integrable turbulence: a review of recent results in optics · wave turbulence in optics...

Integrable turbulence: A review of recent results in optics

Stéphane Randoux and Pierre SuretLaboratoire de Physique des Lasers, Atomes et Molécules

Université de Lille 1, France

in collaboration with Gennady El, Loughborough University (UK)

Miguel Onorato, Universita degli Studi di Torino (Italy)Antonio Picozzi, Université de Bourgogne (France)

Dmitry Agafontsev, Moscow (Russia)Serge Bielawski, Christophe Szwaj, Clément Evain, Lille (France)

Pierre Walczak (PhD), François Gustave (Post Doc), Rebecca El Koussaifi (PhD), Alexey Tikan (PhD)

Wave turbulence in Optics IntroductionWave turbulence in Optics

Lasers 1960s

Nonlinear optics

2nd harmonic generation,

Stimulated Raman/Brillouin scattering …

Statistical Optics/Optical Coherence Theory

Linear Theory (degree of coherence of light)

Optical fibers 1970s

Nonlinear fiber opticsTelecommunications applications(linear operation)Propagation of intense cw/pulsed

coherent light waves in fibers

Optical fibers 1995

Management of fiber dispersion

Supercontinuum generation High power/highly multimode lasers

Nonlinear statistical optics

Nonlinear Statistical Optics : an historical perspective

Photonic crystal fibers

Nonlinear fiber optics

Random fiber lasers…..

PCF

Diffraction grating

IR laser 200 fsSource : University of Bath

Supercontinuum generation in Photonic Crystal fibers

Pump

Supercontinuum

Wavelength (nm)Ranka et al, Opt. Lett 25, 25 (2000)


Solli et al, Optical rogue waves, Nature (2007)


S. Nazarenko, Wave Turbulence, Lecture notes in Physics, vol 825, Springer (2011)

In optics, WT theory has been applied in several circumstances to describe :

- Spectral broadening in supercontinuum generation, see e.g. Barviau et al, Phys. Rev. A79, 063840 (2009)- Spectral properties of Raman fiber lasers, see e. g. Babin et al, J. Opt. Soc. Am. B 24, 1729 (2007) - Spectral properties of random fiber lasers, see e. g. Churkin et al, Nature Comm. 2, 6214 (2015)- Optical wave condensation, see e. g. Connaughton et al, Phys. Rev. Lett. 95, 263901 (2005)

« Wave Turbulence (WT) can be generally defined as out-of-equilibrium mechanics of random nonlinear waves »

For a review, see also, A. Picozzi et al, Phys. Rep. 542, 1-132 (2014)


B. Barviau, B. Kibler and A. PicozziPhys. Rev. A79, 063840 (2009)

Supercontinuum generation is described by generalized 1D-NLS equations

B. Barviau, B. Kibler and A. PicozziPhys. Rev. A79, 063840 (2009)


WT treatment of supercontinuum generation in optical fibers

Non-integrable PDE


At leading order, the nonlinear propagation of waves in single-mode fibers is governed by the integrable 1D nonlinear Schrödinger equation:


The integrable 1D nonlinear Schrödinger equation with random initial conditions and WT theory

Linear dispersion relation

“Turbulence in Integrable Systems,” V. E. Zakharov, Studies in Applied Mathematics, 122, 219 (2009).

Wave turbulence in Optics IntroductionIntegrable Turbulence

« Nonlinear wave systems integrable by Inverse Scattering Method (ISM) 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).

Integrable Turbulence in optics

: Complex random variable

Wave turbulence in Optics Some numerical simulationsIntegrable Turbulence

f0n

randomly distributed between -p and p

Gaussian statistics

Initial condition :

Random Phase (RP) model

In optics P0 represents the wave power


Focusing regime Defocusing regime

Heavy-tailed deviations from the initial gaussian statistics

Low-tailed deviations from the initial gaussian statistics

t t


t

Complex random variable

Another initial condition: plane wave +noise

The statistics is Gaussian at long evolution time.

D. S. Agafontsev and V. E Zakharov, Nonlinearity 28, 2791 (2015)

|h (x)|<<1

Wave turbulence in OpticsIntegrable Turbulence

Outline of the talk

- Weakly nonlinear regime: Wave Turbulence (WT) approach- Strongly nonlinear regime: Dispersive hydrodynamics (semiclassical) approach

- Defocusing regime (Observation of Riemann waves in integrable turbulence- Intermittency phenomenon) - Focusing regime (Fast measurement techniques/ Observation of Peregrine-like coherent structures in integrable turbulence) - Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation

- Conclusions / Future works

- Nonlinear spectral analysis of numerical and experimentalsignals

Wave turbulence in Optics WT theoryIntegrable Turbulence

Weakly nonlinear regime (HNL<<HL), nearly gaussian statistics

WT treatment of the propagation of weakly nonlinear random waves in 1D-NLSE systems

Wave turbulence in Optics WT theoryIntegrable Turbulence

WT Treatment of the propagation of weakly nonlinear random waves in 1D-NLSE systems

P.A.E.M. Janssen, Journal of Physical Oceanography 33, 863 (2003).P. Suret, A. Picozzi, S. Randoux, Opt. Express 19, 17852 (2011).

Very weakly nonlinear regime

WT treatment relevant only to a (very) limited number of experiments in which the spectral broadening is very small

“Quasi-kinetic equations”


Outline of the talk





Wave turbulence in Optics Semiclassical ApproachIntegrable Turbulence

Defocusing propagation regime/Normal dispersion

Madelung Transformation

Shallow water equations

r: fluid height/optical poweru: depth-averaged horizontal velocity/instantaneous frequency

e<< 1 Strongly nonlinear regime

Small dispersion NLS

In the pre-breaking regime


Shallow water equations

Riemann invariants

Propagation equations of two interacting Riemann waves

NLS equation

Defocusing regime

Pre-breaking dynamics

Wave turbulence in Optics Random Riemann wavesIntegrable Turbulence

Integrable turbulence

Numerical simulations of the 1D-NLSE

Hydrodynamical variables (r,U) Riemann invariants (r1,r

2)

The initial statistics (at x=0) is gaussian for y

S. Randoux, F. Gustave, P. Suret and G. El, Phys. Rev. Lett. 118, 233901 (2017)

Defocusing regime/pre-breaking dynamics



The PDF of the random Riemann wave field is invariant with respect to the x evolution



Observation of optical Random Riemann waves in integrable turbulence

Bandwidth of light power fluctuations=4 GHzTime scale of power fluctuations = 250 psPhotodiode detection bandwidth =50 GHz

Oscilloscope bandwidth = 65 GHzOscilloscope sampling rate = 160 Gsa/s (1point every 6.25ps)

LNL

=1.3km

LD=6250km

e=0.014


Defocusing regime


Observation of optical Random Riemann waves in integrable turbulence


Defocusing regimeExperimental results



0

1

2

3

4

5

6

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

6

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

6

-6 -4 -2 0 2 4 6

Pre-breaking regime (random Riemann waves)

Post-breaking regime

Generation of dispersive shock waves (DSWs)

Influence of the interaction among DSWs on the statistics ???


Integrable turbulence in the Defocusing regime


Wave turbulence in Optics Intermittency phenomenonIntegrable Turbulence

Nazarenko et al, J. Fluid. Mech. 642, 395 (2009)

Wave turbulence in Optics Intermittency phenomenonIntegrable Turbulence

Intermittency in integrable turbulence

S Randoux, P Walczak, M Onorato, P Suret

Physical review letters 113 (11), 113902 (2014)

Similar statistical features occur in NLS systems


Outline of the talk





Wave turbulence in Optics Fast measurement techniquesIntegrable Turbulence

Focusing regime

Theoretical challenge : statistical description of nonlinear random wavesA. Picozzi et al., Physics Reports, (2014)

Experimental challenge : sub-picosecond measurement of randomly-fluctuating light (irregular and non reproducible optical signal)

- Asynchronous optical sampling (Characterization of the statistics of the field)

- Time Microscope (Characterization of the power fluctuations of the field)

- Heterodyne Time Microscope (Simultaneous characterization of the phaseand the power fluctuations of the field)

Experimental strategies/tools

Asynchronous Optical SamplingSum frequency generation signal + fs pulses

Temporal resolution : 250 fs

t

t

t

PC ~ 1 W

t ~ 3ps

PC ~ 105 W

τR = 12.5ns

t ~ 140 fs

Signal

Pump

λ = 457nm

λ = 800nm

λ = 1064nm

Sampled signal

BBO

P(t )

(Arb

. Uni

ts)

T i m e ( s )2 . 8 2 . 9 3 3 . 1 3 . 2

a b


P / < P >

1 0 - 1

1 0 - 2

1 0 - 3

1 0 - 4

1 0 - 5

1

1 0 - 6

PD

F[P

/<P

>]

0 1 0 2 0 3 0 4 0 5 0 6 0

z = 0mz = 15m

Exp. : z = 15m

P. Walczak et al, “Optical rogueWaves in integrable turbulence”,Phys. Rev. Lett. 114, 143903 (2015)


Equivalence Time / Space

Characteristics : single shot / 250 fs / 30 ps / 500Hz

P. Suret et al., Nature Communications (2016)

Time microscope


Time microscope

P. Suret et al., Nature Communications (2016)


Is it the Peregrine soliton?

Simultaneous measurement of power and phase: Heterodyne Time Microscope


Heterodyne Time microscope


Heterodyne Time microscope

Experimental observation of a p-phase jump confirming the emergence of Peregrine-like solitons in integrable turbulence

See Tikan et al, arXiv preprint arXiv:1707.07567


Outline of the talk





For analytic initial data (gaussian or sech), dispersive regularization of a gradient catastrophe

Wave turbulence in Optics Peregrine solitonIntegrable Turbulence


e =0.02Numerical simulation of with

Numerical computation of the spectral portraits using the procedure described in“Inverse scattering transform analysis of rogue waves using local periodization procedure” S. Randoux, P. Suret, G. El, Scientific Report 6, 29238 (2016)



Optical fiber Experiments in :Universality of the peregrine soliton in the focusing dynamics of the cubic nonlinear Schrödinger equation

A Tikan et al, Physical review letters 119, 033901 (2017)


Outline of the talk





Wave turbulence in Optics Nonlinear spectral analysisIntegrable Turbulence


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


Nonlinear spectral analysis of the PS recorded in the optical fiber experiment by Kibler et al, Nature Physics 6, 790-795 (2010)

Nonlinear spectral analysis of the PS recorded in the hydrodynamical experiment by Chabchoub et al, PRL 106, 204502 (2011)

S. Randoux, P. Suret, A. Chabchoub, B. Kibler, G. ElManuscript in preparation


Outline of the talk





Wave turbulence in Optics ConclusionIntegrable Turbulence

Dispersive hydrodynamics is the domain of applied mathematics and physicsconcerned with fluid motion in which the internal friction, e. g., viscosity, is negligible relative to wave dispersion.

Dispersive hydrodynamics represents a very relevant framework for the analysis of integrable turbulence


- In the defocusing regime, the statistics of the random wave field before wave breaking can be described in terms of Riemann Waves. The PDF of the random Riemann wave field is invariant with respect to the x evolution (contrary to the hydrodynamical variables that exhibit significant statisticalchanges)

-In the focusing regime, significant recent experimental progresshave shown that the Peregrine soliton represents a coherent structure of fundamental importance in integrable turbulence

-Nonlinear spectral analysis represents a tool of significant interest for the characterization of coherent structures emerging in integrable turbulence


Open questions / Future works

- Connection between Fourier (linear) spectrum and IST (nonlinear) spectrum

- Statistical theory of integrable turbulence in the post-breaking regime

- Role of pertubative effects (dissipation…)

- Soliton gas in Optics and in Hydrodynamics

Thank you for your attention !



One of the key results of [Bertola and Tovbis, Comm. Pure Appl. Math. 66, 678 (2013)] is that when e<<1 (N>>1) the dynamics near the gradient catastrophe universally leads to the generation of a rational PS as a local asymptotic solution of the NLSE.

e =1/N

integrable turbulence: a review of recent results in optics · wave turbulence in optics...

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