Abstract

We provide a theoretical framework for the observed confinement of light modes within a twisted coreless photonic crystal fiber. Asymptotic methods are applied through ray theory and field theory in both the linear and nonlinear regime. We find the modes have a radially symmetric chirp and the envelope will decay away from the axis of propagation. Secondly, we study the stability and singularity formation of unidirectional beams as described by the Schrodinger equation. We propose a novel extension to the modeling equation to include a fractional Laplacian in one spatial dimension and a standard second derivative in a second dimension. The goal is to explore dynamics and stability properties as a function of the degree of fractionality. Numerically, we use a time-splitting Fourier pseudo-spectral method which accounts for nonlocal interactions from the fractional Laplacian and is applicable to the linear and nonlinear cases. We find minimal values of the fractional parameter where singularities form for various power levels and see that symmetry does not always hold near blowup in the fractional case.

Degree Date

Spring 5-2020

Document Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

Advisor

Alejandro Aceves

Subject Area

Mathematics, Applied

Number of Pages

111

Format

.pdf

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

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