Collaborative Research: Mathematical and computational methods for stochastic systems in nonlinear optics

合作研究：非线性光学中随机系统的数学和计算方法

• 批准号: 1108780
• 负责人: Tobias Schaefer
• 金额: $ 14.89万
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108780&HistoricalAwards=false
• 关键词: Collaborative; Research; Mathematical; computational; methods

The objective of this collaborative research project is to develop methods for analysis, modeling, and computation for stochastic partial differential equations arising in nonlinear optics. The study of noise in these systems is complicated by several factors. In particular:(i) the complexity due to the interplay of many physical effects,(ii) the presence of multiple length and time scales, and(iii) the fact that the critical events that represent system failures are, by design, extremely rare. We will address these challenges by (a) creating a set of analytical methods to obtain reductions from the stochastic partial differential equations that govern the system's behavior to a set of low-dimensional stochastic differential equations;(b) developing efficient methods for the study of noise across multiple time and space scales; and (c) implementing hybrid analytical-computational methods that can efficiently target the rare events of interest and ultimately estimate the reliability of these systems.Lightwave systems have a wide range of applications, in particular as lasers and optical communication lines, and their emergence has had a dramatic impact on our lives over the last fifty years. Lasers are used at low powers in applications related to scanning and communications; at higher powers they are employed in surgical and industrial applications. As noise is a major limiting factor in optical systems, further technological progress will depend critically on understanding the impact of noise and on being able to control noise-induced system failures. This research will provide new analytical and computational tools that support the analysis and design of the next generation of optical devices.

这个合作研究项目的目标是开发非线性光学中出现的随机偏微分方程的分析、建模和计算方法。这些系统中的噪声研究由于几个因素而变得复杂。特别是：(i)许多物理效应相互作用的复杂性，（ii）多个长度和时间尺度的存在，以及（iii）根据设计，代表系统故障的关键事件极其罕见的事实。我们将通过(a)创建一套分析方法来解决这些挑战，以获得从控制系统行为的随机偏微分方程到一组低维随机微分方程的约化；(b)发展跨时间和空间尺度研究噪音的有效方法；(c)实现混合分析计算方法，可以有效地针对感兴趣的罕见事件并最终估计这些系统的可靠性。光波系统具有广泛的应用，特别是激光和光通信线路，在过去的五十年中，它们的出现对我们的生活产生了巨大的影响。激光以低功率用于与扫描和通信有关的应用；在更高的功率下，它们被用于外科和工业应用。由于噪声是光学系统的主要限制因素，进一步的技术进步将主要取决于理解噪声的影响，并能够控制噪声引起的系统故障。这项研究将提供新的分析和计算工具，支持下一代光学器件的分析和设计。