Supercontinuum to solitons:

frequentverseUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

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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

1945

1934

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

Rogue waves as solitons
-

supercontinuum generation

Modeling the supercontinuum requires NLSE with additional terms









Essential physics = NLSE + perturbations

Supercontinuum physics

Linear dispersion

SPM, FWM, Raman

Self
-
steepening




Three main processes





Soliton ejection



Raman



shift to long
l




Radiation


shift to short

l

Modeling the supercontinuum requires NLSE with additional terms









Essential physics = NLSE + perturbations

Supercontinuum physics

Linear dispersion

SPM, FWM, Raman

Self
-
steepening




Three main processes





Soliton ejection



Raman



shift to long
l




Radiation


shift to short

l

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 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

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
, 1089
-
1093 (1986)



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 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,

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 form













1938

-
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 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