This paper outlines the design and testing of a z-scan spectrometer capable of measuring the third order refraction index of liquids. The spectrometer underwent multiple redesigns, with each explored in this paper with their benefits and drawbacks discussed. The first design was capable of measuring the third order refraction index for glass, and found a value of 8.43 +- 0.392 x 10^(-16) cm^2/W for the glass sample, with the literature stating glass has a refraction index between 1-100 x 10^(-16) cm^2/W. The second design was capable of measuring the third order refraction index of liquids, and found values of 1.23 $\pm$ 0.121 $\e{-16}$ and 9.43 +- 1.00 x 10^(-17) cm^2/W for water and ethanol respectively, with literature values of 2.7 x 10^(-16) and 5.0 x 10^(-17) cm^2/W respectively. The third design gave inconclusive results due to extreme variability in testing, and and the fourth design outlined has not been tested yet due to time constraints.

This document is a guide that can be used by undergraduate physics students alongside Richard J. Jacob and Professor Emeritus’s Tutorials in the Mathematical Methods of Physics to aid in their understanding of the key mathematical concepts from PHY201 and PHY302. This guide can stand on its own and be used in other upper division physics courses as a handbook for common special functions. Additionally, we have created several Mathematica notebooks that showcase and visualize some of the topics discussed (available from the GitHub link in the introduction of the guide).

In a hypothetical Grand Unified Theory, magnetic monopoles are a particle which would act as a charge carrier for the magnetic force. Evidence of magnetic monopoles has yet to be found and based off of their relatively high mass (4-10 TeV) will be difficult to find with current technology. The goal of my thesis is to mathematically model the magnetic monopole by finding numerical solutions to the equations of motion. In my analysis, I consider four cases: kinks, cosmic strings, global monopoles, and magnetic monopoles. I will also study electromagnetic gauge fields to prepare to include gauge fields in the magnetic monopole case. Numerical solutions are found for the cosmic string and global monopole cases. As expected, the energy is high at small distance r and drops off as r goes to infinity. Currently numerical solutions are being worked towards for electromagnetic gauge fields and the magnetic monopole case.