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Ever since the first laser demonstration in 1960, applications for laser systems have increased to include diverse fields such as: national defense, biology and medicine, entertainment, imaging, and communications. In order to serve the growing demand, a wide range of laser types including solid-state, semiconductor, gas, and dye lasers have been developed. For most applications it is critical to have lasers with both high optical power and excellent beam quality. This has traditionally been difficult to simultaneously achieve in semiconductor lasers. In the mid 1990's, the advent of an optically pumped semiconductor vertical-external-cavity surface-emitting laser (VECSEL) led to the demonstration of high (multi-watt) output power with near diffraction limited (TEM00) beam quality. Since that time VECSELs covering large wavelength regions have been developed. It is the objective of this dissertation to investigate and explore novel cavity designs which can lead to increased functionality in high power, high brightness VECSELs. Optically pumped VECSELs have previously demonstrated their potential for high power, high brightness operation. In addition, the "open" cavity design of this type of laser makes intracavity nonlinear frequency conversion, linewidth narrowing, and spectral tuning very efficient. By altering the external cavity design it is possible to add additional functionality to this already flexible design. In this dissertation, the history, theory, design, and fabrication are first presented as VECSEL performance relies heavily on the design and fabrication of the chip. Basic cavities such as the linear cavity and v-shaped cavity will be discussed, including the role they play in wavelength tuning, transverse mode profile, and mode stability. The development of a VECSEL for use as a sodium guide star laser is presented including the theory and simulation of intracavity frequency generation in a modified v-cavity. The results show agreement with theory and the measurement of the sodium D1 and D2 lines are demonstrated. A discussion of gain coupled VECSELs in which a single pump area accommodates two laser cavities is demonstrated and a description of mode competition and the importance of spontaneous emission in determining the lasing condition is discussed. Finally the T-cavity configuration is presented. This configuration allows for the spatial overlap of two VECSEL cavities operating with orthogonal polarizations. Independent tuning of each cavity is presented as well as the quality of the beam overlap and demonstration of Type II intracavity sum frequency generation. Future applications to this configuration are discussed in the generation of high power, high brightness lasers operating from the UV to far-infrared and even terahertz regimes.

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The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.

This dissertation is devoted to better understanding the role that fluids play in the selection of the shapes and functions of objects and creatures in nature. Toward that end, three specific examples are considered: stalactites, icicles, and species of colonial green algae known as Volvox. In the cases of stalactites and icicles, the object's growth is considered as a free-boundary problem. For stalactites, the coupling of thin-film fluid dynamics with calcium carbonate chemistry leads to a local, geometric growth law that is proportional to the thickness of the water layer covering the surface at any point. Application of this law to a uniformly translating shape allows a universal stalactite form to be derived; the comparison of this shape to images of actual stalactites supports the theory. In the case of icicles, the transport of the latent heat of fusion is coupled with the dynamics of both the thin-film of water encompassing the icicle and a thermally buoyant boundary layer in the immediately surrounding air. The uniformly translating shape solution is found to be parameter-free, and is, in fact, the same shape exhibited by stalactites. A comparison between this shape and icicle images validates the theory. The final example considers how advection of nutrients due to the stirring of water by the flagella of a Volvox colony leads to a metabolite uptake rate that is much greater than would occur by diffusion alone. Moreover, nutrient acquisition by pure diffusion would limit the size of Volvox species to a certain bottleneck radius at the point where diffusional uptake just meets metabolic demands, whereas advection increases the uptake in such a way as to avoid this problem entirely, thus enabling the evolution of the larger Volvox species.

We investigate conditions for the ergodicity of threshold autoregressive time series by embedding the time series in a general state Markov chain and apply a FosterLyapunov drift condition to demonstrate ergodicity of the Markov chain. We are particularly interested in demonstrating V uniform ergodicity where the test function V () is a function of a norm on the statespace. In this dissertation we provide conditions under which the general state space chain may be approximated by a simpler system, whether deterministic or stochastic, and provide conditions on the simpler system which imply V uniform ergodicity of the general state space Markov chain and thus the threshold autoregressive time series embedded in it. We also examine conditions under which the general state space chain may be classified as transient. Finally, in some cases we provide conditions under which central limit theorems will exist for the V uniformly ergodic general state space chain.

NonLinear Principal Component Analysis (NLPCA) using three-hidden-layer feed-forward neural networks can produce solutions that over-fit the data and are non-unique. These problems have been dealt with by subjective methods during the network training. This study shows that these problems are intrinsic due to the three-hidden-layer architecture. A simplified two-hidden-layer feed-forward neural network that has no encoding layer and no bottleneck and output biases is proposed. This new, compact NLPCA model alleviates these problems without employing the subjective methods and is called quasi-objective. The compact NLPCA is applied to the zonal winds observed at seven pressure levels between 10 and 70 hPa in the equatorial stratosphere to represent the Quasi-Biennial Oscillation (QBO) and investigate its variability and structure. The two nonlinear principal components of the dataset offer a clear picture of the QBO. In particular, their structure shows that the QBO phase consists of a predominant 28.4-month cycle that is modulated by an 11-year cycle and a longer-period cycle. The significant difference in variability of the winds between cold and warm seasons and the tendency for a seasonal synchronization of the QBO phases are well captured. The one-dimensional NLPCA approximation of the dataset provides a better representation of the QBO than the classical principal component analysis and a better description of the asymmetry of the QBO between westerly and easterly shear zones and between their transitions. The compact NLPCA is then applied to the Arctic Oscillation (AO) index and aforementioned zonal winds to investigate the relationship of the AO with the QBO. The NLPCA of the AO index and zonal-winds dataset shows clearly that, of covariation of the two oscillations, the phase defined by the two nonlinear principal components progresses with a predominant 28.4-month periodicity, plus the 11-year and longer-period modulations. Large positive values of the AO index occur when westerlies prevail near the middle and upper levels of the equatorial stratosphere. Large negative values of the AO index arise when easterlies occupy over half the layer of the equatorial stratosphere.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

This thesis reports on ultrafast linear and nonlinear optical probing of nanometer thick films. Exfoliated graphene and few-layer graphite are probed through optical second harmonic generation (SHG) with 800 nm, 150 fs pulses. Samples of varying thickness from 1 carbon layer to bulk graphite are deposited onto an oxidized silicon substrate. SHG measurements are taken as a function of azimuthal rotation angle of the films. It is found that the SHG from graphene is much weaker than that from bilayer graphene, and has a qualitatively different azimuthal pattern. As the sample thickness increases from bilayer graphene to bulk graphite, the SHG yield generally decreases. Both of these effects are explained in terms of the symmetry of graphene and graphite, and modeled using multilayer optical transfer matrices, and an identical set of nonlinear susceptibility tensor elements for the front and back surfaces. These tensors are independent of sample thickness. MnAs films of 150 and 190 nm thickness on (001)GaAs are optically excited with 775 nm, 200 fs pump pulses. Specular SHG at 388 nm and first order optical diffraction at ∼ 400 nm are used to probe the samples on timescales up to 2 μs. It is found that the SHG probes the temperature-dependent, spatially averaged, surface strain. This strain reaches a maximum deviation in ∼ 6–100 ps after optical excitation depending on the pump fluence and initial temperature. The strain then recovers in hundreds of picoseconds, a timescale consistent with heat diffusion. The optical diffraction probes the first Fourier component of the paramagnetic–ferromagnetic stripes inherent to MnAs films in the 10–40◦C temperature range. After optical excitation, the diffraction data show highly nonthermal behaviour in the MnAs films. If a sample is excited from the coexistence phase, the diffraction signal shows decaying oscillations with a period of ∼ 335±4 (408±4) ps for the 150 (190) nm films; this is consistent with the release of a standing acoustic wave. Decay occurs on a timescale of ∼ 2 ns consistent with local diffusion through the films. The stripes are restored on a timescale of hundreds of nanoseconds, with a temporal behavior consistent with a diffusion process, possibly thermal in origin.