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

Physics of multimode continuous wave (Raman fiber) lasers Motivations Wave Turbulence in optical fibers Nonlinear Dynamics / statistical Physics Experiments Theory Numerical simulations Nonlinear propagation of incoherent waves

Introduction Wave Turbulence in optical fibers Optical fibers 1970s Nonlinear fiber optics Telecommunications (linear operation) intense cw/pulsed coherent light waves Lasers 1960s Nonlinear optics 2 nd harmonic generation Photonic Crystal fibers 1995 Incoherent nonlinear (fiber) optics supercontinuum

Outlines 1 1 Wave turbulence in optical fibers : some fundamentals Anomalous thermalization (non integrable system) 2 2 Non trivial degeneracy of resonance conditions Wave Turbulence in optical fibers 3 3 Irreversible evolution in Nonlinear Schrödinger equation (NLS 1D) No resonance conditions 4 4 Open questions

Wave Turbulence Theory : incoherent wave weak nonlinearity (H NL << H L ) (~ linear dispersion curve !) closure of moments kinetics equations (  Boltzmann equation) Wave Turbulence Theory : incoherent wave weak nonlinearity (H NL << H L ) (~ linear dispersion curve !) closure of moments kinetics equations (  Boltzmann equation) Hamiltonian systems Wave Thermalization Entropy Microcanonics Energy equipartition Condensation of waves WaveTurbulence Wave Turbulence in optical fibers 1) Principles Hydrodynamics, Mechanics, Plasmas physics, BEC, Optics Dissipatives systems Kolmogorov-Zakharov cascade forcing damping

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

WaveTurbulence Wave Turbulence in optical fibers Vibrating plate 1) Principles

Nonlinear fiber optics : how does it look like in real life? Experiments with optical fibers 1) Principles Wave Turbulence in optical fibers

1) Principles Wave Turbulence in optical fibers What do we observe in optics ? narrow optical spectrum carrier wave t Detector / intensity = low pass filter slow varying Amplitude

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

What is non linear ? 1) Principles Wave Turbulence in optical fibers Electromagnetic waves Dielectric Polarization Maxwell equations (E & B) losses diffractiondispersion Kerr Raman Gain … Generalized NLS Interaction light / matter

core Optical cladding n0n0n0n0 n(r) Single mode optical fiber 1) Principles Wave Turbulence in optical fibers LOW losses (0.1-1% /100m) L single-mode fibers Fiber core diameter 6-9µm 1 or several Waves

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

Strongly multimode Continuous Wave laser = incoherent waves random phases complex dynamics modes ! Optical spectrum of cw lasers Wave Turbulence in Optical systems 1) Principles L ?

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

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

Wave Turbulence Propagation of incoherent waves in optical fiber 1 1 Wave turbulence in optical fibers : some fundamentals 2 2 Anomalous thermalization 3 3 Irreversible evolution in Non Linear Schrödinger equation (NLS 1D)

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

Number of particules global invariants Kinetic Energy Momentum H-Theorem entropy Thermodynamical equilibrium A simple example of waves thermalization Wave Turbulence in optics Anomalous Thermalization Kinetic equations optical spectrum :

Numerical simulations Lagrange parameters / E, P, N j Distribution de Rayleigh-Jeans Wave Turbulence in optics Anomalous Thermalization A simple example of wave thermalization Energy equipartition Rayleigh-Jeans

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

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

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

local equilibrium spectrum H-theorem NO energy equipartition local equilibrium state preserves a memory of the initial condition  : Lagrange parameter associated to New invariant J  LOCAL invariant (for each  ) Distribution de Rayleigh-Jeans Wave Turbulence in optics Anomalous Thermalization Anomalous thermodynamical equilibrium

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

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

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

z = 0 z = L Distribution de Rayleigh-Jeans Wave Turbulence in optics Anomalous Thermalization Experiments

Non-trivial degenerate resonances Local invariant Breakdown of Energy equipartition Memory of the initial condition a general phenomenon another example : Distribution de Rayleigh-Jeans Wave Turbulence in optics Anomalous Thermalization Conclusion Suret et al., PRL 104, (2010) C. Michel et al., Opt. Lett. 35, (2010) C. Michel et al., Lett. In Math. Phys., 96, p 415 (2011)

Wave Turbulence in optical fibers 1 1 Wave turbulence in optical fibers : some fundamentals 2 2 Anomalous thermalization (non integrable system) 3 3 Irreversible evolution in Nonlinear Schrödinger equation (NLS 1D)

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

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

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

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

Wave Turbulence in optics 1D NLS Numerical simulations H NL /H L =0.05 H NL /H L =0.5

Wave Turbulence in optics 1D NLS Numerical simulations H NL /H L =0.05 H NL /H L =0.5

Wave Turbulence in optics 1D NLS Numerical simulations Good approximation N (z) = N (z=0) !! H NL /H L =0.05 H NL /H L =0.5

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

Wave Turbulence in optics 1D NLS Dominant contribution H NL /H L =0.5

Wave Turbulence in optics 1D NLS Damped oscillations toward steady state comparison with numerical integration 1D NLS The period of oscillations is given by the dominant contribution 

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

Complete characterization of the equilibrium state (exponential tails) General wave turbulence theory H NL << H L Which theory at high optical power H NL ~ H L and H NL >> H L 1DNLS (single pass) Open Questions Propagation of incoherent waves in optical fiber

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

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

t =fast time (round trip) «space» |A|2|A|2 Mean-field model t Ginzburg-Landau eq. T= slow time « time » T t Braggs : - ln(R 1 (  )R 2 (  ))=  0 +  2  2 Dispersion / effet Kerr :  2 /  Raman Gain : g losses :  S Open Questions Propagation of incoherent waves in optical fiber

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

Statistical properties of Raman fiber lasers Optical power spectra transmitted by the narrow-bandwidth (5GHz-2pm) optical filter Open Questions

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

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