OP: Collaborative Research: Nonlinear Theory of Slow Light

OP：合作研究：慢光非线性理论

• 批准号: 1615859
• 负责人: Gregor Kovacic
• 金额: $ 23.5万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615859&HistoricalAwards=false
• 关键词: OP; Collaborative; Research; Nonlinear; Theory

This collaborative project expands the research programs of the Principal Investigators on mathematical models of optical phenomena. It comes in response to the NSF initiative on "Optics and Photonics". The interaction between light and optical media is one of the most fruitful areas of study in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. For decades, it has been providing a rich source of new physical phenomena, among the latest being "slow light", the recently-observed slowing-down of light pulses to the speed of a bicycle. Slow light can potentially be used in devices such as optical memory. This project is aimed at understanding the physical mechanisms underlying the slow light phenomenon by using a remarkable, highly accurate mathematical model that can be solved with explicit formulas. The validity of this model and its explicit solutions will be verified using numerical simulations of more realistic models and careful comparisons with experiments. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students, and a lively, challenging research and training environment for both student groups will be established.The slowdown of light pulses is modeled as the interaction between an optical pulse and an active medium with two or three working levels, the latter being a prototypical case known as the Lambda configuration. This interaction is described by completely integrable Maxwell-Bloch equations with non-vanishing boundary conditions, a new twist. This project is a mathematical study of novel dynamics generated by the interaction of light with two-level media and the Lambda-configuration medium, and includes: (i) developing a systematic, completely integrable theory of the dynamics for the two-level and Lambda-configuration Maxwell-Bloch equations with non-zero boundary conditions, (ii) using the analytical results of step (i) to describe phenomena related to slow light, (iii) numerical studies of dynamical phenomena in more general cases in which the two-level and Lambda-configuration Maxwell-Bloch equations are not integrable. The completely-integrable description of slow light involves two new aspects: (1) non-zero boundary conditions, (2) non-trivial evolution of the spectral data. The understanding of the first aspect will be extended from the Nonlinear Schroedinger equation to the Maxwell-Bloch equations by studying scattering and inverse-scattering problems with the spectral parameter on a Riemann surface. The second aspect is complicated by the presence of the former and involves a careful derivation of how spectral data evolves from the initial state of the medium and finding correct cancellations of highly oscillatory terms. In addition to generating new models and descriptions of the dynamics exhibited by light interacting with active optical media, the project will advance the theory of completely integrable systems.

该协作项目扩大了有关光学现象数学模型的主要研究人员的研究计划。 它响应于“光学和光子学”的NSF计划。 光介质和光学介质之间的相互作用是应用物理学中最富有成果的研究领域之一，并提供了诸如激光器和光学放大器之类的基本机制。 几十年来，它一直为新的物理现象提供了丰富的来源，其中最新的是“慢光”，这是最近观察到的轻脉冲速度降低到自行车速度的速度。 慢光可能可能用于光学内存等设备中。 该项目的目的是通过使用可以使用明确的公式来解决的显着，高度准确的数学模型来理解慢光现象的物理机制。 该模型及其显式解决方案的有效性将通过更现实的模型的数值模拟以及与实验进行仔细的比较来验证。 将向研究生和本科生提供应用数学和非线性光学的跨学科培训，并为两个学生群体建立一个活泼的，充满挑战的研究和培训环境。具有两个或三个工作水平的活性培养基，后者是一种原型情况，称为lambda配置。 这种相互作用是通过具有非变化边界条件的完全可整合的Maxwell-Bloch方程来描述的。 该项目是一项数学研究，是对光与两级介质和lambda-configuration介质的相互作用产生的新型动力学的数学研究，其中包括：（i）开发一种系统的，完全可以整合的，可用于两级和lambda的动力学理论 - 具有非零边界条件的Maxwell-Bloch方程，（ii）使用步骤（i）的分析结果来描述与慢光相关的现象，（iii）在更一般的情况下，动态现象的数值研究级别和lambda-configuration Maxwell-Bloch方程不可集成。 慢光的完全集成描述涉及两个新方面：（1）非零边界条件，（2）光谱数据的非平凡演化。 对第一方面的理解将从非线性schroedinger方程扩展到Maxwell-Bloch方程，通过研究散射和逆散射问题的频谱参数在Riemann表面上。 前者的存在使第二个方面变得复杂，并涉及仔细推导光谱数据从介质的初始状态中演变而来并找到正确振动性术语的正确取消。 除了生成新模型以及对光线相互作用的动力学的新模型和描述外，该项目还将推进完全可集成的系统的理论。