Multiscale theory for paraxial waves with applications

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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.