Stability of Nonlinear Waves in Mode-locked Lasers and Nonlinear Optics

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New analytical methods are needed now as novel pulse evolutions in lasers promise to greatly enhance the performance of practical instruments. This project will perform theoretical studies of new approaches to the generation of femtosecond light pulses and optical bullets in mode-locked lasers. The underlying mathematical methods are rooted in nonlinear dispersive wave theory and their dynamics and stability in nonlinear, saturable media. New pulse-shaping mechanisms have the potential for major impact on ultrafast science, but current theoretical understanding is rudimentary because a pulse undergoes large changes in its temporal shape, spectral shape, and phase or frequency as it traverses a laser cavity; these in turn pose severe challenges to mathematical models. Highly-chirped and/or self-similar pulse solutions can exist in the presence of strong dissipation, creating new classes of laser pulses that offer remarkable behavior and performance. Development of improved models proposed here will enable major scientific advances such as the generation of multidimensional solitons and will lead to enhanced instruments for applications. The research aims to be of a truly interdisciplinary nature: combining asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis with models which are based on experimental observations. The impact of the proposed research will extend beyond the understanding of nonlinear pulse propagation in laser systems. The concepts developed in this project will bear on a range of topics, from the fundamental science of nonlinear dynamical systems to commercial laser instruments. Lasers that generate femtosecond-duration optical pulses have great potential for expanding the range of short-pulse optical techniques into real-world applications such as precision micro-machining, nonlinear optical imaging techniques, including multi-photon and Raman microscopies, and ocular surgery. It is very likely that performance advances that result from this work will be implemented in research and commercial laboratories, and there is strong potential for commercial development. The students working on this project will gain experience ranging from analytical solutions of partial differential equations to numerical simulations, understanding of ultrafast nonlinear optics, and exposure to technical aspects of photonics. The close collaboration of professional theorists with physicists, engineers, and industrial scientists will significantly broaden and enhance the students' educational experiences and prepare them for a range of future opportunities in the mathematical sciences.