Wide-view circular polarizer consisting of a linear polarizer and two biaxial films

: A simple wide-view circular polarizer comprising of a linear polarizer and two biaxial films is proposed. Over the ±85 o viewing cone, the produced polarization is almost circular and the light leakage from the crossed circular polarizers is calculated to be less than 8.23×10 -5 , provided that the air-interface surface reflections are ignored. The design tolerance within ±5% of the optimal parameters is analyzed.


Introduction
Circular polarizer is an important component for optical communications, optical remote sensing, and liquid crystal displays (LCDs) [1][2][3][4][5]. Various multi-domain approaches have been proposed to widen the viewing angle and reduce the color shift of LCDs [6]. However, the LC directors twist continuously from domain to domain so that the boundary areas are formed between domains [7][8][9][10][11]. These boundary areas appear dark under crossed linear polarizers and the pixel's transmittance is reduced. One way to remove the dark zones and boost light transmittance is to use a pair of crossed circular polarizers [2][3][4][5]. Nevertheless, a conventional circular polarizer consisting of a linear polarizer and a uniaxial quarter-wave plate produces circular polarization only at normal incidence [2][3][4][5]. Replacing the uniaxial quarter-wave plate with a biaxial quarter-wave plate only slightly improves the acceptance angle [2][3][4][5]. A wide-view circular polarizer using the combinations of optimally designed uniaxial A-plates and C-plates significantly widens the acceptance angle, but its device configuration is too sophisticated [5]. As a result, it is difficult to fabricate and the cost is high. There is an urgent need to develop a simple and low cost wide-view circular polarizer.
Comparing to a uniaxial retardation film, the biaxial film provides an extra degree of freedom so that the device configuration of the wide-view circular polarizer can be simplified [11]. In this paper, we propose a wide-view circular polarizer consisting of a linear polarizer and two biaxial films. The produced states of polarization are very close to the ideal circular polarization over a wide range of incident angles. Over the entire ±85 o viewing cone, the light leakage of the crossed circular polarizers is less than 8.23×10 −5 , provided that the air-interface surface reflections are ignored.
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.

Design of wide-view circular polarizer
A circular state of polarization is achieved when a quarter-wave plate is illuminated by a linearly polarized light vibrating at 45 o with respect to its slow axis [12][13]. If this light is produced by a linear polarizer, the quarter-wave plate together with the linear polarizer forms a conventional circular polarizer. At normal incidence, when the linearly polarized light passes through the quarter-wave plate, the light sustains π/2 phase change while the magnitudes of its orthogonal components remain equal so that the light becomes circularly polarized. However, at oblique angles the quarter-wave plate contributes other than π/2 phase change [2][3][4][5] and the slow axis is not at 45 o with respect to the incoming linear polarization. As a result, the produced polarization state becomes elliptical and a relatively large amount of light leaks through the crossed circular polarizers. Replacing the uniaxial quarter-wave plate with a biaxial quarter-wave plate reduces the variation in the π/2 phase change at oblique incident angles. However, the 45 o angle between the light's vibration direction and the biaxial film's slow axis is not maintained at oblique incidence [5,11]. Therefore, although the produced polarization is relatively close to the circular polarization, the light leakage of the crossed circular polarizers is still large at wide viewing angles [2][3][4].
To preserve circular polarization over a wide range of incident angle, we laminate two biaxial retardation films to the linear polarizer as shown in Fig. 1. Since all of the three Stokes parameters are modified inside a biaxial film, the azimuthal angles and the d·(n x -n y ) as well as the NZ factors of both biaxial films are the key design parameters. Here the azimuthal angle of biaxial film is the angle between the n x axis of the biaxial film and the absorption direction of the polarizer as Fig. 2 shows.
We use genetic algorithm [5,16] to optimize the polarizer design. For a given combination of the above parameters, we first solve the orthogonal components of the electric field using the 4-by-4 matrix method. Then we find the produced polarization and apply it into the following cost function where S 3_(2B) is the S 3 of the produced state of polarization P (2B) . By minimizing the cost function shown in Eq.
(2), we could obtain the optimal design parameters for the biaxial films.  Figure 2 illustrates the polarization states inside this circular polarizer using Poincaré sphere representation. As Fig. 2 depicts, the first biaxial film serves as a wide-view half-wave plate so that the light entering the second biaxial film is almost linearly polarized and vibrating at ~45 o with respect to the n x axis of the second biaxial film. Meanwhile, the second biaxial film performs as a wide-view quarter-wave plate. Therefore, although the polarizations across each biaxial film vary with the incident angle, they compensate each other so that the final polarization remains circular over a wide viewing cone. As demonstrated in Fig. 3, for the proposed right-handed circular polarizer S 3 only increases to −0.997 when the viewing angle increases to 85 o . This is equivalent to having the polarization difference ΔP (2B)−(RCP) less than 0.078 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 a left-handed circular polarizer using above configuration, the NZ factors and d·(n x − n y ) of both biaxial films are not changed but the azimuthal angles of the biaxial films are the negative of their counterparts in the right-handed circular polarizer. Since the produced polarization approaches ideal circular polarization, the light leakage from crossed circular polarizers can be significantly reduced. Figure 4 sketches the configuration of the crossed circular polarizers. The top linear polarizer and the first two biaxial films compose a wide-view right-handed circular polarizer. The bottom analyzer and the last two biaxial films form a second circular polarizer crossed to the first one. The absorption axis of the analyzer and the azimuthal angles of the last two biaxial films are perpendicular to their counter parts in the first circular polarizer. Besides the light leakage of crossed polarizers, for wide-view LCD applications we also need to know the angular-dependent transmittance when the polarizers are open. Results are plotted in Fig. 5(b). The maximum transmittance which occurs at normal incidence is 37.8%. The major optical loss is from the absorption of the dichroic linear polarizers. The isotransmittance contour shown in Fig. 5(b) is relatively symmetric. At ±85 o viewing cone, the transmittance remains greater than 31.1%. Thus, from Figs. 5(a) and 5(b), the calculated extinction ratio (at λ=550 nm) of the circular polarizer remains ~4000:1 at the ±85 o viewing cone.
In order to reduce the interference of the air-polarizer surface reflection, an ideal antireflection (AR) film is assumed during simulations. The ten-layer anti-reflection film is coated on the air interface of both polarizers. This AR film is designed using genetic algorithm and the gradient refractive indices profile is illustrated in Fig. 6(a) [5,16]. The origin represents the air-AR interface. The transmittance of this ten-layer AR film is greater than 0.99 over the ±85 o incident cone within the 450 nm ~ 650 nm spectrum as Fig. 6(b) depicts.

Spectral bandwidth
For some applications such as direct-view LCDs, broad bandwidth is as important as wide viewing angle [1][2][3][4][5][6][7][8][9][10][11]. Although the proposed wide-view circular polarizer is designed at a single wavelength λ=550 nm, however, at normal incidence the retardation of the first biaxial film is almost equal to one half of the designed wavelength, i.e., d·(n x − n y ) 1 = 264.08 nm ≈ 550 nm/2 = λ/2, and the retardation of the second biaxial film is close to a quarter of the which describes the relationship between the azimuthal angles of the half-wave plate and the quarter-wave plate inside a broadband circular polarizer [17]. Therefore, at normal incidence the proposed circular polarizer performs as a broadband circular polarizer so that the light leakage of the crossed circular polarizers is less than 1.64×10 −6 over the 450~650 nm spectral range, as shown in Fig. 7. Within the ±60 o viewing cone, the light leakage is maintained less than 1.90 ×10 −3 over the specified visible spectral range. At ±85 o viewing cone, the maximum light leakage reaches 3.92 ×10 −3 at λ = 450 nm and 1.39 ×10 −3 at λ = 650 nm. For display applications, the light leakage at blue is more forgiven than green and red because human visual system is less sensitive to blue. Moreover, the blue spectral content is the weakest among the three primary colors. Fig. 7. The calculated maximum light leakage of the crossed circular polarizers at different viewing angles as a function of wavelength. The configuration of the crossed circular polarizers is in Fig. 1 and the ten-layer anti-reflection film in Fig. 6 is assumed.

Discussion
Design tolerance is an important concern from manufacturing viewpoint. For the proposed design of a wide-view circular polarizer, we calculate the maximum light leakage of the crossed circular polarizers over the ±85 o viewing cone if the film parameters of the biaxial films deviate from the optimal values by ±5%. The simulation results are plotted in Fig. 8. First we evaluate the influence of film thickness. The blue line in Fig. 8 indicates that a +5% deviation in the first biaxial film thickness boosts the light leakage to 4.1×10 −4 . By contrast, the light leakage is quite inert to the errors of the second biaxial film thickness (dashed blue lines). Next, we study how the NZ factor affects the light leakage. As shown by the black solid line in Fig. 8, the NZ factor of the first biaxial film plays a significant role. A ±5% deviation causes ~7.0×10 −4 light leakage. On the other hand, a +5% error in the NZ factor of the second biaxial film boosts the light leakage to 4.5×10 −4 (black dashed line). The solid and dashed red lines in Fig. 8 almost overlap with the blue dashed lines implying the light leakage is almost invariant when the orientations of both biaxial films vary by ±5%. Thus, from this tolerance analysis we find that the accuracy of the first biaxial film parameters is more critical than that of the second biaxial film and the NZ factors of both biaxial films require a higher accuracy. Fig. 8. Design tolerance of the proposed wide-view circular polarizer. The viewing cone is ±85 o and λ= 550 nm. Ten-layer anti-reflection film is assumed.

Conclusion
We propose a new wide-view circular polarizer consisting of a linear polarizer and two biaxial films. We use phase compensation techniques to constrain the resultant polarization state to the desired circular 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 circular polarizer is less than 8.23×10 −5 over the ±85 o viewing cone at the designed wavelength λ = 550 nm. Within the 450~650 nm spectral range, the light leakage of the crossed circular polarizers is kept below 3.92 ×10 −3 over the entire ±85 o viewing cone. This simple circular polarizer will find useful application for enhancing the transmittance of the multi-domain vertically-aligned liquid crystal displays. The authors are indebted to Toppoly Optoelectronics (Taiwan) for the financial supports.