Photonic crystals are artificial dielectric lattice structures which may be periodic in up to three dimensions. These strongly diffractive structures interact with electromagnetic radiation in such a way that radiation becomes quantized into discrete energy bands. This results in an optical energy range (known as an optical band gap), for which no modes of propagation are supported by the structure. Photonic crystals can be regarded as either strongly scattering sub-wavelength structures, multi-dimensional diffraction gratings, or as quantum optical devices. Methods of computational analysis are similar to those used to model the properties of electron waves propagating in semiconductor materials, or the interference of waves in multi-layer stacks. Until recently photonic crystals have proven impossible to fabricate at optical wavelengths due to the required sub-micron dimensions. This thesis describes many important advances in the field of photonic crystals, both theoretical and practical, culminating in the demonstration of the first ever photonic crystal waveguide devices with polarisation dependent band gaps at visible wavelengths. Firstly a detailed description of the two-dimensional plane wave method of analysis is given by way of introduction to the theory. This has traditionally been applied to model nominally two-dimensionally periodic lattice structures. However, any device which can actually be fabricated will always have finite height. This greatly modifies the photonic properties. For this reason, the plane wave method is extended to perform a rigorous 'three-dimensional' analysis of a nominally two-dimensionally periodic structure. Some of the most useful future applications for photonic crystals are anticipated to require a planar waveguide geometry. For this reason, the three-dimensional plane wave model has been modified to accurately calculate the guided Bloch-mode structure of photonic crystals incorporated within the guiding core layer of a planar optical waveguide. Two different methods are presented, both of which take full account of the highly dispersive dielectric waveguide boundary conditions. In addition, the latter method also allows the accurate evaluation of the effective mode index of the photonic crystal, as seen by any selected confined Bloch mode. This can be used for the optimisation of mode coupling between a photonic crystal and the outside world.
