Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration

doi:10.1364/OE.21.004244

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Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration

Victor Belyaev, Alexey Solomatin, and Denis Chausov »View Author Affiliations

Optics Express, Vol. 21, Issue 4, pp. 4244-4249 (2013)
http://dx.doi.org/10.1364/OE.21.004244

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Abstract

Phase retardation of both extraordinary and ordinary polarized rays passing through a liquid crystal (LC) cell with homogeneous and inhomogeneous LC director distribution is calculated as a function of the LC pretilt angle $θ_{0}$ on the cell substrates in the range $0 \leq θ_{0} \leq 90 °$ . The LC pretilt on both substrates can have the same or opposite direction, thereby forming homogeneous, splay, or bend director configurations. At the same pretilt angle value, the largest phase retardation $Δ Φ$ is observed in splay LC cells, whereas the smallest phase retardation is observed in bend cells. For the $θ_{0}$ values close to 0, 45°, and 90°, analytical approximations are derived, showing that phase retardation depends on LC birefringence variation.

1. Introduction

In many types of modern liquid crystal displays (LCDs), tilted alignment and/or sophisticated configuration of the LC director is used (MVA, OCB, etc.; [1

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A New MVA-LCD by Polymer Sustained Alignment Technology”, in SID’04 Digest, (2004), pp.1200–1203.

–3

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.

]). It can improve the LCD speed of response or viewing angle range [4

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

]. Therefore, investigation of optical properties of such cells, in particular, determination of the influence of the director distribution on the polarized ray propagation through a birefringent material (for example, an LC) is a challenge. This effect is described by the value of phase retardation

Δ Φ

between polarized extraordinary and ordinary waves passing through the LC cell. It is well studied for the cases of homogeneous planar or vertical alignment or TN-mode. However, there are no satisfactory data on LC cells with inhomogeneous LC director distribution. The goal is to calculate the

Δ Φ

values for LC cells with homogeneous and inhomogeneous LC director distribution as a function of the LC pretilt angle

θ_{0}

on the cell substrates.

2. Objects of research

Liquid crystal cells with homogeneous, splay, and bend director distribution are the object of research (Fig. 1 .). The pretilt angle

θ_{0}

varies in the range from 0 to 90°. The phase retardation difference between both extraordinary and ordinary rays passing through such a cell is determined by the expression

ΔΦ = 2π (n_{e}^{eff} - n_{o}) L/λ

(1)

where

L

is the cell thickness,

λ

is the wavelength,

n_{o}

is the refractive index for the ordinary ray, and

n_{e}^{eff}

is the effective refractive index for the extraordinary ray that may depend on the LC director distribution or the electric field strength.

Fig. 1 Schematic of the LC director distribution in LC cells with homogeneous (H), splay (S), and bend (B) configuration:

θ

is the deviation from the substrate plane, and

δ

is the deviation from the normal to the substrate.

The case of the normal incidence of light is considered. The single constant approximation (the Frank elastic constants for a nematic LC

K_{11} {=K}_{33}

) is used to simplify the calculations. In this case, the LC elastic energy is independent of the local tilt angle [5

S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992).

L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, 1996). p.149.

] and the tilt angle variation inside the cell is described by a linear function for every LC director configuration:

θ(z) = θ_{0} = const ​ ​ ​ ​ (H)

(2a)

θ(z) = - {2θ}_{0} z/L ​ ​ ​ ​ (S)

(2b)

θ(z) = \frac{π}{2} + (π - {2θ}_{0}) z/L ​ ​ ​ ​ (B)

(2c)

where

-0 .5 \leq z \leq + 0 .5

. If

θ_{0} \to π/2

, than the replacement

δ_{0} = π/2 - θ_{0}

is convenient for further consideration. Then,

δ(z) = - {2δ}_{0} z/L ​ ​ ​ ​ (B)

(3)

The

θ(z)

and

δ(z)

dependences for a fixed

θ_{0}

value are shown in Fig. 1(b) and 1(c).

3. Results for three main configurations

Dependences of the phase retardation difference

Δ Φ

on the pretilt angle

θ_{0}

are calculated for the LC cells in question. Phase retardation

Δ Φ

for the cells with an arbitrary LC director distribution is described by the expression [6

L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, 1996). p.149.

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

ΔΦ = \frac{2π}{λ} [\int_{0}^{L} \frac{n_{o} n_{e} dz}{{{(n}_{o}^{2} \cos^{2} θ(z) + n_{e}^{2} \sin^{2} θ(z))}^{1/2}} - n_{o} L]

(4)

Let us first consider the case of the homogeneous LC director distribution (Eq. (2a)). The

Δ Φ (θ_{0})

dependence is determined by the expression:

ΔΦ = \frac{2 π n_{o} L}{λ} (\frac{n_{e}}{{{(n}_{o}^{2} \cos^{2} θ_{0} + n_{e}^{2} \sin^{2} θ_{0})}^{1/2}} - 1)

(5)

Equation (5) yields the pretilt angle value for a LC with initial homogeneous configuration:

θ_{0} = arc \cos {\frac{n_{e}^{2}}{n_{e}^{2} - n_{o}^{2}} [1 - {(1 + \frac{ΔΦ}{{ΔΦ}_{max}} \frac{Δn}{n_{o}})}^{- 2}]}^{1 / 2}

(6)

The phase difference parameter

Φ = Δ Φ / Δ Φ_{\max}

is introduced (where

Δ Φ

is reduced to its maximum value

{ΔΦ}_{max} = 2π Δ nL/λ

, and

Δn = n_{e} - n_{o}

is the LC birefringence). The case of

Δ Φ_{\max}

is realized for the cells with the LC planar alignment (

θ_{0} = 0

). For the inhomogeneous director distribution, the dependences of the parameter

Φ

on the pretilt angle

θ_{0}

are calculated for different refractive indices.

Figure 2 shows the dependences of

Φ

θ_{0}

for three director configurations at different values of

n_{e}

and fixed

n_{o} = 1.5

. The

{Φ(θ}_{0})

dependences are compared for all three cell geometries considered in the

θ_{0}

range from 0 to

π/2

, typical for the LC cells in the absence of a voltage or with a slowly changing voltage. The

{Φ(θ}_{0})

value changes from 1 to 0 only in the case of the homogeneous director distribution. For the S-configuration, the

{Φ(θ}_{0})

value decreases down to 0.5-0.55 at

θ_{0} = π/2

. For the B-configuration

{Φ(θ}_{0})

decreases from 0.45 to 0.55 at

θ_{0} = 0

down to 0 at

θ_{0} = π/2

. The total change in the

{Φ(θ}_{0})

dependences for both S- and B-geometries is about 0.5 because of coexistence of the LC cell parts with planar, tilted, or vertical orientation. The

Φ

value at

θ_{0} = 0

for both S- and B-configurations depends on the LC birefringence.

Fig. 2 Dependences of the phase retardation parameter

Φ

on the pretilt angle value

θ_{0}

for three LC director configurations at different

n_{e}

and

n_{o} = 1.5

. For every configuration upper dashed line corresponds to

n_{e} = 1.6

, middle solid line

n_{e} = 1.7

, lower dot-dashed line

n_{e} = 1.8

n_{o}

and

θ_{0}

are fixed and

n_{e}

increases, then

Φ

reduces for all the three configurations (see Section 6). If the birefringence is the same at different

n_{o}

values and fixed angle

θ_{0}

, then

Φ

is almost independent on

n_{o}

in the

n_{o}

range typical for the liquid crystals (1.4-1.7). A simple analysis of Eq. (6) confirms this conclusion.

4. Analytical approximations for the cases of $θ \to 0$ or $θ \to π/2$

Some analytical approximations of the

{Φ(θ}_{0})

dependences are possible if

θ_{0} < < 1

δ_{0} < < 1

. The results of analytical expansions of both

{Φ(θ}_{0})

and

{Φ(δ}_{0})

dependences at fixed

n_{e} = 1.6

and

n_{o} = 1.5

are presented in Table 1 . Numerical estimations of these dependences are compared for the three cases considered in Fig. 3(a) and 3(b).

Table 1 Analytical approximations of

{Φ(θ}_{0})

and

{Φ(δ}_{0})

dependences at

θ_{0} < < 1

δ_{0} < < 1

	$θ_{0} < < 1$
H	$Φ (θ_{0}) = [n_{e} (1 - \frac{θ_{0}^{2}}{4} (\frac{{2n}_{e}^{2}}{n_{o}^{2}} - 1)) - n_{o}] / (n_{e} - n_{o})$	1-5.1 $θ_{0}^{2}$
S	$Φ (θ_{0}) = [n_{e} (1 - \frac{θ_{0}^{2}}{12} (\frac{{2n}_{e}^{2}}{n_{o}^{2}} - 1)) - n_{o}] / (n_{e} - n_{o})$	1-1.7 $θ_{0}^{2}$

$δ_{0} < < 1$
H	${Φ(δ}_{0}) = - \frac{δ_{0}^{2}}{4} (\frac{{2n}_{o}^{2}}{n_{e}^{2}} - 1) \frac{n_{o}}{n_{e} - n_{o}}$	2.8 $δ_{0}^{2}$
B	${Φ(δ}_{0}) = - \frac{δ_{0}^{2}}{12} (\frac{{2n}_{o}^{2}}{n_{e}^{2}} - 1) \frac{n_{o}}{n_{e} - n_{o}}$	0.95 $δ_{0}^{2}$

Fig. 3 Dependences of the phase retardation parameter

Φ

on the pretilt angle value

θ_{0}

n_{o} = 1.5

: (a) the case of a small pretilt angle (

θ_{0} < < 1

) for homogeneous (left) and splay (right) configurations, and (b) the case of a large pretilt angle (

δ_{0} < < 1

) for homogeneous (left) and bend (right) configurations.

{1-n}_{e} = 1 .6

{2-n}_{e} = 1 .7

{3-n}_{e} = 1 .8

Deviation of the

{Φ(θ}_{0})

dependence in the case of

θ \to 0

θ \to π/2

for splay or bend configurations, respectively, is 3 times lower in comparison with the case of the homogeneous director distribution. Therefore, the knowledge of the tilt angle distribution is necessary to avoid an error in experimental measurements of the pretilt angle.

5. Analytical approximation for the case $θ_{0} ~ π/4$

θ_{0} ~ π/4

, then the quasilinear part of the

{Φ(θ}_{0})

dependence for the cell with the homogeneous tilt distribution (Fig. 4 .) is described by the equation:

Φ (θ_{0}) = \frac{1}{n_{e} - n_{o}} [\frac{\sqrt{2} n_{e} n_{o}}{{{(n}_{e}^{2} + n_{o}^{2})}^{1/2}} (1 + \frac{π}{4} \frac{n_{e}^{2} - n_{o}^{2}}{n_{e}^{2} + n_{o}^{2}}) - n_{o} - θ_{0} \frac{\sqrt{2} n_{e} n_{o}}{{{(n}_{e}^{2} + n_{o}^{2})}^{1/2}} \frac{n_{e}^{2} - n_{o}^{2}}{n_{e}^{2} + n_{o}^{2}}]

(7)

When the refractive indices are equal to

n_{e} = 1.6

and

n_{o} = 1.5

, Eq. (7) is transformed to

{Φ(θ}_{0}) = 1 .253-0 {.997θ}_{0}

. The parameter

{Φ(θ}_{0})

is equal to 0.470 at a pretilt angle

θ_{0} = π/4

Fig. 4 Dependences of the phase retardation parameter

Φ

on the pretilt angle

θ_{0}

n_{o} = 1.5

for the homogeneous configuration in the case of small deviation of

θ_{0}

from

π/4~0 .785

{1-n}_{e} = 1 .6

{2-n}_{e} = 1 .7

{3-n}_{e} = 1 .8

Let us investigate the

{Φ(θ}_{0})

dependence for the case of fixed

θ_{0}

and varying n_e values in more details. The dependence of

Φ

on the ratio of the refractive indices can be derived from Eq. (7) at fixed

θ_{0}

. Let us introduce the parameter

x = n_{e} {/n}_{o}

. Then, Eq. (7) can be written in the form

Φ (θ_{0}) = \frac{1}{x - 1} [\frac{\sqrt{2} x}{{{(x}_{}^{2} + 1)}^{1/2}} (1 + \frac{π}{4} \frac{x_{}^{2} - 1}{x_{}^{2} + 1}) - 1 - θ_{0} \frac{\sqrt{2} x (x_{}^{2} - 1)}{{{(x}_{}^{2} + 1)}^{3/2}}]

(8)

Equation (8) confirms the fact that the tilt of the

{Φ(θ}_{0})

curves weakly depends on the birefringence

Δn = n_{e} - n_{o}

. Besides, the

{Φ(θ}_{0})

value decreases at fixed

θ_{0}

Δn

increases (Fig. 2-4). Let us suggest that the Δn value is small or

α < < 1

(

α = x - 1

). Then, Eq. (8) can be written in the form (intermediate stages of derivation are omitted):

Φ (θ_{0}) = \frac{1}{4} [(2 + π - {4θ}_{0})(1 - \frac{1}{2} α)]

(9)

If the birefringence parameter α is equal to 0 and the pretilt angle is

θ_{0} = π/4

, then the phase retardation parameter

{Φ(θ}_{0}) = 1/2

resembles the behavior of the dependence shown in Fig. 4. The

{Φ(θ}_{0},x)

dependence has no discrepancy at

Δn = 0

(

x = 1

), it is a continuous function if the birefringence changes its sign.

Therefore, all the formulas and data shown in Figs. 2-4 are valid for both calamitic (

Δn > 0

) and discotic (

Δn < 0

) liquid crystals.

6. Conclusions

The dependences of the phase retardation difference

Δ Φ

on the pretilt angle

θ_{0}

have been calculated for the LC cells with homogeneous, splay, or bend director configuration. When

θ_{0}

ranges from 0 to

π/2

, the

{Φ(θ}_{0})

value [

{Φ(θ}_{0}) = {ΔΦ(θ}_{0})/Δ Φ_{max}

] reduces from 1 to 0 for homogeneous LC cells, from 1 down to 0.5-0.55 for the S-configuration, and from 0.45 to 0.55 down to 0 for the B-configuration. The

{Φ(θ}_{0})

value decreases with increasing birefringence

Δn

at fixed

θ_{0}

The results can be used to develop the methods of the LC pretilt angle measurements in cells with sophisticated director configuration [8

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13(4), 349–354 (2005). [CrossRef]

–11

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

]. The known methods provide good accuracy for the cells with a homogeneous tilt inside the cell; however, they can give a wrong estimate of the pretilt angle if the director distribution is inhomogeneous. This is due to the fact that the same

{Φ(θ}_{0})

value can be obtained in an experiment in LC cells with different boundary conditions and director distribution [12

C. Dascalu, “Asymmetric electrooptic response in a nematic liquid crystal,” Rev. Mex. Fis. 47, 281–285 (2001).

–15

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, (2012), pp.1422–1425.

The calculation method developed can be used also select the LC director configuration for different purposes. It becomes possible to change the LC cell phase retardation that can be used when designing optical compensators. Besides, the method can be used for different LC cells with a given LC director distribution and symmetric or asymmetric boundary conditions.

Acknowledgments

The work is supported by the Russian Foundation for Basic Research (Grant Nos. 10-07-00385-a, 12-07-90006_Bel-a, 12-07-31172_mol_a, and 12-07-90801-mol_rf_nr) and by President Grants for Government Support of the Leading Scientific Schools (Grant No. VSh-1495.2012.8) and Young Russian Scientists of the Russian Federation (Grant No. MK-1969.2012.9).

References and links

1.	K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A New MVA-LCD by Polymer Sustained Alignment Technology”, in SID’04 Digest, (2004), pp.1200–1203.
2.	P. J. Bos, “Passive Optical Phase Retarders for Liquid Crystal Displays”, in 14th IDRC Proc., (1994), p.118.
3.	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.
4.	D. K. Yang and S. T. Wu, “Fundamentals of Liquid Crystal Devices” (Wiley, NY, 2006).
5.	S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992).
6.	L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, 1996). p.149.
7.	V. G. Chigrinov, “Orientation effects in nematic liquid crystals in electric and magnetic fields,” Sov. Phys. Crystallogr. 27, 245–264 (1982).
8.	A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp. 13(4), 349–354 (2005). [CrossRef]
9.	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 Stat. Nonlin. Soft Matter Phys. 71(6), 061707 (2005). [CrossRef] [PubMed]
10.	Ch. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001). [CrossRef] [PubMed]
11.	G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K₃₃/K₁₁ ratio in nematic liquid crystals,” Crystallogr. Rep. 27, 1019–1021 (1982).
12.	C. Dascalu, “Asymmetric electrooptic response in a nematic liquid crystal,” Rev. Mex. Fis. 47, 281–285 (2001).
13.	X. Nie, “Anchoring energy and pretilt angle effects on liquid crystal response time”, Ph.D. Thesis, University of Central Florida, 2007.
14.	V. V. Belyaev and V. G. Mazaeva, “Green technologies of LC alignment on the base of organosilicon compunds”, in SID’11 Digest (2011), pp.1412–1415.
15.	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, (2012), pp.1422–1425.

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(160.3710) Materials : Liquid crystals

ToC Category:
Optical Devices

History
Original Manuscript: November 22, 2012
Revised Manuscript: December 25, 2012
Manuscript Accepted: January 28, 2013
Published: February 12, 2013

Citation
Victor Belyaev, Alexey Solomatin, and Denis Chausov, "Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration," Opt. Express 21, 4244-4249 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4244

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References

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A New MVA-LCD by Polymer Sustained Alignment Technology”, in SID’04 Digest, (2004), pp.1200–1203.
P. J. Bos, “Passive Optical Phase Retarders for Liquid Crystal Displays”, in 14th IDRC Proc., (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.
D. K. Yang and S. T. Wu, “Fundamentals of Liquid Crystal Devices” (Wiley, NY, 2006).
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992).
L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, 1996). p.149.
V. G. Chigrinov, “Orientation effects in nematic liquid crystals in electric and magnetic fields,” Sov. Phys. Crystallogr.27, 245–264 (1982).
A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp.13(4), 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 Stat. Nonlin. Soft Matter Phys.71(6), 061707 (2005). [CrossRef] [PubMed]
Ch. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express9(13), 780–790 (2001). [CrossRef] [PubMed]
G. A. Beresnev, V. G. Chigrinov, and M. F. Grebenkin, “New method to determine K33/K11 ratio in nematic liquid crystals,” Crystallogr. Rep.27, 1019–1021 (1982).
C. Dascalu, “Asymmetric electrooptic response in a nematic liquid crystal,” Rev. Mex. Fis.47, 281–285 (2001).
X. Nie, “Anchoring energy and pretilt angle effects on liquid crystal response time”, Ph.D. Thesis, University of Central Florida, 2007.
V. V. Belyaev and V. G. Mazaeva, “Green technologies of LC alignment on the base of organosilicon compunds”, in SID’11 Digest (2011), pp.1412–1415.
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, (2012), pp.1422–1425.

Crystallogr. Rep.

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

J. Soc. Inf. Disp.

A. Muravsky, A. Murauski, V. Mazaeva, and V. Belyaev, “Parameters on the LC alignment of organosilicon compound films,” J. Soc. Inf. Disp.13(4), 349–354 (2005). [CrossRef]

Opt. Express

Ch. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express9(13), 780–790 (2001). [CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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 Stat. Nonlin. Soft Matter Phys.71(6), 061707 (2005). [CrossRef] [PubMed]

Rev. Mex. Fis.

C. Dascalu, “Asymmetric electrooptic response in a nematic liquid crystal,” Rev. Mex. Fis.47, 281–285 (2001).

Sov. Phys. Crystallogr.

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

Other

K. Hanaoka, Y. Nakanishi, Y. Inoue, S. Tanuma, and Y. Koike, “A New MVA-LCD by Polymer Sustained Alignment Technology”, in SID’04 Digest, (2004), pp.1200–1203.
P. J. Bos, “Passive Optical Phase Retarders for Liquid Crystal Displays”, in 14th IDRC Proc., (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.
D. K. Yang and S. T. Wu, “Fundamentals of Liquid Crystal Devices” (Wiley, NY, 2006).
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992).
L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, 1996). p.149.
X. Nie, “Anchoring energy and pretilt angle effects on liquid crystal response time”, Ph.D. Thesis, University of Central Florida, 2007.
V. V. Belyaev and V. G. Mazaeva, “Green technologies of LC alignment on the base of organosilicon compunds”, in SID’11 Digest (2011), pp.1412–1415.
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, (2012), pp.1422–1425.

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Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration

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Abstract

1. Introduction

2. Objects of research

3. Results for three main configurations

4. Analytical approximations for the cases of $θ \to 0$ or $θ \to π/2$

5. Analytical approximation for the case $θ_{0} ~ π/4$

6. Conclusions

Acknowledgments

References and links

References

Crystallogr. Rep.

J. Soc. Inf. Disp.

Opt. Express

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

Rev. Mex. Fis.

Sov. Phys. Crystallogr.

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Phase retardation vs. pretilt angle in liquid crystal cells with homogeneous and inhomogeneous LC director configuration

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Abstract

1. Introduction

2. Objects of research

3. Results for three main configurations

4. Analytical approximations for the cases of θ→0 or θ→π/2

5. Analytical approximation for the case θ0~π/4

6. Conclusions

Acknowledgments

References and links

References

Crystallogr. Rep.

J. Soc. Inf. Disp.

Opt. Express

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

Rev. Mex. Fis.

Sov. Phys. Crystallogr.

Other

Cited By

Figures

4. Analytical approximations for the cases of $θ \to 0$ or $θ \to π/2$

5. Analytical approximation for the case $θ_{0} ~ π/4$