John DudleyUniversité de Franche-Comté, Institut FEMTO-ST

CNRS UMR 6174, Besançon, France

Supercontinuum to solitons: extreme nonlinear structures in optics

Goery GentyTampere Universityof TechnologyTampere, Finland

Fréderic DiasENS Cachan FranceUCD Dublin, Ireland

Nail AkhmedievResearch School ofPhysics & 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

19451934

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 noiseamplification 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 overlong and short distances …

We also see importanceof understanding wavecrossing effects

We are considering how muchcan in principle be containedin a 1D NLSE model

(Extreme ocean waves)

Rogue waves as solitons - supercontinuum generation

Rogue waves as solitons - supercontinuum generation

Modeling the supercontinuum requires NLSE with additional terms

Essential physics = NLSE + perturbations

Supercontinuum physics

Linear dispersion SPM, FWM, RamanSelf-steepening

Three main processes

Soliton ejectionRaman – shift to long λRadiation – shift to short λ

Modeling the supercontinuum requires NLSE with additional terms

Essential physics = NLSE + perturbations

Supercontinuum physics

Linear dispersion SPM, FWM, RamanSelf-steepening

Three main processes

Soliton ejectionRaman – shift to long λRadiation – shift to short λ

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. 45 217 (2009)Erkinatalo, Genty, Dudley Eur. Phys J. ST 185 135 (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 filterDeep 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 dynamicvariation 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

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

Whitham, Bespalov-Talanov, Lighthill, Benjamin-Feir (1965-1969)

Akhmediev-Korneev Theor. Math. Phys 69 189 (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 GVDallow the study of quasi-CW MI dynamics

Power-dependence of spectral structure illustratesthree main dynamical regimes

Spontaneous MI sidebands SupercontinuumIntermediate

(breather) regime

Dudley et al Opt. Exp. 17, 21497-21508 (2009)

Breather spectrum explains the “log triangular” wings seen in noise-induced MI

Comparing supercontinuum and analytic breather spectrum

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 form1938-2007

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, 112-114 (2011)

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 breathercollisions

Other systems

Capillary rogue wavesShats et al. PRL (2010)

Financial Rogue WavesYan Comm. Theor. Phys. (2010)

Matter rogue wavesBludov et al. PRA (2010)

Resonant freak microwavesDe Aguiar et al. PLA (2011)

Statistics of filamentationLushnikov et al. OL (2010)Optical turbulence in

a nonlinear optical cavityMontina 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

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 localisationBlue region our experiments the Peregrine regime

•Under induced conditions we enter Peregrine soliton regime