Multiscale theory for paraxial waves with applications

获奖摘要：DMS-0307011，Knut Solna，加州大学欧文分校标题：傍轴波的多尺度理论及其应用当激光束在大气中传播时，它会受到恶劣天气和湍流的影响。同样，无线通信系统、遥感系统和成像算法也受到介质波动的影响。 该建议旨在发展一种理论，用于描述和分析这种现象所需的随机和多尺度介质中的波传播，并将该理论应用于复杂和多尺度介质中涉及波的系统的优化设计。 介质非均匀性采用随机场模型，其目的是分析介质非均匀性的统计特征， 通过介质传播的波。 两种制度被认为是。第一，一个具有清晰的“尺度分离”的区域，其中介质变化发生在微尺度上，例如，比波长小得多的尺度。 第二，正在考虑连续尺度变化的情况，如湍流。 尽管这种状态的重要性，但没有一般的理论来描述介质如何影响这种情况下的波脉冲。 在这里提出的新方法的目的是分析这些制度在一个统一的框架中，介质波动描述的布朗场，然后一个目的是开发一个通用的标度理论，这样的白色噪声模型。也就是说，描述波场的矩以及这些矩如何取决于例如相对传播距离。 该分析涉及由布朗场驱动的随机微分方程，以及如何使用鞅理论在各种制度下描述解的矩，这是随机分析中的一个普遍感兴趣的话题。 在分析中，抛物或前向散射近似，导致一个随机的薛定谔方程以及声波和电磁波将被考虑。 大气中 由气流湍流引起的温度、压力和湿度的小尺度波动 导致复杂的和多尺度的空间和时间变化的波的速度。 因此，湍流导致 光束漂移、光束展宽和强度波动（闪烁）等现象。 所提出的工作，将这种影响的特点是重要的成像和通信算法的设计和几个应用程序将被考虑。 宣传和交流工作 由于激光通信和跟踪等相关应用的重要性，以及由于迄今为止几乎没有可用的理论，湍流大气中的大气湍流是特别重要的。 该理论是必要的，例如最佳通信协议的设计，多输入多输出天线系统的无线通信，安全通信算法的设计和强大的跟踪算法通过大气。 作为这项工作的一部分，Speclab的发展，一个通用的多尺度和湍流数据分析软件包已经开始，它将进一步发展。 到目前为止，它已被用于分析大气湍流数据（克特兰空军基地），也用于分析生物医学数据（哈佛大学）。 这项工作对于指导波动现象的建模和确保我们工作的相关性是重要的。 多尺度介质中的波的研究对于理解波场在非均匀地球、波动和分层的海洋以及生物组织中的传播也很重要。 在这方面，将考虑的一些重要应用涉及医疗成像、通过树叶成像和地雷探测。

Award Abstract: DMS-0307011, Knut Solna, University of California-IrvineTitle: Multiscale theory for paraxial waves with applications When a laser beam propagates through the atmosphere it is affected by bad whether and turbulence. Similarly wireless communication systems, remote sensing systems and imaging algorithms are affected by medium fluctuations. This proposal aims at developing a theory for waves propagating in random and multiscale media that is needed to describe and analyze such phenomena and at applying this theory for optimal design of systems involving waves in complicated and multiscale media. The medium heterogeneity is modeled by a random field and one aims at analyzing the statistical character of the waves that have propagated through the medium. Two regimes are considered. First, a regime with a clear ``scale separation'' where the medium variations take place on a microscale, a scale that is much smaller than the wave length for instance. Second, the case with variations on a continuum of scales, like in turbulence, is being considered. Despite the importance of this regime there is no general theory that describes how the medium affects the wave pulse in this case. In the novel approach set forth here one aims to analyze these regimes in a unified framework in which the medium fluctuations are described by a Brownian field and then one aims to develop a general scaling theory for such a white noise model. That is, a description of moments of the wave field and how these depend on for instance the relative propagation distance. The analysis involves stochastic differential equations driven by the Brownian field and how the moments of the solution can be described in various regimes using martingale theory, this is a topic of general interest in stochastic analysis. In the analysis the parabolic or forward scattering approximation that leads to a random Schr\"odinger equation as well as acoustic and electromagnetic waves will be considered. In the atmosphere small scale fluctuations of temperature, pressure and humidity caused by the turbulence of air velocities lead to complicated and multiscale spatial and temporal variations in the wave-speed. The turbulence results therefore in phenomena like wave beam wander, beam broadening and intensity fluctuation (scintillation). The proposed work that will characterize such effects is important for the design of imaging and communication algorithms and several applications will be considered. The work regarding propagation and communication in the turbulent atmosphere is particularly important due to the significance of associated applications like laser communication and tracking and since so far little theory is available. The theory is needed for design of for instance optimal communication protocols, multiple-input-multiple-output antenna systems for wireless communication, for design of secure communication algorithms and robust tracking algorithms through the atmosphere. As a part of this work the development of Speclab a general purpose software package for analysis of multiscale and turbulent data has been initiated and it will be further developed. It has so far been used for analysis of atmospheric turbulence data (Kirtland Air Force Base) and also for analysis of biomedical data (Harvard University). This work is important to guide the modeling of the wave phenomena and ensure the relevance of our work. The work on waves in multiscale media will also be important for understanding wave-fields propagating in the heterogeneous earth, the fluctuating and stratified ocean and through biological tissue. In these contexts some important applications that will be considered relates to medical imaging, imaging through foliage and mine detection.