John Dudley Université de Franche-Comté, Institut FEMTO-ST CNRS UMR 6174, Besançon, France Supercontinuum to solitons: extreme nonlinear structures in optics

Goery Genty Tampere University of Technology Tampere, Finland Fréderic Dias ENS Cachan France UCD Dublin, Ireland Nail Akhmediev Research School of Physics & Engineering, ANU, Australia Bertrand Kibler, Christophe Finot, Guy Millot Université de Bourgogne, France Supercontinuum to solitons: extreme nonlinear structures in optics

The analysis of nonlinear guided wave propagation in optics reveals features more commonly associated with oceanographic “extreme events” Challenges – understand the dynamics of the specific events in optics – explore different classes of nonlinear localized wave – can studies in optics really provide insight into ocean waves? Context and introduction Emergence of strongly localized nonlinear structures Long tailed probability distributions i.e. rare events with large impact

1974 Extreme ocean waves Drauper 1995 Rogue Waves are large (~ 30 m) oceanic surface waves that represent statistically-rare wave height outliers Anecdotal evidence finally confirmed through measurements in the 1990s

There is no one unique mechanism for ocean rogue wave formation But an important link with optics is through the (focusing) nonlinear Schrodinger equation that describes nonlinear localization and noise amplification through modulation instability Cubic nonlinearity associated with an intensity-dependent wave speed - nonlinear dispersion relation for deep water waves - consequence of nonlinear refractive index of glass in fibers Extreme ocean waves NLSE

Ocean waves can be one-dimensional over long and short distances … We also see importance of understanding wave crossing effects We are considering how much can in principle be contained in a 1D NLSE model (Extreme ocean waves)

Rogue waves as solitons - supercontinuum generation

Modeling the supercontinuum requires NLSE with additional terms Essential physics = NLSE + perturbations Supercontinuum physics Linear dispersionSPM, FWM, RamanSelf-steepening Three main processes Soliton ejection Raman – shift to long Radiation – shift to short 

Modeling the supercontinuum requires NLSE with additional terms Essential physics = NLSE + perturbations Supercontinuum physics Linear dispersionSPM, FWM, RamanSelf-steepening Three main processes Soliton ejection Raman – shift to long Radiation – shift to short 

With long (> 200 fs) pulses or noise, the supercontinuum exhibits dramatic shot-to-shot fluctuations underneath an apparently smooth spectrum Spectral instabilities 835 nm,  150 fs 10 kW, 10 cm Stochastic simulations 5 individual realisations (different noise seeds) Successive pulses from a laser pulse train generate significantly different spectra Laser repetition rates are MHz - GHz We measure an artificially smooth spectrum

Spectral instabilities Stochastic simulations Schematic Time Series Histograms Initial “optical rogue wave” paper detected these spectral fluctuations

Dynamics of “rogue” and “median” events is different Differences between “median” and “rogue” evolution dynamics are clear when one examines the propagation characteristics numerically

Dynamics of “rogue” and “median” events is different Dudley, Genty, Eggleton Opt. Express 16, 3644 (2008) ; Lafargue, Dudley et al. Electronics Lett (2009) Erkinatalo, Genty, Dudley Eur. Phys J. ST (2010) Differences between “median” and “rogue” evolution dynamics are clear when one examines the propagation characteristics numerically But the rogue events are only “rogue” in amplitude because of the filter Deep water propagating solitons unlikely in the ocean

More insight from the time-frequency domain pulse gate pulse variable delay gate Spectrogram / short-time Fourier Transform Foing, Likforman, Joffre, Migus IEEE J Quant. Electron 28, 2285 (1992) ; Linden, Giessen, Kuhl Phys Stat. Sol. B 206, 119 (1998) Ultrafast processes are conveniently visualized in the time-frequency domain We intuitively see the dynamic variation in frequency with time

More insight from the time-frequency domain Ultrafast processes are conveniently visualized in the time-frequency domain pulse gate pulse variable delay gate Spectrogram / short-time Fourier Transform Foing, Likforman, Joffre, Migus IEEE J Quant. Electron 28, 2285 (1992) ; Linden, Giessen, Kuhl Phys Stat. Sol. B 206, 119 (1998)

Median event – spectrogram “Median” Event

Rogue event – spectrogram

An Extreme Case of Continuous Interaction L=12 m Energy of Largest Pulse Temporal Profile (periodic window) Zakharov et al. “One Dimensional Wave Turbulence,” Physics Reports (2004) The Champion Soliton

An Extreme Case of Continuous Interaction L=12 m Energy of Largest Pulse Temporal Profile (periodic window) Survival of the Fittest (1864) Winner takes it All (1980)

The extreme frequency shifting of solitons unlikely to have oceanic equivalent BUT... dynamics of localization and collision is common to any NLSE system What can we conclude? MI

Early stage localization The initial stage of breakup arises from modulation instability (MI) A periodic perturbation on a plane wave is amplified with nonlinear transfer of energy from the background MI was later linked to exact dynamical breather solutions to the NLSE Akhmediev & Korneev Theor. Math. Phys. 69, (1986) Whitham, Bespalov-Talanov, Lighthill, Benjamin-Feir ( ) Akhmediev-Korneev Theor. Math. Phys (1986)

Simulating supercontinuum generation from noise sees pulse breakup through MI and formation of Akhmediev breather (AB) pulses Experimental evidence can be seen in the shape of the spectrum Temporal Evolution and Profile : simulation : AB theory Early stage localization

Experiments Spontaneous MI is the initial phase of CW supercontinuum generation 1 ns pulses at 1064 nm with large anomalous GVD allow the study of quasi-CW MI dynamics Power-dependence of spectral structure illustrates three main dynamical regimes Spontaneous MI sidebands Supercontinuum Intermediate (breather) regime Dudley et al Opt. Exp. 17, (2009)

Breather spectrum explains the “log triangular” wings seen in noise-induced MI Comparing supercontinuum and analytic breather spectrum

Observing an unobservable soliton

The Peregrine Soliton Particular limit of the Akhmediev Breather in the limit of a  1/2 The breather breathes once, growing over a single growth-return cycle and having maximum contrast between peak and background Emergence “from nowhere” of a steep wave spike Polynomial form

Two closely spaced lasers generate a low amplitude beat signal that evolves following the expected analytic evolution By adjusting the modulation frequency we can approach the Peregrine soliton Under induced conditions we excite the Peregrine soliton

Experiments can reach a = 0.45, and the key aspects of the Peregrine soliton are observed – non zero background and phase jump in the wings Temporal localisation Nature Physics 6, 790–795 (2010) ; Optics Letters 36, (2011)

(Optics returns the favor to hydrodynamics) The first soliton was observed as the “wave of translation” by Russell (1834) We have confirmed in optics the existence of a soliton whose prediction was made in hydrodynamics but never observed on the surface of water

Spectral dynamics Signal to noise ratio allows measurements of a large number of modes

Collisions in the MI-phase can also lead to localized field enhancement Such collisions lead to extended tails in the probability distributions Controlled collision experiments suggest experimental observation may be possible through enhanced dispersive wave radiation generation Early-stage collisions Time Distance Single breather 2 breather collisions 3 breather collisions

Other systems Capillary rogue waves Shats et al. PRL (2010) Financial Rogue Waves Yan Comm. Theor. Phys. (2010) Matter rogue waves Bludov et al. PRA (2010) Resonant freak microwaves De Aguiar et al. PLA (2011) Statistics of filamentation Lushnikov et al. OL (2010) Optical turbulence in a nonlinear optical cavity Montina et al. PRL (2009)

Analysis of nonlinear guided wave propagation in optics reveals features more commonly associated with oceanographic “extreme events” Solitons on the long wavelength edge of a supercontinuum have been termed “optical rogue waves” but are unlikely to have an oceanographic counterpart The soliton propagation dynamics nonetheless reveal the importance of collisions, but can we identify the champion soliton in advance? Studying the emergence of solitons from initial MI has led to a re-appreciation of earlier studies of analytic breathers Spontaneous spectra, Peregrine soliton, sideband evolution etc Many links with other systems governed by NLSE dynamics Challenges – understand the dynamics of the specific events in optics – explore different classes of nonlinear localized wave – can studies in optics really provide insight into ocean waves? Conclusions and Challenges

Tsunami vs Rogue Wave Tsunami Rogue Wave

Tsunami vs Rogue Wave Tsunami Rogue Wave

Real interdisciplinary interest

Without cutting the fiber we can study the longitudinal localisation by changing effective nonlinear length Characterized in terms of the autocorrelation function Longitudinal localisation

Localisation properties can be readily examined in experiments as a function of frequency a Define localisation measures in terms of temporal width to period and longitudinal width to period Temporal Longitudinal determined numerically More on localisation

Localisation properties as a function of frequency a can be readily examined in experiments Define localisation measures in terms of temporal width to period and longitudinal width to period Temporal Spatial Spatio- temporal Under induced conditions we enter Peregrine soliton regime

Localisation properties as a function of frequency a can be readily examined in experiments Define localisation measures in terms of temporal width to period and longitudinal width to period Temporal Spatial Spatio- temporal Red region corresponds to previous experiments – weak localisation Blue region – our experiments – the Peregrine regime Under induced conditions we enter Peregrine soliton regime