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.

这个合作项目扩展了主要研究人员对光学现象数学模型的研究计划。 它是响应NSF关于“光学和光子学”的倡议。 光与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，并提供了激光器和光学放大器等设备的基本机制。 几十年来，它一直在提供新的物理现象的丰富来源，其中最新的是“慢光”，最近观察到的光脉冲减速到自行车的速度。 慢光可以潜在地用于诸如光学存储器之类的设备中。 该项目旨在通过使用可以用显式公式求解的显式，高度精确的数学模型来理解慢光现象背后的物理机制。 该模型及其显式解的有效性将通过对更现实的模型进行数值模拟并与实验进行仔细比较来验证。 将为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，并为这两个学生群体建立一个充满活力、具有挑战性的研究和培训环境。光脉冲的减速被建模为光脉冲与具有两个或三个工作水平的活性介质之间的相互作用，后者是被称为Lambda配置的原型情况。 这种相互作用是由完全可积的麦克斯韦-布洛赫方程与非零边界条件，一个新的扭曲。 该项目是对光与两级介质和λ配置介质相互作用产生的新动力学的数学研究，包括：（i）为具有非零边界条件的二能级和λ-组态Maxwell-Bloch方程开发系统的、完全可积的动力学理论，（ii）使用步骤（i）的分析结果来描述与慢光相关的现象，（iii）在更一般的情况下对动力学现象进行数值研究，在这种情况下，二能级和λ组态Maxwell-Bloch方程是不可积的。 慢光的完全可积描述涉及两个新的方面：（1）非零边界条件，（2）光谱数据的非平凡演化。 通过研究Riemann曲面上带谱参数的散射和逆散射问题，将对第一个方面的理解从非线性Schroedinger方程扩展到Maxwell-Bloch方程。 第二个方面是复杂的存在前，涉及到一个仔细推导的光谱数据如何从介质的初始状态演变，并找到正确的取消高度振荡的条款。 除了产生新的模型和描述光与有源光学介质相互作用所表现出的动力学，该项目还将推进完全可积系统的理论。