Tight-Binding Approximations in 1D and 2D Coupled-Cavity Photonic Crystal Structures

Light confinement and controlling an optical field has numerous applications in the field of telecommunications for optical signals processing. When the wavelength of the electromagnetic field is on the order of the period of a photonic microstructure, the field undergoes reflection, refraction, and coherent scattering. This produces photonic bandgaps, forbidden frequency regions or spectral stop bands where light cannot exist. Dielectric perturbations that break the perfect periodicity of these structures produce what is analogous to an impurity state in the bandgap of a semiconductor. The defect modes that exist at discrete frequencies within the photonic bandgap are spatially localized about the cavity-defects in the photonic crystal. In this thesis the properties of two tight-binding approximations (TBAs) are investigated in one-dimensional and two-dimensional coupled-cavity photonic crystal structures We require an efficient and simple approach that ensures the continuity of the electromagnetic field across dielectric interfaces in complex structures. In this thesis we develop \textrm{E} -- and \textrm{D} --TBAs to calculate the modes in finite 1D and 2D two-defect coupled-cavity photonic crystal structures. In the \textrm{E} -- and \textrm{D} --TBAs we expand the coupled-cavity \overrightarrow{E} --modes in terms of the individual \overrightarrow{E} -- and \overrightarrow{D} --modes, respectively. We investigate the dependence of the defect modes, their frequencies and quality factors on the relative placement of the defects in the photonic crystal structures. We then elucidate the differences between the two TBA formulations, and describe the conditions under which these formulations may be more robust when encountering a dielectric perturbation. Our 1D analysis showed that the 1D modes were sensitive to the structure geometry. The antisymmetric \textrm{D} mode amplitudes show that the \textrm{D} --TBA did not capture the correct (tangential \overrightarrow{E} --field) boundary conditions. However, the \textrm{D} --TBA did not yield significantly poorer results compared to the \textrm{E} --TBA. Our 2D analysis reveals that the \textrm{E} -- and \textrm{D} --TBAs produced nearly identical mode profiles for every structure. Plots of the relative difference between the \textrm{E} and \textrm{D} mode amplitudes show that the \textrm{D} --TBA did capture the correct (normal \overrightarrow{E} --field) boundary conditions. We found that the 2D TBA CC mode calculations were 125-150 times faster than an FDTD calculation for the same two-defect PCS. Notwithstanding this efficiency, the appropriateness of either TBA was found to depend on the geometry of the structure and the mode(s), i.e. whether or not the mode has a large normal or tangential component.

Towards Mapping of Rock Walls Using a UAV-Mounted 2D Laser Scanner in GPS Denied Environments

In geotechnical engineering, the stability of rock excavations and walls is estimated by using tools that include a map of the orientations of exposed rock faces. However, measuring these orientations by using conventional methods can be time consuming, sometimes dangerous, and is limited to regions of the exposed rock that are reachable by a human. This thesis introduces a 2D, simulated, quadcopter-based rock wall mapping algorithm for GPS denied environments such as underground mines or near high walls on surface. The proposed algorithm employs techniques from the field of robotics known as simultaneous localization and mapping (SLAM) and is a step towards 3D rock wall mapping. Not only are quadcopters agile, but they can hover. This is very useful for confined spaces such as underground or near rock walls. The quadcopter requires sensors to enable self localization and mapping in dark, confined and GPS denied environments. However, these sensors are limited by the quadcopter payload and power restrictions. Because of these restrictions, a light weight 2D laser scanner is proposed. As a first step towards a 3D mapping algorithm, this thesis proposes a simplified scenario in which a simulated 1D laser range finder and 2D IMU are mounted on a quadcopter that is moving on a plane. Because the 1D laser does not provide enough information to map the 2D world from a single measurement, many measurements are combined over the trajectory of the quadcopter. Least Squares Optimization (LSO) is used to optimize the estimated trajectory and rock face for all data collected over the length of a light. Simulation results show that the mapping algorithm developed is a good first step. It shows that by combining measurements over a trajectory, the scanned rock face can be estimated using a lower-dimensional range sensor. A swathing manoeuvre is introduced as a way to promote loop closures within a short time period, thus reducing accumulated error. Some suggestions on how to improve the algorithm are also provided.