Cross-waves generated by an oscillating wavemaker in a rectangular wave tank are examined when two or more modes are simultaneously unstable. The partial differential equations governing the evolution of the complex amplitude of inviscid cross-waves are shown to be two coupled nonlinear Schrodinger equations with transverse modulations. Energy dissipation in the system is taken into account by the inclusion of a linear viscous damping term into the amplitude equations. A linearized stability analysis is performed on these equations to determine the critical modes, the growth rates and the stability curves. A center manifold analysis is used to reduce the PDE's to a system of ODE's in the neighborhood of a codimension two point where two adjacent spanwise modes are simultaneously nearly marginal. Four possible steady states of the system are found, one of which is a mixed mode state. A Hopf bifurcation from the mixed mode is predicted for a certain region of the parameter plane, suggesting the possibility of energy interchange between the two modes. The stability of the Hopf bifurcation is determined by studying a fifth order problem, where the quintic contributions come from the higher modes as well as the perturbations of damping and detuning.

published_or_final_version / abstract / toc / Mechanical Engineering / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Michigan, 1930. / From Physical review, v. 36, Oct. 1, 1930.

Thesis (Ph. D.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 99-104).

This thesis describes an investigation of the propagation of a pressure wave, through a long straight pipe, in a rarefied atmosphere of Argon. This phenomenon is approximately analogous to thermal wave propagation in the boundary scattering limit. Casimir and Ziman studied this latter phenomenon for the steady state case. P.W. Matthews extended their theory to the non-steady state and found a diffusion coefficient D depending on L. Smoluchowski studied the corresponding phenomenon in gases for the steady state and again P.W. Matthews extended the theory to the non-steady state, and again found a dependence of D on L. Therefore at the low frequency limit a wave pressure velocity would depend on L. An experiment was set up in order to produce a wave pressure and measure its velocity as a function of the frequency and the pressure and hence L. Results have been obtained and no dependence on the pressure has been observed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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Copies of author's previously published articles, inserted. Bibliography: p. 165-179. Reports on the development of a scalable high power laser for gravitational wave interferometry.

Bibliography: p. 165-179. Electronic publication; Full text available in PDF format; abstract in HTML format. Reports on the development of a scalable high power laser for gravitational wave interferometry. Electronic reproduction.[Australia] :Australian Digital Theses Program,2001.

Some modern laser applications require continuous wave (CW) high power (>100 W), and diffraction limited performance near 1.064 um. One such laser application with these, and additional, requirements is gravitational wave interferometry. This thesis will report the development of a scalable high power laser for this application. A high-power, single-transverse-mode laser might be produced by intensely pumping the small mode volume within a stable resonator or by using a resonator that has a large transverse mode. Intensely pumping a small volume can lead to crystal fracture and large thermally-induced wavefront aberrations. Using a large transverse mode would also be difficult if using a stable resonator as these are, in general, not suited to fundamental modes that have large cross-sectional areas. Unstable resonators, by comparison, routinely produce fundamental modes that have large cross-sectional areas. They have been used for decades with high-power, high-gain chemical or gas lasers and provide efficient energy extraction, good mode discrimination and beam quality. However, the low gain of Nd:YAG in combination with the high output coupling associated with unstable resonators would limit the efficiency of such a CW laser. One way to utilize the properties of unstable resonators while reducing the output coupling, and thus increase the efficiency, is to use a stable-unstable resonator. These resonators are stable in one plane and unstable in the orthogonal plane, rather than unstable in both planes. The required output coupling can be further reduced without degrading the beam quality by using a Graded Reflectivity Mirror (GRM) as the output coupler. The soft aperturing of the GRM also eliminates diffraction loss associated with scraper mirrors in hard-edged unstable resonators, and enhances mode discrimination. The stable-unstable resonator reported in this thesis is side-pumped by fibre-coupled diode- lasers and side-cooled. It uses a total internal reflection (TIR) zigzag slab geometry, in which the zigzag is co-planar with the pumping and cooling. The resonator is stable in the plane of the zigzag (horizontal) and unstable in the plane orthogonal to the zigzag. In this configuration the strong thermal lensing in the horizontal direction is averaged out by the zigzag. The vertical thermal lens is controlled by Thermo-Electric Coolers (TECs) which are used to adjust the temperature of the bottom and top surfaces of the slab. To test the performance of the side-pumped, side-cooled laser head it was operated initially with a stable resonator. Efficient operation was achieved and will be reported. Control of the refractive index profile (thermal lens) using the TECs on the bottom and top surfaces results in a vertical thermal lens that could be set to any value between 47 mm and 450 mm. The thermal lens encountered by the zigzag mode in the plane of pumping and cooling is weak (horizontal direction) and independent of TEC current. Thus, the thermal lensing in the horizontal and vertical directions is de-coupled, as is necessary for scalability of the mode volume in the vertical direction. A travelling-wave (for ease of injection locking) stable-unstable resonator was investigated using a Fox-Li model, which assumed a greater pump power and mode volume than used for the laser head presented in this thesis. A strip, n=2 super-Gaussian GRM is shown to be the optimum output coupler for the stable-unstable laser. Furthermore, it is shown that the output coupling loss associated with a resonator magnification of -1.3 could be sustained using pump densities below the crystal fracture limit. Useful operation over a realistic range of thermal lens focal lengths is predicted. The validity of the Fox-Li modelling is confirmed using with a standing-wave stable-unstable resonator. The standing-wave resonator was chosen as it suited the available crystal and pump power used for the work in this thesis. The GRM reflectivity profile used the minimum commercially available profile radius. The vertical thermal lens is varied by adjusting the pump power, and then by adjusting the temperature of the bottom and top surfaces at full pump power. This demonstrated CW operation of the standing-wave laser with M=1.3 and good beam quality. Good qualitative agreement with the Fox-Li model of the standing-wave resonator is thus confirmed. Finally, suppression of the multiple longitudinal modes by injection locking is reported. / Thesis (Ph.D.)--Physics and Mathematical Physics, 2001.