Space - PDF document

JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003DegreeofPolarizationDegradationDuetoCross-PhaseModulationandItsImpactonPolarization-ModeDispersionCompensatorsAlbertoBononi,ArmandoVannucci,Member,IEEE,A.Orlandini,Member,IEEE,E.Corbel,S.Lanne,andS.Bigo,Member,IEEEAnovelanalyticalmodelisproposedtopredictthecross-phasemodulation(XPM)-induceddepolarizationinatwo-channeltransmissionsystem,inwhichtheStokes’vectorofeachchannelrotatesaroundaspace-invariantpivotbyatime-varyinganglewhichdependsonthetotalinstantaneous JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003TheDOPformulaturnsouttobequitesimplewhenthepumpkeying(OOK)modulatedbyasquarewaveemulatinganinfinitealternationofmarksandspaces,whileitisslightlymorecomplexwithrandompumpbitsequences(RBS).AgoodagreementoftheDOPformulawith10-Gb/sexperimentalre-sultsandsimulationsbasedonthebeampropagationmethod(BPM)[17]isfound.SimulationresultsarealsopresentedthatshowtheDOPdegradationinthepresenceofPMD.Wecarriedoutanexperimentofatwo-channel10-Gb/sWDMsystemwhosestatisticalperformanceinthepresenceofXPMwasstudiedbothwithandwithoutfirst-orderDOP-basedPMDcompensation[8].TheresultsshowthatPMDdistortingeffectsareemphasizedbyXPM,andthereforetheOPMDCefficiencyisreduced.Thepaperisorganizedasfollows.InSectionII,thenovelan-alyticalmodelforXPM-inducedDOPdegradationinaWDMsystemisderivedforapumpandprobescheme.InSectionIII,theDOPoftheprobeiscomputedfromalinearfilteringofthepumpchannel,andanexplicitDOPformulainthecaseofa1010modulatedpump,andforanRBSmodulatedpumparederivedandcheckedagainstBPMsimulations.InSectionIV,exper-imentalDOPmeasurementsforvaryinglaunchpowercondi-tionsareshowninordertovalidatetheanalyticalDOPfor-mula.Moreover,theperformanceofaWDMsystemwithPMDandXPMisevaluatedintermsofthestatisticsofthe -factorpenalty,bothwithandwithoutfirst-orderPMDcompensation.InSectionV,wesummarizeourconclusions.Theappendixcon-tainsanalyticalresultsusedinSectionIII.II.TAROUSELInalongWDMsystemwithnegligiblePMD,theevolutionofeachpropagatingchannelinthenonlinearregimecanbeef-fectivelydescribedbytheManakovequation[1],whichtakesintoaccounttheinteractionbetweenthenonlinearityandtherapidlyvaryingfiberbirefringencethroughascalingofthenon-linearcoefficient byafactor8/9.SuchequationisobtainedfromtheCNLSEbyaveragingtheKerreffectoverthePoincarésphereundertheassumptionofcompletemixing,whichisver-ifiedwhenthefibercorrelationlengthismuchshorterthanthenonlinearlength[18].TheManakovequationfortwocompletelypolarizedOOK-probesignals,propagatingatwavelengths and ,yieldsthefollowingsystemofnonlineardifferentialintheproberetarded-timeframe alongthefiber [1] (1)where and arethecomplexenvelopesofprobeandpump,respectively,referredtotheprobefrequency; isthewalk-offparameter[17]; and aretheWeuseheretheengineers’notationforFouriertransforms,asopposedto[1],wherethephysicists’notationisused.GVDcoefficientsat and ,respectively;andthesymbol indicatesthetransposeconjugate.Notethattheenvelopesareunattenuated,asattenuationhasbeenremovedbyachangeofvariableandappearsasanexponentialthatmultipliesthenon-linearcoefficient .Inthefirstequationin(1),werecognize ,wherethe ’sarethePaulimatricesandweusedthestandarddecom-positionoftheprojectormatrix [19],inwhichthepump andtheinstantaneouspumpStokes’vector appear.Byusingsimilarmanip-ulationsinvolvingtheStokes’vectoroftheprobe,(1)becomes wherethetensor isknownasthePaulispin[20].NotethatthenonlineartermsofSPMandscalarXPMareweightedbyanonlinearcoefficientscaledby8/9and4/3,respectively.ThevectorialXPMeffectsaretakenintoac-countbythetracelessHermitianmatrices and NeglectingtheGVDterms ,theJones-domain(2)canbetranslatedintoanequivalentequationofmotionoftheStokes’vectors[21] wherethesymbol standsforvectorcross-product.Equation(3)describesthemotionofthepump/probeStokes’vectorsasalocalrotationaroundatime-and -varyingaxisalongthefiber,wheretherotationspeeddependsonthepowerofthemodulatedchannelsandontheamountofwalk-offamongthem.InthegeneralcaseofOOK-modulatedsignals,thepreviouslyshownequationofmotiondoesnothaveaclosed-formAsaspecialcase,itisknown[1],[12]that,whenbothprobeandpumpareCW,theStokes’vectors and in(3)performarotationaroundatime-and -independentpivot .Theproofissimplyobtainedbyaddingthetwotime-independent(3).Atcoordinate ,therotationangleis wherethemagnitudeofthepivot dependsonthe(fixed)relativepolarizationangle between and andonthepeakpowersoftheprobeandthepump and respectively, istheeffectivefiberlength,andtherotationspeedin(4)decreasesin becauseofthefiberattenuation .Notethatthetwosignalsundergothesamerotationanglearound ;however,ifthetransmittedpowerismuchlargerononechannel,suchchannelpracticallyremainsfixed,sinceits etal.:DEGREEOFPOLARIZATIONDEGRADATIONDUETOXPMStokes’vectorisalmostcoincidentwith .Finally,notethattherotationangledoesnotdependonthewavelengthdistancebetweenpumpandprobe:nomatterhowfarapart,thetwoCWchannelsundergotherotationin(4).AccordingtosuchCWmodel,theWDMchannelsclearlydonotdepolarize.However,theCWassumptionforWDMchan-nelsisreasonableonlyforthosesystemsinwhichchromaticdispersionorchannelspacingaresolargethateachchannelisonlyaffectedbytheaveragepoweroftheremainingchannels.Hence,theCWmodelfailstodescribethedepolarizationduetochannelmodulationandfinitewalk-off.Inordertotacklesuchamoregeneralcase,weadoptthefol-lowingcarouselmodelasanapproximatesolutionof(3),validforcompletelypolarizedinputpumpandprobefields,withanOOK-modulatedpumpandaCWprobe.Assumeat thatthepumpis.AsintheCWmodel,the“carousel”ofthetwoStokes’vectorsstartsrotatingaroundafixedpivot ,where isaunitvector.Assoonasthepumpgoes,thecarouselstopsitsrotationaround andresumesitonlywhenthepumpgoesagain.At ,therotationangleis[6] anddependsonthesectionoftheOOK-modulatedpumpbitsthathavewalkedpasttheprobefromtheinputtocoordinate ,asexpressedbythepumppower intheaboveintegral.ItissuchAtime-varyingrotationanglearoundanaverageangle (duetotheCWcomponentofthepumpandpredictablebytheCWmodel)thatcausesthedepolarizationoftheprobesignal,whoseDOPisthereforedecreased,aswewillshortlyquantify.Itcanbeeasilyverifiedfromthefirstequationin(3)thatthesolutionshownpreviouslyisexactinthelimitingcase andnonreturn-to-zero(NRZ)pumpbits.Wenotethatwiththechangeofvariable ,theinte-gralin(5),canbewrittenasaconvolution operation wherethewalk-offfilterinpulseresponse[22]is (7)where isagatingfunctiontakingvalue1for ,andzeroelse.Notethat,forsimplicity,weassumed ,althoughthecase issimilar[22].Thefrequencyresponseofthewalk-offfilterhaszerosatmultiplesof duetothegatingfunction[22],[23].Hence,aperiodic1010pumpsequence,ofaperiodtwicethebittime ,shouldproduceontheprobeonlyanaveragerotation andnodepolarizationat if isamultipleof ,i.e.,ifthefiber isamultipleof ,with Fig.1.SketchoftheinteractionofaninitiallyCWprobeandaperiodicsquarewaveformpumppredictedbythecarouselmodel,atdifferentpropagationlengths.Seetextforadescription.beingthewalk-offlength,i.e.,thelengthoverwhichthepumpwalkspasttheprobebyonebit.WevisualizesuchacaseinFig.1,wherewerepresentthein-teractionbetweenalinearlypolarizedCWprobeandacircularlypolarizedpumpmodulatedbytheperiodicalternationofmarkandspacebits,inthecase .Wealsoneglecttheattenua-tion.The(real)envelopeoftheprobe andthenormalizedpower ofthetwopropagatingfieldsareportrayedinthetimedomainatmultiples ofhalfthewalk-offlength.Focusontheinteractionbetweenthefirstmarkbitofthepumpandthetime“sections”A,B,C,D,Eoftheprobeattimes ,T/2,T,3T/2,2Trespectively.At theprobeiscompletelypo-larized[Fig.1(a)].At ,A,DandEareunaltered,astheyhavenotinteractedwiththepump,whileBandCareequallyrotatedbythepump[Fig.1(b)].At ,sectionsAandEarestillunaltered,sectionBhasthesamerotationasat sinceitstoppedsensingthepumpbit,whilesectionCisthemostrotated,andsectionDhasthesamerotationassec-tionBsinceitsensedthepumpbitforthesameamountoftime[Fig.1(c)]:thisisthe coordinateatwhichthedepolarizationinducedbyXPMisthelargest.Movingat ,sec-tionsAandEstartrotatingwhilethemostadvancedsectionCstopssensingthepump.Finallyat ,allprobesectionshaverotatedbythesameamountastheysensedthepumpbitbythesameamountoftime,sothattheprobeisagaincompletelypolarized,aspredictedby(6).Ifthepumpperiodincreasesto ,thesamesituationoftotalrepolarizationoftheprobecanbefoundat ,andthemaximumpolarizationrotationatsectionCat isincreasedby times.Suchargumentcanconvinceusthatlongsequencesofmanyconsecutivemarks JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003 Fig.2.SOPevolutiononthePoincarésphereofaninitiallyCWprobe,ofasquare-waveformpump,andoftheirrotationaxisina100-kmlinkwith 10ps/nm/km,0.4nm,and .BPMsolutionofManakovfollowedbymanyspacesonthepumpyieldtheworstdepolar-izingcase.SuchpointwillbefurtherexploredinSectionIIIonDOP.Wenowmovetothelimitsofourmodel.Wealreadyverifiedthatthecarouselmodelgivestheexactsolutionof(3)whenthepumplargelydominatesthesignalandthusremainscompletelypolarized.However,atequalsignalandpumppowers,weshowinFig.2,bydirectBPMsolutionoftheManakov(2),theSOPevolutionalong (forfixedtime )inasituationoflargewalk-off,wheretheSOPsofaninitiallyCW (darkgreypoints),ofapump at10Gb/sby“1010 ”NRZbitpattern(greypoints)andtheSOPoftheir“rotationaxis” (blackpoints)arerepresentedonthePoincarésphere.Thetwochannelspropa-gateina100-kmpolarizationmaintainingfiberwithparameters 10ps/nm/km, 0.2dB/km, 1.34W/kmwithaveragepowerof13dBm,spacedby0.4nmandwithawalk-offof4boverthelink.Startingfrom ,duringthefirstmarkofthepump,thetwoSOPsrotatearoundthefixedpivot(blackcentralpoint)describingthetwoarcs .Then,whenthepumpisofftherotationstops,asexpectedfromthecarouselmodel.Afterthat,sincetheincomingmarkbitofthepumphasalreadybeendepolarizedbyitspreviouspropagation,theSOPsoftheprobeandofthepumpmove,respectively,onthedarkgreyandgreyarcs ,rotatingaroundapivotthatisnotfixedanymore,butmovesontheblacktrajectory .Thesameexpla-nationcanbegivenforwhathappensontrajectory Notethat,ifthepumpisnotaperfectlysquareNRZwave-form,evenintheabsenceofpumpdepolarization,thepivot varieswithtimeduringamarkbit.Morelimitsofourmodelarethefollowing:i)sinceitstemsfromtheManakovequation,itdoesnottakeintoaccountthedepolarizationduetoSPM[10];ii)thepulsedistortionduetoGVDisneglected;andiii)PMDisneglected.TheeffectofPMDcanbeunderstoodasfollows.Ifonethinksofarealfiberasaconcatenationofrandomlyorientedbirefrin-gentplates,thenwithineachplatepumpandproberotatebydifferentanglesaroundthelocalbirefringencevector.HencetheneteffectofPMDistorandomlyvarytherelativepolarization andtheorientationofthepivot duringpropagation.III.DOPCALCULATIONInaWDMsystem,thedepolarizingeffectsofXPMcanbeevaluatedbynumericalintegrationoftheManakov(2)(extendedtothecaseof channels[1])bymeansoftheBPM.However,theaccuracyoftheBPMispaidwithalongcomputationaltime,especiallyinlonglinkswherelargepowersarelaunched.WeprovidehereafasterbutaccurateDOPcalculationbasedonthecarouselmodel,whichpostulatesthatatthefiberoutput theSOPoftheprobedepictsintimeacirculartrajec-toryaroundthepivot,witharotationanglethatswingsintimearoundanaveragevalue byanamount ,whoseexpressionisobtainedfrom(6).Withoutlossofgenerality,wechooseareferenceframeoftheStokes’spaceinwhichthepivotisalignedwiththethirdStokes’ ,andthecomponentoftheaverageprobeoutputSOP iszero,sothatwecanexpressthetime-dependentoutputprobeSOPas wherethezero-meanprocess representstheSOP’sazimuth,and istheanglebetweentheprobeandthepivot.Suchanglecanbeobtainedfrom andthepump-probepower throughtheanalyticalrelationship .Using(8)inthedefinitionofDOPyields whereallweneedtoevaluateisthetimeaverages and ,with filteringofthenormalizedOOKpumppower.Wecanalreadydrawsomeconclusionsfromsuchequations.First,thelargertheswingangle ,thesmallertheDOP.Therefore,sincethewalk-offfilterisalow-passfilter,thelowerthepumpmodulationfrequency,thelowertheDOP,thusconfirmingthesameobservationalreadymadeaboutFig.1.Next,ifpumpandprobeStokes’vectorsareinitiallyalignedorcounteraligned,then ,sothattheDOPisunity.Unfortunately,thepresenceofPMDpreventssuchinitialalignmenttobekeptduringpropagation,sothatdepolarizationoccurs.Finallynotethat,byinvokingtheergodictheorem[24],onecancomputethetimeaveragesin(9)byusingthecharacteristicfunction(CF)oftherandomvariable(RV) where denotesexpectation. etal.:DEGREEOFPOLARIZATIONDEGRADATIONDUETOXPMIfonenowwantstoavoidanyfurtheranalysis,onemaynotethat(10)and(9)canbeusedforafastsemianalyticalMonte-CarlosimulationoftheDOP.Althoughsuchmethodreadilypro-videsuswiththecorrectDOPvalues,wenextlookforexplicitexactDOPformulasthathighlightthefunctionaldependenciesoftheDOP.A.PeriodicPumpWestartbyobservingthat(9)canbegivenaclosedformifthepumpisaperiodic1010 modulatingsignal.Considerthecaseofapump modulatedwithaperiodicalternationofNRZpulses,with .Aperiodicandskew-sym-metricsignal results,withperiod Ifweassumethathigherorderharmonicsarefilteredoutbythewalk-offfilter,onlythefirstharmonic ofitsFourierseriesexpansioncontributesto ,with , thefrequencyofthe“1010”sequence,and thebitrate.Hence,from(10),weget (11)where isthefrequencyresponseofthewalk-offfilter, thephaseof ,and forlongfiberswith [22],[23].ThemaximumswinganglefortheprobeSOP creasesforincreasingwalk-offs andincreasingbitrates,untiltheeffectofattenuation becomesdominant.Inotherwords,alargerwalk-offlength theamountofdepolarizationontheprobesignal,asexpectedac-cordingtothediscussionaboutFig.1.Toevaluatethetime-av-eragesin(9),weexpand and inFourierseriesas (12)where istheBesselfunctionofthefirstkindoforder Byaveragingoveratimemuchlongerthan ,onegets[25] IfthesupportingpulsesofthepumparenotNRZ,(13)canbeeasilygeneralizedbyinsertingthefirstFouriercoefficient theconsideredpulseshapeintheargumentoftheBesselfunc- ,althoughthecarouselmodelisslightlylessaccurateinsuchcase,asalreadynoted.Propagationon perfectlycom-pensatedspanshasbeenaccountedforbymultiplying by .Moreover,iftheextinctionratio oftheconsideredsystemisnotzero,acorrectivefactor thatmultipliesthepivot mustbeintroducedintheargumentof .Nowusing(13)in(9)gives wherethedependenceoftheDOPontherelativepump-probepolarizationangle isimplicitin and .From(14)and(13)wecanconcludethat,ifpolarizationcontrolofthesignals or cannotbeachieved,e.g.,duetoPMD,thebasiccountermeasureagainstDOPdegradationistoincreasethebitwalk-offbyfurtherspacingthechannelsorbyusingamoredispersivefiber,soastoreducetheargumentoftheBesselfunction.Clearly,alsoincreasingthebitrate impliesareductionofXPM-inducedDOPdegradation.B.RBS-ModulatedPumpAnempiricalDOPformulafortheRBScasewasderivedin[25],basedonanextensionof(13).Hereinstead,weprovideanexactDOPformula.TheappendixdetailsamethodfortheevaluationoftheCFoflinearfilteringofapulse-amplitude-modulatedsignal.Inourcasetheoutput isgivenin(10)andthePAMsymbolsarebinary,with .Theconvolution in(20)canbemadeexplicitwithourexponentialfinitememorywalk-offfilter in(7)whenthesupportingpulse iszero-meanNRZ.Assuming ,i.e.,largewalk-offperspan,onegets if if if sothattheCFin(20)becomeshere wherealltermswith disappearintheinfiniteproductbythecausalityof .Withthechangeofvariable andposing ,thissimplifiesto (17)where istheeffectivefiberlength.Theinfiniteproductcanbesafelytruncatedto .Since in(17)isreal,itcoincideswith ,andthus .HenceagaintheDOPformulaisgivenby(14).Ausefulandtightlowerboundisfoundbysetting intheintegrandin(17) (18) JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003 Fig.3.DOPcurvesversusobtainedwithBPMnumericalsimulations(dashedlinewithcrosses)andanalyticalDOP(14)(solidline)fortwoindependentRBSmodulatedchannelswithNRZpulses,propagatingona100-kmlinkofTeraLightfiber,with 3dBm,inthetwocasesPR=9.3dBandPR=11.8dB.Greylines:periodic1010...pump;blacklines:RBS-modulatedpump.fromwhichitisclearthatverysimilarconsiderationsabouttheeffectofsystemparametersasin(13)apply.Inordertovalidatesuchformulas,inFig.3,wecomparedtheanalyticalDOPcurvesobtainedby(14)(solidline)tothoseevaluatedwiththeManakovequationthroughtheBPM(dashedlinewithcrosses)whenchannelsaremodulatedwithNRZsupportingpulsesat10Gb/s,withnopoweronzeros ThesystemconsistedofthreeperfectlycompensatedlinksofTeraLightfiber( 0.2dB/km, 1.68W/km,and ps/nm/km)of 100kmeach,withspacing 0.8nm.DOPisplottedversustheangle betweeninputSOPs,foranaverageprobepower dBmandtwodifferentvaluesofpowerratio 9.3dBand 11.8dB.Thegreycurvesrefertotheperiodicallymodulatedpump,whiletheblackcurvesrefertotheRBScase.IntheBPMsimulations,thelocaldistor-tionduetoGVDwasincluded( in(1)).IntheRBSpumpcase,weobserveanexcellentagreementbetweentheoryandsimulation.ThesmalldiscrepanciesaroundthevaluesofminimumDOPcanbeattributedmostlytotheassumptionofa -independentpivotofthecarouselmodel(therewasawalk-off 6.4bovertheentirelink)andinminorparttothefactthatalsotheprobeismodulated.Intheperiodicpumpcase,(14)givesalessaccurateapproximationofthesimulatedcurves.WeverifiedthatthisisduetotheGVDdistortingef-fectonthepumpbits,whichwereneglectedinourcalculations.Intheanomalouspropagationinoursystem,theinteractionofGVDandnonlinearityinfactcompressestheNRZpumppulses,sothatin(13)thefirstFourierharmonicofthepumpshouldbeactuallygivenbya coefficientlowerthan .Forthisreason,theanalyticalresultsoverestimatetheactualprobede-Inallcurves,wenotethatdepolarizationisnotpresentwhentheinputStokes’vectorsareeitheralignedorcounteraligned,asexpected,whileDOPisminimumwhentheyareroughlyor-thogonal,sincethisimpliesamaximumvelocityonthecircular Fig.4.AnalyticalworstcaseDOPversusbitrateforthesamesystemconsideredinFig.3,withRBSpumpandPR=11.8dB.trajectory,whichiseasilyseentobeproportionalto WenoteaverylowDOPminimumofabout20%causedbyXPMatthelargeaveragepumppowerof14.8dBm(RBSpumpcase),whileweremarkthatdepolarizationduetoPMDaloneneverdecreasestheDOPbelow50%forNRZunchirpedpulses,sincetheCWcomponentcannotbedepolarizedbyPMD[10].Notealsothat(14),althoughvalidinthepump-probecase,alsoappliestotheworstcaseofan -channelWDMsysteminwhichallchannelshavethesameaveragepower ,theinter-feringchannelsarecopolarized,andtheirvectorsumhasarela-tivepolarizationangle withrespecttotheprobe.Insuchcase,thenumberofchannels dependsonthepowerratio as Fig.4showstheworst-caseDOPat versusbit ,calculatedwiththeanalyticalformula(14)foranRBSmodulatedpumpwithNRZpulses,forthepreviouslydescribedthree-spansystem,bothforachannelspacing 0.8nm(solidline)andfor 0.4nm(dot-dashedline).Inthe 0.4,aminimumDOPvalueisvisiblearound 13.5Gb/s.Aroughanalyticalapproximationofsuchminimum-DOPvalueof canbeobtainedfrom(18)as .Notethatat40Gb/s,wherethePMDdistortingeffectsbecomestronger,theXPMdepolarizingef-fectsontheTeraLightproduceaDOPof70%at 0.8nm,whileamuchlower at 0.4nm.Toconcludethissection,weshowbysimulationtheeffectonDOPoftwofactorsthatwereneglectedintheanalysis,namelypolarization-dependentloss(PDL)andPMD.Fig.5(a)showstheDOPforthesamesystemusedtoderiveFig.3,withapump-proberatio 11.8dB,butheresignalshadanextinctionratioof 10dB ,andapolarizerwithaPDLofeither1or2dBandadiattenuationaxisalignedeitherwith orwith wasinsertedbeforethereceiver.Firstfocusonthesolidlinecurve(noPDL).WenoteahighervalueoftheDOPminimumascomparedwithFig.3,duetoafiniteextinctionratio.Next,wenotethatPDLmarkedlybreaksthesymmetryoftheDOPcurvesandshiftstheminimumawayfrom . etal.:DEGREEOFPOLARIZATIONDEGRADATIONDUETOXPM Fig.5.DOPcurvesversusobtainedbyBPMsimulationforthesamesystemstudiedinFig.3inthecaseofRBS-modulatedchannelswithPR=11.8dB,andwithanextinctionratio10dB.(a)WithapolarizeratthereceiverandnoPMD;greydashedline:PDL=1dB;blackdashedline:PDL=2dB;solidline:withoutPDL.(b)WithoutPDLbutwithPMD;diamonds:DGD=16ps;circles:DGD=41ps;solidline:withoutPMD. Fig.6.ExperimentalsetupformeasurementsofoutputprobeDOP.Inset:Dispersionmap.Fig.5(b)showsthesame3 100-kmsystemofFig.5(a)withoutPDLandfortwodifferentfiberrealizations,thefirstwithatotaldifferentialgroupdelay(DGD)of16ps(diamonds),andthesecondof41ps(circles).SuchcurveswereobtainedbyBPMsimulationoftheManakov-PMDequation[18],andtookaboutadayofsimulationtime.WenotethatPMDfurtherenhancestheXPM-induceddepolarization,theworstreductionoccurringforthelargestrotationspeedcase 0 ,andthesmallestforthelowesttotalDOPcase 180 ,asalreadynotedin[11].IV.EInthissection,wefirstgivesomeexperimentalresultsonDOPmeasurementthatvalidatethenewanalyticalformula(14).ThenweshowhowtheinteractionofPMDandXPMinatwo-channelsystemcandegradetheperformanceofafirst-orderPMDcompensatorwhosefeedbacksignalisrepresentedbytheoutputprobeDOP[8].A.ExperimentalMeasurementsofDOPWeperformedDOPmeasurementsonthedispersion-man-aged3 100-kmlinkdepictedinFig.6.Thedispersionmapisshownintheinset.WeusedTeraLightasthetransmissionfiberwhosetotalmeasuredDGDonthelinkisbelow2ps,sothatPMDcanbesafelyneglected.Pumpandprobearespacedby 0.8nmandareNRZmodulatedat10Gb/sbyindepen- pseudo-RBS.Weperformedfivesetsof500mea-surementsofboththetotalinputDOPandtheoutputDOP,afterfilteringtheprobechannel,randomlychangingthepolarizationcontroller(PC)eachtime.Fig.7reportsthemeasuredprobeDOP(dots)versus alongwithBPMsimulationresults(trian-gles),andthetheoreticalDOPcurvesobtainedby(14)(solidline).Theaverageprobepowerisfixedat3dBm,whilethepowerratio isvariedforeachsetofmeasurements,being 6.3,9.3,11.8dBinFig.7(a),(b),(c)respectively.Therelativepolarizationangle canbeeasilycalculatedfromthetotalmeasuredinputDOPandthepowerratio byapplyingtheanalyticalrelationship .Theshapeofthetrans-mittedpump intheexperimentscanbereasonablyrepro-ducedintheanalysisbyaraisedcosineinpower,withroll-off andwithanextinctionratioaround dB.Insuchcase,theanalyticalDOP(14)hasbeencalculatedbyusing(16)withthecorrectpulseshapeandbymultiplying byafactor WeusedifferentDOPscalestohighlightthecasesinwhichthepumppowerissmaller.Thespreadinthemeasurementpointsismainlyduetotheamplifiersnoise,whichisthemainsourceofdepolarizationwhenXPMisnegligible.Sucheffectisnottakenintoaccountintheoryandsimulations.SomePDL,estimatedin0.5dB,wasalsopresentintheexperiment,whichjustifiestheslightasymmetryoftheaveragemeasuredDOPcurve[recallFig.5(a)].Forincreasing ,whensignificantXPMeffectsactonthepropagatingsignals,areasonablematchamongexperimental,analyticalandnumericalresultsareshown.NoticethatallplotsinFig.7(a)–(c)havethesameV-shape,andtheirminimumisfoundaround90 ,asexpected.Foreachconsidered value,theWDMsystemmaybeinterpretedasaprobesignalpropagatingwithfour,eight,orsixteeninterferingchannels,respectively,for 6.3,9.3,and11.8dBm,whereallchannelshavethesameaveragepowerof3dBmandalltheinterferingchannels’inputSOPs JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003 (a) (b) Fig.7.OutputprobeDOPafterthreespansversusrelativepolarization,forseveralpump-probepowerratios:measurements(dots),BPMsimulations(triangles),analysisasper(14)(solidline).arealignedasaworstcaseforprobeDOP.Finally,inFig.8,BERmeasurementsobtainedforthesamecaseofFig.7(c)arereported.Withoutgivingdetailsonthereceiver,thepurposeofthisfigureistoshowthat,as increasesfrom0 to180 ,BERimproves.ThereasonisthatthedominantsystemimpairmentisherethescalarXPM-to-intensity-modulation(XPM-IM)conversionoperatedbyGVD[14],whichismaximumforalignedStokes’vectors,andminimumforcounteralignedvectors.SuchtwoextremecasescanberegardedasscalarpropagationcasesoftheManakovequation,whosebehavior Fig.8.ExperimentallymeasuredBERinthesamecaseofFig.7(c).canbequicklyinferredfrom(1),wheretheXPMcontributioninthefirstequationisproportionalto weclearlyseethatthesecondtermisproportionaltothefirstwhenthechannelsarecopolarized(maximumXPM),whileitdisappearswhentheyareorthogonal(i.e., ),givingaminimumXPM.NotethatalargePDLatthereceivercouldchangethepreviousconclusions,asdiscussedin[26].Therefore,ifinputpolarizationcontrolisfeasible,oneshouldalwayschoose 180 forbestperformance,althoughthepresenceofPMDreducestheinputorthogonalityandincreasestheXPM-IM[11].B.PMDCompensatorPerformanceinPresenceofXPMandHigh-OrderPMDInthissection,wequantifytheperformancedegradationin-ducedbyXPMonasystemthatusesafirst-orderPMDcompen-satorbasedontheDOPasafeedbacksignal[2].Severalotherpapersalreadyreportedthedegradationincompensatedsystemsthatusedifferentfeedbacksignals[5],[9].WeconsiderthesameWDMsystemdescribedintheprevioussection,inwhichahigh-orderPMDemulatorandasingle-stageDOP-basedPMDcompensatorwereintroduced,asshowninFig.9.Inordertoreproducetheactualinteractionbetweenhigh-orderPMDandXPM,adistributedmultisectionemulatorwasemployed,whereeachsectionconsistedofapolarizationcon-trollerandaPMF.Onesection,thatintroducesonlyfirst-orderPMD,wasinsertedbeforethefirstspanofTeraLight,thenthreesectionswereaddedateachinterstage,beforethechromaticdispersioncompensatingfiber(DCF).Inthisway,aconditionofmixingofthePMDeffectswithrespecttothefibernonlin-earitywasachievedaccordingtoasortofexperimentallyimple-mentedBPM.Inthelinearregime,PMDdepolarizestheoutputsignalanditsSOPchangesduringpropagation.Consequently,thepolarizationangle betweentwocopropagatingchannelsisrandomlyperturbedbyPMDduringpropagation.Theem-ulatorintroducedanaveragetotaldifferentialgroupdelayof30ps.ThePMDcompensatorwasasingle-stagecompensatorinwhichapolarizationcontrollerandaPMFwereused,andafeedbackloopwasappliedtomaximizetheDOPoftheoutput etal.:DEGREEOFPOLARIZATIONDEGRADATIONDUETOXPM (a) Fig.9.ExperimentalsetupasinFig.6,withinsertionofadistributedPMDemulatorandafirst-orderPMDcompensator. Fig.10.CCDFofrandom-factorpenalty,with(dashedline)andwithout(solidline)PMDmitigation,withaverageDGD30ps,(a)inthelinearregimeand(b)inthenonlinearregimewith ,(c)inthenonlinearregimewith ,and(d)withrandomprobe[2].ThefixedDGDintroducedbythecompensatorPMFwassetatthevalueof53.1ps,whichisoptimumintheabsenceofXPM[2].ThesystemperformanceisgiveninFig.10intermsofthecomplementarycumulativedistributionfunction(CCDF)oftherandom -factorpenalty ,definedas dB .The -factorpenaltywasmeasuredataconstantreceivedopticalsignal-to-noiseratio 24dB(over0.1nm),withrespecttothelineartransmissionwhereneitherPMDnornonlineareffectswerepresent.Thepoweratthepreamplifierwassetsoastoobtain inthelinearcase.Fig.10showsthemeasuredCCDFbothinacompensated(dashedline)andinanon-compensated(solidline)system,bothinthelinearregime[Fig.10(a)]andinthenonlinearregimewheredifferentvalues havebeenconsidered,namely 180 [Fig.10(b)], 0 [Fig.10(c)],andarandom [Fig.10(d)].Inthelattercase,therelativepolarizationanglewasrandomizedoverthePoincarésphereusingthePCintheprobetransmitterarm.Cross-andcopolarizedinputchannelswereobtainedduetoapolarizationbeamsplitterandapolarizationbeamcoupler,respectively.Ineachconsideredsituation,morethan1000 JOURNALOFLIGHTWAVETECHNOLOGY,VOL.21,NO.9,SEPTEMBER2003PMDconditionsweredrawn,sothattheCCDFwasaccuratelyevaluateddownto .Fortherandom case,foreachPMDrealizationadifferentvalueof wasdrawn.Inthewholeexperiment,thepoweroftheprobechannel,equalto3dBm,waskeptconstantsothattheinterplayamongPMDandthenonlineareffectsofSPMwasnegligible.Thepowerofthepumpchannelwas3dBminthelinearregime[Fig.10(a)],whileitwasswitchedto12dBminthenonlinearregime[Fig.10(b)–(d)].OSNRoftheprobechannelbeforefil-teringwaskeptconstantinallcasesto24dB/0.1nm.Notethata -factorpenaltyof0.5dBwithonlyXPMwasmeasuredforthebestrelativepolarizationangle 180 ,andof0.8dBintheworstcase 0 ,whichjustifiesthefactthattheCCDFsreachvalue1forsuchlowpenaltyvalues.Com-paringFig.10(b)or(c)withFig.10(a)inthenoncompensatedcase,itcanbeobservedthatPMD-inducedpenalty,ataCCDFof cannotbesimplyaddedtotheXPMone.Moreover,acom-parisonamongFig.10(b)–(d)forthenoncompensatedsystemshowsthatthe -factorpenaltyismorerelevantwhennonlinearcouplingismostefficient 0 ,evenifPMDmixestherel-ativeangle amongthepropagatingchannels.Arandomchoice isveryclosetotheworstcase.NowmakingacomparisonamongtheperformanceofthenoncompensatedandcompensatedsystemshowninFig.10,areductionofPMDcompensationefficiencyinpresenceofXPMeffectsisobserved.Inthelinearcase,thedifferencebetweenthe -factorpenaltywithandwithoutPMD,respectively,is2dBfora .Inthecases 0 and 180 gainreducesto1dBand1.2dB,respectively.When isran-domized,thereductionofPMDcompensatorefficiency(downto0.7dBofgainwithrespecttothenoncompensatedsystem)isthemoststrikingbecausealargerXPM-induceddepolarizationismorelikelytooccur,asexpected.AsimplifiedbutconvincingexplanationofthecooperationofPMDandXPMintheperformancedegradationofPMDcom-pensatedsystemshasbeenprovidedin[9]intermsofXPM-in-ducedPMD-mediatedintensitynoiseonthereceivedprobe,al-thoughwebelievethatamorecompletesystemmodelthatcap-turesthefinedetailsofsuchinteractionisyettobefound.V.CAcarouselmodelthatallowspredictionoftheXPM-in-duceddepolarizationinatwo-channelWDMsystemhasbeenproposed.ThemodelpostulatesthattheStokes’vectorsoftheprobeandthepumpperformarotationaroundafixedpivotbyabit-pattern-dependentanglethatincreaseswiththetotallaunchedpower.Accordingtothemodel,theworstcaseDOPdegradationisinducedbylongsequencesof“1”pumpbits,whilelargewalk-offsarefoundtoreducetheXPM-induceddepolarization.ThecarouselmodelleadstoanewDOPfor-mulabasedonasimplewalk-offlinearfilteringofthepumpchannel,whichisingoodagreementwithbothsimulationsandexperimentalresultsat10Gb/s.ComparedwiththelongcomputationaltimeofBPMsimulations,suchanewformulayieldsafastDOPevaluationtool,whichisaccuratewhenPMDandPDLaresmall.ThecarouselmodelhelpsidentifythekeyphysicalfactorsthatdeterminetheXPM-inducedperformancedegradationinOPMDCbasedonDOPfeedbackcontrol.Finally,theperformanceofafirst-orderOPMDCina10-Gb/spump-probe3 100-kmsystemhasbeenexperimentallyevaluated,andtheexpectedreducedefficiencyinPMDcom-pensationcausedbyXPMhasbeenstatisticallyquantified.ItisverifiedthatPMDandXPMcooperatetodegradeperformance.Countermeasurestoreducesuchdegradationrelyonincreasingthechannels’relativewalk-off,aswellascontrollingtheinputSOPssoastominimizethetotalDOPoftheinputWDMcomb.Inthisappendix,wederiveaclosed-formexpressionofthecharacteristicfunctionofanylinearfilteringofapulseampli-tudemodulated(PAM)signalwithindependentsymbols. beastationaryPAMrandomprocess,with anRVuniformover[0, andindependentandidenticallydistributed(IID)random (independentof ),feedingalinear,time-invariantfilterwithimpulseresponse .Theoutputofsuchfilteris ,where TheCFoftheRV isfound,foranyreal ,as[24] wheretheexternalexpectationisperformedwithrespecttothe .SincethesymbolsareIID,then where isthecommonCFoftheRV’s .Recallingthat isuniform,wehave whichisthesoughtclosed-formCF.Wenotethatsincetheinputprocessisstationaryandthefiltertime-invariant,alsotheoutput isstationary,sothatany valuein(19)willdo,for .Thefunction canbeeffi-cientlyevaluatedwithafast-Fourierroutine,whiletheexternalaveragingcanbeeasilyevaluatedbytakingtheaverageoftheintegrandforafinitenumberof values,uniformlysamplingtheinterval[0, Inthespecialcaseofbinary,equallylikelysymbols ,onegets andthus (20) etal.:DEGREEOFPOLARIZATIONDEGRADATIONDUETOXPM[1]D.WangandC.R.Menyuk,“PolarizationevolutionduetotheKerrnonlinearityandchromaticdispersion,”J.LightwaveTechnol.,vol.17,pp.2520–2529,Dec.1999.[2]F.Roy,C.Francia,F.Bruyère,andD.Penninckx,“Asimpledynamicpolarizationmodedispersioncompensator,”inProc.OFC1999,SanDiego,CA,Feb.1999,PaperTuS4,pp.275–278.[3]F.Heismann,D.Fishmann,andD.Wilson,“Automaticcompensationoffirstorderpolarizationmodedispersioncompensator,”inProc.,Madrid,Spain,Sept.1998,pp.529–530.[4]R.Noé,D.Sandel,M.Y.Dierolf,S.Honz,V.Mirvoda,A.Schöplin,C.Glingener,E.Gottwald,C.Scheerer,G.Fisher,T.Weyerauch,andW.Haase,“Polarizationmodedispersioncompensationat10,20and40Gbit/swithvariousopticalequalizers,”J.LightwaveTechnol.,vol.17,pp.1602–1616,Sept.1999.[5]R.Khoshravani,Y.Xie,L.-S.Yan,Y.W.Song,A.E.Willner,andC.R.Menyuk,“LimitationstofirstorderPMDmitigationcompensationinWDMsystemsduetoXPM-inducedPSPchanges,”inProc.OFC2001Anaheim,CA,2001,PaperWAA5.[6]Z.Pan,Q.Yu,A.E.Willner,andY.Arieli,“FastXPM-inducedpolariza-tion-statefluctuationsinWDMsystemsandtheirmitigation,”inProc.OFC2002,Anaheim,CA,2002,PaperThA7,pp.379–381.[7]L.Möller,P.Westbrook,S.Chandrasekhar,R.Dutta,andS.Wielandy,“SetupfordemonstrationofcrosschannelinducednonlinearPMDinWDMsystem,”Electron.Lett.,vol.37,no.5,pp.306–307,Mar.2001.[8]E.Corbel,J.-P.Thiéry,S.Lanne,S.Bigo,A.Vannucci,andA.Bononi,“ExperimentalstatisticalassessmentofXPMimpactonopticalPMDcompensatorefficiency,”inProc.OFC2003,Atlanta,GA,Feb.2003,PaperThJ2.[9]J.H.Lee,K.J.Park,C.H.Kim,andY.C.Chung,“EffectsofnonlinearcrosstalkinopticalPMDcompensation,”IEEEPhoton.Technol.Lett.vol.14,pp.1082–1084,Aug.2002.[10]N.Kikuchi,“Analysisofsignaldegreeofpolarizationdegradationusedascontrolsignalforopticalpolarizationmodedispersioncompensa-J.LightwaveTechnol.,vol.19,pp.480–486,Apr.2001.[11]L.Möller,S.Chandrasekhar,andL.L.Buhl,“DependenceofnonlineardepolarizationontheoverallpolarizationofPMDdistortedWDMsig-nals,”inProc.ECOC’01,Amsterdam,TheNetherlands,Sept.2001,PaperTuA3.6,pp.214–215.[12]B.C.CollingsandL.Boivin,“Nonlinearpolarizationevolutioninducedbycross-phasemodulationanditsimpactontransmissionsystems,”IEEEPhoton.Technol.Lett.,vol.12,pp.1582–1584,Nov.2000.[13]A.Cartaxo,“Impactofmodulationfrequencyoncross-phasemodula-tioneffectintensitymodulation-directdetectionWDMsystems,”Photon.Technol.Lett.,vol.10,pp.1268–1270,Sept.1998.[14]G.Bellotti,M.Varani,C.Francia,andA.Bononi,“Intensitydistor-tioninducedbycross-phasemodulationandchromaticdispersioninop-tical-fibertransmissionswithdispersioncompensation,”IEEEPhoton.Technol.Lett.,vol.10,pp.1745–1747,Dec.1998.[15]C.R.S.Fludger,V.Handerek,andR.J.Mears,“PumptosignalRINtransferinRamanfiberamplifiers,”J.LightwaveTechnol.,vol.19,pp.1140–1148,Aug.2001.[16]K.-P.Ho,“StatisticalpropertiesofstimulatedRamancrosstalkinWDMJ.LightwaveTechnol.,vol.18,pp.915–921,July2000.[17]G.P.Agrawal,NonlinearFiberOptics.NewYork:Academic,1989.[18]D.Marcuse,C.R.Menyuk,andP.K.A.Wai,“ApplicationoftheMan-akov-PMDequationtostudiesofsignalpropagationinopticalfiberswithrandomlyvaryingbirefringence,”J.LightwaveTechnol.,vol.15,pp.1735–1745,Sept.1997.[19]S.Huard,Polarizationdelalumière.Paris,France:Masson,1994.[20]J.P.GordonandH.Kogelnik,“PMDfundamentals:Polarizationmodedispersioninopticalfibers,”Proc.Nat.AcademySci.,vol.97,no.9,pp.4541–4550,Apr.2000.[21]N.J.Frigo,“Ageneralizedgeometricalrepresentationofcoupledmodetheory,”IEEEJ.QuantumElectron.,vol.QE-22,pp.2131–2140,Nov.[22]A.Bononi,C.Francia,andG.Bellotti,“Impulseresponseofcross-phasemodulationfiltersinmulti-spantransmissionsystemswithdispersionProc.OpticalFiberTechnology4,pp.371–383,1998.[23]T.-K.Chiang,N.Kagi,M.E.Marhic,andL.G.Kazovsky,“Cross-phasemodulationinfiberlinkswithmultipleopticalamplifiersanddispersionJ.LightwaveTechnol.,vol.14,pp.249–259,Mar.1996.[24]A.Papoulis,Probability,RandomVariables,andStochasticProcesses3rded.NewYork:McGraw-Hill,1991.[25]A.Vannucci,A.Bononi,A.Orlandini,E.Corbel,J.-P.Thiéry,S.Lanne,andS.Bigo,“AsimpleformulaforthedegreeofpolarizationdegradedbyXPManditsexperimentalvalidation,”inProc.OFC2003,Atlanta,GA,Feb.2003,PaperThJ1.[26]M.R.PhilipsandD.M.Ott,“CrosstalkduetoopticalfibernonlinearitiesinWDMCATVlightwavesystems,”J.LightwaveTechnol.,vol.17,pp.1782–1792,Oct.1999.AlbertoBononi,photographandbiographynotavailableatthetimeofArmandoVannucci(S’95–M’01),photographandbiographynotavailableatthetimeofpublication.A.Orlandini(S’00–M’01),photographandbiographynotavailableatthetimeofpublication.E.Corbel,photographandbiographynotavailableatthetimeofpublication.S.Lanne,photographandbiographynotavailableatthetimeofpublication.S.Bigo(A’99–M’99),photographandbiographynotavailableatthetimeof