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 θ0 on the cell substrates in the range 0θ090°. 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 ΔΦ is observed in splay LC cells, whereas the smallest phase retardation is observed in bend cells. For the θ0 values close to 0, 45°, and 90°, analytical approximations are derived, showing that phase retardation depends on LC birefringence variation.

©2013 Optical Society of America

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.; [13]). It can improve the LCD speed of response or viewing angle range [4]. 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 ΔΦ 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 ΔΦ values for LC cells with homogeneous and inhomogeneous LC director distribution as a function of the LC pretilt angle θ0 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 θ0 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=(neeffno)L/λ
where L is the cell thickness, λ is the wavelength, no is the refractive index for the ordinary ray, and neeff 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.

Download Full Size | PPT Slide | PDF

The case of the normal incidence of light is considered. The single constant approximation (the Frank elastic constants for a nematic LC K11=K33) is used to simplify the calculations. In this case, the LC elastic energy is independent of the local tilt angle [5,6] and the tilt angle variation inside the cell is described by a linear function for every LC director configuration:

θ(z)=θ0=const(H)
θ(z)=0z/L(S)
θ(z)=π2+(π0)z/L(B)
where -0.5z+0.5. If θ0π/2, than the replacement δ0=π/2θ0 is convenient for further consideration. Then,
δ(z)=0z/L(B)
The θ(z) and δ(z) dependences for a fixed θ0 value are shown in Fig. 1(b) and 1(c).

3. Results for three main configurations

Dependences of the phase retardation difference ΔΦ on the pretilt angle θ0 are calculated for the LC cells in question. Phase retardation ΔΦ for the cells with an arbitrary LC director distribution is described by the expression [6,7].

ΔΦ=λ[0Lnonedz(no2cos2θ(z)+ne2sin2θ(z))1/2noL]
Let us first consider the case of the homogeneous LC director distribution (Eq. (2a)). The ΔΦ(θ0) dependence is determined by the expression:
ΔΦ=2πnoLλ(ne(no2cos2θ0+ne2sin2θ0)1/21)
Equation (5) yields the pretilt angle value for a LC with initial homogeneous configuration:
θ0=arccos{ne2ne2no2[1(1+ΔΦΔΦmaxΔnno)2]}1/2
The phase difference parameter Φ=ΔΦ/ΔΦmax is introduced (where ΔΦ is reduced to its maximum value ΔΦmax=ΔnL/λ, and Δn=neno is the LC birefringence). The case of ΔΦmax is realized for the cells with the LC planar alignment (θ0=0). For the inhomogeneous director distribution, the dependences of the parameter Φ on the pretilt angle θ0 are calculated for different refractive indices.

Figure 2 shows the dependences of Φ on θ0 for three director configurations at different values of ne and fixed no=1.5. The Φ(θ0) dependences are compared for all three cell geometries considered in the θ0 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(θ0) value changes from 1 to 0 only in the case of the homogeneous director distribution. For the S-configuration, the Φ(θ0) value decreases down to 0.5-0.55 at θ0=π/2. For the B-configuration Φ(θ0) decreases from 0.45 to 0.55 at θ0=0 down to 0 at θ0=π/2. The total change in the Φ(θ0) 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 θ0=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 θ0 for three LC director configurations at different ne and no=1.5. For every configuration upper dashed line corresponds to ne=1.6, middle solid line ne=1.7, lower dot-dashed line ne=1.8.

Download Full Size | PPT Slide | PDF

If no and θ0 are fixed and ne increases, then Φ reduces for all the three configurations (see Section 6). If the birefringence is the same at different no values and fixed angle θ0, then Φ is almost independent on no in the no 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 θ0 or θπ/2

Some analytical approximations of the Φ(θ0) dependences are possible if θ0<<1 or δ0<<1. The results of analytical expansions of both Φ(θ0) and Φ(δ0) dependences at fixed ne=1.6 and no=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).

Tables Icon

Table 1. Analytical approximations of Φ(θ0) and Φ(δ0) dependences at θ0<<1 or δ0<<1.

 

Fig. 3 Dependences of the phase retardation parameter Φ on the pretilt angle value θ0 at no=1.5: (a) the case of a small pretilt angle (θ0<<1) for homogeneous (left) and splay (right) configurations, and (b) the case of a large pretilt angle (δ0<<1) for homogeneous (left) and bend (right) configurations. 1-ne=1.6, 2-ne=1.7,3-ne=1.8.

Download Full Size | PPT Slide | PDF

Deviation of the Φ(θ0) dependence in the case of θ0 or θπ/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 θ0~π/4

If θ0~π/4, then the quasilinear part of the Φ(θ0) dependence for the cell with the homogeneous tilt distribution (Fig. 4 .) is described by the equation:

Φ(θ0)=1neno[2neno(ne2+no2)1/2(1+π4ne2no2ne2+no2)noθ02neno(ne2+no2)1/2ne2no2ne2+no2]
When the refractive indices are equal to ne=1.6 and no=1.5, Eq. (7) is transformed to Φ(θ0)=1.253-0.997θ0. The parameter Φ(θ0) is equal to 0.470 at a pretilt angle θ0=π/4.

 

Fig. 4 Dependences of the phase retardation parameter Φ on the pretilt angle θ0 at no=1.5 for the homogeneous configuration in the case of small deviation of θ0 from π/4~0.785. 1-ne=1.6, 2-ne=1.7,3-ne=1.8.

Download Full Size | PPT Slide | PDF

Let us investigate the Φ(θ0) dependence for the case of fixed θ0 and varying ne values in more details. The dependence of Φ on the ratio of the refractive indices can be derived from Eq. (7) at fixed θ0. Let us introduce the parameter x=ne/no. Then, Eq. (7) can be written in the form

Φ(θ0)=1x1[2x(x2+1)1/2(1+π4x21x2+1)1θ02x(x21)(x2+1)3/2]
Equation (8) confirms the fact that the tilt of the Φ(θ0) curves weakly depends on the birefringence Δn=neno. Besides, the Φ(θ0) value decreases at fixed θ0 if Δn increases (Fig. 2-4). Let us suggest that the Δn value is small or α<<1 (α=x1). Then, Eq. (8) can be written in the form (intermediate stages of derivation are omitted):
Φ(θ0)=14[(2+π0)(112α)]
If the birefringence parameter α is equal to 0 and the pretilt angle is θ0=π/4, then the phase retardation parameter Φ(θ0)=1/2 resembles the behavior of the dependence shown in Fig. 4. The Φ(θ0,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 ΔΦ on the pretilt angle θ0 have been calculated for the LC cells with homogeneous, splay, or bend director configuration. When θ0 ranges from 0 to π/2, the Φ(θ0) value [Φ(θ0)=ΔΦ(θ0)/ΔΦmax] 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(θ0) value decreases with increasing birefringence Δn at fixed θ0.

The results can be used to develop the methods of the LC pretilt angle measurements in cells with sophisticated director configuration [811]. 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(θ0) value can be obtained in an experiment in LC cells with different boundary conditions and director distribution [1215].

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.

References

  • View by:
  • |
  • |
  • |

  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.

2005 (2)

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]

2001 (2)

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,” Crystallogr. Rep. 27, 1019–1021 (1982).

Belyaev, V.

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]

Beresnev, G. A.

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).

Chigrinov, V.

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]

Chigrinov, V. G.

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,” Crystallogr. Rep. 27, 1019–1021 (1982).

Dascalu, C.

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

Fercher, A.

Goetzinger, E.

Grebenkin, M. F.

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).

Hitzenberger, Ch.

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

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

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(4), 349–354 (2005).
[Crossref]

Murauski, A.

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]

Muravsky, A.

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]

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]

Pircher, M.

Sticker, M.

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

Crystallogr. Rep. (1)

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. (1)

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 (1)

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

Rev. Mex. Fis. (1)

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

Sov. Phys. Crystallogr. (1)

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

Other (9)

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.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
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.
Fig. 2
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 .
Fig. 3
Fig. 3 Dependences of the phase retardation parameter Φ on the pretilt angle value θ 0 at 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 .
Fig. 4
Fig. 4 Dependences of the phase retardation parameter Φ on the pretilt angle θ 0 at 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 .

Tables (1)

Tables Icon

Table 1 Analytical approximations of Φ(θ 0 ) and Φ(δ 0 ) dependences at θ 0 <<1 or δ 0 <<1 .

Equations (11)

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

ΔΦ=( n e eff n o )L/λ
θ(z)= θ 0 =const(H)
θ(z)= 0 z/L(S)
θ(z)= π 2 +( π 0 )z/L(B)
δ(z)= 0 z/L(B)
ΔΦ= λ [ 0 L n o n e dz (n o 2 cos 2 θ(z)+ n e 2 sin 2 θ(z)) 1/2 n o L ]
ΔΦ= 2π 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+ ΔΦ ΔΦ max Δn 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 x1 [ 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 0 )(1 1 2 α) ]

Metrics