John Dudley

1
Supercontinuum to solitons extreme nonlinear
structures in optics
John Dudley Université de Franche-Comté,
Institut FEMTO-ST CNRS UMR 6174, Besançon, France
2
Supercontinuum to solitons extreme nonlinear
structures in optics
Goery Genty Tampere University of
Technology Tampere, Finland
Fréderic Dias ENS Cachan France UCD Dublin,
Ireland
Bertrand Kibler, Christophe Finot,Guy Millot
Université de Bourgogne, France
Nail Akhmediev Research School of Physics
Engineering, ANU , Australia
3
Context and introduction

• 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?

• Emergence of strongly localized nonlinear
  structures
• Long tailed probability distributions i.e.
  rare events with large impact

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Extreme ocean waves

• 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

1934
1974
1945
Drauper 1995
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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
  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

NLSE
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(Extreme ocean waves)

• 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

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Rogue waves as solitons - supercontinuum
generation
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Rogue waves as solitons - supercontinuum
generation
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Supercontinuum physics

• Modeling the supercontinuum requires NLSE with
  additional terms
• Essential physics NLSE perturbations

Linear dispersion
SPM, FWM, Raman
Self-steepening
Three main processes Soliton
ejection Raman shift to long l Radiation
shift to short l
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Supercontinuum physics

• Modeling the supercontinuum requires NLSE with
  additional terms
• Essential physics NLSE perturbations

Linear dispersion
SPM, FWM, Raman
Self-steepening
Three main processes Soliton
ejection Raman shift to long l Radiation
shift to short l
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Spectral instabilities

• With long (gt 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
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Spectral instabilities

• Initial optical rogue wave paper detected these
  spectral fluctuations

Schematic
Stochastic simulations
Time Series
Histograms
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Dynamics of rogue and median events is
different

• Differences between median and rogue
  evolution dynamics are clear when one examines
  the propagation characteristics numerically

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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)

15
More insight from the time-frequency domain

• Ultrafast processes are conveniently visualized
  in the time-frequency domain
• We intuitively see the dynamicvariation in
  frequency with time

Spectrogram / short-time Fourier Transform
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)

16
More insight from the time-frequency domain

• Ultrafast processes are conveniently visualized
  in the time-frequency domain

Spectrogram / short-time Fourier Transform
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)

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Median event spectrogram

• Median Event

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Rogue event spectrogram
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An Extreme Case of Continuous Interaction

• Temporal Profile (periodic window)

• Energy of Largest Pulse

• L12 m

The Champion Soliton

• Zakharov et al. One Dimensional Wave
  Turbulence,
• Physics Reports 398 1-65 (2004)

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An Extreme Case of Continuous Interaction

• Temporal Profile (periodic window)

• Energy of Largest Pulse

• L12 m

• Survival of the Fittest
  (1864) Winner takes it All (1980)

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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
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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)

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Early stage localization

• 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

simulation ------ AB theory
Temporal Evolution and Profile
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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
Intermediate (breather) regime
Supercontinuum

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

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Comparing supercontinuum and analytic breather
spectrum

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

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Observing an unobservable soliton
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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
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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

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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 , 790795 (2010) Optics
Letters 36, 112-114 (2011)
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(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

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Spectral dynamics

• Signal to noise ratio allows measurements of a
  large number of modes

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Early-stage collisions

• 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

3 breathercollisions
2 breather collisions
Single breather
Distance
Time
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Other systems

• Statistics of filamentation
• Lushnikov et al. OL (2010)

• Capillary rogue waves
• Shats et al. PRL (2010)

Optical turbulence in a nonlinear optical
cavity Montina et al. PRL (2009)

• Matter rogue waves
• Bludov et al. PRA (2010)

Resonant freak microwaves De Aguiar et al. PLA
(2011)

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

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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

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Tsunami vs Rogue Wave
Tsunami
Rogue Wave
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Tsunami vs Rogue Wave
Tsunami
Rogue Wave
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Real interdisciplinary interest
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Longitudinal localisation

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

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• 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

40

• 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

41

• 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