Simulation of Refraction, Retardation and Transmission in Liquid Crystal Displays with Slow Lateral Variations K. Neyts, K. Vermeirsch, S. Vermael, H. De Vleeschouwer, F. Bougrioua, S. Rozanski, D. De Boer * , J. van Haaren * , S. Day + ELIS Department, Ghent University, BELGIUM * Philips Research Laboratories, Eindhoven, THE NETHERLANDS + University College London, UK Abstract Slow lateral variations in the liquid crystal properties distort the shape of an incident wavefront. The lateral variation in the phase, obtained with the extended Jones calculus, is used to determine refraction effects. Refraction depends on the polarization state of the light and the resulting transmission through the liquid crystal may be very different from what is obtained with the Jones calculus. Introduction In several recently developed liquid crystal display (LCD) technologies, the director orientation varies as a function of the lateral position on the glass substrate. The resulting inhomogeneous dielectric tensor distorts the wavefront of an incident plane or spherical wave and this may influence the transmission characteristics of the LCD considerably. The aim of this paper is to investigate to what extent small lateral variations in the director orientation will influence the transmission in realistic liquid crystal devices. An accurate treatment of this problem depends largely on the length scale of the lateral inhomogeneity. If the scale of the variation in index of refraction is sufficiently large compared to the wavelength of the light, it is acceptable to determine the transmission at a given location with the Jones calculus, using local values for the director orientation. When on the other hand the scale of a periodic lateral variation is much smaller than the wavelength, the wavefront will basically react to the average of the permittivity tensor [1]. In these two extreme cases, incoming plane waves are transformed into outgoing plane waves with the same k-vector. In the intermediate case, where the scale of the variations is similar to the wavelength of the light source, light is scattered and diffraction effects play an important role. In this paper we present a method to study the effect of lateral variations on a scale which is relatively large compared to the wavelength, which makes it possible to consider diffraction effects as a perturbation on the Jones calculus. Diffraction and Refraction Figure 1 shows the setup of a spherical wave incident on a thin layer with lateral variations in the index of refraction n(x). With this example we will illustrate the approximation of ‘slow lateral variation’ of the index of refraction with respect to x. light source ∆φ(x) D x z θ t d x* k t n(x) n i n t Fig. 1 Setup for diffraction and refraction. According to the Fresnel theory for diffraction [2], the complex amplitude is modulated as a function of the transmission direction: [ ] ∫ ∞ ∞ - - ∆ + + = dx x jk j x D n jk k U xt i xt φ 2 2 exp ) ( (1) with k xt the x component of the k-vector of the transmitted light: ) sin( 2 t t xt n k θ λ π = The first term in the exponent gives the phase delay before the layer is reached; ∆φ is the phase delay in the layer. The above diffraction formula suggests that the entire layer contributes to the transmitted light for a particular k xt . However, in the case that the variation of ∆φ along the x axis is small, the main contribution in the integral is from the x-interval where the phase is stationary. Setting the derivative of the phase equal to zero yields: dx d x D x n k k i xt φ ∆ + + = 2 2 (2) For a given k xt , this equation has a unique solution x* © 2000 SID ISSN1083-1312/00/2001-0225-$1.00 + .00 227 225