## Abstract

An optical compensation principle of the crossed circular polarizers is developed to widen the viewing angle of high-transmittance multi-domain vertical-alignment liquid crystal displays (MVA-LCDs). The optical properties of a biaxial film are analyzed by the Berreman 4×4 matrix method, and the analytical solution for the slow-axis orientation of a biaxial film is calculated to obtain the compensation principle of the crossed circular polarizers. Based on this compensation principle, the high-transmittance MVA-LCD theoretically has a complete 80° viewing cone for contrast ratio (CR)>100:1 and experimental results reveal that the compensated high-transmittance MVA-LCD can achieve a viewing angle of over the entire 80° viewing cone for CR>20:1. Practical application as a mobile display is emphasized.

© 2008 Optical Society of America

## 1. Introduction

High contrast ratio (CR), high brightness and wide viewing angle are the critical requirements of a liquid-crystal display (LCD). The multi-domain vertical-alignment liquid crystal display (MVA-LCD) exhibits an excellent contrast ratio at the normal viewing direction and a wide viewing angle. However, the transmittance of the MVA-LCD is relatively poor, resulting in low brightness. A circular polarizer-type multi-domain vertical-alignment LCD is recently developed to achieve a high transmittance for mobile applications [1]. For this high-transmittance MVA-LCD, however, the off-axis light leakage, resulting from the combination of LC and crossed circular polarizers, results in a low contrast ratio in the off-axis direction. Several methods have been proposed to compensate for the viewing angle of the high-transmittance MVA-LCD [2–4]. These proposed methods provide a wide viewing angle, but the complicated configurations or the specific parameters of compensation films cause fabrication difficulties and high cost. Our previous study presented a greatly simplified compensation method with uniaxial films based on the elimination of the off-axis light leakage, which is caused by LC and crossed circular polarizers, to enable the high-transmittance MVA-LCD to have a viewing angle that exceeds the entire 80° viewing cone [5]. However, the negative A plates that were adopted in our previous proposed compensation method are difficult to be fabricated, which fact limits the practical applicability of the compensated high-transmittance MVA-LCD. There is an urgent need to develop practical compensation methods to compensate for the viewing angle of the high-transmittance MVA-LCD.

This study proposes a compensation principle for the crossed circular polarizers to widen the viewing angle of the high-transmittance MVA-LCD. The proposed compensation principle can be utilized to obtain practical compensation methods for the high-transmittance MVA-LCD. We begin with analyzing the optical properties of a biaxial film by using the Berreman 4×4 matrix method [6], and then deriving and analyzing the analytical solution for the slow-axis orientation of a biaxial film, and minimizing the light leakage of the crossed circular polarizers. Finally, the compensation principle of the crossed circular polarizers is developed to widen the viewing angle of the high-transmittance MVA-LCD. Based on the proposed principle, the high-transmittance MVA-LCD is predicted to have a complete 80o viewing cone for CR>100:1. The experimental results demonstrate that the compensated high-transmittance MVA-LCD can achieve a wide viewing angle over the 80o viewing cone for CR>20:1.

## 2. Design of wide view circular polarizers for liquid crystal displays

Figure 1 depicts the basic cell configuration of the proposed compensation principle for the high-transmittance MVA-LCD. The proposed structure comprises crossed circular polarizers, consisting of crossed linear polarizers and crossed biaxial λ/4 plates, one negative C plate that is inserted between the λ/4 plate and the VA-LC to compensate exactly for LC off-axis light leakage in the dark state, as described in our previous work [5], and one biaxial λ/2 plate is inserted on the inner side of the bottom polarizer with its slow axis perpendicular to the absorption axis of the bottom polarizer.

In Fig. 1, the crossed biaxial λ/4 plates are used to prevent the leakage of light from the crossed λ/4 plates that is caused by the shift in the azimuthal angle of the slow axes of crossed λ/4 plates for off-axis light. To investigate the slow-axis orientation of the biaxial λ/4 plates, we first analyze the optical properties of optical films. For an anisotropic medium such as an optical film, the optical properties can be calculated by the Berreman 4×4 matrix method [6–8]. The Berreman 4×4 matrix method which takes the reflected light from boundary into account is superior to the 2×2 matrix method [9, 10]. For a plane monochromatic wave inside the anisotropic medium, the Maxwell equation can be transformed into the Berreman equation which is given as [6]

where *ψ*=(E_{x}, H_{y}, E_{y}, -H_{x})^{T}, k=ω/c and ** D**(z) is the Berreman matrix. For a nonmagnetic medium with nontilted principal axes, the Berreman matrix

**(z) is given by**

*D*with

$${D}_{21}={n}_{1}^{2}{\mathrm{cos}}^{2}\varphi +{n}_{2}^{2}{\mathrm{sin}}^{2}\varphi ,$$

$${D}_{23}=\left({n}_{1}^{2}-{n}_{2}^{2}\right)\mathrm{sin}\varphi \mathrm{cos}\varphi ,$$

$${D}_{43}={n}_{1}^{2}{\mathrm{sin}}^{2}\varphi +{n}_{2}^{2}{\mathrm{cos}}^{2}\varphi -{\chi}^{2}.$$

where n_{1}, n_{2} and n_{3} are the principal refractive indices for the principal axes of the anisotropic medium X, Y and Z, respectively, ϕ is the orientation angle of the principal axes with respect to laboratory coordinate system (x, y, z), and χ=sinθ_{i} where θ_{i} is the incidence angle. For homogeneous media, the matrix ** D** does not depend on z and the solution of Eq. (1) can be written as

where *q _{j}* and Ψ

^{j}are the eigenvalues and the eigenstates of the Berreman equation. By substituting Eq. (3) into (1), the Berreman equation leads to an eigenvalue equation for

**which is written as**

*D*Equation (4) implies that the eigenstates of the incident light inside the homogeneous media can be obtained by calculating the eigenstates of the Berreman matrix ** D**. Since

**is a 4×4 matrix, the characteristic equation of**

*D***is a quartic equation. By solving this quartic equation, the eigenvalue of the Berreman matrix**

*D***are**

*D*with

$$\phantom{\rule{.8em}{0ex}}\phantom{\rule{.2em}{0ex}}-\frac{2{\chi}^{2}\left({n}_{1}^{2}-{n}_{2}^{2}\right)\left({n}_{1}^{2}{\mathrm{cos}}^{2}\varphi -{n}_{2}^{2}{\mathrm{sin}}^{2}\varphi -{n}_{3}^{2}\mathrm{cos}2\varphi \right)}{{n}_{3}^{2}}.$$

For the Berreman matrix ** D**, the expression of the eigenstates with eigenvalue can be written as [7]

where *A _{j}* is the normalization constant. By substituting Eqs. (2), (5) and (6) into Eq. (7), the eigenstates of the matrix

**are written as**

*D*and

where *ψ*
^{+}
_{e} and *ψ*
^{-}
_{e} are the eigenstates of the forward propagating light and the backward propagating light, respectively, for the extraordinary ray and *ψ*
^{+}
_{o} and ψ^{-}
_{o} are the eigenstates of the forward propagating light and the backward propagating light, respectively, for the ordinary ray. After normalization, we can see that the eigenstates of the backward propagating light and the forward propagating light, given in Eqs. (8) and (9), have the same component in the electric field and the magnetic field for both the extraordinary ray and ordinary ray. That is the backward propagating light (reflected light) and the forward propagating light (incident light) have the same eigenstates (with different eigenvalues) inside the medium with nontilted principal axes. Notably, the same eigenstates for the reflected light and the incident light is just a consequence of the medium with nontilted principal axes. In the general case, the relations of the eigenstates between the reflected light and the incident light are not so simple.

The above eigenstates, calculated from the Berreman 4×4 matrix method, are the tangential components of the electromagnetic field. Thus, the complete eigenstates of the propagating light inside the anisotropic medium such as an optical film cannot be obtained directly from Eqs.(8) and (9). Based on the above analysis, the reflected light and the incident light have the same eigenstates inside the medium with nontilted principal axes. Thus, the complete eigenstates of the anisotropic medium can be obtained by only considering the forward propagating light (incident light) in our study. The two allowed eigenstates of the anisotropic medium represent the slow axis and the fast axis of the anisotropic medium for the incident light [11]. For the incident light, the eigenstates of the anisotropic medium can be obtained by calculating the eigenstates of the impermeability tensor. In the principal coordinates, the impermeability tensor of a biaxial film *η* is given by

A coordinate transformation that causes the Z-axis to be the propagation direction of the light, yields a transformed impermeability tensor *η _{t}*, in the new coordinates, that can be simplified as a 2×2 tensor, which can be written as

where θ_{o} is the refraction angle of incident light inside the biaxial film, as shown in Fig. 2(a). Calculating the eigenstates of the transformed impermeability tensor *η _{t}* yields the analytical solution for the slow-axis orientation of a biaxial film for off-axis light. To eliminate efficiently the off-axis light leakage of the crossed λ/4 plates, we first investigate the orientation of the slow axis of a biaxial film at

*ϕ*=45°, which is the direction of the maximum shift angle of the slow axis of a biaxial film for off-axis light from our calculations (not shown here). For off-axis light, the analytical solution for the slow-axis orientation of a biaxial film at

*ϕ*=45° is derived as

where A, B, C and D are as follows

*A*=2(*n*
^{2}
_{1}-*n*
^{2}
_{2})*n*
^{2}
_{3}cos*θ _{o}*

*B*=2*n*
^{2}
_{1}
*n*
^{2}
_{2}sin^{2}
*θ _{o}*-sin

^{2}

*θ*(

_{o}*n*

^{2}

_{3}

*n*

^{2}

_{2}+

*n*

^{2}

_{1}

*n*

^{2}

_{3})

*C*=2*n*
^{2}
_{2}
*n*
^{2}
_{1}sin^{2}
*θ _{o}*+(cos

^{2}

*θ*+1)(

_{o}*n*

^{2}

_{3}

*n*

^{2}

_{1}+

*n*

^{2}

_{3}

*n*

^{2}

_{2})

*D*=8*n*
^{2}
_{1}
*n*
^{2}
_{2}
*n*
^{2}
_{3}(*n*
^{2}
_{1}sin^{2}
*θ _{o}*+

*n*

^{2}

_{2}sin

*+2*

^{2}θ_{o}*n*

^{2}

_{3}cos

^{2}

*θ*)

_{o}Notably, the ψ, given in Eq. (12), is defined as the included angle between the slow axis of the optical film and the plane of incidence. For a C plate, its two in-plane refractive indices are equal, i.e., n_{1}=n_{2}. By substituting n_{1}=n_{2} into Eq. (12), the ψ is zero regardless of the refraction angle. This is because the slow axis of the C plate, which is a plate of uniaxially birefringent medium with its optic axis normal to the plate surfaces [11], is always parallel to the plane of incidence regardless of the incident angle. A cell of LC with a vertical alignment is a good example of the C plate. Additionally, the slow-axis orientation ψ of the biaxial film affects the final polarization state of the incident light. The basic concept of the optical compensation for LCDs is to keep the polarization state of the off-axis incident light to be the same as that of the normal incident light in the dark state of LCDs by designing the proper ψ and the phase retardation of the biaxial film [11–13].

For the optical compensation of a LCD, the biaxial factor Nz is used to characterize the optical properties of the biaxial film [14]. Considering both the difference and ratio of n_{1}, n_{2} and n_{3}, yields a modified biaxial factor as

Based on Eqs. (3) and (4), the slow-axis shift angle of the biaxial film Δψ, shown in Fig. 2(a), which is defined as Δψ=ψ(θ_{o})-ψ(θ_{o}=0°), is calculated with different Nz for off-axis light at ϕ=45°, as shown in Fig. 2(b). From Fig. 2(b), Δψ is a negative value for the compensation film with Nz>0.5 and a positive value for the compensation film with Nz<0.5, indicating that the slow axis shifts toward the normal to the incident plane when the compensation film has Nz>0.5 and toward a direction parallel to the incident plane when the compensation film has Nz<0.5 for off-axis light. Furthermore, the compensation film with Nz=0.5 has a nearly constant slow-axis orientation, Δψ~0, and the Δψ curves for different Nz, plotted in Fig. 2(b), exhibit almost mirror symmetry with respect to the curve of Nz=0.5, such that the Δψ curves for Nz>0.5 and Nz<0.5 have the same magnitude but different signs. The same tendency of the slow axis to shift is obtained at ϕ=-45°.

Based on the above analysis for the slow-axis orientation of the biaxial film, the compensation principles for the crossed λ/4 plates and the crossed linear polarizers are developed to eliminate the off-axis light leakage of crossed circular polarizers. For the crossed λ/4 plates, the off-axis light leakage can be significantly reduced under the condition Nz_{_λ/4}+Nz’_{_λ/4}=1, where Nz_{_λ/4} and Nz_{’_λ/4} are the Nz values of the top and bottom λ/4 plates, respectively. Figure 3(a) explains this proposed compensation principle for the crossed λ/4 plates. Figure 3(a) is the orientation of the two slow axes of the crossed λ/4 plates at ϕ=45° and ϕ=-45° using a Poincaré sphere representation. Based on the above analysis, the slow axis of the λ/4 plate with its orientation at ϕ=45° shifts from A to C for Nz>0.5 (A to E for Nz<0.5) as the incident angle increases, while the slow axis of the other λ/4 plate at ϕ=-45° shifts from B to F for Nz<0.5 (B to D for Nz>0.5) as shown in Fig. 3(a). Accordingly, the two slow axes of the crossed λ/4 plates shift in the same direction on the equator of the Poincaré sphere for off-axis light when one λ/4 plate with Nz>0.5 and one λ/4 plate with Nz<0.5 are used. Moreover, the mirror symmetry of Δψ curves with respect to the curve of Nz=0.5 in Fig. 2(b) is such that, while Nz_{_λ/4}+Nz’_{_λ/4}=1 is satisfied, the shift angles Δψ_{1} and Δψ_{2} (Δψ_{1}’ and Δψ_{2}’), shown in Fig. 3(a), are almost equal, indicating that the slow axes of the crossed λ/4 plates, which are satisfied with Nz_{_λ/4}+Nz’_{_λ/4}=1 remain crossed for off-axis light, and then the off-axis light leakage, which is caused by the crossed λ/4 plates, is eliminated. Additionally, when both of the λ/4 plates with Nz=0.5 are used, the orientation of the two slow-axes of the crossed λ/4 plates remain at A and B, shown in Fig. 3(a), for off-axis light, indicating that the slow axes of the crossed λ/4 plates also remain crossed for off-axis light when both of the λ/4 plates with Nz=0.5 are used. Figure 3(b) plots the calculations of the included angle of the slow axes of the crossed λ/4 plates based on the proposed principle for off-axis light. Figure 3(b) reveals that the maximum deviation of the included angle of the slow axes of the crossed λ/4 plates with various combinations of Nz, as determined by the proposed principle, is less than 2° at θ_{o}=70°, which angle represents a significant improvement over that, ~52°, of conventional crossed λ/4 plates.

A biaxial λ/2 plate with a biaxial factor Nz_{_λ/2}=0.5 is used to eliminate the off-axis light leakage by crossed linear polarizers. The compensation mechanism of the biaxial λ/2 plate with Nz_{_λ/2}=0.5 for the off-axis light leakage of the crossed linear polarizers comes from the fact that the biaxial film of Nz=0.5 has a nearly unchanged slow-axis orientation for off-axis light [13]. Figure 2(b) indicates that the maximum shift angle of the slow axis of the biaxial film with Nz_{_λ/2}=0.5, defined by the proposed Eq. (4), is ~0.9° at θ_{o}=70°. This value is less than ~1.8°, which is the slow-axis shift of the biaxial film with Nz_{_λ/2}=0.5 by the conventional definition
${N}_{z}=\frac{\left({n}_{1}-{n}_{3}\right)}{\left({n}_{1}-{n}_{2}\right)}$
with *n*
_{1}>*n*
_{2}). Accordingly, the off-axis light leakage of the crossed linear polarizers can be greatly reduced by compensation using a biaxial λ/2 plate with the proposed Nz_{_λ/2}=0.5. Following the compensation for the crossed λ/4 plates and the crossed linear polarizers based on the proposed principle, Nz_{_λ/4}+Nz’_{_λ/4}=1 and _{Nz_λ/2}=0.5, the off-axis light leakage of the crossed circular polarizers can be effectively eliminated, thus widening the viewing angle of the high-transmittance MVA-LCD.

## 3. Simulation and experimental results

Based on the proposed compensation principle, the following refractive indices and film thickness of the LC, the C plate, the biaxial λ/2 plate and the λ/4 plates are employed in the proposed structure to widen the viewing angle of the high-transmittance MVA-LCD; *n _{e_LC}*=1.5897 and

*n*=1.4874 with

_{o_LC}*d*=4µm,

_{LC}*n*=1.5089 and

_{e_-c}*n*=1.5124 with

_{o_-c}*d*=107.94µm,

_{-c}*n*=1.5095,

_{1_λ/2}*n*=1.5095 and

_{2_λ/2}*n*=1.5102 with

_{3_λ/2}*d*=184µm and

_{λ/2}*n*=1.5095,

_{1_λ/4}*n*=1.5011,

_{2_λ/4}*n*=1.5095~1.511 for

_{3_λ/4}*Nz*=0~1, with

_{_λ/4}*d*=92µm. The Berreman 4×4 matrix method is used for the calculation of the viewing angle. The calculations reveal that all of the iso-CR curves of the compensated high-transmittance MVALCD, based on the proposed compensation principle, are similar, and the viewing angle exceeds the entire 80° viewing cone for CR>100:1. Figures 4(a), (b) and (c) plot the calculations for the crossed circular polarizers with (Nz

_{λ/4}_{_λ/2}, Nz

_{_λ/4}, Nz’

_{_λ/4})=(0.5, 0, 1), (0.5, 0.3, 0.7), and (0.5, 0.5, 0.5), respectively.

Presently, the compensation film with Nz=0.5 is used widely for the optical compensation of LCDs. Accordingly, in the experiment, the polarizer sample with (Nz_{_λ/2}, Nz_{_λ/4}, Nz’_{_λ/4})=(0.5, 0.5, 0.5) was prepared (Nitto Denko). The high-transmittance MVA-LCD cell with a cell gap of 4µm was filled with a commercial LC mixture, MJ-042783 (Merck), with Δn=0.1 at 550nm. The diameter and the height of the circular protrusion are 15µm and 1.5µm, respectively, the sub-pixel size is 47µm×141µm, and the gap between two adjacent pixel electrodes is 8µm. Figure 4 (d) plots measured iso-CR curves of the compensated high-transmittance MVA-LCD sample, indicating that the compensated high-transmittance MVALCD sample, based on the proposed principle, can achieve a viewing angle over the entire 80° viewing cone for CR>20:1. The experimental iso-CR curves, shown in Fig. 4(d), are lowered because the ideal parameters may not be precisely controlled. Furthermore, the variation in thickness of the compensation films and the LC alignment distortion near the spacers, the protrusions and the pixel edges also reduce the contrast ratio. In practice, a light-shielding metal layer that is formed during the thin film transistor array process can be designed under the pixel edges, the spacers, and the protrusions to prevent light leakage to improve the iso- CR curves.

For the applications of the LCD, wide spectral bandwidth is as important as wide viewing angle. Although, in the proposed cell configuration, shown in Fig. 1, only single λ/4 plate is used for each of the circular polarizer. However, the slow axes of the two λ/4 plates in the proposed structure are designed to be crossed which leads to the self-compensation effect at the normal viewing direction. Additionally, based on the proposed compensation principle, the slow axes of these two λ/4 plates remain crossed for off-axis light, this indicates that the self-compensation effect occurs over full viewing cone. Because of this self-compensation effect, the proposed compensation films tend to have a wide spectral bandwidth so that the light leakage of the crossed circular polarizers is less than 3.59×10^{-5} over the 450~650 nm spectral range at the normal viewing direction for different (Nz_{_λ/2}, Nz_{_λ/4}, Nz’_{_λ/4}) designs, as shown in Fig. 5 (calculated by the 4×4 matrix method). The maximum light leakage is ~4×10^{-3} at λ=450 nm and ~1.47×10^{-3} at λ=650 nm at 80° viewing cone. This is sufficient for LCD applications because the human visual system is insensitive to blue, thus, the light leakage at blue is more acceptable than red and green. It should be noted that for some LCD applications such as transflective type LCDs, the color dispersion and the viewing angle in the reflective part cannot be compensated by using the proposed simple compensation method. This is because that the reflected light of the transflective type LCD only passes through the top circular polarizer and the self-compensation effect no more occurs under this condition. A more complicated configuration of the crossed circular polarizers is needed to widen the viewing angle and the spectral bandwidth of the transflective type LCDs. Further study on the broadband wide-view circular polarizers for the transflective type LCDs is now underway.

## 4. Conclusions

This study analyzes the optical properties of the biaxial film by using the Berreman 4×4 matrix method and obtains the analytical solution for the slow-axis orientation of the biaxial film to develop an optical compensation principle of crossed circular polarizers for widening the viewing angle of the high-transmittance MVA-LCD. Based on this principle, the high-transmittance MVA-LCD theoretically and experimentally has a complete 80° wide viewing cone. Because of the self-compensation effect, the light leakage of the proposed crossed circular polarizers is maintained below 4×10^{-3} over the 450~650 nm spectral range at 80° viewing cone. We believe the proposed principle of the crossed circular polarizers will be a powerful method not only in practical applications of high-transmittance MVA-LCD but also in devices with crossed circular polarizers.

## Acknowledgment

The authors would like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract Nos. NSC 96-2112-M-110-014-MY2.

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