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