Dynamics of Light Interacting with Active Media

这项工作将解决的动力学中产生的三个新的影响所产生的光与光学介质中的λ-配置的相互作用的数学描述。 第一个是随机偏振开关的光脉冲传播通过一个双折射配置介质，其描述引起了罕见的组合，同时完全可积和随机偏微分方程。 第二个是“光停止”现象，这需要发展一种新的逆散射变换技术和孤子理论来研究无穷远处非零边界条件的Maxwell-Bloch方程。第三个是光传播通过一个组合，双折射配置的超材料，这需要推导出一个新的模型进行研究，通过精确的解决方案，渐近和数值技术的组合，并可能会引起描述的新现象，如同时颜色和方向切换和相关的非线性版本的安德森本地化，以及机制的损失补偿超材料。 更广泛地说，这项工作将推进完全可积系统的理论，特别是对超材料中损耗补偿的成功描述，可能会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。光与有源光学介质（这些介质的这种相互作用特别强）之间的相互作用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。 它仍然是新物理现象的丰富来源，其中最新的是“光停止”，在此期间，特别准备的光脉冲被减慢到光速的一小部分，并可用于设计光学存储器。 此外，这种相互作用还可以用于减少光学超材料（具有以前无法实现的光学特性的人造纳米复合材料，例如完美的透镜聚焦）的损失，从而有助于将其从当前的概念验证实验发展到最终的实用光学器件。 由于它所展示的物理现象种类繁多，光与有源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，将开发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，以及最后一种罕见的光传播通过高度无序介质的精确描述。

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.