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
Number of Pages
111
Format
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License
Recommended Citation
Copeland, Austin, "Nonlinear Photonics in Twisted and Nonlocal Structures" (2020). Mathematics Theses and Dissertations. 6.
https://scholar.smu.edu/hum_sci_mathematics_etds/6
