CHAPTERIVNUMERICALSOLUTIONSTOTHENONLINEARSCHRÖDINGEREQUATION

4.1IntroductionIngeneral,analyticalsolutionstothefullMaxwellwaveequationforanonlinearopticalsystemdonotexist.Evennumericalsolutionstothewaveequationareextremelydifficulttoimplementduetothedimensionalityoftheproblem.Thevectorformofthewaveequationisafour-dimensional(threespatial,onetemporal),second-orderpartialdifferentialequation.Thus,approximationsbasedonpropagationconditionsandexperimentalresultsareneededinordertosolveanapproximatescalarformofthewaveequation,i.e.thenonlinearSchrödingerequation.However,theapproximationslistedinthepreviouschapterdolimitthegeneralityandvalidityofthesolutions.Forexample,theconditionextremenonlinearity,asforthecaseinsupercontinuumgeneration,isapropagationregimewhereslowlyvaryingenvelopeapproximationmaybeviolated.ThepurposeofthischapteristoprovideanintroductiontoaverypowerfulmethodinnumericallysolvingtheNLSE,knownasthesplit-stepFouriermethod(SSFM)[15].ThechapterwillbeginwithalistpointingtheadvantagesoftheSSFMcomparedtofinite-differencemethods.Then,theSSFMthesymmetricSSFMwillbeintroduced.ThechapterwillthendetailtheinclusionoftheRamaneffectinthenumericalsolution.

4.2WhyusetheSplit-StepFourierMethod?TheSSFMisthetechniqueofchoiceforsolvingtheNLSEduetoitseasyimplementationandspeedcomparedtoothermethods,notablytime-domainfinite-differencemethods[73].ThefinitedifferencemethodsolvestheMaxwell’swaveequationexplicitlyinthetime-domainundertheassumptionoftheparaxialapproximation.TheSSFMfallsunderthecategoryofpseudospectralmethods,whichtypicallyarefasterbyanorderofmagnitudecomparedtofinitedifferencemethods[74].Themajordifferencebetweentime-domaintechniquesandtheSSFMisthattheformaldealswithallelectromagneticcomponentswithouteliminatingthecarrierfrequency.Asshowninthepreviouschapter,thecarrierfrequencyisdroppedfromthederivationoftheNLSE.Thus,finitedifferencemethodscanaccountforforwardandbackwardpropagatingwaves,whiletheNLSEderivedfortheSSFMcannot.Sincethecarrierfrequencyisnotdroppedintheformoftheelectricfield,finite-differencemethodscanaccuratelydescribepulsepropagationofnearlysingle-cyclepulses.WhilethefinitedifferencemethodmaybemoreaccuratethantheSSFM,itisonlyatthecostofmorecomputationtime.Inpractice,themethodchosentosolvetheNLSEdependsontheproblemathand.Forpulsepropagationfortelecommunicationapplications(~100pspulsesthrough80kmoffiberwithdispersionandSPM)theSSFMworksextremelywellandproducesresultsthatareinexcellentagreementwiththeexperiments[75,76].However,theSSFMwouldnotworkformodelingfiberBragggratingswherethereexistsaforwardandbackwardpropagatingwave.ThisthesisdemonstratesthattheSSFMalsoworksefficientlyandaccuratelyfordescribingpulsepropagationinmicrostructurefiber.

4.3TheSplit-StepFourierMethodThemathematicaltermsduedispersionandnonlinearityareseparateanddecoupledintheNLSE.ItisthisfactthatallowstheuseoftheSSFMforsolvingtheNLSE.BylookingatNLSE,theoperatorsˆDandˆNcanbewrittentocorrespondtothedispersive(andabsorptive)andnonlineartermsrespectively(ignoringtheRamaneffect),

112ˆ=22mmmmm

m

iDt−−=α∂−−β∂¦(4.1)

and()22

0

2ˆ(,)(,)(,)(,)i

NiEztEztEztEztt§·∂=γ+¨¸ω∂©¹(4.2)

whereE(z,t)isthecomplexfieldenvelopeatstepzandtimet.TheNLSEthencanbewrittenintheoperatorformas

(,)ˆˆ()(,).

Ezt

DNEzt

z

∂=+

∂(4.3)where()ˆˆ(,)exp((1),),EjhthDNEjhtªº=+−¬¼(4.4)

isasolutiontothedifferentialequationatstepz=jh(jisaninteger).NotethattheˆNoperatormultipliesthefieldsolutionandisafunctionofthesolutionE(z,t).TheˆD

operatorisadifferentialoperatorexpressedintermsoftimederivativesthatoperateonE(z,t).Toreducethecomputationaltime,theoperationofˆDisperformedinthefrequencydomain;thistransformsthederivativesinthetimedomaintoamultiplicationinthefrequencydomain.AftertakingtheFouriertransformofˆDthemultiplicativeoperatorinthefrequencydomainisobtained,

111122ˆˆ(){}().2222mmmmmmmmm

mm

iiDiDi

t

−−

−−==

­½α∂α

ω≡=−−β=−−βω®¾

∂¯¿

¦¦))

(4.5)TheSSFMisaniterativeprocessthatdeterminesthefieldsolutionforspatialstepsofh.Thisisperformedstep-by-stepfortheentirelengthofthefiber.TheprocedureduringonestepisillustratedinFigure4.1.AdielectricmediumoflengthLisbrokenintosL=L/hstepsoflengthh.ThefieldpropagationsolutionE(jh,t)perspatialstephatthestepjh(j=1,2,….sL)fortheentirelengthoffiberusingrelation

()(){}{}1ˆˆ(,)exp()exp((1),),EjhthDiFhNEjht−≈ω−)

(4.6)