Majid Taki Mardi 11 juin 2013 Les sminaires CEMPI Groupe
NLSE
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Notions lmentaires danalyse de stabilit linaire AvecParamtre de
contrle Solution stationnaire et uniforme Relation de
dispersion
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Stabilit Marginale La solution stationnaire est stable sipout
tout Elle est instable sil existe unavec La situation est marginale
(stable) si Pourfix
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Contexte et position du problme La structure de la fibre
optique Les pertes linaires Absorption des impurets ( ). La
diffusion de Rayleigh. La rsonance IR.
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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).
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2 0 Basses voyagent moins vite 2 0, dispersion normale
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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.
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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
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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
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Rponse non linaire de la fibre optique: La rponse non
instantane Raman.
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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
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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.
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Le gain spectral en puissance dans la fibre optique Signal
bruite lentre Train dimpulsions la sortie Frquences gain maximum
MI
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Spectre exprimental MI NLSE
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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
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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 !!!
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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)
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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 !!!
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NLS model: From Modulational instability to Akhmediev
Breather
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Rogue wave management NLSE Usama Al Khwaja, and Majid Taki,
Rogue waves management by external potentials soumis Phys. Lett.
A
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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
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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
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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
!!!
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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
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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
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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
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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
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Numerical simulations : longitudinal evolution Supercontinuum
formation from simulationsstandard event of previous simulations
Solitons Dispersive waves Spectral domain
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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)
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Experiments Supercontinuum Continuous pumping Experimental
evidence of rogue events
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Probability Density Functions (PDFs): Experiments
Supercontinuum Continuous pumping First approach : Statistics
Signature of rogue events Log(PDF) Experimental evidence of rogue
events
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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
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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)
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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
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Numerical simulations : comparison Convective system vs
absolute system Drift (convection) important ingredient for
generating rare and strong optical waves Same scale!!
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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
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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
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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
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Spectrograms Numerics Supercontinuum Continuous pumping
Mechanism of formation of rogue events