Development and Analysis of High Accuracy Numerical Methods for Computational Optics

Abstract

This work involves the development and application of high accuracy numerical methods for computational optics. Three main topics form the core of the text. First is an application of finite difference methods to the simulation of a tapered slab coupled to an Erbium /Ytterbium doped fiber. Light pumped into the device is modeled using the paraxial beam propagation equation. The pump efficiency is measured and compared with a simple linear estimate. The results show agreement between the estimated and measured efficiencies over a few centimeters for low absorption rates and random initial conditions. In the second topic, I have applied the Weeks method for the numerical inversion of the Laplace transform to the matrix exponential. This has involved the extension of the scalar theory to matrix functions parameterized by time, a detailed error analysis, and a study of automated selection algorithms for the method's two tuning parameters. By calculating the exponential of both pathological matrices and those from the method-of-lines applied to the nonparaxial beam propagation equation, it has been possible to compare the parameter selection algorithms and demonstrate the method's high accuracy. The third topic is the implementation of a multidomain pseudospectral approach to the derivatives in Schr\"odinger type equations. The method is based on a cellular decomposition of the domain and Legendre pseudospectral differentiation within each cell. A symmetric high-order finite difference stencil for boundary points is found to be stable, accurate, and conservative. The application of this algorithm is the numerical analysis of a procedure for determining the effects of high intensity light on the nonlinear susceptibility coefficients of the hydrogen atom. With this approach, I have been able to resolve the solutions to the underlying equations and accurately compute the ionization probability and polarization. The full susceptibility and one which depends solely on the electronic bound-bound transitions are derived from these quantities. The analysis reveals the accuracy of the numerical solution procedure, that saturation of the susceptibility occurs without including bound-continuum transitions, and that a linear extrapolation of the total susceptibility versus the ionization probability to obtain the instantaneous susceptibility is quantitatively unreliable.