Recent developments around wave turbulence in nonlinear fiber optics Stéphane Randoux 1, Antonio Picozzi 2, Pierre Suret 1 1: Laboratoire de Physique des Lasers, Atomes et Molécules Université de Lille 1, France 2: Institut Carnot de Bourgogne Université de Bourgogne, Dijon, France

Wave turbulence in Optics Introduction: Nonlinear optics with incoherent waves One-dimensional (1D) optical wave turbulence Conservative systems: Integrable wave systems ( spectral broadening in optical fibers ) Non-Integrable wave systems ( anomalous thermalization, Supercontinuum generation ) Optical wave condensation ( liquid crystal ) Dissipative systems: Wave turbulence in highly multimode fiber lasers Optical rogue wavesSingle-pass propagation in optical fibers (supercontinuum generation, fiber amplifier..) Fiber lasers (connections with hydrodynamics? ) Two-dimensional (2D) optical wave turbulence Optical wave condensation in graded-index fibers

Wave turbulence in Optics Introduction Wave turbulence in Optics Historical viewpoint Lasers 1960s Nonlinear optics 2 nd harmonic generation, Stimulated Raman/Brillouin scattering … Statistical Optics/Optical Coherence Theory Linear Theory (degree of coherence of light) Optical fibers 1970s Nonlinear fiber optics Telecommunications applications (linear operation) Propagation of intense cw/pulsed coherent light waves Optical fibers 1995 Management of fiber dispersion Supercontinuum generation High power/highly multimode lasers Incoherent nonlinear fiber optics

Wave turbulence in Optics Introduction Wave turbulence in Optics Incoherent nonlinear (fiber) optics Wave Turbulence (WT) ? Use of WT theory for the understanding/description of some « standard » optical experiments (spectral broadening in fiber lasers, supercontinuum generation…) Design of optical experiments to investigate problems from WT theory (optical wave condensation, cascades…)

Wave turbulence in Optics PEOPLE Wave turbulence in Optics Antonio Picozzi Dijon (France) Theory G. Millot, C. Finot, B. Kibler, J. Fatome Dijon (France) Nonlinear fiber optics Experiments S. Randoux, P. Suret Lille (France) Fiber lasers, nonlinear fiber optics Exp/Theory S. Nazarenko Coventry (UK) Theory S. Residori, U. Bortolozzo Nice (France) Liquid crystal Experiments S. Turitsyn, D. Churkin Birmingham (UK) Fiber lasers, fiber telecommunication Exp/Theory S. Babin, E. Podivilov Novossibirsk (Russia) Fiber lasers Exp/theory

Wave turbulence in Optics Introduction Wave turbulence in Optics Hydrodynamics : Time (t) evolution of a signal/pattern Incoherent wave Optics : Space (z) evolution of the incoherent wave z t

L 1D wave turbulence in single-mode fibers Wave turbulence in Optics Introduction Wave turbulence in Optics Optical power spectra ~1 ps Fiber core diameter 6-9µm 2D wave turbulence in multimode fibers Fiber core diameter µm

Absorption Dispersion Non linearity: Kerr, Raman Wave turbulence in Optics Introduction Wave turbulence in Optics Nonlinear propagation of 1D waves in single_mode fibers ZDW Anomalous dispersion Nomal dispersion Fiber parameters ( , ,  ) are known with good precision Accurate numerical simulations of nonlinear propagation can be easily performed PCFs : the dispersion curve can be engineered

Wave turbulence in Optics Introduction Wave turbulence in Optics 2 nd order dispersion Kerr Raman Optical waves with narrow-bandwidth spectra Nonlinear propagation of 1D waves in single_mode fibers : Bandwidth of the optical spectrum In a lot of experiments : and stimulated Raman scattering occurs when

Wave turbulence in Optics Introduction Wave turbulence in Optics Nonlinear fiber optics : how does it look like in real life?

Wave turbulence in Optics Introduction: Nonlinear optics with incoherent waves One-dimensional (1D) optical wave turbulence Conservative systems: Integrable wave systems ( spectral broadening in optical fibers ) Non-Integrable wave systems ( anomalous thermalization, Supercontinuum generation ) Optical wave condensation ( liquid crystal ) Dissipative systems: Wave turbulence in highly multimode fiber lasers Optical rogue wavesSingle-pass propagation in optical fibers (supercontinuum generation, fiber amplifier..) Fiber lasers (connections with hydrodynamics? ) Two-dimensional (2D) optical wave turbulence Optical wave condensation in graded-index fibers

Wave turbulence in Optics Spectral broadening in fibers Wave turbulence in Optics Narrow-bandwidth (1nm) optical power spectra far from zero dispersion wavelength (ZDW) WT theory : No change in the power spectrum of the optical wave !!

Wave turbulence in Optics NLS equations (not integrable) Kinetic equation local equilibrium spectrum ANOMALOUS THERMALIZATION PHENOMENON Suret et al, Phys. Rev. Lett. 104, (2010 ) Non-Integrable 1D optical wave systems Isotropic (spun) fiber 1.6 meters A1A1 A2A2

Wave turbulence in Optics Non-Integrable 1D optical wave systems B. Barviau, B. Kibler and A. Picozzi Phys. Rev. A79, (2009)

Wave turbulence in Optics Non-Integrable 1D optical wave systems B. Barviau, B. Kibler and A. Picozzi Phys. Rev. A79, (2009)

Wave turbulence in Optics Optical wave turbulence in 1D wave systems U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori Optical wave turbulence and the condensation of light J. Opt. Soc. Am. B, 26, 2280 (2009) 1D light propagation in a Liquid Crystal (LC) cell

Wave turbulence in Optics Optical wave turbulence in 1D wave systems U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori Optical wave turbulence and the condensation of light J. Opt. Soc. Am. B, 26, 2280 (2009) WT treatment Intensity spectrum

Wave turbulence in Optics Optical wave turbulence in 1D wave systems U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori Optical wave turbulence and the condensation of light J. Opt. Soc. Am. B, 26, 2280 (2009)

Wave turbulence in Optics Introduction: Nonlinear optics with incoherent waves One-dimensional (1D) optical wave turbulence Conservative systems: Integrable wave systems ( spectral broadening in optical fibers ) Non-Integrable wave systems ( anomalous thermalization, Supercontinuum generation ) Optical wave condensation ( liquid crystal ) Dissipative systems: Wave turbulence in highly multimode fiber lasers Optical rogue wavesSingle-pass propagation in optical fibers (supercontinuum generation, fiber amplifier..) Fiber lasers (connections with hydrodynamics? ) Two-dimensional (2D) optical wave turbulence Optical wave condensation in graded-index fibers

Wave turbulence in Optics Optical wave turbulence in 1D wave systems Single mode optical fiber Ytterbium fiber laser λsλs λpλp L~500 m R~99%R~80% Wavelength (nm) FBG rflectivity R1(λ)R1(λ) R2(λ)R2(λ) Opical power spectrum 300 GHz (iii) Fibre Bragg Grating (FBG) mirrors P. Suret and S. Randoux, Opt. Comm. 237, 201 (2004) Cavity Free Spectral Range ~ 100 kHz Laser linewidth ~ 100 GHz ~ 10 6 cavity modes 1100 nm 1160 nm Raman fiber lasers

Wave turbulence theory Mean field approximation FBGs mirrorsparabolic losses in frequency space 2 nd order dispersion from cavity fiber Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser S. A. Babin, D. V. Churkin, A. E. Ismagulov, S. I. Kablukov, E. V. Podivilov JOSA B 24, 1729 (2007) Wave turbulence in Optics Optical wave turbulence in 1D wave systems

Numerical simulations (Mean field approximation) Influence of the sign of 2 nd order dispersion (  2 ) Turitsyna et al, Phys. Rev. A 80, (R) (2010) Formation of the optical power spectrum in Raman fiber lasers Changes in the shape of the laser spectrum NOT described from WT theory

Wave turbulence in Optics Introduction: Nonlinear optics with incoherent waves One-dimensional (1D) optical wave turbulence Conservative systems: Integrable wave systems ( spectral broadening in optical fibers ) Non-Integrable wave systems ( anomalous thermalization, Supercontinuum generation ) Optical wave condensation ( liquid crystal ) Dissipative systems: Wave turbulence in highly multimode fiber lasers Optical rogue wavesSingle-pass propagation in optical fibers (supercontinuum generation, fiber amplifier..) Fiber lasers (connections with hydrodynamics? ) Two-dimensional (2D) optical wave turbulence Optical wave condensation in graded-index fibers

PCF Diffraction grating IR laser 200 fs Source : University of Bath Wave turbulence in Optics Optical rogue waves Wave turbulence in Optics Supercontinuum generation in Photonic Crystal fibers Pump Supercontinuum Wavelength (nm) Ref: Ranka etal, Opt. Lett 25, 25 (2000)

Wave turbulence in Optics D. R. Solli, C. Ropers, P. Koonath, B. Jalali Nature 450, 1054 (2007) Rogue waves in optical supercontinuum

Wave turbulence in Optics J. M. Dudley, G. Genty, and B. J. Eggleton Optics Express 16, 3644 (2008) Raman fiber amplifier: K. Hammani, C. Finot, J. M. Dudley, G. Millot; Opt. Express 16, (2008) Parametric fiber amplifier: K. Hammani, C. Finot, G. Millot; Opt. Lett 34, 1138 (2009) Rogue waves in optical supercontinuum

Wave turbulence in Optics Granularity and inhomogeneity are joint generators of optical rogue waves F. T. Arrechi, U. Bortolozzo, A. Montina, and S. Residori Phys. Rev. Lett. 106, (2011) Optical rogue waves

Wave turbulence in Optics Non gaussian statistics and extreme waves in a nonlinear optical cavity A. Montina, U. Bortolozzo, S. Residori and F. T. Arrechi Phys. Rev. Lett. 103, (2009) Optical rogue waves

Wave turbulence in Optics Narrow-bandwidth Electrical filter Central frequency : 6Hz Bandwidth : 1Hz Extreme-type statistics in hydrodynamic experiments P. Dennissenko, S. Lukaschuck, S. Nazarenko Phys. Rev. Lett. 99, (2007)

Wave turbulence in Optics Extreme-type statistics in Raman Fiber lasers Slicing/Filtering of the intracavity Stokes spectrum in a Raman fiber laser Single mode optical fiber Ytterbium fiber laser λsλs λpλp L~500 m R~99%R~80% 1100 nm 1160 nm 300 GHz Bandwidth of the optical filter: 5GHz S. Randoux and P. Suret, Opt. Lett. 37, 500 (2012)

Wave turbulence in Optics Raman fiber laser Extreme-type statistics in Raman Fiber lasers Slicing/Filtering of the intracavity Stokes spectrum in a Raman fiber laser Narrow-bandwidth optical filter

Optical power spectra transmitted by the narrow-bandwidth (5GHz-2pm) optical filter Wave turbulence in Optics 300 GHz Extreme-type statistics in Raman Fiber lasers S. Randoux and P. Suret, Opt. Lett. 37, 500 (2012)

Dynamics and Statistics at the output of the narrow-bandwidth optical filter Centered filter Off-centered filter Wave turbulence in Optics Extreme-type statistics in Raman Fiber lasers S. Randoux and P. Suret, Opt. Lett. 37, 500 (2012)

Wave turbulence in Optics Introduction: Nonlinear optics with incoherent waves One-dimensional (1D) optical wave turbulence Conservative systems: Integrable wave systems ( spectral broadening in optical fibers ) Non-Integrable wave systems ( anomalous thermalization, Supercontinuum generation ) Optical wave condensation ( liquid crystal ) Dissipative systems: Wave turbulence in highly multimode fiber lasers Optical rogue wavesSingle-pass propagation in optical fibers (supercontinuum generation, fiber amplifier..) Fiber lasers (connections with hydrodynamics? ) Two-dimensional (2D) optical wave turbulence Optical wave condensation in graded-index fibers

Wave turbulence in Optics Optical wave condensation Dyachenko et. al, Physica D (1992) Connaughton et. al, PRL (2005) Düring et. al, Physica D (2009) kxkxkxkx kykykyky 0 0 n(k x,k y ) χ (3) xyz kxkxkxkx kykykyky 0 0 n(k x,k y )

core Optical cladding n(r)r n0n0n0n0 n1n1n1n1 Core refractive index cœur Wave turbulence in Optics Optical wave condensation

V(r) ra V0V0V0V0 Wave turbulence in Optics Optical wave condensation Some theoretical works -Condensation and thermalization of classical optical waves in a waveguide P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi Phys. Rev. A 83, (2011) -Wave turbulence in Bose-Einstein Condensates Y. Lvov, S. Nazarenko, R. West Physica D 184, 333 (2003) But still an experimental challenge !!

Recent developments around wave turbulence in nonlinear fiber optics Stéphane Randoux 1, Antonio Picozzi 2, Pierre Suret 1 1: Laboratoire de Physique des Lasers, Atomes et Molécules Université de Lille 1, France 2: Institut Carnot de Bourgogne Université de Bourgogne, Dijon, France