Supercontinuum to solitons:
extreme nonlinear structures in optics
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
Nail Akhmediev
Research School of
Physics & Engineering,
ANU , Australia
Fréderic Dias
ENS Cachan France
UCD Dublin, Ireland
Bertrand Kibler,
Christophe Finot,
Guy Millot
Université de
Bourgogne, France
Context and introduction
The analysis of nonlinear guided wave propagation in optics reveals features
more commonly associated with oceanographic “extreme events”
• Emergence of strongly localized
nonlinear structures
• Long tailed probability distributions
i.e. rare events with large impact
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?
Extreme ocean waves
Rogue Waves are large (~ 30 m) oceanic surface waves that represent
statistically-rare wave height outliers
1934
1945
1974
Anecdotal evidence finally confirmed through measurements in the 1990s
Drauper 1995
Extreme ocean waves
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
NLSE
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)
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
Rogue waves as solitons - supercontinuum generation
Rogue waves as solitons - supercontinuum generation
Supercontinuum physics
Modeling the supercontinuum requires NLSE with additional terms
Linear dispersion
Self-steepening
SPM, FWM, Raman
Essential physics = NLSE + perturbations
Three main processes
Soliton ejection
Raman – shift to long l
Radiation – shift to short l
Supercontinuum physics
Modeling the supercontinuum requires NLSE with additional terms
Linear dispersion
Self-steepening
SPM, FWM, Raman
Essential physics = NLSE + perturbations
Three main processes
Soliton ejection
Raman – shift to long l
Radiation – shift to short l
Spectral instabilities
With long (> 200 fs) pulses or noise, the supercontinuum exhibits dramatic
shot-to-shot fluctuations underneath an apparently smooth spectrum
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
835 nm, 150 fs 10 kW, 10 cm
Spectral instabilities
Initial “optical rogue wave” paper detected these spectral fluctuations
Schematic
Stochastic simulations
Time Series
Histograms
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
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
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)
More insight from the time-frequency domain
Ultrafast processes are conveniently visualized in the time-frequency domain
Spectrogram / short-time Fourier Transform
•gate
•pulse
pulse variable delay gate
We intuitively see the dynamic
variation in frequency with time
Foing, Likforman, Joffre, Migus IEEE J Quant. Electron 28 , 2285 (1992) ; Linden, Giessen, Kuhl Phys Stat. Sol. B 206, 119 (1998)
More insight from the time-frequency domain
Ultrafast processes are conveniently visualized in the time-frequency domain
Spectrogram / short-time Fourier Transform
•gate
•pulse
pulse variable delay gate
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
What can we conclude?
The extreme frequency shifting of solitons unlikely to have oceanic equivalent
BUT ... dynamics of localization and collision is common to any NLSE system
MI
Early stage localization
The initial stage of breakup arises from modulation instability (MI)
Whitham, Bespalov-Talanov, Lighthill, Benjamin-Feir (1965-1969)
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 69 189 (1986)
Early stage localization
Simulating supercontinuum generation from noise sees pulse breakup
through MI and formation of Akhmediev breather (AB) pulses
Temporal Evolution and Profile
: simulation
------ : AB theory
Experimental evidence can be seen in the shape of the spectrum
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
Intermediate
(breather) regime
Supercontinuum
Dudley et al Opt. Exp. 17, 21497-21508 (2009)
Comparing supercontinuum and analytic breather spectrum
Breather spectrum explains the “log triangular” wings seen in noise-induced MI
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
1938
-2007
Under induced conditions we excite the Peregrine soliton
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
Temporal localisation
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
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
Early-stage collisions
Collisions in the MI-phase can also lead to localized field enhancement
3 breather
collisions
2 breather collisions
Single breather
Distance
Time
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
Other systems
Optical turbulence in
a nonlinear optical cavity
Montina et al. PRL (2009)
Matter rogue waves
Bludov et al. PRA (2010)
Statistics of filamentation
Lushnikov et al. OL (2010)
Capillary rogue waves
Shats et al. PRL (2010)
Financial Rogue Waves
Yan Comm. Theor. Phys. (2010)
Resonant freak microwaves
De Aguiar et al. PLA (2011)
Conclusions and Challenges
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
Tsunami vs Rogue Wave
Tsunami
Rogue Wave
Tsunami vs Rogue Wave
Tsunami
Rogue Wave
Real interdisciplinary interest
Longitudinal localisation
Without cutting the fiber we can study the longitudinal localisation by
changing effective nonlinear length
Characterized in terms of the autocorrelation function
•More on 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
•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
temporal
•
Spatial
Spatio-
•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