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.

Genomic consequences of mating system evolution in the Pacific coastal dune endemic, Abronia umbellata (Nyctaginaceae)

The advent of next-generation sequencing has significantly reduced the cost of obtaining large-scale genetic resources, opening the door for genomic studies of non-model but ecologically interesting species. The shift in mating system, from outcrossing to selfing, has occurred thousands of times in angiosperms and is accompanied by profound changes in the population genetics and ecology of a species. A large body of work has been devoted to understanding why the shift occurs and the impact of the shift on the genetics of the resulting selfing populations, however, the causes and consequences of the transition to selfing involve a complicated interaction of genetic and demographic factors which are difficult to untangle. Abronia umbellata is a Pacific coastal dune endemic which displays a striking shift in mating system across its geographic range, with large-flowered outcrossing populations south of San Francisco and small-flowered selfing populations to the north. Abronia umbellata is an attractive model system for the study of mating system transitions because the shift appears to be recent and therefore less obscured by post-shift processes, it has a near one-dimensional geographic range which simplifies analysis and interpretation, and demographic data has been collected for many of the populations. In this study, we generated transcriptome-level data for 12 plants including individuals from both subspecies, along with a resequencing study of 48 individuals from populations across the range. The genetic analysis revealed a recent transition to selfing involving a drastic reduction in genetic diversity in the selfing lineage, potentially indicative of a recent population bottleneck and a transition to selfing due to reproductive assurance. Interestingly, the genetic structure of the populations was not coincident with the current subspecies demarcation, and two large-flowered populations were classified with the selfing subspecies, suggesting a potential need for re-evaluation of the current subspecies classification. Our finding of low diversity in selfing populations may also have implications for the conservation value of the threatened selfing subspecies.