Bragg grating rogue wave

Physics Letters A 379 (2015) 1067–1070

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Bragg

grating

rogue

wave

Antonio Degasperis

a

,

Stefan Wabnitz

b

,

∗

,

Alejandro

B. Aceves

c

a_{DipartimentodiFisica,“Sapienza”UniversitàdiRoma,P.leA.Moro2,00185Roma,Italy}

b_{DipartimentodiIngegneriadell’Informazione,UniversitàdegliStudidiBresciaandINO-CNR,viaBranze38,25123Brescia,Italy} c_{SouthernMethodistUniversity,Dallas,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21January2015 Accepted23January2015 Availableonline28January2015 CommunicatedbyV.M.Agranovich Keywords: Nonlinearwaves Nonlinearoptics Opticalfibers Periodicmedia Waterwaves

We derive the rogue wave solution ofthe classical massive Thirring model, that describes nonlinear opticalpulsepropagationinBragg gratings.Combiningelectromagneticallyinduced transparencywith Braggscatteringfour-wavemixing mayleadtoextremewavesatextremelylowpowers.

©2015ElsevierB.V.All rights reserved.

1. Introduction

Extreme wave phenomenon appears in a variety of scientific and social contexts, ranging from hydrodynamics and oceanog-raphy to geophysics, plasma physics, Bose–Einstein condensation (BEC), financial markets and nonlinear optics [1–5]. Historically, thefirstreportedmanifestationofextremeorrogue wavesisthe sudden appearance in the open sea of an isolated giant wave, with height and steepness much larger than the average values ofoceanwaves.Auniversalmodelfordescribingthedynamicsof rogue wave generation in deep water with a flat bottom is the one-dimensionalnonlinearSchrödinger(NLS)equationinthe self-focusingregime.ThemechanismleadingtotheappearanceofNLS roguewavesrequires nonlinearinteractionandmodulation insta-bility(MI) ofthe continuous wave (CW) background[6].Indeed, the nonlinear development of MI may be described by families of exact solutions such as the Akhmediev breathers [7], which are recognized as a paradigm for rogue wave shaping. A special memberofthissolutionfamilyisthefamousPeregrinesoliton

[8]

, whichdescribesawavethatappearsfromnowhereanddisappears withouta trace.Extreme wavesthat maybe well representedby thePeregrine solitonhaverecentlybeenexperimentally observed inopticalfibers

[9]

,inwater-wavetanks

[10]

andinplasmas

[11]

. Movingbeyond the one-dimensional NLS model, it is impor-tanttoconsiderextremewavephenomenonineither

multidimen-*

Correspondingauthor.

E-mailaddress:[email protected](S. Wabnitz).

sional or multicomponent nonlinear propagation. Vector systems arecharacterizedby thepossibilityofobservingacouplingof en-ergyamongtheirdifferentdegreesoffreedom,whichsubstantially enrichesthecomplexityoftheirrogue-wavefamilies.Recent stud-ies haveunveiled theexistence of extremewave solutionsin the vector NLS equation or Manakov system[12–15], thethree-wave resonant interaction equations[16], thecoupled Hirota equations

[17]andthelong-wave–short-waveresonance

[18]

.

InthisLetter,wepresenttheroguewavesolutionofthe classi-calmassiveThirringmodel(MTM)

[19]

,atwo-component nonlin-ear waveevolutionmodelthat iscompletelyintegrablebymeans ofthe inversescatteringtransform method[20–22]. Theclassical MTM is a particularcase ofthe coupled mode equations(CMEs) thatdescribepulsepropagationinperiodicorBraggnonlinear op-ticalmedia

[23–27]

.Furthermore,the CMEs also appearin other physical settings.In particular andrelevant torogue waves,they describe ocean waves in deep water for a periodic bottom [28]. As such, the search for novel solution forms of these equations includingrogue waves,provides understanding ofnonlinear phe-nomenonandleadstoapplicationsbeyondopticalsystems.Inthis respect, benefiting fromthe result[25,29] that many MTM solu-tions(including singleandmulti-solitonsandcnoidal-waves)may be mapped into solutions of the CMEs, provides a tool used in severalworksincludingoceanwaves

[30]

,BEC

[31]

and metama-terials[32].

AfterdiscussingtheanalyticalroguewavesolutioninSection2, inSection3wenumericallyconfirmitsstability,andshowthatit mayalsobeappliedtodescribethegenerationofextremeevents http://dx.doi.org/10.1016/j.physleta.2015.01.026

1068 A. Degasperis et al. / Physics Letters A 379 (2015) 1067–1070

inthemoregeneralcontextoftheCMEs. Finally,inSection 4we discussthephysicalimplementationofMTMroguewavesbyusing coherenteffectsinresonantnonlinearmedia,suchas electromag-neticallyinducedtransparency (EIT),which maylead tothegiant enhancementofcross-phase modulation(XPM)withthe simulta-neoussuppressionofself-phasemodulation(SPM).

2. Analytical solution

Letus expressthe MTM equations for the forwardand back-wardwaveswithenvelopesU andV ,respectively,as

Uξ

= −

i

ν

V

−

i

ν

|

V

|

2_U Vη

= −

i

ν

U

−

i

ν

|

U

|

2_V

_.

₍₁₎

Here thelight-cone coordinates

ξ

,

η

are relatedtothe space co-ordinate z and time variablet bythe relations

∂

ξ

= ∂

t

+

c

∂

z and

∂

η

= ∂

t

−

c

∂

z,wherec

>

0 isthelineargroupvelocity.Eventhough

thearbitraryrealparameter

ν

canberescaled tounity,wefindit convenienttokeepitfordimensionalreasons.

TheroguewavestraveloverthefollowingCWbackground

U0

=

aeiφ

,

V0

= −

beiφ (2)

where,withnolossofgenerality,theconstantamplitudesa andb

arereal,andthecommonphase

φ (ξ,

η

)

is

φ

=

α

ξ

+ β

η

,

α

=

b



ν

a

−

b

ν



,

β

=

a



ν

b

−

a

ν



.

(3)

Uptothispointweconsiderthetwoamplitudesa,b asfree back-groundparameters.It canbeprovedthat roguewavesolutions of Eqs.(1)existifandonlyifthetwoamplitudesa

,

b satisfythe in-equality

0

<

ab

<

ν

2

.

(4)

ByapplyingtheDarbouxmethodtotheMTM

[33]

,oneobtainsthe followingroguewavesolution

U

=

aeiφ

μ

∗

μ



1

−

4iq ∗ 1q2

μ

∗



,

V

= −

beiφ

μ

μ

∗



1

−

4iq ∗ 1q2

μ



(5)

withthefollowingdefinitions

q1

= θ

1

(

1

+

iq

)

+

q

θ

2

,

q2

= θ

2

(

1

−

iq

)

+

q

θ

1

,

q

=

a

χ

∗

η

+

b

χ

∗

_ξ

₍₆₎ and

μ

= |

q1

|

2

+ |

q2

|

2

+ (

i

/

p

)



|

q1

|

2

− |

q2

|

2



,

p

=



ν

2 ab

−

1

>

0

.

(7)

Intheexpression

(5)

,whichistheanalogofthePeregrinesolution ofthefocusingNLSequation,thefreeparametersarethetworeal backgroundparametersa

,

b,whicharehoweverconstrainedbythe condition (4), and the two complex parameters

θ

1

,

θ

2,while the

parameter

χ

isgivenbytheexpression

χ

=

b

ν

(

1

+

ip

)

=

ν

a

(

1

−

ip

)

.

(8)

Expression

(5)

oftheroguewavesolutionmaybesimplifiedby fix-ingthereferenceframeofthespace–timecoordinates.Thegeneral solution (5) maythen be obtained by applying tothis particular solutionaLorentztransformation.

Accordingtothelastremarkabove,wenowprovidetherogue wave solution in terms of the space z

=

c

(ξ

−

η)

and time t

=

(ξ

+

η)

coordinatesdirectly.ByrewritingtheCWphase

(3)

inthese coordinates,oneobtains

φ

=

kz

−

ω

t

,

k

=

ν

2c



1

−

ab

ν

2



b a

−

a b



,

ω

= −

ν

2



1

−

ab

ν

2



a b

+

b a



,

(9)

wherek isthewavenumberofthebackgroundCW.Settinga

=

b

means choosing the special frame of reference such that k

=

0. Note that the other possibility a

= −

b doesnot satisfy the con-dition that p is real (see (7)). From a physical standpoint, the CW background solution with a

=

b corresponds to a nonlinear wave whose frequency

ω

= −

ν(

1

−

a2

/ν

2

₎

_{enters deeper inside}

the (linear) forbidden band-gap

ω

2

_<

_ν

2 _{as its intensity grows}

larger.AlinearstabilityanalysisoftheCWbackgroundsolution(2)

showsthat it ismodulationallyunstablefor perturbationswitha wavenumberk2

_<

_4a2

_/

_c2_{(fordetails,see}

_[34]

_).Notethat

modula-tion instability gain extends all the way to arbitrarily long-scale perturbations (albeit with a vanishing gain), a condition which hasbeenrefereedto as“basebandinstability”, andthat isclosely linked with the existence condition of rogue waves in different nonlinearwavesystems(e.g.,theManakovsystem,seeRef.[15]).It isalsointerestingtopointoutthat,outsidetherangeofexistence oftheroguewavesolution

(5)

,thatisfora2

>

ν

2_{,thebackground}

isunstablewithrespecttoCWperturbationwithafinite(nonzero) gain(seeRef.[34]).

Byusingtranslationinvariancetoeliminate theparameters

θ

1,

θ

2,onefinally endsupwiththefollowingexpressionoftheMTM

roguewavesolution

U

=

ae−iωt

μ

∗

μ



1

−

4

μ

∗q ∗

₍

_q

₊

_i

₎



_,

V

= −

ae−iωt

μ

μ

∗



1

−

4

μ

q ∗

₍

_q

₊

_i

₎



₍₁₀₎ where

ω

= −

ν



1

−

a 2

ν

2



,

q

= −

a 2

ν

c

ip

(

z

−

z0

)

−

c

(

t

−

t0

)

,

p

=



ν

2 a2

−

1

,

μ

=

2

|

q

|

2

_{+ (}

₁

₊

_{2 Im q}

₎



1

−

i p



,

(11)

where z0 andt0 arearbitraryspace andtime shifts,respectively.

Notethatafurthersimplificationmaycomefromrescalingz,t,U , V byusingthelengthscalefactorS

= −

ν

c

/

a2_.

In Fig. 1 we show the dependenceon space and time of the intensities

|

U

|

2 _and

_|

_V

_|

2_{oftheforwardandbackwardcomponents}

oftherogue wave

(10)

.Here wehaveset

ν

= −

1,c

=

1,a

=

0

.

9,

t0

=

2 andz0

=

3

.

5.Ascanbe seen,theinitial spatialmodulation

att

=

0 evolves intoan isolated peakwitha maximumintensity ofaboutninetimes larger thantheCWbackground intensity.The correspondingcontourplotoftheseintensitiesisshownin

Fig. 2

.

3. Numerical results

In order to verify the spatio-temporal stability of the rogue wave solution (10) over a finite spatial domain, we numerically solved Eqs.(1) with

ν

= −

1, c

=

1,and using theinitial (i.e.,at

A. Degasperis et al. / Physics Letters A 379 (2015) 1067–1070 1069

Fig. 1. Evolution in time and space of the intensities in the forward and backward components of the rogue wave solution(10).

Fig. 2. Contourplotofthe intensitiesoftheforwardand backwardwavesas in Fig. 1.

Fig. 3. Numerical solution corresponding to the analytical solution inFig. 1.

theexactexpression

(10)

,withthesamesolutionparametersasin

Figs. 1–2,andwithL

=

5



4S.

Fig. 3displaysthenumericallycomputedintensitiesofthe for-wardandbackwardwaves:asitcanbeseen,thereisanexcellent agreementwiththeanalyticalsolution.Thisconfirms thestability andobservabilityoftheroguewavesolution

(10)

,inspiteofinthe presenceofthecompetingbackgroundMI.

At this point it is quite natural, and interesting, to numeri-cally check whether the initial conditions ofthe rogue wave

so-Fig. 4. NumericalsolutionofCMEswithσ=0.3 andinitialandboundaryconditions asgivenbytheanalyticalsolutionofFig. 1.

lution (10) may induce also the generation of a rogue wave as modeled by the CMEs for pulse propagation in nonlinear Bragg gratings [25] or ocean waves with periodic bottom [28]. Indeed, theCMEs Uξ

= −

i

ν

V

−

i

ν



|

V

|

2

+

σ

|

U

|

2



U

,

Vη

= −

i

ν

U

−

i

ν



|

U

|

2

+

σ

|

V

|

2



V (12)

differfrom theMTM Eqs.(1)by an additional SPMterm, its rel-ativestrengthbeingexpressed bythe coefficient

σ

. Wehave nu-mericallysolved Eqs.(12)withinitialandboundaryconditionsas given by expression (10). As shown by Fig. 4 and Fig. 5, quite surprisingly even in the case of a comparatively large SPM con-tribution, one still observes the generation of an extreme peak with nearly the same intensity and time width as in the MTM case. However now the peak no longer disappears, but it leaves behind tracesintheformofdispersivewaves,owingtothe non-integrability nature of the CMEs. InFig. 4 we set

σ

=

0

.

3, while qualitativelyverysimilarresultsarealsoobtainedforlargervalues of

σ

(e.g.,for

σ

=

0

.

5,whichcorrespondstothecaseofnonlinear fibergratings).

Ontheotherhand,in

Fig. 5

weshowthecasewith

σ

= −

0

.

5, thatisrelevanttowaterwave propagationinoceanswitha peri-odicbottom

[28]

.Interestingly,inthehydrodynamic casethe for-mation ofafirstrogue peak wherebothcomponentshavenearly thesameamplitudeasintheMTMcase

Fig. 3

,isfollowedbywave

1070 A. Degasperis et al. / Physics Letters A 379 (2015) 1067–1070

Fig. 5. As inFig. 4, withσ= −0.5.

breakingintosecondary, andyetintensepeaksineachofthetwo generatedpulsesthattravelinoppositedirections.

4. Discussion and conclusions

ForthestrictapplicabilityoftheanalyticalMTMroguewave so-lution

(10)

ofSection2,itisnecessarythatSPMcanbeneglected withrespecttoXPM.Thissituation occurswheneverthe forward and backward wave envelopes have differentcarrier frequencies, say,

ω

U and

ω

V, and their frequency difference

ω

=

ω

U

−

ω

V

is close to a resonant frequency of the nonlinear medium [35]. Whenever the forward and backward waves are coherently cou-pled in a four-level atomic system by a CWpump field, a giant enhancement ofthe strength ofXPM isalso possible viathe EIT effect

[36,37]

,withnocompetingSPM.Inaddition,EITbringsthe important benefitof zero linearabsorption losses, andit is only limitedby theresidual two-photon absorption,whichcan be ne-glectedforrelativelyshortinteractiondistances.

For maximizing the interaction length of the counter-propag-ating waves, diffraction effects can be suppressed by using an atomic vapor cell containing an hollow-core photonic band gap fiber (PBGF). A scheme of Bragg soliton generation in a coher-ent medium exhibiting EIT has been previously discussed, how-evercounter-propagatingpulsesatthesameprobefrequencywere considered [38]. A dynamic refractive index grating for coupling forwardand backward signal waves at differentfrequencies may be induced by the standing wave generated fromthe beating of twocounter-propagatingpumpwaves,withafrequencydifference equalto

ω

.ThisprocessisknownasBraggscatteringfour-wave mixing(BS-FWM)

[39]

, andit enablesnoise-free frequency trans-lation. Indeed, BS-FWM in the co-propagation geometry and at microwattpump powerlevels hasbeenrecentlydemonstratedin Rubidiumvapor,confinedtoafewcentimeterslongPBGF

[40]

.

Notethatthesolution

(10)

mayalsoapplytodescribeextreme waveemergence intheco-propagationoftwo signalsatdifferent carrierfrequencies [41], bysimply interchanging therole oftime andspacevariablesinEqs.(1).Inthiscase,adynamicgratingmay be inducedby thebeating oftwopump wavesin theorthogonal polarization

[42]

.

In short summary, we obtained the rogue wave solution of the classical MTM, which extends the Peregrine soliton solution oftheNLSequation tothecaseofwavepropagationinaperiodic

nonlinearmedium.Animplementationisproposedusingcoherent resonant wavemixing:weenvisage thatBraggsolitonsandrogue wavesmayultimatelybe observedatsub-milliwatt pumping lev-els by usingchip-scalewaveguides that areevanescently coupled toatomicvapors

[43]

.

Acknowledgements

The present research was supported by Fondazione Cariplo, grant No. 2011-0395, and the Italian Ministry of Education, Re-searchandUniversities (MIUR)(grantcontract2012BFNWZ2).

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