A short tutorial on optical rogue waves John M Dudley 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
Oceanic rogue waves Large ocean waves that appear in an otherwise calm sea Large (~ 30 m) surface waves that represent statistical outliers Measurements in 1990 s have established long-tailed statistics 1995 1945 1974 C. Kharif et al. Rogue Waves in the Ocean, Springer (2009)
The 2008 scientific context 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 birth of nonlinear fiber optics Reliable techniques for fabricating small-core waveguides allows tailored linear guidance (dispersion) and controlled nonlinear interactions
The link with light extreme nonlinear propagation
The link with light extreme nonlinear propagation Numerical Model
Stable clocks Low noise supercontinuum generation allows the stabilisation of the carrier oscillations underneath a femtosecond laser pulse History of Clocks There is much interest in understanding these optical instabilities
Origin of the optical-ocean analogy 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) A is surface elevation of wave group Optical and water waves have same nonlinearity speed depends on intensity
Noisy supercontinuum spectra are also interesting Modelling reveals that the supercontinuum can be highly unstable Stochastic simulations 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 J. M. Dudley, G. Genty, S. Coen, Rev. Mod. Phys. 78 1135 (2006)
Experiments are always better than theory Experiments reveal that these instabilities yield long-tailed statistics Stochastic simulations Time series Histogram Power Frequency Time Power
These rare soliton events are optical rogue waves Experiments reveal that these instabilities yield long-tailed statistics Time series Histogram Power Frequency Time Power
Insight from the time-frequency domain The time-frequency domain allows convenient visualisation of complex wave envelope dynamics in optics Spectrogram / short-time Fourier Transform gate pulse variable delay gate pulse
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 Identical parameters except for different quantum noise 5 ps, 100 W peak power, typical supercontinuum with 1 µm zero dispersion fiber
Turbulence and «Champion Solitons» We have identified important links with turbulence theory Emergence of a champion Collisions and turbulence in optical rogue wave formation Phys. Lett. A 374 989-996 (2010) Rogue waves, rational solitons and wave turbulence theory Phys. Lett. A 375, 3149-3155 (2011)
What can we conclude? Inelastic collisions lead to the emergence of a champion soliton This clarifies the origin of the supercontinuum rogue waves Solitons can be observed on deep water but there have been no systematic observations in the natural environment The role of this class of soliton as an ocean rogue wave candidate remains an open question
The NLSE admits other families of soliton Solitary Waves Pulses on a zero background Periodic Explode-Decay Solitons or Breathers Energy exchange between localised peaks and a background
What about the emergence phase? The initial phase of propagation of an optical supercontinuum shows the appearance of these localized breather states Spontaneous MI sidebands Intermediate (breather) regime Supercontinuum
Experimental confirmation of breather solutions Analytic predictions for the spectrum are confirmed by experiments Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation Optics Express 17, 21497 (2009)
Exciting the Peregrine Soliton Optical technology enables experiments in optical hydrodynamics
Exciting the Peregrine Soliton Optical technology enables experiments in optical hydrodynamics 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
Raw data Optical technology enables experiments in optical hydrodynamics 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
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 Confirms Sears et al Phys. Rev. Lett. 84 1902 (2000) Erkintalo, Genty, Kibler et al. Phys Rev Lett 107 253901 (2011)
Essential Conclusions Optical fiber propagation shows noise properties qualitatively similar to those seen in the study of wave propagation on deep water The solitons of the white light supercontinuum in optics may be present in deep water but there is not clear experimental evidence 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