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

随着激光器中新的脉冲演化有望大大提高实用仪器的性能，现在需要新的分析方法。 本计画将对锁模雷射中产生飞秒光脉冲及光子弹的新方法进行理论研究。基本的数学方法是植根于非线性色散波理论和他们的动力学和稳定性在非线性，饱和介质。新的脉冲整形机制有可能对超快科学产生重大影响，但目前的理论理解是初步的，因为脉冲在穿过激光腔时，其时间形状，光谱形状和相位或频率会发生很大变化;这些反过来又对数学模型提出了严峻的挑战。 高啁啾和/或自相似脉冲解决方案可以存在于强耗散的情况下，从而产生提供卓越行为和性能的新型激光脉冲。本文提出的改进模型的开发将实现重大科学进步，例如多维孤立子的产生，并将导致应用仪器的增强。该研究的目的是一个真正的跨学科性质：结合渐近和扰动方法，科学计算，严格的数学分析与模型是基于实验观察。拟议研究的影响将超出对激光系统中非线性脉冲传播的理解。在这个项目中开发的概念将涉及一系列主题，从非线性动力系统的基础科学到商业激光仪器。产生飞秒持续时间的光脉冲的激光器具有将短脉冲光学技术的范围扩展到实际应用的巨大潜力，例如精密微加工、非线性光学成像技术（包括多光子和拉曼显微术）以及眼科手术。这项工作所带来的性能进步很可能会在研究和商业实验室中得到实施，并且具有很强的商业开发潜力。从事该项目的学生将获得从偏微分方程的解析解到数值模拟的经验，了解超快非线性光学，并接触光子学的技术方面。专业理论家与物理学家，工程师和工业科学家的密切合作将显着扩大和增强学生的教育经验，并为他们在数学科学的一系列未来机会做好准备。

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.