During the last decade, the fringe-field switching (FFS) liquid crystal (LC) mode has been widely used for mobile phones, tablets, and high-end notebook displays, owing to its high transmittance and low power consumption required for ultra-high resolution devices [1]. In particular, a single-domain FFS LC mode has good transmittance efficiency as compared with a multi-domain FFS LC mode that exhibits inevitable reduction in the transmittance efficiency owing to the LC disclination lines caused by discontinuous LC distributions in the boundary of domains [2-4]. Thus, the single-domain FFS LC mode has been widely used for high-resolution displays, particularly in the small pixels for the mobile displays that require ultra-high resolution and low power consumption. However, a pretilt angle in the polar direction is unavoidable during the rubbing process for aligning the LC molecules. The presence of the pretilt angle in planar LC modes causes the asymmetry in viewing angle in terms of the light leakage and grayscale inversion. This asymmetry negatively affects the viewing angle properties such as the light leakage, color shift, and grayscale inversion in the specific viewing angle direction [5, 6]. To enhance the viewing angle properties of the single-domain planar mode, many optical compensation techniques have been proposed [7-15]. These optical compensation methods can reduce the light leakage and remove the grayscale inversion. However, these techniques do not eliminate the viewing angle asymmetry that persists owing to a nonzero pretilt angle in the single-domain FFS LC mode. Despite some clear advantages such as high transmittance, high aperture ratio, and low power consumption, the asymmetric performance of the single-domain FFS LC mode has been an obstacle to its adoption for use in mobile displays.

In previous work, Oh-e reported the viewing angle dependence on the pretilt angle in the in-plane switching (IPS) LC mode [5]. In the FFS mode, because LC directors on the surface are affected by the vertical component of the fringe-field, the viewing angle dependence on the pretilt angle does not exactly match the previous result reported for the IPS mode [6]. For the FFS mode, we employed the parallel rubbing method to reduce the tilt distribution of the bulk LC layer [16]. Although the asymmetric luminance in the field-off state and in the low grayscale level could be effectively reduced, we could not achieve a perfect solution owing to the presence of a pretilt angle on the surface. In our recent work, to improve the viewing angle properties of the single-domain FFS LC mode, we proposed a promising method to realize the zero pretilt angle by employing a reactive mesogen (RM)-stabilized polystyrene (PS) layer [6]. The photo alignment technology has been mature for manufacture and can be applied to realize the zero pretilt angle as well [17]. For the clear understanding of the viewing angle results in the single-domain FFS LC mode, we need a comprehensive study of the evolving polarization state of the light passing through an LC layer with a pretilt angle.

In this study, we simulated the viewing angle properties of the light leakage and grayscale inversion depending on the pretilt angle in the single-domain FFS LC mode. To understand the tendency of the light leakage and grayscale inversion depending on the viewing direction, we performed the Póincare sphere analysis and investigated the origin of the asymmetric viewing angle problems. Finally, we fabricated a single-domain FFS cell with the zero pretilt angle and confirmed the simulation results.

To understand the pretilt angle effect on the off-axis transmittance in the single-domain FFS LC mode, we performed numerical calculations using commercial software (Techwiz LCD 3D, Sanayi System Co., Ltd.). Figure 1 shows the schematic structure and optic axes of the single- domain

FFS LC mode for the simulation in detail ( θ_k: polar angle of the viewing direction, ϕ_k: azimuthal angle of the viewing direction, and α : surface pretilt angle). The physical properties of the employed LC material are as follows: dielectric anisotropy of Δε = 7.9; elastic constants of K₁₁ = 10.2 pN, K₂₂ = 6.9 pN, and K₃₃ = 13.6 pN; and refractive indices of n_e = 1.5885 and n_o = 1.4859. We conducted the simulation for 0°, 2°, and 4° pretilt angles while keeping the other parameters unchanged. We simulated the viewing angle properties for the azimuthal angle ϕ ranging from 0° to 355° in steps of 5°, and for the polar angle θ ranging from 0° to 80° in steps of 10°. The induced voltage ranged from 0 V to 10 V, sufficient for obtaining maximum transmittance in all simulation conditions. From the simulation results, we could determine the severest viewing angle direction as θ_k = 70° and ϕ_k = 40°, regardless of the pretilt angle. This is the same result as a reported one [14]. The off-axis light leakage and the grayscale inversion were analyzed at this severest viewing angle direction and voltage conditions. In addition, to explain the simulation results in detail, we analyzed the polarization state of the light passing through the LC layer by extracting the corresponding Stokes parameters and its trace depending on the voltage on the Póincare sphere. Finally, we fabricated single-domain FFS LC cells and evaluated their viewing angle performance for different pretilt angles, and compared these results to the simulation results.

In the crossed polarizer condition, light leakage is an inevitable phenomenon in oblique view due to the effective absorption angle change. The angle between the absorption axes of the two polarizers is changed into the effective angle, φ, as follows [14]:

where ϕ_p and ϕ_a are the azimuthal absorption angles of the polarizer and the analyzer, which are 173° and 83°, respectively, in our simulation structure, as shown in Fig. 1(b), and ϕ_a − ϕ_p is π/2 in the crossed polarizer condition. Eq. (1) also implies that φ cannot be 90° if θ_k is not zero. This means that the off-axis light leakage in the field-off state occurs in the oblique viewing condition, and it is related to the origin of the grayscale inversion.

The extent of the light leakage in the oblique viewing angle varies depending on the surface pretilt angle and the positional asymmetry of the light leakage becomes more severe as the pretilt angle increases, as we reported in our previous paper [6]. We reported the viewing angle properties of the single-domain FFS LC mode in terms of voltage-dependent transmittance and contrast ratio depending on the pretilt angle of the alignment layer [6]. In that study, as the pretilt angle increased, the voltage (V_inv) that causes the maximum grayscale inversion in a viewing angle of θ_k = 70° and ϕ_k = 40° increased as well, and the viewing angle asymmetry became more pronounced. Figure 2(a) shows the simulated transmittance variation as a function of applied voltage at θ_k = 70°, ϕ_k = 40° and 130° in this study. The horizontal dashed lines correspond to the initial transmittance levels in the field-off state (T_0V) for each pretilt angle condition. For ϕ_k = 40°, the bounce in the transmittance variation becomes more severe and V_inv increases with increasing the pretilt angle, as shown in Fig. 2(a) (open symbols connected by solid lines). For ϕ_k = 130°, the bounce in the transmittance variation is not observed, and the light leakage increases with increasing the pretilt angle, as shown in Fig. 2(a) (closed symbols connected by dashed lines). The values of T_0V and transmittance at V_inv (T_inv) for the different pretilt angle conditions are summarized in Table 1.

Figures 3(a) to 3(c) show the transmittance variations in the field-off (closed circles) and V_inv (open inverted triangles) in the different pretilt angle conditions, as a function of the azimuthal viewing angle direction. In all plots, shaded areas indicate the azimuthal viewing angle range in which the grayscale inversion occurs. As shown in Fig. 3, the transmittance asymmetry in the field-off state and at V_inv, and the grayscale inversion asymmetry at V_inv become more severe with increasing the pretilt angle. The results in Fig. 2(a) and Figs. 3(a) to 3(c) are in an excellent agreement with our previous results [6].

Figure 2(b) and Figs. 3(d) to 3(f) show the results applying the optical compensation method. We used a biaxial film (n_x: 1.521, n_y: 1.519, n_z: 1.520; thickness: 138 μm) stacked on a negative C plate as a compensation film [12]. As the negative C plate, we used a triacetyl cellulose (TAC) film (n_x: 1.4793, n_y: 1.47962, n_z: 1.47890; thickness: 40 µm). As shown by the horizontal dashed lines in Fig. 2(b) and by the closed circles in Figs. 3(d) to 3(f), the optical compensation method helps to reduce the off-axis light leakage and transmittance bounce irrespective of the pretilt angle. However, the T_inv asymmetry in the azimuthal viewing angle direction in the nonzero pretilt angle condition is not eliminated, as shown by the open triangles in Figs. 3(e) and 3(f). This asymmetry cannot be eliminated even by using the optical compensation technique owing to the presence of the surface pretilt angle. To validate the pretilt angle effect on the viewing angle properties of the single-domain FFS LC mode, we need to analyze in detail using the Póincare sphere the polarization state passing through an LC layer with a nonzero pretilt angle.

4.1.1. Pretilt Angle = 0°

Figure 4 shows the Póincare spheres for the optical analysis of the viewing angle properties depending on the pretilt angle, the viewing angle direction, and the applied voltage. P, A, and T indicate the transmission axes of the polarizer, the analyzer, and the polarization state of the incident light positioned before the analyzer in the normal viewing condition, respectively. In the field-off state, there is no light leakage in the normal viewing condition because T and A are always located on opposite sides of each other. However, in the oblique viewing directions of ϕ_k = 40° and 220°, which are indistinguishable between the head and tail of the LC director in the zero pretilt angle condition, the effective transmission axes of both P and A move toward the horizontal direction. Therefore, P and A move to the new transmission axes of P_t and A_t, respectively, and the angle between the absorption axes of the two polarizers is changed into the effective angle, φ, as shown in Eq. (1). For the oblique viewing directions of ϕ_k = 40° and 220°, P_t does not coincide with the analyzer absorption axis A_a, and from Eq. (1), the effective polarizer angle φ is calculated as 105° at θ_k = 70°, which produces the light leakage in the field-off state, as shown in Fig. 4(a).

Moreover, when the light propagates into a uniaxial medium such as an LC layer in oblique incidence, the phase retardation is induced by the azimuthal angle mismatch between P_t and the LC optic axis. In general, when the optic axis of a uniaxial medium is oriented at the tilting angle α and the azimuthal angle ϕ_n, the phase retardation of the uniaxial medium in oblique incidence of light, can be expressed as [18]:

where ε _xz=(n_e²-n_o²)sinαcosαcos(ϕ_n−ϕ_k) and ε_xz=n_o²+(n_e²−n_o²)sin²α. As shown in Eq. (2), the phase retardation depends on the orientation of the optic axes α and ϕ_n. When the pretilt angle is zero in the LC layer, i.e., when the optic axis α becomes 0° in the initial state, the slow axes Ψ_ϕ=40° and Ψ_ϕ=220° of the oblique viewing directions of θ_k = 70° at ϕ_k = 40° and 220° are 90.5°, which coincides with A_a, and the phase retardations are 1.27π in both cases, when calculated by using Eq. (2). The consequent Stokes parameters (S₁, S₂, S₃) are (−0.8437, −0.3757, −0.3834) for the polarization state of T_{ϕ=40°, ϕ=220°} in Fig. 4(b). As a result, after passing the LC layer, the extent of light leakage is the same for the oblique viewing directions of ϕ_k = 40° and ϕ_k = 220° because the deviation of polarization state T from P_t is also the same for ϕ_k = 40° and ϕ_k = 220°, as shown in Fig. 4(b).

4.1.2. Pretilt Angle ≠ 0°

In the field-off state, the pretilt angle affects the phase retardation of the incident light, as shown in Eq. (2), as well as the incident light angle, owing to the viewing direction dependency on the birefringence of the LC director. Therefore, the slow axes and the phase retardation of the incident light are formed a little differently between the oblique viewing directions of ϕ_k = 40°, which is observed from the head position of the LC director, and ϕ_k = 220°, which is observed from the tail position of the LC director. The slow axes Ψ_ϕ=40° and Ψ _ϕ=220° of the oblique viewing directions of θ_k = 70° at ϕ_k = 40° and ϕ_k = 220° are formed in the counterclockwise position based on P_t in the Póincare sphere, and are calculated as Ψ_ϕ=40° = 91.6° (which is larger than that of the zero pretilt angle owing to the head-up viewing effect of the LC director) and Ψ _ϕ=220° = 89.3° (which is smaller than that of the zero pretilt angle owing to the tail-down viewing effect of the LC director) when the pretilt angle is 2°. Based on Eq. (2), the phase retardations for each case are calculated as 1.23π and 1.32π, and these values are different from that of the zero pretilt angle condition owing to the dependence of the viewing angle direction on the birefringence of the LC director. Hence, incident light passing through the LC layer exhibits different polarization states between T_ϕ=40° and T_ϕ=220°, as shown in Fig. 4(c), with the Stokes parameters (−0.8005, −0.4821, −0.3560) for the polarization state of T_ϕ=40° and (−0.8772, −0.2704, −0.3967) for the polarization state of T_ϕ=220° in Fig. 4(c), respectively. The extent of the light leakage is determined by the variation in the distance of T_ϕ=40° or T_ϕ=220° from A_a, respectively. Therefore, the light leakage in the viewing direction of ϕ_k = 40° is larger than that in the viewing direction of ϕ_k = 220° because T_ϕ=40° is farther from A_a than T_ϕ=220° in the non-zero pretilt angle. This result is in a good agreement with the simulation result in Fig. 3(b), which shows larger light leakage at the head position of the LC director.

Based on P_t, both the slow axes Ψ_ϕ=40° and Ψ_ϕ=220° are formed in the opposite direction to the rotating direction of the LC easy axis, which is driven by the applied voltage and is shown by thick blue arrows in Figs. 4(b) and 4(c). Therefore, when the applied voltage increases, both the slow axes Ψ_ϕ=40° and Ψ_ϕ=220° rotate in the clockwise direction. Consequently, T_ϕ=40° and T_ϕ=220° become closer to and then pass A_a at a voltage larger than V_inv. This means that the grayscale inversion occurs in both directions as the applied voltage increases. Besides, the grayscale inversion and V_inv in the viewing direction of ϕ_k = 40° are larger than those in the viewing direction of ϕ_k = 220° because T_ϕ=40° has to rotate farther from A_a than T_ϕ=220° owing to the difference between the positions of Ψ_ϕ=40° and Ψ_ϕ=220°, as shown in Fig. 4(c). These are in a good agreement with the simulation result in Fig. 2(a) and Figs. 3(a) to 3(c), which show a more severe grayscale inversion asymmetry and higher V_inv with increasing the pretilt angle.

As a result, the asymmetric light leakage and the transmittance distribution are caused by the presence of the LC pretilt angle in the viewing directions of ϕ_k = 40° and 220°, and the grayscale inversion is particularly more severe in the viewing direction of ϕ_k = 40°.

4.2.1. Pretilt Angle = 0°

In the oblique viewing directions of ϕ_k = 130° and 310°, P_t is also not coincident with A_a, as shown in Fig. 4(d), but is located in the opposite direction with respect to A_a compared to the orientation in Fig. 4(a), which corresponds to the viewing directions of ϕ_k = 40° and 220°. This follows because the effective polarizer angle φ is calculated as 75° at θ_k = 70°, based on the condition in Eq. (1). Therefore, this mismatch between P_t and A_a also induces the light leakage in the field-off state, as in the case of ϕ_k = 40° and 220°.

In addition, when the pretilt angle is zero in the LC layer, i.e., the optic axis α becomes 0° in the initial state, the slow axes Ψ_ϕ=130° and Ψ_ϕ=310° of the oblique viewing directions of θ_k = 70° at ϕ_k = 130° and 310° are 75.5°, which is coincident with A_a, and the phase retardations are 1.32π in both cases, as calculated from Eq. (2). The consequent Stokes parameters are (−0.6004, 0.6771, 0.4255) for the polarization state of T_{ϕ=130°, ϕ=310°} in Fig. 4(e). As a result, after passing through the LC layer, the extent of light leakage is almost the same for the θ_k = 70° oblique viewing directions of ϕ_k = 130° and ϕ_k = 310°, which is similar to the cases involving the oblique viewing directions of ϕ_k = 40° and ϕ_k = 220°, as shown in Fig. 4(b).

4.2.2. Pretilt Angle ≠ 0°

The analysis for the non-zero pretilt angle in the viewing directions of ϕ_k = 130° and 310° in the field-off state is similar to the case in Fig. 4(c), which corresponds to the viewing directions of ϕ_k = 40° and 220° in terms of inducing variations in the slow axes. However, the slow axes Ψ_ϕ=130° and Ψ_ϕ=310° of the oblique viewing directions of θ_k = 70° at ϕ_k = 130° and ϕ_k = 310° are formed in the clockwise position based on P_t in the Póincare sphere, and are calculated as Ψ_ϕ=130° = 73.8° (which is smaller than that of the zero pretilt angle, owing to the head-up viewing effect of the LC director) and Ψ_ϕ=310° = 76.3° (which is larger than that of the zero pretilt angle, owing to the tail-down viewing effect of the LC director) when the pretilt angle is 2°. The phase retardations of each case are calculated as 1.28π and 1.36π from Eq. (2), and these values are different from that of the zero pretilt angle condition, owing to the viewing direction dependency on the birefringence of the LC director. Hence, for each case, the incident light passing through the LC layer exhibits different polarization states T_ϕ=130° and T_ϕ=310°, as shown in Fig. 4(f), and the consequent Stokes parameters are (−0.5128, 0.7531, 0.4121) for the polarization state of T_ϕ=130° and (−0.6777, 0.5995, 0.4258) for the polarization state of T_ϕ=310° in Fig. 4(f). In addition, the light leakage in the viewing direction of ϕ_k = 130° is larger than that in the viewing direction of ϕ_k= 310°, because the deviation of T_ϕ=130° from A_a is farther than that of T_ϕ=310°. This is in a good agreement with the simulation result in Fig. 3(b), which shows larger light leakage at the head position of the LC director.

Furthermore, the grayscale inversion does not occur because, based on P_t, the induced slow axes Ψ_ϕ=130° and Ψ_ϕ=310° are formed in the same direction as the rotating direction of the LC easy axis, which is driven by the applied voltage and is shown by thick blue arrows in Figs. 4(e) and 4(f). Therefore, T_ϕ=130° and T_ϕ=310° followed by Ψ_ϕ=130° and Ψ_ϕ=310° are already formed in the clockwise direction of A_a in the Póincare sphere, and do not pass A_a when the applied voltage increases, unlike T_ϕ=40° and T_ϕ=220° in Figs. 4(b) and 4(c). As a result, the only difference between the viewing directions between ϕ_k = 130° and ϕ_k = 310° is the extent of light leakage and the grayscale inversion does not occur at both the viewing directions.

Figure 5 shows the Póincare sphere analysis of the single-domain FFS LC mode using a biaxial optical compensation film. The polarization state of the light passing through the biaxial compensation film, marked by the open circle in Fig. 5, is close to the analyzer absorption axis A_a and the LC slow axis Ψ [6]. Therefore, the polarization state passing through the LC layer, denoted by T, does not strongly deviate from A_a [6, 12]. As a result, because the extent of light leakage is determined by the distance of T from A_a, the light leakage can be dramatically reduced in the oblique viewing direction, as shown by the horizontal dashed lines in Fig. 2(b) and closed circles in Figs. 3(d) to 3(f). In addition, because the polarization state in the field-off state is close to A_a, the transmittance bounce, which occurs at the low grayscale level, is also reduced, as shown by open symbols in Fig. 2(b). However, in the nonzero pretilt angle condition, because the slow axes and the phase retardations of the incident light are slightly different between the oblique viewing directions of ϕ_k = 40° in Fig. 5(b) (or 130° in Fig. 5(d)), which is observed from the head position of the LC director, and ϕ_k = 220° in Fig. 5(b) (or 310° in Fig. 5(d)), which is observed from the tail position of the LC director, the incident light passing through the LC layer also exhibits different polarization states as shown in Figs. 5(b) and 5(d). As the applied voltage increases, the slow axes rotate as shown by the thick blue arrows in Figs. 5(b) and 5(d), and the difference between two polarization states becomes obvious. This leads to the transmission difference in the grayscale level and to the asymmetric viewing angle properties when comparing between ϕ_k = 40° and 220°, ϕ_k = 130° and 310°. Although the light leakage and grayscale inversion can be improved by using the compensation method, the viewing angle asymmetry in the grayscale level in the nonzero pretilt angle condition cannot be eliminated by using only the optical compensation method.

From the Póincare sphere analysis, we confirmed the surface pretilt angle-related difference between the viewing angle properties in the single-domain FFS LC mode. In addition, even though a compensation method is applied, we verified that the surface LC pretilt angle has to be zero for characterizing the symmetrical transmittance in the field-off state and at the low grayscale level as well as for minimizing the grayscale inversion.

To verify the simulation result related to the pretilt angle effect, we fabricated single-domain FFS cells for two types of pretilt angle conditions and evaluated the viewing angle performance of the fabricated cells. For the nonzero pretilt angle condition, we used a conventional polyimide layer with the pretilt angle of 2°. To obtain the zero pretilt angle condition, we employed a polystyrene layer stabilized with UV curable reactive mesogen (RM). These two types of single-domain FFS cells were fabricated as described previously [6].

To demonstrate obviously the dependence on the oblique viewing angle direction in the measured result, we cross-sectioned the viewing planes along the diagonal direction of ϕ_k = 40° through 220° (A₁−A₂) and ϕ_k = 130° through 310° (B₁−B₂). In Fig. 6(a), for the A₁−A₂ direction, the light leakage increased at T_0V and the grayscale inversion became severe at T_inv for the oblique viewing angle direction, irrespective of the alignment layers. In particular, for the single-domain FFS LC cell with the PI layer, the asymmetric transmittance distribution and more serious grayscale inversion in the direction of ϕ_k = 40° appeared as in nonzero pretilt angle simulations that were described in the previous section. Moreover, as shown in Fig. 6(b), for the B₁−B₂ direction, the asymmetric transmittance distribution of the single-domain FFS LC cell with the PI layer was maintained at V_inv owing to the initial asymmetry of light leakage at 0 V; however, there was no grayscale inversion, irrespective of the alignment layers. These results on the cross-sectioned viewing planes of A₁−A₂ and B₁−B₂ were in a good agreement with our simulation results.

Figure 6(c) shows the contrast ratio obtained by dividing T_inv by T_0V as a function of the polar viewing angle on the cross-sectioned viewing planes of A₁−A₂ and B₁−B₂. The value of CR (ϕ = 310°)/CR (ϕ = 130°) was maintained at around 1, irrespective of the polar viewing angle and the pretilt angle (that is, the alignment layers). However, the value of CR(ϕ = 220°)/CR(ϕ = 40°) for the single-domain FFS LC cell with the PI layer increased with increasing the polar viewing angle, while that of the single-domain FFS LC cell with the RM-stabilized PS layer was maintained at around 1. This implies that the asymmetric transmittance in the oblique viewing direction is seriously affected by the pretilt angle, especially for the azimuthal viewing directions of ϕ = 40° and ϕ = 220° in the single- domain FFS LC mode.

In the single-domain FFS LC mode, the phase retardation and the effective angle influenced by the pretilt angle affect the light leakage and the grayscale inversion in the oblique view. In this paper, we used the Póincare sphere for conducting a detailed analysis of the asymmetric viewing angle properties of the light leakage and the grayscale inversion in relation to the pretilt angle in the single-domain FFS LC mode. The light leakage and the grayscale inversion in relation to the applied voltage could be estimated with the positions of the analyzer absorption axis (A_a) and the polarization state of the light passing through the LC layer (T) on the Póincare sphere, and their positional comparison.

Although the light leakage was dramatically reduced by using the optical compensation method irrespective of the pretilt angle, the asymmetry in grayscale level of the nonzero pretilt angle condition could not be eliminated even by using the optical compensation method. To enhance the asymmetric viewing angle properties in the single-domain FFS LC mode, the zero pretilt angle condition is essential, and we validated this conclusion by measuring the viewing angle properties of a fabricated single-domain FFS cell with zero pretilt angle

The analysis using the Póincare sphere employed in this study is very helpful for understanding the influence of the pretilt angle on the polarization state of the light passing through LC layers, and is likely to become an important tool for estimating the optical performance of LC devices.