin, M. F.

V. V. Belyaev, S. A. Ivanov, and M. F. Grebenkin, “Temperature dependence of rotational viscosity of nematic liquid crystals,” Sov. Phys. Crystallogr. 30, 674–682 (1985).

V. V. Belyaev and M. F. Grebenkin, “Rotational viscosity of mixtures of nematic liquid crystals,” Sov. Phys. Crystallogr. 29, 483–484 (1984).

G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K33/K11 ratio in nematic liquid crystals,” Sov. Phys. Crystallogr. 27, 1019–1021 (1982).

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

M. F. Grebenkin and A. V. Ivashchenko, Liquid Crystalline Materials (Khimiya, 1983) (in Russian).

Grebyonkin, M. F.

M. F. Grebyonkin, G. A. Beresnev, and V. V. Belyaev, “Visco-elastic properties of liquid crystalline mixtures,” Mol. Cryst. Liq. Cryst. 103, 1–18 (1983).
[CrossRef]

Grigos, V. I.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Wiley Series in Pure and Applied Optics (Wiley, 2009).

Han, K. Y.

K. Y. Han, T. Miyashita, and T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

Hanaoka, K.

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A new MVA-LCD by polymer sustained alignment technology,” in SID’04 Digest (Society for Information Display, 2004), pp. 1200–1203.

Harris, F. W.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Hinov, H. P.

H. P. Hinov and S. Sainov, “Total internal reflection from nematic liquid crystals,” Rev. Phys. Appl. 15, 1307–1321 (1980).
[CrossRef]

Hitzenberger, C.

Ho, J.

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Inoue, Y.

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A new MVA-LCD by polymer sustained alignment technology,” in SID’04 Digest (Society for Information Display, 2004), pp. 1200–1203.

Ivanov, S. A.

V. V. Belyaev, S. A. Ivanov, and M. F. Grebenkin, “Temperature dependence of rotational viscosity of nematic liquid crystals,” Sov. Phys. Crystallogr. 30, 674–682 (1985).

Ivaschenko, A. V.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

Ivashchenko, A. V.

M. F. Grebenkin and A. V. Ivashchenko, Liquid Crystalline Materials (Khimiya, 1983) (in Russian).

Jing, J. A.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Kalashnikov, A. Y.

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

King, B. M.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Koike, Y.

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A new MVA-LCD by polymer sustained alignment technology,” in SID’04 Digest (Society for Information Display, 2004), pp. 1200–1203.

Kovalenko, V. I.

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

Kovshev, E. I.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Kozunov, V. A.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

Kravchuk, R.

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

Kuo, C.-L.

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

Kuo, T.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Kwok, H. S.

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Kwok, H.-S.

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

Lavrentovich, O. D.

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

Li, H.

Loseva, M. V.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Mazaeva, V.

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

Mazaeva, V. G.

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

V. V. Belyaev and V. G. Mazaeva, “Green technologies of LC alignment on the base of organosilicon compunds,” in SID’11 Digest (Society for Information Display, 2011), 1412–1415.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

V. V. Belyaev, V. G. Mazaeva, S. N. Timofeyev, and A. A. Min’ko, “Liquid crystal alignment by linear and cyclic organosiloxanes with molecular microrelief,” in Proceedings of 28th International Display Res. Conf. EuroDisplay’09, Rome, Italy, 14–17 September (2009), pp. 139–142.

Mi, X.-D.

X.-D. Mi, M. Xu, D.-K. Yang, and P. J. Bos, “Effects of pretilt angle on electro-optical properties of Pi-cell LCDs,” in SID’99 Digest (1999), pp. 24–27.

Min’ko, A. A.

V. V. Belyaev, V. G. Mazaeva, S. N. Timofeyev, and A. A. Min’ko, “Liquid crystal alignment by linear and cyclic organosiloxanes with molecular microrelief,” in Proceedings of 28th International Display Res. Conf. EuroDisplay’09, Rome, Italy, 14–17 September (2009), pp. 139–142.

Miyashita, T.

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

K. Y. Han, T. Miyashita, and T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

Molochko, V. A.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Murauski, A.

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Murausky, A. A.

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

Muravskii, A. A.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

Muravsky, A.

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Muravsky, A. A.

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

Nakanishi, Y.

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A new MVA-LCD by polymer sustained alignment technology,” in SID’04 Digest (Society for Information Display, 2004), pp. 1200–1203.

Natsyuk, S. N.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

Nie, X.

X. Nie, “Anchoring energy and pretilt angle effects on liquid crystal response time,” Ph.D. thesis (University of Central Florida, 2007).

Opara, T.

T. Opara, J. W. Baran, and J. Zmija, “Interferential method for determining the inclination angle of molecules in plane-parallel liquid crystalline layers,” Cryst. Res. Technol. 23, 1073–1083 (1988).
[CrossRef]

Oton, J. M.

Peralta, S.

M. Warenghem and S. Peralta, “Nematic film under electric field: total internal reflection, surface tilt angle and anchoring energy,” Mol. Cryst. Liq. Cryst. 375, 553–562 (2002).
[CrossRef]

Petrov, V. F.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, V. I. Grigos, and V. A. Kozunov, USSR Patent No. 1295732, 28January1992.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Pircher, M.

Qiu, L.

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

Quintana, X.

Sainov, S.

H. P. Hinov and S. Sainov, “Total internal reflection from nematic liquid crystals,” Rev. Phys. Appl. 15, 1307–1321 (1980).
[CrossRef]

Sevostyanov, V. S.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Shoshin, V. M.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

Shtykov, N. M.

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

Sobolevsky, M. V.

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

Solomatin, A.

Solomatin, A. S.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, V. M. Shoshin, Y. P. Bobylev, A. A. Muravskii, A. A. Muravskii, V. G. Mazaeva, S. N. Natsyuk, and A. A. Gorbunov, “Different polarity NLC alignment on organosilicon films of varying molecular structure,” in Proceedings of Thirty-first IDRC Eurodisplay 2011, Arcachon, France, 19–22 September 2011 (Society for Information Display, 2011), pp. 17–19.

V. V. Belyaev, A. S. Solomatin, D. N. Chausov, and A. A. Gorbunov, “Measurement of the LC pretilt angle and polar anchoring in cells with homogeneous and inhomogeneous LC director configuration and weak anchoring on organosilicon aligning films,” in SID’12 Digest (Society for Information Display, 2012), pp. 1422–1425.

Song, F.

Sticker, M.

Suzuki, M.

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

Tanuma, S.

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A new MVA-LCD by polymer sustained alignment technology,” in SID’04 Digest (Society for Information Display, 2004), pp. 1200–1203.

Timofeyev, S. N.

V. V. Belyaev, V. G. Mazaeva, S. N. Timofeyev, and A. A. Min’ko, “Liquid crystal alignment by linear and cyclic organosiloxanes with molecular microrelief,” in Proceedings of 28th International Display Res. Conf. EuroDisplay’09, Rome, Italy, 14–17 September (2009), pp. 139–142.

Tsvetkov, V. A.

M. I. Barnik, S. V. Belyaev, M. F. Grebenkin, V. A. Tsvetkov, and N. M. Shtykov, “Electrical, optical, and viscoelastic properties of a liquid-crystal mixture of azoxy compounds,” Sov. Phys. Crystallogr. 23, 460–464 (1978).

Uchida, T.

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

K. Y. Han, T. Miyashita, and T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

Warenghem, M.

M. Warenghem and S. Peralta, “Nematic film under electric field: total internal reflection, surface tilt angle and anchoring energy,” Mol. Cryst. Liq. Cryst. 375, 553–562 (2002).
[CrossRef]

Wu, S. T.

D. K. Yang, and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Xu, M.

X.-D. Mi, M. Xu, D.-K. Yang, and P. J. Bos, “Effects of pretilt angle on electro-optical properties of Pi-cell LCDs,” in SID’99 Digest (1999), pp. 24–27.

Yang, D. K.

D. K. Yang, and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Yang, D.-K.

X.-D. Mi, M. Xu, D.-K. Yang, and P. J. Bos, “Effects of pretilt angle on electro-optical properties of Pi-cell LCDs,” in SID’99 Digest (1999), pp. 24–27.

Yang, X.

Yaroshchuk, O.

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Wiley Series in Pure and Applied Optics (Wiley, 2009).

Yeung, F. S. Y.

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Yudina, G. I.

V. F. Petrov, M. F. Grebenkin, M. I. Barnik, V. V. Belyaev, E. I. Kovshev, A. V. Ivaschenko, M. V. Loseva, N. I. Chernova, G. I. Yudina, V. S. Sevostyanov, and V. A. Molochko, USSR Patent No. 1302683, 28January1992.

Zhang, D.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Zhang, Y.

Zheng, X. J.

F. W. Harris, D. Zhang, X. J. Zheng, T. C. Germroth, T. Kuo, J. A. Jing, and B. M. King, “Optical compensation films with disk groups for liquid crystal display,” U.S. patent 8,377,558, (19February2013).

Zmija, J.

T. Opara, J. W. Baran, and J. Zmija, “Interferential method for determining the inclination angle of molecules in plane-parallel liquid crystalline layers,” Cryst. Res. Technol. 23, 1073–1083 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C.-L. Kuo, T. Miyashita, M. Suzuki, and T. Uchida, “Crucial influences of K33/K11 ratio on viewing angle of display mode using a bend-alignment liquid-crystal cell with a compensator,” Appl. Phys. Lett. 68, 1461–1463 (1996).
[CrossRef]

Cryst. Res. Technol. (1)

T. Opara, J. W. Baran, and J. Zmija, “Interferential method for determining the inclination angle of molecules in plane-parallel liquid crystalline layers,” Cryst. Res. Technol. 23, 1073–1083 (1988).
[CrossRef]

J. Display Technol. (1)

J. Soc. Inf. Disp. (2)

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13, 349–354 (2005).
[CrossRef]

A. A. Muravsky, A. A. Murausky, V. G. Mazaeva, and V. V. Belyaev, “Parameters of LC alignment on films of organosilicon compounds,” J. Soc. Inf. Disp. 13, 349–356 (2005).
[CrossRef]

Liq. Cryst. (3)

O. Yaroshchuk, R. Kravchuk, A. Dobrovolskyy, L. Qiu, and O. D. Lavrentovich, “Planar and tilted uniform alignment of liquid crystals by plasma treated substrates,” Liq. Cryst. 31, 859–869 (2004).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

A. Murauski, V. Chigrinov, and H.-S. Kwok, “New method for measuring polar anchoring energy of nematic liquid crystals,” Liq. Cryst. 36, 779–786 (2009).
[CrossRef]

Mol. Cryst. Liq. Cryst. (4)

K. Y. Han, T. Miyashita, and T. Uchida, “Accurate measurement of pretilt angle in the liquid crystal cell by an improved crystal rotation method,” Mol. Cryst. Liq. Cryst. 241, 147–157 (1994).
[CrossRef]

M. Warenghem and S. Peralta, “Nematic film under electric field: total internal reflection, surface tilt angle and anchoring energy,” Mol. Cryst. Liq. Cryst. 375, 553–562 (2002).
[CrossRef]

M. F. Grebyonkin, G. A. Beresnev, and V. V. Belyaev, “Visco-elastic properties of liquid crystalline mixtures,” Mol. Cryst. Liq. Cryst. 103, 1–18 (1983).
[CrossRef]

M. V. Sobolevsky, V. G. Mazaeva, V. I. Kovalenko, V. V. Belyaev, and A. Y. Kalashnikov, “Manufacture and physical properties of silicon-organic films for LC alignment,” Mol. Cryst. Liq. Cryst. 329, 293–304 (1999).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (1)

A. Murauski, V. Chigrinov, A. Muravsky, F. S. Y. Yeung, J. Ho, and H. S. Kwok, “Determination of liquid-crystal polar anchoring energy by electrical measurements,” Phys. Rev. E 71, 061707 (2005).
[CrossRef]

Rev. Phys. Appl. (1)

H. P. Hinov and S. Sainov, “Total internal reflection from nematic liquid crystals,” Rev. Phys. Appl. 15, 1307–1321 (1980).
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Figures (7)

Fig. 1.
Fig. 1.

Schematic LC director distribution in LC cells with homogeneous (H), splay (S), and bend (B) configurations. θ , deviation from the substrate’s plane; δ , deviation from the normal to the substrate (dashed line).

Fig. 2.
Fig. 2.

Tilt angle θ ( z ) distribution in LC cells with splay (S, left) and bend (B, right) configurations.

Fig. 3.
Fig. 3.

Φ ( θ 0 ) dependences for homogeneous, splay, and bend configurations at n o = 1.5 and varying n e values.

Fig. 4.
Fig. 4.

Φ ( θ 0 ) dependences for homogeneous configuration at n o = 1.5 and varying n e values in the case where θ 0 π / 4 .

Fig. 5.
Fig. 5.

Transmission-voltage dependence for a ZhK-1289 ( Δ ε > 0 ) LC cell with polyimide PAK-14 aligning film. L = 23 μm , λ = 0.635 μm , and T = 25 ° C . θ 0 < 0.1 ° . The voltage axis is shown in the logarithmic scale.

Fig. 6.
Fig. 6.

Transmission-voltage dependence for a ZhK-807 ( Δ ε > 0 ) LC cell with an OC aligning film. L = 23 μm , λ = 0.635 μm , and T = 25 ° C . θ 0 = 45 ° . The voltage axis is shown in the logarithmic scale.

Fig. 7.
Fig. 7.

Transmission-voltage dependence for a ZhK-440 ( Δ ε < 0 ) LC cell with olygomethylhydridsiloxane aligning film. L = 23 μm , λ = 0.635 μm , and T = 25 ° C . θ 0 = 9 ° . The voltage axis is shown in the logarithmic scale.

Tables (2)

Tables Icon

Table 1. Analytical Approximations of Both Φ ( θ 0 ) and Φ ( δ 0 ) Dependences for H , S and B Configurations in the Case of θ 0 1 or δ 0 1 . n e = 1.6 and n o = 1.5

Tables Icon

Table 2. Formulas for the Calculation of the Total Phase Retardation Difference Change Δ ϕ / π at the Bias Voltage U Change From U to the Zero Valuea

Equations (20)

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Δ Φ = 2 π ( n e eff n o ) L / λ ,
θ ( z ) = θ 0 = const ( H ) ,
θ ( z ) = 2 θ 0 z / L ( S ) ,
θ ( z ) = π 2 + ( π 2 θ 0 ) z / L ( B ) ,
δ ( z ) = 2 δ 0 z / L ( B ) .
Δ Φ = 2 π λ [ L / 2 + L / 2 n o n e d z ( n o 2 cos 2 θ ( z ) + n e 2 sin 2 θ ( z ) ) 1 / 2 n o L ] .
Δ Φ = 2 π n o L λ ( n e ( n o 2 cos 2 θ 0 + n e 2 sin 2 θ 0 ) 1 / 2 1 ) .
θ 0 = arccos { n e 2 n e 2 n o 2 [ 1 ( 1 + Δ Φ Δ Φ max Δ n n o ) 2 ] } 1 / 2 .
Φ ( θ 0 ) = 1 n e n o [ 2 n e n o ( n e 2 + n o 2 ) 1 / 2 ( 1 + π 4 n e 2 n o 2 n e 2 + n o 2 ) n o θ 0 2 n e n o ( n e 2 + n o 2 ) 1 / 2 n e 2 n o 2 n e 2 + n o 2 ] .
Φ ( θ 0 ) = 1 x 1 [ 2 x ( x 2 + 1 ) 1 / 2 ( 1 + π 4 x 2 1 x 2 + 1 ) 1 θ 0 2 x ( x 2 1 ) ( x 2 + 1 ) 3 / 2 ] .
Φ ( θ 0 ) = 1 4 [ ( 2 + π 4 θ 0 ) ( 1 1 2 α ) ] .
I ( U ) = I 0 sin 2 ( Δ Φ ( U ) / 2 ) ,
Δ Φ ( U = 0 ) = N π + 2 arcsin ( I / I 0 ) 1 / 2 .
Δ Φ ( U = 0 ) = ( N + 1 ) π 2 arcsin ( I / I 0 ) 1 / 2 .
N = Entier [ Δ Φ / ( U ) / π ] = 2 Δ n L / λ 2 π 1 arcsin ( I ( U ) / I 0 ) 1 / 2
N = Entier [ Δ Φ ( U ) / π ] = 2 Δ n L / λ 1 + 2 π 1 arcsin ( I ( U ) / I 0 ) 1 / 2
Δ I ( U ) / I 0 = ( I 2 ( U ) / I 0 I 1 ( U ) / I 0 ) = sin 2 [ ( Δ Φ + δ ( Δ Φ ) ) / 2 ] sin 2 ( Δ Φ ) / 2 ] .
Δ I ( U ) / I 0 = sin [ δ ( Δ Φ ) / 2 ] { sin Δ Φ + sin [ δ ( Δ Φ ) / 2 ] cos 2 ( Δ Φ ) / 2 ] } .
Δ I ( U ) / I 0 = sin [ δ ( Δ Φ ) / 2 ] × sin Δ Φ .
Δ I ( U ) / I 0 = sin 2 [ δ ( Δ Φ ) / 2 ] × cos 2 ( Δ Φ / 2 ) sin 2 [ δ ( Δ Φ ) / 2 ] .

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