86
gain cross section g/N [10 cm]2-20420-2-4-6-897098099010001010102010301040105010601070Nu/N= 0 (-σa) 0.1 0.2 0.3 0.4 0.5 1.0 (σe)Wavelength [nm]Fig.2.Yb:KYWgaincrosssectionsatvariousinversionsNu/N(seelegend),correspondingtoemissioncrosssectionσeforNu/N=1and(negative)absorptioncrosssection?σaforNu/N=0(allforthenmdirection).Theexperimentallaserspectraand(negative)pumpspectrumarealsogiven(thickblacklines).
zeroandtheincidentpumppower.ThefunctionisbasedonBrent’smethod,aderivative-freecombinationofbisection,secant,andinversequadraticinterpolationmethods[14].Thismini-mizationprocessisfairlydemandingoncomputationaltime,sinceeachevaluationsteprequiresaniterativesolutionofadifferentialequationforalldiscretizedspatialpoints.
Forthenumericalsimulations,thecomputerprogramMatlabwasused.Thediscretizationofthespacecoordinatesrandzwasdoneina15by15grid,withrtruncatedatr=2.5wp.Thefollowingexperimentalinputdatawereusedforthemodelling.Thelifetimeusedwasτ=236μs,determinedfor5%dopingfrom[15],andaKYWrefractiveindexofn=2wasassumed.
Todeterminethecrosssectionsinthenmpolarizationdirection,directmeasurementsonthelasercrystalweremade.Below1010nm,wheretheabsorptionishigh,theabsorptionandemissioncrosssectionswereobtainedthroughthereciprocitymethod[13]fromdirecttransmissionlossmeasurementswithatunableTi:sapphirelaseratlowintensity.Atemperatureof300K,5%dopingconcentration,aKYWdensityof6.5g/cm2[16],andtheintramanifoldenergysublevelsreportedin[17]wereassumed.Above1010nm,wheretheabsorptionisweak,the?uorescenceinducedfrompumpingat980nmwasusedto?ndthecrosssectionsthroughthereciprocitymethodandtheF¨uchtbauer-Ladenburgmethod[18].The?uorescencedatawerecalibratedbya?tatthe1010nmjunctionpoint.TheemissionandabsorptioncrosssectionsaredepictedinFig.2,togetherwiththegaincrosssections,givenby(3),atvariouslevelsofinversion.Themeasuredvalueofthepeakabsorptioncrosssectionat981nmof7.1·10?20cm2
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(C) 2008 OSA28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6448
volumeBragggratingpumpM3Yb:KYWM2M4laserFig.3.Lasersetup,showingthelaserbeam(solidred)andthepumpbeam(dashedblue).MirrorsM2andM3arehighlyre?ectivewitharadiusofcurvatureof50mmandmirrorM4is?atwithvariablere?ectivityR.
canbecomparedtopreviouslyreportedvalues.In[19],avalueof3.7·10?20cm2inthenmdirectionisreported,whileavalueof16·10?20cm2isgivenin[17],althoughinthea-direction,at～15?degreestonm.Ascanbeseen,the?uctuationsarelarge,possiblybecauseofuncertaindopingconcentrations.Sinceinthispaper,thesamecrystalwasusedfordeterminingthecrosssectionsandinthelaserexperiments,theaccuracyinthesimulationsshouldstillbegood.3.Experimentalsetup
Thelasercavitywasformedinabow-tiesetup,asdepictedinFig.3.ItconsistedofavolumeBragggratingasinputcoupler,twobroadbandhighlyre?ective50mmradius-of-curvaturemirrorssurroundingtheYb:KYWlasercrystal,and?nallya?atoutputcoupler.Thelaserwasend-pumpedbyaTi:sapphirelaserdeliveringamaximumpowerof～900mWinaclosetodiffraction-limitedbeam.Thepumppowerwascontrolledbyawaveplate-polarizerarrange-mentandtoavoidfeedbackintotheTi:sapphirelaser,anopticalisolatorwasused.Thepumpwascollimatedwhenpassingthroughtheinputcoupler,withabeamradiusof280μmatmirrorM2,yieldingafocusof27μmradiusinthelasercrystal.Formaximumabsorption,thepumpwaspolarizedinthenmdirectionoftheYb:KYWcrystalandtunedtoawavelengthof980nm,seeFig.2.Sincethelaseroutputcouplershadabroadbandcoating,theyalsore?ectedalargefractionofthepumplightforasecondpassthroughthecrystal,whichincreasedtheabsorbedpumplight.However,onlyapartofthesecondpasspumplightisbelievedtobeproperlyfo-cussedandpositionedinthecrystaltobeofuse.ThisissincethemirrorsM3andM4wereprimarilyadjustedtooptimizethelasercavity,withthedoublepasspumpingasabene?cialside-effect.
ThreedifferentvolumeBragggratings,at990.2nm,997.2nmand1066.0nm,wereusedasinputcouplers(OptigrateInc.).The990gratinghadafullwidthathalfmaximumbandwidthof0.22nmandapeakre?ectivityof99.5%,the997gratingabandwidthof0.25nmandare?ectivityof99.7%andthe1066gratingabandwidthof0.22nmandare?ectivityof98.8%.Thegratingshadabroadbandanti-re?ection(AR)coating.Forcomparison,aconventionaldielectricinputcouplerwasalsoused,withhightransmission>98?low980nmandhighre?ectivity>99.9?ove1020nm.Thelasercrystalwas5at.%Yb:KYWthatwasb-cutwithalengthof3mmandAR-coatedsurfaces.
Toadjustthelasermodesizeinthelasercrystalforoptimalperformance,mirrorM3andthelasercrystalcouldbetranslatedalongthecavityaxis.Toavoiddetrimentaleffectsfromobliqueincidenceonacurvedmirror,theincidentangleformirrorsM2andM3shouldbekeptsmall.Intheexperiments,anangleof2.4?wasused,whichgaveslightlydifferentmoderadiiforthetangentialandsagittaldirections.Asanexample,atotaldistancebetweenthetwocurvedmirrorsof(53.0mm,52.0mm,51.7mm)yieldscavitymodewaistradiiof(28μm;22μm;18μm)inthetangentialand(28μm;21μm;16μm)inthesagittaldirection.Inthe
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(C) 2008 OSA28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6449
1.00.9absorption0.8 1 pass pumping 2 pass pumping 990 nm lasing 997 nm lasing 1029 nm lasing 1066 nm lasing0.70.601002003004005006007008009001000incident pump power [mW]Fig.4.Pump-inducedsaturationofpumpabsorptionforvarioussituations,showingexper-imentalmeasurements(points)andtheoreticalsimulations(lines).
experiments,thecavitywasoptimizedforthe990nmoperationandwaskeptwiththatsettingfortheotherwavelengths.ThedistancesvolumeBragggrating–M2andM3–M4wereboth100mm.Afterthelasercavity,theresidualtransmittedpumpat980nmwasseparatedfromthegeneratedlaserbyaslightlytiltedvolumeBragggratingat982nmfornormalincidence,withare?ectivityof99%.
Variousoutputcouplerre?ectivitieswereinvestigatedforallthedifferentinputcouplers,rangingfrom35%to99%.Formostoftheresultsreportedbelowandthe990,997andcon-ventionalinputcouplers,respectively,are?ectivityof～85%wasused(withpumpre?ectivityof83%).Duetothelowergainforthe1066inputcoupler,a95%re?ectivitywasused(pumpre?ectivity93%).
4.Experimentalandnumericalresults
ForthevariousvolumeBragggratings,thelaserwaslockedtolaseatthegratingwavelength,asshowninFig.2.Forthe990-grating,thequantumdefectisonly1.0%,amongthelowestvaluesreportedever.Infact,bytuningthepumpwavelengthfrom980.0nmto981.8nm,itwasstillpossibletolaseat990.2nm,withaquantumdefectof0.85%,althoughtheef?ciencywaslowduetodecreasingpumpabsorption,withalaseroutputpowerofaround10mW.Forthefree-runninglaserwithaconventionalinputcoupler,theloss-gainbalancedeterminedawavelengthof～1029nmfor85%and90%outputcouplers.ThiscanbecomparedtothegaincrosssectionsinFig.2,indicatingthatinacrystalwithhomogenousinversion,thecorrespond-inginversioniswith～10%ofthepopulationintheupperlaserlevel.ThelaserbandwidthsfortheBragg-grating-lockedversionswerebelowthe0.07nmresolutionofthegrating-basedspectrumanalyzer,whiletheconventionallasershowedafewpeakswith～0.5nmseparation.ByobservingtheTi:sapphirepumplaserspectrumbeforeitenteredtheYb-lasercavity,itwascon?rmedthattheobservedradiationatthevolumeBragggratingwavelengthswasnotcausedbyspectrallockingoftheTi:sapphirepump,butindeedoriginatedfromtheYb-laser.Inallexperiments,thelaserwasfoundtobelinearlypolarizedalongthenmdirectionoftheKYWcrystal.
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(C) 2008 OSA28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6450
1000incident pump [mW]800600400 990 nm 997 nm 1029 nm 1066 nm20000.30.40.50.60.70.80.91.0reflectivityFig.5.Laserthresholdforvariousoutputcouplerre?ectivities,experimentaldata(points)andtheoreticalsimulations(lines).
Fora?rstcomparisonbetweenexperimentsandtheory,thesaturationofthepumpabsorptionwithincreasedpumppowerwasinvestigated,inotherwordsthebleachingofthelasercrystal,withoutlaseroperation.Inthisthree-level-laser,thebleachingisanindicationofhowmanyionsarepumpedtotheupperlaserlevel,i.e.theinversioninthelaser.First,thesinglepump-passabsorptionwithoutmirrorM4wasmeasured,asshowninFig.4(emptystars).Tocomparewiththetheoryandestablishtheeffectivepumpbeamradiuswp,thisparameterwasusedasa?tparameter,yieldingavalueof26μm,thatwasusedinsubsequentcalculations.This?tisinverygoodagreementwiththeexperimentalvalueof27μm.ThetheoreticallypredictedabsorptionisshownwithadashedlineinFig.4.WithmirrorM4inplaceandaligned,buttheinputcouplermisalignedtopreventlasing,theabsorptionfordoublepasspumpingwasalsoevaluated(emptycirclesinFig.4),forvariousinputcouplersandoutputcouplerre?ectivities.Acomparisonwiththetheoreticalpredictionfordoublepasspumping(dottedline),showsthat,assuspected,thesecondpumppassisnotpreciselyaligned,somethingwhichdecreasestheeffectofthedoublepasspumping.Nevertheless,insubsequentcalculations,perfectdoublepasspumpingisassumedinordernottocomplicatethings.
Thepumpabsorptionwasalsomeasuredunderlasingconditions.Theoutputcouplerhad～85%re?ectivityforallwavelengthsexceptat1066nm,where95%re?ectivitywasused.BoththeexperimentaldataandthetheoreticalpredictionsareshowninFig.4.Here,thetheo-reticalpredictionsaremadeunderthesameassumptionsasforthelaserthresholdandpowercalculationsdescribedbelow.
Thelaserthresholdatthedifferentwavelengthsandlosseswasinvestigatedexperimentallybytryingoutdifferentoutputcouplers.TheresultsarereportedinFig.5.Hereitshouldbenotedthatfortheconventionalinputcoupler,thereisadriftofthelaserwavelengthwithoutputcouplerre?ectivity.Toavoidtoomuchdrift,onlyexperimentswithlowoutputcouplingwereperformed.
TheexperimentallaseroutputpoweratdifferentwavelengthsisshowninFig.6.Themax-imumoutputpowersobtainedwere70mW,160mW,370mWand140mWatwavelengthsof990nm,997nm,1029nmand1066nm,respectively.Here,theoutputcouplerhad～85%
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(C) 2008 OSA28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6451
400350300 990 nm, slope = 23% 997 nm, slope = 30% 1029 nm, slope = 54% 1066 nm, slope = 31%laser power [mW]25020015010050002004006008001000incident pump power [mW]
Fig.6.Laserpower,comparisonofexperiments(points)andtheory(lines)forvariouswavelengths.Thelegendgivestheexperimentalslopeef?ciency.
re?ectivityforallwavelengthsexceptat1066nm,where95%re?ectivitywasused.Theexper-imentalslopeef?ciencyatthedifferentwavelengthsisshowninthelegendofFig.6.
InFig.5andFig.6,theresultsofthetheoreticalsimulationsofthelaserthresholdandoutputpower,respectively,arealsoshown,asdescribedinSec.2,assumingdouble-passpumping.Forthesecalculations,assumptionsmustbemadeaboutthelasermodebeamwaistandposition,aswellasthepassiveroundtriplossLinthecavity.Byacomparisonwiththeexperimentalresultsat990nm,abeamwaistwl0=16μmandawaistlocationzl0=d/2(inthemiddleofthecrystal)werededuced,aswellasalossof7%.Sincethesamecavitysetupwasusedfortheotherwavelengths,thelaserbeamparameterswerekept?xed,whilethelosswasadjusted.Inthisway,alossof4%wasinferredfor997nmand2%lossat1066nm.AscanbeseeninFig.6,theexperimentaloutputpowerfortheconventionaloutputcouplerrunningat1029nmishigherthanthetheoreticallypredictedone,whyno?tofthelosscouldbemade.Insteadalossof1%wasassumed.Furthermore,forthe1066nmthresholdsimulations,amodi?cationoftheexperimentallymeasuredcrosssectionshadtobemade.Sincethegainislowthere,asmallerrorinthecrosssectionmeasurementscanhavelargeeffectsonthemodellingoutcome.Inthepresentcase,thetheorypredictedthelasertobebelowthresholdatallpumppowers,whichisnotveryinteresting,andindisagreementwiththelaserexperiments.Instead,a?tofthecrosssectionvalueswasmadetotheexperimentalthresholdat95%re?ectivity.Thisyieldedanincreaseofthecrosssectionsof38%comparedtotheseparatelymeasureddata,avalueusedinthesubsequentcalculations.Thereasonforthedisagreementisbelievedtobeduetoarelativelylargeerrorinthemeasuredemissioncrosssectionatthiscomparativelylowvalue,perhapsduetoamiscalibrationofthe?uorescencedata.
Todeterminethetransversepropertiesofthelaser,thebeamradiuswasmeasuredjustbeyondtheoutputcouplerforthelaserat1029nm,withacollimatedbeamof450μmby540μminthetangential/sagittaldirection.Fromthesemeasurements,abeamwaistinthelasercrystalcanbededucedof18μmby15μminthetangential/sagittaldirection.Thereasonfortheellipticbeamisasomewhatlargeincidenceangleonthecurvedmirrors,somethingtoimproveinfutureexperiments.TheM2ofthebeamwasmeasuredtobe<1.05inbothdirections.
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(C) 2008 OSA28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6452
(二极管三极管)2008_实验和理论研究了一种体布拉格光栅锁相Yb,KYW激光器在选定波长下的性能
