Wavelet-based algorithms are developed for solution of quantum and electromagnetic differential equations. Wavelets offer orthonormal localized bases with built-in multiscale properties for the representation of functions, differential operators, and multiplicative operators. The work described here is part of a series of tools for use in the ultimate goal of general, efficient, accurate and automated wavelet-based algorithms for solution of differential equations. The most recent work, and the focus here, is the elimination of operator matrices in wavelet bases. For molecular quantum eigenvalue and dynamics calculations in multiple dimensions, it is the coupled potential energy matrices that generally dominate storage requirements. A Coefficient Product Approximation (CPA) for the potential operator and wave function wavelet expansions dispenses with the matrix, reducing storage and coding complexity. New developments are required, however. It is determined that the CPA is most accurate for specific choices of wavelet families, and these are given here. They have relatively low approximation order (number of vanishing wavelet function moments), which would ordinarily be thought to compromise both wavelet reconstruction and differentiation accuracy. Higher-order convolutional coefficient filters are determined that overcome both apparent problems. The result is a practical wavelet method where the effect of applying the Hamiltonian matrix to a coefficient vector can be calculated accurately without constructing the matrix. The long-familiar Lanczos propagation algorithm, wherein one constructs and diagonalizes a symmetric tridiagonal matrix, uses both eigenvalues and eigenvectors. We show here that time-reversal-invariance for Hermitian Hamiltonians allows a new algorithm that avoids the usual need to keep a number Lanczos vectors around. The resulting Conjugate Symmetric Lanczos (CSL) method, which will apply for wavelets or other choices of basis or grid discretization, is simultaneously low-operation-count and low-storage. A modified CSL algorithm is used for solution of Maxwell's time-domain equations in Hamiltonian form for non-lossy media. The matrix-free algorithm is expected to complement previous work and to decrease both storage and computational overhead. It is expected- that near-field electromagnetic solutions around nanoparticles will benefit from these wavelet-based tools. Such systems are of importance in plasmon-enhanced spectroscopies.

<p>Complex artificial materials (metamaterials) strongly interact with light and can be used to fabricate structures which mimic a material response that has no natural equivalent. Classical tools for the design of optical or radio frequency devices are often ill-suited to utilize such media or have shortcomings in their ability to capture important physics in the device behavior. Recently it has been demonstrated that the structure of Maxwell's equations can be used to exploit this newly available freedom. By leveraging the `form-invariance' of Maxwell's equations under coordinate transforms, it is possible to develop material distributions in which light will behave as though flowing through warped coordinates. This design process is termed `transformation optics' and has inspired the creation of many novel electromagnetic structures such as the invisibility cloak.</p><p>In this dissertation the tools used in the field of transformation optics will be explored and expanded. Several new designs are discussed, each of which expands upon the ideas that have previously been employed in the field. To begin, I show that the explicit use of a transformation which extends throughout all space may be used to reduce the overall size of an optical device without changing its optical properties. A lens is chosen as a canonical device to demonstrate this behavior. For this work I provided the original idea for a compressing transformation as well as its dielectric-only implementation. I then mentored Dan Roberts as he confirmed the device properties through simulation. I further demonstrate that currents may be succesfully employed within the framework of transformation optics-resulting in novel antenna designs. For this work I suggested handling the sheet currents as the limit of a volumetric current density. I also demonstrated how an intermediate coordinate system could be used to easily handle the types of transformatios which were being explored.</p><p>For a particular functionality the choice of transformation is, in general, not unique. It is natural, then, to seek optimized transformations which reduce the complexity of the final structure. It was recently demonstrated that for some transformations a numerical scheme could be employed to find quasi-conformal transformations for which the requisite complex material distribution could be well approximated by an isotropic, inhomogeneous media. This process was previously used to demonstrate a carpet cloak-a device which masks a bump in a mirror surface. Unlike the more common transformation optical media, which exhibit strong losses at high frequencies, isotropic designs can be readily made to function at infrared or even optical frequencies.</p><p>The prospect of leveraging transformation optics in devices which operate at high frequencies, into the infrared and visible, motivates the use of quasi-conformal transformations in lens design. I demonstrate how transformation optics can be used to take a classical lens design based on spherical symmetry, such as a Luneburg lens, and warp it to suit the requirements of a planar imaging array. I report on the experimental demonstration of this lens at microwave frequencies. In the final design a lens is demonstrated in a two-dimensional field mapping waveguide to have a field of view of ~140 degrees and a bandwidth exceeding a full decade. In this work I proposed the idea of using the inverse of the quasi-conformal transform to arrive at the lens index profile. I performed all necessary simulations and wrote ray tracing code to confirm the properties of the lens. I proposed the metamaterial realization of the lens and performed the necessary retrievals for material design. I wrote code which would create the layout for an arbitrary gradient index structure in a standard computer aided drafting format. I fabricated three lenses-two of which are described in this thesis-and took all of the data shown in the thesis.</p><p>The most well known example of a transformation optical device is the invisibility cloak. Despite the great deal of attention paid to the cloak in the literature, the most natural way in which to quantify the efficacy of the cloak-its cross-section-has never been experimentally determined. This measurement is of practical interest because the cloak provides a useful canonical example of a medium which relies on the unique properties of metamaterials-strong anisotropy, inhomogeneity and both magnetic and electric response. Thus, a cloaking cross-section measurement provides a useful way to quantify advancements in the effective medium theories which form the basis for metamaterials. I report on the first such measurements, performed on the original microwave cloaking design. The experiments were carried out in a two-dimensional TE waveguide. Explicit field maps are used to determine the Bessel decomposition of the scattered wave. It is found that the cloak indeed reduces the scattering cross-section of a concealed metal cylinder in a frequency band from 9.91 to 10.14 GHz. The maximum cross-section reduction was determined to be 24%. The total cross-section and the Bessel decomposition of the scattered wave are compared to an analytical model for the cloaking design which assumes a discrete number of loss-less, homogenized cylinders. While the qualitative features of the cloak-a reduced cross-section at the cloaking frequency-are realized, there is significant deviation from the homogenized calculation. These deviations are associated with loss and inaccuracies of the effective-medium-model for metamaterials. In this work I proposed of direct integration of the fields to perform cross-section measurements. I worked out the necessary formulas to determine the coefficients in the Bessel expansion and the resulting scattering cross-section. I mentored an undergraduate student, Dan Gaultney, who scripted the application of the cross-section analysis and took the necessary data. All of the data in this thesis, however, is based on my own implementation of the data analysis.</p> / Dissertation

Process generated particle contamination on unpatterned silicon wafers etched in an SF6/argon plasma using a Tegal MCR-1 etcher in the plasma triode-1 mode was characterized using response surface methodology. Particle deposition was observed to be a predictable function of plasma parameter space, which can be determined by relatively few statistically designed experiments. A model of particle deposition as a function of 13.56 MHz chamber electrode rf power, chamber pressure, gas flow rate, etch time and 100 kHz wafer electrode power was constructed. It is found that particle deposition depends linearly on etch time and both 13.56 MHz and 100 kHz power. In addition, particle deposition increased with gas flow rate at low flow rate, reaches a maximum, then decreased as flow rate increased further. Moreover, there was no observable effect on particle deposition due to pressure variation in the pressure range explored. Auger chemical analysis showed that the particles contained elemental sulfur, fluorine, silicon, aluminum, carbon and oxygen. Most particles were typically less than 2 μm in diameter.

The postulated characteristics of double-negative (DNG) materials--i.e., materials with simultaneously negative permittivity and negative permeability (ε < 0, μ < 0)--and recent attempts to realize those characteristics with synthetic metamaterials are briefly reviewed. Investigations into the causality of signal propagation in a DNG medium are then presented. Previous research in this topic is examined, and it is verified that a DNG medium must be dispersive in order to be causal. An accurate time-domain description of propagation in a DNG medium is shown to be elusive due to the presence of dispersion, though approximate solutions and recommendations for future analytical research are provided. The results of numerical investigations into this topic are then discussed, and the anticipated combination of causal signal transmission and a negative phase shift are observed in the numerical data. Potential applications of DNG metamaterials to antenna systems are then presented. A DNG shell is observed to reduce the intrinsic reactance sensed by an infinitesimal electric dipole, thereby increasing the dipole's radiated power. Analytical expressions for the fields in the dipole--DNG shell system are derived, and numerical results for a variety of DNG shell configurations are discussed. The presence of a DNG shell is shown to increase the dipole's radiated power by orders of magnitude in some cases. A circuit model of the dipole--DNG shell system is additionally presented and used to interpret the system's physical behavior. The scattering properties of nested metamaterial shells are then analyzed. Various layering combinations of DNG, double-positive (DPS) and single-negative (SNG) shells are observed to produce resonant scattering of an incident, fundamental radial transverse-magnetic (TMᵣ) wave. Reciprocity between the metamaterial configurations that exhibit TMᵣ scattering resonances and those shown to maximize the power radiated by the infinitesimal electric dipole is demonstrated. Several additional metamaterial configurations are shown to produce both resonant TMᵣ scattering and resonant dipole radiation. A resonant configuration with one epsilon-negative (ENG) shell is especially appealing due to its manufacturability. The effects of a DNG layer on the creeping waves scattered by a small metal sphere are also discussed as a minor yet curious offshoot to the scattering analyses.

Modeling and evaluation of communication and computing systems is an important undertaking. In many cases, large-scale systems are designed in an ad-hoc manner, with validation (or disappointment regarding) system performance coming only after an implementation is made. This does not need to be the case. Modern modeling tools and techniques can yield accurate performance predictions that can be used in the design process. Stochastic activity networks (SANs), stochastic Petri nets (SPNs) and analytic solution methods permit specification and fast solution of many complex system models. To enhance the modeling power of SANs (SPNs), new steady-state analysis methods have been proposed for SAN (SPN) models that include non-exponential activities (transitions). The underlying stochastic process is a Markov regenerative process (MRP) when at most one non-exponential activity (transition) is enabled in each marking. Time-efficient algorithms for constructing the Markov regenerative process have been developed. However, the space required to solve such models is often extremely large. This largeness is due to the large number of transitions in the MRP. Traditional analysis methods require all these transitions be stored in memory for efficient computation. If the size of available memory is smaller than that needed to store these transitions, a time-efficient computation is impossible using these methods. To use this class of SANs to model real systems, the space complexity of MRP analysis algorithms must be reduced. In this thesis, we propose a new steady-state analysis method that is time and space efficient. The new method takes advantage of the structure of the underlying process to reduce both computation time and required memory. The performance of the proposed method is compared to existing methods using several SAN examples. In addition, the ability to model real systems using SANs that include exponential and deterministic activities is demonstrated by modeling and evaluating the performability of a group communication protocol, called Psync. In particular, we study message stabilization time (the time required for messages to arrive at all hosts) under a wide variety of workload and message loss probabilities. We then use this information to suggest a modification to Psync to reduce message stabilizing time. Another important issue we consider is the dependability modeling and evaluation of fault-tolerant parallel and distributed systems. Because of the inherent component redundancy in such systems, the state space size of the underlying stochastic process is often very large. Reduced base model construction techniques that take advantage of symmetries in the structure of such systems have the potential to avoid this state space growth. We investigate this claim, by considering the application of SANs together with reduced base model construction for the dependability modeling and evaluation of three different systems: a fault-tolerant parallel computing system, a distributed database architecture, and a multiprocessor shared-memory system.

A two-step methodology for predicting the radiated fields from lines radiating in the presence of conductor-backed substrates is presented. The method employs the use of transmission line theory to find the current distributions on the lines forming the interconnects of a circuit. These currents are used to evaluate the far-fields of the circuit through the use of dipole theory and superposition. The method was tested and validated by comparison to full-wave models. Investigations established that radiation from common-mode currents, which are not accounted for by the circuit analysis, are found to be dominated by the radiation due to differential mode currents, and thus EMI prediction based on the two-step methodology is found to have good engineering accuracy. A full-wave method based on the Finite-Difference Time-Domain (FDTD) is presented for the evaluation of radiation from structures of such geometrical complexity that the transmission line model is not applicable. The Perfectly Matched Layer truncation scheme is implemented in the FDTD and investigated for radiating structures found in printed circuit boards (PCBs). Proximity effects of the PML dictate careful attention to the proper implementation of this absorbing boundary condition. Also, the FDTD subcell model for thin wires is investigated for modelling thin microstrip interconnect lines. To evaluate the far-fields from radiating structures found in multilayer electronic packages, a novel near-to-far field transform at a single frequency is developed and implemented for sources in stratified medium. This transform is validated and investigated with regard to PML and structure proximity. The near-to-far field transform is also implemented in a methodology for obtaining the radiated emissions from a radiating structure. This methodology is used to address important concerns regarding the grounding of heat sinks, "floating" conducting planes, and the electromagnetic behavior of split ground planes.

In many applications it is necessary to determine coupling from a line current source to a nearby wire. Applications include current coupling in high-speed interconnects and wire interaction with a charged particle beam. A common physical configuration occurs when the source and wire are perpendicular to each other. In this dissertation, we investigate the scattered field and coupled current that result from such a configuration. We solve the problem for three different sources: a dipole, an array of dipoles, and a continuous line current. We detail the solution for the line current source where we obtain the scattered field by numerical integration, the far-zone approximation using steepest descents, and the excited current by numerical integration. We also show that the solution of an infinite number of phased dipoles approaches the continuous line source excitation. For the continuous line source case, we assume an infinite traveling wave line current. We also assume that the current magnitude and phase are not affected by the existence of the nearby wire. The current travels with a speed less than the speed of light in the surrounding medium. The wire is infinitely long and infinitesimally thin, and is located a distance d from the line source. We solve for the scattered field both numerically and approximately using steepest descents. We then add corrections to the saddle point approximation through two different approaches. We also solve numerically for the coupled current on the wire. Finally, we produce plots that allow us to compare the levels of the field with and without the wire present. Our problem could serve as a prelude to investigation of a traveling wave of current and an array of parallel wires. However, such a problem is quite different since the physical configuration would then allow the presence of guided waves.

The formation and propagation of hotspots in a cylindrical medium that is undergoing microwave heating is studied in detail. A mathematical model developed by Garcia-Reimbert, C., Minzoni, A. A. and Smyth, N. in Hotspot formation and propagation in Microwave Heating, IMA, Journal of Applied Mathematics (1996), 37, p. 165-179 is used. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading order temperature solution of a singular perturbation analysis is used to reduce the system to a free boundary problem. This approximation accurately predicts the steady-state solutions for the temperature and electric fields in closed form. These solutions are valid for arbitrary values of the electric conductivity, and thus extend the previous (small conductivity) results of Garcia-Reimbert et.al. A time-dependent approximate profile for the electric field is used to obtain an ordinary differential equation for its relaxation to the steady-state. This equation appears to accurately describe the time scale of the electric field's evolution even in the absence of a temperature front (with zero coupling to the temperature), and can be of wider interest than the model for microwave heating studied here. With sufficiently small thermal diffusivity and strong coupling, the differential equation also accurately describes the time evolution of the temperature front's location. A closed form expression for the time scale of the formation of the hotspot is derived for the first time in the literature of hotspot modeling. Finally, a rigorous proof of the existence of steady-state solutions of the free boundary problem is given by a contraction mapping argument.

Six thunderstorms that occurred at the NASA Kennedy Space Center, Florida, have been studied in an attempt to characterize their electrical structure and electrification. Ground-based measurements of the cloud electric fields, the locations of lightning VHF radio sources, cloud-to-ground lightning strike points, and dual-polarization radar data were used in this study. Changes in the electric field due to lightning were used to determine the locations and magnitudes of changes in cloud charge. The fields themselves were used to compute displacement current densities following lightning flashes. The altitudes of negative charge regions were between 6.5 and 8.5 km and were almost constant. The altitude of upper positive charge exhibited more variability, and usually increased as cells developed. Amounts of charge removed by lightning increased during each cell in large storms but were nearly constant during the early part of small storms. A lower positive charge center (LPCC) usually appeared in the fields before any other charge regions could be detected at the ground. A LPCC appeared to be involved in the initiation of the majority of CG flashes. During periods of lightning, a LPCC was sometimes created by a flash, but more typically, LPCCs were produced by a cloud charge separation process. Displacement current densities were used to estimate charge accumulation rates in the cloud. The rates derived for the main negative and upper positive charge regions were compared to the average rate of charge removal by lightning. The generation rates and average lightning currents each had values ranging from 0.2 to 1.5 A and were approximately equal within expected errors in single-cell storms. Once the storm was multicellular, however, the lightning current was larger than the cloud charging rate, possibly because lightning was removing residual charge from older cells. The cloud charging rates and average lightning currents were compared with the currents computed using a non-inductive ice-graupel charging mechanism and radar-derived cloud microphysical data. This mechanism provided currents that were comparable to the observed charging rates and lightning currents and appeared to be capable of producing the LPCC.

In this work we solve two new matrix models, using standard and new techniques. The two models are based on matrix ensembles not previously considered. They are represented by special forms of antisymmetric matrices and are classified in the DIII generator ensemble. It is shown that, in the double scaling limit, their free energy has the same behavior as previous models describing oriented and unoriented surfaces. We also found an additional solution for the chapter 3 model.