Majid Taki Mardi 11 juin 2013 Les séminaires CEMPI Groupe NLSE.

Majid Taki Mardi 11 juin 2013 Les sminaires CEMPI Groupe NLSE

Notions lmentaires danalyse de stabilit linaire AvecParamtre de contrle Solution stationnaire et uniforme Relation de dispersion

Stabilit Marginale La solution stationnaire est stable sipout tout Elle est instable sil existe unavec La situation est marginale (stable) si Pourfix

Contexte et position du problme La structure de la fibre optique Les pertes linaires Absorption des impurets ( ). La diffusion de Rayleigh. La rsonance IR.

avec La dispersion chromatique Vitesse de phase: Analytiquement pour prendre en compte les effets dispersifs on fait un DL de Taylor autour de la frquence de central du paquet dondes : est relie la vitesse de groupe. la dispersion de vitesse de groupe (GVD). la pente de la dispersion de vitesse de groupe (TOD).

2 0 Basses voyagent moins vite 2 0, dispersion normale

avec Les effets non linaires Leffet Kerr Optique Leffet Kerr optique est la rponse instantane lectronique des molcules de silice aux champs incidents. Il conduit de nombreux phnomnes non-linaires comme lautomodulation de phase, la modulation de phase croiss et le mlange 4 ondes. Dans les matriaux centro-symtriques (la silice), est nul en raison de la symtrie dinversion au niveau molculaire. La contribution dominante de la polarisation non linaire vient donc de la susceptibilit d'ordre trois. Lindice de rfraction non linaire 2.6x10 -20 m/W dans la silice.

La diffusion Raman stimule Dans la silice, la bande des frquences amplifies s'tend jusqu' 40 THz avec un maximum de gain -13.2 THz. L'effet Raman dpend de la partie imaginaire de, elle est considre comme la rponse des noyaux de la molcule de silice aux champs incidents, son temps de rponse est de lordre de 60-70 fs dans les fibres de silice. Processus inlastique

L'quation non linaire de Schrdinger gnralise (GNLSE) Lquation de propagation des ondes sous formes vectorielle Lquation de propagation de lenveloppe lentement variable des impulsions dans la fibre optique i.La polarisation non linaire doit tre traite comme une perturbation de la polarisation linaire (les fibres optiques faiblement non linaires). ii.Le champ optique est suppos maintenir sa polarisation le long de laxe de propagation de la fibre. iii.Le champ lectrique est quasi-monochromatique ( ), ce qui est vrifi pour des ondes continues ou pour des impulsions de dure infrieure la picoseconde. La dispersion La rponse non linaire Lenveloppe lentement variable

Rponse non linaire de la fibre optique: La rponse non instantane Raman.

avec Instabilit modulationnelle (MI) dans la fibre optique La MI est interprte physiquement comme un quilibre entre les effets non linaires et la dispersion linaire au cours de la propagation dun champ optique. La solution stationnaire: La stabilit de cette solution stationnaire est tudie en la soumettant des fluctuation de la forme o Le problme linaris autour de la solution stationnaire est : perturbations

Instabilit linaire standard On cherche des solutions non nulles du problme linaris. La condition de solvabilit (ici simplement le dterminant non nul) nous donne la relation de dispersion suivante : Linstabilit est uniquement possible en rgime de dispersion anormale: Gain spectral en puissance: Le problme linaris prend une forme plus simple : Ici on fait un choix crucial : on prend u et v de la forme et reprsentent respectivement la pulsation et le nombre donde de la perturbation.

Le gain spectral en puissance dans la fibre optique Signal bruite lentre Train dimpulsions la sortie Frquences gain maximum MI

Spectre exprimental MI NLSE

Rogue waves or freak waves Gigantic wall of water of about 30 m height But very dangerous !! More information.. http://www.youtube.com/watch?v=0aKy9xSUCN4&feature=related A Book Or a BBC report Extremely rare

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

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)

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

NLS model: From Modulational instability to Akhmediev Breather

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

Optical Intensity Time Optical rogue wave Intensity bins (arb. units) Number of events Statistical Characterization Oceanic rogue wave From oceanic rogue waves to optical rogue waves Oceanic rogue waves Optical rogue waves Defined by: Maxima/minima Amplitude Rarity

From oceanic rogue waves to optical rogue waves Oceanic rogue waves Optical rogue waves High power laser + Optical Intensity Time Optical rogue wave Supercontinuum Oceanic rogue wave pulsed Defined by: Maxima/minima Amplitude Rarity

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(-H 2 /H S 2 ) A rogue wave of AI = 3 (H = 3 H S ) may occur once in 20 years !!!

Open Problems Extreme sensitivity to noise and/or initial conditions Asymmetry of Rogue waves (Lo 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

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 Appearance/disappearance of optical rogue waves Mechanisms involved in the formation This work

Numerical simulations Case of an absolute system ( 3 =0 and no Raman effect) 100 simulations with different initial conditions Output depends on initial conditions Statistic different from the L shape

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 ( 3, 5, 7 .) Self steepening GNLSE is a convective system Explain why Rogue waves are extremely sensitive to initial conditions

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

Experimental results : output spectrum 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, 17010 (2009)

Experiments Supercontinuum Continuous pumping Experimental evidence of rogue events

Probability Density Functions (PDFs): Experiments Supercontinuum Continuous pumping First approach : Statistics Signature of rogue events Log(PDF) Experimental evidence of rogue events

Probability Density Functions (PDFs): Experiments Supercontinuum Continuous pumping Signature of rogue events Log(PDF) Experimental evidence of rogue events 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

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

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

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

White dots track the most intense pulse Highest intensity tracking Numerics Mechanism of formation of rogue events Supercontinuum Optical rogue wave Evolution of the highest intensity

Spectrograms Rogue event (Rogue Soliton) collision between two giant solitons A. Mussot et al., Opt. Exp. 17, 17010 (2009) Very fast appearance and disappearance 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 Numerics Supercontinuum Continuous pumping Mechanism of formation of rogue events

Spectrograms Rogue event (Rogue Soliton) collision between two giant solitons A. Mussot et al., Opt. Exp. 17, 17010 (2009) Very fast appearance and disappearance Numerics Supercontinuum Continuous pumping Very importance of asymmetric drift dynamics ( 3 and Raman) Mechanism of formation of rogue events

Spectrograms Numerics Supercontinuum Continuous pumping Mechanism of formation of rogue events

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Majid Taki Mardi 11 juin 2013 Les séminaires CEMPI Groupe NLSE.

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