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This paper is a continuation of Melenk et al., "Stability analysis for electromagnetic waveguides. Part 1: acoustic and homogeneous electromagnetic waveguides" (2023), extending the stability results for homogeneous electromagnetic (EM) waveguides to the non-homogeneous case. The analysis is done using perturbation techniques for self-adjoint operators eigenproblems. We show that the non-homogeneous EM waveguide problem is well-posed with the stability constant scaling linearly with waveguide length L. The results provide a basis for proving convergence of a Discontinuous Petrov-Galerkin (DPG) discretization based on a full envelope ansatz, and the ultraweak variational formulation for the resulting modified system of Maxwell equations, see Part 1.
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This content will become publicly available on September 1, 2026
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(-ikz) with a user-defined wavenumber 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 solutions to the vectorial DPG Maxwell system in fibers that are 1000x longer 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.
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- Award ID(s):
- 2103524
- PAR ID:
- 10633354
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Computers & Mathematics with Applications
- Volume:
- 193
- Issue:
- C
- ISSN:
- 0898-1221
- Page Range / eLocation ID:
- 34 to 53
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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