Ray-optics analysis of inhomogeneous optically anisotropic media

Abstract

When the optical behavior of light in a medium depends on the direction in which light is traveling, the medium is called optically anisotropic. Light is an electromagnetic wave and in this thesis, we discuss the electromagnetic theory on optical anisotropy. We do this with the assumption that the wavelength of light approaches zero. The field in optics in which this approach is applied is called geometrical optics. Then the wave character of light is not taken into account. In addition, we define a light wave as a set of rays, each with a certain direction and polarization state. The polarization state of a light ray defines the direction and the phase of the oscillating electric field of the light. In general, the light path of a light ray in an anisotropic medium depends on both the direction and the polarization state. The study of optical systems by means of calculating ray paths of polarized light rays is called polarized ray tracing. Optical anisotropy in the geometrical-optics approach is a classical problem, and most of the theory has been known for more than a century. Since the 1970s optical anisotropy is frequently discussed in the literature due to the rapid advances in liquid-crystal applications, such as the Liquid-Crystal Display (LCD). Liquid crystal is attractive for high-tech applications since it has the material properties of a fluid and the optical properties of an anisotropic crystal. Moreover, the optical properties of liquid crystal can be controlled with electric or magnetic fields. In the past few years Philips Research has had several activities in the field of liquid crystal. Novel liquid-crystal devices and applications have been investigated and developed into proof-of-principle demonstration models. In 2004 Philips Research introduced an auto-stereoscopic display technique based on liquid-crystal technology. Other examples are liquid-crystal-based backlight architectures for LCDs, liquid-crystal lenses and beam steering devices. Most of these liquid-crystal technologies are characterized by inhomogeneous material properties. This means that the optical properties depend on the position inside the liquid crystal. In view of the development of these technologies it is desired to understand and predict the propagation of light in inhomogeneous optically anisotropic media. This is the main goal of this thesis. Optical anisotropy exists in two forms, namely uniaxial and biaxial anisotropy. Uniaxially anisotropic media are characterized by one optical axis and biaxially anisotropic media by two optical axes. The optical axis is a local direction of symmetry in the optical properties. The literature frequently discusses uniaxial anisotropy in the geometrical-optics approach at the boundary between two different media. However, the literature does not provide a straightforward procedure to calculate ray paths of light rays in the bulk of inhomogeneous uniaxially anisotropic media. Moreover, the literature is nearly silent about the propagation of light rays in inhomogeneous biaxially anisotropic media. In this thesis we provide a general and rigorous overview of the classical theory on the propagation of light through inhomogeneous anisotropic media, either uniaxial or biaxial. The literature provides the fundamental principle for the ray-tracing process in inhomogeneous media in geometrical optics, called the Hamiltonian principle. Building further on the Hamiltonian principle, we introduce general ray-tracing equations for inhomogeneous anisotropic media. These equations are new and define the state-of-the-art in the field of polarized ray tracing. We apply our ray-tracing equations to study the optical properties of a number of liquid-crystal applications. For example, we study the residual lens action of a switchable lens structure for application in auto-stereoscopic three-dimensional (3D) displays. Auto-stereoscopic 3D displays enable a viewer to perceive depth in an image without any additional appliances, such as 3D glasses. With the help of a liquid-crystal lens structure, images can be switched between a normal (2D) mode and a 3D mode. With the help of our ray-tracing procedure, we propose an improved anisotropic lens design to minimize residual lens actions in the 2D mode, without compromising the performance of the 3D mode. The desired lens effect in 3D displays can also be achieved with so-called liquid-crystal gradient-index (GRIN) lenses. The lens effect in these type of lenses is due to gradients in the material properties rather than a physically curved lens surface. For the first time in the literature, we simulate the angular-dependent optical properties of an advanced GRIN lens structure for application in auto-stereoscopic 3D displays. With our advanced ray-tracing method, we have also studied the optical properties of liquid-crystal micro structures for application in for example side-lit LCD backlights. With this exercise, we have shown that our ray-tracing method can be applied to assess complex anisotropic optical configurations. Finally, we formulate a criterion for the applicability of geometrical optics to typical in-plane liquid-crystal configurations in terms of the inhomogeneous material properties. In general, we can conclude that we have worked out the electromagnetic theory of inhomogeneous anisotropic media in the geometrical-optics approach. This has resulted in an advanced ray-optics analysis procedure. This procedure has been applied to study the optical properties of novel liquid-crystal applications. In contrast with other methods addressing optical anisotropy, the advanced ray-tracing procedure provides solid physical insight into the subject, is able to handle large computational domains and can be applied relatively easy to assess complex anisotropic electro-optical devices.