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T HERMALIZATION OF R ANDOM N ONLINEAR W AVES: A PPLICATION TO O PTICAL W AVES

T HERMALIZATION OF R ANDOM N ONLINEAR W AVES: A PPLICATION TO O PTICAL W AVES. Self-organization induced by thermalization towards the most disordered state. A.Picozzi, Institut Carnot de Bourgogne (ICB), Dijon. Theory. Experiments & Simulations.

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T HERMALIZATION OF R ANDOM N ONLINEAR W AVES: A PPLICATION TO O PTICAL W AVES

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  1. THERMALIZATIONOF RANDOM NONLINEAR WAVES: APPLICATION TO OPTICAL WAVES Self-organization induced by thermalization towards the most disordered state A.Picozzi,Institut Carnot de Bourgogne (ICB), Dijon Theory Experiments & Simulations G. Millot, S. Pitois, B. Barviau, S. Lagrange, B. Kibler (ICB) P. Suret, S. Randoux (PhLam, Lille) J. Fleischer (Princeton Un.) S. Rica (LPS, ENS) H.R. Jauslin (ICB)

  2. Condensation of incoherent waves Hamiltonian Reversible (z - z) 10-4 0 0 kx y ky z (3) x Dyachenko, Newell, Pushkarev, Zakharov, Physica D (92) Nazarenko, Zakharov, Physica D (05) Connaughton, Josserand, Picozzi, Pomeau, Rica PRL (05)

  3. 10-4 0 0 kx ky 0 0 kx ky y z (3) x Condensation of incoherent optical waves Hamiltonian Reversible (z - z) Dyachenko, Newell, Pushkarev, Zakharov, Physica D (92) Nazarenko, Zakharov, Physica D (05) Connaughton, Josserand, Picozzi, Pomeau, Rica, PRL (05)

  4. Weak turbulence theory e = << RPA: closure of the hierarchy of moments equation: H / H 1 nl kin - Irreversible kinetic equation (H-theorem) ˜ Spectrum: Forced NLS equation: Non-equilibrium stationary solut. Zakharov-Kolmogorov spectra - Rayleigh-Jeans spectrum V. Zakharov, V. L'vov, G. Falkovich Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992) V. Zakharov, F. Dias, A. Pushkarev One dimensional wave turbulence Phys. Rep., 398, 1 (2004) Maximum S[nk] Coll[nkeq] = 0

  5. Condensation of incoherent waves Kinetic theory 2D 3D

  6. Small-condensate amplitude: Weak Turbulence (FWI) N0/N High-condensate amplitude: Bogoliubov regime (TWI) H/V

  7. Quantitative agreement theory/numerics D = 3 D = 2 Wave turbulence Bogoliubov Beyond thermodyn. limit Düring, Picozzi, Rica, Physica D (09)

  8. Quantitative agreement at higher dimensions… D=4 D=5

  9. Towards the experiment: Saturable nonlinearity D = 2 Preliminary experimental results J. W. Fleischer’s group (Princeton Univ.) Work in progress…

  10. Quantum kinetic equation (bosons) nk «1 nk »1 Weak-turbul. kin. eq. Boltzmann kin. eq. RPA NLS equation B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991) Yu. Kagan, B. Svistunov, PRL 79, 3331 (1997) M. J. Davis, R. J. Ballagh, K. Burnett, J. Phys. B 34, 4487 (2001) M. J. Davis, S. A. Morgan, K. Burnett, PRL 87, 160402 (2001)

  11. Recent experimental results on optical wave thermalization…

  12. Velocity locking of incoherent wave-packets u1 Ω1 Velocity Locking optical fiber u2 Ω2 FWI-TWIin 2D or 3D An isolated system at equilibrium can only exhibit a uniform motion of translation as a whole, while any macroscopic internal motion is not possible. S. Pitois, S. Lagrange, H. Jauslin, A. Picozzi, PRL (2006) S. Lagrange, H. Jauslin, A. Picozzi, EPL (2007)

  13. Velocity-locking: Experimental results A2, u2 n2 A1, u1 n1 S. Pitois, S. Lagrange, H. Jauslin, A. Picozzi, PRL (2006)

  14. Thermalization does not necessarily occur…

  15. Partial thermalization of 1D random waves Manakov Sov. Phys. JETP (1974) Zakharov & Schulman Physica 4D (1982) Integrable: We consider the non-integrable case: No energy equipartition

  16. Suret, Picozzi, Jauslin, Randoux (submitted) H-theorem for: Local maximum of entropy Theory Initial condition Numerics: NLS

  17. for RPA: Langmuir WT eq.: Musher, Rubenchik, Zakharov, Phys. Rep. 252, 177 (1995) w w Noninstantaneous nonlinearity : Spectral incoherent solitons

  18. Spectral incoherent solitons: Experiment • Incoherent wave source with a spectral width of 20 THz (at l=630nm): • → ASE of a dye amplifier pumped by frequency-doubled Nd:YAG laser (5ns – 25Hz) • Use of a conventional PM optical fiber (loss~2.3km-1) Comparison with numerics L=300m P=8W P=14W Picozzi, Pitois, Millot, PRL (08)

  19. Weak-turbulence description of Supercontinuum generation White light generation from laser light

  20. Experimental signature of optical thermalization NLS RJ vg(w1) = vg(w2) RJ RJ spectrum NLS without Raman NLS with Raman Experiment Barviau, Kibler, Kudlinski, Mussot, Millot, Picozzi, Optics Express (2009) Barviau, Kibler, Picozzi, PRA (2009)

  21. -Condensation of incoherent waves - Velocity locking - Partial thermalization -Spectral incoherent solitons -Wave turbulence approach of Supercontinuum generation

  22. Evolution of Q(3) = F(Q(5), Q(1)Q(4), Q(2)Q(3), Q(1)Q(1)Q(3))

  23. High-condensate amplitude: Bogoliubov regime (TWI)

  24. A.Picozzi, S.Pitois,and G.Millot, Phys. Rev. Lett. 101 (2008) EF4.1 Friday 8:30 ROOM B0.R1 Spectral incoherent solitons B.kibler Simulation of GNLSE with Raman Simulation of GNLSE without Raman Fiber length (m) Time (ps) 1. Context : What happen when P increase ? P = 1kW injected No coherent solitons: Turbulent behaviour !

  25. defining RPA Phase matching Entropy Conserved quantities maximum entropy Equilibrium spectrum T: temperature µ: chemical potential h:…velocity locking 2. Statistical analysis of supercontinuum generation: S.Dyachenko, A.C.Newell, A.Pushkarev, and V.E.Zakharov, Physica D 57, 96 (1992) V.E.Zakharov, V.S.L’vov, and G.Falkovich, « kolmogorov spectra of turbulence » (springer, 1992) B.Barviau, B.Kibler, S.Coen and A.Picozzi Optics Letters 33, 2833 (2008)

  26. No trapping 3. Interpretation in term of potential : k(w) B.Barviau, B.Kibler, S.Coen and A.Picozzi Optics Letters 33,2833 (2008)

  27. -hw Maxima of neq Minima of F(w): k’(w) = - h vg(w1) = vg(w2) = 1/<k’(w)> since Unique solution (w1,w2) 4. Velocity locking : B.Barviau, B.Kibler and A.Picozzi Physical Review A. in press (2009)

  28. with (ii) With Self-steepening: (i) Without Self-steepening: 4. Role of self steepening in wave thermalization: Introduce a transformation:

  29. Wave turbulence theory: condensation, spontaneous polarization… A.Picozzi, Optics Express 16,17171 (2007) 5. Experimental demonstration: Experimental setup: 660ps pulses @1064nm, rate 7.7kHz estimated injected peak power:3.5kW Experiment: cut back method Experimental signature of optical wave thermalization Thermodynamic interpretation of the double peaked spectrum as a velocity-locking B.Barviau, B.Kibler, A.Kudlinski, A.Mussot, G.Millot and A.Picozzi Optics Express 17, 7392 (2009)

  30. Second experiment: Supercontinuum generation • Coherent wave source (at l0=1064nm): • → passively Q-switched Nd:YAG laser (600ps - 7.5kHz) • Use of a customized PCF with 2 ZDW around l0 (A. Kudlinski, IRCICA, Lille) Spectrogram analysis of the spectral incoherent soliton (GNLSE simulation of CW) • The spectral incoherent soliton do not exhibit a confinement in the temporal domain, but exclusively in the frequency domain. • (Similar observation in a different context) B. Barviau et al., Opt. Exp. 17, 7392 (2009) A.V. Gorbach and D.V. Skryabin, Opt. Lett. 31, 3309 (2006)

  31. with (ii) With Self-steepening: (i) Without Self-steepening: 4. Role of self steepening in wave thermalization: Introduce a transformation:

  32. An increase of disorder (entropy) requires the generation of the coherent plane-wave HNonLin H HKin

  33. Coherence absorption T nkeq = k2- (theorem of energy equipartition) Bose gases? ideal Bose gas A. Picozzi, S. Rica, EPL (2008)

  34. Coherence absorption T nkeq = k2- (theorem of energy equipartition) ideal Bose gas

  35. Influence of advection? u1 Ω1 … optical fiber uM ΩM Velocity Locking j=1,…M

  36. Velocity-locking (M=3) n - - 1 1 w = ¶ ¶ w = + a w v ( ) k / u 2 j j j j

  37. Velocity-locking of incoherent wave-packets

  38. Partial thermalization of 1D random waves

  39. Kinetic theory of the vector NLS eq. RPA - Irreversible kinetic equation (H - theorem) - Thermodynamic equilibrium equilibrium

  40. Thermalization velocity-locking S. Pitois, S. Lagrange, H. Jauslin, A. Picozzi, PRL (2006) S. Lagrange, H. Jauslin, A. Picozzi, EPL (2007)

  41. Experimental results A2, u2 n2 A1, u1 n1

  42. Coherence transfer ≈ Coherent U optical fiber Thermal equilibrium ? Incoherent V Kinetic theory Tu=Tv

  43. H-theorem of entropy growth S=Su+Sv Sw=∫ln(wk)dk 1 ≈ 2 Qu Q Qv Qw= ∫k2wkdk H-theorem violated S z A. Picozzi - PRL (2006)

  44. Velocity-locking: Extension to(2) in 2D (3D) S. Lagrange, H. Jauslin, A. Picozzi, EuroPhys. Lett. (2007)

  45. Velocity-locking: Extension to(2) in 2D (3D)

  46. To regularize the divergence of neqfrequency cut-off kc 2D 3D condensation no condensation

  47. Bogoliubov’s transformation

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