BISTABLE twisted nematic (BTN) liquid crystal displays

JOURNAL OF DISPLAY TECHNOLOGY, VOL 1, NO 2, DECEMBER 2005 217 Analytic Solution of Film Compensated Bistable Twisted Nematic Liquid Crystal Displays Fushan Zhou and Deng-Ke Yang Abstract In this paper, we studied the optics of transmissive bistable nematic liquid crystal displays by using the Poincaré sphere approach. We derived analytical solutions of the polarization state of the outcoming light for the two bistable states. We have found the optimum modes of film compensated bistable twisted nematic liquid crystal displays. Our results show that the number of optimized modes at a fixed wavelength of film-compensated bistable twisted nematic (BTN) is infinite. We identified the optimized modes for displays operated in the whole visible light wavelength region. We calculated the transmission spectra of the two bistable states of the optimized modes. Index Terms Bistable twisted nematic (BTN), display, liquid crystal, Mueller matrix. I. INTRODUCTION BISTABLE twisted nematic (BTN) liquid crystal displays [1], [2] have received considerable attention since it offers power-off memory, no crosstalking, and good viewing angle. The BTN display is operated between two liquid crystal configurations, which are stable at zero field, for a chosen boundary condition. If it is required that the two configurations be topologically equivalent, then the twist angles in the two states have to differ by, and the displays are referred to as BTN displays. BTN displays with quite good performance have been made using this bistability [3], but the two states are only metastable [4], [5]. There is another type of BTN, referred to as BTN, in which the twist angles of the two bistable states differ by. BTN can be truly bistable and may give faster response [6], [7]. Since, in this case the two states are not topologically equivalent, the switching must be done by either defect propagation, layer melting or anchoring breaking. Attempts to obtain the optimized configurations of transmissive and reflective BTN using the Poincaré sphere method [8] [10], Mueller matrix approach [11] [13], parameter space approach [14] [18] are quite successful so far. However, film compensation of BTN is still limited to a few special configurations such as using quarter-wave plates [19], [20], half-wave plates or integer-wave plates [21], [22]. In this paper, we consider the general case of a film-compensated transmissive BTN, its design principles and plausible configurations. Using Stokes parameters and Mueller matrix approach, we obtained a 3 3 matrix for a uniformly twisted nematic film, which corresponds to a rotation on the Poincaré Manuscript received June 13, 2005; revised July 12, 2005. The authors are with the Chemical Physics Interdisciplinary Program, Liquid Crystal Institute, Kent State University, Kent, OH 44242 USA (e-mail: fzhou@kent.edu; dyang@lci.kent.edu). Digital Object Identifier 10.1109/JDT.2005.858870 Fig. 1. Parameters that specify the polarization state. is the azimuth angle and is the ellipticity angle. sphere. Based on a theorem that rotation preserves angular separation of two vectors on the Poincaré sphere, we derived an equation that gives the sufficient and necessary conditions for a film compensated transmissive BTN. By solving this equation, we obtained a family of solutions which show that the number of plausible optimized modes is infinite. We calculated the electro-optical performance of some of the film compensated BTN modes. We derived the conditions under which BTN displays can be optimized. II. THEORY Consider an elliptically polarized light propagating along the axis of a Cartesian coordinate system, which is described by a Jones vector Generally, the end point of the vector traces out an ellipse in the - plane, as shown in Fig. 1. The major and minor axis of the polarization ellipse are and, respectively. The polarization state can also be characterized by another two parameters: one is the azimuthal angle of the major axis with respect to the axis, the other is the ellipticity angle, which is defined by. The Stokes vector [23] of the elliptically polarized light is defined as (1) (2) 1551-319X/$20.00 2005 IEEE

218 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005 Therefore, must be unitary, namely,. We consider the anglular separation between two polarization vectors, say and on the Poincaré sphere. (6) After the two light pass through an optical film whose Mueller matrix is, the polarization vectors change to and. Because is unitary (7) Fig. 2. Poincaré sphere representation. The poles C and C are left- and right-handed circularly polarized light; the equator is the locus of all linearly polarized light. L is linearly polarized along x and L is linearly polarized along y. where,defined by is a unit vector, which can also be used to characterize the polarization state of light. In Poincaré sphere representation [24], the three orthogonal axes are and. The polarization state of the light is represented by a unit vector from the origin O to the point P on the spheric surface,. The longitude of this vector is and the latitude is, as shown in Fig. 2. The equator is the locus of all linearly polarized light. The north pole and the south pole represent left- and right-circularly polarized light. All the other points on the sphere are elliptically polarized. In the Poincaré sphere representation, the polarization state of a completely polarized light is represented by the unit vector with 3-components. The effect of an optical element is represented by the simplified Mueller (3 3) matrix if there is no absorption [25]. The polarization state vectors of light before and after the optical element are related to each other by The effect of is to rotate the polarization vector on the Poincaré sphere. The length of a polarization vector is 1 and does not change under the action of any lossless optical element. (3) (4) Therefore, the angular separation between sthe polarization vectors on the Poincaré sphere is invariant under the action of any lossless optical element. The simplified Mueller matrix in the laboratory frame for a retarder with the retardation angle when its slow axis makes the angle with the axis of the laboratory frame is shown in the equation at the bottom of the page. We now consider a uniform twisted nematic liquid crystal film. The slow axis of the liquid crystal at the entrance plane is parallel to the axis of the laboratory frame. The light propagating is in the direction. In the local frame, the axis is parallel to the slow axis of the liquid crystal, the Mueller matrix of the twisted nematic liquid crystal film with the total twist angle, thickness, and birefringence is given by where (9) (10) (11) where is the wavelength of the incident beam. If the light propagating direction is reversed, the matrix changes to (12) The local reference frame is used to simplify the analysis. In many practical applications, we are concerned with the ellipticity but not the azimuth angle of the polarization state. In the following discussion, unless otherwise specified, the local reference frame is used. The evolution of the polarization vector on the Poincaré sphere under the action of a uniformly twisted nematic film is a rotation, and is given by (5) (13) (8)

ZHOU AND YANG: ANALYTIC SOLUTION OF FILM-COMPENSATED BISTABLE TWISTED NEMATIC LCDs 219 Fig. 4. Orientations of optical elements of the film compensated BTN in the laboratory frame. ~n shown is the liquid crystal director at the top of the film. Fig. 3. where Schematic structure of the film compensated BTN. (14) which is a unit vector. The physical meaning of (14) is that the effect of the twisted liquid crystal film is to rotate the vector by the angle around the axis represented by on the Poincaré sphere. Now consider a film compensated transmissive bistable nematic liquid crystal display. Fig. 3 gives the schematic structure of the BTN display which consists of two pixels. The liquid crystal in one of the pixels is in the stable state with the twist angle. The liquid crystal in the other pixel is in the other stable state with the twist angle for for (15) The liquid crystal layer is sandwiched between two compensation films. At the top and bottom are two linear polarizers. The backlight, which is under the bottom polarizer, is not shown in the figure. The polarization vector on the Poincaré sphere after light goes through each optical element is also shown in the figure and is expressed in the local reference frames. and are the rotation matrices of the liquid crystal film in the two stable states, respective. So we have (16) (17) Because the twist angles between the two stable states differ by either or, the local reference frames for and are the same. In order to make the discussion easy, we also constructed Fig. 4 where we choose the lab frame such that light propagates along the axis and the rubbing direction at the lower substrate of the liquid crystal cell is along the axis. All the angles are defined with respect to the axis of the laboratory frame. and are the angles of the transmission direction of the bottom and top polarizers, and is the angle of the rubbing direction of the upper substrate of the liquid crystal cell, or the total twist angle of the liquid crystal in one of the stable states. and are the angles of compensation film and, respectively. Now we examine the condition for the BTN display to have optimum performance. After light goes through the bottom polarizer, the polarization state is linear. In order to get the optimum performance, and are required to be linear and orthogonal to each other in real space. For example, we choose the transmission axis of the top polarizer parallel to. Then the light with can pass the polarizer (bright state) and the light cannot pass the polarizer (dark state). Consequently, on the Poincaré sphere, the angular separation of the two polarization vectors is 180. In order to find the relations between and,we need to use the following two facts [26]. 1) The angular separation of two polarization vectors on the Poincaré sphere remains the same under the action of any birefringent film. 2) For any polarization state, it can always be brought to linear by an appropriate compensation film and vice versa. From these facts we can see that in order to make the angular separation between and to be 180 on the Poincaré sphere, the angular separation between the two vectors and on the Poincaré sphere must also be 180. We can state this in a different way; if the angular separation between the two vectors and is 180, then we can always find a compensation film so that and are both linear and orthogonal to each other in the real space. On the Poincaré sphere, this condition is equivalent to, (18) which means that and lie on the same diameter but pointing in opposite directions. Using (16) and (17), we can claim that a film compensated BTN can be optimized if and only if the following condition is met: (19) 1) Optimized Configurations: When the twist angles of the film-compensated BTN are and, (19) can be changed to (20) Using (9), it follows as shown in (21) at the bottom of the next page, where (22) (23)

220 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005 Equation (21) can be put into the following form: (24) The value of and can be found by comparing (24) with (21). From (24), we can draw the conclusion that a transmissive BTN can be optimized if and only if the following condition is met: (25) After some manipulation, (25) becomes (26) It can be shown that when (26) holds, the rank of the matrix in (25) is 1. As a result, there is an infinite number of solutions for. The solution family of is either a unit circle or a part of it on the Poincaré sphere. For example, some solutions to (26) are Fig. 5. The solution family of all optimized film compensated BTN mode (A) and modes without film compensation (B), labeled as a, b, c and d. (Color version available online at http://ieeexplore.ieee.org.) (27) (28) This means that (29) (30) where and are integers and. In fact, this is the solution when there is no compensation film [11]. When and satisfy (26), the display is optimized, i.e., one of the bistable states has the transmittance of 0% and the other bistable state has the transmittance of 100% when the appropriate compensation film is used. Equation (26) can be rearranged as. The retardation for optimized modes can be calculated as a function of, and the results are shown in Figs. 5 and 6 for BTN and BTN displays, respectively. In the figures, only solutions with are shown because solutions with high are not attractive because of their large wavelength dependence. The curves have the following features: (1) they are symmetric about a vertical line (at for BTN display and for BTN display), (2) if is too small ( for BTN display and for BTN display), there is no solution. The optimized modes without film compensation, which correspond to Fig. 6. The solution family of all optimized film compensated 2BTN mode (A and B) and the modes without film compensation (C), labeled as a, b, c, d, e and f. (Color version available online at http://ieeexplore.ieee.org.) several discrete points labeled as a, b, c, d, (e and f), are shown in the figures. For BTN display, it cannot be optimized when the twist angles are and for the two bistable states. The versus figures also provide information on the thickness- and wavelength-sensitivity of the displays. In practice, the twist angle can be controlled precisely while the cell thickness is difficult to control, and the wavelength varies in visible light region. The retardation is given by, and, therefore, any change of or results a change of. For a given point on the - curve, if the slope of the curve is (21)

ZHOU AND YANG: ANALYTIC SOLUTION OF FILM-COMPENSATED BISTABLE TWISTED NEMATIC LCDs 221 Fig. 7. Projection of P ~ and P ~ for all BTN displays on the s 0 s plane. Points on the equator, labeled as a, b, c and d, are modes without any compensation films. (Color version available online at http://ieeexplore.ieee.org.) small, a change of results a large deviation from the optimum condition; if the slope of the curve is large, a change of results a small deviation from the optimum condition. We can see that for a BTN display, when the twist angle is 21.7 where, the display is the most insensitive to the retardation variation. When is where, the display is very sensitive to retardation variation. For a BTN display, when the twist angle is, or,or,or, where, the displays are the most insensitive to retardation variation. When is or 35.5, where, the displays are very sensitive to retardation variation. So far the discussion has been general. Now we consider some special cases. One special case is that there is no compensation film between the bottom polarizer (no ) and the liquid crystal, where is linearly polarized., where is the angle between the transmission axis of the bottom polarizer and the liquid crystal director at the bottom of the liquid crystal film. The solution to (21) is (31) The physical meaning of (31) may be stated as that for a given pair of and, the required value of to achieve optimization can be calculated by (31). It can also be stated as that for a given value of and must satisfy (31) in order to optimize the display. In order to visualize the rotation of polarization vectors under the action of optical elements, we examine the projection of polarization vectors on the plane. Because [from (18)], we only consider and show the projections of and. Because is always linearly polarized (when no compensation film ), its projection is a circle with the radius of 1 (the equator of the Poincaré sphere). The projection of in the BTN and BTN displays is shown in Figs. 7 and 8, respectively. In Fig. 7, we see that there are 4 points, labeled a, b, c and d on the unit circle which correspond to linear polarizations. In these cases, because and are linearly polarized and orthogonal, the display is optimized without a compensation film (F ) between the liquid crystal and the top polarizer. The modes represented by these points are also shown in Fig. 5 Fig. 8. Projection of ~ P and ~ P of all 2BTN displays on the s 0 s plane. Points on the equator, labeled as a, b, c, d and e, are for the modes without any compensation films. (Color version available online at http://ieeexplore.ieee.org.) and labeled by the same letters. Similarly, in Fig. 8, there are 5 points, labeled a, b, c, d, and e, on the unit circle, which correspond to linear polarizations. These are the modes which do not need compensation film. The modes represented by these points are also shown in Fig. 6 and labeled by the same letters. It should be mentioned that for each optimized mode discussed above, is also an optimized mode. 2) Bistable TN Displays With a Quarter-Wave Plate Compensation Film: Now we consider the case where the compensation film is a quarter-wave plate (and there is no ). For agiven, the required for optimization is calculated by using (26). Once and are known, the angle of the bottom polarizer is calculated by (31). Once and are known the polarization state of the light after passing through the liquid crystal film can be calculated, which is elliptically polarized in general. (32) If the compensation film is a quarter wave plate with its slow axis making the angle with the axis, it s simplified Mueller matrix is (33) The polarization vector of the light after passing the compensation film is (34)

222 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005 Fig. 9. Angles of transmission directions of the bottom and top polarizers, and, and the angles of the slow axis of the QWP, as a function of for the BTN display. The labeled points a, b, c, and d are the modes when there is no compensation film. (Color version available online at http://ieeexplore.ieee.org.) When the angle of the quarter wave plate is chosen to be, then, which is linearly polarized. The elliptically polarized light after the liquid crystal film can always be converted into linear polarization by a (broadband) quarter-wave plate (QWP) with the appropriate angle,, of the slow axis of the quarter wave plate. The angle of the linearly polarized light is determined by the polarization of the light after the liquid crystal film and the angle of the quarter wave plate. The top polarizer must be oriented in such a way that its transmission axis is parallel to the polarization direction of the linearly polarized light after the quarter wave plate, therefore. As an example, when the state with the twist angle is chosen as the bright state, the calculated values of and as a function of the twist angle are shown in Figs. 9 and 10 for the BTN and BTN displays, respectively. The retardation is wavelength dependent, and thus the transmittance of the two bistable states are also wavelength dependent. For an optimized BTN display, the transmittance of the bright state is 100% and the transmittance of the dark state is 0% only at one wavelength which is chosen to be 550 nm in our calculation. The transmission spectra of the two stable states of all the optimized BTN displays with a quarterwave plate compensation film are shown in Fig. 11. From this figure, it can be seen that should be around, where color dispersion of the dark state is small, and therefore high contrast ratio can be achieved for the integrated transmittance. The transmission spectra of the two stable states of all the optimized BTN displays with a quarter wave plate compensation film are shown in Fig. 12, where only the branches with smaller retardation values are shown. From the figure, it can be seen that should be around 360, where color dispersion of the dark state is small, and therefore high contrast ratio can be achieved for the integrated transmittance. Fig. 10. Angles of transmission directions of the bottom and top polarizers, ;, and the angles of the slow axis of the QWP, as a function of for 2BTN. Only braches with smaller retardation were shown. The labeled points a, b, c, d and e are the modes when there is no compensation film. (Color version available online at http://ieeexplore.ieee.org.) Fig. 11. Transmission spectra of the bistable states of all the optimized BTN display with the quarter-wave plate compensation film. (Color version available online at http://ieeexplore.ieee.org.) 3) Bistable TN With Non-Quarter-Wave Plate Compensation Film: Non-quarter-wave plates can also be used as the compensation film. If we have an elliptically polarized light shown in (32), it can be converted into linear polarization by a compensation film with the retardation angle of and its slow axis at, whose Mueller matrix is shown in the equation at the bottom of the page.the polarization vector after the compensation film will be. The display is optimized when the angle of the top polarizer is chosen to be. As an example, the state with twist angle is chosen to be the bright state. The angles, and, of the transmission axis of the bottom and top polarizers, the angle of the slow axis and retardation angle of the compensation film as a function of the twist angle of the BTN and BTN displays are shown in Figs. 13 and 14, respectively. (35)

ZHOU AND YANG: ANALYTIC SOLUTION OF FILM-COMPENSATED BISTABLE TWISTED NEMATIC LCDs Fig. 12. Transmission spectra of the bistable states of all the optimized 2 BTN display with the quarter wave plate compensation film. (Color version available online at http://ieeexplore.ieee.org.) Fig. 13. Angles of transmission axis of the bottom and top polarizers, ;, the angle of the optic axis of the compensation film,, and its phase retardation of the BTN display. The points labeled as a, b, c, and d are the modes when there is no compensation film. (Color version available online at http://ieeexplore.ieee.org.) 223 Fig. 15. Transmission spectra of the bistable states of all the optimized BTN display with the nonquarter wave plate compensation film. (Color version available online at http://ieeexplore.ieee.org.) Fig. 16. Transmission spectra of the bistable states of all the optimized 2 BTN display with the non-quarter-wave plate compensation film. (Color version available online at http://ieeexplore.ieee.org.) be seen from Fig. 16 that for BTN display when is close to, the color dispersion of the dark state is small. This is similar to the case where the compensation film is a quarter wave plate. III. CONCLUSION We derived the general situation of film compensated transmissive bistable nematic liquid crystal displays. There is an infinite number of optimized modes. This gives more freedom in improving other properties of the display, such as viewing angle and contrast. REFERENCES Fig. 14. Angles of transmission axis of the bottom and top polarizers, ;, the angle of the optic axis of the compensation film,, and its phase retardation of the 2 BTN display. The points labeled as a, b, c, d and e are the modes when there is no compensation film. (Color version available online at http://ieeexplore.ieee.org.) The transmission spectra of all the optimized BTN with the nonquarter wave plate are shown in Figs. 15 and 16. It can be seen from Fig. 15 that for BTN display when is close to, the color dispersion of the dark state is small. It can also [1] D. W. Berreman and W. R. Heffner, New bistable cholesteric liquidcrystal display, Appl. Phys. Lett., vol. 37, pp. 109 111, Jul. 1980., New bistable liquid crystal twist cell, J. Appl. Phys., vol. 52, pp. [2] 3032 3039, Apr. 1981. [3] T. Tanaka et al., A bistable twisted nematic (BTN) LCD driven by a passive-matrix addressing, in Asia Display, vol. 26, 1995, pp. 259 262. [4] C. D. Hoke and P. J. Bos, A bistable twist cell exhibiting long-term bistability suitable for page-sized displays, in SID 00 Dig. Tech. Papers, 1998, pp. 854 857. [5] G. P. Bryan-Brown, C. V. Brown, and J. C. Jones, GB Patent 9 521 106.6, 1995. [6] I. Dozov, M. Nobili, and G. Durand, Fast bistable nematic display using mono-stable surface switching, Appl. Phys. Lett., vol. 70, pp. 1179 1181, Mar. 1997.

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