Rebecca Bryant | PhD in Mathematical Sciences Dissertation Defense

Title: The Method of Periodic Averaging Applied to Reduced Coupled Mode Theory Models for Fiber Laser Amplifiers

Abstract:

Fiber laser amplifier (FLA) models are often implemented without rigorous mathematical justification or thorough numerical validation. Without a proper theoretical basis for assumptions and approximations, or a technical analysis of model performance, there is significant uncertainty about the limitations of any given reduced model and its suitability for an application. This research aims to address the lack of comprehensive assessment of FLA models by directly comparing distinct models and recommending a mathematical alternative to replace heuristic model-reduction techniques. The work in this dissertation is divided into two projects: a comparative study that uses existing FLA models to assess the validity of model approximations for predicting relative changes in the transverse mode instability (TMI) thresholds, and the derivation of an innovative reduced-order FLA model based on mathematically sound approximations.

The TMI comparison project studies the performance of three unique FLA models that can be characterized as high-, mid-, and low-fidelity, respectively. The high- and mid-fidelity models exhibit exceptional similarity in the stable regime before the onset of TMI. As the amplifier transitions from the stable regime to the TMI regime, all three models produce power spectral densities (PSDs) with offset frequencies within the same bandwidth at which TMI is observed. This alignment informs a novel interpretation of the TMI nonlinearity as a phase modulation effect. The reduced model development project derives an efficient FLA model based on the mid-fidelity model using the method of averaging from dynamical systems. A generalized periodic averaging theorem is applied to the mid-fidelity model, removing all higher frequencies and yielding the reduced model. The new reduced model is evaluated for its increased computational efficiency and for the limitations of the averaging method as a robust reduction technique.