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Wave Turbulence in nonintegrable and integrable optical (fiber) systems

Wave Turbulence in nonintegrable and integrable optical (fiber) systems. Pierre Suret et Stéphane Randoux Phlam, Université de Lille 1 Antonio Picozzi laboratoire Carnot, Université de Bourgogne, Dijon Séminaire CEMPI , Décembre 2012. Wave Turbulence in optical fibers. Motivations.

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Wave Turbulence in nonintegrable and integrable optical (fiber) systems

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  1. Wave Turbulence in nonintegrable and integrable optical (fiber) systems Pierre Suret et Stéphane Randoux Phlam, Université de Lille 1 Antonio Picozzi laboratoire Carnot, Université de Bourgogne, Dijon Séminaire CEMPI, Décembre 2012

  2. Wave Turbulence in opticalfibers Motivations Physics of multimodecontinuouswave (Raman fiber) lasers Nonlinear Dynamics / statisticalPhysics Nonlinear propagation of incoherent waves Experiments Theory Numerical simulations

  3. Wave Turbulence in opticalfibers Introduction Lasers 1960s Nonlinearoptics 2ndharmonicgeneration Optical fibers 1970s Telecommunications(linearoperation) Nonlinearfiberoptics intense cw/pulsedcoherent light waves Photonic Crystal fibers1995 supercontinuum Incoherentnonlinear (fiber) optics

  4. Wave Turbulence in opticalfibers Outlines Wave turbulence in opticalfibers : somefundamentals 1 Anomalousthermalization (non integrable system) 2 Non trivial degeneracy of resonance conditions Irreversible evolution in Nonlinear Schrödinger equation (NLS 1D) 3 No resonance conditions Open questions 4

  5. Wave Turbulence in opticalfibers WaveTurbulence 1) Principles Hydrodynamics, Mechanics, Plasmas physics, BEC, Optics Wave Turbulence Theory : incoherentwave weaknonlinearity (HNL << HL) (~ linear dispersion curve !) closure of moments kineticsequations ( Boltzmann equation) damping forcing Dissipatives systems Kolmogorov-Zakharov cascade Hamiltoniansystems WaveThermalization Entropy Microcanonics Energyequipartition Condensation of waves

  6. Wave Turbulence in opticalfibers WaveTurbulence 1) Principles Example in mechanics vibrating plate Miquel B., N. Mordant Phys. Rev. Lett. 107(3), 034501 (2011) Cobelli P. et al. Phys. Rev. Lett. 103 204301 (2009)

  7. Wave Turbulence in opticalfibers WaveTurbulence 1) Principles Vibrating plate

  8. Wave Turbulence in opticalfibers Experimentswithopticalfibers 1) Principles Nonlinear fiber optics : how does it look like in real life?

  9. Wave Turbulence in opticalfibers What do we observe in optics ? 1) Principles narrowopticalspectrum slow varying Amplitude carrier wave Detector / intensity = lowpassfilter t

  10. Wave Turbulence in opticalfibers Optical (power) spectrum 1) Principles Optical spectrum : « measurement » of fast dynamics Optical power ~1 ps Slow detector Dispersive setup Optical spectrum = power spectral density = number of particles (kinetic equation)

  11. Wave Turbulence in opticalfibers Whatis non linear ? 1) Principles Interaction light / matter Dielectric Polarization Maxwell equations (E & B) Electromagneticwaves Generalized NLS Kerr Raman Gain … diffraction dispersion losses

  12. Wave Turbulence in opticalfibers Single mode opticalfiber 1) Principles Optical cladding LOW losses (0.1-1% /100m) core n0 n(r) single-mode fibers Fiber core diameter 6-9µm L 1 or several Waves

  13. z t |A|2 t Wave Turbulence in opticalfibers Propagation in single mode fiber (1D) 1) Principles Spatiotemporal system (scalar 1D Nonlinear Schrödinger equations) group velocity dispersion Kerr effect Distance z « time » «space» Time t

  14. StronglymultimodeContinuousWave laser = incoherentwaves Wave Turbulence in Optical systems Optical spectrum of cw lasers 1) Principles complexdynamics 101-106 modes ! random phases ? L

  15. Wave Turbulence in opticalfibers Wave Turbulence / kinetic Theory 1) Principles Weak nonlinear interaction among spectral components (3) / Four Waves Mixing Initial condition 101-108 modes ! incoherent waves / random phases Phase matching conditions (

  16. Wave Turbulence in Optical systems Wave Turbulence Theory Motivations, context Thermodynamics ex gas : collisions KineticTheory of gases Nonlinear Optics Four Waves Mixing Wave Kinetic theory

  17. Wave Turbulence Propagation of incoherentwaves in opticalfiber Wave turbulence in opticalfibers : somefundamentals 1 Anomalousthermalization 2 Irreversible evolution in Non Linear Schrödinger equation (NLS 1D) 3

  18. Wave Turbulence in optics A simple example of wave thermalization Anomalous Thermalization XPM Cross 4 Waves mixing () Energy Phase-Matching

  19. Wave Turbulence in optics A simple example of waves thermalization Anomalous Thermalization Kineticequations H-Theorem entropy Thermodynamicalequilibrium optical spectrum : Number of particules global invariants Momentum KineticEnergy

  20. Wave Turbulence in optics A simple example of wave thermalization Distribution de Rayleigh-Jeans Anomalous Thermalization Energyequipartition Rayleigh-Jeans Numerical simulations Lagrange parameters / E, P, Nj

  21. Wave Turbulence in optics Numerical simulations Anomalous Thermalization Rayleigh-Jeans NO energy equipartition

  22. Wave Turbulence in optics Phase-matching conditions Anomalous Thermalization 4 waves mixing () degeneracy

  23. Wave Turbulence in optics Anomalous thermodynamical equilibrium Distribution de Rayleigh-Jeans Anomalous Thermalization

  24. Wave Turbulence in optics Anomalous thermodynamical equilibrium Distribution de Rayleigh-Jeans Anomalous Thermalization New invariant Jw LOCAL invariant (for eachw) H-theorem local equilibrium spectrum  : Lagrange parameter associated to NO energyequipartition local equilibrium state preserves a memory of the initial condition

  25. Wave Turbulence in optics Anomalous thermodynamical equilibrium Distribution de Rayleigh-Jeans Anomalous Thermalization particular case

  26. Wave Turbulence in optics Experiments Distribution de Rayleigh-Jeans AnomalousThermalization Polarization maintaining fiber (PMF) WDM Raman QWP Q-Switch Nd/YAG Laser l=1064 nm Isotropic (spun) fiber 1.6 meters QWP HWP trigger electronics OSA 10ns AOM t

  27. Wave Turbulence in optics Experiments Distribution de Rayleigh-Jeans Anomalous Thermalization

  28. Wave Turbulence in optics Experiments Distribution de Rayleigh-Jeans AnomalousThermalization z = 0 z = L z = 0 z = L

  29. Wave Turbulence in optics Conclusion Distribution de Rayleigh-Jeans Anomalous Thermalization Non-trivial degenerateresonances Local invariant Breakdown of Energyequipartition Memory of the initial condition a general phenomenon anotherexample : Suret et al., PRL 104, 054101 (2010) C. Michel et al., Opt. Lett. 35, 2367-2369 (2010) C. Michel et al., Lett. In Math. Phys., 96, p 415 (2011)

  30. Wave Turbulence in opticalfibers Wave turbulence in opticalfibers : somefundamentals 1 Anomalous thermalization (non integrable system) 2 Irreversible evolution in Nonlinear Schrödinger equation (NLS 1D) 3

  31. Wave Turbulence in optics Experiments / numerical simulations 1D NLS Irreversibleevolutiontoward a steady state defocusing case : no BF B. Barviau, S. Randoux, and P. Suret , Optics Letters, 31, pp. 1696-1698 (2006)

  32. Wave Turbulence in optics Trivial interaction ! 1D nonlinear Schrödinger equation 1D NLS (3) / 4 waves interaction 2>0 normal dispersion (<1300nm) : no modulationnalinstability NLS1D : integrable equation infinity of motion constants quasi-periodicbehavior usual Wave Turbulence theory

  33. Wave Turbulence in optics What does the kinetic theory say ? 1D NLS 1D NLS

  34. Wave Turbulence in optics What does the kinetic theory say ? 1D NLS Oscillatory terms neglected in the usual treatment 1DNLS : NO Phase matched interactions Oscillatoryterms ? Transientregime ?

  35. Wave Turbulence in optics Numerical simulations 1D NLS HNL/HL=0.5 HNL/HL=0.05

  36. Wave Turbulence in optics Numerical simulations 1D NLS HNL/HL=0.5 HNL/HL=0.05

  37. Wave Turbulence in optics Numerical simulations 1D NLS HNL/HL=0.5 HNL/HL=0.05 Good approximation N(z) = N(z=0) !!

  38. Wave Turbulence in optics A last approximation… 1D NLS Dominant contributions

  39. Wave Turbulence in optics Dominant contribution 1D NLS HNL/HL=0.5

  40. Wave Turbulence in optics Damped oscillations toward steady state 1D NLS comparison with numerical integration 1D NLS  The periodof oscillations isgiven by the dominant contribution

  41. Wave Turbulence in optics Experiments 1D NLS =1064nm Experiments / numerical simulations (NLS) Simulations (NLS / Kinetic equations) 0.15nm

  42. Propagation of incoherent waves in optical fiber 1DNLS (single pass) Open Questions • Complete characterization • of the equilibrium state • (exponentialtails) • General wave turbulence theory • HNL << HL • Whichtheoryathighoptical power • HNL ~ HL and HNL >> HL

  43. Propagation of incoherent waves in optical fiber 1DNLS (single pass) Open Questions - Gas of solitons ? - Numerical Inverse ScatteringTransform(incoherent initial conditions) Periodic defocusing 1D NLS equation Hyperelliptic functions Eigenvalues of the IST problem

  44. Propagation of incoherent waves in optical fiber Open Questions Non-local / non-instantaneous (experiments / NLS ?) Single pass : towardthermodynamicalequilibrium ? Initial conditions ? 2D (multimode fibers) Dissipative systems Gain Optical cavity (forcing / losses)

  45. T T= slow time « time » t Propagation of incoherent waves in optical fiber Open Questions |A|2 «space» t =fast time (round trip) t Ginzburg-Landau eq. Mean-field model Braggs : -ln(R1()R2())=0+22 Dispersion / effet Kerr : 2 /  Raman Gain : g losses : S

  46. Statistical properties of Raman fiber lasers Dissipative systems : open Questions Recent experimental results about extreme statistics (Raman fiber lasers) S. Randoux and P. Suret, Opt. Lett. 37, 500-502 (2012) Off-centered filter Model /Numerical integration : ok Which theory (mechanisms, PDF) ?

  47. Statistical properties of Raman fiber lasers Open Questions Optical power spectratransmitted by the narrow-bandwidth (5GHz-2pm) opticalfilter

  48. Statistical properties of Raman fiber lasers Dynamics at the output of the narrow-bandwidth optical filter Optical filter at central Stokes wavelength Off-centered optical filter (detuned from 1.5 nm from the central Stokes wavelength) Total (not filtered) Stokes power

  49. Statistical properties of Raman fiber lasers Statistics at the output of the narrow-bandwidth optical filter Centered filter Off-centered filter

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