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A numerical investigation of pulse and beam propagation in nonlinear optical media using the full adaptive wavelet transform (FAWT).

机译：使用完全自适应小波变换（FAWT）的非线性光学介质中脉冲和束传播的数值研究。

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摘要

Depending upon the particular application, most problems of practical interest in nonlinear optics can be described by various forms of the traditional nonlinear Schrodinger (NLS) equation. The purpose of this research is to investigate a unique numerical scheme for solving the complex partial differential equations (PDEs) associated with nonlinear optics applications. Most numerical schemes for resolving nonlinear PDEs typically employ either a strict finite difference method or some form of pseudo-spectral technique, such as the split-step Fourier method. While the split-step method is generally faster when compared to finite differences, it may be used only after several simplifying assumptions which allow for separation of the linear and nonlinear components. While in general these simplifying assumptions are usually justified, this is not always the case. This dissertation describes a unique numerical scheme for solving any nonlinear PDE using an adaptive wavelet transform, done entirely in the wavelet domain, and referred to as the full adaptive wavelet transform (FAWT). This technique differs from previous wavelet solutions in that these previous works typically used a "split-step wavelet" method in which the nonlinear portion was solved using a collocation or finite-difference scheme. The FAWT is a spectral technique, based upon the method of weighted residuals, and as such, uses a larger time step than does a strict finite difference, making it faster than a finite difference scheme. This unique numerical scheme takes advantage of the scaling and shifting properties associated with the wavelet transform in order to solve the complex nonlinear partial differential equations associated with pulse and beam propagation. As such, it is highly adaptive in its ability to track steep gradients in the numerical solution by switching to higher and higher (i.e., narrower and narrower) wavelet levels as these steep gradients develop. This adaptive ability becomes critical in studies involving self-steepening of Gaussian pulses and self-focusing of Gaussian beams.
机译：根据特定的应用，可以通过各种形式的传统非线性薛定inger（NLS）方程来描述非线性光学中许多实际感兴趣的问题。这项研究的目的是研究解决与非线性光学应用相关的复杂偏微分方程（PDE）的独特数值方案。解决非线性PDE的大多数数值方案通常采用严格的有限差分法或某种形式的伪谱技术，例如分步傅里叶方法。尽管与有限差分相比，分步方法通常更快，但只有在允许线性和非线性分量分离的几个简化假设之后，才可以使用它。通常，这些简化的假设通常是合理的，但并非总是如此。本论文描述了一种使用自适应小波变换求解任何非线性PDE的独特数值方案，该方案完全在小波域内完成，称为全自适应小波变换（FAWT）。该技术与先前的小波解决方案不同之处在于，这些先前的工作通常使用“分步小波”方法，其中非线性部分是通过搭配或有限差分方案求解的。 FAWT是一种基于加权残差方法的频谱技术，因此，与严格的有限差分相比，FAWT使用的时间步长更大，因此它比有限差分方案要快。这种独特的数值方案利用了与小波变换相关的缩放和平移特性，以便求解与脉冲和波束传播相关的复杂非线性偏微分方程。这样，随着这些陡峭梯度的发展，通过切换到越来越高的（即，越来越窄的）小波水平，它在数值解中跟踪陡峭梯度的能力具有高度适应性。这种适应能力在涉及高斯脉冲的自加深和高斯光束的自聚焦的研究中变得至关重要。

著录项

  • 作者

    Stedham, Mark Anthony.;

  • 作者单位

    The University of Alabama in Huntsville.;

  • 授予单位 The University of Alabama in Huntsville.;
  • 学科 Engineering Electronics and Electrical.
  • 学位 Ph.D.
  • 年度 2000
  • 页码 238 p.
  • 总页数 238
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 TS97-4;
  • 关键词

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