Designs of wide-view and broadband circular polarizers

: A novel methodology for designing wide view circular polarizers is proposed. Both single wavelength and broadband wide-view circular polarizers are discussed. Over the ±85 o viewing cone, the light leakage from the crossed circular polarizers is less than 2.87×10 -4 using the proposed single wavelength circular polarizers ( λ =550 nm) and less than 1.7×10 -3 using the proposed broadband circular polarizer ( λ =450~650 nm). An example of using the designed broadband, wide-view circular polarizers for enhancing the optical efficiency of a direct-view liquid crystal display is elucidated.


Introduction
Circular polarizer is an important optical component with many useful applications, such as optical communications, optical remote sensors, and liquid crystal displays (LCDs) [1][2][3][4]. Two methods have been commonly used to generate a circularly polarized light: Bragg reflection using a cholesteric liquid crystal (CLC) film and a linear polarizer laminated with a quarter-wave film. In the former approach, a right-handed CLC film would reflect the righthanded circularly polarized light and transmit the left-handed component. A drawback is that blue shift occurs at oblique angles [1,5]. In the latter approach a quarter-wave film is laminated to a linear polarizer [6]. In the normal incidence, a very good circular polarization is produced. However, at oblique angles the produced state of polarization becomes elliptical resulting in light leakage through the crossed circular polarizers. Wide-view circular polarizers using biaxial retardation films have been proposed for improving the light efficiency of LCDs [2][3][4]. However, the reported contrast ratio is limited to ~10:1 at 60 o viewing cone because of the still large light leakage. For direct-view LCDs, broad bandwidth is as important as wide viewing angle [1][2][3][4][7][8][9][10][11].
Phase compensation methods have been widely applied in LCDs for reducing the dark state light leakage and thus increasing the contrast ratio at wide viewing angles [2][3][4][7][8][9][10][11]. To obtain a wide-view circular polarizer, a straightforward approach is to combine a wide-view linear polarizer [7][8][9][10][11] with a wide-view quarter-wave film [2][3][4]. However, this approach is difficult to obtain a pure circular polarization state, especially at a large incident angle. In this paper, we apply the phase compensation methods to develop wide-view circular polarizers for both single wavelength and broadband white light. The produced state of polarization is very close to the ideal circular state of polarization over a wide range of incident angles. Over the entire ±85 o viewing cone, after reducing the air-interface surface reflection, the light leakage from the crossed single-wavelength circular polarizers is <2.87×10 -4 at λ=550 nm and <1.7×10 -3 over the 450~650 nm spectrum for the crossed broadband circular polarizers. This device is particularly useful for enhancing the optical efficiency of direct-view LCDs.

Stoke parameters
The state of polarization can be represented by Stokes parameters (S 1 , S 2 , and S 3 ) and plotted on Poincaré sphere [6] after the parallel and perpendicular components of the electric field are solved using the 4-by-4 matrix method [12]. If the state of polarization is represented by vector P = (S 1 , S 2 , S 3 ), then the polarization difference between two states of polarization P (1) and P (2) can be described by where S 1_(1) , S 2_(1) , S 3_(1) , S 1_(2) , S 2_ (2) , and S 3_(2) are the Stokes parameters of P (1) and P (2) , respectively. P (LCP) = (0, 0, 1) denotes the left-handed circular polarization and P (RCP) = (0, 0, −1) gives the right-handed circular polarization. S 3 equals to zero for the linear polarization and |S 3 | is neither zero nor one for the elliptical polarization. Since S 1 , S 2 and S 3 satisfy the relationship that 1 2 3 (1) can be simplified so that the polarization difference between P (X) and P (RCP) is related to the S 3 of P (X) by where P (X) = (S 1_(X) , S 2_(X) , S 3_(X) ). Once S 3_(X) descents to −1, ΔP (X)−(RCP) approaches zero and P (X) becomes P (RCP) . In this paper, the linear polarizer is modeled as a lossy uniaxial material. The employed refractive indices of the polarizer, positive birefringence uniaxial A-plate and C-plate, and negative birefringence uniaxial A-plate and C-plate are as follows: n e_pol = 1.5 + i×3.251×10 -3 , n o_pol = 1.5 + i×2.86×10 -5 , n e_p_A-plate = 1. 5124, n o_p_A-plate = 1.5089, n e_p_C-plate = 1.5124, n o_p_Cplate = 1.5089 , n e_n_A-plate = 1.5089, n o_n_A-plate = 1. 5124, n e_n_C-plate = 1.5089, and n o_n_C-plate = 1.5124. The thickness of the polarizer is 210 μm. The A-plate and C-plate with negative dΔn can be realized by negative birefringence A-plate and C-plate. We assume the color dispersions of linear polarizer, A-plate, and C-plate are negligible. On both sides of the absorptive polarizer, the protective Tri-Acetyl-Cellulose (TAC) films exhibit a small birefringence and act as negative birefringence C-plates. The phase change due to the TAC film can be minimized if we laminate a positive birefringence C-plate to the exit protective film. The phase retardation of this C-plate compensates for the adjacent protective film so that the C-plate effect of the linear polarizer is negligible.

Single-wavelength wide-view circular polarizers
A conventional circular polarizer consists of a linear polarizer and a quarter-wave plate. The quarter-wave plate is laminated on the light emerging side of the linear polarizer and its slow axis is oriented at 45 o with respect to the absorption direction of the polarizer. At normal incidence, the light emerging from the linear polarizer sustains π/2 phase change from the quarter-wave plate so that it becomes circularly polarized light. However, at oblique angles, the phase change contributed by the λ/4 plate is different from π/2 [2][3][4] so that the produced polarization state becomes elliptical as Fig. 1(a) illustrates. If a pair of crossed circular polarizers is constructed as Fig. 2(a) depicts, the polarizer and the first quarter-wave plate form a circular polarizer, and the analyzer and the second quarterwave plate form a crossed circular polarizer. Figure 2(b) plots the iso-transmittance contour of light leakage. Although the light leakage is almost zero at normal viewing direction, it increases to 0.098 at θ ι = 85 o because of the resulted elliptical polarization. The light leakage is the strongest at near bisectors (φ ι = 40 o , 130 o , 220 o , and 320 o ) since the produced S 3 peaks at these angles, as depicted in Fig. 1(b). During simulations, an ideal anti-reflection (AR) film is assumed in order to reduce the interference of the air-polarizer surface reflection. The ten-layer anti-reflection film is coated on the air interface of both polarizers. This AR film is designed using genetic algorithm [13] and the gradient refractive indices profile is illustrated in Fig. 3(a). The origin represents the air-AR interface. The transmittance of this ten-layer AR film is greater than 0.97 over the ±85 o incident cone for λ= 450 ~ 650 nm as Fig. 3(b) illustrates. To produce circular state of polarization at a large incidence angle, we laminate one uniaxial C-plate to the quarter-wave plate as Fig. 4 drafts. This positive birefringence C-plate contributes phase retardation at oblique angles [1] so that the produced polarization is closer to an ideal circular polarization, while the normal incidence angle performance of conventional circular polarizer is not compromised.   To find the dΔn of this C-plate for minimizing S 3 over the ±85 o viewing cone, Fig. 5(b) illustrates that the produced S 3 decreases to its minimum when the dΔn of the C-plate is gradually increased from 0 to 59.9 nm. Further increasing the dΔn of the C-plate increases the produced S 3 . By exhaustive search we can find when the dΔn of the C-plate equals 59.90 nm, the S 3 of the produced state of polarization is less than −0.952 (ΔP (λ/4+1C)−(RCP) ≤ 0.31) over the entire ±85 o viewing cone as Fig. 6(a) shows. Due to this additional C-plate, the produced S 3 remains −1 at normal incidence and slowly increases to −0.952 as the viewing angle increases to 85 o , which is significantly reduced in contrast to a conventional circular polarizer. This decreases the light leakage of the crossed circular polarizers to 0.027 over the ±85 o viewing cone, as demonstrates in Fig. 6 Since the C-plate does not change the S 1 of the polarization state [8], the produced S 3 remains as high as −0.952 at 85 o viewing angle and cannot be further reduced as we observed in Figs. 5 and 6. Whereas all of the three Stokes parameters are changed inside an A-plate [8], to further improve the viewing angle performance we could laminate an extra A-plate to the circular polarizer shown in Fig. 4 and obtain a new design shown in Fig. 7. The C-plate is laminated between two A-plates. Over wide incident angles, the combination of these A-plates and C-plate is expect to be equivalent to the quarter-wave plate in the conventional circular polarizer at normal incidence. Due to the presence of the additional A-plate, the azimuthal angle and the dΔn of the quarter-wave plate must be redesigned so that the produced state of polarization remains circular at normal incidence. The azimuthal angles of both A-plates as well as the dΔn of all A-plates and C-plate are the subject of design. Using genetic algorithm [13], by minimizing the cost function where S 3_(2A+1C) is the S 3 of the produced state of polarization P (2Α+1C) , we obtain the parameters of the phase retardation films. For this design, the azimuthal angles of A-plates are:  Inside this circular polarizer, all of the S 1 , S 2 and S 3 are modified so that the compensations between the retardation films are further improved as Fig. 8(a) demonstrates. The two A-plates not only reduce S 3 to −1 but also reduce the transmitted S 1 and S 2 to zero. On the other hand, the C-plate tempers the transmitted state of polarization to further reduce the viewing angle sensitivity. Therefore, the produced S 3 is only slightly increased to −0.991 when the viewing angle increases to 85 o as depicted in Fig. 8(b). This is equivalent to having the polarization difference ΔP (2A+1C)−(RCP) less than 0.134 over the ±85 o viewing cone. The produced S 3 remains at −1 at normal angle as in the conventional right-handed circular polarizer. For the left-handed circular polarizer using the configuration shown in Fig. 7, the dΔn of all A-plates and C-plate are not changed but the azimuthal angles of the A-plates are the negative of their counterparts in the right-handed circular polarizer. From above discussions, an extra phase retardation film gives an extra degree of freedom to improve the viewing angle of a circular polarizer. By laminating an additional A-plate and a C-plate to the above circular polarizer as Fig. 9 depicts, the polarization difference between the produced polarization and the desired circular polarization can be further reduced. In this configuration, A-plates are interlaced with C-plates. By minimizing the cost over the ±85 o viewing cone using genetic algorithm [13], we obtain the design parameters of A-plates and C-plates, where S 3_(3A+2C) is the S 3 of the produced state of polarization P (3A+2C) .
From this design, the azimuthal angles of A-plates are: φ ne_A_1st = 78.55 o , φ ne_A_2nd = −28.71 o , and φ ne_A_3rd = 42.46 o ; the dΔn of all retardation films are: dΔn _A_1st = 75.69 nm, dΔn _A_2nd = 24.30 nm, dΔn _A_3rd = 128.96 nm, dΔn _C_1st = 106.56 nm, and dΔn _C_2nd = −21.08 nm. The additional A-plate and C-plate significantly reduce the viewing angle sensitivity of the circular polarizer because of the extra compensations between retardation films. As illustrated in Fig. 10(a), all of the S 1 , S 2 and S 3 are subtly modified in the retardation films so that the produced S 3 remains less than −0.999 (ΔP (3A+2C)−(RCP) ≤ 0.045) over the entire ±85 o viewing cone, which can be seen in Fig. 10(b). Variation in the produced S 3 is further reduced so that the produced polarization is nearly circular at any incident angle within ±85 o . figures, the configuration of the circular polarizer is shown in Fig. 9. λ=550 nm.
Since the produced polarization approaches the ideal circular polarization, the light leakage of the crossed circular polarizers is less than 2.87×10 -4 over the ±85 o viewing cone as Fig. 11(a) shows. Although the light leakage is more pronounced at φ ι ~ 10 o , 100 o , 190 o , and 280 o , it is still less than 1.72×10 -4 at other azimuthal angles when the incident angle is within ±85 o . As compared to the case of using conventional circular polarizer, as shown in Fig. 2(b), our results are significantly improved despite the increased cost. Figure 11(b) depicts the configuration of the crossed circular polarizers. The polarizer and the first three A-plates together with the first two C-plates form a wide-view right-handed circular polarizer. The analyzer and the last three A-plates together with the last two C-plates form a second circular polarizer crossed to the first one. The arrangement of the A-plates and C-plates are reversed for the crossed polarizers so that the state of polarization emerging from the last A-plate is linear along the absorption direction of the analyzer, thus the light leakage is small. The ideal anti-reflection film in Fig. 3(a) is assumed and coated on the air interface of both polarizers. For a left-handed circular polarizer using the configuration in Fig. 9, the dΔn of all A-plates and C-plates are not changed but the azimuthal angles of the A-plates are the negative of their counterparts in the right-handed circular polarizer.

Broadband wide-view circular polarizers
In the above designs, the produced states of polarization are very close to the ideal circular polarization over a wide viewing cone, however, only at a single wavelength. As the incident wavelength deviates from the designed one, the phase retardations of the A-plates and Cplates will walk off from the designed values. As a result, the produced polarization state is no longer circular as Fig. 12 demonstrates. In Fig. 12, the conventional broadband circular polarizer (red line) is indeed quite insensitive to the wavelength in the 450-650 nm spectral range, but only at normal incident angle. All the other three designs (black, blue and green curves) based on a single wavelength are rather sensitive to the wavelength. In this section we will focus on the designs of broadband and wide-view circular polarizers. A commonly used broadband circular polarizer is comprised of laminating a half-wave plate between the linear polarizer and the quarter-wave plate as illustrated in Fig. 13(a) [14]. When the azimuthal angles of the half-wave plate and the quarter-wave plate satisfy the following relationship the produced state of polarization is very close to the ideal circular polarization over a broad spectrum at normal incidence as observed from Fig. 12. However, at oblique angles, the relationship in Eq. (4) is no longer satisfied on the wave plane and the phase retardations of the half-wave plate and the quarter-wave plate are changed. Thus, the produced polarization is no longer circular and varies significantly with the incident spectrum [2][3][4]. As Fig. 14 shows, the produced S 3 from a conventional broadband circular polarizer (black curve) is larger than −0.5 at 85 o incident angle over the spectrum of 450 ~ 650 nm. The red line in Fig. 14 shows the produced S 3 from a broadband wide-view circular polarizer that we are going to discuss.  The above designs of single-wavelength circular polarizer show that replacing the quarter-wave plate in the conventional circular polarizer with the combination of A-plates and C-plates significantly reduces the viewing angle sensitivity of the produced state of polarization. Likewise, if both half-wave plate and quarter-wave plate in the conventional broadband circular polarizer can be replaced by multi-layer equivalent plates, then the resulted circular polarizer would be broadband and wide-view. In this case, over wide viewing angle and broad spectrum, the multi-layer equivalent plates should produce similar states of polarization as their single-layer counterparts do at normal incidence.
To design the multi-layer equivalent half-wave plate, we first derive the states of polarization P λ/2 emerging from the half-wave plate at θ ι = 0 o and φ ι = 0 o ~ 360 o for λ = 450 ~ 650 nm. Then we use the combination of two A-plates and one C-plate to replace the half-wave plate as Fig. 13 we find the phase retardation film parameters, where ΔP (2A+1C_λ/2)−( λ/2) is the polarization difference between the state of polarization P (2A+1C_λ/2) emerging from the equivalent λ/2 plate and the state of polarization P λ/2 emerging from the single-layer half-wave plate. With the multi-layer equivalent half-wave plate, the quarter-wave plate in the conventional broadband circular polarizer is replaced by the combination of three A-plates and two C-plates as Fig. 13(b)  Unlike above single wavelength circular polarizers, the variation in the produced state of polarization over a broad spectrum is similar to the conventional broadband circular polarizer at normal incidence angle. This can be seen in Fig. 14, in which the red line shows that, over the ±85 o viewing cone, the produced S 3 is less than −0.963 at λ = 450 nm and decreases to −0.995 at λ = 530 ~ 650 nm. The wavelength sensitivity is reduced by satisfying the relationship in Eq. (4) and the viewing angle sensitivity is reduced by the above multi-layer equivalent plates. Thus the light leakage of the crossed circular polarizers is suppressed below 1.7×10 -3 over the ±85 o viewing cone within the 450~650 nm spectral range, as Fig. 15 depicts. Fig. 15. The calculated maximum light leakage from three-types crossed circular polarizers over the ±85 o viewing cone as a function of wavelength. The ten-layer anti-reflection film is assumed.
In Fig. 15, with the broadband wide-view circular polarizer, the light leakage of the crossed circular polarizers is not only kept below 1.7×10 -3 in the visual spectrum but also less than 3.79×10 -4 at λ = 550 nm. This is preferred in the liquid crystal displays since human visual system is more sensitive to the green light so that the green color requires a higher contrast ratio. In contrast, using single wavelength circular polarizers the light leakage increases dramatically when the incident spectrum deviates from the designed wavelength.
To form the crossed broadband circular polarizer, the polarizer and the first five A-plates together with the first three C-plates compose a right-handed circular polarizer as Fig. 13(b) sketches. The analyzer and the other five A-plates together with the other three C-plates form the second circular polarizer. The arrangement of the A-plates and C-plates laminating to the analyzer are in reverse order and the azimuthal angles of the A-plates are at ninety degree with respect to their counter parts laminating to the polarizer. The ideal anti-reflection film in Fig. 3(a) is assumed and coated on the air interface of both polarizers. For a left-handed circular polarizer using the configuration in Fig. 13(b), the dΔn of all A-plates and C-plate are not changed but the azimuthal angles of the A-plates are the negative of their counterparts in the right-handed circular polarizer.

Applications
A pair of crossed polarizers is of key components in many transmissive mode LCDs [1]. If an LC cell is laminated between two crossed linear polarizers, to achieve maximum transmittance in the bright state the LC directors should be reoriented to the bisectors of the crossed linear polarizers [1- 4,15]. Vertical alignment (VA) has been used in many LCDs because of its excellent contrast ratio. In a VA-LCD, in order to have uniform image quality over all the azimuthal angles, four domains are formed along the bisectors of the crossed linear polarizers [15,16]. However, due to the continuity the LC directors twist continuously from domain to domain so that the boundary areas are formed between domains [1-4, 15]. These boundary areas become dark areas under crossed linear polarizers so that the transmittance of the whole pixel is reduced. Nevertheless, under crossed circular polarizers, the transmittance of LCD only depends on the phase retardation (δ) of the LC layer: Hence, the azimuthal angles of the LC directors are not necessary to be at the bisectors. As a result, the use of circular polarizers greatly enhances the bright state transmittance [2][3][4].
However, the light leakage from the crossed conventional circular polarizer is large at wide viewing angles so that the contrast ratio of VA-LCDs would be low [2][3][4]. To improve the light efficiency without sacrificing the contrast ratio at wide viewing angles, we can apply above designed wide-view circular polarizers to the multi-domain VA-LCDs. Figure 16  On each side of LC layer, five A-plates and three C-plates are laminated to the adjacent linear polarizer to form a broadband wide-view circular polarizer as illustrated in Fig. 13(b). The arrangement of the A-plates and C-plates are reversed on two sides and the azimuthal angles of the A-plates are at ninety degree with respect to their counter parts on the other side. Since the vertical alignment LC layer is not considered in the design of above-mentioned wide-view circular polarizers, to compensate the phase retardation of the LC layer in the dark state, two C-plates with equal thickness are laminated on both sides of the LC layer. The summation of the phase retardations of these two C-plates is the negative of the phase retardation of the LC layer. We should emphasize that, in the dark state the light at the center of the LC layer is circularly polarized at wide viewing angles. We use the finite difference method to simulate the bright state LC director distributions [17,18] and then use the 4-by-4 matrix method [12]  nm. Here the color dispersion is assumed to be weak and not considered. The thickness of the polarizer is 210 μm and LC cell gap is 4 μm. We also assume the backlight is uniform within the ±85 o viewing cone. The color filters are not considered during calculations. To reduce airpolarizer surface reflection, an ideal anti-reflection film in Fig. 3(a) is assumed and coated on the air interface of both polarizers. Figure 17 depicts the optical characteristics of this LCD at λ = 450, 550 and 650 nm. The maximum transmittance is higher than 0.34 (c.f. maximum transmittance is 0.37) at normal viewing angle for the green and blue light, but for red it is ~0.30. Over the entire ±85 o viewing cone, the minimum bright state transmittance remains ~68% and ~90% of the maximum transmittance for the green and red light, respectively. Further more, in all cases the transmittance is uniform over all the azimuth angles. Among these three colors, the green light has the highest contrast ratio, which is greater than 420:1 over the ±85 o viewing cone. Although the contrast ratio is lower for the red light, it is still higher than 115:1 over all the viewing angles. By using the proposed broadband wide-view circular polarizer, the contrast ratio of the multi-domain VA-LCD is greatly improved while the high light efficiency is maintained, and furthermore, high angular uniformity is achieved.
In a real display panel, the actual contrast ratio could be lowered because the abovementioned ideal parameters may not be controlled precisely. Moreover, the lower extinction ratio of linear polarizer, imperfect LC alignment, variation and non-uniformity of the compensation film thickness, color dispersions of optical components, as well as the stress birefringence from films and substrates could also reduce the contrast ratio. Other than the above reasons, the actual angular brightness uniformity could also be lowered because of the non-ideal anti-reflection film. At the same time, some LC directors around the domains' boundaries are not reoriented in the bright state because of the discontinuities between LC directors in different domains [2][3][4]15]. This decreases the actual bright state transmittance. Color filters in a real display panel further reduce the actual bright state transmittance.

Discussion
Design tolerance is an important concern for display manufacturing. For the design of a single-wavelength circular polarizer shown in Fig. 9, we calculate the maximum light leakage of the crossed circular polarizers over the ±85 o viewing cone if the dΔn of the A-plates or Cplates varies by ±5%. As depicted in Fig. 18(a), while the light leakage is insensitive to the errors in the second C-plate, a −5% error in the first C-plate increases the light leakage to 1.43×10 -3 . Figure 18(b) depicts the maximum light leakage if the orientations of A-plates vary by ±5%. The light leakage rises to 1.61×10 -3 with a +5% error in the first A-plate. However, the light leakage is almost invariant when the orientation of the second A-plate varies by ±5%. Thus, for this design, the accuracy in the first A-plate and the first C-plate are more critical. For the broadband circular polarizer shown in Fig. 13(b), we calculate the maximum light leakage of the crossed circular polarizers if the dΔn of the A-plates or C-plates varies by ±5%. Results are plotted in Fig. 19(a). Although the light leakage is increased to 1.21×10 -3 with a +5% error in the first two A-plates and increased to 1.49×10 -3 with a −5% error in the first Cplate, it is less than 5.58×10 -4 in all other cases. Figure 19(b) shows the maximum light leakage if the orientations of the A-plates are deviated by ±5%. Although a +5% error in the first A-plate increases the light leakage to 1.49×10 -3 , the light leakage is almost invariant with ±5% errors in the second and the third A-plats. The first two A-plates together with the first C-plate compose the equivalent half-wave plate and the other A-plates and C-plates compose the equivalent quarter-wave plate. This circular polarizer is more sensitive to the errors in the equivalent half-wave plate but insensitive to the errors in the equivalent quarter-wave plate. Comparing Figs. 18 and 19, in the least favorable case, the maximum light leakage is still less than 1.61×10 -3 and 1.49×10 -3 for the single-wavelength and broadband circular polarizers, respectively.

Conclusion
We demonstrate a novel methodology for designing wide-view circular polarizers. Both single wavelength and broadband circular polarizers are discussed. We use phase compensation techniques to reduce the difference between the produced state of polarization and the desired circular state of polarization over a wide range of viewing angle. The phase retardation film parameters are designed using genetic algorithm. The light leakage from the crossed circular polarizers using the proposed single-wavelength circular polarizer is less than 2.87×10 -4 over the ±85 o viewing cone at λ= 550 nm. Using the proposed broadband circular polarizer, the light leakage is predicted to be less than 1.7×10 -3 over the ±85 o viewing cone for the visual white light and it is lower than 3.79×10 -4 at λ= 550 nm. To highlight its potential applications, we apply the wide-view circular polarizer to a multi-domain VA LCDs. The maximum transmittance is predicted to be greater than 90% and the contrast ratio is higher than 420:1 for the green light. Over the entire visual spectrum the maximum transmittance is greater than 81% and the contrast ratio is higher than 115:1. The uniformity of better than 68% in the bright state transmittance is achievable over the ±85 o viewing cone. The authors are indebted to Toppoly Optoelectronics for the financial supports.