Optical Rogue Waves in Integrable Turbulence

Pierre Suret et Stéphane Randoux

Postdoc : F. Gustave PhD : R. El Koussaifi, A. Tikan, P. Walczak (2016) Collaborators : Fast Measurement : S. Bielawski, C. Szwaj, C. Evain (Phlam) Oceanography : M. Onorato (Turin, Italie) Integrable Systems : G. El (Loughbourough, UK) 1. Context

2. Experiments

3. Theoretical approach Optical Rogue Waves in Wave Turbulence Integrable Turbulence

ü Waves ubiquitous phenomena in Physics

ü Dispersion

ü Nonlinearity

ü Often, randomness plays a significant role

Wave turbulence = nonlinear dispersive random waves

hydrodynamics, oceanography, optics, mechanics, plasma physics… Optical Rogue Waves in Nonlinear Statistical OpticsIntegrable Turbulence

Statistical Optics (linear)

ü XXth century Random / partially coherent light (spatial or temporal fluctuation, coherence / spectrum…)

Nonlinear Statistical (fiber) Optics

ü 1960s Lasers / nonlinear optics

ü 1990s Supercontinuum (ps, cw) : nonlinear + incoherent

A.Kudlinski and A. Mussot, Opt. Lett., 33, (2008) J. M. Dudley et al., Opt. Express 17, (2009)

ü 2000s Challenge : statistical description of nonlinear random waves A. Picozzi et al., Optical wave turbulence : Toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics Physics Reports, (2014) Optical Rogue Waves in “Turbulence” in Optical fibersIntegrable Turbulence

Optical fibers : tabletop laboratories of hydrodynamical phenomena

ü Turbulence E.G. Turitsyna et al., « The laminar–turbulent transition in a fibre laser », Nat. Photonics, 7 (2013) ü Rogue waves

~26m

~12m

D. R. Solli et al., « Optical rogue waves », Nature 450, 1054-1057 (2007) Optical Rogue Waves in The 1D Nonlinear SchrödingerIntegrableEquation (1DNLSE)Turbulence

@ @2 i = 2 2 @z 2 @t2 | | ü Spatio-Temporal dynamics at leading order of numerous systems

BEC, nonlinear optics, water waves… Narrow band approximation

ü Deep water waves (focusing) / Single mode fiber

water wave elevation electric field

i(k z ! t) i(k0z !0t) ⌘ = ( e 0 0 ) E(x, y, z, t)= A(x, y) (z,t) e < < A η y Optical Rogue Waves in 1D-NLSE is integrable Integrable Turbulence

ü Inverse scattering transform

ü Exact solutions Ø Solitons Ø Solitons on finite background (SFB) Peregrine solitons, Akhmediev breathers… Ø SFB : proptotype of rogue waves ? N. Akhmediev et al., Physics Letters A, 373, 2009

ü Experiments with coherent initial conditions Ø Optical fibers B. Kibler et al., Nature Physics 6, 790, (2010) B. Kibler et al., Scientific Reports, 2, 790, (2012) B. Frisquet et al., Phys. Rev. X, 3, 041032, (2013)

Ø Water tank Chabchoub et al., Phys. Rev. Lett. 106, 204502, (2011) B. Kibler et al., Phys. Rev. X, 5, 041026 (2015) Optical Rogue Waves in Integrable Turbulence Integrable Turbulence

ü Random initial conditions + integrable system (1D-NLSE) V.E. Zakharov, Turbulence in Integrable Systems, 2009 D.S. Agafontsev and V.E. Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity, 2015 J. Soto-Crespo et al., Integrable Turbulence and Rogue Waves : Breathers or Solitons ?, Phys. Rev. Lett., 2016 ü Non resonant four waves mixing (FWM)

i!pt (t)= ^(!p) e Collisions in gas 1 3 p E E E E X 1 + 2 = 3 + 4     p + p = p + p ](!) 2 1 2 3 4 | | € 2 4 ω3 ω1 ω2 ω4 € ω

k(ω1) + k(ω2 ) = k(ω3 ) + k(ω4 ) € € € € ω1 + ω 2 = ω 3 +€ω 4 €

€ Optical Rogue Waves in Integrable Turbulence Integrable Turbulence

i!pt (t)= ^(!p) e Collisions in gas 1 3 p E E E E X 1 + 2 = 3 + 4     p + p = p + p ](!) 2 1 2 3 4 | | € 2 4 ω ω ω ω 3 1 2 4 € ω @ @2 i = 2 2 @z 2 @t2 | | k(ω1) + k(ω2 ) = k(ω3 ) + k(ω4 ) € € € € k(!)= 2 !2 ω1 + ω 2 = ω 3 +€ω 4 2 €

€ Optical Rogue Waves in Principle of experimentsIntegrable Turbulence

Initial partially coherent waves

i!pt ü Linear superposition of independent waves (t)= ^(!p) e p X ü Probability Density Functions (PDF) Ø Field : Gaussian statistics (central limit theorem) PDF ( ) =exp ( )2 < < Ø Power : exponential ⇥ ⇤ ⇥ ⇤ PDF(P/ < P >)=exp( P/ < P >) 2 Nonlinear propagation described by @ 2 @ 2 The focusing 1D-NLSE i = @z 2 @t2 | | 2. Experiments Fast measurement

Statistics (Optical Sampling) Dynamics (Time lens – Time Microscope) Optical Rogue Waves in Experimental challengesIntegrableOpticalTurbulence (power) spectrum

ü Time scale <1ps

t 1ps ⇡

ü Standard photodetectors / Oscilloscope Dt<20 ps

ü Irregular / random evolution (non reproductible) Optical Rogue Waves in Fast measurement of statistics (random light)Integrable Turbulence

ü Asynchronous Optical Sampling Sum frequency generation signal + fs pulses Temporal resolution : 250 fs

P. Walczak et al., Phys. Rev. Lett., 2015 a b Signal

t ~ 3ps

PC ~ 1 W

λ = 1064nm t BBO

Pump

P(t) (Arb. Units) τR = 12.5ns λ = 457nm t

5 t ~ 140 fs PC ~ 10 W Sampled signal

λ = 800nm t 2.8 2.9 3 3.1 3.2 Time ( s) Optical Rogue Waves in Fast measurement of statisticsIntegrable Turbulence

Strong Deviation from the Gaussian statistics

1 z = 0m 10-1 z = 15m exp(-P/

) 10-2

10-3

10-4 PDF[P/

] 10-5

10-6

0 10 20 30 40 50 60 P/

Occurrence of P=50

? 1 / 1010 seconds @ z=0m 1 / 10-6 seconds @ z=15m

P. Walczak et al., Phys. Rev. Lett., 2015 Optical Rogue Waves in Numerical simulations Integrable Turbulence

Emergence of strongly non Gaussian statistics

1 z = 0m 10-1 z = 15m i!pt (t)= ^(!p) e -2 Exp. : z = 15m p 10 X 2 10-3 @ 2 @ 2 i = 2 10-4 @z 2 @t | | PDF[P/

] 10-5

10-6

0 10 20 30 40 50 60 P. Walczak et al., Phys. Rev. Lett., 2015 P/

Optical Rogue Waves in Influence of the initial conditionIntegrable Turbulence

0 10 ü Noise driven modulational instability (a)

−2 D.S. Agafontsev and V.E. Zakharov, Nonlinearity, 2015 10

−4 10 ) 2 |

Ψ −6 2 10 |y(t)| Stationnary state : P(| −8 Gaussian statistics 10 −10 10

0 5 10 15 20 25 t |Ψ|2

ü Strongly fluctuating initial condition P. Walczak et al., Phys. Rev. Lett., 2015 1 z = 0m 10-1 z = 15m

-2 |y(t)|2 Stationnary state : 10 strongly non Gaussian 10-3

10-4 PDF[P/

] statistics 10-5 t 10-6

0 10 20 30 40 50 60 P/

Open theoretical question : transition between the two cases J. Soto-Crespo et al., Integrable Turbulence and Rogue Waves : Breathers or Solitons ?, Phys. Rev. Lett., 2016 Optical Rogue Waves in Influence of the initial conditionIntegrable Turbulence

ü Noise driven modulational instability Dudley et al. Nat. Photon. 8, 75 (2014)

500

12 490 10

480 8

Time (ps) Time 470 P(t)/ 4 460 2

450 0 0 1 2 3 4 5 z (km)

ü Strongly fluctuating initial condition

300 70 290 60

280 50

40 270

Time (ps) Time 30 260 P(t)/ 20 250 10

240 0 0 1 2 3 4 5 z (km) Ø movie Optical Rogue Waves in Observation of the fast dynamicsIntegrable Turbulence

ü Single shot recording (irregular, non reproductible fluctuations)

ü “Large” temporal window 25-30 ps

ü Ultrafast measurement resolution 250fs

ü Strategy : Time Microscope (1 Time lens + Spectral observation)

Collaboration : C. Evain, C. Szwaj, S. Bielawski (Phlam)

Kolner et al., Opt. Lett. 14, 630 (1989) Bennett and Kolner, Opt. Lett. 24, 783 (1999) Foster et al., Nature 456, 81 (2008)

First single shot observation of optical rogue waves

P. Suret et al., Nature Communications (2016) Optical Rogue Waves in Time Microscope Integrable Turbulence

ü Equivalence Time / Space Ø Dispersion / Diffraction Ø Lens : Quadratic phase Ø Optical spectrum / far field

ü Characteristics : single shot / 250 fs / 30 ps / 500Hz Optical Rogue Waves in Time Microscope Integrable Turbulence Optical Rogue Waves in Time Microscope Integrable Turbulence Optical Rogue Waves in Time Microscope Integrable Turbulence Optical Rogue Waves in Influence of the initial conditionIntegrable Turbulence

|y(t)|2

t 3. Theoretical approaches

Wave Turbulence Theory Exact Relation Inverse Scattering Transform Semi-Classical approach Optical Rogue Waves in Toward a theory of IntegrableIntegrableturbulence ?Turbulence

ü Wave turbulence theory (kinetic theory) Ø Weakly nonlinear regime A. Picozzi et al., Physics Reports, (2014)

ü Exact relation between spectrum and statistics M. Onorato, et al. Phys. Lett. A, 380 , 39, (2016)

ü Semiclassical limit of 1D-NLSE Ø Strongly nonlinear regime @ @2 i/✏ i✏ + ✏2 + 2 =0 = p⇢e @z @t2 | | Ø Collaboration with Gennady El Ø Hydrodynamical formulation of nonlinear optics Ø Random Riemann waves S. Randoux et al., arXiv:1702.00006, 2017 Optical Rogue Waves in Semiclassical limit of 1D-IntegrableNLSE Turbulence

ü Universal mechanism : Peregrine soliton M. Bertola and A. Tovbis , “Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point : Rational Breathers and Poles of the Tritronquée Solution to Painlevé I”, Comm. on Pure and Applied Mathematics, (2013)

ü Ingredient of integrable turbulence

ü Collaboration with J. Dudley, G. Genty et al.

Ø Pulse propagation / Peregrine soliton

Ø Measurement of Intensity and phase A. Tikan et al., arXiv:1701.08527, 2017 Optical Rogue Waves in Inverse scattering transformIntegrable Turbulence

ü Nonlinear analysis of local structures emerging in integrable turbulence S. Randoux, P. Suret, G. El, Scientific Report 6, 29238 (2016) “Inverse scattering transform analysis of rogue waves using local periodization procedure,”

-360 CS1 20 -37060 |ψ(x,t)| -380 15 -39040 x -400 CS3 10 2 -41020 CS4 CS2 5 -420 -4300 0 0 5 10 15 t 20 25 30 9 2 2 Best fit by PS: 6 1 blue lines 3 0

|ψ(x,t)| -1 0 Arg[ψ(x,t)] -2 0 2 -2 0 2 x x 9 1

2 CS1 6 Im 0 (ξ) 3 |ψ(x,t)| -1 0 -10 0 10 -30 0 30 -0.5 0 0.5 x x Re(ξ) Periodic IST Optical Rogue Waves in Conclusion Integrable Turbulence

ü Front edge fast measurement techniques in optics

Ø Time lens / time microscope

ü Open new fundamental and theoretical questions

Ø rogue waves / integrable turbulence Ø Influence of initial statistics Ø Role of non resonant interaction in turbulence

ü Perspectives

Ø Application of fast measurement (pulsed lasers, ultrafast imaging…)

Ø New mathematical problems • Integrable / non integrable (interdisciplinary CNRS project - CEMPI) • Inverse Scattering Transform / perdiodic boundary conditions Optical Rogue Waves in Conclusion Integrable Turbulence

Articles

[1] S. Randoux , P. Walczak, M. Onorato and P. Suret, Phys. Rev. Lett., 113, 113902, (2014) [2] A. Picozzi et al., Physics Reports, (2014) [3] P. Walczak, S. Randoux and P. Suret, Opt. Lett., 40, 3101-3104, (2015) [4] P. Walczak, S. Randoux and P. Suret, Phys. Rev. Lett., 114, 143903, (2015) [5] M. Onorato and P. Suret, Natural Hazards, 84 , 541-548, (2016) [6] S. Randoux, P. Walczak, M. Onorato and P. Suret, Physica D : Nonlinear Phenomena, 333, (2016) [7] M. Onorato, D. Proment, G. El, S Randoux and P. Suret, Phys. Lett. A, 380 , 39, 3173 - 3177, (2016) [8] S Randoux, P. S u r e t a n d G . El, Scientic Reports, 6 , 29238, (2016) [9] Faure, E. et al. Nat. Commun. 7, 8674 (2016). [10] Suret P. et al. Nat. Commun. 7, 13136 (2016).

Chapters in book

Rogue and Shock Waves in Nonlinear Dispersive Media Lecture Notes in Physics, Springer International Publishing, 926 pp1-22, 2016 [1] Hydrodynamic and Optical Waves : A Common Approach for Unidimensional Propagation M. Onorato, F. Baronio, M. Conforti, A. Chabchoub, P. Suret and S. Randoux [2] Integrable Turbulence with Nonlinear Random Optical Waves S. Randoux and P. Suret