In many applications such as beam shaping, mode conversion, and phase encoding it is necessary to alter the spatial phase profile of a beam via a phase mask. Conventional techniques to accomplish this either involve surface relief profiling in thin films such as PMMA or refractive index modulation in bulk photorefractive crystals such as lithium niobate. These materials have been used extensively for the past several decades and perform admirably in low power conditions. However, in high power systems these materials will be destroyed, requiring a new means of producing phase masks. In this dissertation a method for producing robust phase masks in the bulk of photo-thermo-refractive glass is developed and successfully demonstrated. Three main applications of phase masks were studied in detail. The first is mode conversion, where binary phase masks convert a Gaussian beam to higher order modes. The second is beam shaping, where phase masks are used as focusing elements and for optical vortex generation. Near-theoretical conversion efficiency was achieved for all elements in these cases. The third application is aberration analysis and correction. Here the degradation of volume Bragg gratings recorded in an aberrated holographic system was modeled, with the simulations indicating that correcting elements are generally necessary for high-quality production of gratings. Corrective phase masks are designed which can selectively correct one or multiple aberrations of varying magnitudes are shown. A new type of optical element is also developed in which a phase mask is encoded into a transmitting Bragg grating. This technique combines the local phase modulation of a phase mask with the multiplexing ability of transmitting Bragg gratings, allowing for multiple phase masks to be recorded in a single element. These masks may be used at any wavelength iii satisfying the Bragg condition, increasing the useful wavelength regime of a single element by orders of magnitude.

This thesis is principally concerned with the conditions under which a feedback oscillator with a single non-linear element can support two component waves, These component waves are required to have unrelated frequencies. A theory is produced to predict the oscillation frequencies and amplitudes and examine their stability. The conclusions reached in this thesis are then compared with those reached by previous workers in the same field. The concluding parts of the thesis contain an examination of the possible approximations which can be made in order to represent various non-linear elements mathematically and also a demonstration of a system for examining frequency relationships without recourse to direct numerical measurement. / Thesis / Master of Engineering (ME)

We obtain an upper bound for decay rate of p-adic oscillatory integrals of with analytic phase function and constructible amplitude map. / Thesis / Doctor of Philosophy (PhD)

The goal of this research is to measure viscosity via the analysis of amplitude response of a piezo driven vibrating cantilevers partially immersed in a viscous medium. As a driving frequency is applied to a piezoceramic material, the external forces acting on the system will affect its maximum amplitude. This thesis applies this principle through experimental and analytical analyses of the proportional relationship between viscosity and the amplitude response of the first natural frequency mode of the sinusoidal vibration. Currently, the few cantilever-based viscometer designs that exist employ resonant frequency response as the parameter by which the viscosity is correlated. The proposed piezoelectric viscometer employs amplitude response in lieu of resonant frequency response. The goal of this aspect of the research was to provide data confirming amplitude response as a viable method for determining viscosity. A miniature piezoelectric plate was mounted to a small stainless-steel cantilever beam. The tip of the cantilever was immersed within various fluid test samples. The cantilever was then swept through a range of frequencies in which the first frequency mode resided. The operating principle being as the viscosity of the fluid increases the amplitude response of cantilever vibration will decrease relatively. What was found was in fact an inversely exponential relationship between dynamic viscosity and the cantilever beam's vibrational amplitude response. The experiment was performed using three types of cantilevers as to experimentally test the sensitivity of each.

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Estimating local site effects and modifying the uncertainty in ground motion predictions are two indispensable parts of partially non-ergodic site-specific PSHA. Local site effects can be estimated using site response simulations or recorded ground motions at the site. When such predictions are available, the aleatory variability of ground motions used in PSHA can be changed to the single station sigma value. However, in these cases, the epistemic uncertainty in predicting site effects must be incorporated into the hazard analyses. This research focuses on the challenges specific to conducting partially non-ergodic site-specific PSHA using recorded ground motions or site response analysis. The main challenge in estimating local site effects using recorded data is whether ground motions collected in a relatively short time can be used to estimate site effects for long return period events. We first develop a database for recorded ground motions at the KiK-net array to investigate this question and use this database to develop a predictive model for the Fourier Amplitude Spectra of ground motions. The ground motion model (GMM) residuals are used to investigate the stability of site terms across different tectonic regimes. We observe that empirical site terms are stable across different tectonic regimes. This observation allows the use of ground motions from any tectonic regime (whether they belong to the tectonic regime that controls the hazard or not) to estimate local site effects. Moreover, in Fourier amplitude, site effects are not dependent on event magnitude and source to site distance; therefore, estimates of site effects from low magnitude events can be easily extrapolated to larger events. The Fourier amplitude GMM developed in this study adds to the library of Fourier amplitude models to be used in future partially non-ergodic site-specific PSHAs. In practice, one of the most common tools for simulating wave propagation is 1-D site response analysis. Two central assumptions in 1-D site response analysis are that the soil profile is comprised of horizontal soil layers of infinite extent and that the vertically propagating SH-waves control the horizontal component of ground motion. SH-waves tend to propagate vertically near the surface because as earthquake waves hit softer layers traveling from the source to the site, they refract until the path becomes steeply inclined. The validity of both assumptions in 1-D site response depends on the geological setting at the site and the geology between the earthquake source and the site, raising the question of which sites are suitable for 1-D site response analysis and what the model error in 1-D site response analysis is. We use the GMM developed for FAS to estimate observed and empirical site terms. The empirical site effects are then compared with the theoretical site effects to determine whether sites are amenable to 1-D site response analyses, and to quantify the model error in the analyses. / Doctor of Philosophy / It is impossible to predict future earthquake-induced ground motions due to randomness in the process and a lack of knowledge. In fact, there are significant uncertainties not only in predicting the location, time, and magnitude of a future earthquake but also in predicting the intensity of ground motion induced by a given future earthquake. Therefore, assessing the safety of the human environment against earthquake hazards requires a method that considers all sources of uncertainties. To this end, Earthquake Engineers have developed Probabilistic Seismic Hazard Analysis(PSHA) framework. Structural engineers use the results of PSHA to design a new structure or assess the safety of an existing building. The accuracy of PSHA estimations leads to designs that are both safe and cost-efficient. The distribution of possible ground motions induced by a given earthquake scenario significantly controls the result of PSHA. This distribution should consider the effect of source, source to site path, and local site effects. This research focuses on improving PSHA results by estimating local site effects using recorded ground motions or simulating wave propagation in the site. In estimating local site effects using recorded data, the local site effect observed in ground motions collected in a relatively short time window is used to estimate hazards from all scenarios. However, the collected ground motions usually belong to frequent low magnitude events that are different from large magnitude events that control the hazard. This difference requires either using a measure of local site effect that is independent of the magnitude and distance of the earthquake or considering the effect of magnitude and distance on the local site effect estimate. Moreover, since frequent events sample different sources and paths than large events, we need to make sure the local site effect is consistent across different sources and paths. This research develops Ground Motion Models(GMMs) for Fourier amplitude, a linear function of ground motion times series, using Japanese ground motions. The ratio of Fourier amplitude at the surface over bedrock is a measure of local site effect that is not dependant on magnitude and distance. The model is then used to see if the trade-off between source and site effect and path and site effect is significant or not. In practice, one of the most common tools for simulating wave propagation is 1-D site response analysis. Two central assumptions in 1-D site response analysis are that the soil profile comprises horizontal soil layers of infinite extent and that the vertically propagating horizontal shear waves (SH-waves) control the horizontal component of ground motion. SH-waves tend to propagate vertically near the surface because as earthquake waves hit softer layers traveling from the source to the site, they refract until the path becomes vertically inclined. The validity of both assumptions in 1-D site response depends on the geological setting at the site and the geology between the earthquake source and the site, raising the question of which sites are suitable for 1-D site response analysis and what the model error in 1-D site response analysis is. We use the GMM developed for FAS to estimate empirical local site effects. The empirical site effects are then compared with the theoretical site effects to determine whether sites are amenable to 1-D site response analyses and quantify the model error in the analyses.

No / This paper presents a novel and hardware-efficient architecture for power-of-two FFT processors. The proposed design is based on the phase-amplitude splitting technique which converts a DFT to cyclic convolutions and additions. The cyclic convolutions are implemented with a filter-like structure and the additions are computed with several stages of butterfly processing units. The proposed architecture requires no multiplier, and comparisons with other designs show it can save up to 39% total equivalent gates for an 8-bit 16-point FPGA-based FFT processor.