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

现在需要新的分析方法，因为激光中新的脉冲演化有望极大地提高实际仪器的性能。该项目将对在锁模激光器中产生飞秒光脉冲和光学子弹的新方法进行理论研究。基本的数学方法植根于非线性色散波理论及其在非线性饱和介质中的动力学和稳定性。新的脉冲整形机制有可能对超快科学产生重大影响，但目前的理论理解还处于初级阶段，因为脉冲在穿过激光腔时，其时间形状、光谱形状以及相位或频率都会发生巨大变化；这些反过来又对数学模型构成了严峻的挑战。高度啁啾和/或自相似的脉冲解决方案可以在强耗散存在的情况下存在，从而产生提供卓越行为和性能的新类型的激光脉冲。这里提出的改进模型的开发将使重大科学进步成为可能，例如多维孤子的产生，并将导致应用的增强仪器。这项研究的目标是真正的跨学科性质：将渐近和微扰方法、科学计算和严格的数学分析与基于实验观察的模型相结合。这项研究的影响将超越对激光系统中非线性脉冲传输的理解。在这个项目中提出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。产生飞秒脉冲的激光具有巨大的潜力，可以将短脉冲光学技术的范围扩展到实际应用中，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镜，以及眼科手术。这项工作带来的性能进步很可能将在研究和商业实验室中实施，商业开发的潜力很大。从事这个项目的学生将获得从偏微分方程的解析解到数值模拟的各种经验，了解超快非线性光学，以及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大大拓宽和增强学生的教育经验，并为他们未来在数学科学领域的一系列机会做好准备。

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.