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

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.