A Vectorial Envelope Maxwell Formulation for Electromagnetic Waveguides with Application to Nonlinear Fiber Optics

This article presents an ultraweak discontinuous Petrov-Galerkin (DPG) formulation of the time-harmonic Maxwell equations for the vectorial envelope of the electromagnetic field in a weakly-guiding multi-mode fiber waveguide. This formulation is derived using an envelope ansatz for the vector-valued electric and magnetic field components, factoring out an oscillatory term of \(exp(-i \mathsf{k}z)\) with a user-defined wavenumber \(\mathsf{k}\), where \(z\) is the longitudinal fiber axis and field propagation direction. The resulting formulation is a modified system of the time-harmonic Maxwell equations for the vectorial envelope of the propagating field. This envelope is less oscillatory in the \(z\)-direction than the original field, so that it can be more efficiently discretized and computed, enabling solution of the vectorial DPG Maxwell system for \(1000\times\) longer fibers than previously possible. Different approaches for incorporating a perfectly matched layer for absorbing the outgoing wave modes at the fiber end are derived and compared numerically. The resulting formulation is used to solve a 3D Maxwell model of an ytterbium-doped active gain fiber amplifier, coupled with the heat equation for including thermal effects. The nonlinear model is then used to simulate thermally-induced transverse mode instability (TMI). The numerical experiments demonstrate that it is computationally feasible to perform simulations and analysis of real-length optical fiber laser amplifiers using discretizations of the full vectorial time-harmonic Maxwell equations. The approach promises a new high-fidelity methodology for analyzing TMI in high-power fiber laser systems and is extendable to including other nonlinearities.