Three -dimensional visualization and analysis of a dual -disk resonator with dielectric misalignment and surface anomaly using edge finite elements

The search for life in other planets and solar systems by scientists and engineers brings about an effort to design and develop equipment of high standards which extend the capability to listen for signals which have been traveling in space many light years. In this study the purpose was to provide a more realistic and illustrative scientific understanding of one such piece of precision equipment, the dielectric resonator, which designers seek to extend its frequency stability below 10−15. At such tolerances special cryogenic cooling procedures are required. Due to its accuracy it can be used to set short term time and frequency standards to correct the atomic clock. A theoretical means of studying this type of resonant device is necessary. One contribution made in extending the current understanding of such a device is the scientific tool developed specifically for this dissertation. It applies the Minimum Theorem from variational calculus using edge finite elements for numerical modeling. The use of quasi-linear vector basis functions allowed an implementation of Helmholtz\u27s three-dimensional equation without a penalty term. Furthermore, the intermixing of spurious solutions with the true ones due to a nodal basis was eliminated. Calculation of the average edge electric fields was made possible by applying the Rayleigh-Ritz criterion. Model enclosure was provided by a cylindrical metal shield situated in a rectangular coordinate system. Linear, homogeneous, nonmagnetic, lossless, uniaxial, and anisotropic media were considered. Integration of NASA\u27s Unix Lanczos eigensolver permitted the accurate estimation of the smaller eigenvalues and associated vectors for large matrices on workstations and personal computers in relatively short computational times. Calculation of the lower frequency modes demonstrated the ability to address device imperfections for two selected cases. Both were influenced by problems encountered in the use of crystals constrained by cost, or because of imperfections in the manufacturing process. The first, considered dielectric properties not aligned with the model axis. And, the second inserted a surface variation on the crystal. Numerically, the general decrease in mode frequencies due to misalignment and overall increase due to a surface anomaly were graphed. Visually, the resulting deformation of electromagnetic mode structures with a high energy concentration in the altered material were shown. Those not visibly affected were modes with less energy in the modified material. An expected effect on their quality factors was given. The research presented provides illustrative scientific understanding using models that are more realistic. Moreover, it aids in resonator design by predicting electromagnetic behavior due to media imperfections