A practical numerical theory for light propagation in erbium-doped waveguide amplifiers

Abstract

The complex transverse waveguide geometries twinned with the transcendental rate equations governing Erbium-Doped Waveguide Amplifiers (EDWA) warrant the application of intricate Numerical Methods. To date all models approaching the problem have incorporated a finite element scheme for the modal analysis of the waveguide in conjunction with either a Beam Propagation Method (BPM) or an explicit Runge-Kutta procedure towards solving the channel propagation equations. In deploying these models, a certain ambiguity lies in their practical utilisation enforcing a trade-off between simulation time and demands on computer memory. The underlying theory of the finite element method (FEM) for optical waveguide simulations comprises of a multitude of approaches to solving the scalar Helmholtz equation. Past FEM techniques were based on a combination of matrix sparsity manipulations and iterative techniques but today, even modest computers can be utilised in more direct but denser methods for eigenvalue solutions. This paper aspires to demonstrate an efficient modal analysis technique founded on an appropriate mesh with due consideration for both computational complexity and available Random Access Memory requirements. It is then trivial to continue from this modal field definition to the solution of the aforementioned rate equations for channel gain, noise or nonlinear losses including upconversion and cross relaxation in the EDWA by implementing a suitable boundary value problem approach.