Two-Dimensional Optical Simulation Tool for Use in Microdisplay Analysis C. M. Titus, P. J. Bos, J. R. Kelly Liquid-crystal Institute, Kent State University, Kent, OH
Abstract Optical performance of small pixel structures utilized in liquid crystal microdisplays may be affected by diffraction and multidimensional variations in liquid crystal director orientation. Modeling such structures may require more robust computational methods than existing onedimensional techniques. We present an implementation of the Finite-Difference Time-Domain method suitable for analysis of these structures.
Keywords: Modeling, FDTD, Simulation, Optics, Diffraction, Multidimensional
Titus, Charles M. The Liquid-crystal Institute Kent State University Kent, OH 44242 Phone: (330)672-3999 x222 FAX: (330)672-2796 email:
[email protected]
Bos, Philip J. The Liquid-crystal Institute Kent State University Kent, OH 44242 Phone: (330)672-2511 FAX: (330)672-2796 email:
[email protected]
Kelly, Jack R. The Liquid-crystal Institute Kent State University Kent, OH 44242 Phone: (330)672-2633 FAX: (330)672-2796 email:
[email protected]
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Two-Dimensional Optical Simulation Tool for Use in Microdisplay Analysis C. M. Titus Objective and Background Mature liquid crystal display technologies utilize simple structures such as the twisted-nematic. Because of the use of relatively large pixels, the liquid crystal director (and thus the dielectric tensor) can be regarded as possessing variations in only one dimension, normal to the display surfaces. Several methods, such as the extended Jones method1 and the Berreman matrix method2, have been satisfactorily used for optical analysis of these relatively simple liquid crystal displays. These methods are restricted to or designed specifically for simulations of structures with variations of the liquid crystal director in only one-dimension. Recent developments in the areas of liquid crystal microdisplays require the use of smaller pixels. As the pixel size is reduced, the impact of diffractive effects on optical performance may become significant. In addition, the reduction in pixel size results in increased significance of non-uniformity of liquid crystal director orientation due caused by fringing fields. One or both of these problems can also be encountered in diffractive liquid crystal devices and in direct-view display structures such as in-plane switching displays. With these display technology advances there is an increasing need for a more sophisticated, multidimensional optical modeling tool. For these and other reasons, the Finite-Difference Time-Domain (FDTD) computation of Maxwell’s equations has recently been considered for use as an optical simulation tool for liquid crystal structures with generally inhomogeneous director orientation3. The FDTD method was introduced by Yee4 for isotropic media. It makes use of direct discretization of Maxwell’s equations. Because of this, the FDTD method can accommodate multidimensional inhomogeneity of the dielectric tensor. In this paper, we will review our recent analysis5 of the accuracy of FDTD simulation of liquid crystal structures. Then the results of a two-dimensional FDTD computation, applied to a small two-dimensional liquid crystal structure, will illustrate diffractive effects not seen in one-dimensional calculations. At the heart of the FDTD method is the discretization of Maxwell’s equations, which can be written, for sourceless inhomogeneous anisotropic media:
∂ E (r ) −1 = å(r ) ⋅ [∇ × H (r )] ∂t −
∂ H (r ) −1 = µ 0 ⋅ [∇ × E (r )] ∂t
(1)
(2)
where ε-1(r) is the inverse of the spatially varying dielectric tensor. Space and time derivatives are approximated with central finite difference formulae. The structure to be analyzed is discretized spatially and placed on a computational grid. An oscillating wave source is introduced into the grid and the entire grid is time-stepped until steady state oscillations are obtained at all points on the grid. The previous FDTD simulations of liquid crystals employed this technique3,5 In order to produce an accurate solution to Maxwell’s equations, the results must be a solution over an infinite space. However, practical limitations of computer power and memory require the termination of the computational grid. Any such termination method must not affect the computations inside the finite computational grid. In other words, the grid termination must not result in reflection of propagating waves back into the grid. For this reason, the “Perfectly Matched Layer” (PML) absorbing layer6 was used for the simulations in this study. The PML is implemented as an absorbing layer which surrounds the inner computational domain. It consists of a fictitious lossy medium which incorporates magnetic as well as electric conductivity. Impedance matching of the PML with the adjacent inner computational domain makes that interface very transmissive to waves radiating from the inner grid. As a result, radiation reaching the boundary passes into the PML and decays rapidly. Illumination of liquid crystal structures in this study was provided by means of the separate-field formulation7. In this technique, the inner computational domain is divided into two regions. In the inner region the total field, the superposition of the source and the scattered fields is calculated. In the outer region, only the scattered
Two-Dimensional Optical Simulation Tool for Use in Microdisplay Analysis C. M. Titus field is calculated. The source is introduced at the interface between these two regions, added into Maxwell’s equations on the inner side of the interface and subtracted on the outer side. This technique can produce diffractionfree plane waves in simulations for which periodic boundary conditions are not appropriate. The separate-field formulation also permits easy extension of this method to off-axis incident light. Two other matters are of importance in FDTD simulations. First, limiting the size of the time step is necessary to prevent growth of numerically induced oscillations. Second, inaccuracies can result from the temporal and spatial discretization of the problem. Any finite difference approximation of the first derivatives appearing in Maxwell’s equations will be somewhat less than exact. Such discretization can manifest itself in numerical dispersion7, which is the departure of numerical phase velocity from that which would occur in continuous space. In particular, when using second-order finite difference approximations to first derivatives, spectral data generated by FDTD simulations will be shifted to longer wavelengths. As the grid is refined, this shift is reduced. Results Some recent work submitted for publication5 compared FDTD and analytic solutions for several wellknown liquid crystal problems. The results of that study are summarized here. All derivatives in that study were approximated with second-order-accurate finite differences. The first test case was the transmission of light through a twisted-nematic (TN) cell, following the analysis of Gooch and Tarry8. The second case covered Bragg reflection from a planar cholesteric layer9. The previous application of FDTD to liquid crystal simulations presented a qualitative illustration of FDTD-computed polarization rotation by a TN cell3. Quantitative FDTD analysis of a TN cell in normally dark mode showed that some attention to the node spacing was required to produce accurate numerical results. Earlier work by Gooch and Tarry8 provided an analytical expression for transmitted intensity as a function of ∆nd/λ. FDTD computations of the same structure were conducted with node spacings of λ/20 and λ/40. For the coarser grid (figure 1), FDTD results were shifted to longer wavelengths, which is caused by numerical dispersion. Reducing the node spacing by one-half, to λ/40, reduced this shift to a negligible amount Uniform cholesteric liquid crystal layers are known to exhibit Bragg reflection of wavelengths comparable to the cholesteric pitch. An exact analytical solution for this problem at normal incidence can be found in ref. 9. Numerical dispersion, evidenced by the shift of numerical results to longer wavelengths, was apparent for computation using the λ/20 coarser grid (figure 2). Substantially reduced deviation between analytical and numerical reflectivity required a node spacing of λ/60, finer than required for the twisted-nematic problem. The study concluded that, without resorting to higher-order finite-differences, choice of node spacing was the only simple means of achieving accurate FDTD results for liquid crystal structures. As the twisted-nematic case above illustrated, accurate FDTD results can be obtained at moderate node spacing for liquid crystal problems with slowly varying dielectric tensor. As spatial variation of the dielectric tensor increased (i.e. moving from TN to cholesteric layer), finer node spacing was required in order to obtain accurate FDTD computation. Accurate FDTD simulations of liquid crystal structures containing significant defects or high birefringence cholesteric materials may require a very fine grid, perhaps as small as λ/60. Of course, as node spacing is reduced for a fixed geometry FDTD problem, memory requirements increase and more time steps are required to reach steady state. For instance, if the node spacing in a two-dimensional simulation is cut in half, the problem must be represented by four times as many nodes, or four times as much memory. And since the maximum stable time step size is proportional to node spacing, cutting the node spacing in half also requires twice as many time steps to reach a steady state solution. This two-dimensional refined grid calculation would thus take eight times as long and four times as much memory, making accurate two-dimensional FDTD computation inaccessible to most desktop PC’s. Since our previous study, this problem has been addressed by replacing the second-order finite difference approximation of the spatial derivatives with a fourth-order approximation. FDTD computation of the same cholesteric layer produced accurate results forλ/20 node spacing (figure 3). The FDTD method was then applied to a two-dimensional simulation of a simple liquid crystal structure. The grid was organized as shown in figure 4. The liquid crystal structure consisted of a matrix of undisturbed
Two-Dimensional Optical Simulation Tool for Use in Microdisplay Analysis C. M. Titus 1.5µm-thick first-interference minimum TN, into which was placed a 25mm-wide pixel with some field applied. For this initial test, the pixel’s director structure was one-dimensional. The entire liquid crystal slab was illuminated with linearly polarized light, and an ideal polarizer was implemented mathematically at the exit surface, aligned parallel to the incident linearly polarization. A profile of the transmitted intensity is shown in figure 5. Except for the outer edges of the slab and the region in and around the central pixel, the first-interference-minimum TN structure transmits nothing, as expected. Transmission in and around the central pixel exhibits a transmitted intensity characteristic of near-field (Fresnel) diffraction through a slit aperture. This near-field diffraction is not duplicated by a one-dimensional Jones calculation (figure 6). Work currently in progress includes insertion of true two-dimensional director configurations and the transformation of the transmitted near-field intensity profile into the far field; those results will be presented. Impact The FDTD method provides an avenue toward optical simulation of liquid crystal devices with small pixel size and/or multi-dimensional inhomogeneities of director orientation. These conditions occur in liquid crystal microdisplays, some diffractive liquid crystal devices, and in existing liquid crystal displays such as those employing in-plane switching. In particular, FDTD appears to be capable of uncovering near-field diffractive effects not seen when using existing one-dimensional methods. In future comparisons with far-field results of onedimensional optical techniques applied to these problems, the one-dimensional methods may well prove equal to the task. However, without very precise experimental methods or a tool such as FDTD, that cannot be known. References 1.
A. Lien, Appl. Phys. Lett.; Extended Jones Matrix Representation for the Twisted Nematic Liquid-Crystal Display at Oblique Incidence; 57 (1990) 2767.
2.
D. W. Berreman; Optics in Stratified and Anisotropic Media: 4 X 4 Matrix Formulation; J. Opt. Soc. Am. 62 (1972) 502.
3.
B. Witzigmann, P. Regli, W. Fichtner; Rigorous Electromagnetic Simulation of Liquid Crystal Displays; J. Opt. Soc. Am. A 15 (1998) 753.
4.
K. S. Yee; Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media; IEEE Trans. Ant. Prop. AP-14 (1966) 302.
5.
C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland; Comparison of Analytical Calculations to Finite-Difference Time-Domain Simulations of One-Dimensional Spatially Varying Anisotropic Liquid Crystal Structures; submitted (October 1998) for publication in the Japanese Journal of Applied Physics.
6.
J-P. Berenger; A Perfectly Matched Layer for the Absorption of Electromagnetic Waves; J. Comp. Phys. 114 (1994) 185.
7.
A. Taflove; Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
8.
C. H. Gooch, H. A. Tarry; The Optical Properties of Twisted Nematic Liquid Crystal Structures with Twist Angles < 90 degrees; J. Phys. D 8 (1975) 1575.
9.
H. Wohler, G. Haas, M. Fritsch, D. A. Mlynski; Faster 4 X 4 Matrix Method for Uniaxial Inhomogeneous Media; J. Opt. Soc. Am. A 5 (1988) 1554. Prior Publications
Reference 5 above refers to similar work performed here at Kent State. That prior work was restricted to onedimensional analysis, and was intended to serve as the foundation for the work described here. The work presented in this paper adds the use of fourth-order-accurate finite differencing and applies the method to two-dimensional liquid crystal problems.
Inner Computational Domain
Analytical
0.15
Source Line
wavelength/20 wavelength/40
Liquid Crystal Matrix
PML
0.10
Liquid Crystal Matrix
Liquid Crystal Pixel
PML
Transmitted Intensity
PML
0.20
0.05 Plane wave
0.00 0.0
1.0
2.0
3.0
4.0
∆ nd/λ Figure 1: Transmission through normally dark TN. Comparison of FDTD results at two different grid resolutions to Gooch and Tarry curve.
PML
Figure 4: Layout of FDTD grid employed in two-dimensional computations
1.0
1.0 0.8 0.6 Exact
0.4 0.2
λ/20 λ/60
Transmitted Intensity
Reflectivity (R)
1.2
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.0 0.47 0.49 0.51 0.53 0.55 0.57 0.59 Wavelength, µ m Figure 2: Reflection from planar cholesteric layer. Comparisoon of FDTD results at two different grid resolutions to analytical calculation.
0.0 0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Lateral Position on Exit Surface, µm
Figure 5: Transmitted intensity along exit surface of TN layer containing 25 µm-wide partially activated pixel. 1.0 0.9
1.0 0.8 0.6 0.4
Exact λ/20
0.2
Transmitted Intensity
Reflectivity (R)
1.2
0.8 0.7 0.6 0.5 0.4 FDTD
0.3
1-D Jones
0.2 0.1
0.0 0.47 0.49 0.51 0.53 0.55 0.57 0.59 Wavelength, µ m Figure 3: Reflection from planar cholesteric layer. Comparisoon of FDTD results using fourth-order spatial finite differences to analytical calculation.
0.0 20
25
30
35
40
45
50
55
Lateral Position on Exit Surface, µ m Figure 6: Transmitted intensity along exit surface of TN layer containing 25 µm-wide partially activated pixel.
