- 1Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR CNRS-Université Bourgogne Franche-Comté, Dijon, France
- 2Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia
- 3Skolkovo Institute of Science and Technology, Moscow, Russia
- 4Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW, Australia
- 5Marine Studies Institute, The University of Sydney, Sydney, NSW, Australia
- 6Landau Institute for Theoretical Physics RAS, Chernogolovka, Russia
- 7Department of Mathematics, University of Arizona, Tucson, AZ, United States
Mutual interaction of localized nonlinear waves, e.g., solitons and modulation instability patterns, is a fascinating and intensively-studied topic of nonlinear science. Here we report the observation of a novel type of breather interaction in telecommunication optical fibers, in which two identical breathers propagate with opposite group velocities. Under controlled conditions, neither amplification nor annihilation occurs at the collision point and most interestingly, the respective envelope amplitude, resulting from the interaction, almost equals another envelope maximum of either oscillating and counterpropagating breather. This ghost-like breather interaction dynamics is fully described by an N-breather solution of the nonlinear Schrödinger equation.
Introduction
The study of both, formation and interaction of localized waves has been a central task in nonlinear physics during the last decades, including plasma physics, fluid dynamics, Bose-Einstein condensates and photonics. Among different types of nonlinear localized waves, solitons are the most representative and ideal testbed to investigate nonlinear wave interactions due to their intrinsic particle-like properties during propagation [1–4]. A generic and relevant case of study for various fields of research is the elastic and nonlinear interaction of envelope solitons, which can be described by the focusing one-dimensional nonlinear Schrödinger equation (NLSE). In this conservative and integrable system, the possible collision of solitons with different velocities does not affect their shape or velocity after interaction, and their main physical properties keep unchanged. In general, the interaction-induced displacement in position and phase shift are independent on the relative phases of the envelope solitons. However, collision dynamics in the interaction region strongly depends on the relative phases. Consequently, in the simplest case of two-soliton collision with opposite velocities, as shown in Figures 1A1–D1 the two solitons appear to attract with each other and cross (forming a transient peak) in the in-phase configuration, while they seem to repel each other and as such stay apart in the out-of-phase case. The wave magnitude at the central point of collision then evolves from the sum of the two solitons’ amplitudes (i.e., amplification) to their difference (i.e., annihilation), respectively. A large range of theoretical descriptions, numerical simulations and experimental observations of such soliton interactions and their possible synchronization have been reported earlier [5–12].
FIGURE 1. Typical temporal evolution of soliton-pair interaction (first line) and breather-pair interaction (second line). (A1) Dependence of amplitude at the soliton collision point
Besides solitons, breather solutions of the NLSE are also exciting examples to investigate nonlinear wave interactions because of the salient complexities of breather synchronization in relation to their self-oscillating properties. From this point of view, phase-sensitive breather interactions are now widely studied [13–22]. More particularly, for co-propagative breathers, breather molecules can be formed when group velocity and temporal phase of breathers are perfectly synchronized, while for counter-propagating breathers, the phase-sensitive collision process exhibits various dynamical behaviors. Two of them have been studied in detail in the context of rogue wave formation, namely amplification and annihilation cases that resemble soliton collisions. The above interactions are fully described by N-breather solutions of the NLSE [16, 18]. However, the two-breather collision has been recently found to provide a peculiar third configuration for particular phases, neither of the above-mentioned cases, the later leads to a peak amplitude at the central point of collision equivalent to the single breather amplitude before or after the collision. Phenomenologically, it seems that one breather mysteriously disappears in the nonlinear interaction region, but it then appears after that. That is why this intriguing breather interaction was vividly termed by “ghost interaction” [19]. Its generalization to the N-breather interaction is still under investigation. However, both, detailed analysis and experimental confirmation of this remarkable dynamics for the simplest two-breather collision are still to emerge into light.
To address this scientific gap, we present the observation of ghost interaction of two breathers in a single-pass telecommunication optical fiber experiment. By means of the Fourier-transform pulse shaping technique applied to an optical frequency comb, we generate the initial condition for two counter-propagating breathers with desired temporal phases. The experimental results are in excellent agreement with the exact two-breather solution of the NLSE. We confirm that this peculiar phase-sensitive breather interaction is strictly different to the well-known soliton interactions. Our study paves the way for novel directions of investigation in the rich landscape of complex nonlinear wave dynamics [23–26].
Method
Theoretical Model and Breather Solutions
Our theoretical framework and starting point is based on the dimensionless form of the self-focusing 1D-NLSE:
where subscripts stand for partial differentiations. Here,
where
and
In these expressions,
with
The parameters
Figures 1A2–D2 presents the interaction of a pair of counter-propagating breathers when
We emphasize that such ghost interaction of breathers as illustrated in Figure 1D2 cannot occur for the soliton counterpart (see Figures 1A1–D1). To clarify this point, we compare systematically the phase-dependent soliton collision and the phase-dependent breather collision. Similarly, we consider a pair of counter-propagating solitons with the amplitudes
In this expression,
Again, the
Experimental Setup
In order to validate these theoretical predictions with respect to ghost interaction of breathers, we have performed experiments with light waves propagating in high-speed telecommunication-grade components, as depicted in Figure 2. The main challenge here is the arbitrary wave shaping to establish the specific initial excitation of counter-propagating breathers with desired phases in the
FIGURE 2. Experimental setup and generation of initial conditions. (A) Schematic diagram of the experimental setup. EDFA: erbium-doped fiber amplification; SMF: single mode fiber; OSA: optical spectral analyzer; OSO: optical sampling oscilloscope. Shaded-green box represents the home-made frequency comb source with a repetition rate of 20 GHz. (B–C) Designed initial conditions at 20 GHz repetition rate for a pair of contra-propagative breathers in both temporal and spectral domains. Solid blue lines are theoretical curves; Solid red lines are experimental measurements. Here breather parameters are:
To this end, a 20 GHz optical frequency comb passes through a programmable optical filter (wave-shaper) to precisely control both amplitude and phase characteristics of each comb line. As a result, we can synthesize any arbitrary perturbation of a continuous wave background in a time-periodic pattern whose frequency is equal to the comb spacing. This temporal pattern is then amplified by erbium-doped fiber amplifier (EDFA) to achieve the exact excitation of the two-breather solution in terms of average power for nonlinear propagation into our single-mode optical fiber (SMF). The corresponding temporal and spectral power profiles of the light-wave are presented in Figures 2B–C. Note that the initial condition for the breather pair is time-periodic with a period of
Results
We present our experimental results on the nonlinear space-time evolution of the breather pair studied in the above theoretical section, for the specific temporal phases
Figures 3A1–A2 presents the concatenation of temporal (amplitude) profiles and power spectra which were recorded at the output of the distinct fiber segments with increasing length. The careful control of phases allows to observe the ghost interaction between the counter-propagating breathers. The full space-time dynamics is indeed in excellent agreement with theory shown in Figures 3B1–B2. We can notice the five mains peaks appearing during the whole evolution studied in Figure 3A1: two peaks at
FIGURE 3. Experimental observation of ghost interaction of two breathers. Color maps showing the evolution of temporal amplitude (A1) and power spectrum (A2) for the two breathers observed in experiment. Dashed white lines indicate the position of local maximum amplitudes, which are also the position of largest spectral broadenings, before collision (
Discussion
As shown above, during the ghost interaction of two breathers, only a single breather peak remains occurrent at the collision point. The reason for this intriguing phenomenon is related to the fact that there is a continuous and varying power exchange between the background and each localized perturbation all along the propagation, which is an intrinsic property of breathers. Therefore, when these two breathers nonlinearly interact near the collision point, for a given particular phase-collision-interaction, one of the breather peaks appears to be almost hidden in the background and then emerges again after the collision by following the energy conservation. Moreover, the breather pair keeps the spatial and temporal symmetry during the whole evolution. It is also worth to mention that such peculiar ghost interaction does not occur in conventional soliton-soliton collision scenarios because of the lack of pulse-background energy exchange [see Figures 1A1–D1].
In summary, we performed a systematic theoretical comparison between the phase-sensitive soliton-soliton collisions and breather-breather collisions. All different configurations are fully described by the exact N-breather solution of the NLSE. More importantly, we provided the first experimental observation of the very fascinating type of ghost interaction of breathers, which confirms our theoretical predictions. We also point out that our study is here restricted to the interaction of two identic counter-propagating breathers, while much more complicated many-body interactions of breathers with asymmetric conditions, including different amplitudes and/or oscillating frequencies, still require further investigations. Our current results concede a novel step toward the understanding of interactions between localized waves in nonlinear physics. These may naturally lead to encourage further relevant experimental studies and theoretical investigations in various fields of nonlinear wave physics.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author Contributions
All authors listed have made significant contributions to the theoretical and experimental developments, data acquisition, results interpretation and manuscript writing.
Funding
French National Research Agency (PIA2/ISITE-BFC, Grant No. ANR-15-IDEX-03, “Breathing Light” project). Theoretical part of the work was supported by Russian Science Foundation (Grant No. 19-72-30028).
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Keywords: nonlinear waves, modulation instability, breathers, solitons, nonlinear fiber optics
Citation: Xu G, Gelash A, Chabchoub A, Zakharov V and Kibler B (2020) Ghost Interaction of Breathers. Front. Phys. 8:608894. doi: 10.3389/fphy.2020.608894
Received: 22 September 2020; Accepted: 10 November 2020;
Published: 07 December 2020.
Edited by:
Manuel Asorey, University of Zaragoza, SpainReviewed by:
Haci Mehmet Baskonus, Harran University, TurkeyAbdullahi Yusuf, Federal University, Nigeria
Copyright © 2020 Xu, Gelash, Chabchoub, Zakharov and Kibler. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Bertrand Kibler, bertrand.kibler@u-bourgogne.fr