Analysis of laser radiation using the Nonlinear Fourier transform

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.

Neural networks for computing the continuous nonlinear Fourier spectrum in focusing nonlinear Schrodinger equation: NFT-Net

“…Finally, we note that, recently, the interest in using the NFT as a signal-processing tool has risen in fields that are not directly relevant to optical transmission. In particular, the NFT was applied in the so-called integrable turbulence to monitor the appearance of coherent structures, such as breathers, solitons, and rogue waves 46,47 , to the optical microresonators regime analysis 48 , to the optical frequency combs characterisation 49 , and to the analysis of laser regimes and the emergence of dissipative coherent nonlinear structures [50][51][52] . The analysis of NFT modes' evolution for such systems often appears to be more informative and convenient than dealing with the conventional Fourier modes.…”

Characterization of coherent structures in dissipative systems using nonlinear Fourier transform

“…The NFT based on the Zakharov-Shabat spectral problem makes it possible to simplify the analysis of the NLSE and reduce complex nonlinear dynamics to simple evolution in a certain basis -the so-called nonlinear spectrum that consists of a continuous and discrete part. Recently we have introduced a concept of using NFT for analysis of the evolution of dissipative, non-integrable systems, 6,7 see also. [8][9][10] Nonlinear cubic Ginzburg-Landau equation (CGLE), that is a particularly important example in the context of modeling laser systems was considered in.…”

Nonlinear Fourier transform for analysis of optical spectral combs

“…The initial idea (e.g. presented in [19][20][21][22]) behind using IST/NFT beyond the traditional integrable systems, was to exploit the fact that for some non-integrable (e.g. dissipative) models the Hamiltonian part of these equations is NLSE, and, thus, one can expect that the IST/NFT might still be a useful tool for analysis of the whole (non-Hamiltonian) systems.…”

“…Second, Fourier transform might be useful in simplifying the description of complex objects by presenting them via spectral harmonics. It was shown in [19,21,22], that in a similar manner IST/NFT can be employed not only for solving integrable equations, but also for the characterisation of localised coherent structures in dissipative systems in the anomalous dispersion regime. Note that in [23] periodic NFT was applied for analysis of static (output) optical comb profiles in the LLE model.…”

“…In this Letter, we demonstrate novel features and applications of NFT compared to the initial idea [19][20][21]. We advance this emerging signal processing technique with the following novel applications of NFT to the characterisation of the dynamics of coherent structures during generating of optical combs: (i) for the first time we apply NFT based on ZSSP1 (with β = −1 in ZSSP (2)) to the analysis of dissipative dark solitons in Eq.…”