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Accelerated Coupled Mode Model for Fiber Laser Amplifiers as an Averaged Dynamical System

Rebecca Bryant, Jacob Grosek, Jay Gopalakrishnan

TL;DR

This paper develops a mathematically rigorous reduced model for fiber laser amplifiers by applying a generalized averaging theorem to a complex, multi‑mode coupled mode theory (CMT) formulation. By explicitly extracting periodic terms and nondimensionalizing, the authors form an autonomous averaged model (ACM) whose accuracy is guaranteed within an $\mathcal{O}(\varepsilon)$ bound and that enables substantially larger longitudinal steps. For a representative Yb‑doped fiber amplifier, the ACM reproduces key metrics such as output power and amplification efficiency with negligible error while delivering about a 4,000× speedup over the full CMT solver, and it remains accurate for steady‑state temperature predictions. The results suggest the technique generalizes to a wider class of amplifier configurations via a first‑order Taylor expansion of the gain around the DC component, enabling broader applicability beyond ytterbium and two‑tone scenarios.

Abstract

We apply a known theorem for simplifying dynamical systems with bounded error to a specific optical fiber waveguide problem, supplementing the physical intuition and heuristics used in the optics community with proper mathematical justification. Using techniques from averaging theory of dynamical systems, a reliable accelerated model based on the coupled mode theory (CMT) approach for a common fiber laser amplifier application is derived. Computational testing reveals that this accelerated model achieves an ${\sim}4000$x increase in computational speed compared to the CMT model while preserving a high accuracy in key figures-of-merit such as output power and amplification efficiency. Further, we argue that by adopting our recommended approximations within the reduced model framework enables the model to be applied a wider set of amplifier types and configurations than can the current (comparable) reduced models found in the literature.

Accelerated Coupled Mode Model for Fiber Laser Amplifiers as an Averaged Dynamical System

TL;DR

This paper develops a mathematically rigorous reduced model for fiber laser amplifiers by applying a generalized averaging theorem to a complex, multi‑mode coupled mode theory (CMT) formulation. By explicitly extracting periodic terms and nondimensionalizing, the authors form an autonomous averaged model (ACM) whose accuracy is guaranteed within an bound and that enables substantially larger longitudinal steps. For a representative Yb‑doped fiber amplifier, the ACM reproduces key metrics such as output power and amplification efficiency with negligible error while delivering about a 4,000× speedup over the full CMT solver, and it remains accurate for steady‑state temperature predictions. The results suggest the technique generalizes to a wider class of amplifier configurations via a first‑order Taylor expansion of the gain around the DC component, enabling broader applicability beyond ytterbium and two‑tone scenarios.

Abstract

We apply a known theorem for simplifying dynamical systems with bounded error to a specific optical fiber waveguide problem, supplementing the physical intuition and heuristics used in the optics community with proper mathematical justification. Using techniques from averaging theory of dynamical systems, a reliable accelerated model based on the coupled mode theory (CMT) approach for a common fiber laser amplifier application is derived. Computational testing reveals that this accelerated model achieves an x increase in computational speed compared to the CMT model while preserving a high accuracy in key figures-of-merit such as output power and amplification efficiency. Further, we argue that by adopting our recommended approximations within the reduced model framework enables the model to be applied a wider set of amplifier types and configurations than can the current (comparable) reduced models found in the literature.

Paper Structure

This paper contains 19 sections, 1 theorem, 113 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized Averaging Theorem and Example
  3. Generalized Periodic Averaging Theorem
  4. A Complex-Valued Vectorial Averaging Example
  5. Coupled Mode Theory (CMT) Model
  6. CMT Acceleration Via Averaged Dynamics: ACM Model
  7. Explicit Dependence on Periodic Terms
  8. Extracting $\varepsilon$
  9. Summary of the CMT and ACM Model Governing Equations
  10. Model Comparison Results
  11. Steady-State Temperature Solution
  12. Points Per Mode Beat Length Study
  13. Conclusion
  14. References
  15. Steady-State Gain Formulas
  16. ...and 4 more sections

Key Result

Theorem 2.1

Suppose, for $i \in \{1, ..., N \}$, that $\mathbf{f}_{i}$ is Lipschitz continuous and that the following two criteria hold: Then there exists a constant $c > 0$, independent of $\varepsilon$, such that for $0 \leq \varepsilon \leq \varepsilon_{0}$ and $0 \leq t \leq \frac{S}{\varepsilon}$.

Figures (17)

  • Figure 1: Power evolution along a two-tone configured fiber laser amplifier.
  • Figure 2: Magnitude of the solutions to the original \ref{['vecODE']} and averaged \ref{['AvgVecODE']} ODEs.
  • Figure 3: Frequencies of the solutions to the original \ref{['vecODE']} and averaged \ref{['AvgVecODE']} ODEs.
  • Figure 4: Solutions to the original \ref{['vecODE']} and averaged \ref{['AvgVecODE']} ODEs for various values of $\varepsilon$.
  • Figure 5: The maximum span of time ($S$) for which the relative error \ref{['eq:z_error']} remains below $5\%$ for various values of $\varepsilon$.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Theorem 2.1