Analytical and computational methods for femtosecond lasers

非线性光学最重要的最新进展之一是发展了新一代非常稳定的短脉冲激光器，典型的脉冲持续时间只有几飞秒。这样的激光器有很多用途。其中，它们为光谱学和频率计量提供了精确的来源，是光学原子钟的重要组成部分。然而，由于存在许多物理效应，涉及的时间尺度也非常不同，因此，对这些系统的行为及其基本性能限制进行定量预测是一项具有挑战性的任务。该项目将通过开发新的数学模型来应对这些挑战，该模型旨在描述这些激光器的行为，并附带一套分析和计算工具。新的模型将明确考虑到这些系统的内在复杂性和存在的多个时间尺度，然后将用于分析激光腔中脉冲的锁模和非线性动力学。此外，这些新模型将指导统计计算算法的发展，这些算法可以准确量化噪声的影响，并计算限制这些系统的精度和最终可靠性的罕见事件的概率。飞秒激光具有广泛的适用性。除了已经提到的计量学和原子钟，重要的应用是新一代全球定位系统，探测化学反应期间发生的瞬变，以及肿瘤的消融。因此，该项目不仅将影响科学界，而且将影响整个社会。作为该项目的一部分，新的数学模型和计算方法将大大推进飞秒激光器的理论，它们将使准确和有效地预测这些激光器的性能成为可能。因此，它们将为设计和建造这些系统的科学家和工程师提供有用的工具。该项目的一个重要部分也将是对学生和初级研究人员的培训：通过这种跨学科的努力，学生将被训练使用其主要学科之外的概念、方法和技术，从而极大地丰富他们的教育和专业经验。

One of the most important recent advances in nonlinear optics has been the development of a new generation of very stable short-pulse lasers, with typical pulse durations of just a few femtoseconds. Such lasers have numerous applications. Among others, they provide precise sources for spectroscopy and frequency metrology, and are an essential component in optical atomic clocks. Because of the many physical effects present and the vastly different timescales involved, however, making quantitative predictions about the behavior of these systems and their fundamental performance limits is a challenging task. This project will address these challenges by developing new mathematical models aimed at describing these lasers' behavior and an accompanying set of analytical and computational tools. The new models, which will explicitly take into account the inherent complexity of these systems and the multiple time scales present, will then be used to analyze the mode-locking and nonlinear dynamics of pulses in the laser cavity. Moreover, these new models will guide the development of statistical computational algorithms that can accurately quantify the impact of noise and calculate the probability of the rare events that limit the accuracy and ultimate reliability of these systems.Femtosecond lasers have a wide range of applicability. Beyond metrology and atomic clocks, already mentioned, important applications are a new generation of global positioning systems, the probing of transients occurring during chemical reactions and the ablation of tumors. Thus, this project will impact not only the scientific community, but also society at large. The new mathematical models and computational methods that will be derived as part of this project will substantially advance the theory of femtosecond lasers, and they will make it possible to accurately and efficiently predict the performance of these lasers. As a result, they will provide useful tools for the scientists and engineers who design and build these systems. An important part of the project will also be the training of students and junior researchers: through this interdisciplinary effort, students will be trained to use concepts, methods and techniques outside their main discipline, thus greatly enriching their educational and professional experience.