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

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.; [1–3]). 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 $\Delta \Phi$ 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 $\Delta \Phi$ values for LC cells with homogeneous and inhomogeneous LC director distribution as a function of the LC pretilt angle ${\text{θ}}_{\text{0}}$ 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 ${\text{θ}}_{\text{0}}$ 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

 

Fig. 1 Schematic of the LC director distribution in LC cells with homogeneous (H), splay (S), and bend (B) configuration: $\text{θ}$ is the deviation from the substrate plane, and $\text{δ}$ is the deviation from the normal to the substrate.

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The case of the normal incidence of light is considered. The single constant approximation (the Frank elastic constants for a nematic LC ${\text{K}}_{\text{11}}{\text{=K}}_{\text{33}}$) 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:

3. Results for three main configurations

Dependences of the phase retardation difference $\Delta \Phi$ on the pretilt angle ${\text{θ}}_{\text{0}}$ are calculated for the LC cells in question. Phase retardation $\Delta \Phi$ for the cells with an arbitrary LC director distribution is described by the expression [6,7].

Figure 2 shows the dependences of $\Phi$ on ${\text{θ}}_{\text{0}}$ for three director configurations at different values of ${\text{n}}_{\text{e}}$ and fixed ${\text{n}}_{\text{o}}=1.5$. The ${\text{Φ(θ}}_{\text{0}}\text{)}$ dependences are compared for all three cell geometries considered in the ${\text{θ}}_{\text{0}}$ range from 0 to $\text{π/2}$, typical for the LC cells in the absence of a voltage or with a slowly changing voltage. The ${\text{Φ(θ}}_{\text{0}}\text{)}$ value changes from 1 to 0 only in the case of the homogeneous director distribution. For the S-configuration, the ${\text{Φ(θ}}_{\text{0}}\text{)}$ value decreases down to 0.5-0.55 at ${\text{θ}}_{\text{0}}=\text{π/2}$. For the B-configuration ${\text{Φ(θ}}_{\text{0}}\text{)}$ decreases from 0.45 to 0.55 at ${\text{θ}}_{\text{0}}=0$ down to 0 at ${\text{θ}}_{\text{0}}=\text{π/2}$. The total change in the ${\text{Φ(θ}}_{\text{0}}\text{)}$ 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 $\Phi$ value at ${\text{θ}}_{\text{0}}=0$ for both S- and B-configurations depends on the LC birefringence.

 

Fig. 2 Dependences of the phase retardation parameter $\Phi$ on the pretilt angle value ${\text{θ}}_{\text{0}}$ for three LC director configurations at different ${\text{n}}_{\text{e}}$ and ${\text{n}}_{\text{o}}=1.5$. For every configuration upper dashed line corresponds to ${\text{n}}_{\text{e}}=1.6$, middle solid line ${\text{n}}_{\text{e}}=1.7$, lower dot-dashed line ${\text{n}}_{\text{e}}=1.8$.

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If ${\text{n}}_{\text{o}}$ and ${\text{θ}}_{\text{0}}$ are fixed and ${\text{n}}_{\text{e}}$ increases, then $\Phi$ reduces for all the three configurations (see Section 6). If the birefringence is the same at different ${\text{n}}_{\text{o}}$ values and fixed angle ${\text{θ}}_{\text{0}}$, then $\Phi$ is almost independent on ${\text{n}}_{\text{o}}$ in the ${\text{n}}_{\text{o}}$ 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 $\text{θ}\to \text{0}$ or $\text{θ}\to \text{π/2}$

Some analytical approximations of the ${\text{Φ(θ}}_{\text{0}}\text{)}$ dependences are possible if ${\text{θ}}_{\text{0}}<<1$ or ${\text{δ}}_{\text{0}}<<1$. The results of analytical expansions of both ${\text{Φ(θ}}_{\text{0}}\text{)}$ and ${\text{Φ(δ}}_{\text{0}}\text{)}$ dependences at fixed ${\text{n}}_{\text{e}}=1.6$ and ${\text{n}}_{\text{o}}=1.5$ 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 ${\text{Φ(θ}}_{\text{0}}\text{)}$ and ${\text{Φ(δ}}_{\text{0}}\text{)}$ dependences at ${\text{θ}}_{\text{0}}<<1$ or ${\text{δ}}_{\text{0}}<<1$.

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Fig. 3 Dependences of the phase retardation parameter $\Phi$ on the pretilt angle value ${\text{θ}}_{\text{0}}$ at ${\text{n}}_{\text{o}}=1.5$: (a) the case of a small pretilt angle (${\text{θ}}_{\text{0}}<<1$) for homogeneous (left) and splay (right) configurations, and (b) the case of a large pretilt angle (${\text{δ}}_{\text{0}}<<1$) for homogeneous (left) and bend (right) configurations. ${\text{1-n}}_{\text{e}}=\text{1}\text{.6}$, ${\text{2-n}}_{\text{e}}=\text{1}\text{.7}$,${\text{3-n}}_{\text{e}}=\text{1}\text{.8}$.

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Deviation of the ${\text{Φ(θ}}_{\text{0}}\text{)}$ dependence in the case of $\text{θ}\to \text{0}$ or $\text{θ}\to \text{π/2}$ 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 ${\text{θ}}_{\text{0}}~\text{π/4}$

If ${\text{θ}}_{\text{0}}~\text{π/4}$, then the quasilinear part of the ${\text{Φ(θ}}_{\text{0}}\text{)}$ dependence for the cell with the homogeneous tilt distribution (Fig. 4 .) is described by the equation:

 

Fig. 4 Dependences of the phase retardation parameter $\Phi$ on the pretilt angle ${\text{θ}}_{\text{0}}$ at ${\text{n}}_{\text{o}}=1.5$ for the homogeneous configuration in the case of small deviation of ${\text{θ}}_{\text{0}}$ from $\text{π/4~0}\text{.785}$. ${\text{1-n}}_{\text{e}}=\text{1}\text{.6}$, ${\text{2-n}}_{\text{e}}=\text{1}\text{.7}$,${\text{3-n}}_{\text{e}}=\text{1}\text{.8}$.

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Let us investigate the ${\text{Φ(θ}}_{\text{0}}\text{)}$ dependence for the case of fixed ${\text{θ}}_{\text{0}}$ and varying n_e values in more details. The dependence of $\text{Φ}$ on the ratio of the refractive indices can be derived from Eq. (7) at fixed ${\text{θ}}_{\text{0}}$. Let us introduce the parameter $\text{x}={\text{n}}_{\text{e}}{\text{/n}}_{\text{o}}$. Then, Eq. (7) can be written in the form

Therefore, all the formulas and data shown in Figs. 2-4 are valid for both calamitic ($\text{Δn}>\text{0}$) and discotic ($\text{Δn}<\text{0}$) liquid crystals.

6. Conclusions

The dependences of the phase retardation difference $\Delta \text{Φ}$ on the pretilt angle ${\text{θ}}_{\text{0}}$ have been calculated for the LC cells with homogeneous, splay, or bend director configuration. When ${\text{θ}}_{\text{0}}$ ranges from 0 to $\text{π/2}$, the ${\text{Φ(θ}}_{\text{0}}\text{)}$ value [${\text{Φ(θ}}_{\text{0}}\text{)}={\text{ΔΦ(θ}}_{\text{0}}\text{)/Δ}{\Phi }_{\text{max}}$] 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 ${\text{Φ(θ}}_{\text{0}}\text{)}$ value decreases with increasing birefringence $\text{Δn}$ at fixed ${\text{θ}}_{\text{0}}$.

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 ${\text{Φ(θ}}_{\text{0}}\text{)}$ 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.