Song, Jung Heon2020-10-262020-10-262020-07https://hdl.handle.net/11299/216825University of Minnesota Ph.D. dissertation. July 2020. Major: Mathematics. Advisors: Mitchell Luskin, Matthias Maier. 1 computer file (PDF); viii, 65 pages.This thesis develops numerical frameworks for the simulation of light-guiding structures involving atomically thin 2D materials. Particular electromagnetic waves of interest are the surface plasmon polaritons (SPPs), which are slowly decaying electromagnetic waves that are highly confined to the 2D material. Characterized by subwavelength confinement, effective manipulation of SPPs are of great importance in nanophotonics, and in turn, in the design of an optical wavegudie. To that end, this dissertation studies how to manipulate intrinsic properties of waveguides to control its effectiveness. In particular, we discuss analytically and numerically the propagation and energy transmis- sion of electromagnetic waves caused by the coupling of SPPs between two spatially separated layers of 2D materials, such as graphene, at subwavelength distances. We construct an adaptive finite element method to compute the distance at which the optimal tranmission is attained. Next, a more generalized structural setup is laid out by introducing spatially dependent dielectrics and arbitrarily shaped interior conducting interface(s). This leads to a quartic eigen- value problem with mixed transverse electric (TE) and transverse magnetic (TM) modes, and strongly coupled electric and magnetic fields. We derive a weak formulation of the quartic eigenvalue problem and introduce a numerical solver based on a quadratification approach, in which the quartic eigenvalue problem is transformed to a spectrally equivalent linear eigenvalue problem. As a practical example, we demonstrate how an improved quality factor (defined by the ratio of the real and the imaginary part of the computed eigenvalues) can be obtained for a family of spatially dependent host materials with internal conducting interfaces. We outline how this result lays the groundwork for solving related shape optimization problems.enThesis or Dissertation