High Harmonic Generation (HHG) enables the production of bursts of coherent soft x-rays with attosecond pulse duration. This process arrises from the nonlinear interaction between intense infrared laser pulses and an ionizing gas medium. Soft x-ray photons are used for spectroscopy of inner-shell electron correlation and exchange processes, and the availability of attosecond pulse durations will enable these processes to be resolved on their natural time scales. The maximum or cutoff photon energy in HHG increases with both the intensity as well as the wavelength of the driving laser. It is highly desirable to increase the harmonic cutoff as this will allow for the generation of shorter attosecond pulses, as well as HHG spectroscopy of increasingly energetic electronic transitions. While the harmonic cutoff increases with laser wavelength, there is a corresponding decrease in harmonic yield. The first part of this thesis describes the experimental measurement of the wavelength scaling of HHG efficiency, which we report as lambda^(-6.3) in xenon, and lambda^(-6.5) in krypton. To increase the HHG cutoff, we have developed a 1.8 um source, with stable carrier envelope phase and a pulse duration of <2 optical cycles. The 1.8 um wavelength allowed for a significant increase in the harmonic cutoff compared to equivalent 800 nm sources, while still maintaing reasonable harmonic yield. By focusing this source into neon we have produced 400 eV harmonics that extend into the x-ray water window. In addition to providing a source of photons for a secondary target, the HHG spectrum caries the signature of the electronic structure of the generating medium. In krypton we observed a Cooper minimum at 85 eV, showing that photoionization cross sections can be measured with HHG. Measurements in xenon lead to the first clear observation of electron correlation effects during HHG, which manifest as a broad peak in the HHG spectrum centred at 100 eV. This thesis also describes several improvements to the HHG experiment including the development of an ionization detector for measuring laser intensity, as well as an investigation into the role of laser mode quality on HHG phase matching and efficiency.

Abstract: Nonlinear multicontrast microscopy is established in this study as an important tool for understanding biological structure and function of muscle cells. Second harmonic generation, third harmonic generation and multi-photon excitation fluorescence are acquired simultaneously in order to establish the origin of nonlinear signal generation in myocytes, and investigate myocyte structure and functionality during muscle contraction. Using structural cross-correlation image analysis, an algorithm developed specifically for this research, for the first time, third harmonic generation is shown to originate from the mitochondria in myocytes. The second harmonic, which is generated from the anisotropic bands of the sarcomeres, is further shown to be dependent on the crystalline order of the sarcomeres, thereby providing a potential diagnostic tool to evaluate disorder in muscle cells. The combination of the second and third harmonic provides complementary information that can be used to further elucidate the basic principles of muscle contraction. Time-lapse nonlinear microscopic imaging showed structural and functional dynamics in the myocytes. The second harmonic contrast revealed nonsynchronized nanocontractions of sarcomeres in relaxed, non-contracting, cardiomyocytes and Drosophila muscle samples, providing insight into the asynchronous behaviour of individual sarcomeres. Furthermore, macrocontracting samples were found to exhibit a synchronization of nanocontractions, providing new evidence for how muscles contract. Dynamic image correlation analysis, another algorithm developed specifically for this investigation, is used to reveal networks of mitochondria, which show fluctuations of multi-photon excitation fluorescence and third harmonic generation signals. The intensity fluctuations in the networks reveal both slow and fast dynamics; phase shifts of the slow dynamics between different networks are observed. Fast dynamics appear only in the inner networks, suggesting functional difference between interfibrillar and subsarcolemma mitochondria. The groundwork for studying bioenergetics of mitochondria in cardiomyocytes with nonlinear multimodal microscopy is fully developed in this work. The origin of the nonlinear signals and the development of the image analysis techniques provide a solid foundation to further study of muscle contractility and bioenergetics.

Conventional step-index silica fibers do not possess a second-order optical nonlinearity due to symmetry concerns. However, through the process of poling, the generation of a frozen-in DC field $E^{DC}$, and in turn, a non-zero second-order nonlinearity $\chi^{(2)} = 3\chi^{(3)}E^{DC}$, can be created in optical fibers. In this thesis, I measure the individual $\chi^{(2)}$ tensor elements of birefringent periodically poled fiber via second-harmonic generation and sum-frequency generation experiments. The symmetry of the $\chi^{(2)}$ tensor is consistent with that of the $\chi^{(3)}$ for isotropic media. This is the first study that characterizes all the $\chi^{(2)}$ tensor elements in birefringent poled fiber. Furthermore, I investigate the intermix of the $\chi^{(2)}$ tensor elements by twisting the fiber, which results in the generation of new second-harmonic signals not observed in untwisted fiber. The conversion efficiencies and spectral positions of these new signals can be varied by twisting the fiber.

The three wave mixing process was modeled in GaAs:AlGaAs superlattices using two new modeling tools that were developed in the course of this work: A 2D beam propagation tool for optimizing quasi-phase matching gratings, and a 1D iterative beam propagation tool for determining the output powers and threshold of optical parametric oscillators of arbitrary geometries. The 2D tool predicts close to 80% enhancement of conversion e ciency by phase matching near 800 nm compared to 775 nm, which was the originally designed operation wavelength. The model also predicts resonant behaviour for an abrupt grating pro le. The 1D tool was used to determine the threshold conditions for para- metric oscillation for di erent geometries. The performances of di erent phase matching approaches in AlGaAs were quantitatively compared. The model also indicated the need for pulsed operation to achieve reasonably low threshold powers in AlGaAs waveguides.

Conventional step-index silica fibers do not possess a second-order optical nonlinearity due to symmetry concerns. However, through the process of poling, the generation of a frozen-in DC field $E^{DC}$, and in turn, a non-zero second-order nonlinearity $\chi^{(2)} = 3\chi^{(3)}E^{DC}$, can be created in optical fibers. In this thesis, I measure the individual $\chi^{(2)}$ tensor elements of birefringent periodically poled fiber via second-harmonic generation and sum-frequency generation experiments. The symmetry of the $\chi^{(2)}$ tensor is consistent with that of the $\chi^{(3)}$ for isotropic media. This is the first study that characterizes all the $\chi^{(2)}$ tensor elements in birefringent poled fiber. Furthermore, I investigate the intermix of the $\chi^{(2)}$ tensor elements by twisting the fiber, which results in the generation of new second-harmonic signals not observed in untwisted fiber. The conversion efficiencies and spectral positions of these new signals can be varied by twisting the fiber.

The three wave mixing process was modeled in GaAs:AlGaAs superlattices using two new modeling tools that were developed in the course of this work: A 2D beam propagation tool for optimizing quasi-phase matching gratings, and a 1D iterative beam propagation tool for determining the output powers and threshold of optical parametric oscillators of arbitrary geometries. The 2D tool predicts close to 80% enhancement of conversion e ciency by phase matching near 800 nm compared to 775 nm, which was the originally designed operation wavelength. The model also predicts resonant behaviour for an abrupt grating pro le. The 1D tool was used to determine the threshold conditions for para- metric oscillation for di erent geometries. The performances of di erent phase matching approaches in AlGaAs were quantitatively compared. The model also indicated the need for pulsed operation to achieve reasonably low threshold powers in AlGaAs waveguides.

Abstract: Nonlinear multicontrast microscopy is established in this study as an important tool for understanding biological structure and function of muscle cells. Second harmonic generation, third harmonic generation and multi-photon excitation fluorescence are acquired simultaneously in order to establish the origin of nonlinear signal generation in myocytes, and investigate myocyte structure and functionality during muscle contraction. Using structural cross-correlation image analysis, an algorithm developed specifically for this research, for the first time, third harmonic generation is shown to originate from the mitochondria in myocytes. The second harmonic, which is generated from the anisotropic bands of the sarcomeres, is further shown to be dependent on the crystalline order of the sarcomeres, thereby providing a potential diagnostic tool to evaluate disorder in muscle cells. The combination of the second and third harmonic provides complementary information that can be used to further elucidate the basic principles of muscle contraction. Time-lapse nonlinear microscopic imaging showed structural and functional dynamics in the myocytes. The second harmonic contrast revealed nonsynchronized nanocontractions of sarcomeres in relaxed, non-contracting, cardiomyocytes and Drosophila muscle samples, providing insight into the asynchronous behaviour of individual sarcomeres. Furthermore, macrocontracting samples were found to exhibit a synchronization of nanocontractions, providing new evidence for how muscles contract. Dynamic image correlation analysis, another algorithm developed specifically for this investigation, is used to reveal networks of mitochondria, which show fluctuations of multi-photon excitation fluorescence and third harmonic generation signals. The intensity fluctuations in the networks reveal both slow and fast dynamics; phase shifts of the slow dynamics between different networks are observed. Fast dynamics appear only in the inner networks, suggesting functional difference between interfibrillar and subsarcolemma mitochondria. The groundwork for studying bioenergetics of mitochondria in cardiomyocytes with nonlinear multimodal microscopy is fully developed in this work. The origin of the nonlinear signals and the development of the image analysis techniques provide a solid foundation to further study of muscle contractility and bioenergetics.

In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

<p>A National Aeronautics and Space Administration computational fluid dynamics code, OVERFLOW 2, was modified to utilize a harmonic balance solution method. This modification allows for the direct calculation of the nonlinear frequency-domain solution of a periodic, unsteady flow while avoiding the time consuming calculation of long physical transients that arise in aeroelastic applications.</p><p>With the usual implementation of this harmonic balance method, converting an implicit flow solver from a time marching solution method to a harmonic balance solution method results in an unstable numerical scheme. However, a relatively simple and computationally inexpensive stabilization technique has been developed and is utilized. With this stabilization technique, it is possible to convert an existing implicit time-domain solver to a nonlinear frequency-domain method with minimal modifications to the existing code.</p><p>This new frequency-domain version of OVERFLOW 2 utilizes the many features of the original code, such as various discretization methods and several turbulence models. The use of Chimera overset grids in OVERFLOW 2 requires care when implemented in the frequency-domain. This research presents a harmonic balance version of OVERFLOW 2 that is capable of solving on overset grids for sufficiently small unsteady amplitudes.</p> / Dissertation