Optical-Feedback-Effects-on-the-Spectral

IEEEJOURNALOFQUANTUMELECTRONICS,VOL.34,NO.9,SEPTEMBER19981717OpticalFeedbackEffectsontheSpectralLinewidthofSemiconductorLaserSensorsUsingSelf-MixingInterferenceGr´egoryMourat,No¨elServagent,andThierryBoschAbstract—Thespectralcharacteristicsofasemiconductorlaseraresignificantlyaffectedbyopticalfeedbackintheactivecavityofthediode.Inthispaper,theinfluenceofweakopticalfeedbackonthelinewidthisexaminedandanewrelationshipproposed.Feedback-inducedchangesinthepowerspectraldensityarealsodeterminedbyatheoreticalanalyticalmodel,ingoodagreementswithexperiments.Theseresultsarethendiscussedforsensingap-plicationsusingself-mixinginterference,asthemaximumrangeofdistanceanddisplacementsensorscanbelimitedbyhalfthecoherencelengthofthelaserdiodemodifiedbytheopticalfeedback.IndexTerms—Lasermeasurements,opticalfeedback,semicon-ductorlasers.I.INTRODUCTIONSPECTRALcharacteristicsofasemiconductorlaserarestronglyaffectedbyopticalfeedback[1]–[4].Seriousproblemsariseinpracticebecauseofunwantedopticalfeed-back.Forexample,theperformanceoflaserscanbedegradedforsomeapplicationslikeopticalfibertransmissionorlaserdiscreaders.Inthesecases,theoperatingcharacteristicsofthelasercanbemodifiedbytheseunintentionalfeedback-inducedchanges,leadingtochaoticbehavior.However,thisopticalfeedbackcanbeofpracticaluseforlinewidthreduction,forthelongitudinalmodeselectionenhancement[1]–[5],orforsensingapplications.Insuchasetup,thelaserbeamback-scatteredfromamirror-likeoraroughtargetinfrontofthelaserisaddedcoherentlywiththefieldinsidetheactivecavity.Variationsoftheopticaloutputpowerduetothisinterferencearemonitoredbythephotodiodewithinthelaserdiodepackageinordertodeterminedisplacements[6]–[8],distances[9]–[11],orvelocities[12].Likeclassicalinterference,thisphysicalphenomenoncanonlybeusedforatargetatadistancesmallerthanhalfthecoherencelength.Insuchacase,bothlaserandtargetwillbe-haveasalaserwithacompoundcavity.However,self-mixinginterferenceisquitedifferentfromclassicalinterferenceasthecoherencepropertiesofthelasersourcedependonthedistancebetweenthelaserandthetargetandonthereflectioncoefficientoftheconsideredtarget[2].ManuscriptreceivedJanuary22,1998;revisedApril14,1998.TheauthorsarewiththeDepartmentofAutomaticControl,EcoledesMinesdeNantes,F-44307Nantescedex3,France.PublisherItemIdentifierS0018-9197(98)06241-1.Currently,thelinewidthbehaviorofthesemiconductorlasercanberepresentedbyananalyticalmodelobtainedfromthelinearrateequations[13].However,thismethodgivesaccurateresultsonlyinthecaseofveryweakfeedback.Inothercases,themoresuitablemethodistodeducethebehaviorofthelaserbysimulatingtherateequations[3].Inthispaper,theeffectofmoderateopticalfeedbackonthepowerspectraldensityisconsideredtheoretically.Thisanalysis—basedontherateequations—ispresentedtodeduceanewanalyticalrelationshipofthesemiconductorlaserlinewidthforsensingapplicationsusingtheself-mixinginterference.Asamatteroffact,thislinewidthbehaviorhastobedeterminedtodeducethecoherencelengthofself-mixingtypesensors,i.e.,themaximumrangeofdistance,velocity,anddisplacementsensors.Experimentalresultsarepresentedandcomparedwiththeoreticalanalysis.II.THEORETICALANALYSISAschematicarrangementforasolitarysingle-modesemi-conductorlasercanberepresentedasatwo-facetFabry–Perotcavityinordertodelineateanactivelayeroflength.Thesefacetscanbedefinedbytheirreflectioncoefficientandwithrespecttotheelectricfieldamplitude[Fig.1(a)].Byconsideringanexternaltargetofreflectioncoefficientsmallerthanandandlocatedatadistancesmallerthanthehalf-coherencelengthoftheopticalsource,bothlaserandtargetcanberepresentedasathree-mirrorFabry–Perotcavity.AnequivalenttheoreticalmodelcanbeintroducedbyconsideringthecomplexeffectivereflectioncoefficienttakingthetargetasanexternalFabry–Perotcavity[Fig.1(b)](1)whereistheround-triptimeintheexternalcavity.Thegainandtheopticalfrequency,fromwhichthegenerallaserconditionscanbededuced,aregivenby(2)(3)where(4)0018–9197/98$10.001998IEEE1718IEEEJOURNALOFQUANTUMELECTRONICS,VOL.34,NO.9,SEPTEMBER1998(a)(b)Fig.1.(a)Theoreticalmodelofthesemiconductorlaserwithexternalfeedback.(b)Equivalentmodelofthetwo-mirrorcavity.isthefractionofthelightbackscatteredfromthetarget;isthethresholdgain,thelasercavityround-triptime,andthefeedbackcoefficientgivenby(5)whereisthelinewidthenhancementfactor.Fromtheclassicalrelationship(3),thefeedbacklevelcanbeclassifiedandthelongitudinalmodebehaviorforeachleveldeduced[2],[4],[14].Ithasbeendemonstratedthattheparameterrepresentstheexternalfeedbackcoefficientstrenghinordertodefinethesinglelongitudinalmodeorthemultimodelasingoperation.Forthelasercanoscillateonseverallongitudinalmodesdependingonthephasefeedback.Inthiscase,itispossiblebutdifficulttodeducedisplacementsbecauseofthehysteresisoccurringinthelaserdiode[15].Therefore,inthefollowinganalysis,thecaseofwillonlybeconsidered.Toobtaintheexpressionofthepowerspectraldensityofthelaser,therateequationsofthesemiconductorlaserwithopticalfeedbackhavetobeconsideredbyfollowingtheLangevinapproach.Theserateequationscanbewrittenasfollows[16]:(6)withthephotonnumber,thecarrierdensity,thephaseof