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Supercontinuum generation in optical fibers: 40 years of nonlinear optics in

one experiment

Goëry Genty

Optics Laboratory, Tampere University of Technology, Finland

TAMPERE UNIVERSITY OF TECHNOLOGYInstitute of Physics

Part I: back to basics

● Optical fibers for supercontinuum generation

● Ultrafast pulse propagation and supercontinuum

generation ● Visualizing supercontinuum dynamics

● Coherence properties

Part II: recent developments

● New fibers, new materials and new spectral regions

● Supercontinuum instabilities & optical rogue waves

● Bridging the gap between traditional nonlinear optics and supercontinuum generation

● Future steps and challenges

A historical note

Supercontinuum timeline Supercontinuum generation in bulk silica (R. Alfano)

1980 1990 2000

Nonlinearity in silica fiber (R. Stolen & Lin)

Experimental Fiber solitons (L. Mollenauer)

1970

Femtosecond Ti:Sapphire (Sibbett)

Solid core photonic crystal fiber (P. Russell)

Supercontinuum generation in photonic crystal fiber (J. Ranka et al.)

Nobel prize Hall & Hänsch

Today

Supercontinuum generation in silica fiber (R. Stolen & Lin)

Photonics@be

2005

The esthetic slide

Why the excitement?

Supercontinuum has been around for decades

There are some issues though

● Limitations - Walk off - Diffraction - Strong dispersion, limited broadening

● Need very high energy

- Damage

Optical fibers are more convenient

● Pulse propagation in optical fibers – No diffraction, long interaction length

● Dispersion/nonlinearity can be controlled: crucial! – propagation dynamics depend on pump wavelength relative to

fiber zero dispersion wavelength (ZDW)

Small core, high doping Photonic crystal fibers Tapered fibers Non-silica materials

“NEW” FIBERS

Dispersion

nonlinearity

Confinement

PARAMETER CONTROL APPLICATIONS

Supercontinuum Frequency conversion Pulse compression Regeneration Amplification

Optical fibers for nonlinear applications ● Chemistry and geometry both contribute to the refractive

index profile – Determine modal confinement (nonlinearity) and dispersion

characteristics – Wide range of possibilities

Photonic crystal fibers

● Generally single material with a high air-fill fraction

● ZDW displaced to shorter wavelengths: wavelength of

high-power short pulse sources ● Large nonlinearity (x100 compared to standard fibers)!

⇒ nonlinear effects dramatically enhanced

Taper/microfibers

● Tapering standard fibers: similar properties to photonic crystal fibers

Key aspects for supercontinuum

Micro/nano- structured waveguides: engineered nonlinearity and dispersion

● Dispersion shifts soliton regime to shorter wavelengths ● Long interaction lengths ● Light tightly confined very large intensities!

Pulse propagation in optical fibers

E(r, z, t) = F(r)A(z, t)eiω0t

F(r) modal distributionA(z,t) time variation

ω0 =2πcλ0

carrier frequency

!

"

###

$

###

t

A(z,t)

● Modal distribution does not vary with propagation ● Consider only the fundamental mode LP01 (Gaussian

distribution)

● Only temporal effects matters, no diffraction

Nonlinear pulse propagation in optical fibers

),(),(),(with

),(),(1),( 2

2

02

2

2

trPtrPtrPttrP

ttrE

ctrE

NLL +=∂

∂−=

∂

∂+×∇×∇ µ

...),(),(),(),( 3)3(0

2)2(0

)1(0

)0(0 ++++= trEtrEtrEtrP χεχεχεχε

● Pulse propagation in optical fiber obeys :

● In silica: χ(2)=0 (centro-symmetric material)

● Only THIRD-ORDER nonlinear effects

Dispersion Nonlinearity

The Kerr effect

● Light intensity changes the refractive index: n = n0+ n2I = n0 + (n2/Aeff)P

– n2 proportional to χ(3) : nonlinear refractive index

– Aeff effective area of the fiber mode

● Important parameter: nonlinear coefficient γ = n2ω/cAeff – Nonlinear effects scales with γ$

● In silica, n2≈ 3×10-20 m2/W: weak nonlinear material! But

silica has the lowest losses…

● Need small Aeff

Nonlinear effects in optical fibers

● Pump wavelength is crucial!

● Generalized nonlinear Schrödinger Equation (GNLSE)

● Can include noise, frequency-dependent mode area,

polarization…etc

● Validity: down to single cycle regime

Self-Steepening term SPM, FWM, Raman

Intensity-dependent ref. index

Raman response

Dispersion

Modeling pulse propagation

Loss

● Generalized nonlinear Schrödinger Equation (GNLSE)

● Can include noise, frequency-dependent mode area,

polarization…etc

● Validity: down to single cycle regime

Self-Steepening term SPM, FWM, Raman

Intensity-dependent ref. index

Raman response

Dispersion

Modeling pulse propagation

Loss

Nonlinear envelope equation modeling ofsub-cycle dynamics and harmonicgeneration in nonlinear waveguides

G. GentyHelsinki University of Technology, Metrology Research Institute, FIN-02015 HUT, Finland

P. KinslerImperial College, Blackett Laboratory, Imperial College London, SW7 2BW, United Kingdom.

B. Kibler and J. M. DudleyInstitut FEMTO-ST, Department of Optics, Universite de Franche-Comte, Besancon, France.

Abstract: We describe generalized nonlinear envelope equation modelingof sub-cycle dynamics on the underlying electric field carrier duringone-dimensional propagation in fused silica. Generalized envelope equationsimulations are shown to be in excellent quantitative agreement with thenumerical integration of Maxwell’s equations, even in the presence ofshock dynamics and carrier steepening on a sub-50 attosecond timescale.In addition, by separating the effects of self-phase modulation and thirdharmonic generation, we examine the relative contribution of these effectsin supercontinuum generation in fused silica nanowire waveguides.

© 2007 Optical Society of AmericaOCIS codes: 060.7140 Ultrafast processes in fibers, 190.7110 Ultrafast nonlinear optics

References and links1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).

2. M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycledurations in photonic nanowires,” Opt. Express 13, 6848–6855 (2005).

3. V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys.Rev. A 62, 011804(R) (2000).

4. N. Karasawa, “Computer simulations of nonlinear propagation of an optical pulse using a finite-difference in thefrequency-domain method,” IEEE J. Quantum Electron. 38, 626–629 (2002).

5. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,”IEEE J. Quantum Electron. 25, 2665–2673 (1989).

6. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78,3282–3285 (1997).

7. N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Compari-son between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulsesin a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).

8. P. Kinsler and G.H.C. New, “ Wideband pulse propagation: single-field and multi-field approaches to Ramaninteractions,”Phys.Rev. A 72, 033804 (2005).

9. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photoniccrystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).

10. A. V. Husakou and J. Herrmann, “Supercontinuum generation, four-wave mixing, and fission of higher-ordersolitons in photonic-crystal fibers,” J. Opt. Soc. Am. B 19, 2171–2182 (2002).

11. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirec-tional equations,” Phys. Rev. E 70, 036604 (2004).

#79783 - $15.00 USD Received 5 Feb 2007; accepted 2 Apr 2007; published 18 Apr 2007(C) 2007 OSA 30 Apr 2007 / Vol. 15, No. 9 / OPTICS EXPRESS 5382

● Generalized nonlinear Schrödinger Equation (GNLSE)

● Important considerations – window size (avoid wrapping around) – temporal/spectral resolution – step size (FWM artifacts) – DO NOT use linear Raman model – Disp. coeff. must be changed if you change the pump wavelength

Modeling pulse propagation

● Good agreement with experimental results

● Success in modeling: physics behind supercontinuum generation now well-understood

Modeling supercontinuum generation

Supercontinuum generation is easy!

But its dynamics can be rather complex…

● 30 fs, 10 kW input pulses

● Evolution can be divided in 3 stages

– Initial higher-order soliton compression (spectral broadening), – soliton fission and dispersive wave generation – Raman self-frequency shift

Deconstructing supercontinuum dynamics

● Evolution can be divided in 3 stages

– Initial higher-order soliton compression (spectral broadening), – soliton fission and dispersive wave generation – Raman self-frequency shift

Better in color Spectral Evolution Temporal Evolution

SOLITON FISSION

DISPERSIVE WAVE BACKGROUND

DIS

PER

SIVE

WAV

E ED

GE

RA

MA

N S

ELF

FREQ

UEN

CY

SHIF

T SOLITONS

● Understanding the complex dynamics: study separately the different stages

● Soliton propagation dynamics – Fundamental solitons – Higher-order solitons – Effect of perturbation: higher-order dispersion and Raman scattering

Deconstructing supercontinuum dynamics

● Fundamental soliton is invariant upon propagation (except for a constant nonlinear phase-shift)

Fundamental soliton

Frequency (THz) Time (ps)

Spectrum evolution Time evolution

z/z s

ol

z/z s

ol

( ) 0 00, sech( / )A z T P T T= = ( ) 0 0, sech( / ) j zA z T P T T e= solk

γP0/2 Soliton number 2

0 0 2/ / | | 1d nlN L L PTγ β= = =

dB

Requirements

Higher-order solitons

dB

z/z s

ol

z/z s

ol

● Higher-order soliton is periodic upon propagation A(z+zsol,T) = A(z,T)

( ) 0 00, sech( / )A z T P T T= = 2

sol 0 2/ | |2 2dz L Tπ π

β= =

Soliton period 20 0 2/ / | | 2,3,4...d nlN L L PTγ β= = =

Quantized!

Frequency (THz) Time (ps)

Spectrum evolution Time evolution

Requirements

N = 2

Higher-order solitons

dB

z/z s

ol

z/z s

ol

● Higher-order soliton is periodic upon propagation A(z+zsol,T) = A(z,T)

( ) 0 00, sech( / )A z T P T T= = 2

sol 0 2/ | |2 2dz L Tπ π

β= =

Soliton period 20 0 2/ / | | 2,3,4...d nlN L L PTγ β= = =

Quantized!

Frequency (THz) Time (ps)

Spectrum evolution Time evolution

Requirements

N = 3

Cutter in 3D

Perturbations of solitons

● Solitons (fundamental/higher-order) – Solutions of the pure nonlinear Schrödinger equation (only GVD

and Kerr nonlinearity)

● Higher-order dispersion and Raman scattering perturb the ideal evolution of solitons – Soliton self-frequency shift – Dispersive wave generation – Soliton fission

∂A∂z

+ i β22∂2A∂T 2 = iγ A

2 A

∂A∂z

+ i β22∂2A∂T 2 −

β36∂3A∂T 3 +... = iγ (1− fR ) A

2 A+ fRA hR ∗ A2( )

HOD Raman

Raman perturbation of N=1 soliton ● Fundamental soliton propagating in the presence of

stimulated Raman scattering

● Frequency of soliton shifts towards lower frequencies (longer

wavelengths) with propagation = soliton SELF-frequency shift

Frequency (THz) -30 -13.2 0

Sees gain

Pump

ν0!

Frequency shifts scales as T0

-4

● Frequency of soliton shifts towards lower frequencies (longer wavelengths) with propagation = soliton SELF-frequency shift

● Parabolic/linear trajectory in the time/frequency domain

Raman perturbation of N=1 soliton

HOD perturbation of N=1 soliton

Wavelength

Dis

pers

ion

ZDW

Anomalous Normal

● Dispersion depends on wavelength/frequency

● Soliton near the zero-dispersion wavelength is strongly perturbed: part of its spectrum extends in the normal dispersion regime!

Soliton allowed

Soliton forbidden

● Higher-order dispersion: fundamental soliton sheds energy as a dispersive wave

● Dispersive wave located in the normal dispersion regime – phase-matching condition: phase(soliton) = phase(disp. wave)

sol sol 1 sol 0 DW DW 1 sol( ) ( ) / 2 ( ) ( )Pβ ω ω β ω γ β ω ω β ω− + = −

HOD perturbation of N=1 soliton Δϕ

0

Spe

ctru

m

Ω=ω -ωsol

Dispersive Wave

N =1 Soliton

ANOMALOUS DISPERSION

NORMAL DISPERSION

2

0 322

3

3 | |3Pγ ββ

β βΩ ≈ +

Ω!2 33 02 ...

2 6 2Pβ γβ

Ω + Ω + =

HOD perturbation of N=1 soliton

● Higher-order dispersion: fundamental soliton sheds energy as a dispersive wave

● Dispersive wave located in the normal dispersion regime – phase-matching condition: phase(soliton) = phase(disp. Wave)

HOD perturbation of N=1 soliton

● Longer soliton wavelength = shorter dispersive wave wavelength

● Energy of dispersive wave inversely proportional to soliton detuning with respect to ZDW

Long means short…

β 2(λ

)

0

Dis

p. W

ave

Wav

elen

gth

(nm

)

Soliton wavelength (nm)

ANOMALOUS DISPERSION

NORMAL DISPERSION

ZDW

Soliton fission

● Higher-order N-soliton is unstable, sensitive to perturbations N-soliton breaks up into N fundamental solitons

Soliton fission

● Higher-order N-soliton is unstable, sensitive to perturbations N-soliton breaks up into N fundamental solitons

Raman perturbation of N-Soliton

Fission of N-soliton

● Fission length: Lfiss≈Ld/N=N Lnl

● Soliton fission + dispersive wave radiation + soliton self-frequency shift = supercontinuum generation

Let’s put everything together

● Red shifting soliton interact with the dispersive wave through XPM Dispersive wave “trapping” which is blue-shifted

Increasing the bandwidth

● Continuously redshifting soliton induces a continuous blueshift of the dispersive wave

Dispersive wave trapping

● Solitons only exist in anomalous dispersion regime: different dynamics

● Self-phase modulation, four-wave mixing

Normal dispersion regime

● Initial stages of supercontinuum generation strongly depends on pump pulses duration

● Short pulses (typically T0≤150 fs): soliton dynamics

● Long pulses: modulation instability (four-wave mixing) followed by soliton dynamics

Pump duration

● Initial dynamics: noise-seeded modulation instability

● Random noise “chaotic” field

Supercontinuum in the incoherent regime

Fiber with two zero-dispersion wavelengths

G. Genty et al. Opt. Express 12, 3471-3480 (2004)

Time-frequency analysis

● Ultrafast dynamics can be conveniently visualized in the time-frequency domain

Foing, Likforman, Joffre, Migus IEEE J Quant. Electron 28 , 2285 (1992) ; Linden, Giessen, Kuhl Phys Stat. Sol. B 206, 119 (1998)

Time-frequency analysis

Experimental implementation

One more thing

Reference pulse duration determines the resolution!

€

ΔtΔf ≥1/2

● Time-spectrum representation allows to conveniently identify dynamics

Spectrogram of supercontinuum

Coherence of supercontinuum

● Spectral coherence: “How spectra differs from shot to shot?”

Coherence of supercontinuum

● Spectral coherence: “How spectra differs from shot to shot?”

Coherence of supercontinuum

● BUT supercontinuum is not necessarily coherent!

Short pulse regime Long pulse regime

Anomalous dispersion

- Soliton dynamics

- Dispersive waves

- Modulation instability

- Soliton dynamics

Normal dispersion

- Self-phase modulation

- Four-wave mxing

- Soliton dynamics

- Four-wave mixing

- Raman scattering

- Soliton dynamics

Regimes of coherence

Incoherent

Inco

here

nt

Measuring supercontinuum coherence

tr

Movable Mirror

Fixed Mirror Beam splitter

200 fs

Ti:Sapphire Laser

+ Spectrometer

ω$

Interferogram

Delay tr+τ

● Michelson interferometer

● Spectral interference

● Fringes visibility gives the coherence function

● Fibers with normal dispersion at all wavelengths: ANDI

● Allows for high coherence and stability, flat spectra

● Can reach octave-spanning

Fibers with all normal dispersion

Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal

dispersion photonic crystal fibers Alexander M. Heidt,1,2* Alexander Hartung,2 Gurthwin W. Bosman,1 Patrizia Krok,1

Erich G. Rohwer,1 Heinrich Schwoerer1, and Hartmut Bartelt2 1Laser Research Institute, Physics Department, University of Stellenbosch, Private Bag X1,

Matieland 7604, South Africa 2Institute of Photonic Technology (IPHT), Albert-Einstein-Str. 9, 07745 Jena, Germany

Abstract: We present the first detailed demonstrations of octave-spanning SC generation in all-normal dispersion photonic crystal fibers (ANDi PCF) in the visible and near-infrared spectral regions. The resulting spectral profiles are extremely flat without significant fine structure and with excellent stability and coherence properties. The key benefit of SC generation in ANDi PCF is the conservation of a single ultrashort pulse in the time domain with smooth and recompressible phase distribution. For the first time we confirm the exceptional temporal properties of the generated SC pulses experimentally and demonstrate their applicability in ultrafast transient absorption spectroscopy. The experimental results are in excellent agreement with numerical simulations, which are used to illustrate the SC generation dynamics by self-phase modulation and optical wave breaking. To our knowledge, we present the broadest spectra generated in the normal dispersion regime of an optical fiber. © 2011 Optical Society of America OCIS codes: (320.6629) Supercontinuum generation; (060.5295) Photonic crystal fibers.

References and links 1. J. M. Dudley, and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press,

2010). 2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.

78(4), 1135–1184 (2006). 3. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J.

Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88(17), 173901 (2002).

4. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90(11), 113904 (2003).

5. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).

6. X. Gu, M. Kimmel, A. Shreenath, R. Trebino, J. Dudley, S. Coen, and R. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11(21), 2697–2703 (2003).

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12(6), 1045–1054 (2004).

8. E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thøgersen, and S. R. Keiding, “Broadband multiplex coherent anti- Stokes Raman scattering microscopy employing photonic- crystal fiber,” J. Opt. Soc. Am. B 22(9), 1934–1938 (2005).

9. P. Falk, M. H. Frosz, and O. Bang, “Supercontinuum generation in a photonic crystal fiber with two zero-dispersion wavelengths tapered to normal dispersion at all wavelengths,” Opt. Express 13(19), 7535–7540 (2005).

10. A. M. Heidt, “Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal fibers,” J. Opt. Soc. Am. B 27(3), 550–559 (2010).

#139710 - $15.00 USD Received 16 Dec 2010; revised 24 Jan 2011; accepted 25 Jan 2011; published 11 Feb 2011(C) 2011 OSA 14 February 2011 / Vol. 19, No. 4 / OPTICS EXPRESS 3775

Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal

dispersion photonic crystal fibers Alexander M. Heidt,1,2* Alexander Hartung,2 Gurthwin W. Bosman,1 Patrizia Krok,1

Erich G. Rohwer,1 Heinrich Schwoerer1, and Hartmut Bartelt2 1Laser Research Institute, Physics Department, University of Stellenbosch, Private Bag X1,

Matieland 7604, South Africa 2Institute of Photonic Technology (IPHT), Albert-Einstein-Str. 9, 07745 Jena, Germany

Abstract: We present the first detailed demonstrations of octave-spanning SC generation in all-normal dispersion photonic crystal fibers (ANDi PCF) in the visible and near-infrared spectral regions. The resulting spectral profiles are extremely flat without significant fine structure and with excellent stability and coherence properties. The key benefit of SC generation in ANDi PCF is the conservation of a single ultrashort pulse in the time domain with smooth and recompressible phase distribution. For the first time we confirm the exceptional temporal properties of the generated SC pulses experimentally and demonstrate their applicability in ultrafast transient absorption spectroscopy. The experimental results are in excellent agreement with numerical simulations, which are used to illustrate the SC generation dynamics by self-phase modulation and optical wave breaking. To our knowledge, we present the broadest spectra generated in the normal dispersion regime of an optical fiber. © 2011 Optical Society of America OCIS codes: (320.6629) Supercontinuum generation; (060.5295) Photonic crystal fibers.

References and links 1. J. M. Dudley, and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press,

2010). 2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.

78(4), 1135–1184 (2006). 3. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J.

Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88(17), 173901 (2002).

4. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90(11), 113904 (2003).

5. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).

6. X. Gu, M. Kimmel, A. Shreenath, R. Trebino, J. Dudley, S. Coen, and R. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11(21), 2697–2703 (2003).

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12(6), 1045–1054 (2004).

8. E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thøgersen, and S. R. Keiding, “Broadband multiplex coherent anti- Stokes Raman scattering microscopy employing photonic- crystal fiber,” J. Opt. Soc. Am. B 22(9), 1934–1938 (2005).

9. P. Falk, M. H. Frosz, and O. Bang, “Supercontinuum generation in a photonic crystal fiber with two zero-dispersion wavelengths tapered to normal dispersion at all wavelengths,” Opt. Express 13(19), 7535–7540 (2005).

10. A. M. Heidt, “Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal fibers,” J. Opt. Soc. Am. B 27(3), 550–559 (2010).

#139710 - $15.00 USD Received 16 Dec 2010; revised 24 Jan 2011; accepted 25 Jan 2011; published 11 Feb 2011(C) 2011 OSA 14 February 2011 / Vol. 19, No. 4 / OPTICS EXPRESS 3775

● Use of materials with low losses in the mid-IR

Novel glasses for mid-IR

Fluoride (ZBLAN) fibers (not PCFs)

Swiderski et al. Opt. Express 21, 7851-7857 (2013) Xia et al. Opt. Express 15, 865-871 (2007)

Material ZDW 1.6 microns n2= n2

silica

● Enhanced nonlinearity

● Engineer dispersion

Novel glasses for mid-IR

Tellurite fibers (PCFs) Material ZDW 2.2 microns n2= n2

silicax10-20

Savelii et al. Opt. Express 20, 27083-27093 (2012)

Novel glasses for mid-IR

Tellurite fibers (PCFs) Material ZDW 2.2 microns n2= n2

silicax10-20

Savelii et al. Opt. Express 20, 27083-27093 (2012) Domachuk et al. Opt. Express 16, 7161-7168 (2008)

● Enhanced nonlinearity

● Engineer dispersion

Novel glasses for mid-IR

Sulfide/Chalcogenide fibers (PCFs or not)

Material ZDW 5 microns n2= n2

silicax100

Savelii et al. Opt. Express 20, 27083-27093 (2012)

● Enhanced nonlinearity

● Engineer dispersion

Novel glasses for mid-IR

Sulfide/Chalcogenide fibers (PCFs or not)

Material ZDW 5 microns n2= n2

silicax100

Savelii et al. Opt. Express 20, 27083-27093 (2012)

● Enhanced nonlinearity

● Engineer dispersion

● Many pumping schemes and SC characteristics possible

State-of-the-art

Towards on-chip supercontinuum sources

● Initial stages of supercontinuum generation strongly depends on pump pulses duration

● Short pulses (typically T0≤150 fs): soliton dynamics

● Long pulses: modulation instability (four-wave mixing) followed by soliton dynamics

Pump duration

● Long pulse regime

● Early stage: noise seeded modulation instability – Growth of symmetrical sidebands around the pump – Pulse breaks into a random train of shorter pulses

Supercontinuum instabilities

● Noise-seeded MI: large pulse to pulse instabilities

● Commonly seen for > 150 fs pulses

● Leads to artificial spectral smoothness

Supercontinuum instabilities

Already known in early 2000…

Fluctuations in supercontinuum generation

● 2007: real-time detection of these fluctuations on the long wavelengths

● Dispersion maps wavelength to time

● Filtering selects long wavelength

edge

Dudley et al., RMP (2006) Demircan et al., Appl. Phys. B (2007)

Corwin et al., PRL (2003)

Gu et al., OL (2002) Dudley et al., JSTQE (2002)

Fluctuations in supercontinuum generation

Fluctuations in supercontinuum generation

● “Optical rogue waves” by analogy with hydrodynamics

Presence of extreme events

Mind the filter…

Mind the filter…

● Numerically possible to detect ALL solitons from ALL shots

Insight from numerics

● Only few events satisfy hydrodynamic rogue wave criterion

Erkintalo et al., EPJ Special Topics 185, 135 (2010)

Collisions are more extreme

The subtle role of collisions

● In fact, the filtering technique indirectly capture collision events (to some extent)

● Collisions of solitons lead to energy exchange and enhanced redshift (the larger sucks energy from the smaller)

Rogue soliton

Rogue soliton

Collision

BTW phase matters

● Soliton-collision is not necessarily extreme

● Collision is phase-dependent

Collision

Antikainen et al., Nonlinearity 25, R73 (2012)

Capturing single shot spectra

● Real-time measurement of single shot is spectra is possible

● Use dispersive time-to-frequency transformation

● At large distance in the dispersive fiber, the stretched temporal trace corresponds to the spectrum: analog to far-field diffraction

Goda et al., Nature Photonics 7, 102–112 (2013)

Capturing single shot spectra

● Real-time measurement of single shot confirms large fluctuations

Capturing single shot spectra

● Direct access to spectral fluctuations at ALL wavelengths

Wetzel et al., Scientific Reports 2, 882 (2012)

Link with traditional discrete nonlinear optics

● We should be able to interpret SC generation in terms of traditional discrete wave mixing dynamics

● Dispersive wave generation: only phase matching condition, energy conservation?

Link with traditional discrete nonlinear optics

● Two CW pumps lead to the generation of multiple discrete sidebands through cascaded four-wave mixing

ω-p ω+p

ω

Δ

Cascaded four-wave mixing

● Two CW pumps lead to the generation of multiple discrete sidebands through cascaded four-wave mixing

ω-p ω+p

ω

Δ

Cascaded four-wave mixing

Δ

ω1 ω+p + ω+p ω-p + ω1

● Two CW pumps lead to the generation of multiple discrete sidebands through cascaded four-wave mixing

● The net result is a χ(2n+1) Process!

ω-p ω+p

ω

Δ

Cascaded four-wave mixing

Δ

ω1 ω+p + ωn-1 ω-p + ωn

Δ

ω2

Δ

ωn ωn-1

ω+p

ω-p

ωn

ω-p ω-p

ω+p

ω+p

ω+p

(n+1)ω+p nω-p BUT: only efficient energy transfer if the process is phase-matched

(n+1)ω+p nω-p + ωn

● Such cascaded four-wave mixing is naturally phase-matched in optical fibers due to the symmetry breaking!

● Leads to asymmetry in the cascaded spectrum

Cascaded four-wave mixing

(n+1) β(ω+p) = n β(ω-p) + β(ωn)!

Taylor-series expansion ωn = ω0 + 3|β2|/β3

(n+1)ω+p nω-p + ωn

2

when the direct higher-order process is phase-matched,i.e. when:

�±n(z) = (n+ 1)�±p(z)� n�⌥p(z), (1)

where �±p and �±n denote the pump and signal phases,respectively. From Eq. (1) we can determine the cor-responding phase-matching condition in terms of thepump wave and fiber properties. In particular, assum-ing that the pump waves experience only self-phase andmutual cross-phase modulation (SPM, XPM) (i.e. areuna↵ected by the weak sidebands), the pump phasesevolve as: �±p(z) = [�(!±p) + �P±p + 2�P⌥p] z, where�(!) is the frequency-dependent propagation constantof the fiber mode, � is the nonlinear coe�cient, andP±p the pump powers. Correspondingly, the dominantcontribution to the nonlinear phase shift experienced bythe sidebands arises from XPM from the pumps suchthat: �±n(z) = [�(!±n) + 2�P

+p + 2�P�p] z. Combin-ing these conditions with Eq. 1 yields a general phase-matching condition for the amplification of a particularhigher-order component at !±n:

�(!±n) =± n [�(!±p)� �(!⌥p)] + �(!±p)

+ n� [P⌥p � P±p]� �P±p. (2)

Under typical experimental conditions, the nonlinearterms in Eq. (2) can be neglected, and phase-matching at!±n is determined only by the linear mismatch ��± =(n + 1)�(!±p) � n�(!⌥p) � �(!±n). Expanding �(!)about !

0

up to third-order yields that ��± = 0 is ful-filled for frequency separations �± > 0:

�± = ⌥ 3�2

�3

(n+ 1/2), (3)

where �2

and �3

denote respectively the second- andthird-order dispersion coe�cients at !

0

. For conventionalfibers �

3

> 0 and the phase-matched frequency is thengiven by

!±n = !0

± 3|�

2

|�3

. (4)

At this stage it is important to note that: (i) phase-matched amplification requires both pump waves to ex-perience the same sign of �

2

; (ii) the phase-matched fre-quency !±n always lies in the opposite dispersion regimeto that of the pumps; (iii) amplification at !

+n (upshiftedfrom the pump) or !�n (downshifted from the pump) cor-responds to the pumps lying in the anomalous or normaldispersion regime, respectively.

The theoretical treatment above has been confirmedusing numerical simulations based on the extended non-linear Schrodinger equation. We model propagation in80 m of dispersion-shifted fiber (DSF) with zero dis-persion wavelength (ZDW) �zdw = 1553.8 nm, nonlin-earity � = 2.5 W�1km�1 and �

3

= 1.6 ⇥ 10�4 ps3/m

at the ZDW. For completeness, fourth-order dispersion�4

= �7.0 ⇥ 10�7 ps4/m is also included in the modelbut has negligible influence. These parameters corre-spond to our experiments below, and we wish to remarkthat because we expand the propagation constant at theZDW we have �

2

= 0. Of course, for an arbitrary fre-quency �

2

(!) = �3

(! � !zdw) + �4

/2(! � !zdw)2, with!zdw = 2⇡c/�zdw the zero-dispersion frequency.Figure 1(a,b) shows numerical results illustrating the

spectral features of cascaded phase-matched amplifica-tion of the (n = ±2) sideband associated with an equiva-lent �5 nonlinearity for pumps in the anomalous (Fig.1(a)) and normal dispersion regime (Fig. 1(b)). In(a) the central frequency is !

0

/2⇡ = 188.5 THz (�0

=1590.5 nm) and �/2⇡ = 5.4 THz (�� ⇠ 46 nm); in (b)!0

/2⇡ = 196.6 THz (�0

= 1524.9 nm and �/2⇡ = 4.3THz (�� ⇠ 33 nm)). We see how the spectrum afterpropagation becomes asymmetric as a result of the am-plification of the phase-matched component on the oppo-site side of the ZDW to the pumps, with 15 dB enhance-ment relative to other generated components. This is atfirst sight counter-intuitive because the strength of cas-caded processes is expected to diminish with order, yetit is precisely this result that highlights the significanceof the phase-matching condition in yielding amplificationof a higher-order component.

!20 !10 0 10 20

!60

!40

!20

0

(!!!0)/(2") (THz)

Sp

ect

rum

(d

B) (b) !

!p!

+p

!!2

NA

!20 !10 0 10 20

!60

!40

!20

0

(!!!0)/(2") (THz)

Sp

ect

rum

(d

B) (a) !!p

!+p

!2

NA

0 20 40 60 800

0.2

0.4

0.6

0.8

Distance (m)Sid

eb

an

d p

ow

er

(W)

(c)

!1

!2

SimulationAnalytical (undepleted)

0 20 40 60 80

18

20

22

Distance (m)

Pu

mp

po

we

r (W

) (d)

!+p

!!p

FIG. 1. Simulation results showing asymmetric output spec-tra for pumping in (a) anomalous and (b) normal dispersionregime. (c) and (d) show sideband and pump power evolu-tion respectively for anomalous dispersion pumping. P+p =P�p = 20W. Labels A and N indicate regions of anomalousand normal dispersion, respectively, and the dashed verticalline marks the zero-dispersion frequency.

To gain more insight into the dynamics, Fig. 1(c)and (d) plot the evolution of the sidebands !

1

and !2

and pump waves as indicated for anomalous dispersionpumping (similar results are obtained for normal disper-sion pumping). Fig. 1(c) shows how the amplitude ofthe non-phase-matched !

1

sideband remains small andoscillates at the coherence length L

coh

= 2⇡/1

⇠ 3.1 m.Here

1

= 2�+p � ��p � �

1

is the elementary mismatchassociated with the direct generation of the n = 1 side-

2 pumps: anomalous dispersion Phase-matched side-band: normal dispersion

Only phase-matched through the cascade!

● Precisely the phase-matching condition for sispersive wave generation by solitons!

● Cascaded four-wave mixing: discrete frequency-domain

description of dispersive wave generation by solitons

ω0 ω

ωDW

Soliton Dispersive

wave

Link with dispersive wave generationb

ωn = ω0 + 3|β2|/β3 ωDW = ωS + 3|β2|/β3

Nonlinear Fiber β2 & β3

CW ω-p

CW ω+p

ω-p ω+p

ω0 ω

Nonlinear Fiber β2 & β3

fs ω0

Experimental evidence

Erkintalo et al., PRL 109, 223904 (2012)

fs pulse

Future challenges

Future challenges

It’s all in there…

● Similar NLS equation governs deep water waves dynamics

● Perturbations also lead to soliton fission!

Cherenkov radiation

Supercontinuum in water waves

4

1 2 3 4 5 6 7 8 9 1010−15

10−10

10−5

100

f (Hz)

S(f

)(cm

2s)

0 5 10 15 20 25 30 35 40 45 50−1

0

1

t (s)

⌘(cm)

0 5 10 15 20 25 30 35 40 45 50−1

0

1

t (s)

⌘(cm)

1 m

10 m

10 m1 m

(c)

(b)

(a)

FIG. 6. (Color online) Fission of the four-soliton solution forthe carrier parameters a0 = 1 mm and " = 0.04, generated atx = 0. (a) Wave-gauge measurement of the surface elevationat a position 1 m from the flap. (b) Wave-gauge measurementof the surface elevation at a position 9 m from the flap. (c)Spectra of the corresponding measurements, confirming thegeneration of supercontinuum.

account in the weakly-nonlinear approach, described bythe NLS. In addition, striking broadening of the soliton-spectra as well as measurement-spectra during the evo-lution in the flume confirm the observation of super-continuum in water waves. Furthermore, soliton-fission,also engendering the supercontinuum, is also reportedin this work. As a next step, numerical simulationsshould be performed in order to analyze the soliton-fission-phenomenon in more detail.

... N. A. is supported by the Alexander von HumboldtFoundation.

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[3] V. E. Zakharov and A. B. Shabat, Interaction betweensolitons in a stable medium, Soviet Phys. JETP 37 823–828 (1973), (transl. of Zh. Eksp. Teor. Fiz. 64, 1627–1639,(1973)).

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[5] H. C. Yuen, & B. M. Lake, Advances in Applied Mechan-ics, 22, 67–229 (1982).

[6] E.A. Kuznetsov, Solitons in a parametrically unstableplasma, Sov. Phys. Doklady 22, 507–508, (1977)

[7] Y.C. Ma, The perturbed plane wave solutions of the cu-bic nonlinear Schrodinger equation, Stud. Appl. Math.60, 43–58 (1979).

[8] N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Gen-eration of a periodic sequence of picosecond pulses inan optical fibre: exact solutions, Zh. Eksp, i Teor. Fiz.89, 1542–1551 (1985), English translation in: [Sov. Phys.

JETP, 61, 894–899 (1985)].[9] D. H. Peregrine, Water waves, nonlinear Schrodinger

equations and their solutions, J. Austral. Math. Soc. Ser.B 25, 16–43 (1983).

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[11] F. Baronio, A. Degasperis, M. Conforti, and S. Wab-nitz, Solutions of the Vector Nonlinear Schrodinger Equa-tions: Evidence for Deterministic Rogue Waves, Phys.Rev. Lett. 109, 044102 (2012).

[12] A. Osborne, Nonlinear ocean waves and the inverse scat-tering transform, (Elsevier, Amsterdam, 2010).

[13] C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue wavesin the ocean, (Springer, Heidelberg - NY, 2009).

[14] A. Chabchoub, N. P. Ho↵mann, and N. Akhmediev,Rogue wave observation in a water wave tank, Phys. Rev.Lett. 106, 204502 (2011).

[15] A. Chabchoub, N. Ho↵mann, M. Onorato, and N.Akhmediev, Super rogue waves: observation of a higher-order breather in water waves, Phys. Rev. X 2, 011015(2012).

[16] G. Clauss, M. Klein, and M. Onorato, Formationof Extraordinarily High Waves in Space and Time,OMAE2011-49545 2, 417–429 (2011).

[17] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G.Genty, N. Akhmediev, and J. M. Dudley, The Peregrinesoliton in nonlinear fibre optics, Nat. Phys. 6, 790–795(2010).

[18] B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B.Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, Ob-servation of Kuznetsov-Ma soliton dynamics in opticalfibre, Scientific Reports 2, 463 (2012).

[19] H. Bailung, S. K. Sharma, and Y. Nakamura, Observa-tion of Peregrine solitons in a multicomponent plasmawith negative ions, Phys. Rev. Lett. 107, 255005 (2011).

[20] N. Akhmediev and A. Ankiewicz, Solitons: Non-linearpulses and beams (Optical and Quantum Electronics),(Chapman and Hall, London, 1997)

[21] N. Akhmediev and A. Ankiewicz, Spatial Soliton X-Junctions and Couplers, Optics Communications 100186–192 (1993).

[22] J. Satsuma and N. Yajima, Initial value problems of one-dimensional self-phase modulation on nonlinear waves indispersive media, Progr. Theor. Phys. Suppl. 55 284–306(1974).

[23] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Ex-perimental Observation of Picosecond Pulse Narrowingand Solions in Optical Fibers, Phys. Rev. Lett. 45, 1095(1980).

[24] J. M. Dudley, G. Genty and S. Coen, Supercontinuumgeneration in photonic crystal fiber, Rev. Mod. Phys. 78,11351184 (2006)

[25] Y. Kodama and A. Hasegawa, Nonlinear pulse propa-gation in a monomode dielectric guide, IEEE PhotonicsTechnol. Lett. QE-23, 510-524 (1987).

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Generation of multi-solitons and of hydrodynamic supercontinuum

A. Chabchoub1, N. Ho↵mann1, M. Onorato2,3,G. Genty4, J. M. Dudley5, and N. Akhmediev61 Mechanics and Ocean Engineering, Hamburg University of Technology, D-21073 Hamburg, Germany⇤

2Dipartimento di Fisica, Universita degli Studi di Torino, IT-10125 Torino, Italy3 Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, IT-10125 Torino, Italy

4 Tampere University of Technology, Optics Laboratory, FI-33101 Tampere, Finland5 Institut FEMTO-ST, UMR 6174 CNRS- Universite de Franche-Comte, FR-25030 Besancon, France and

6 Optical Sciences Group, Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia

We report first observations of exact breather solutions of the nonlinear Schrodinger equation(NLS) on zero background in water waves. Excitation of these breathers is an important confirmationof the ability of the NLS to describe extreme localization in water waves under conditions of higher-amplitude initial conditions. But in addition, we also study the evolution of the solutions beyond thepoint of NLS validity where we report the first detailed experimental data showing the appearanceof a broad and continuous wave spectrum. Adopting vocabulary from optics, we refer to this asthe first controlled observation of hydrodynamic supercontinuum. We expect this to point to newdirections of research in the study of nonlinear ocean waves.

The evolution equation describing the propagation inspace and in time of weakly nonlinear wave packets isthe nonlinear Schrodinger equation (NLS) [1, 2]. Experi-ments on stationary solutions [3] in deep-water confirmedthe validity of the NLS describing pulse soliton evolu-tion in water waves [4, 5]. Another class of solutions arebreather solutions. Here, we distinguish two kinds of pul-sating solutions: Breathers on finite background [6–11],which are under intensive study since they are consid-ered to be appropriate models to describe oceanic roguewaves [12–16] as well as extreme events in other non-linear dispersive media [17–19], and breathers on zerobackground [20–22]. Here, we present first observationsof multi-soliton solutions, also known as Satsuma-Yajimabreathers, starting from zero background amplitude, in awater wave tank. The focusing time NLS [12], appropri-ate for the analysis of experimental data measured withrespect to time t, describing the evolution of deep-waterwave packets in space x can be written as:

i

✓@A

@x

+2k0!0

@A

@t

◆� k0

!

20

@

2A

@x

2 � k

30 |A|2 A = 0. (1)

The liner dispersion !0 =pgk0 relates the wave fre-

quency !0 and the wavenumber k0 of the carrier wave,where g denotes the gravitational acceleration. Deep-water wave packets propagate with the group velocity:

cg := d!dk

���k=k0

= !02k0

. The surface elevation of the water

⌘(x, t), taking into account the second Stokes harmonic,can be computed from the NLS solution A(x, t) and is:

⌘(x, t) =1

2

✓A (x, t) ei# +

1

2k0A

2 (x, t) e2i# + c.c.

◆, (2)

where c.c. denotes complex conjugate and # =(k0x� !0t). Scaling the NLS (1) gives the standard form:

i X +1

2 TT + | |2 = 0, (3)

Applying the transformations X �! 2T and T �! X

transforms the scaled time NLS into a scaled space NLS:

i T + XX + 2 | |2 = 0. (4)

The observation of pulse soliton solutions, derived usingthe inverse scattering technique [3], confirmed the valid-ity of the NLS for deep-water waves [4]. Another promis-ing technique is the Darboux-transformation [11, 20].This method allows to write multi-soliton solutions ina general form. A periodic two-soliton solution of Eq.(4) was derived by [22]:

2 (X,T ) =4 (cosh (3X) + 3 cosh (X) exp (8iT ))

cosh (4X) + 4 cosh (2X) + 3 cos (8T )(5)

⇥ exp (iT ) ,

and was first experimentally reported in [23]. At T = 0this breather solution on zero background takes the pulsesoliton form 2 sech(X). The exact expressions of higer-order solutions of this kind are too long and cumbersometo be reproduced here. Generally, a periodic N -solitonsolution, also referred to as Satsuma-Yajima breather,starts at T = 0 from an N sech(X) shape. Fig. 1 showsthe two-soliton and the three-soliton Satsuma-Yajima so-lution. Excitation of these breathers is of course an im-portant confirmation of the ability of the NLS to describeextreme localization in water waves under conditions ofhigher-amplitude initial conditions. But furthermore, wealso study the evolution of the solutions beyond the pointof NLS validity where we report the first detailed experi-mental data showing the appearance of a broad and con-tinuous wave spectrum. In optics, we refer to this as thefirst controlled observation of supercontinuum in waterwaves, and we anticipate that this will reflect much moreclosely the formation of rogue waves in their natural en-vironment. Significantly, the fact that we are still ableto qualitatively interpret the features of the continuumin terms of specific solutions of the NLS is similar to the

Generation of multi-solitons and of hydrodynamic supercontinuum

A. Chabchoub1, N. Ho↵mann1, M. Onorato2,3,G. Genty4, J. M. Dudley5, and N. Akhmediev61 Mechanics and Ocean Engineering, Hamburg University of Technology, D-21073 Hamburg, Germany⇤

2Dipartimento di Fisica, Universita degli Studi di Torino, IT-10125 Torino, Italy3 Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, IT-10125 Torino, Italy

4 Tampere University of Technology, Optics Laboratory, FI-33101 Tampere, Finland5 Institut FEMTO-ST, UMR 6174 CNRS- Universite de Franche-Comte, FR-25030 Besancon, France and

6 Optical Sciences Group, Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia

We report first observations of exact breather solutions of the nonlinear Schrodinger equation(NLS) on zero background in water waves. Excitation of these breathers is an important confirmationof the ability of the NLS to describe extreme localization in water waves under conditions of higher-amplitude initial conditions. But in addition, we also study the evolution of the solutions beyond thepoint of NLS validity where we report the first detailed experimental data showing the appearanceof a broad and continuous wave spectrum. Adopting vocabulary from optics, we refer to this asthe first controlled observation of hydrodynamic supercontinuum. We expect this to point to newdirections of research in the study of nonlinear ocean waves.

The evolution equation describing the propagation inspace and in time of weakly nonlinear wave packets isthe nonlinear Schrodinger equation (NLS) [1, 2]. Experi-ments on stationary solutions [3] in deep-water confirmedthe validity of the NLS describing pulse soliton evolu-tion in water waves [4, 5]. Another class of solutions arebreather solutions. Here, we distinguish two kinds of pul-sating solutions: Breathers on finite background [6–11],which are under intensive study since they are consid-ered to be appropriate models to describe oceanic roguewaves [12–16] as well as extreme events in other non-linear dispersive media [17–19], and breathers on zerobackground [20–22]. Here, we present first observationsof multi-soliton solutions, also known as Satsuma-Yajimabreathers, starting from zero background amplitude, in awater wave tank. The focusing time NLS [12], appropri-ate for the analysis of experimental data measured withrespect to time t, describing the evolution of deep-waterwave packets in space x can be written as:

i

✓@A

@x

+2k0!0

@A

@t

◆� k0

!

20

@

2A

@x

2 � k

30 |A|2 A = 0. (1)

The liner dispersion !0 =pgk0 relates the wave fre-

quency !0 and the wavenumber k0 of the carrier wave,where g denotes the gravitational acceleration. Deep-water wave packets propagate with the group velocity:

cg := d!dk

���k=k0

= !02k0

. The surface elevation of the water

⌘(x, t), taking into account the second Stokes harmonic,can be computed from the NLS solution A(x, t) and is:

⌘(x, t) =1

2

✓A (x, t) ei# +

1

2k0A

2 (x, t) e2i# + c.c.

◆, (2)

where c.c. denotes complex conjugate and # =(k0x� !0t). Scaling the NLS (1) gives the standard form:

i X +1

2 TT + | |2 = 0, (3)

Applying the transformations X �! 2T and T �! X

transforms the scaled time NLS into a scaled space NLS:

i T + XX + 2 | |2 = 0. (4)

The observation of pulse soliton solutions, derived usingthe inverse scattering technique [3], confirmed the valid-ity of the NLS for deep-water waves [4]. Another promis-ing technique is the Darboux-transformation [11, 20].This method allows to write multi-soliton solutions ina general form. A periodic two-soliton solution of Eq.(4) was derived by [22]:

2 (X,T ) =4 (cosh (3X) + 3 cosh (X) exp (8iT ))

cosh (4X) + 4 cosh (2X) + 3 cos (8T )(5)

⇥ exp (iT ) ,

and was first experimentally reported in [23]. At T = 0this breather solution on zero background takes the pulsesoliton form 2 sech(X). The exact expressions of higer-order solutions of this kind are too long and cumbersometo be reproduced here. Generally, a periodic N -solitonsolution, also referred to as Satsuma-Yajima breather,starts at T = 0 from an N sech(X) shape. Fig. 1 showsthe two-soliton and the three-soliton Satsuma-Yajima so-lution. Excitation of these breathers is of course an im-portant confirmation of the ability of the NLS to describeextreme localization in water waves under conditions ofhigher-amplitude initial conditions. But furthermore, wealso study the evolution of the solutions beyond the pointof NLS validity where we report the first detailed experi-mental data showing the appearance of a broad and con-tinuous wave spectrum. In optics, we refer to this as thefirst controlled observation of supercontinuum in waterwaves, and we anticipate that this will reflect much moreclosely the formation of rogue waves in their natural en-vironment. Significantly, the fact that we are still ableto qualitatively interpret the features of the continuumin terms of specific solutions of the NLS is similar to the

Generation of multi-solitons and of hydrodynamic supercontinuum

A. Chabchoub1, N. Ho↵mann1, M. Onorato2,3,G. Genty4, J. M. Dudley5, and N. Akhmediev61 Mechanics and Ocean Engineering, Hamburg University of Technology, D-21073 Hamburg, Germany⇤

2Dipartimento di Fisica, Universita degli Studi di Torino, IT-10125 Torino, Italy3 Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, IT-10125 Torino, Italy

4 Tampere University of Technology, Optics Laboratory, FI-33101 Tampere, Finland5 Institut FEMTO-ST, UMR 6174 CNRS- Universite de Franche-Comte, FR-25030 Besancon, France and

6 Optical Sciences Group, Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia

We report first observations of exact breather solutions of the nonlinear Schrodinger equation(NLS) on zero background in water waves. Excitation of these breathers is an important confirmationof the ability of the NLS to describe extreme localization in water waves under conditions of higher-amplitude initial conditions. But in addition, we also study the evolution of the solutions beyond thepoint of NLS validity where we report the first detailed experimental data showing the appearanceof a broad and continuous wave spectrum. Adopting vocabulary from optics, we refer to this asthe first controlled observation of hydrodynamic supercontinuum. We expect this to point to newdirections of research in the study of nonlinear ocean waves.

The evolution equation describing the propagation inspace and in time of weakly nonlinear wave packets isthe nonlinear Schrodinger equation (NLS) [1, 2]. Experi-ments on stationary solutions [3] in deep-water confirmedthe validity of the NLS describing pulse soliton evolu-tion in water waves [4, 5]. Another class of solutions arebreather solutions. Here, we distinguish two kinds of pul-sating solutions: Breathers on finite background [6–11],which are under intensive study since they are consid-ered to be appropriate models to describe oceanic roguewaves [12–16] as well as extreme events in other non-linear dispersive media [17–19], and breathers on zerobackground [20–22]. Here, we present first observationsof multi-soliton solutions, also known as Satsuma-Yajimabreathers, starting from zero background amplitude, in awater wave tank. The focusing time NLS [12], appropri-ate for the analysis of experimental data measured withrespect to time t, describing the evolution of deep-waterwave packets in space x can be written as:

i

✓@A

@x

+2k0!0

@A

@t

◆� k0

!

20

@

2A

@x

2 � k

30 |A|2 A = 0. (1)

The liner dispersion !0 =pgk0 relates the wave fre-

quency !0 and the wavenumber k0 of the carrier wave,where g denotes the gravitational acceleration. Deep-water wave packets propagate with the group velocity:

cg := d!dk

���k=k0

= !02k0

. The surface elevation of the water

⌘(x, t), taking into account the second Stokes harmonic,can be computed from the NLS solution A(x, t) and is:

⌘(x, t) =1

2

✓A (x, t) ei# +

1

2k0A

2 (x, t) e2i# + c.c.

◆, (2)

where c.c. denotes complex conjugate and # =(k0x� !0t). Scaling the NLS (1) gives the standard form:

i X +1

2 TT + | |2 = 0, (3)

Applying the transformations X �! 2T and T �! X

transforms the scaled time NLS into a scaled space NLS:

i T + XX + 2 | |2 = 0. (4)

The observation of pulse soliton solutions, derived usingthe inverse scattering technique [3], confirmed the valid-ity of the NLS for deep-water waves [4]. Another promis-ing technique is the Darboux-transformation [11, 20].This method allows to write multi-soliton solutions ina general form. A periodic two-soliton solution of Eq.(4) was derived by [22]:

2 (X,T ) =4 (cosh (3X) + 3 cosh (X) exp (8iT ))

cosh (4X) + 4 cosh (2X) + 3 cos (8T )(5)

⇥ exp (iT ) ,

and was first experimentally reported in [23]. At T = 0this breather solution on zero background takes the pulsesoliton form 2 sech(X). The exact expressions of higer-order solutions of this kind are too long and cumbersometo be reproduced here. Generally, a periodic N -solitonsolution, also referred to as Satsuma-Yajima breather,starts at T = 0 from an N sech(X) shape. Fig. 1 showsthe two-soliton and the three-soliton Satsuma-Yajima so-lution. Excitation of these breathers is of course an im-portant confirmation of the ability of the NLS to describeextreme localization in water waves under conditions ofhigher-amplitude initial conditions. But furthermore, wealso study the evolution of the solutions beyond the pointof NLS validity where we report the first detailed experi-mental data showing the appearance of a broad and con-tinuous wave spectrum. In optics, we refer to this as thefirst controlled observation of supercontinuum in waterwaves, and we anticipate that this will reflect much moreclosely the formation of rogue waves in their natural en-vironment. Significantly, the fact that we are still ableto qualitatively interpret the features of the continuumin terms of specific solutions of the NLS is similar to the

Nonlinearity GVD

Hydrodynamic supercontinuum