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 on the cell substrates in the range . 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 is observed in splay LC cells, whereas the smallest phase retardation is observed in bend cells. For the 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
In many types of modern liquid crystal displays (LCDs), tilted alignment and/or sophisticated configuration of the LC director is used (MVA, OCB, etc.; [1–3]). It can improve the LCD speed of response or viewing angle range . 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 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 values for LC cells with homogeneous and inhomogeneous LC director distribution as a function of the LC pretilt angle 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 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
The case of the normal incidence of light is considered. The single constant approximation (the Frank elastic constants for a nematic LC ) 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:Fig. 1(b) and 1(c).
3. Results for three main configurations
Dependences of the phase retardation difference on the pretilt angle are calculated for the LC cells in question. Phase retardation for the cells with an arbitrary LC director distribution is described by the expression [6,7].Eq. (2a)). The dependence is determined by the expression:Equation (5) yields the pretilt angle value for a LC with initial homogeneous configuration:
Figure 2 shows the dependences of on for three director configurations at different values of and fixed . The dependences are compared for all three cell geometries considered in the range from 0 to , typical for the LC cells in the absence of a voltage or with a slowly changing voltage. The value changes from 1 to 0 only in the case of the homogeneous director distribution. For the S-configuration, the value decreases down to 0.5-0.55 at . For the B-configuration decreases from 0.45 to 0.55 at down to 0 at . The total change in the 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 value at for both S- and B-configurations depends on the LC birefringence.
If and are fixed and increases, then reduces for all the three configurations (see Section 6). If the birefringence is the same at different values and fixed angle , then is almost independent on in the 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 or
Some analytical approximations of the dependences are possible if or . The results of analytical expansions of both and dependences at fixed and are presented in Table 1 . Numerical estimations of these dependences are compared for the three cases considered in Fig. 3(a) and 3(b).
Deviation of the dependence in the case of or 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
If , then the quasilinear part of the dependence for the cell with the homogeneous tilt distribution (Fig. 4 .) is described by the equation:Eq. (7) is transformed to . The parameter is equal to 0.470 at a pretilt angle .
Let us investigate the dependence for the case of fixed and varying ne values in more details. The dependence of on the ratio of the refractive indices can be derived from Eq. (7) at fixed . Let us introduce the parameter . Then, Eq. (7) can be written in the formEquation (8) confirms the fact that the tilt of the curves weakly depends on the birefringence . Besides, the value decreases at fixed if increases (Fig. 2-4). Let us suggest that the Δn value is small or (). Then, Eq. (8) can be written in the form (intermediate stages of derivation are omitted):Fig. 4. The dependence has no discrepancy at (), it is a continuous function if the birefringence changes its sign.
The dependences of the phase retardation difference on the pretilt angle have been calculated for the LC cells with homogeneous, splay, or bend director configuration. When ranges from 0 to , the value  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 value decreases with increasing birefringence at fixed .
The results can be used to develop the methods of the LC pretilt angle measurements in cells with sophisticated director configuration [8–11]. 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 value can be obtained in an experiment in LC cells with different boundary conditions and director distribution [12–15].
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.
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.