OP: Collaborative Research: Nonlinear Theory of Slow Light

该合作项目扩展了首席研究员在光学现象数学模型方面的研究项目。这是对美国国家科学基金会“光学与光子学”倡议的回应。光与光介质之间的相互作用是应用物理学中最具成果的研究领域之一，它为激光和光放大器等器件提供了基本的机制。几十年来，它为新的物理现象提供了丰富的来源，其中最新的是“慢光”，即最近观察到的光脉冲减慢到自行车的速度。慢光可以潜在地应用于光存储器等设备。该项目旨在通过使用一个卓越的、高度精确的数学模型来理解慢光现象背后的物理机制，这个数学模型可以用明确的公式来求解。该模型及其显式解的有效性将通过更实际模型的数值模拟和与实验的仔细比较来验证。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训练，并为两个学生群体建立一个充满活力，具有挑战性的研究和训练环境。光脉冲的减速被建模为光脉冲与具有两个或三个工作能级的有源介质之间的相互作用，后者是称为Lambda配置的典型情况。这种相互作用由具有非消失边界条件的完全可积麦克斯韦-布洛赫方程描述，这是一种新的扭曲。本项目是对光与两能级介质和λ -构型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两能级和λ构型麦克斯韦-布洛赫方程开发了一个系统的、完全可积的动力学理论；（ii）利用第(i)步的分析结果来描述与慢光有关的现象；（iii）在两能级和λ构型麦克斯韦-布洛赫方程不可积的更一般情况下的动力学现象的数值研究。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。通过研究黎曼曲面上具有谱参数的散射和反散射问题，将对第一方面的理解从非线性薛定谔方程扩展到麦克斯韦-布洛赫方程。第二个方面由于前者的存在而变得复杂，涉及到光谱数据如何从介质的初始状态演变的仔细推导，并找到高振荡项的正确消去。除了产生光与有源光介质相互作用的新模型和动力学描述外，该项目还将推进完全可积系统的理论。

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.