Rare Events in a Mode-Locked Fiber Laser

Sanford, Nathan Len

doi:https://doi.org/10.21985/n2-65b0-ge30

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Rare Events in a Mode-Locked Fiber Laser

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Optical fibers utilize nonlinear effects to help transmit soliton or near soliton pulses in a variety of contexts including optical communication systems and fiber lasers. Fiber lasers produce ultra-short pulses, down to a few femtoseconds in duration, via a process called mode-locking where modes of the optical cavity are synchronized via a variety of physical mechanisms to produce stable solitons. Active mode-locking, where an optical device such as a modulator facilitates the synchronization, is a scheme popular for the level of control it gives to the mode-locking process. Even with such control, however, pulses in an actively mode-locked laser can occasionally slip relative to the timing signal leading to fluctuations in the pulse repetition rate. Monte Carlo simulations can be used to capture such errors, but are ineffective when the events are rare as the number of simulations required to determine their probability is too computationally expensive. In modern nonlinear optical fibers, error rates are typically below $10^{-10}$ so traditional methods are not within the realm of possibility. Importance sampling modifies Monte Carlo simulations to make them feasible by more efficiently sampling from regions in phase space that give rise to rare events via the introduction of biasing distributions. The identification of important regions is usually accomplished via the solution of an optimization problem. In nonlinear optics, the formulation of such problems is frequently guided by soliton perturbation theory. Here, quantification of position slip error rates in a model of a soliton-based mode-locked laser are obtained using importance-sampled Monte Carlo simulations where the physical effects of the active mode-locking are incorporated as small perturbations. Position slips are studied in two distinct cases: an overdamped regime where they are primarily direct, and an underdamped regime where they typically involve oscillations. The qualitative distinction of direct versus oscillatory biasing paths between these regimes motivates the use of different importance sampling techniques. Quantification of the slip rate is shown to be more straightforward in the overdamped regime. Dynamic importance sampling, where error pathways are dynamically recomputed mid-simulation to maximize error yield, is found to be necessary to accurately and efficiently capture error rates. Dynamic importance sampling is shown to be more difficult to implement in the underdamped regime due to the existence of multiple routes by which position slip errors can occur. This motivates the development of a new importance sampling method blending dynamic importance sampling and multiple importance sampling, where multiple biasing distributions are used simultaneously. This algorithm is first applied to a toy model with multiple paths, a two-dimensional random walk past a transverse wall obstacle, and then is applied to the underdamped laser model. In both cases, this dynamic multiple importance sampling method is found to be far superior to using either dynamic importance sampling or multiple importance sampling in isolation.

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