Optical hysteresis of ridge waveguide magnetic garnet films and cavities in two-dimensional magneto-photonic crystal slabs

Magnetic iron garnets as well as magnetic photonic crystals are of great interests in magneto-optic applications such as isolators, current captors, circulators, TE-TM mode conversion, wavelength accordable filters, optical sensors and switches, all of which provide a promising platform for future integrated optical circuits. In the present work, two topics are studied based on magnetic iron garnet films. In the first part, the characteristics of the magnetization are investigated for ridge waveguides fabricated on (100) oriented iron garnet thin films. The magnetic response in magneto-optic waveguides patterned on epitaxial magnetic garnet films depends on the crystallographic orientation of the waveguides and the magnetic anisotropy of the material. These can be studied by polarization rotation hysteresis loops, which are related to the component of magnetization parallel to the light propagation direction and the linear birefringence. Polarization rotation hysteresis loops for low birefringence waveguides with different orientations are experimentally investigated. Asymmetric stepped curves are obtained from waveguides along, due to the large magnetocrystalline anisotropy in the plane. A model based on the free energy density is developed to demonstrate the motion of the magnetization and can be used in the design of magneto-optic devices. The second part of this thesis focuses on the design and fabrication of high-Q cavities in two-dimensional magneto-photonic crystal slabs. The device consists of a layer of silicon and a layer of iron garnet thin film. Triangular lattice elliptical air holes are patterned in the slab. The fundamental TM band gap overlaps with the first-order TE band gap from 0374~0.431(a/λ) showing that both TE and TM polarization light can be confined in the photonic crystals. A nanocavity is designed to obtain both TE and TM defect modes in the band gaps. Additional work is needed to overlap the TE and TM defect modes and obtain a high-Q cavity so as to develop miniaturized Faraday rotators.

Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

UNDERSTANDING FARMERS’ PERCEPTIONS AND THE EFFECTS OF SHEA TREE Vitellaria paradoxa DISTRIBUTION IN AGROFORESTRY PARKLANDS OF THE UPPER WEST REGION, GHANA

Agroforestry parklands represent a vast majority of the agricultural landscape under subsistent-oriented farming in semi-arid West Africa. Parklands are characterized by the growth of well- maintained trees (e.g., shea) on cultivated fields as a result of both environmental and human influences. Shea (Vitellaria paradoxa) provides a cultural and economic benefit to the local people of Ghana, especially women. Periods between traditional fallow rotation systems have reduced recently due to agricultural development and a demand for higher production. As a result, shea trees, which regenerate during fallow periods, has decreased over the landscape. The aim of this study was to determine beneficial spatial distributions of V. paradoxa to maintain high yields of staple crops, and how management of V. paradoxa will differ between male and female farmers as a result of farmer based needs and use of shea. Vegetation growth and grain yield of maize (Zea mays) associated with individual trees, clumped trees, and open fields were measured. Soil moisture and light availability were also measured to determine how V. paradoxa affected resource availability of maize in either clumped or scattered distributions of V. paradoxa. As expected, light availability increased as measurement locations moved farther away from all trees. However, soil moisture was actually greater under trees in clumps than under individual trees. Maize stalk height and cob length showed no difference between clumped and single trees at each measurement location. Grain yield per plot and per cob increased as measurement locations moved farther from single trees, but was actually greater near clumped trees that in the open field subplots. Cob length and maize stalk height increased with greater light availability, but grain yield per cob or per plot showed no relationship with light, but were not affected by soil moisture. Conversely, grain yield increased with increasing soil moisture, but had no relationship with light availability. Initial farming capital is the largest constraint to female farmers; therefore the collection of shea can help provide women with added income that could meet their specific farming needs. Our data indicate that overall effects of maintaining clumped distributions of V. paradoxa provided beneficial microclimates for staple crops when compared to single trees. It is recommended that male and female farmers allow shea to grow in clumped spatial distributions rather than maintaining scattered, individual trees.