Mathematical Sciences: Geometric Models and Methods in Nonlinear Optics

数学科学：非线性光学中的几何模型和方法

• 批准号: 9626306
• 负责人: Nicholas Ercolani
• 金额: $ 7.22万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626306&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Models; Methods

9626306 Ercolani In the projects proposed here, the PI intends to concentrate on two particular areas where geometric methods are playing a fundamental role. The methods to be applied here are, first, soliton and near-integrable systems theory and, second, geometric singularity theory.The first concerns the modelling of dispersive difference schemes. The equations of nonlinear optics frequently operate in regimes where dispersive effects tend to dominate over those of viscosity. A particularly relevant example of this is the effects of numerical dispersion in modelling carrier wave shocking of femtosecond pulses. An appropriate model that can provide a benchmark for these effects is the integrable discrete nonlinear Schrodinger equations. The integrability of this model enables one to entertain a precise analysis of continuum limits for this system. The PI proposes to use the underlying Kahler geometry of this model to deduce convexity estimates which should enable one to derive a valid characterization of these limits including the appearance of modulational instabilities. These characterizations will then be compared to modulation equations derived by formal averaging as well as to numerical simulations of non-integrable discretizations of the nonlinear Schrodinger (NLS) pde. The second topic concerns pattern formation in nonlinear optics. Experiments reveal a rich variety of transverse patterns for coherent fields in a nonlinear optical cavity. The Nonlinear Optics group of the Arizona Center for Mathematical Sciences has derived mean field models for a cavity filled with an isotropic, nonlinear Kerr medium, and driven by a linearly polarised input field. The equations for this model have the form of a damped, driven coupled pair of defocussing NLS equations and have been numerically demonstrated to produce patterns and defects of the type seen experimentally. The PI and his collaborators are particularly interested to understand the formation and evolut ion of defects in pattern forming evolution equations and have developed order parameter equations which model the behaviour of modulated roll paterns in a variety of such evolution equations. The PI proposes to derive and study such modulation equations for the above mentioned mean field model. The PI has also developed a general approach, based on geometric singularity theory, for the description of singularties in systems of pde's. The goals of this project will be first, to classify generic types of defects for the nonlinear optics model and assess their temporal stability; second, to control the formation of defects to the end of either eliminating them or using them to encode information in patterns. Expectations concerning this second goal are supportied by recent sucess in controlling pattern irregularities in one dimensional models. %%% Nonlinear optics, which may be defined as the study of the interaction of intense light with matter, was born with the invention of the laser in the 1960's. In those early days such interactions required very large intensities; however, nowadays such systems can be constructed with only modest power requirements. Along with this has come the possibility of constructing large, so-called wide aperture, devices such as semiconductor lasers. Such devices will play an important role in many areas of technology such as satellite communications systems with an ultrafast switching capacity. With the introduction of these larger physical scales one expects to see the formation of patterns as well as defects in these patterns (analogous to crystalline defects) which can be modelled by the equations developed in this proposal. Understanding how such patterns arise and evolve can play an important role in efficiently operating and controlling these devices. In the second project of this proposal oscillatory solutions in a special class of interacting lattice systems will be investigated. These systems have modelling app lications which range from theoretical biology to the numerical methods used in numerically simulating optical systems. The goal is to obtain, rigorously, a continuum description, given by partial differential equations and valid on long spatial and temporal scales, of the microscopic variations in these lattice systems. Such models can help to explain how a microscopic system, such as an alpha-helix protein molecule or a coupled laser array, can behave collectively, and hence macroscopically, in order to contribute to phenomena on scales much larger than those orignally present in the microscopic system. ***

小行星9626306 在这里提出的项目中，PI打算集中在两个特定的领域，几何方法发挥着重要作用。 这里应用的方法是，首先，孤子和 近可积系统理论，第二，几何奇异性理论。第一个涉及色散差分格式的建模。非线性光学方程经常在色散效应往往超过粘性效应的情况下工作。一个特别相关的例子是模拟中数值分散的影响 飞秒脉冲的载波冲击。可以为这些效应提供基准的适当模型是可积离散 非线性薛定谔方程该模型的可积性使人们能够对该系统的连续极限进行精确的分析。 PI建议使用该模型的基本Kahler几何来推导凸性估计，这应该使人们能够得出这些限制的有效表征，包括调制不稳定性的出现。这些特征，然后将比较正式平均以及数值模拟的非积分离散的非线性薛定谔（NLS）PDE调制方程。 第二个主题是关于非线性光学中的图形形成。 实验揭示了非线性光学腔中相干光场的丰富的横向斑图。亚利桑那州数学科学中心的非线性光学小组已经推导出填充有各向同性， 非线性克尔介质，并由线性偏振输入场驱动。 该模型的方程具有阻尼、驱动耦合的形式， 对散焦NLS方程，并已被数值证明产生的模式和实验中看到的类型的缺陷。 PI和他的合作者特别是 有兴趣了解缺陷的形成和演变， 形成演化方程，并发展了序参量 方程，其在各种这样的演变方程中对调制的滚动参数的行为进行建模。PI建议推导并研究上述平均场模型的调制方程。PI还开发了一种通用的方法，基于几何奇异性理论，用于描述偏微分方程系统中的奇异性。的 本计画的目标将是：第一，为非线性光学模型分类一般类型的缺陷，并评估其时间稳定性; 2第二，控制缺陷的形成，以达到消除缺陷或利用缺陷将信息编码成图案的目的。 关于第二个目标的期望是由最近的成功控制模式的不规则性在一维模型。 %%% 非线性光学，可以定义为研究强光与物质相互作用的学科，随着20世纪60年代激光的发明而诞生。在那些早期的日子里，这种相互作用需要非常大的强度;然而，现在这样的系统可以构建只有适度的功率要求。 沿着而来的是建造大的、所谓的宽孔径的装置，例如半导体激光器的可能性。这种装置将在许多技术领域发挥重要作用，例如具有超快交换能力的卫星通信系统。随着这些更大的物理尺度的引入 人们期望看到图案的形成以及这些图案中的缺陷（类似于晶体缺陷）， 提议了解这些模式是如何产生和演变的，可以在有效操作和控制这些设备方面发挥重要作用。 在第二个项目中， 一类特殊的相互作用格点的振动解 系统将进行调查。 这些系统具有建模应用程序 从理论生物学到数值方法， 数值模拟光学系统。我们的目标是严格地， 连续描述，由偏微分方程给出，有效 在长的空间和时间尺度上，这些晶格系统的微观变化。这样的模型可以帮助解释一个微观系统，如α-螺旋蛋白质分子或耦合激光阵列，如何能够集体行为，因此宏观上，为了有助于在尺度上比那些原始存在于微观系统中的现象大得多。 ***