Blinded by the Light: An Investigation of the Wave Propagation of Vector Modes of Light in a Spherically Symmetric Refractive Index Profile

Preston Pozderac

2017

We investigated the two vector mode solutions to the spherically symmetric wave equation for a step function refractive index profile. The investigation began with a discussion on the classification and boundary conditions of partial differential equations. We proceeded with the derivation of the numerical finite difference method for different types of partial differential equations. In addition to this numerical technique, we presented the analytical separation of variables method for cartesian and spherical coordinates. We then analyzed our physical system, a glass microsphere surrounded by another medium, by dividing the spherically symmetric wave equation into an unperturbed Hamiltonian and a perturbative correction term. This separation allowed us to determine the two families of unperturbed solutions, the vector modes, analytically and to perform first order perturbation theory on the numerically computed wavenumbers. We studied the dependence of the waves’ energy and propagation properties on the quantum numbers, vector modes, and different surrounding media. We finished our study of our wave equation by assuming our unperturbed solutions were factorable in their spin and orbital degrees of freedom. Applying perturbation theory, we determined the vector basis that diagonalized the perturbation Hamiltonian.