modulation instability

The nonlinear stage of modulation instability (MI) is extremely rich. For periodic perturbations multiple recurrences occurs according to a complex homoclinic structure that represents the continuation of MI in the depleted stage. When the perturbation becomes localized the MI recurrences break down and different scenarios are possible. A quite universal scenario is the development of an auto-modulation, i.e. a strongly oscillating structure within a characteristic wedge-shaped region that smoothly connect to the background. However, for sufficiently generic perturbations the auto-modulation can be accompanied by the emission of breather pairs. In this talk we discuss how to predict the parameters of such breathers in terms of simple formulas.

The study of instabilities that drive extreme events is central to nonlinear science. Perhaps, the most canonical form of nonlinear instabilities is modulation instability (MI) describing the exponential growth of a weak perturbation on top of a continuous background. In optical fibres, when driven initially by small-amplitude noise, MI has been shown to lead to the emergence of localized temporal breathers with random statistics. It has also been suggested that these dynamics may be associated with the emergence of extreme events or rogue waves [1,2]. However, direct measurement in the time-domain of the breather properties is extremely challenging, requiring complex time-lens systems that typically suffer from drastic experimental constraints [3,4]. Real-time spectral measurement techniques such as the dispersive Fourier transform (DFT) on the other hand are commonly used to measure ultrafast instabilities [5]. Although relatively simple to implement, the DFT only provides spectral information. Here, we show how machine learning can overcome this restriction to study time-domain properties of optical fibre modulation instability based only on spectral intensity measurements. Specifically, we demonstrate that it is possible to train a supervised neural network to correlate the spectral and temporal properties of modulation instability using numerical simulations, and then apply the trained neural network to the analysis of high dynamic range experimental MI spectra and yield the temporal probability distribution for the highest peaks in the instability field [6].

[1] D.R. Solli, C. Ropers, P. Koonath and B. Jalali, "Optical rogue waves", Nature 450, 1054-057 (2007).

[2] J.M. Dudley, F. Dias, M. Erkintalo, and G. Genty, "Instabilities, breathers and rogue waves in optics," Nat. Photonics 8, 755–764 (2014).

[3] K. Goda and B. Jalali, "Dispersive Fourier transformation for fast continuous single-shot measurements", Nat. Photon. 7, 102-112 (2013).

[4] M. Närhi, et al. "Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability," Nat. Commun. 7, 13675 (2016).

[5] P. Suret et al., "Single-shot observation of optical rogue waves in integrable turbulence using time microscopy," Nat. Commun. 7, 13136 (2016).

[6] M. Narhi et al., ''Machine learning analysis of extreme events in optical fibre modulation instability,'' Nat. Commun. 9, 4923 (2018)

We report an original method enabling a non invasive characterization in phase and intensity of the longitudinal evolution of the main spectral components involved in the Fermi Pasta Ulam recurence process. We will show that it allows to evidence the symmetry breaking of the process. Future prospects and recent results will be presented.