Statistical and spectral properties of the modulation instability: experiments and modelling by using soliton gas Pierre Suret Laboratoire de Physique des Lasers, Atomes et Molécules (Phlam), Univ. de Lille, France Stéphane Randoux, François Copie Alexey Tikan (PhD), Rebecca El Koussaifi (PhD), Adrien Kraych (PhD), Alexandre Lebel (PhD) Christophe Szwaj, Clément Evain, Serge Bielawski Gennady El, Thibault Congy (Postdoc), Giacomo Roberti (PhD), Northumbria univ., UK Andrey Gelash, Novosibirsk, Russia Dmitry Agafontsev, Moscow, Russia Vladimir Zakharov , Landau Institute for Theoretical Physics, Chernogolovka, Russia Antonio Picozzi , Dijon, France Miguel Onorato , Torino, Italy Eric Falcon, Annette Cazaubiel (PhD) MSC, Univ. Paris Diderot, France Guillaume Michel, Gaurav Prabhudesai (ENS, PhD) , Ecole Norm. Sup., France Félicien Bonnefoy, Guillaume Ducrozet, Ecole Centrale de Nantes, France Amin Chabchoub, Univ. Of Sydney, Australia WCST2019, Norwich (UK), 30 Oct-1st Nov. 2019
Optical Rogue Waves in Integrable Turbulence Modulation Instability Ø Benjamin-Feir instability (1967) Ø Deep Water waves Ø Sideband instability (breathers) N. Akhmediev et al. ,, Sov. Phys. JETP 62, 894 (1985). N. Akhmediev and V. Korneev,, Theor. Math. Phys. 69, 1089 (1986). N. Akhmediev, et al. Phys. Lett. A 373, 675 (2009). from Nobuhito Mori, reuse with permission Benjamin, T. Brooke; Feir, J.E. (1967). Journal of Fluid Mechanics. 27 (3) p.417–430 Benjamin, T.B. (1967). Proceedings of the Royal Society of London. A. 299 (1456) p.59–76
<latexit sha1_base64="6h8u34QXlZ72DuewKVyquldTxY=">ACR3icbVDLSsNAFJ3UV62vqEs3wSIQklS8bEQim5cVrAPaNIymU7q0MmDmYlQ0/ydG7fu/AU3LhRx6SQNalsPDHPm3HO5c48TUsKFr8ohYXFpeWV4mpbX1jc0vd3mnyIGIN1BA9Z2IMeU+LghiKC4HTIMPYfiljO8Sute8w4CfxbMQqx7cGBT1yCoJBST+1aLoMotkLIBIFUs0JOkt/nQ3IxMRhJbCbT3q456xZdMzki41Qd8306qlvaJn0OaJkZMyFHvqc9WP0CRh32BKOS8Y+ihsON0AKI4KVkRxyFEQzjAHUl96GFux1kOiXYglb7mBkweX2iZ+rcjh7nI8+RTg+KOz5bS8X/ap1IuGd2TPwEthHk0FuJBcOtDRUrU8YRoKOJIGIEflXDd1BGZaQ0ZeyEM5TnPysPE+aZsWoVqo3x+XaZR5HEeyBfXAIDHAKauAa1EDIPAIXsE7+FCelDflU/maWAtK3rMLplBQvgHPHLRV</latexit> Optical Rogue Waves in Integrable Turbulence Focusing 1D nonlinear Schrodinger equation τ ∂ 2 ψ ∂ψ ∂ z = 1 ∂ t 2 + i | ψ | 2 ψ 2 T 0 Ø Nonlinear optics (e.g. fibers) A ( x, y ) ψ ( z, t ) e i ( k 0 z − ω 0 t ) � � E ( x, y, z, t ) = < T0 ~ 5 fs τ ~ ps L ~ 0.1-1 km Ø Deep water waves T0 ~ s ψ ( z, t ) e i ( k 0 z − ω 0 t ) � � η ( z, t ) = < τ ~ 5s L ~ 0.1 km
<latexit sha1_base64="6h8u34QXlZ72DuewKVyquldTxY=">ACR3icbVDLSsNAFJ3UV62vqEs3wSIQklS8bEQim5cVrAPaNIymU7q0MmDmYlQ0/ydG7fu/AU3LhRx6SQNalsPDHPm3HO5c48TUsKFr8ohYXFpeWV4mpbX1jc0vd3mnyIGIN1BA9Z2IMeU+LghiKC4HTIMPYfiljO8Sute8w4CfxbMQqx7cGBT1yCoJBST+1aLoMotkLIBIFUs0JOkt/nQ3IxMRhJbCbT3q456xZdMzki41Qd8306qlvaJn0OaJkZMyFHvqc9WP0CRh32BKOS8Y+ihsON0AKI4KVkRxyFEQzjAHUl96GFux1kOiXYglb7mBkweX2iZ+rcjh7nI8+RTg+KOz5bS8X/ap1IuGd2TPwEthHk0FuJBcOtDRUrU8YRoKOJIGIEflXDd1BGZaQ0ZeyEM5TnPysPE+aZsWoVqo3x+XaZR5HEeyBfXAIDHAKauAa1EDIPAIXsE7+FCelDflU/maWAtK3rMLplBQvgHPHLRV</latexit> Spontaneous (noise induced) Modulation Instability Optical Rogue Waves in Integrable Turbulence ∂ 2 ψ ∂ψ ∂ z = 1 ∂ t 2 + i | ψ | 2 ψ 2 time time frequency frequency p g ( ω ) = | ω | 4 − ω 2 Ø Local emergence of breathers Numerical simulations | y (t)| 2 Numerical simulations Toenger, S., et al. J. M. Scientific reports, 5, 10380 (2015)
<latexit sha1_base64="MQU/z3vnwbqJNmDCaDaFc+pg3uQ=">ACBXicbVC7TsNAEDyHVwgvAyUFhFSaCI7IEZQUMZJPKQYhOdz+vklDvbujuDIicNDb9CQwFCtPwDHX/D5VFAwkgrjWZ2tbvjJ4xKZdvfRm5peWV1Lb9e2Njc2t4xd/caMk4FgTqJWSxaPpbAaAR1RWDViIAc59B0+9fjf3mPQhJ4+hWDRLwO5GNKQEKy1zMOh+0ADUJQFkLmJpCU35tDFJ6PhXaVjFu2yPYG1SJwZKaIZah3zyw1iknKIFGFYyrZjJ8rLsFCUMBgV3FRCgkfd6GtaYQ5SC+bfDGyjrUSWGEsdEXKmqi/JzLMpRxwX3dyrHpy3huL/3ntVIUXkajJFUQkemiMGWiq1xJFZABRDFBpgIqi+1SI9LDBROriCDsGZf3mRNCpl57RcuTkrVi9nceTRATpCJeSgc1RF16iG6oigR/SMXtGb8WS8GO/Gx7Q1Z8xm9tEfGJ8/yGeYvA=</latexit> Optical Rogue Waves in Integrable Turbulence Spontaneous modulation instability: statistics Ø Transient regime: oscillations Agafontsev, D. S., & Zakharov, V. E. Integrable turbulence and formation of rogue waves, Nonlinearity, 28 ,(8), 2791. (2015) Ø Long-term statistics : normal law ! P(| Ψ | 2 ) 10 10 10 10 10 10 − 10 − 8 − 6 − 4 − 2 0 0 5 10 | Ψ | 2 15 20 simulations Numerical 25 (a) | ] 〈 H d 〉 , 〈 H 4 〉 Ø Stationary spectrum ψ ( ω ) | 2 I k − 1.5 − 0.5 0.5 − 1 10 10 10 10 10 10 0 1 0 − 4 − 6 − 5 − 4 − 3 − 2 − 1 50 − 2 100 t k 0 simulations Numerical 150 2 simulations Numerical 200 4 (a) (a)
Integrable Turbulence Optical Rogue Waves in Integrable Turbulence ü Random initial conditions + integrable system (1D-NLSE) ∂ 2 ψ i ∂ψ ∂ z = β 2 ∂ t 2 − γ | ψ | 2 ψ 2 NLS, KdV, Sine-Gordon ü Universal equations ü Inverse Scattering Transform (IST) ü Solitons, breathers … “Nonlinear wave systems integrable by Inverse Scattering Method could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence -Turbulence in Integrable Systems” Turbulence in Integrable Systems , V.E. Zakharov, Studies in Applied Mathematics, 122 , 219 (2009) D.S. Agafontsev and V.E. Zakharov, Nonlinearity, (2015) No Resonances … but stationary state J. Soto-Crespo et al., Phys. Rev. Lett., 2016 P. Walczak et al., Phys. Rev. Lett., 114 , 143903, (2015) S. Randoux et al, Physica D : Nonlinear Phenomena, 333 , (2016) P. Suret et al . Nat. Commun. 7, 13136 (2016). A. Tikan, et al ., Nat. Photon., 12 , 228 (2018)
Experiments in optical fibers and in water tank Optical (power) spectrum Ø Phlam, University of Lille, France Ø Ecole Centrale of Nantes, France
Modulation Instability in optical fiber experiments Heterodyne time lens Measurements Recirculating (phase and amplitude) loop fiber techniques Tikan et al. Nat. Photon. 12 (2018) Dispersive Fourier 2019 Transform Kolner et al. Opt Lett. (1989) 2018 2016 Time-lens 2012 1989 0 1 3 4 5 6 7 8 2 500 EXP (a) 1986 Temporal imaging applied to 400 nonlinear fiber optics Solli et al. Nat. Photon. 6 (7) (2012) P. Suret et al., Nat. Commun. 7 , 13136 (2016) Distance (km) 300 MI observation Närhi et al. Nat. Comm. 7 (2016) 200 100 0 0 1000 2000 Tai et al. PRL 56 (2) (1986) Time (ps) Spatio-Temporal Single Shot Measurement First observation of dynamics of spectra MI in optical fiber A. E. Kraych et al. , Phys. Rev. Lett. 123 , 093902 (2019)
Ultrafast measurement in optical fiber experiments ? ü P=10mW / Time scale ~100ps : fast photodetectors and Oscilloscope ü P=1W / Time scale ~1ps ü Temporal imaging (SEAHORSE) ü + Phase single-shot spectrum analyser 800nm χ (2) 528nm 1550nm Time Sum frequency generation (SFG) w pump + w signal = w SFG space ! P. Suret et al., Nat. Commun. 7 , 13136 (2016) A. Tikan et al., Nat. Photon. 12 (2018)
Spontaneous modulation instability in optical fiber experiments Experimentals results : real-time observation of MI generated nonlinear structures Ø Phase and Amplitude ultrafast measurement Exp P. Suret et al., Nat. Commun. 7 , 13136 (2016) One A. Tikan et al., Nat. Photon. 12 (2018) frame Zoom Exp Num z Observations of quasi-periodic structures close to Akhmediev’s Breather solution t • 500m of SMF-28 (γ = 1,3/W/km, β 2 = -21.7 ps/km²) , P ≈7W , L ≈ 5 x L NL
Long term evolution of spontaneous modulation instability (optical fiber experiments) New experiments from A. Lebel et al. Note, see also : Närhi et al. Nat. Comm. 7 (2016) Exp Stationary (statistical) state Optical Spectrum z Num t Exp Exp Probability Density Function (PDF) of | ψ | 2
<latexit sha1_base64="Um6q8jsLNwOPk/W0xA2cLeSCSac=">AB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EohePFewHtmvJptk2NJsNSVYo2/4Lx4U8eq/8ea/MW3oK0PBh7vzTAzL5CcaeO6305uZXVtfSO/Wdja3tndK+4fNHScKELrJOaxagVYU84ErRtmOG1JRXEUcNoMhjdTv/lElWaxuDcjSf0I9wULGcHGSg+1q3FHajZ+rHSLJbfszoCWiZeREmSodYtfnV5MkogKQzjWu250vgpVoYRTieFTqKpxGSI+7RtqcAR1X46u3iCTqzSQ2GsbAmDZurviRHWo+iwHZG2Az0ojcV/PaiQkv/ZQJmRgqyHxRmHBkYjR9H/WYosTwkSWYKGZvRWSAFSbGhlSwIXiLy+TRqXsnZUrd+el6nUWRx6O4BhOwYMLqMIt1KAOBAQ8wyu8Odp5cd6dj3lrzslmDuEPnM8fOaGQnw=</latexit> Long term evolution of spontaneous modulation instability (optical fiber experiments) Power autocorrelation at stationnary state Ø Autocorrelation of power (second order coherence) P = | ψ | 2 Nonlinear stage of MI : oscillations of g (2) around unity <-> period of modulation instability (note : random waves : g (2) > 1)
Noise-driven Modulation instability (optical fiber experiments) A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, Phys. Rev. Lett. 123 , 093902 (2019) OSA CW EDFA AOM 1550 nm 90/10 Initial condition Recirculating fi ber loop OSC 200 ns WDM WDM Raman Pump 1450 nm 8 km 2 km 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Optical Spectra 10 0 (e) 500 10 -1 EXP SIMU 10 -2 (a) (b) 10 -3 10 -4 400 10 -5 40 20 0 20 40 Frequency (GHz) Optical Spectra 10 0 Distance (km) (d) 300 10 -1 10 -2 10 -3 10 -4 200 10 -5 40 20 0 20 40 Frequency (GHz) Optical Spectra 10 0 (c) 100 10 -1 10 -2 10 -3 10 -4 0 10 -5 0 1000 2000 0 1000 2000 40 20 0 20 40 Frequency (GHz) Time (ps) Time (ps)
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