First Advisor

Jay Gopalakrishnan

Date of Award

8-2025

Document Type

Thesis

Department

Mathematics and Statistics

Language

English

DOI

10.15760/math_honors.01

Abstract

In an optical waveguide, only finitely many electric field intensity profiles travel without loss, and these are aptly named guided modes. This paper uses fundamental concepts from the two paradigms of interpreting light, geometrical and physical optics, to derive guided modes axiomatically. First, basic principles of geometrical optics are demonstrated in a slab waveguide to explain characteristics of optical waveguides such as total internal reflection and numerical aperture. Dispersion curves are assembled from the same principles and used to show that only finitely many guided modes exist for a given waveguide. Next, Maxwell’s equations are used to derive the famous electromagnetic wave equation. PDE theory is used to investigate the resulting Helmholtz eigenproblem, with comparison drawn to the Schrödinger equation of quantum physics. Afterward, semi-analytical means of calculating both guided mode field profiles and their corresponding propagation constants for step-index optical fibers are developed. In the final section, this semi-analytical method is compared to a numerical finite element solution of the aforementioned Helmholtz eigenproblem on the fiber cross-section.

Rights

Copyright 2025 by Mason Spears.

Licensed under CC BY 4.0

This license allows others to distribute, remix, adapt, and build upon the work, even commercially, as long as they credit you for the original creation

Comments

An undergraduate honors project submitted in partial fulfillment of the requirements for the Fariborz Maseeh Department of Mathematics and Statistics Honors Track.

Persistent Identifier

https://archives.pdx.edu/ds/psu/44077

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