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Experimental realization of fermi- pasta-Ulam-tsingou recurrence in a long-haul optical fiber transmission system

Abstract : The integrable nonlinear Schrödinger equation (NLSE) is a fundamental model of nonlinear science which also has important consequences in engineering. The powerful framework of the periodic inverse scattering transform (IST) provides a description of the nonlinear phenomena modulational instability and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It associates the complex nonlinear dynamics with invariant nonlinear spectral degrees of freedom that may be used to encode information. While optical fiber is an ideal testing ground of its predictions, maintaining integrability over sufficiently long distances to observe recurrence, as well as synthesizing and measuring the field in both amplitude and phase on the picosecond timescales of typical experiments is challenging. Here we report on the experimental realization of FPUT recurrence in terms of an exact space-time-periodic solution of the integrable NLSE in a testbed for optical communication experiments. The complex-valued initial condition is constructed by means of the finite-gap integration method, modulated onto the optical carrier driven by an arbitrary waveform generator and launched into a recirculating fiber loop with periodic amplification. The measurement with an intradyne coherent receiver after a predetermined number of revolutions provides a non-invasive full-field characterization of the space-time dynamics. The recurrent space-time evolution is in close agreement with theoretical predictions over a distance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinear spectrum. The space-time scale exceeds that of previous experiments on FPUT recurrence in fiber by three orders of magnitude. The NLSE is an important exactly solvable model for the study of nonlinear phenomena. An example is modu-lational instability (MI) 1 , an exponential amplification of periodic random fluctuations at the expense of a pump wave that has been suggested as a possible mechanism for the generation of rogue waves 2. The reversal of this process can give rise to repeated cycles of growth and decay, which constitute a realization of FPUT recurrence 3,4. In the framework of the IST, these phenomena find a description in terms of exact solutions 5,6 associated with conserved nonlinear spectral degrees of freedom. From an engineering perspective, the prospect of encoding information in the invariant nonlinear spectrum is of high interest for optical communication systems, which today are limited by nonlinear interference 7. Various predictions of the underlying analytical NLSE theory have been observed in optical fiber experiments, including solitons 8 , Akhmediev breathers 9 and their collisions 10 , the Peregrine soliton 11 , and the Kuznetsov-Ma soliton 12. Such experiments are not without challenges. They are typically conducted at average signal powers up to a few Watts 10,12-17. The dynamics take place over distances of several hundred meters up to few kilometers and on the picosecond scale. At such timescales, the generation of arbitrary initial conditions is difficult and has been approximated by amplitude modulation based on dual-frequency excitation 9,18 , by beating of two narrow-linewidth lasers to create a low-frequency modulation 11 , or excitation of superpositions of complex expo-nentials with tuned relative phases and amplitudes. The latter can be obtained from an optical frequency comb shaped with a programmable optical filter 10. The observation of the spatial dynamics has been achieved with fiber cutback experiments 12. Simultaneous observation of amplitude and phase information can be realized by frequency-resolved optical gating (FROG) 11 or nonlinear digital holography 14 .
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Jan-Willem Goossens, Hartmut Hafermann, Yves Jaouen. Experimental realization of fermi- pasta-Ulam-tsingou recurrence in a long-haul optical fiber transmission system. Scientific Reports, Nature Publishing Group, 2019, 9 (1), ⟨10.1038/s41598-019-54825-4⟩. ⟨hal-02395372⟩

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