## Dissertations and Theses

#### Author

Jay Gopalakrishnan

Fall 2021

12-9-2021

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

#### Department

Mathematics and Statistics

English

#### Subjects

Mathematics, Fiber optics

#### DOI

10.15760/etd.7710

#### Physical Description

1 online resource (xi, 178 pages)

#### Abstract

In this work, the finite element method and the FEAST eigensolver are used to explore applications in fiber optics. The present interest is in computing eigenfunctions u and propagation constants β satisfing [sic] the Helmholtz equation Δu + k2n2u = β2u. Here, k is the freespace wavenumber and n is a spatially varying coefficient function representing the refractive index of the underlying medium. Such a problem arises when attempting to compute confinement losses in optical fibers that guide laser light. In practice, this requires the computation of functions u referred to as guided modes and leaky modes. For guided modes, the location of the corresponding propagation constants in the complex plane is known in the optics literature, making the FEAST algorithm an ideal candidate to tackle this problem. In practice, one solves the Helmholtz equation by prescribing zero Dirichlet boundary conditions on a bounded, circular domain representing the fiber cross-section. In this work, we compute numerical solutions to this problem using the FEAST algorithm with the Discontinuous Petrov-Galerkin method for the underlying discretization. To compute leaky modes, we must find complex-valued propagation constants β with corresponding outgoing functions u. To compute such quantities, we employ a Perfectly Matched Layer (PML) to force exponential decay of the solutions, rendering the problem computationally tractable on a bounded domain. A frequency-dependent approach is taken for the PML: This transforms the weak formulation of the Helmholtz problem into a polynomial eigenvalue problem, thus motivating our the adaptation of the FEAST algorithm to solve such eigenvalue problems. We verify the results of our algorithm by finding leaky modes and propagation constants of a step-index fiber, where one can compare against known analytic solutions. Our algorithm is then applied to the task of computing confinement losses of a microstructure fiber, where lossy modes are expected. Confinement losses are computed from the imaginary parts of the corresponding propagation constants, and quantify the loss of power as light travels through an optical fiber. Our numerical results show that in practice, there is a large preasymptotic regime prior to convergence of confinement losses, suggesting that confinement losses reported in the literature could be matched in this regime. In addition, we show that our computed confinement losses remain stable as the strength of the decay in the PML region and the PML region's size are varied. Our results also show that computed confinement losses are extremely sensitive to minor perturbations in fiber geometry.