Chapter 2 Theory and Numerical Simulation
2.1 Theories of deformations and optical properties of liquid
2.2.2 Beam propagation method
Jones matrix method mentioned above as well as other extended matrix-type solvers [22,23] are based on the assumption of a stratified medium. Fig. 2.7 (a) sketches the concept of these methods. As shown in the figure, the LC layer is divided into N×M grids.
Each grid is a slab of homogenous anisotropy material. Therefore, the light scattering and diffraction effects are excluded in these calculations. They are reliable only in the regime where the liquid crystal profiles are maintained uniformly over a scale far exceeding the optical wavelength.
In LCOS devices, the pixel pitches can be comparable to the wavelength, which means that the diffraction effect may have severe influences to the optical performance. The beam propagation method (BPM) has been proven useful in this condition [24,25]. This method is a numerical approach to the solution of Maxwell’s curl equations. Unlike the matrix-type methods, the BPM considers the scattering and diffraction effects during the light propagation. Furthermore, the computational efficiency of BPM is much higher than that of the sophisticated finite-difference time domain (FDTD) method [26-28] and their results are almost identical when the liquid crystal (LC) layer is illuminated within ±30°[25]. Fig. 2.7 (b) sketches the concepts of BPM. Basically, it develops an operational relationship between the field components at two planes separated by a sufficiently small propagation step ∆z. The excitation field is propagated from the entrance to the exit plane by the repeated application of this operational relationship.
In this section, the concept and capabilities of BPM are reviewed.
Scalar, Paraxial BPM
We firstly restrict the incident wave to be a scalar field. The wave equation can be given based on the well-known Helmholtz equation:
0
Here the scalar electric field is expressed as E(x,y,z,t)=Φ(x,y,z)e-iωt and k is the spatially dependent wavenumber. Other than the scalar assumption, this equation is exact. The field can be written as the product of a slowly varying envelope times a fast varying reference oscillating term:
*z
where k* is a constant to represent the average phase variation of the field Φ, and is called the reference wavenumber. Inserting Eq. (2.55) into Eq. (2.54), we obtain
0
Assuming the variation of u with z is sufficiently slow so that the first term in Eq. (2.56) can be neglected. It is also referred to as the paraxial approximation. Therefore, Eq. (2.56) can be rewritten as:
⎟⎟⎠
This is the basic three-dimensional (3D) BPM equation.
For numerical solution, the finite-difference approach based on the Crank-Nicholson scheme is often adopted [24]. The field in the transverse xy plane is represented at discrete points on discrete planes along the longitudinal or propagation direction, i.e. z-direction.
The purpose is to derive the numerical equations that determine the field at the next z plane.
For simplicity, the approach is limited in two-dimensional (2D) analyses, i.e. omitting the y-dependent term.
Let u denotes the field at transverse grid point i and longitudinal plane r. The grid ri points and planes are equally spaced by ∆x and ∆z, respectively. Eq. (2.57) can be written in the discrete form based on the Crank-Nicholson method:
2 can be rearranged into a standard tridiagonal matrix equation for the unknown field uri+1 in terms of known quantities as:
i
where a, b, c, and d are known coefficients. Therefore, the solutions can be obtained by using the lower-upper (LU) decomposition [24].
Wide-Angle BPM
The restriction of paraxiality on BPM is due to neglecting the ∂2u/∂z2 term. If the ∂u/∂z term is treated as a variable, one can obtain a first-order differential equation in z by solving Eq. (2.56):
This equation is exact that there is no paraxial approximation. However, the differential operator P in the radical introduces difficulties for solving the equation. To solve this problem, one approach is to use a Taylor expansion. Another method is to use Padè approximation which was reported to be more accurate than the Taylor expansion for the same order of terms [29]. Using Padé approximation, Eq. (2.60) can be modified as:
(P) u
where Nm and Dn are polynomials of P, and (m,n) is the order of approximation. The higher order it is, the more accurate result can be obtained in wide-angle problem. Table (2.3) lists
the first few terms of the Padé series.
Padé Order (m,n)
1+3P/4+P2/16
Padé Order (m,n)
Wide-Angle BPM for Liquid Crystal Device
When light propagates in a liquid crystal device, the spatially dependent dielectric constant ε(x,z) can be written as a tensor:
2 where θ and φ are the tilt and azimuthal angle of LC director, respectively. In this case, the scalar assumption in Eq. (2.54) is not appropriate for solving the wave equation. The polarization of light has to be included by recognizing that the electric and magnetic fields are vectors, i.e. E=Exx+Eyy+Ezz and H=Hxx+Hyy+Hzz. From Maxwell’s Equations, a set of coupled equations can be given as:
~ 0
Table 2.3 The first few terms of the Padé series.
0
Under the slowly varying assumption, the field components can be written as:
( )
Substituting Eq. (2.66) in Eqs. (2.64) and (2.65), we have the following equations:
0 0
⎥⎥
The wide-angle scheme can be formed by using the Padé recurrence relation as mentioned previously. From Eq. (2.67), we can have the following operational relation:
[ ]
This equation suggests the recurrence relation as:
[ ]
When setting n=0, Eq. (2.76) gives the result identical to that under paraxial assumption (i.e.
the second-order differential term is omitted). When n=1, we have ∂/∂z|1=-([R]+[S])-1[A].
The contribution of matrix [S] is weak compared to [R] and is negligible in this case.
Therefore, Eq. (2.67) can expressed by
⎥ ⎦
After applying the Crank-Nicolson finite-difference scheme with a weight parameter α
(0.5<α<1), the discrete form of Eq. (2.77) is given as:
and superscript r=1, 2,…n indicates the propagation step of the discretized axial plane, i.e., z=r∆z assuming that z is the propagation direction of the light. By unfolding Eq. (2.78), a sparse linear system can be formed to represent the relation between the known value of Eyr, Hyr and the unknown value Eyr+1, Hyr+1. In order to prevent artificial reflections from the boundary, transparent boundary condition (TBC) is utilized to ensure that the outgoing radiation passes through the boundary freely [30]. Finally, the following sparse linear system can be expressed as followed:
⎥⎥
Each of the sub-blocks [Xi.j] with i,j=1,2 corresponds to a tridiagonal matrix. Both B1 and B2 are functions related to Eyr and Hyr. By solving Eq. (2.80) with an initial condition of a plane wave propagating along the z direction, we can obtain information that includes the amplitude and phase of the propagating light wave. The incident angle θi can be changed by modifying the initial condition, that is
)]
Extended Wide-Angle BPM for Reflective Liquid Crystal Devices
The reflective propagation process of the light wave can be derived by sending the light into the mirror image of the LC molecular distribution, as shown in Fig. 2.5. Therefore, the reflective transverse fields, E’y and H’y, of the LCOS device can be described as follows:
where Rp and Rg represent the reflectance of the reflective pixels and the interpixel gap, respectively. From Eq. (1), we obtain
(
x y z)
So far we have obtained the information of all the components of the reflective wave.
Further analyses on 2D reflectance profile and angular spectrum can be given as will be mentioned in Chapter 5.
References
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[15] J. E. Anderson, P. E. Watson and P.J. Bos, “Shortcomings of the Q Tensor Method for Modelling Liquid Crystal Devices.” Soc. Information Display Tech. Digest, paper 16.3 (1999).
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Simulation Software and Technology Guide (Artech House, Boston, 2001).
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[18] autronic-MELCHERS GmbH, http://www.autronic-melchers.com.
[19] R. C. Jones, J. Opt. Soc. A. 31, pp.488 (1941).
[20] A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
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[22] D. W. Berreman, J. Opt. Soc. Am. 62, pp.502 (1972).
[23] A. Lien: Appl. Phys. Lett. 57, pp.2767 (1990).
[24] E. E. Kriezis and S. J. Elston: J. Mod. Opt. 46, pp.1201 (1999).
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[26] B. Witzigmann, P. Regli and W. Fichtner: J. Opt. Soc. Am. A 15, pp.753 (1998).
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[29] G. R. Hadley, Opt. Lett. 17, pp.1426 (1992).
[30] G. R. Hadley: IEEE J. Quantum Electron. 28 pp.363 (1992).
Chapter 3
Fringing-Field Effects of LCOS Devices
3.1 Introduction
As mentioned in Chapter 1, the fringing fields generated by voltage difference between neighboring pixels are critical to the optical performance of LCOS devices. Many parameters of LC cell are influential to the fringing-field-effect. In this chapter, we present detailed analyses of liquid crystal panels in various molecular alignment conditions, polarizer angles, and LC phase retardation values. These parameters determine the optical mechanisms employed, including birefringence effect and polarization rotation effect. The LC operation modes can be classified as twisted and non-twisted modes depending on their initial director profile. Here in the twisted category, the commonly employed mixed-mode twist nematic (MTN) [1] and the twist nematic (TN) modes [2] are studied, while in the non-twisted category we analyze the film-compensated homogenous (FCH) [3] and vertically aligned (VA) modes [4]. The influence of cell structures including the effects of electrode slope, cell thickness and pixel pitch are also compared. Furthermore, the fringing-field induced distortions of LC directors may cause delay in moving images. The dynamic behaviors of LCOS panel when switched from the dark-bright-dark state to the all-bright pixel configuration are discussed in the last part of this chapter.
3.2 Liquid crystal operation modes
In this section, we simulate the static electro-optical properties in eight LC operation modes. Figure 3.1 shows the cell structure used for the two-dimensional simulations of the LC director distributions and light efficiency. The cell gap is adjusted to obtain the required d∆n for each mode, where d is the cell gap and ∆n is the LC birefringence. The pixel size is 15 µm and the inter-pixel gap is 0.9 µm. A passivation lever is deposited on the Al electrode
for protection and leveling the inter-pixel gap. The LC director orientations are calculated by the commercial software 2dimMOS. Afterwards, the voltage-dependent reflectance is obtained by Jones matrix method with the mirror image of the director profiles. Here, we assume that the reflectance of the reflective pixels (Rp) and the inter-pixel gap (Rg) are 1 and 0, respectively. The LC parameters used for simulations are listed in Table 3.1.
Silicon substrate
For a twisted LC cell with a large value of phase retardation d∆n compare to the wavelength λ, the Mauguin’s condition as mentioned in Chap. 2 can be satisfied, that is d∆n >> λ; d=φt/q0,
ω>>2πq0c/(φt∆n)>>2q0c/∆n,
Fig. 3.1 The LC cell structure used for the two-dimensionalsimulations.
7.8 Vertically aligned
mode (VA) homogenous aligned
mode (FCH) Twist nematic
mode (TN) Mixed-mode twist
nematic (MTN) Vertically aligned
mode (VA) homogenous aligned
mode (FCH) Twist nematic
mode (TN) Mixed-mode twist
nematic (MTN)
Table. 3.1 The LC parameters used for the simulations.
where φt is the twist angle. Thus, the eignemodes are linearly polarized light which follow the twist of the LC directors reasonably well. This is so-called the polarization rotation effect, which is commonly utilized in TN-LCDs. When d∆n becomes smaller, the polarization rotation effect is incomplete and the phase retardation effect must be imposed to enhance the light modulation efficiency. Base on the light modulation mechanism, the twisted LC operation modes can be subdivided into two categories [5]:
(1) TN modes. In these modes, the d∆n values are large and the angle β (referring to Fig.
2.6) is normally zero. Therefore, the corresponding light modulation mechanism in such cells is polarization rotation effect. TN modes are often operated at normally black (NB) sheme, which gives a dark state when no voltage is applied on the panel.
(2) MTN modes. The d∆n values in such cells are smaller and usually β≠0. Both polarization rotation and phase retardation effects are present. MTN modes are often operated at normally white (NW) scheme, which gives a bright state when no voltage is applied on the panel.
In the followings, we will consider three MTN modes: 90°-MTN [1,6], film-compensated 63.6°-MTN [7], and film-compensated 45°-MTN [8]; three TN modes: 63.6°-TN [9], 45°-TN [10] and 52°-TN [11].
3.2.1.1 Mixed-mode twisted nematic (MTN)
NW 90
°
-MTN ModeThis mode has twist angle φt=90° and β=20°. The required phase retardation d∆n is about 240 nm for normally white operation. Figure 3.2 (a) shows the one-dimensional (1D) simulation results on the voltage-dependent normalized reflectance using broadband red (R:620-680 nm), green (G:520-560 nm) and blue (B:420-480 nm) lights. As shown in the figure, the maximum reflectance of this mode is only about 88%. Based on the curve, we let the turned-on voltage (Von) to be 5 volts and the turned-off voltage (Voff) to be 0.7 volts.
Figure 3.2 (b) shows the calculated iso-contrast contour viewing diagram for the broadband green channel. The polar angle corresponding to 1000:1 contrast ratio exceeds 8°. Although the property of viewing angle is less critical in projection displays, a wide acceptance angle allows a small F-number projection lens to be used which greatly improves the display brightness [12].
Fig. 3.2 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of NW 90°-MTN mode with β=20° and d∆n==240 nm.
Figure 3.2 (c) shows the 2D simulation results of the LC director profile and the reflectance calculated by Jones matrix method for the dark-bright-dark pixel configuration with 2.8 µm cell gap and 2° pretilt angle at green band (540 nm). In this condition, the fringing fields are the strongest. As shown in the 2D LC director profile, the fringing fields at the pixel edges will penetrate into the adjacent pixels and induce light leakage in the dark pixels. This effect will degrade the contrast ratio as well as the image sharpness.
The advantages of this mode are the high contrast ratio owing to the natural surface phase compensation of the two orthogonal boundary layers and the relatively small fringing-field effect. The calculated 2D flat field contrast ratio exceeds 6000:1 for the normally incident light. The shortcoming of this 90o MTN mode is that its maximum optical filled factor is ~84.7% at the dark-bright-dark configuration. Here we define the optical filled factor as
∫
= Rbright
F S1
, (3.1)
where S denotes the dimension of the turned-on pixels and Rbright. is the normalize reflectance within the bright pixel area. Changing the twist angle to 80° would boost the filled factor to ~95%. However, the contrast ratio decreases drastically because of incomplete surface phase compensation.
NW Film-Compensated 63.6
°
-MTN ModeThe normalized reflectance R⊥ of a reflective twisted LC cell under crossed-polarizer condition can be given as*:
( ) d∆n/λ= 2/4. This mode has the advantage of high reflectance; however, a very high voltage is needed to obtain a good dark state due to the absence of intrinsic phase compensation. In order to operate at lower voltage, a uniaxial film is employed. Figure 3.3 (a) shows the 1D simulated results of the voltage-dependent reflectance curve under broadband incident light when a uniaxial film with α=110° (referring to Fig. 2.6) and (d∆n)film=24 nm is applied. The required phase retardation d∆n of the LC cell is 203 nm and entrance polarizer angle is set at β=0°. As shown in Fig. 3.3 (a), the maximum reflectance slightly decreases to 97% due to the effect of the compensation film. Figure 3.3 (b) plots the calculated results of iso-contrast viewing diagram under broadband green light with Von=5 V and Voff=0.7 V. The contour line for 1000:1 contrast ratio reaches 10°
viewing cone. Figure 3.3 (c) demonstrates the 2D simulated results of the LC director profile and the reflectance for the dark-bright-dark pixel configuration with 2.4 µm cell gap.
The light leakages still exists at the dark pixels due to the influence of fringing fields.
Fig. 3.3 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of 63.6°-MTN mode with β=0°, d∆n==203 nm, (d∆n)film=24 nm, and 110° film anglerelated to x-axis.
In addition, the optical filled factor is only about 92.7% since the maximum R⊥ can not reach 100%. When setting β=4°, d∆n=212 nm, (d∆n)film=15 nm, and α=136°, the maximum R⊥ can be boosted up to 99%. In this circumstance, more birefringence effect is introduced to enhance the reflectance. However, a stronger fringing field occurs and results in a lower F. Thus, this tradeoff is not worth taking.
NW Film-Compensated 45
°
-MTN ModeThe film-compensated 45° MTN cell has twist angle φt=45° and β=78°. The required phase retardation d∆n is only 195 nm for normally white operation. To obtain a good dark state at lower voltage, a uniaxial film with α=110° and (d∆n)film=27 nm is added. Figure 3.4 (a) shows the 1D simulated results of the voltage-dependent reflectance curve under broadband R, G and B lights. The maximum reflectance of this mode can reach 100% and on-state voltage is 5V. Figure 3.4 (b) shows the calculated results of iso-contrast viewing diagram under broadband green light with Von=5.0 V and Voff=0.7 V. We can see that the contour line for 1000:1 contrast ratio exceeds 10° polar angle in all directions. This large viewing cone results from the small d∆n value. Figure 3.4 (c) demonstrates the 2D simulation results of the LC director profile and the reflectance for the dark-bright-dark pixel configuration with 2.3 µm cell gap. It is shown that its response to fringing fields is similar to other two MTN modes as we have discussed. Among them, 90°-MTN cell has the smallest and 63.6°-MTN has the largest light leakages in the dark-pixel areas.
The advantages of the film-compensated 45°-MTN mode are twofold: high optical filled factor (up to 96.7%) and low d∆n value which, in turn, leads to a thin cell gap and fast response time. Therefore, this mode is particularly attractive for color sequential displays using a single LC panel. However, the dark state of this mode is not as good as that of 90°-MTN mode since the complete compensation only occurs at certain wavelengths.
Besides, the required compensation film has a relatively small d∆n value which is not easy to fabricate. The film needs to be laminated onto the surface of the polarizing beam splitter.
Any artifacts or bubbles during the lamination process would be magnified and projected to the screen. Moreover, the film has to withstand high power illumination from the arc lamp.
Fig. 3.4 Simulated results of (a) the broad band RV curve, (b) the broad band green light iso-contrast viewing diagram, and (c) the LC director and the reflectance profiles of 45°-MTN mode with β=78°, d∆n==195 nm, (d∆n)film=27 nm, and 110° film angle related to x-axis.
3.2.1.2 Twisted nematic modes (TN)
NB 63.6
°
-TN ModeAs in 63.6°-MTN mode, this mode also has twist angle φt=63.6° and beta angle β=0°, but without any compensation film. When d∆n=512 nm, it can be operated in normally black mode, however, its bright state is greatly affected by the d/P0 ratio. Figure 3.5 (a) shows the 1D simulated results of the voltage-dependent reflectance curve with d/P0=0.6.
Unlike the MTN devices, NB TN mode has the advantage of relatively low on-state voltage because the LC molecules only need to tilt up slightly to reach the maximum reflectance.
As shown in Fig. 3.5 (a), R⊥ =1 is reached when the applied voltage is merely 3.3 V.
Another advantage of the NB 63.6°-TN mode is that it does not require a compensation film. Hence, the cost is reduced. Figure 3.5 (b) shows the calculated results of the iso-contrast viewing diagram under broadband green light with Von=3.3 V and Voff=0.7 V.
The 1000:1 contrast ratio contour line exceeds 12° viewing cone.
When considering the 2D LC director profile, the chiral dopant plays an important role in the fringing-field effect. Figure 3.5 (c) demonstrates the 2D simulated results with d/P0=0.6 and d=4.8 µm. We find that the chiral dopant reorients the LC molecules and induces severe LC distortions. These distortions extend from the bright-pixel to the dark-pixel areas, which optically generate many unwanted light fringes. Hence, the contrast ratio is very poor in this condition. In order to obtain high reflectance and high contrast ratio simultaneously, we can reduce the cell gap to minimize the fringing-field effect.
Adjusting the d/P0 ratio can also achieve a better performance in this mode. Figure 3.6 shows the dependency of contrast ratio on d/P0 ratio in the 63.6°-TN mode with d=2.1 µm.
As shown in this figure, the largest contrast ratio (~110:1) occurs at around d/P0=0.45.
Besides, a too large or too small d/P0 ratio will deteriorate the contrast ratio.