Generation of Caustics and Rogue Waves from Nonlinearity

CanadaExcellenceResearchChairs

Chaires d'excellenceenrecherche duCanada

Akbar Safari1, Robert Fickler1, Miles J Padgett2 and Robert W. Boyd1,2,3

1University of Ottawa, Canada2University of Glasgow, UK

3University of Rochester, USA

1

2

Researchinterest:Nonlinearoptics,quantumoptics,integratedphotonics,meta-materials,etc.

CanadaExcellenceResearchChair(CERC)inNonlinearQuantumOptics

Oceanic rogue waves

3

Sailors: weseegiganticwaves.Scientists: itisafairytail!

OceanwavesfollowGaussiandistribution.

2AP e-µProb

ability

Amplitude2

Before1995

Oceanic rogue waves

4

First scientific observationof rogue waves in Draupneroil platform (1995):

Characteristics of rogue waves

5

Wave amplitude2

Prob

abili

ty (l

og-s

cale

)

Normaldistribution(decayexponentially)

Roguewaves(Heavy-tailed)

Probabilitydistributioninroguesystems:

§ Roguewavesappearfromnowhereanddisappearwithoutatrace.

§ Roguewaves≠accidentalconstructiveinterference

§ Theyoccurmuchmorefrequentlythanexpectedinordinarywavestatistics.

§ Not limited to ocean: Observed in many other wavesystems including optics.

1D vs 2D systems

6

NonlinearSchrödingerequationexplainsthewavedynamicsinoceanaswellasoptics.

In1Dsystems,suchasopticalfibersarestudiedextensively.

J.M.Dudleyetal,Nat.Photon,8,755(2014)

Twofocusingeffectsin2Dsystems:• Linear:Spatial(geometrical)focusing• Nonlinear:Selffocusing

Waterwavesarenot1D.

Optical caustics

7

Swimmingpool

Coffeecup

• Causticsaredefinedasenvelopeofafamilyofrays

• Singularitiesinrayoptics

• Catastrophetheoryisrequiredtoremovesingularity

Books:J.F.Nye, NaturalFocusingandFineStructureofLight.Y.A.Kravtsov,Caustics,CatastrophesandWaveFields.O.N.Stavroudis,TheOpticsofRays,Wavefronts,andCaustics.

Raypicture

Generation of optical caustics

8Asharpcausticisformedonlyifthephasevariationsarelarge

Phasevariations:

Correspondingintensityvariations:

Statistics of caustics

9

1000differentpatternsforeachΔ

( )CAExp B I-

Intensitydistributionswithfitto

A.Mathis,L.Froehly,S.Toenger,F.Dias,G.Genty &J.Dudley.ScientificReports5,1(2015).

Causticsexhibitlong-tailedprobabilitydistribution

Caustics in ocean waves

10

M.V.Berry,Focusedtsunamiwaves,Proc.R.Soc.A(2007)

Causticoftsunamifocusedbyanunderwaterislandlens

Simulatedlinearpropagationoftsunamiwaves,usingrealoceanfloorbathymetry:

Oceanbathymetry

H.Degueldre,J.Metzger,T.GeiselandR.Fleischmann,NaturePhysics 12,259(2016).

Nonlinear focusing

11

Selffocusing:

n = n0 + n2 IRefractiveindexdependsonintensity:

RubidiumcellRubidiumvaporsshowlargenonlineareffects

Afterlinearpropagation:

Afternonlinearpropagation:

12

Effect of nonlinearity on caustics

Phasevariations:

( )CAExp B I-Intensitydistributionswithfitto

Linearpropagation:

Afternonlinearpropagation:

13

Statistics of caustics

14

LinearpropagationwassimulatedbyFFTbeampropagation

Experiment:

Simulation:

Simulation – Linear propagation

15

NLSE: Atomicpolarization:

OurRb modelincludes:• Allhyperfinetransitions• Dopplerbroadening• Powerbroadening• Collisional broadening• Opticalpumping

Simulation – Rb model

16

Experiment:

Simulation:

NonlinearpropagationwassimulatedbyFFTbeampropagationandsplit-step

Simulation – Nonlinear propagation

A.Safari,R.Fickler,M.Padgett,R.Boyd,PhysicalReviewLetters119,203901(2017)

17

§ Caustics are important:• Natural way of focusing energy of a wave• Can generate large amplitude waves

§ Generation of caustics in linear space requires largefluctuation

§ Nonlinear instability can generate caustics from smallfluctuations

Conclusion

Linear Nonlinear

18

Robert Fickler Miles PadgettAkbar Safari

Thank you for your attention

Thanks to:

Tsunami wave

19

20

Scintillationindex:

“Caustics,CatastrophesandWaveFields”,Kravtsov,Orlov (1993)

Uniformdistribution

Simulation – Nonlinear propagation

21

1DNLSE

Transformtheframeandreplacetimewithspaceinoceanwavebasedon

22

2 2

12

A Aik i A Ax t

g¶ ¶+ =

¶ ¶Optics:

223

2 2

A k Ai k A Ax tw¶ ¶

- =¶ ¶

Ocean:

22

1 2 2

12

A A Ak ik i A Ax t t

g¶ ¶ ¶+ + =

¶ ¶ ¶Optics:

2 22

2 28 2gA A A ki v A At x k x

w w¶ ¶ ¶æ ö+ = +ç ÷¶ ¶ ¶è øOcean:

gx v t=

22

2DNLSE

222 Aik A i A Ax

g^

¶+Ñ =

¶Optics:

Ocean:2 2 2

22 2 2 28 4 2

A A A ki A At k x k y

w w w¶ ¶ ¶- + =

¶ ¶ ¶

1D vs 2D systems

23

NonlinearSchrödingerequationexplainsthewavedynamicsinoceanaswellasoptics.

Modulationalinstability

In1Dsystems(studiedextensively):

J.M.Dudleyetal,Nat.Photon,8,755(2014)

In2Dsystems:• Spatial(geometrical)focusing• Nonlinearfocusing

• Nonlinearmodulationalinstability