Computation and stability analysis of periodically stationary pulses in a short pulse laser




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Short pulse lasers generate a regular train of ultrashort pulses by balancing gain and loss, dispersion and nonlinearity. The spectrum of such a train of pulses is called a frequency comb, which has a wide range of applications in time and frequency metrology. Modern short pulse lasers generate periodically stationary pulses that change shape as they propagate around the laser, returning to the same shape each round trip. Soliton lasers generate stationary pulses which can be studied with averaged models. How- ever, with each subsequent generation of short pulse laser, there has been a dramatic increase in the amount by which the pulse varies over each round trip. Therefore, lumped models are required to accurately compute the periodically stationary solutions generated by these lasers. A lumped model consists of various components modeled either as input-output de- vices or using partial differential equations, each having a different effect on the pulse. The key issue in the modeling of short pulse lasers is to determine those regions in parameter space in which the laser operates stably. In this thesis, we first describe an evolutionary approach to informally assess the stability of periodically stationary pulses in an experimental stretched pulse laser. We then develop a two-stage dynamical approach to more rigorously determine the stability of periodically stationary pulses in lumped laser models. In the first stage of the dynamical approach, we numerically compute periodically stationary pulses by minimizing a Poincar ́e map functional using a gradient based optimization method. We derive a formula for the gradient of the Poincar ́e map functional that is used in the numerical optimization method. In the second stage of the dynamical approach, we study the linear stability of these periodically stationary pulses. To do so, we present a method based on Floquet theory, in which the stability of periodically stationary pulses is characterized by the spectrum of the monodromy operator, which is obtained by linearizing the laser system about the periodic solution. We establish an existence, uniqueness, and regularity theorem for the monodromy operator under reasonable regularity and decay hypotheses on the periodically stationary pulse. We derive a formula for the essential spectrum of the monodromy operator, which quantifies the growth rate of continuous waves far from the pulse. We provide a rigorous proof of this formula using spectral theory and the theory of evolution semigroups. We implement the dynamical approach using a lumped model of the stretched pulse laser and compare its results with the evolutionary approach. We present results showing excellent agreement between the essential spectrum obtained using a matrix discretization of the monodromy operator and the formula. We use the dynamical approach to study bifurcations from stable to unstable periodically stationary pulses by varying the parameters in the laser. In particular, we demonstrate how the effects of fast saturable loss and slow saturable gain can be combined to generate a stable periodically stationary pulse. Our work represents the first spectral stability analysis of periodically stationary pulses in a realistic lumped model of an experimental short pulse laser.



Mathematics, Physics, Optics