Günter Steinmeyer1, Simon Birkholz1, Carsten Brée2, Ayhan Demircan3

1 Max Born Institute, Berlin, Germany

2 Weierstrass Institute, Berlin, Germany 3 Institut für Quantenoptik, University of Hannover

steinmey@mbi-berlin.de

Nonlinear correlation analysis in rogue wave data

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Big Data & Real-time Analytics in Photonics UCLA, Los Angeles, CA

March 27, 2015

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Outline

• Ocean rogue waves may cause serious damage to ships

• Qualitatively similar behavior observed in optical physics

• Exact reasons for ocean rogue waves unknown

• Predictability of rogue events?

• Tracing the reasons: looking for determinism in measured data

• Nonlinear correlation analysis

• Numerical complexity O(N2): N~105: 1h on Desktop PC N~107: 1yr on desktop PC

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What Exactly is a Rogue Wave?

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Rogue Waves – Towards a Unifying Concept? Eur. Phys. J. Special Topics 185 (2010).

Criteria generally agreed upon:

1. Sparsity: rogue events are extremely rare

2. Extremeness: rogue events surpass any prediction derived from observation of previous events

3. Statistical unlikelihood: rogue waves defy regular Gaussian statistics

4. Unpredictability: rogue waves appear w/o warning and disappear w/o a trace

Difficult to quantify

Only criterion so far

New criterion

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Optical Fiber Rogue Waves

Solli et al., Nature 450, 1054 (2007).

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Is this a Rogue Wave?

J.H.E. Cartwright & H. Nakamura, What kind of a wave is Hokusai's Great wave off Kanagawa ? Notes Rec. R. Soc. 63, 119-135 (2009).

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Are Mountains Actually Rogue Waves?

Source: Geo Elevation Data Mathematica, Wolfram Research

Earth surface sampled at 0.1O steps 6,400,000 data points

Sea level

This is where rogue starts

This is where we are 125 m = 400 ft

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Multifilament Rogue Scenario

xenon cell (2 bar)

Ti:sapphire laser 1 kHz, 5 mJ, 40 fs

1 kHz linescan or 100 Hz 2D camera

dynamic beam profile

f =0.75m

Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)

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Spatio-temporal isolation

Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)

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The „Birkholz“ event

Draupner oil rig, North Sea January 1, 1995, 3:20pm

MBI labs, Berlin April 2012

Ref.: Dysthe et al. “Oceanic Rogue Waves” Annu. Rev. Fluid Mech. 40, 287 (2008).

Birkholz

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Statistical Analysis Laser Single

Filament Multi

Filament

Events at 10x significant wave height F1/3

Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)

W. Weibull, J. Appl. Mech.-Trans. 18, (1951)

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Predictability of rogue events

1. Select segment Si,i+L 2. Compute ||Si,i+L-Sj,j+L||2 for all i,j

P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).

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Histogram representation

||Si,i+L-Sj,j+L||2

Cou

nts

most segments are fairly dissimilar

S. Birkholz et al., submitted to PRL

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Surrogates

Time

Am

plitu

de

Refs.: D. Prichard & J. Theiler PRL 73, 951 (1994). T. Schreiber & A. Schmitz PRL 77, 635 (1996) True random numbers from www.random.org

1. Maintain histogram

2. Maintain spectrum

3. Conserve linear correlation function

4. Use „true“ random numbers

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Determinism of multifilament data

surrogates

original

several thousand “déjà vus“ in the original

200 ms segment length = 15x lin. correlation time

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Analysis for original Draupner event

20s segment length

Data is clearly deterministic !

Original data of Draupner event courtesy M. Olagnon IFREMER, Brest, France and Statoil ASA, Norway

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Analysis for fiber Rogue waves

Original data courtesy D. Solli, UCLA D. Solli et al., Nature 450, 1054 (2007)

Original data more random than surrogates

Positively no determinism !!!

1 µs segment length

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Can we actually predict rogue events?

Total data 60,000 camera lines

400 pxls each

Contains 920 spatially

and temporally isolated

rogue events

1 min

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A new type of surrogate analysis

1. Select 100ms long segment before each rogue wave

2. Select identical number of random segments across data set

3. Redo surrogate 100x

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Predictability of rogue waves?

surrogates

originals

Specific dynamics herald the

impact of rogue waves

100ms look ahead time (~10s equivalent for Draupner)

Distance

Cou

nt

J. Martinerie et al., “Epileptic seizures can be anticipated by non-linear analysis,“

Nature Medicine 4, 1173 (1998).

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Conclusions Filament rogue waves behave like ocean rogue

waves concerning several aspects Heavytail-distributions (actually more extreme

than the ocean) Deterministic behavior → predictability Main driver: atmospheric turbulence (as in the

ocean) Is it really true that rogue waves appear w/o any

warning?

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Acknowledgments Simon Birkholz, MBI Rogue wave experiments

Erik Nibbering, MBI 15 year experience w/ filaments

Carsten Brée, WIAS Numerical Simulations

Stefan Skupin, Univ. Bordeaux Propagation Code

Ayhan Demircan, WIAS Theory support

Goëry Genty, TUT, Finland Original idea

Fedor Mitschke, Rostock Grassberger & Procaccia, PhD thesis advisor

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Microscopic origin of rogue events • What is the origin of this sudden intensity/fluence spikes

in the filaments?

• Clamping intensity ???

• We are not measuring inside the cell, beams were allowed to diffract

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Some thoughts on the role of mergers • Can we control these mergers?

• Is it possible to combine two or more filament strings?

• No violation of clamping intensity

• Increased energy / fluence in combined filament

• Control prior to filament formation futile!

• Control during multifilament propagation

• Maybe such experiments work in solid or liquid medium?

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lateral dimension 0 6.5 mm

Freq

uenc

y

The Input Profile – Calm Waters

Gaussian

rogue

average intensity

peak intensity

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Single Filamentation – Choppy Seas

lateral dimension 0 10 mm

peak intensity ≈ 2x average intensity

peak intensity ≈ average intensity

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Multiple Filamentation

lateral dimension 0 10 mm

peak intensity > 10x average intensity

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Numerical Simulations

x

y

0

30 cm 200 µm

2D+1 numerical solution of the NLSE code developed by S. Skupin

0

π

2π Phase

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The Role of Breathers

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Rogue waves arise from mergers C

ount

s

Intensity

all filaments

mergers only

L. Cavaleri et al., “Rogue waves in crossing seas: The Louis Majesty accident,” J. Geophys. Research: Oceans 117, C5 (2012).

V. P. Ruban, “Giant waves in weakly crossing sea states,” J. Exp. Theo. Phys. 110, 529 (2010).

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Isolated breather event

Input: 2 filament strings w/ identical phase

Filaments attract due to XPM

z

Reseparation in orthogonal plane

Further breathing after collapse

• Incidental interference of 2 waves

• Phase plays dice here

• No rogues w/o nonlinearity

• Similarity w/ 1D breathers

• The universal mechanism found?

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Where does the determinism come from?

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Macroscopic origin of rogue waves Experiments with different parameters: • No rogue waves with water as a medium

--- despite higher nonlinearity • 10x reduction of repetition rates with chopper

→ no rogue waves • xenon and SF6 work (2 bar pressure, 1 kHz rep. rate)

Thermal convection mechanism

inside cell Y.-H. Cheng et al. "The effect of long timescale gas

dynamics on femtosecond filamentation," Opt. Express 21, 4740 (2013)

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Turbulent thermal convection

K. Noto et al. J. Thermophys. Heat Transfer 13, 82 (1999). S. Grafsrønningen et al., Int. J. Heat Mass Transfer 57, 519 (2013).

Local Grashof Number

Natural convection Irregular horizontal cylinder array

Transition from laminar to turbulent heat exchange

2x108 < Gry,Q < 2x109 Transition regime Possible to meet for xenon

(low heat conductivity and viscosity) Water requires 1000x higher powers

y

z

Nonlinear correlation analysis in rogue wave dataOutlineWhat Exactly is a Rogue Wave?Optical Fiber Rogue WavesIs this a Rogue Wave?Are Mountains Actually Rogue Waves?Multifilament Rogue ScenarioSpatio-temporal isolationThe „Birkholz“ eventStatistical AnalysisPredictability of rogue eventsHistogram representationSurrogatesDeterminism of multifilament dataAnalysis for original Draupner eventAnalysis for fiber Rogue wavesCan we actually predict rogue events?A new type of surrogate analysisPredictability of rogue waves?ConclusionsAcknowledgmentsMicroscopic origin of rogue eventsSome thoughts on the role of mergersThe Input Profile – Calm WatersSingle Filamentation – Choppy SeasMultiple FilamentationNumerical SimulationsThe Role of BreathersRogue waves arise from mergersIsolated breather eventWhere does the determinism �come from?Macroscopic origin of rogue wavesTurbulent thermal convection