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

Abstract

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.

Figures (9)

• Figure 1: Illustration of a step-index fiber waveguide.
• Figure 2: Electric field magnitude of the lowest-order transverse core-guided LP modes in a weakly-guiding step-index fiber. The $\text{LP}_{01}$ mode is the fundamental mode (FM) and has no cutoff frequency. All other LP modes are higher-order modes (HOMs) that can only propagate above their respective cutoff frequency. Asymmetric HOMs such as $\text{LP}_{11}$ or $\text{LP}_{21}$ have multiple rotations. In a symmetric (i.e. not birefringent) step-index fiber, these rotated modes all have the same cutoff frequency.
• Figure 3: Irradiance plotted in a longitudinal slice (normal to the $y$-axis) illustrating the mode beat between the FM and the HOMs. The modal interference pattern oscillates at a much longer length scale $(\mathcal{O}(\text{mm}))$ than the optical wavelength $(\mathcal{O}(\mu\text{m}))$.
• Figure 4: The cost of discretization for resolving the frequencies (spatially in $z$) of the original propagating field $E$, illustrated in (a), is proportional to the maximum frequency $\mathcal{O}(k_\mathrm{core})$; the envelope ansatz shifts the frequencies by the envelope wavenumber $\mathsf{k}$, re-centering them nearer the zero frequency ($k=0$), as portrayed in (b), thereby reducing the cost of discretization to the maximum frequency of the envelope which is at most $\mathcal{O}(k_\mathrm{core} - k_\mathrm{clad})$. In the weakly-guiding fiber, this frequency shift reduces the maximum frequency by a factor proportional to $k_\mathrm{core} / (k_\mathrm{core} - k_\mathrm{clad}) = \mathcal{O}(1000)$.
• Figure 5: Envelope PML formulation 1: single-mode propagation in a step-index fiber waveguide of about $0.7$ mm length ($\sim 1000$ wavelengths). Using an envelope wavenumber of $\mathsf{k} = 8.5\ \mu\text{m}^{-1}$, somewhat smaller than the effective wavenumber $k_\mathrm{eff} = 8.56833\ \mu\text{m}^{-1}$ of the propagating $\text{LP}_{01}$ mode, the envelope is slowly varying along the $z$ direction (ca. $125$ slower than the original field). Inside the PML region (starting at $z=0.384$ mm and stretching about three envelope beats), the slowly oscillating envelope decays exponentially.

Theorems & Definitions (5)

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