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Les séminaires CEMPI Groupe NLSE

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1 Les séminaires CEMPI Groupe NLSE
Instabilité modulationnelle dans l’Equation de Schrödinger Non Linénaire (NLSE) Majid Taki Mardi 11 juin 2013

2 Notions élémentaires d’analyse de stabilité linéaire
Avec Paramètre de contrôle Solution stationnaire et uniforme Relation de dispersion

3 Stabilité Marginale Pour fixé La solution stationnaire est stable si
pout tout Elle est instable s’il existe un avec La situation est marginale (stable) si

4 Contexte et position du problème
La structure de la fibre optique Les pertes linéaires Absorption des impuretés ( ). La diffusion de Rayleigh. La résonance IR.

5 La dispersion chromatique
Vitesse de phase: Analytiquement pour prendre en compte les effets dispersifs on fait un DL de Taylor autour de la fréquence de central du paquet d’ondes : avec est reliée à la vitesse de groupe la dispersion de vitesse de groupe (GVD). la pente de la dispersion de vitesse de groupe (TOD).

6 b2>0 b2<0 Entrée Sortie Sortie
La dispersion de vitesse de groupe (GVD) Le coefficient de dispersion: b2>0, dispersion normale b2<0, dispersion anormale b2>0 b2<0 Basses l voyagent moins vite Hautes l voyagent moins vite Entrée Sortie Sortie

7 Les effets non linéaires
Dans les matériaux centro-symétriques (la silice), est nul en raison de la symétrie d’inversion au niveau moléculaire. La contribution dominante de la polarisation non linéaire vient donc de la susceptibilité d'ordre trois L’effet Kerr Optique L’indice de réfraction non linéaire 2.6x10-20 m²/W dans la silice. avec L’effet Kerr optique est la réponse instantanée électronique des molécules de silice aux champs incidents. Il conduit à de nombreux phénomènes non-linéaires comme l’automodulation de phase, la modulation de phase croisés et le mélange à 4 ondes.

8 La diffusion Raman stimulée
Processus inélastique L'effet Raman dépend de la partie imaginaire de , elle est considérée comme la réponse des noyaux de la molécule de silice aux champs incidents, son temps de réponse est de l’ordre de fs dans les fibres de silice. Dans la silice, la bande des fréquences amplifiées s'étend jusqu'à 40 THz avec un maximum de gain à THz.

9 L'équation non linéaire de Schrödinger généralisée (GNLSE)
L’équation de propagation des ondes sous formes vectorielle L’enveloppe lentement variable L’équation de propagation de l’enveloppe lentement variable des impulsions dans la fibre optique La polarisation non linéaire doit être traitée comme une perturbation de la polarisation linéaire (les fibres optiques faiblement non linéaires). Le champ optique est supposé maintenir sa polarisation le long de l’axe de propagation de la fibre. Le champ électrique est quasi-monochromatique ( ), ce qui est vérifié pour des ondes continues ou pour des impulsions de durée inférieure à la picoseconde. La dispersion La réponse non linéaire

10 Réponse non linéaire de la fibre optique:
La réponse non instantanée Raman.

11 Instabilité modulationnelle (MI) dans la fibre optique
La MI est interprétée physiquement comme un équilibre entre les effets non linéaires et la dispersion linéaire au cours de la propagation d’un champ optique. La solution stationnaire: La stabilité de cette solution stationnaire est étudiée en la soumettant à des fluctuation de la forme perturbations avec Le problème  linéarisé autour de la solution stationnaire est :

12 Instabilité linéaire standard
Ici on fait un choix crucial : on prend u et v de la forme et représentent respectivement la pulsation et le nombre d’onde de la perturbation. Le problème linéarisé prend une forme plus simple : On cherche des solutions non nulles du problème linéarisé. La condition de solvabilité (ici simplement le déterminant non nul) nous donne la relation de dispersion suivante : L’instabilité est uniquement possible en régime de dispersion anormale: Gain spectral en puissance:

13 MI Le gain spectral en puissance dans la fibre optique
Fréquences à gain maximum Signal bruitée à l’entrée Train d’impulsions à la sortie MI

14

15

16 Spectre expérimental MI NLSE

17 Rogue waves or freak waves
Gigantic wall of water of about 30 m height Extremely rare But very dangerous !! More information….. Or a BBC report… A Book

18 A quantitative measure for Rogue waves:
AI=HRW / HS > 2 AI: Abnormality Index AI = 3 for The New Year Wave (registered on January 1, 95) 26 m high with a period of 12s !!!

19

20 Deterministic approach
for Rogue Waves Can rogue waves be predicted within linear theories? No They appear from nowhere and disappear without a trace They have a very high amplitude Only a nonlinear approach can predict the occurrence of these giant waves Need of instability (Modulationnal Instability) Nonlinear compensation of linear effects (mostly dispersion)

21 Are oceanic rogue waves Akhmediev Breathers ?
NLS model: From Modulational instability to Akhmediev Breather Akhmediev Breather: Rational solution of NLSE AI= 2.4 !!! N. Akhmediev, A. Ankiewicz, M. Taki, waves that appear from nowhere and disppear without a trace, Phys. Lett. A 373, 675 (2009)

22 NLS model: From Modulational instability to Akhmediev Breather

23 Rogue wave management NLSE
Usama Al Khwaja, and Majid Taki, Rogue waves management by external potentials soumis à Phys. Lett. A

24 Statistical Characterization
From oceanic rogue waves to optical rogue waves Oceanic rogue waves Optical rogue waves Defined by: Maxima/minima Amplitude Rarity Number of events Intensity bins (arb. units) They were evidenced using statistical approach, L shape Statistical Characterization Optical Intensity Time Oceanic rogue wave Optical rogue wave

25 From oceanic rogue waves to optical rogue waves
Defined by: Maxima/minima Amplitude Rarity pulsed + High power laser Supercontinuum The name of rogue waves in optics was introduced by Solly as a counterpart of oceanic rogue waves, because they share the same properties. They evidence experimentally the occurrence of ORW in the fiber output intensity in a regime of supercontinuum. New domain since first experimental evidence three years ago. ORW is an example of extreme event, it means that for instance its amplitude is very much greater that the significant one They share common properties and a same governing equation in the weakly nonlinear regime. Optical Intensity Time Oceanic rogue wave Optical rogue wave

26 Statistical approach for Rogue waves
Sensitive dependence on initial conditions Incomplete information about the initial state random wave dynamics Gaussian statistics fails: P(H)~ exp(-H2 /HS2) A rogue wave of AI = 3 (H = 3 HS ) may occur once in 20 years !!!

27 Open Problems Extreme sensitivity to noise and/or initial conditions
Asymmetry of Rogue waves (Léo et al. PRL 2013) Non-Gaussian statistics. Needs to go beyond NLS An original approach that combines deterministic and statistical methods Optical rogue waves can help understanding the mechanism of rogue waves formation

28 Comparison with the ocean difficult?
Optical Rogue waves Focus on optical rogue waves… care must be taken to establish a formal comparison Results published obtained with pulsed pumps Comparison with the ocean difficult? Optical rogue waves generated with a continuous wave pump Calm ocean??? Optical rogue waves originates from convective instabilities This work Appearance/disappearance of optical rogue waves Mechanisms involved in the formation

29 Numerical simulations
Case of an absolute system (b3=0 and no Raman effect) 100 simulations with different initial conditions Output depends on initial conditions Statistic different from the L shape

30 Minimal Model Odd derivatives induce a drift
Slope of the dispersion curve Raman effect Raman effect and the slope of the dispersion induce convective instabilities Generalization to all odd terms presents in the GNLSE All even dispersion orders (b3, b5, b7….) Self steepening GNLSE is a convective system Explain why Rogue waves are extremely sensitive to initial conditions

31 Numerical simulations : longitudinal evolution
Supercontinuum formation from simulations standard event of previous simulations Spectral domain Dispersive waves Solitons

32 Experimental results : output spectrum
Excellent agreement numerics/experiments A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, M. Taki, Observation of extreme temporal events in CW-pumped supercontinuum, Optics Express 17, (2009)

33 Experimental evidence of rogue events
Experiments Experimental evidence of rogue events Supercontinuum Continuous pumping So how to evidence the occurrence of EV in this temporal evolution? The pump laser used in our experiments is a CW ytterbium-doped fiber laser delivering up to 20 W average power at the wavelength λp of 1064 nm with a line width of 0.5 nm. The output of the fiber laser was collimated and launched into a 400 m long PCF with a zerodispersion wavelength of 1062 nm and a nonlinear coefficient γ of 10 /W/km. The output spectrum measured for a launched power of 10 W is displayed in Fig. 1(a). It spans from about 900 to 1300 nm, with an average output power of 2.2 W. It is now commonly accepted in the literature that for λ > λp, the spectrum is mainly composed of solitons (in fibers with a single zero-dispersion wavelength), and for λ < λp, it is mainly made of dispersive and trapped dispersive waves [10,12,13].

34 Signature of rogue events
Experiments Experimental evidence of rogue events Supercontinuum Continuous pumping First approach : Statistics Probability Density Functions (PDFs): First characterization is statistical. PDF. Low probas so signature of RE. Log(PDF) Signature of rogue events

35 Signature of rogue events
Experiments Experimental evidence of rogue events Supercontinuum Continuous pumping The most powerful peak amplitudes are very much larger than 2 times the significant peak height (Hs) which is one of the feature of oceanic rogue waves. Their probability is extremely low Probability Density Functions (PDFs): First characterization is statistical. PDF. Low probas so signature of EV. Log(PDF) Signature of rogue events

36 Nonlinear Schrödinger
Numerics Experiments Minimal model Supercontinuum Continuous pumping Minimal model: Nonlinear Schrödinger Slope of the dispersion curve Raman effect Probability Density Functions (PDFs): We found the minimal model that reproduces our experimental events/PDF. It is a NLS + beta3 + Raman. Excellent agreement! The important result : Only up to beta3, no need up to betan with n = 8,9… Log(PDF) Log(PDF) Excellent agreement + L-shape of PDFs

37 Nonlinear Schrödinger
Numerics Experiments Model Supercontinuum Continuous pumping Minimal model: Nonlinear Schrödinger Slope of the dispersion curve Raman effect Probability Density Functions (PDFs): L shape. PDF Log(PDF) Excellent agreement + L-shape of PDFs

38 Numerical simulations : comparison
Convective system vs absolute system Same scale!! Drift (convection) important ingredient for generating rare and strong optical waves

39 White dots track the most intense pulse
Numerics Mechanism of formation of rogue events Supercontinuum Evolution of the highest intensity Optical rogue wave White dots track the most intense pulse To identify the mechanism of formation of EV we track the most intense pulse versus fiber length, this allow to identify EV like at 300 m fibre length. Highest intensity tracking

40 Mechanism of formation of rogue events
Numerics Mechanism of formation of rogue events Supercontinuum Continuous pumping Spectrograms What we see is that the rogue wave occurs when the 2 solitons are colliding. Very fast appearance and disappearance which was not observed with pulsed pumpings. Rogue event (Rogue Soliton)  collision between two giant solitons Very fast appearance and disappearance A. Mussot et al., Opt. Exp. 17, (2009) P. Peterson et al., Nonlinear Process Geophys. 10, 503 (2003). Soliton interaction as a possible model for extreme waves in shallow water N. Akhmediev et al., Phys. Lett. A 373, 2137 (2009). Collision of two Akmediev breathers M. Erkintalo et al., Opt. Lett. 35, 658 (2010). Giant dispersive waves generation through soliton collision

41 Mechanism of formation of rogue events
Numerics Mechanism of formation of rogue events Supercontinuum Continuous pumping Spectrograms We see here the very importance of the convective nature of the system without which solitons would not drift with different velocities and so would not collide. Rogue event (Rogue Soliton)  collision between two giant solitons Very fast appearance and disappearance A. Mussot et al., Opt. Exp. 17, (2009) Very importance of asymmetric drift dynamics (b3 and Raman)

42 Mechanism of formation of rogue events
Numerics Mechanism of formation of rogue events Supercontinuum Continuous pumping Spectrograms To understand what happens at this length we represent the dynamics with the help of a spectrogram. 2 solitons = 2 moving red big dots

43 Merci de votre attention


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