Analysis of the Discontinuous Petrov Galerkin Method as a Transverse Mode Solver For Optical Fibers in Conjunction with Generalized Polynomial Chaos Theory
Abstract
Analysis of the Discontinuous Petrov Galerkin Method as a Transverse Mode Solver For Optical Fibers in Conjunction with Generalized Polynomial Chaos Theory. This project is to develop the first discontinuous Petrov Galerkin (DPG) finite element method (FEM) solver for finding the transverse modes in optical fibers. These transverse modes are found by solving a time independent Helmholtz eigen-equation; starting with the scalar equation, and possibly exploring the vectorial equation later on. The DPG strategy is a higher-order method that has already been proven to find and utilize approximate optimal test functions and an ultra-weak variational formulation that guarantee stability in the FEM for well-posed linear mathematical problems [1]. However, as yet DPG has never been applied to eigenvalue problems, and there is still much to be discovered about the formulation and mathematical properties of the DPG method when solving an eigen-equation. Part of our plan is to use the DPG method to approximate the operators of the FEAST algorithm [2]. This entails the computation of the eigenvalues by approximating an operator-valued contour integral, which surrounds each eigenvalue of interest, in the complex plane. However, when appropriate, we will also calculate the associated eigenvectors, which are the transverse modes of the given optical fiber. Another significant part of this project will be to apply uncertainty quantification (UQ) techniques to the transverse mode solver problem, especially when considering microstructure optical fibers. The fabrica- tion of microstructure fibers is technically challenging, time consuming, and expensive. Also, it is often difficult to precisely reproduce [3]. Fabrication defects and inconsistencies sometimes have significant effects on the guiding properties and the quality of the fiber (see Fig. 1). Furthermore, for some applica- tions important to the Air Force, experimentalists wish to operate microstructure fibers at the edge of their transmission bandgaps, in single-mode operation at certain wavelengths that maximize laser brightness while avoiding pernicious amplified spontaneous emission (ASE). The complexity, time, and expense of optimizing microstructure designs necessitates computational modeling to ensure that the final fiber will perform as required. A UQ analysis will be used to find stable and robust fiber designs. Our study will focus on the implementation of the generalized Polynomial Chaos (gPC) strategy to UQ [5]. No one has ever applied gPC theory to an eigenvalue problem. This research effort will also cover how the DPG method can be leveraged in conjunction with the gPC technique. It will be important to understand the stability, convergence, and robustness of such an approach, especially in the view of how, in certain cases, small fabrication errors can lead to significant changes to the guiding characteristics of the fiber. We plan on comparing the gPC method to the standard Monte Carlo approach with sophisticated sampling techniques. Finally, it is worth noting that this research not only has the potential to develop new theory and applications for the DPG and gPC methods, but the final product, the transverse mode solver, will be a deliverable code to the AFRL Laser Division, Fiber Laser Beam Combining Program, in order to enable their power scaling efforts through optimally designed fibers. The fact that this solver will be designed to be robust and generalized means that it can be applied to a wide variety of fibers under a various realistic conditions, including fibers that experience thermal lensing, fiber bending (conformal mapping of the refractive index), and possibly stress or strain. Also, conventional, photonic bandgap, photonic crystal, hollow-core, and leaky channel fibers, etc., used for active and passive optical amplifiers, delivery fibers, and gas-filled fiber lasers, can be studied with this novel transverse mode solver, and are of interest to the AFRL.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 31, 2019
- Source ID
- FA94511810034
Entities
People
- Leszek F. Demkowicz
Organizations
- Air Force Research Laboratory
- University of Texas at Austin