A short tutorial on optical rogue waves

Institut FEMTO-ST CNRS-Université de Franche-Comté Besançon, France

Experiments in collaboration with the group of Guy Millot Institut Carnot de Bourgogne (ICB) CNRS-Université de Bourgogne, Dijon, France

John M Dudley

Large ocean waves that appear in an otherwise calm sea

• Large (~ 30 m) surface waves that represent statistical outliers

1974

1945

1995

• Measurements in 1990’s have established long-tailed statistics

C. Kherif et al. Rogue Waves in the Ocean, Springer (2009)

Oceanic rogue waves

The long tailed statistics reflects the unexpected nature of large waves

Rogue waves are large and unexpected

The study of oceanic rogue waves was recognized as an important field of study, requiring new research into the ways propagating wave groups on the ocean surface can attain states of high localization Studying rogue waves in their natural environment is problematic A 2007 Nature paper made a bold proposal that analogous effects could in fact be observed in optical fiber waveguides

The 2008 scientific context

• Reliable techniques for fabricating small-core waveguides allows tailored linear guidance (dispersion) and controlled nonlinear interactions

The birth of nonlinear fiber optics

The link with light – extreme nonlinear propagation

There was immediate interest and impact

An octave-spanning spectrum allows pump carrier-envelope phase and the comb position to be readily stabilized

Pulsed lasers generate a frequency ruler or frequency comb

Example: precision spectroscopy

Molecular fingerprinting S. Diddams et al. Nature 445, 627 (2007) Human breath analysis M. J. Thorpe et al. Opt. Express 16, 2387 (2008)

Who would have predicted this ?

Example: planetary discovery

Periodic Doppler shift of stellar spectral lines is perturbed by planetary motion

The link with light – extreme nonlinear propagation

Numerical Model

Modelling reveals that the supercontinuum can be highly unstable

5 individual realisations, identical apart from quantum noise

Successive pulses from a laser pulse train generate significantly different spectra

We measure an artificially smooth spectrum, but the noise is still present

Stochastic simulations

J. M. Dudley, G. Genty, S. Coen, Rev. Mod. Phys. 78 1135 (2006)

Noisy supercontinuum spectra are also interesting

Experiments reveal that these instabilities yield long-tailed statistics

Stochastic simulations

Time series Histogram

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Experiments are always better than theory …

Time series Histogram

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These rare soliton events are optical rogue waves

Experiments reveal that these instabilities yield long-tailed statistics

Deep water ocean wave groups and ultrashort envelopes in optical fibres are both described by the same propagation equation

• Ocean waves can be 1D over large scales

• Nonlinear Schrödinger equation (NLSE)

• Optical and water waves have same nonlinearity – speed depends on intensity

Origin of the optical-ocean analogy

A is surface elevation of wave group

Analogy is valid for moderate nonlinear strengths before wave breaking

Wave breaking imposes a limit

/ 7H λ<

Insight from the time-frequency domain

pulse

gate pulse variable delay gate

Spectrogram / short-time Fourier Transform

The time-frequency domain allows convenient visualisation of complex wave envelope dynamics in optics

Gabor and the time-frequency domain

J. IEE (London), 93(III):429-457 (1946)

Clarification of the rogue wave mechanism We see the emergence of localized soliton envelopes emerging from low amplitude noise on a longer input pulse

5 ps, 100 W peak power, typical supercontinuum with 1 µm zero dispersion fiber

Clarification of the rogue wave mechanism

5 ps, 100 W peak power, typical supercontinuum with 1 µm zero dispersion fiber

Identical parameters except for different quantum noise

Solitary Waves Periodic Explode-Decay Solitons or Breathers

The NLSE admits other families of soliton

Pulses on a zero background

Energy exchange between localised peaks and a background

Optical technology enables experiments in “optical hydrodynamics”

Experiments

Where do the waves come from?

The initial phase of propagation of an optical supercontinuum shows the appearance of these localized breather states

Frequency

Time

Spectral structure agrees with theory

The spectral wings seen in experiment correspond to the theoretical prediction for the shortest temporal pulses

Kibler et al. Nature Phys. 6 790 (2010)

Hammani et al Opt. Lett. 36 112 (2011) Rogue Waves in a Water Tank Chabchoub et al. Phys. Rev. Lett. 106 204502 (2011)

The Peregrine soliton is an explode-decay rogue wave prototype, measured in optics 20 years after its prediction in hydrodynamics

Specific forms of rogue waves can also be stimulated

New analysis of an old instability Wetzel et al. SPIE Newsroom (2011)

Raw data

The Peregrine soliton in nonlinear fibre optics Nature Physics 6 790 (2010)

The Peregrine soliton in a standard telecommunication fiber Optics Letters 36, 112 (2011)

Optics in 2011

Optical technology enables experiments in “optical hydrodynamics”

Rogue waves can split into self-similar replicas

Experiments

Erkintalo, Genty, Kibler et al. Phys Rev Lett 107 253901 (2011)

Rogue waves can split into self-similar replicas

Experiments

Erkintalo, Genty, Kibler et al. Phys Rev Lett 107 253901 (2011)

Confirms Sears et al Phys. Rev. Lett. 84 1902 (2000)

Why is the control of optical rogue waves interesting?

Essential Conclusions

Optical fiber propagation shows noise properties qualitatively similar to those seen in the study of wave propagation on deep water The coherent structures that can be excited from specific initial conditions such as the Peregrine soliton can be seen in optics and hydrodynamics The goals of MULTIWAVE are to explore this analogy in detail