Abstract
Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions.
Highlights
Several of the operational regimes highlighted above possess a common characteristic, wherein localised structures interact with background dispersive waves
In the case of using the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem for signal processing of localised waveforms one can expect pulses to be approximated by a limited number of discrete eigenvalues, simplifying description compared to use of a large number of linear spectral harmonics
We show that the coherent features revealed by the NFT map one-to-one to localised structures observed in the laser intensity spatio-temporal dynamics, how this mapping can be used to track the appearance of localised structures in realtime, use it for representation and classification of lasing regimes as eigenvalue distribution functions, and under certain conditions how the NFT can be used to derive an approximate model for the laser
Summary
Several of the operational regimes highlighted above possess a common characteristic, wherein localised (in time) structures interact with background dispersive waves. We examine application of the NFT approach based on the Zakharov-Shabat spectral problem for description of laser radiation with a mixture of pulses and dispersive waves. In the case of using the NFT based on the Zakharov-Shabat spectral problem for signal processing of localised waveforms one can expect pulses to be approximated by a limited number of discrete eigenvalues, simplifying description compared to use of a large number of linear spectral harmonics. NFT components can be computed with round-trip time resolution, revealing the characteristic distribution of embedded localised modes and respective real-time pulse evolution features. In this regard, we address the important question of normalisation. The first entails normalising the system using the averaged fibre parameters, similar to ref
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