In this thesis, two aspects of string theory are discussed, tensionless strings and supersymmetric sigma models. The equivalent to a massless particle in string theory is a tensionless string. Even almost 30 years after it was first mentioned, it is still quite poorly understood. We discuss how tensionless strings give rise to exact solutions to supergravity and solve closed tensionless string theory in the ten dimensional maximally supersymmetric plane wave background, a contraction of AdS(5)xS(5) where tensionless strings are of great interest due to their proposed relation to higher spin gauge theory via the AdS/CFT correspondence. For a sigma model, the amount of supersymmetry on its worldsheet restricts the geometry of the target space. For N=(2,2) supersymmetry, for example, the target space has to be bi-hermitian. Recently, with generalized complex geometry, a new mathematical framework was developed that is especially suited to discuss the target space geometry of sigma models in a Hamiltonian formulation. Bi-hermitian geometry is so-called generalized Kähler geometry but the relation is involved. We discuss various amounts of supersymmetry in phase space and show that this relation can be established by considering the equivalence between the Hamilton and Lagrange formulation of the sigma model. In the study of generalized supersymmetric sigma models, we find objects that favor a geometrical interpretation beyond generalized complex geometry.

In the present thesis, computational methods for theanalysis and design of photonic bandgap structure areconsidered. Many numerical methods have been used to study suchstructures. Among them, the plane wave expansion method is veryoften used. Using this method, we show that inclusions ofelliptic air holes can be used effectively to obtain a largercomplete band gap for two-dimensional (2D) photonic crystals.An optimal design of a 2D photonic crystal is also consideredin the thesis using a combination of the plane wave expansionmethod and the conjugate gradient method. We find that amaximum complete 2D band gap can be obtained by connectingdielectric rods with veins for a photonic crystal with a squarelattice of air holes in GaAs. For some problems, such as defect modes, the plane waveexpansion method is extremely time-consuming. It seems that thefinite-difference time-domain (FDTD) method is promising, sincethe computational time is proportional to the number of thediscretization points in the computation domain (i.e., it is oforderN). A FDTD scheme in a nonorthogonal coordinate systemis presented in the thesis to calculate the band structure of a2D photonic crystal consisting of askew lattice. The algorithmcan easily be used for any complicated inclusion configuration,which can have both the dielectric and metallic constituents.The FDTD method is also applied to calculate the off-plane bandstructures of 2D photonic crystals in the present thesis. Wealso propose a numerical method for computing defect modes in2D crystals (with dielectric or metallic inclusions). Comparedto the FDTD transmission spectra method, our method reduces thecomputation time and memory significantly, and finds as manydefect modes as possible, including those that are not excitedby an incident plane wave in the FDTD transmission spectramethod. The FDTD method has also been applied to calculateguided modes and surface modes in 2D photonic crystals using acombination of the periodic boundary condition and theperfectly matched layer for the boundary treatment. Anefficient FDTD method, in which only real variables are used,is also proposed for the full-wave analysis of guided modes inphotonic crystal fibers. / QC 20100629

The absence of appropriate media to cultivate photons efficiently at the micro or nano scale has hindered taking the full advantage of processing information with light. The proposal of such a medium for light, known as photonic crystals (PCs)--multi-dimensional artificially periodic dielectric media--brings the possibility of a revolution in communications and sensing much closer. In such media, one can manipulate light at a scale on the order of the wavelength or even shorter. Applications of PCs other than in communication include bio-sensing because of the peculiar properties of PCs such as the capability of enhance field-matter interaction and control over the group velocity. As a result, PC waveguide (PCW) structures are of interest and it is expected that PC sensors offer the feasibility of multi-analyte and compact sensing schemes as well as the ability of the detection of small absolute analyte quantities (nanoliters) and low-concentration samples (picomoles), which may be advantages over conventional approaches such as fiber optic and slab waveguide sensors. Depending on the nature of the analyte, either dispersive or absorptive sensing schemes may be implemented. Light propagation is controlled fully only with 3D PCs. One of the problems arising due to reducing the dimension to 2D is that PCs become strongly polarization sensitive. In many cases, one wants to implement polarization insensitive devices such that the PC provides a full band gap for all polarizations. To address this problem, a novel type of PC called annular PC is proposed and analyzed. The capability of tuning the TE and TM polarizations independently within the same structure provides great flexibility to produce polarization-independent or polarization-dependent devices as desired. PCW bends are expected to be the essential building blocks of photonic integrated circuits. Sharp corners having small radii of curvature can be obtained. To enhance the low-loss and narrow-band transmission through these bends, PC heterostructures waveguide concept is introduced. We show that in PCWs formed by joining different types of PCs in a single structure, light can flow around extremely sharp bends in ways that are not possible using conventional PCWs based on a single type of PC.

Les diagnostics cliniques des maladies cardio-vasculaires sont principalement effectués à l’aide d’échographies Doppler-couleur malgré ses restrictions : mesures de vélocité dépendantes de l’angle ainsi qu’une fréquence d’images plus faible à cause de focalisation traditionnelle. Deux études, utilisant des approches différentes, adressent ces restrictions en utilisant l’imagerie à onde-plane, post-traitée avec des méthodes de délai et sommation et d’autocorrélation. L’objectif de la présente étude est de ré-implémenté ces méthodes pour analyser certains paramètres qui affecte la précision des estimations de la vélocité du flux sanguin en utilisant le Doppler vectoriel 2D. À l’aide d’expériences in vitro sur des flux paraboliques stationnaires effectuées avec un système Verasonics, l’impact de quatre paramètres sur la précision de la cartographie a été évalué : le nombre d’inclinaisons par orientation, la longueur d’ensemble pour les images à orientation unique, le nombre de cycles par pulsation, ainsi que l’angle de l’orientation pour différents flux. Les valeurs optimales sont de 7 inclinaisons par orientation, une orientation de ±15° avec 6 cycles par pulsation. La précision de la reconstruction est comparable à l’échographie Doppler conventionnelle, tout en ayant une fréquence d’image 10 à 20 fois supérieure, permettant une meilleure caractérisation des transitions rapides qui requiert une résolution temporelle élevée. / Clinical diagnosis of cardiovascular disease is dominated by colour-Doppler ultrasound despite its limitations: angle-dependent velocity measurements and low frame-rate from conventional focusing. Two studies, varying in their approach, address these limitations using plane-wave imaging, post-processed with the delay-and-sum and autocorrelation methods. The aim of this study is to re-implement these methods, investigating some parameters which affect blood velocity estimation accuracy using 2D vector-Doppler. Through in vitro experimentation on stationary parabolic flow, using a Verasonics system, four parameters were tested on mapping accuracy: number of tilts per orientation, ensemble length for single titled images, cycles per transmit pulse, and orientation angle at various flow-rates. The optimal estimates were found for 7 compounded tilts per image, oriented at ±15° with 6 cycles per pulse. Reconstruction accuracies were comparable to conventional Doppler; however, maintaining frame-rates more than 10 to 20 times faster, allowing better characterization of fast transient events requiring higher temporal resolution.

This thesis considers two different aspects of string theory, the tensionless limit of the string and supersymmetric sigma models with extended supersymmetry. First, the tensionless limit is used to find a IIB supergravity background generated by a tensionless string. The background has the characteristics of a gravitational shock-wave. Then, the quantization of the tensionless string in a pp-wave background is performed and the result is found to agree with what is obtained by taking a tensionless limit directly in the quantized theory of the tensile string. Hence, in the pp-wave background the tensionless limit commutes with quantization. Next, supersymmetric sigma models and the relation between extended world-sheet supersymmetry and target space geometry is studied. The sigma model with N=(2,2) extended supersymmetry is considered and the requirement on the target space to have a bi-Hermitean geometry is reviewed. The Hamiltonian formulation of the model is constructed and the target space is shown to have generalized Kähler geometry. The equivalence between bi-Hermitean geometry and generalized Kähler follows, in this context, from the equivalence between the Lagrangian- and Hamiltonian formulation of the sigma model. Then, T-duality in the Hamiltonian formulation of the sigma model is studied and the explicit T-duality transformation is constructed. It is shown that the transformation is a symplectomorphism, i.e. a generalization of a canonical transformation. Under certain assumptions, the amount of extended supersymmetry present in the sigma model is shown to be preserved under the T-duality transformation. Next, extended supersymmetry in a first order formulation of the sigma model is studied. By requiring N=(2,2) extended world-sheet supersymmetry an intriguing geometrical structure arises and in a special case generalized complex geometry is found to be contained in the new framework.

Typical ultrafast plane-wave ultrasound imaging involves: 1) insonifying the medium with several plane-wave pulses emitted at different angles by a linear transducer array, 2) sampling the returning echo signals, after each plane-wave emission, with the same transducer array, 3) beamforming the recorded angle-specific raw data frames, and 4) compounding the beamformed data frames over all angles to form a final image. This thesis attempts to address the following question: Given a set of available plane-wave emission angles, which ones should we select for acquisition (i.e., which angle-specific raw data frames should we sample), to achieve adequate image quality at low cost associated with both sampling and computation? We propose a simple similarity-driven angle selection scheme and evaluate its several variants that rely on user-specified similarity measurement thresholds guiding the recursive angle selection process. Our results show that the proposed scheme has a low computational overhead and can yield significant savings in terms of the amount of sampled raw data. / Graduate

The purpose of this work was to examine the effect of the use the assumption κω2K/ΕCM «1 in calculating K-shell ionization cross sections in the plane wave Born approximation (PWBA) where κω2K is the observed binding energy of the K-shell and ECM is the energy of the incident particle in the center of mass system. Avoiding this assumption produces a threshold for ionization at Ecm = κω2K. Calculations employing the assumption, which leads to the use of approximate limits of integration, do not go to zero for even the .Lowest values of the incident energy.

Coherent plane-wave compounding (CPWC) ultrasound is an important modality enabling ultrafast biomedical imaging. To perform CWPC image reconstruction for a stratified (horizontally layered) medium, one needs to know how the speed of sound (SOS) varies with the propagation depth. Incorrect sound speed and layer thickness assumptions can cause focusing errors, degraded spatial resolution and significant geometrical distortions resulting in poor image reconstruction. We aim to determine the speed of sound and thickness values for each horizontal layer to accurately locate the recorded reflection events to their true locations within the medium. Our CPWC image reconstruction process is based on phase-shift migration (PSM) that requires the user to specify the speed of sound and thickness of each layer in advance. Prior to performing phase-shift migration (one layer at a time, starting from the surface), we first estimate the speed of sound values of a given layer using a cosine similarity metric, based on the data obtained by a multi-element transducer array for two different plane-wave emission angles. Then, we use our speed estimate to identify the layer thickness via end-of-layer boundary detection. A low-cost alternative that obtains reconstructed images with fewer phase shifts (i.e., fewer complex multiplications) using a spectral energy threshold is also proposed in this thesis. Our evaluation results, based on the CPWC imaging simulation of a three-layer medium, show that our sound speed and layer thickness estimates are within 4% of their true values (i.e., those used to generate simulated data). We have also confirmed the accuracy of our speed and layer thickness estimation separately, using two experimental datasets representing two special cases. For speed estimation, we used a CPWC imaging dataset for a constant-speed (i.e., single-layer) medium, yielding estimates within 1% of their true values. For layer thickness estimation, we used a monostatic (i.e., single-element) synthetic-aperture (SA) imaging dataset of the three-layer medium, also yielding estimates within 1% of their true values. Our evaluation results for the low-cost alternative showed a 93% reduction in complex multiplications for the three-layer CPWC imaging dataset and 76% for the three-layer monostatic SA imaging dataset, producing images nearly similar to those obtained using the original PSM methods. / Graduate

Sound pressure equalization of audio signals using digital signal processors has been a subject of ongoing study for many years. The traditional approach is to equalize sound at a point in a listening environment, but because of its specific dependence on the room frequency response between a source and receiver position, this equalization generally causes the spectral response to worsen significantly at other locations in the room. This work presents both a time-invariant and a time-varying implementation of an adaptive acoustic energy density equalization filter for a one-dimensional sound field. Energy density equalization addresses the aforementioned challenge and others that relate to sound equalization. The theory and real-time implementation of time-invariant sound pressure and energy density equalizers designed using the least-squares method are presented, and their performances are compared. An implementation of a time-varying energy density equalizer is also presented. Time-invariant equalization results based on real-time measurements in a plane-wave tube are presented. A sound pressure equalizer results in a nearly flat spectral magnitude at the point of equalization. However, it causes the frequencies corresponding to spatial nulls at that point to be undesirably boosted elsewhere in the sound field, where those nulls do not exist at the same frequencies. An energy density equalization filter identifies and compensates for all resonances and other global spectral effects of the tube and loudspeaker. It does not attempt to equalize the spatially varying frequency nulls caused by local pressure nodes at the point of equalization. An implementation of a time-varying energy density equalizer is also presented. This method uses the filtered-x filter update to adjust the filter coefficients in real-time to adapt to changes in the sound field. Convergence of the filter over time is demonstrated as the closed end of the tube is opened, then closed once again. Thus, the research results demonstrate that an acoustic energy density filter can be used to time-adaptively equalize global spectral anomalies of a loudspeaker and a one-dimensional sound field.

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