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.

翻译：本文针对弱导多模光纤波导中电磁场的矢量包络，提出了一种时谐麦克斯韦方程的超弱间断彼得罗夫-伽辽金（DPG）变分形式。该形式通过引入矢量电场和磁场分量的包络假设推导得出，其中提取了一个振荡项 $exp(-i \mathsf{k}z)$，$\mathsf{k}$ 为用户定义的波数，$z$ 为光纤纵向轴及场传播方向。所得公式是描述传播场矢量包络的修正时谐麦克斯韦方程组。该包络在 $z$ 方向上的振荡性较原始场更弱，因此能够更高效地进行离散化和计算，使得求解矢量DPG麦克斯韦系统的光纤长度可达以往能力的 $1000$ 倍。本文推导了多种引入完美匹配层以吸收光纤端面出射波模式的方法，并进行了数值比较。利用所得公式，结合包含热效应的热传导方程，求解了掺镱有源增益光纤放大器的三维麦克斯韦模型。该非线性模型随后被用于模拟热致横向模式不稳定性（TMI）。数值实验表明，通过对完整矢量时谐麦克斯韦方程进行离散化，对实际长度的光纤激光放大器进行仿真分析在计算上是可行的。该方法为分析高功率光纤激光系统中的TMI提供了一种新的高保真方法，并可扩展至包含其他非线性效应。