Dynamics of Light Interacting with Active Media

这项工作将解决在lambda-构型中光与光介质相互作用产生的三种新效应中产生的动力学的数学描述。第一个是光脉冲在lambda-configuration介质中传播的随机偏振开关，它的描述产生了同时完全可积和随机偏微分方程的罕见组合。第二种是“光停止”现象，这需要发展一种新的反散射变换技术和具有无穷远处不消失边界条件的麦克斯韦-布洛赫方程的孤子理论。第三种是光通过组合的λ构型超材料传播，这需要通过精确解、渐近和数值技术的结合来研究新模型的推导，并可能产生新现象的描述，例如同时颜色和方向切换以及相关的非线性安德森局域化，以及超材料中的损耗补偿机制。更广泛地说，这项工作将推进完全可积系统的理论，特别是对超材料损耗补偿的成功描述，可能对实际的非线性光学产生影响。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。光与有源光介质之间的相互作用（这种相互作用特别强的介质）是应用物理学中最富有成果的领域之一，并提供了激光器和光学放大器等设备的基本机制。它一直是新的物理现象的丰富来源，其中最新的是“光停止”，在此期间，特殊制备的光脉冲被减慢到光速的一小部分，这可以用于设计光存储器。此外，这种相互作用还可以用于减少光学超材料（具有以前无法获得的光学特性的人造纳米复合材料，如完美的透镜聚焦）的损耗，从而有助于推动它们从目前的概念验证实验发展到最终的实用光学设备。由于它所表现出的物理现象的多样性，光与活性介质的相互作用也引起了对其动力学的广泛数学描述。在这项工作的过程中，将发展出三种新颖的数学描述，包括“停止光”，超材料中某种潜在的损失补偿类型，以及最后对光在高度无序介质中传播的罕见精确描述。

This work will address mathematical descriptions of the dynamics arising in three novel effects generated by the interaction of light with an optical medium in the lambda- configuration. The first is random polarization switching of light pulses propagating through a lambda-configuration medium, whose description gives rise to the rare combination of a simultaneously completely integrable and stochastic partial differential equation. The second is the "light stopping" phenomenon, which requires the development of a new inverse-scattering-transform technique and soliton theory for the Maxwell-Bloch equations with nonvanishing boundary conditions at infinity. The third is light propagating through a combined, lambda-configuration metamaterial, which requires the derivation of a new model to be studied through a combination of exact solutions, asymptotic, and numerical techniques, and may give rise to descriptions of novel phenomena such as simultaneous color and direction switching and a related nonlinear version of Anderson localization, as well as mechanisms for loss compensation in metamaterials. More broadly, the work will advance the theory of completely-integrable systems, and especially a successful description of loss compensation in metamaterials, may have impact on practical nonlinear optics. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students.Interaction between light and active optical media (these are media for which such interaction is particularly strong) is one of the most fruitful areas in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. It has continued to be a rich source of new physical phenomena, among the latest being "light stopping," during which specially prepared optical pulses are slowed down to a fraction of the light speed, and which could be used in designing optical memory. In addition, this interaction could also be used to reduce losses in optical metamaterials (artificial nano-composites with previously unattainable optical properties such as perfect lens focusing) and thus help advance their development from the current proof-of-concept experiments to eventual practical optical devices. Due to the great variety of physical phenomena it exhibits, light interaction with active media has also given rise to a broad range of mathematical descriptions of their dynamics. Three novel such mathematical descriptions will be developed in the course of this work, including those of "stopped light," a certain potential type of loss compensation in a metamaterial, and finally a rare exact description of light propagating through a highly disordered medium.