Wave Propagation through Photonic Crystal Slabs

Firstly, we explain the fundamental concepts and the construction of the unconditionally stable method to solve the time dependent Maxwell’s equations. We describe the spatial discretization procedure and the product-formula approach to perform the time integration as well as its stability. Starting with the study of wave propagation in conventional dielectric materials, we generalize the concepts for homogeneous isotropic dielectric media with simultaneously negative permeability and permittivity, also known as negative index materials. We describe the phenomenon of negative refraction of a plane wave falling onto an interface between a conventional dielectric medium and a negative index material. Photonic crystals are considered as a good candidate of negative index material. We demonstrate that for certain frequencies two-dimensional photonic crystals can behave as if they have a negative refractive index which is not limited by the refractive index of the constituents but is determined by the photonic band structure. In some cases this effective refractive index can be negative. We perform a systematic investigation of negative refraction in triangular lattice photonic crystals. We study both photonic crystal structures made from dielectric cylindrical pillars organized on a triangular lattice and photonic crystal structures made from a dielectric material with circular holes drilled in a triangular lattice configuration. We demonstrate that for both photonic crystal structures negative refraction can be observed and that the negative angles of refraction obtained from the finite-difference time domain (FDTD) simulations are in good agreement with the angles predicted by the wave vector diagram analysis method. For the photonic crystal structures made from a dielectric material with circular holes drilled in a triangular lattice configuration, frequencies can be found for which the effective refractive index is approximately -1. For those frequencies we perform an elaborate study of the imaging properties of a photonic crystal slab made of this material. We investigate the effects of the surface terminations of the photonic crystal slab, the source slab distance, the slab thickness and the mirror symmetry of the slab on the image position and image quality. We also consider the position of the internal focus. We demonstrate that for some frequencies the photonic crystal slab behaves in many respects as a slab made from a homogeneous medium with an effective refractive index of approximately -1 and thus operates as a negative-index lens. As an application of the photonic crystal lens we study the image formation of two point sources placed in front of the lens and the image transfer by a cascaded stack consisting of two and three triangular-lattice photonic crystal slabs separated by a layer of air. We also investigate the imaging properties of square lattice photonic crystals. We demonstrate that neither photonic crystal slabs made from dielectric cylindrical pillars organized on a square lattice, nor photonic crystal slabs made from a dielectric material with circular holes drilled in a square lattice configuration, behave as a negative-index lens. At the end of the thesis, we present a simple and efficient procedure to compute the spontaneous emission rate from short-time FDTD simulation data of the electromagnetic field energy. As an illustration, we apply this procedure to two-dimensional photonic crystals. For comparison, we calculate the local radiative density of states employing an unconditionally stable FDTD method that is without solving the eigenvalue problem and integrating over the (first) Brillouin zone. We demonstrate that both methods yield the same predictions about the enhancement or suppression of the spontaneous emission rate by photonic crystals.