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Difficulty
Difficult
PreparationTime
1Hour
ExecutionTime
2Hours
RecommendedGroupSize
2Students
HeliumNeonLaser(ItemNo.:P2260701)
CurricularRelevance
AdditionalRequirements: ExperimentVariations:
Keywords:
Principleandtasks
Principle
RelatedTopics
StimulatedemissioninaHelium-Neongasdischargeasalightamplifier:thegainfactor'sdependenceonlightfrequencyanddischargecurrent,amplifierbandwidth;propertiesofanopticalresonatorcavitybetweenmirrors:stableresonatormodes,naturalfrequenciesandbeamgeometryindependenceonresonatorgeometry;spontaneousoscillationsofanamplifiercombinedwitharesonator;utilizedvolumeoftheamplifyingmedium;laseroutputpowerindependenceonresonatorproperties,outputpowerindependenceondischargecurrent;laserwavelengthdeterminationwithanopticalgrating.
Principle
BasicpropertiesofanopenHeNelasersystemsuchasamplifierbehavior,resonatorstability,outputpowerandlaserwavelengthareexploredanddiscussed.Theinfluenceoftheresonatorcavityonbeamdiameteranddivergenceisobserved.
Fig.1:Fundamentalset-up
Tasks
1. AdjustingandstartingtheHeNelasersystem
2. Stabilitycriteriaofanopticalresonator
3. Beamdiameteranddivergenceindependenceonresonatorproperties
4. Laseroutputpowerindependenceonresonatorproperties
5. Laseroutputpowerindependenceondischargecurrent
6. Determinationofthelaserwavelengthwithanopticalgrating
AreaofExpertise:
Physics
EducationLevel:
University
Topic:
ModernPhysics
Subtopic:
LaserPhysics-
Photonics
Experiment:
HeliumNeonLaser
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Introduction
Laseristheacronymforlightamplificationbystimulatedemissionofradiation.Normallylasersareunderstoodtobesourcesofcoherentradiationoflowdivergencethustobespontaneousoscillators.Butanamplifieralonedoesnotmakeanoscillator.Backfeedingbringsanamplifiertooscillatespontaneously.Ifthebackfeedingisbroad-bandedinfrequency,theamplifierwillthenoscillatewiththefrequencyofit'smaximumgain.Ifthegainprofileoverfrequencyoftheamplifierhasseveralmaximums,thebehaviordependsontheprincipleoftheenergysupplyoftheamplifiertoanexistingoscillation.Anybackfeedoscillationwillreachtheoutputlevelwheretheamplifierreachessaturation,i.e.wheretheamplifyingprocesscannotsupplymoreenergytothatoscillation.
Homogenousandheterogenousgainprofile
Iftheenergyissuppliedtodifferentfrequenciesbydifferentpaths,severalfrequenciesmayoscillatesimultaneously.Inthiscasethecorrespondinggainprofileisreferredtoasinhomogeneousgainprofile.Elsetheoscillationwiththegreatestgainwilltakealltheenergyprovidedbytheamplifierandthegainprofileiscalledhomogeneous.ThebackfeedingsystemoftheHeNelaserisanopticalresonatorwhichhasaverypronouncedfrequencyresponseandneedscloserexamination.SotheoscillationsoccurringintheHeNelasersystemdependonboththepropertiesofamplifierandresonator.Thetotalgain,theproductofbothamplifiergainforagivenfrequencyandresonatorqualityfactorforthatfrequency,determinesthesystem'sbehavioratthatfrequency.
Stimulatedemission
TheamplifieroftheHeNelasersystem,thelaserinstrictsense,isaDClowpressuregasdischargeinaglasstubeof1mmdiameter.Amplificationofincominglightoccurswhenanincomingphoton'sfrequencymatchesthefrequencyofanopticaltransitionwithpopulationinversionandprovokesanelectronofthemoredenselypopulatedupperstateofthattransitiontodecaytothelesspopulatedlowerstateofthattransitionunderemissionofaphotonofthesamequantumstateastheincomingphoton.Thisiscalledstimulatedemission.Theamplifiersaturateswhenforthattransitiontherateofstimulatedemissionreachesthefillingprocessrateoftheupperstatethatprovidesforthepopulationinversionofthattransition.
CreationofpopulationinversionintheHeNegasdischarge:–Collisionstypes
InthermodynamicequilibriumapopulationinversionisimpossiblesotheprocessleadingtopopulationinversioninaHeNegasdischargeneedsexplanation:Thegasdischargeisaplasma,partoftheatomsareionized.Theionsandelectronsareacceleratedintheelectricfieldandloosetheenergyextractedfromthefieldincollisionswiththeneutralatomsandthecapillarywalls.Partlythescatteringontheneutralatomsisinelasticconvertingsomeoftheimpactenergyinelectronexcitationofthecollisionpartners.Soalsotheneutralatomsinthegasdischargegetexited,mainlybyelectrons.Excitationenergyofoneatomicspeciesmaybetransferredtoanotherduringacollisionwithahighprobabilityiftheexcitationenergyofthefirstspeciesmatchesanexitedstateoftheother.Inelasticcollisionswherepartoftheimpactenergyisconvertedtointernaldegreesoffreedomarereferredtoascollisionsofthesecondtype.
HeandNeexitedstates
Thegasmixtureinthegasdischargeconsistsof80%Heliumand20%Neon.Heliumatomswiththeirtwoelectronscanbeinasingulettstatewithelectronspinsantiparallelorinatriplettstatewithelectronspinsparallel.TheHeliumexitedstateswithlowestexcitationenergy,2 with19.820eVand2 with20.616eVcomparedtothegroundstate1 ,aremetastable.Theymaynotdecayopticallyto1 becausetransitionsbetweenS-levelsareforbiddenforallelectromagneticformsofradiation.SotheHeliumatomsexitedbyelectroncollisionexisttoagoodfractioninthemetastableexitedstates.
ThemetastableHeliumexitedstatesdecaythroughcollisionwiththecapillarywallorthroughcollisionswithotheratoms.IncaseofcollisionwithNeonatomsthefollowinghappens:The2 statewith20.616eVofHeliumeasilytransfersit'senergyto
aground-stateNeonatomexitingittotheelectronicconfiguration .Thisconfigurationexistswithfoursub-states,ofwhichmainlytheupperonewith20.663eVispopulatedinthissortofcollision.Theexcessenergyof47meVisprovidedby
thermalenergykT.The2 stateofHeliumwith19.820eVtransfersit'senergytooneofthefour configurationsofNeonwithenergiesof19.664,19.688,19.761and19.780eV.TheseprocessesefficientlyleadtoNeonexcitationandHeliumdeexcitation.
TheNeonexcited and statesdecayto configurations.ThesedecaysarethelasertransitionsusedinHeNelasersystems.ThelifetimeoftheNeon and statesistentimeslongerthanthatofthe statesthatdecaytometastable
emittingultravioletlight.Soifthe and statesarepopulatedbycollisionsofthesecondtype,apopulationinversionoccurs.The decayemitsinthevisibleregionandthe decayemitsinthenearinfrared.Theemptyingof
the levelstogroundstate isopticallyforbiddenandhastobeeffectedbycollisionswiththecapillarywall.Thisprocessmaybeabottlenecktothelaserprocessbecausepopulationinversionisonlyfeasibleifthelowerlaserlevelsgetsemptiedfastenough.Sothecapillarysizehastobeacompromise:Awidersizeallowsforagreateractivemedium'svolumebuttheratiowallsurfacefordeexcitationtoactivemediumvolumegetslowerforawiderdiameterthuslimitinglaserpower,too.
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Fig.2:Neonexcitationsceme
Fig.3:Lasertransitions
Gainprofileoverfrequencyoftheopticalamplifier
SothegainprofileoftheHeNeopticalamplifierconsistsofalltheNeontransitionlineswherecollisionsofthesecondtypebetweenHeliumandNeonhaveeffectedpopulationinversion.Thesetransitionsareallintheinfraredandvisiblerange.EachofthetransitionlineshasanaturallinewidthduetolifetimeofthehigherlevelandalinebroadeningcausedbyDopplereffectduetoatomspeeddistributioninthelineofsight.SinceheretheDopplerlinewidthisconsiderablygreaterthanthenaturallinewidth,setsofatomswithdifferentspeedcanbediscernedandinteractseparatelywiththeradiationfield.Aninteractionwithlightofaspecificfrequencywillonlyalterthepopulationinversionofasetofatomswiththecorrespondingspeed.Thusthegainprofileisinhomogeneous.
Amplifierefficiency
Itshouldbekeptinmindthattheoverallenergyconversionfromelectricenergyfromtheplasmatolaserlightisprincipallyinefficient:ManyatomicexcitationsmaydecaybyincoherentlightemissionandtheprocessofNeonexcitationiscomplexandmanyinelasticscatteringprocessesleadonlytocapillarywallheating.Inthelaserprocessonlyoneofseveraltransitionswithpopulationinversionisused.Andoftheapprox.20eVNeonexcitationenergyonly2eVareusedforthevisiblelasertransition.
SotheefficiencyofHeNelasersdoesnotexceedthe percentrate.
Otherlasertubecharacteristics,Polarizationofthelaserlight
Photonscanbedistinguishednotonlybytheirwavelengthbutalsobytheirpolarizationandthespatialdistributionoftheirdetectionprobability.Principallythesensitivityoftheatomsinthegasdischargeisisotropicforallpossiblepolarizations.Theatom'svelocitydistributionisnearlyisotropic,too.Soavolumeelementofthegasdischargehasanspatiallyisotropicgain.Butthegasdischarge'sgeometryisalengthycylinderalongthecapillaryleadingtoanenhancedgainalongthecapillary'saxiscomparedtoanyotherdirection.Additionallythegasdischargehastobeconfinedfromthesurroundingatmosphere.Sinceagasdischargehasalowdensity,theopticalgainisaltogetherquitelow(some%permeter).Soreflectiononthelasertube'sconfinementhastobeavoided.ThisisdonebyopticalwindowsintheBrewsteranglewhichhavezeroreflectionforaspecificpolarizationbutdohavereflectionlossesforanyotherpolarization.Thusthelasertube'sgaindependsonpolarizationandaHeNelaserbeamofalasertubewithBrewsterwindowsisfullylinearlypolarized.
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Fig.4:Lasertubesceme
Fig.5:Brewsterwindoworientation
Gainprofileofthefeedbacksystem
UsuallytheopticalcavityforaHeNelaserisaFabry-Perotresonator(or:interferometer)consistingoftwooppositemirrorswiththelasertubeinbetween.ThetransmissionofaFabry-Perotresonatorisafrequencycombwithspikesthatareequidistantinfrequencyandthefrequenciesofwhichareintegermultiplesofthefirstnaturalfrequency.Theheightofthespikesismodulatedoverfrequencywiththemirror'sfrequency-dependentreflectivity.HerethefeedbackfortheopticalamplifierishighincaseofthesameinterferenceconditionsthatleadtoahightransmissionoftheFabry-Perotresonator,i.e.whenastandingcontinuouswaveisestablishedinsidetheresonatorandtheamplifier'soutputiscoupledbackintotheamplifierwithcorrectphase.Sincethegainofthelasertubeislow,themirrorsmusthaveahighreflectivitytomakespontaneousoscillationpossible.Spontaneousoscillationisalsoreferredtoaslaserignitionorlasingcondition.SotheFabry-Perotresonatorisofhighqualityandfineness.
Fineness
Thefinenessorqualityfactorisameasurefortheratiooffullwidthathalfmaximum(FWHM)ofonepeaktofrequencydistanceoftwoadjacentpeaksorinotherwordsthefrequencyresolutionoftheFabry-Perotresonator.
NaturalfrequenciesofaFabry-Perotcavity
Naturalfrequenciesbetweentwomirrorswithspacingdinbetweenoccur,whenthephaseafterpassingthewholeopticalpathway isthesameagain,i.e.integermultiplesof correspondingtoawavelength .So
or
with aninteger>0, speedoflightan thelight'sfrequency.
Resonatormodescombinedwithamplifiergain
With , , is thefrequencydistanceofadjacentresonatormodeswhilethe
frequencyofthered633nmvisibleHeNelaserlightis ,makingannintherangeof1.6million.TheDopplerlinewidthwouldbehereabout FWHMsoseveralresonatormodeswouldbeinsidetheline'sgainprofile.Ashortresonatorlengthreducesthepossiblenumberofmodesandausual5"=127mmresonatorlengthlaserrunsinasingleresonatormode.
Aphotonstateinaresonatorisfullydefinedbytheresonatormodeitbelongsto.SofaronlylongitudinalresonatormodeshavebeendiscussedherebuttheremayalsobetransversalmodesinaFabry-Perotresonatordependingonmirrorcurvature,diameteranddistance.Severaltransversalmodesatnearlythesameenergymaybelongtoasinglelongitudinalmodenumber.Thetransversalmodenumbersareusuallylowonedigitintegerswhilethelongitudinalmodenumberisinthemillionsandneveractuallydetermined.Thestableresonatormodesaretobediscussedlateron.
Airyfunction
ThepeakformoftheresonatorisgivenbytheAiry-functionforthetransmissionoftheresonator,thetransmissionagainbeingameasureforthebackfeedingcapabilityoftheresonator:
where denotestheintensityand isthereflectivityofthemirrors.Thehigherthereflectivity,thesharpergetthepeaks.Ifthemirrorsdonothavethesamereflectivitybuthavethereflectivity and ,thenthegeometricmeanvalueofthereflectivitiesdoesapply:
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.
Propertiesofthemirrors
Thereflectivityofamirrorisalwaysafunctionoflightfrequency, ,butthevariationsin arelong-rangedcompared
tothefrequencydistanceofadjacentlongitudinalresonatormodesortransitionlinewidths.
ForacquiringthehighreflectivityneededforHeNelaseroperation,dielectricmirrorsareused.Dielectricmirrorslikeinterferencefiltersconsistofasequenceoflayersofalternatinghighandlowrefractiveindex.HerelayerthicknessistunedtoconstructiveinterferenceoftheFresnel-reflectedlightfromthelayersurfaces.Aprecisematchexistsonlyfordiscretefrequencies.Duetothefactthatthelayersarethinandusuallycontainonlyonequarterofthedesignwavelength,thereflectivityishighforawholefrequencyrange,stillitisfrequencydependent.Ahighreflectivemirrorforvisiblelightisusuallynothighreflectiveinthenearinfrared.Sothelasersysteminuseheremayoperateatseveralvisibletransitionsbutisnotsuitableforinfraredoperation,thoughthelasertube'sgainisprincipallyhigherintheinfraredrange.
Equipment
PositionNo. Material OrderNo. Quantity
1 ExperimentsetHelium-NeonLaser 08656-93 1
2 Photoelement 08734-00 1
3 DMM,autorange,NiCr-Nithermocouple 07123-00 1
4 Scale,l=750mm,onrod 02200-00 1
5 Diffractiongrating,600lines/mm 08546-00 1
6 Plateholder 02062-00 1
7 Slidemountforopticalbenchexpert 08286-00 1
8 Dangersign-laser- 06542-00 1
9 BarrelbasePHYWE 02006-55 1
10 Verniercalliperstainlesssteel0-160mm,1/20 03010-00 1
11 Slidingdevice,horizontal 08713-00 1
12 Connectionbox 06030-23 1
13 Resistor10Ohm2%,2W,G1 06056-10 1
14 Resistor100Ohm2%,1W,G1 06057-10 1
15 Resistor1kOhm,1W,G1 39104-19 1
16 Resistor10kOhm,1W,G1 39104-30 1
17 Resistor100kOhm,1W,G1 39104-41 1
18 Connectingcord,32A,750mm,red 07362-01 1
19 Connectingcord,32A,750mm,blue 07362-04 1
20 ProtectionglassesHeNe-laser 08581-10 1
21 Acetone,chemicalpure,250ml 30004-25 1
Set-upandprocedure
Task1
Notes
InthistaskleftandrightisdefinedaccordingtotheperspectiveofFig.5.
Allfixingscrewsontheopticalrailshouldalwaysbetightenedwellsonocomponentscanunintentionallymove,butwithoutexertingexaggeratedforce!
Nevertouchnorscratchanyoftheopticalsurfaceslikewindowsormirrors!
Handletheopticalcomponentswithcare!Nevertouchadielectricmirrorsurfacewithfingersandkeeptheopticalsurfacesclean.Cleaninghastobedonewithacetoneandspeciallenscleaningtissue.Forcleaningrefertotheoperationmanual.
Amainobjecitveofthepresentexperimentsistogainafinegraspofthehandlingofopticalcomponentsandadjustmenttechniqueswhicharenecessarytosuccessfullyexecutethem.Sotheprinciplesofadjustmentandmovementofcomponentswhileoperatingthelaserandofbeamwalkingshouldberehearsedtosomeextent.
a.Adjustingtheopticalaxis
Placetheopticalbenchonafirmsurfaceandadjustitslevelingscrewssuchthatthebenchcannotwobble.
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Mounttheadjustmentlasertoitsholderandtheleftdiaphragminfrontofit,mountbothattheleftendoftherail.
Mounttherightdiaphragmintoaholderandmountittotherightendoftherail.
TurntheAdjustmentlaseron.
Adjusttheadjustmentlaserpositionifnecessarysothatthebeampassestheholesofbothdiaphragmsandtheinterferencepatternontherightdiaphragmisaspreciselycenteredaspossible.
b.MountingtheHeNelaser
Mountthelasertubeontotherailrememberingtheorderoftighteningofthetwoscrewsthatfixthelasertubeholdertotherail.Whenremovingormovingthelasertubeholderusethesameorderoftighteningtheholdingscrewsagainwhenfixingthelasertubeholderbackontherail.
Leftendofthetubeshouldbeabout300mmawayfromtheleftdiaphragm.
Adjustthelasertubepositionsuchthatthealignmentbeampassesthroughitwithouttouchingthecapillary.Watchthereflectionsonthecapillary.Thevisiblelightspotontherightdiaphragmshouldnotbedistorted.
Thelasertubemayslightlydeflectthealignmentbeam.Itiscrucial,thatthelightpassescompletelythroughthetubeandnodistortionofthebeamisvisible,thedeflectionmaybeignored.
Removethelasertubeagainnotalteringit'sadjustment.
Inserttheconcavehighreflective(HR)flat/1000mmmirrorintoanadjustableholder,theHRsidefacingleft.
Settheholder900mmawayfromtheleftdiaphragmontotherailandadjustthereflectionpositionpreciselycentreoftheleftdiaphragm.
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Fig.6:GoodalignmentoftheadjustmentbeamFig.7:Badlasertubealignment
Fig.8:Goodlasertubealignment
Thereflexesoffrontandbacksideofthemirrorcanbedistinguished:Whenmovingthemirroralongtherailtheusefulfrontreflexshowsfocusingwhiletheuninterestingbackreflexdoesnot.
InsertHRflat/flatmirrorintoanadjustableholder,theHRsidefacingright.
Placetheholder200mmawayfromtheleftdiaphragmandadjustthereflectedbeampositionpreciselycentreoftheleftdiaphragm.
Insertthelasertubeatitsformerposition,turnonitspowersupplyandsetthetubecurrentto6.5mA.
Wobbletheadjustingscrewsontherightmirror–onefastandoneslowscanningthex-y-rangeofthereflectedalignmentbeamwatchingthereflectionofthealignmentbeamontheleftdiaphragm:Thereflectedalignmentbeamshouldbeseenpassingbackthroughthetubeandtheleftmirrorontheleftdiaphragm.
Whenthealignmentbeamiscenteredontheleftdiaphragm,thelasershouldigniteatsomepoint.Ifnot,continuethex-y-scanningwithsomepatience.
Afterseeingthelaserignite,optimizethelaseroutputpowerbysuccessivelyadjustingeveryelementofthelasersystem:tubeposition,leftmirrorandrightmirror.
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Iteratetheoptimizingprocessuntilnomoreintensityimprovementcanbeachieved.
Thebeamdiameterontheleftdiaphragmisclearlysmallerthanontherightone.
c.Beamwalkthelasertotheopticalaxis
Turntheadjustmentscrewsontheleftmirrorandwatchthebehaviorofthelaserlightspotontherightdiaphragm.
Letthelaserspotmovetowardsthecentreofthediaphragmasfaraspossiblewithoutextinguishingthelaser.
Readjustallcomponentsbuttheleftmirrortooptimizethelaserintensityanditeratethisreadjustmentuntiloptimumisreached.
Turntheleftmirroragaininthedesireddirectionwithoutextinguishingthelaserandgothroughtheoptimizingprocessoncemore.
Repeattheprocedureuntilthebeamispreciselycenteredontherightdiaphragm.
Continuebeamwalkingwiththespotontheleftdiaphragmturningtherightmirroruntillaserbeamandopticalaxisarewellaligned.
Especiallyalterthelasertubealignmentinthedesireddirection.
Goodalignmentofthelaserparalleltotheopticalbenchiscrucialbecausecomponentshavetobemovedalongtherailtoagoodextentinthefollowingexperiments.Withoutproperalignmentthereadjustingaftereachstepofmovementbecomespainful.
Fig.9:Beamcenteredontherightdiaphragmwithghostbeamvisibleleftofit
Fig.10:Afterbeamwalkingthebeampassesalmostcompletelytheholeintheleftdiaphragm
Fig.11:Set-up
Task2
Inthistaskthestabilityofanopticalresonatorwillbeinvestigated.Basicallywearelookingifthelaserigniteornotataspecialresonatorgeometry.Theresonatorgeometryisspecifiedbyasetofvaluesfortheresonatorparameterslikedistancebetweenandcurvatationofthemirrors.Weidentifyanopticalresonatorwithacertaingeometryasstableifthelaserignite.Thuswe
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determinetheresonatorstabilitybythestabilityofthelaserprocess.
Buttwoquestionarise:
1. Howistheresonatorgeometryrelatedwiththeignitionofthelaser?
2. Whatmeansresonatorstabilityingeometricalterms?
Thecrucialpointinthelaserprocessisthatphotonsoncecreatedbystimulatedemissionwillproducefurtherphotonsinsubsequentprocessesofstimulatedemission.Butthelasercanonlyigniteiftheefficiencyofthisfeedbackishighenough.Thereforemostofthe(approximatelyall)lightproducedinthelaserprocessshouldstayintheactivelasermediuminsideoftheresonatortokeepthelaserprocessstable.Thus,roughlyspeaking,anopticalresonatorissaidtobestableiflightpropagatinginsideoftheresonatorstaysinsideoftheresonator.Wecanbemorepreciseusingtheconceptoflightpathsconstructedaccordingtothelawsofgeometricaloptic.Inthistermsaresonatorisstableifthewholelightpathisinsidetheresonator.Inthetheorypartallusedquantitieswillbedefinedpreciseandexpressedinmathematicalterms.Ananalyticalstabilitycriterionincorporatingtheresonatorparameterswillbeexplainedandderived.
InthistaskleftandrightisdefinedaccordingtotheperspectiveofFig.12.
Note:Aftereachchangeintheset-up,thelaseroutputistobeoptimisedagain!Aftertheset-upandalignmentprocessdescribedinTask1,thelaserisequippedwithahighreflective(HR)flat/flatmirrorontheleftsideandaconcaveHRflat/1000mmmirrorontherightsidewithamirrorspacingabout700mm.Thisisahemisphericalresonatorconfiguration.
a.Movethelasertotheleftendoftheopticalbench:
Removethealignmentlaserandtheleftdiaphragmfromtheopticalbench.
Movethelasertubeneartotheflat/flatmirrorontheleftandoptimizethelaserpowerbyaligningallcomponentsagain.
Movetheleftmirrorfurthertotheleft–asfaraspossiblesuchthatthelaserignitesagainaftertighteningthemirrorholder'sfasteningscrewtotheopticalbenchandoptimizethelaserpoweroncemore.
Movethelasertubetowardstheleftmirrorandrepeatoutputoptimization.
Movetherightmirrortowardsthelasertube,repeatoptimization.
Repeatthesestepsuntiltheleftmirrorisattheleftendoftheopticalbenchandthelasertuberightinfrontofit.
Alignthelaserbeamwiththeopticalaxisbybeamwalkingcenteringthelightspotontherightdiaphragm.
b.Determinethemaximummirrorspacinginwhichthelaserworksforthismirrorcombination:
Movetherightmirrorinstepsasfaraspossibletotherightsothatthelasercanstillignite–alwaysoptimizinglaserintensityaftereachstep.
Watchlightspotsizeonbothmirrorsandthinkabouttheimplicationsforthenecessarylasertubepositionsincethecapillaryshouldnotfilternorreflectlaserintensity.
Atthecriticaldistancethestepsmayneedtobeassmallassomemillimeters.
Fig.12:Lasermovedtotheleftendoftheopticalrail
Fig.13:Setupwithrightmirrorholderturnedaround
c.Alterthemirrorcombinationtoahemisphericresonatorwithadifferentradiusofcurvature:
Lowerthemirrordistanceto850mmwhilelaseroperates.
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ExchangetherightmirrorwiththeconcaveHRflat/1400mmmirror:Turnthemirrorholderaroundsothatthecavitylengthisextendedandmountthemirrorwiththehighreflectivesidefacingtotheinnerofthemirrorholderandfacingleft.Themirrorsurfacepositionisnow45mmfromtheleftan5mmfromtherightcornerofthemirrorholder.
Startthelaserupagainscanningthex-y-rangewiththerightmirror,spinningonescrewfastandtheotherslow.
Iftheroomwhereyouworkisdarkened,thereflectionofthetube'slightcomingfromthemirrorseenontheendofthetubehousingcanbeusedtodeterminethecorrectmirrorangleprovidedthelaseriswellalignedwiththerail.
Ifthelasertubepositionisnotveryfarleftandsothereflectedlightspotisnotinfocusandinvisible,thentheleftmirrormayberemovedandreplacedbythealignmentlasertopreadjusttherightmirror.Thenreplacethealignmentlaseragainwiththeleftmirrorwithoutalteringtheleftmirror'salignment.
Ifthelasertubepositionisnotveryfarleftandsothereflectedlightspotisnotinfocusandinvisible,thentheleftmirrormayberemovedandreplacedbythealignmentlasertopreadjusttherightmirror.Thenreplacethealignmentlaseragainwiththeleftmirrorwithoutalteringtheleftmirror'salignment.
Scantherightmirrorx-y-rangetostartupthelaser.
Determinethemaximummirrorspacinginthesamemannerforthismirrorcombination.
Removetheholderwiththerightdiaphragmwhennecessary.
d.Alterthemirrorcombinationtoaconfocal/concentricsetup:
Lowerthemirrordistanceto600mmkeepingthelaseroperating.
Movethelasertubeneartherightmirror.
ExchangetheleftHRflat/flatmirrorwiththeconcaveHRflat/1000mmmirror,theholderpositionsuchthattheholderprotrudesacentimeterfromtherailtotheleft.
Reignitethelasernowscanningwiththeleftmirror.
Checkonthealignmentoftheopticalaxiswiththerailwithhelpoftherightdiaphragm,correctbybeamwalkingifnecessary.
Watchthebeamdiameteronbothmirrors,keepthelasertubenearthemirrorwherethediameterissmall.
Startthelaserandrisethemirrordistanceagainuntilthelasergetsunstable,keepingthelasertubeinfrontofthemirrorwiththesmallbeamdiameter.
Nowtrytoreachasecondareaofstabilitywherethebeamdiameterissmallontheleftmirror:Bringthelasertubeasfaraspossibleneartheleftmirrorwhilelaseroperating.
Puttherightmirrorholdertotherightendoftheopticalrail,theholderprotrudingonecentimeterfromtherail'sendtotheright.Mirrordistance isthen1465mm.
Watchthetubelight'sreflectionfromtherightmirrortoadjusttherightmirrororientation:Thereflectionistomatchthelightbeamcomingoutofthetube,checke.g.withwhitesheetofpaper.Withalittlescanningyoushouldbeabletoignitethelaserinthisposition.
Ifnotsuccessfulcheckbeamalignmentwithlowcavitylengthandrightdiaphragm–correctbybeamwalkingwithleftmirrorandtubealignment.
Trytolowerthemirrordistanceasfaraspossible.
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Fig.14:Laserworkingatfulllengthofopticalrail
Fig.15:Lasertubenearrightmirror
Task3
Note:InthistaskleftandrightisdefinedaccordingtotheperspectiveofFig.22.
Nowthegeometricalpropertiesofthebeamwillbeinvestigated.Thedependenceonthegeometricalpropertiesoftheresonatorwillbeanalysed.Practicallywewillmeasurethebeamdiameterinfrontoftheleftandtherightresonatormirror(thusinsideoftheresonator)andattheendoftheopticalrail.Thiswillbedonefordifferentresonatorlengthsandfordifferentmirrorcurvatations.TheexperimentalprocedureofthediametermeasurementsinsideoftheresonatorissimilartotheprocedureofTask2:Withhelpoftheoutsidejawsofaverniercaliperaslitofspecificwidthispreset.Theseslitwillbemovedperpendiculartothebeamaxisinfrontoftheleft(right)mirror.Thiswillbedonefordifferentslitwidths.Weidentifythebeamdiameterinfrontoftheleft(right)mirrorasthesmallestslitwidthsatwhichthelaserignite.WecandosoforthesamereasonasinTask2:Onlyifthe(approximately)wholelightofthelaserbeamcanpasstheslittheefficiencyofthefeedbackinthelaserprocessishighenoughtoignitethelaser.Thediameterofthebeamattheendoftheopticalrail(i.e.outsideoftheresonator)willbedeterminedinadifferentway:Withhelpofaphotoelementthephotocurrentinducedbythelaserbeamwillbemeasured.Thedependencyofthecurrentfromthedistanceofthephotoelementfromthebeamaxisreflectsthetransversalbeamprofile.ThebeamprofilecanbefittedwithaGaussianbell-shapedcurveandthebeamwillbedeterminedbythewidthofthebellshapedcurve.
EquipthelaserwiththeHRflat/flatmirrorontheleftsideandtheHRflat/1000mmmirorontherightside.
Setthetubecurrentto5mAandbringthelaserintooperation.
Removethealignmentlaserandtheslidemountwiththerightdiaphragm.
Movethelaserunderoperationtotheoutmostleftendoftheopticalrail.
Setthemirrordistancedto450mm,optimizelaseroutput.
Thelasertubeistobeneartheleftmirrorbutwithenoughspacebetweenmirrorandtubetomeasurethebeamdiametertherewiththecaliper.
Mountthehorizontalslidedevicetotheslidemountwithoutcolumnontheoutmostrightendoftheopticalrail.
Mountthesiliconphotoelementontotheslidedeviceandconnectittoa10kΩresistoronthetheconnectionbox.
Connectthedigitalmultimetertomeasurethevoltagedropontheresistorwiththecurrentfromthephotoelementpassingtheresistor(seeFig.2andcircuitdiagramm).
Useforexampleblackadhesivetapeorblackadhesivelabelstoshortentheslitofthefaceplateforthephotoelementtoacenteredsquare(0.3mmx0.3mm)andmountthefaceplateontothephotoelement.
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Fig.22:Set-upforbeamgeometryexamination
Fig.23:Slidingdevicewithphotoelement
Oneturnoftheadjustingknoboftheslidedeviceishalfamillimeteroflateralmovement;thespindlecountsupto15turns,thatis7.5mm,beforestartingatzeroagain.
Mindthelostmotion,movetheslidedevicealwaysonlyfromonesidetothedesiredpositionwhilemeasuring.
Checkonthelasermodewithawhitepieceofpaperinfrontofthephotoelement.Thelaserlightspotistoappearroundandsymmetriccorrespondingtothe –mode,thatisthefundamentaltransversalmode,belongingtoaGaussianbeam.Ifthelaseroscillatesina –mode,thatisthefirstnon-fundamentaltransversalmode,oranothernon-fundamentalmodeifbrightest,theopticsneedcleaning.
Usethedigitalmultimetertodeterminetheactualresistancevalue.Besuretheinputresistanceofthemultimeterinuseisatleasttentimeshigherthantheusedresistors.
Registerthephotoelementcurrent(notvoltage!)whichisproportionaltothelightintensityinaprofileperpendiculartotheopticalaxisandthroughthecentreofthelaserbeam.
CalculatethecurrentbyOhm'slawfromthevoltageontheresistor.
Ifthevoltageexceeds60mV,usealowerresistancevalue.
Recordaboutonemeasuringpointperhalfmillimeter,thatisoneturnoftheknob.
Measurethebeamdiameterinfrontoftheleftmirror andinfrontoftherightmirror withthecaliper:Setthe
calipertoaspecificwidthandcheckifthelaserflashesifthecaliperslitmovesperpendiculartotheopticalaxisthroughtheresonatorandtheslitmatchessomewheretheopticalaxis–usethethinpartofthecaliperslit,trywithdifferentanglesbetweencaliperscaleandopticalaxis;notethelowestcaliperspacingforwhichalaserflashcanbeseenasbeamwidth(seeFig.24).
Setthemirrordistance todifferentvalueslessthan1000mmwherethelaserisstillstableandrepeatthemeasurement.
Notes
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Aghostbeammaybeseenasaresultofthewedgeformofthemirror:Multiplereflectionsoccurinsidethemirrorglassbecauseofimperfectantifreflectioncoatingofthemirror'sbackside.Thesereflectionsinterferewiththemainlaserbeam.Toreducethedisturbancecausedbythem,themirrorismadeinawedgeshape,thatisfrontandbacksidearenotparallel,sothereflectionsdifferindirectionfromthemainbeam.Theintensitydistributionisdistortedbytheghostbeam.Thisdistortionistobeneglected.
Intheidealcaseofinfiniteresistancethephotoelementvoltageisindependentonthelightintensityandequaltothebandgapoftheelement.Itdecreaseswithlowerresistancetogreaterextentthelessincidentlightisavailable.
Foraccurateproportionalityofphotoelementcurrenttoincidentphotonrateareversebiascouldbeappliedtothephotoelement.Thenanonlytemperaturedependentdarkcurrentfromintrinsicconductivityoftheelementmaybepresentandhastobesubtractedfromthereadings.Suchanaccuracyisnotneededinthisexperiment.
Thelaserpowercanbecalculatedwiththesensitivityofthephotoelementof0.48A/W.
Fig.24:Beamwidthmeasurementwithacaliper
Task4
Note:InthistaskleftandrightisdefinedaccordingtotheperspectiveofFig.1.
SetthelaserupasinTask1,thatiswithHRflat/flatasleftmirrorandHRflat/1000mmasrightmirror.
Setthetubecurrentto5.5mA.
Movethelaserset-uptotheleftendoftheopticalbenchremovingtheadjustmentlaserandadjustlaseraxistoopticalbenchaxisbybeamwalking.
Movetherightmirrorto420mmdistance totheleftmirror,maximizelaseroutput.
Removetherightdiaphragmfromitsslidemountandinsertthesiliconphotoelementwithouttheblindthere.
Usethedigitalmultimetertodeterminetheactualresistancevalueofa1kΩresistor.
Besuretheinputresistanceofthemultimeterinuseisatleasttentimeshigherthantheusedresistorvalue.
Connectthephotoelementtothis1kΩresistorontheconnectionboxandalignitsuchthatitrecievesallthelaseroutput(s.Fig.31andcircuitdiagram).
Measurethevoltageontheresistor.
Ifthevoltageexceeds60mV,usealowerresistancevaluewhichyoucheckwiththedigitalmultimeter.
Connectthedigitalmultimetertomeasurethevoltagedropontheresistorwiththecurrentfromthephotoelementpassingtheresistor.
CalculatethecurrentbyOhm'slawfromthevoltageontheresistor.
Registerthemaximumachievablephotoelementcurrentfordifferentmirrorspacingswiththelasertubeinfrontoftheleftmirrorandnotethevisiblemodeshape.
Ifthemodeshapeisnotround,themirrorsandbrewsterwindowsneedcleaning.
Alwaysnotedownthecurrentfrombackgroundlightwithnolaserbeam,thebeaminterruptedwithapieceofpaperbetweenleftmirrorandleftendofthelasertube,thetubelightandmaybeotherstraylightfromtheroomstillilluminatingthephotodiode;darkentheroomifexessivebackgroundispresent.
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Withafixedmirrordistanceregisterthemaximumlaseroutputfordifferentlasertubepositionsinsidethecavity,forexamplefor =850mmand =700mm.Youmayuseaspecificpointofthetubeholderasreferencefortheposition.
ExchangetherightmirrorwiththeOCflat/1400mmmirrorandrepeatthemeasurements,tubepositionvariationonlyfor =1050mm.
ExchangetherightmirrorwiththeHRflat/1400mmmirrorandrepeatthemeasurements,tubepositionvariationonlyfor =1050mm.
Thesensitivityofthephotoelementisassumedtobe0.48A/Wtocalculatethelaserpower.
Fig.30:Setupforlaserpowermeasurementwithdifferentresonatorgeometries
Fig.31:Photoelementset-up
Task5
SetthelaserupasinTask1,thatiswithHRflat/flatasleftmirrorbutexchangetherightmirrorwiththeHRflat/1400
mm.
Setthemirrordistanceto =450mm,optimizethelaseroutputaftereachstepofmodificationofthesetup.
Exchangetherightdiaphragmontheslidemountwiththesiliconphotoelementwithouttheblind.
Usethedigitalmultimetertodeterminetheactualresistancevalueoftheusedresistors.
Connectthephotoelementtoa1kΩresistorontheconnectionboxandalignitsuchthatitrecievesallthelaseroutput.
Besuretheinputresistanceofthemultimeterinuseisatleasttentimeshigherthantheusedresistorvalue.
Connectthedigitalmultimetertomeasurethevoltagedropontheresistorwiththecurrentfromthephotoelement
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passingtheresistor(s.Fig.35andcircuitdiagram).
Varythelasertubecurrentatthelasercontrolunitmeasuringthevoltageontheresistorforcurrentsettingsfromlowestpossibleupto10mA.
Observetheuncoherentlightintensitychangesfromthegasdischargebyeye.
Measurethevoltagefromthebackgroundlightfromthegasdischargeandtheroomlightwheninterruptingthelaserbeamleftofthelasertubewithapieceofpaperandsubtractitfromthemeasuredvoltagewithlaseronforeachcurrentvalue.
ExchangetherightmirrorwiththeOCflat/1400mmmirror,optimizelaseroutput.
Insertthe100Ωresistorinsteadofthe1kΩresistor.
Repeatthemeasurement.
Youmayalsotryahighoutputconfiguration(left:HRflat/1000mm,right:OCflat/1400mm, =420mm)andanearlyunstableconfiguration(left:HRflat/flat,right:HRflat/1000mm, =950mm)
Donotexposethelasertubetocurrentsexceeding6.5mAforextendedtime,turnthecurrentdownto6.5mAagainatonceafterthemeasurementpointsarerecorded.
Keepinmindthatthetubetemperatureriseswithrisingcurrentleadingtouncomparableresultswhenturningthecurrentdownagainuntilthetubecooledoff.Thelaserpoweristemperature-dependentsincethedopplerlinewidthandgaspressurearetemperature-dependent.
CalculatethecurrentbyOhm'slawfromthevoltageontheresistor.
Tocalculatethelaserpowerthesensitivityofthephotoelementisassumedtobe0.48A/W.
Notes:
Youalsomayanalyzethetime-dependenceofthelaseroutputpowerwithaLF-amplifyerandaloudspeaker(PhyweNr.13625.93and13765.00)orwithanoscilloscope(Phywe-Nr.11459.95).
ForlaseroutputpowermeasurementinthemWrangethephotoelementcurrentmaybedirectlymeasuredwiththedigitalmultimeter07123.00intheμAsettingwithoutconnectionboxandresistors.
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Fig.35:Photoelementconnection
Fig.36:Experimentalset-up
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Task6
Note:InthistaskleftandrightisdefinedaccordingtotheperspectiveofFig.39.
SetthelaserupasinTask1.
Movelasertubeandthecavitymirrorsnearthealignmentlaser.
Observethelightemanatingfromthegasdischargethroughthegratingbyeye.
Puttheplateholderintotheslidingmountwithoutcolumnrightoutsidethelasercavity.
Insertthediffractiongratingwith600linespermillimeterintotheplateholder,gratinglineshorizontalsothediffractionisvertical.
Insertthescaleonrodintotheslidingdevicewheretherightdiaphragmwas.
Measurethedistancebetweengratingandscale withhelpofthescaleontheopticalbenchandthecaliper.
Measurethedistancebetweentheundiffractedlaserlightspotandthefirstorderdiffractedspots ontheverticalscaleforthegreenandredlaserlight.
Foreliminationofstatisticalerrorsmeasureatdifferentdistancesbetweengratingandverticalscale.
Fig.39:Set-upforthewavelengthmeasurementwithopticalgrating
Theoryandevaluation
Task2
Foropticalcavityanalysisusuallyastabilityfactorgisintroducedcomprisingtheratioofresonatorlengthdtoradiusofmirrorcurvature with
Thestabilitycriterionforalaserresonatortobederivedlateronis
Here,mirrorradiusispositivewithconcavemirrorsandnegativewithconvexmirrorsasseenfromtheinneroftheresonatorcavity.
Inthisexperimentstabilitywastestedforthreecombinationsofmirrorcurvature:
1.
2.
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3.
Inthislasersetupthemirrordistancedisatleastthelasertubelengthof420mmandatmostwithbothmirrorholdersfacingrightandtheholdersprotrudingeach10mmfromtherail1465mm.Fig.16plotsthestabilitycriterion overmirrordistanced.Eachshowncurvecorrespondstoaspecialresonatorgeometry.Thestabilitycriterionisfullfilledaslongasthecurveslieintheareadefinedbythe[0,1]intervalonthey-axisandthewholex-axis.Foroneflatandonecurvedmirrorthestabilitycriterionislinearin andthegraphisthesteeperthesmallertheradiusofcurvature is.Sowithsmallerradiusofcurvaturetheregionofstabilitygetssmallerandisofthesizeoftheradiusofcurvature.Fortwocurvedmirrorsthestabilitycriterionisaparabola.Iftheradiiarenotequalthisresultsintworegionsofstabilitythatareseparatedbyaregionofinstabilityofthesizeofthedifferenceoftheradii.
Ifonemirrorwasconvexandtheotherconcave,thegraphwouldbeaparabolathatopensdownwardsandstartsatunityfor.Thestabilitycriterioncanalsoberepresentedinthe planeasshowninFig.17.Theareasintheplanewherethe
stabilitycriterionisfullfilledareshaded.Withaflat–concaveresonatoronetravelsalongarrow(1)inthe planestartingfrom(1;1)whenincreasingmirrorspacingdfromzero.Withthe1000mm–1400mmconcave–concaveresonatoristhejourneyalongarrow(2).
Table1:Somespecialtypesofresonators
Type Mirrorradius Stabilityparameter
Confocal
Concentric
Symmetric
Sym.confocal
Sym.concentric
semiconfocal
plane
Typicalmeasurementresultsmaybe:Maximalmirrorspacingdmaxformirrorcombination1):
Maximalmirrorspacingdmaxformirrorcombination2):
Maximalmirrorspacingdmax,1formirrorcombination3)andthefirstareaofstability:
Minimalmirrorspacingdmin,2formirrorcombination3)andthesecondareaofstability:
Thedeviationsfromtheorticalvaluesdependon
a)dampingbydirtandimperfectionsontheopticsincaseofimpossiblelasingthoughinsidestabilityrangeb)presenceofraysnotintheplaneoftheopticalaxisasindoughnut-shapedorothertransversalmodesincaseoflasingthoughoutsidestabilityrange.
Fig.16:Thestabilitycriterionintheg1–g2plane
Fig.17:stabilitycriteriong1·g2overmirrorspacingd
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Thestabilitycriterionistobederivedinthefollowing:Toexaminethestabilityoftheopticalresonatorthebehaviorofarayinthesenseofgeometricalopticsafteraroundtripintheresonatorisdetermined.
Theresonator(seeFig.17)istoconsistofaleftmirrorwithfocallength andradiusofcurvature andarightmirrorwithfocallength andradiusofcurvature .The axisistoshowtotherightandthe axisistopointupwards.Theleftmirrorissituatedat =0andtherightat . and areconsideredpositiveforconcaveandnegativeforconvexmirrorsasseenfromtheinneroftheresonator .
Becauseofrotationsymmetryoftheresonatortheazimuthangleoftherayisomittedandonlyraysinaplanewiththeopticalaxisareconsidered.ThismaynotbedonefordetailedmodeexaminationofdiffractionlimitedbeamsthataredescribedwithHermitepolynomialsasintensitydistribution.Let'sconsiderageometricalraystartingattheleftmirror withanarbitraryinitialangle totheopticalaxisandanarbitrarydistancetotheopticalaxis . isconsideredsmallenoughsothatimagingrulesapply. shouldbereasonablysmallsuchthat ≈ holds.Afterpassingthedistance therayimpingesupontherightmirror .Theangleisstill butthedistancetotheopticalaxisisnow(seeFig.17):
.
Afterreflection, isunchangedbuttheangleisnowwith(seeFig.18):
;
,thus
Nowthebeamtravelsintheotherdirectionalongthedistance withtheangle ,thusanegativesignwith .Whiletheangle
isconstantthedistancefromtheaxischangesfrom tofinally(seeFig.19)
(1)
Tocompletetheroundtrip,thefinalangleafterreflectionontheleftmirrorhastobeaccountedfor(seeFig.20):
; (2)
Ingeometricopticsintheplaneoftheopticalaxisabeamiscompletelydescribedbyit's -and -values,theyarecoordinatestodescribethebeam.Afteroneroundtripintheresonatortheinitialvalues and aremappedontothefinalvalues and
byatransformation .
Thesetoflinearequations(1)and(2):
definethistransformation
.
Stabilityanalysisoftheresonatormeanstofindconditionswherefiniteeigenvaluesexistforthistransformationiteratedn
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times.Sincetheeigenvalue ofthe -timesiteratedtransformationequalsthe -thpowerofthe(ingeneral)complexeigenvalue ofthetransformation,i.e.therelation isvalid,itissufficienttoinvestigatetheeigenvaluesof
thetransformation.Soithastobeevaluatedwheretheeigenvalueequation
,orelse
with\overline{\overline{\text{I}}}the2x2identitymatrix hassolutionsfor withabsolutevalue1(iftheabsolute
valueisunequal1 \lambda^ngoestoinfinityortozerowithincreasing ).
Eigenvaluesexistif
sothecharacteristicequation
hastobesolvedfor withtraceofamatrixthesumofthediagonalelements,
andthedeterminantofa2x2matrix ,sohere
(whichyoumayproveasanadditionaltask)thus
Usuallyastabilityfactor isintroducedcomprisingtheratioofresonatorlength toradiusofmirrorcurvature with
,sothat
andthecharacteristicequationbecomes
Thisquadraticequationissolvedby
with realnumbers.Theeigenvalue after passesthroughtheresonatoristostaylimited,so hastoeitherbeunityorcompriseanimaginarypart. =1doesnotreallyrepresentstabilitysinceany(physicallyalwaysexisting)deviationingileadstolimitlessgrowthordecreaseof .So
tomakethesquarerootimaginary,thus
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isthestabilitycriterionforalaserresonator.Theeigenvalueisaperiodicsolutionoftheform
or and .
Additionaltheory
Thiswayofexploringpropertiesofanopticalwaveguideconsistingoflenses,mirrorsandfreespacealonganopticalaxiscanbesimplified,ifforeachelementoftheopticalwaveguidearaytransfermatrixisspecified.Thebehaviorofseveralelementsinseriesisdescribedbythematrixproductofthematricesofeachelement.Eachmatrixmapstheangle anddistance oftheinputrayoftheopticalelementtotheoutputray'sangle toopticalaxisanddistance fromopticalaxis.
Forexamplethematrix forathinlenswithfocallength iswithpositive forconverginglenses
=
andthematrix\overline{\overline{\text{S}}}forfreebeampropagationalongthedistance is
= .
Sincethebehaviorofaresonatorwithconcavemirrorsindistance issimilartoasetofconverginglenseswiththesamefocal
lengthinthesamedistance,theabovematrix is
= .
tobereadas" after after after ".Thesequenceofthematricesisofimportancesincethematrixproductisnotcommutativeasinphysicalrealityitisofcourseofimportancewhichlenscomesfirste.g.whichendofthetelescopeyouholdtoyoureye.Thebehaviorofthesystemisthendescribedbytheequation
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Fig.18:Raystartingfromleftmirrortotheright
Fig.19:Rayreflectedfromtherightmirror
Fig.20:Distanceoftheraytotheopticalaxiswhenarrivingontheleftmirroragain
Fig.21:Reflectionoftherayontheleftmirror
Task3
Theory
Theactualelectro-magneticfielddistributioninsidethelasercavitycannotbefullydeterminedbygeometricalopticssincethelaserbeamisdiffractionlimited.Onemoreconditiontobefulfilledforastableresonatormodeisthatthemirrorsurfacesbeplanesofconstantphaseofthelightwavesforfundamentallongitudinalmodesormayonlyhavesomenodelinesonthemirrorsurfacefornon-fundamentaltransversalmodes.Inacalculationthisisdonebyintroducingcontinuitycriteriafortheelectricfieldatthemirrorsurfaceasboundaryconditionforthedifferentialequationsdescribingtheelectricalfield.Alsofortheresonatorfinesseithastobekeptinmindthatresonatorqualityisreducedduetodiffractiononaperturesasdischargecapillaryandmirrorrimswhichlessenthelightintensityinthecavitybycouplingsomelightoutoftheresonator.
AsolutionforthefielddistributionfulfillingtherequirementsofbothgeometricalopticsanddiffractioninthecavityforafundamentallongitudinalmodeisalwaysprovidedbyaGaussianbeam.SotheactualbeampropertiesareherecomparedtothetheoreticalpropertiesofaGaussianbeam.
AGaussianbeamischaracterizedbythefollowingproperties:
AGaussianbeamhasrotationalsymmetryaboutit'saxisofpropagation.
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AnideallensormirroralwaystransformsoneGaussianbeamintoanother.
Thetransversalprofile,thatistheradial(in -direction)intensityprofileperpendiculartotheopitcalaxis,ofaGaussianbeampropagatingalongtheopticalaxisinz-directionisaGaussianbell-shapedfunction,thenormaldistributionfunction.Thushasthebeamnoactualboundariesbutislaterallyextendedtoinfinity.Thebeamwidthwisdefinedastheradius,wheretheintensityhasdecreasedby ≈0.36788.Practicallytheexponentialdecayofintensitywithgrowingdistance
toopticalaxis providesthattherimsofmirrorsandlensescanbeneglectedinsomeormostcases.
Forthelongitudinalprofilealongtheopticalaxisin directioncanbesaid(s.Fig.25):AGaussianbeamhasabeamwaistat = withaminimalspotsize .Inthefollowingtheoriginofthe -axis =0issettothepositionofthewaist,=0.Thebeamwidth dependson as
(1)
withtheRayleighrangeordepthoffocus
. (2)
AtadistancefromthewaistequaltotheRayleighrange ,thewidthwofthebeamis
ThewavefrontsinaGaussianbeam,thatistheplanesofconstantphase,haveacurvature
. (3)
Especiallythewavefrontisflatinthecentreofthebeamwaist,thatis isindefinite.Seenfromthebeamwaist,the
curvatureofthewavefrontsisalwaysconcave(seeFig.25),soifthisformulaistobeapplied,theradiusofcurvatureisdefined(incontrasttoTask2)negativeifconcaveisseenfromright,thatisfromthepositiveendofthez-axisallalongtheopticalrail.
Fig.25:DiagramofaGaussianbeamwaist,z-axiscoincidewiththeopticalaxis
FarawayfromthebeamwaistorRayleighrange theGaussianbeamhasaconstantdivergence,whichmeansthe
beamopenslikeaconewithopeningangle , theanglebetweenconesurfaceandopticalaxis
. (4a)
andwithhelpof(1), and(2)
. (4b)
ThecentralphaseontheopticalaxisshiftsfrominsidetheRayleighrangetooutsidewiththeGouyphase
. (5)
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Thiscontributestoaphaseshiftof for to .
Toassessthequalityoflaserbeams,theiractualpropertiesarecomparedtothepropertiesofaGaussianbeamasthebestachievablebeamquality.Thequalityoflaserbeamsisoftendescribedbythebeamparameterproduct(BPP).TheBPPistheproductofdivergenceangle andbeamwaistradius ,BPP= .ThelowertheBPP,thebetterthebeamquality.ItisminimalforaGaussianbeamwithBPP= .ThemeasuredBPPdividedby isaccordingtoISO11146ameasureforlaser
beamqualityandcalled factorwith .
InaHeNelaser factorsnearunitymaybeachievedandthebeamisthencalleddiffractionlimitedorGaussian.IdeallyaGaussianbeamestablishesinthelasercavitysuchthatthecurvatureofthewavefronts matchesthemirrorcurvatures.
Sofromtheknowledgeofmirrorspacingandcurvaturethepositionofbeamwaist ,beamwaistradius and,angleofdivergence maybedetermined.Let'sputthe -axisorigintotheleftmirrorwithcurvature andasecondmirrorwithcurvature to .Assimplestcaseonemirrorisflatwith = andsothebeamwaistissituatedontheleftmirror,
.
So(3)becomes
with andwith(2)solvedfor :
or .(6)
Thebeamradiusontherightmirror isthenaccordingto(1)
(7)
Theangleofdivergenceaisthenaccordingto(4)
.(8)
Evaluation
Westarttheevaluationwithanoverviewoftherelevantquantitiesoftheexperiment:
: wavelengthofthelaserlight.
: beamwaistinfrontoftheleftmirror.
beamwaistinfrontoftherightmirror.
beamwaistattherightendoftheopticalrail.
distancebetweentheleftandtherightresonatormirror.
curvatationoftherightmirror.
standarddeviationoftheGaussiandistribution.
openingangleofthelaserlightcone.
Fig.26showsaplotfor and for = mand =1000mm.Thereisamaximalbeamwaistradius of0.317mmfor =500mm.
Fig.27showstheintensitydistributionofthelaserbeammeasuredattheendoftheopticalrailindependenceofdistancetotheopticalaxis1400mmawayfromthebeamwaistontheflat(left)mirror.Fittingagaussianscalednormaldistributiontoeachseriesofmeasurementdatayieldsanestimationofthestandarddeviation andthusthebeamradiusforeachsetofdata.Withthistheangleofdivergencecanbecalculated,seeTable2:
Fig.28showsaplotforafor = mand =1000mm.Thereisaminimaltheoreticalangleofdivergenceaof0.00063rador0.036°or2arcminutesfor =500mm.
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Fig.26:Calculatedbeamradiusontherightmirrorwdandbeamwaistradiusw0accordingto(6)and(7)independenceonmirrorspacingdinmmwithmeasurementvaluesfromTable1.
Fig.27:Laserlightintensitymeasuredperpendiculartotheopticalaxis
Fig.28:Calculateddivergenceangleaaccordingto(8)independenceonmirrorspacingdinmmwiththemeasuredvaluesofTable2
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Table1:Measuredbeamradiusvaluesinsidetheresonator.
/mm /mm /mm /mm /mm
450 0.75 0.375 1.00 0.500
500 0.75 0.375 1.05 0.525
550 0.75 0.375 1.10 0.550
650 0.75 0.375 1.25 0.625
750 0.70 0.350 1.40 0.700
800 0.65 0.325 1.45 0.725
850 0.55 0.275 1.55 0.775
875 0.55 0.275 1.60 0.800
900 0.50 0.250 1.70 0.850
925 0.45 0.250 1.95 0.975
950 0.50 0.225 2.10 1.050
Table2:Anglemeasurementresults:sisdeterminedbythefittingprocedureofthetransversalbeamprofileandw1400=√2s.alphaiscalculatedwithhelpof(4a)outof
thevaluesw1400.
/mm /mm /mm /rad
450 0.67 0.95 0.00068
500 0.57 0.80 0.00057
550 0.58 0.82 0.00059
650 0.60 0.84 0.00060
750 0.63 0.89 0.00063
800 0.66 0.93 0.00066
850 0.70 0.99 0.00071
875 0.73 1.03 0.00074
900 0.80 1.13 0.00080
925 0.90 1.27 0.00091
950 1.03 1.46 0.00104
Table3:MeasuredM^2factorsaccordingtoTable1and2withlambda=633nm.
/mm /rad /mm
450 0.00068 0.375 1.28
500 0.00057 0.375 1.06
550 0.00059 0.375 1.10
650 0.00060 0.375 1.12
750 0.00063 0.350 1.10
800 0.00066 0.325 1.07
850 0.00071 0.275 0.96
875 0.00074 0.275 1.01
900 0.00080 0.250 0.99
925 0.00091 0.250 1.13
950 0.00104 0.250 1.16
ItcanbeseenherethatthebeampropertiesaresomewhatintherangeofaGaussianbeam.Onemeasurementerrorhereisaprincipalerrorconnectedwithintra-cavitybeamwidthmeasurementwithacaliper.Thisoverestimatesthebeamwidthsincethelaserdoesnotigniteifonlysomepercentofthepowerarefilteredperresonatorroundtripofthelight.Sothemeasuredvaluedoesnotexactlymatchtheradial1/ intensitydropdefinedasbeamradius.Furthermorethetotallasergaindecreaseswithdecreasingresonatorstabilitywithincreasingmirrorspacing .Soforanearlyunstablelaseronlyasmallerportionofthebeamdiametermaybecutbythecalipertostillignitethelaser.
Additionaltheory
ThefielddistributioninaGaussianbeamisthesolutionoftheHelmholtzequationinparaxialapproximation–thetimeandphasedependentpartoftheHelmholtzequationalreadyseparatedwiththemethodofseparationofvariables:
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.
withthetimeandphaseindependentamplitude oftheelectricfieldandthetransverseformoftheLaplacianoperator
.
incylindercoordinates and .ParaxialapproximationisasinTask2:
or and
Thesolutionforcomplexelectricfieldamplitude atpositionvector includingphaseinformationbutignoringtime
dependencyisthennotdependenton
andtheintensitydistribution,whichisnotphase-sensitive,is
.
isthevacuumimpedanceorapproximately377Ohm.Todiscussthe -dependencyof wechooseanarbitrarybut
fixed value .Forevery isafunctionwithrespectto thatdescribesthefieldintensitytransversaltothe
directionofpropagation.ComparedtotheGaussiannormaldistribution
,
itis whichhastobekeptinmindwhencalculatingbeamwidth fromaGaussianscalednormaldistributionfittedtothemeasurementdata.
Weusethephaseplanecurvatationtodiscuss dependencyof
.
Ifthebeamwaistisnotatamirror,thatisbothmirrorsarecurved,then
.
Thenthephaseplanecurvature isnegative for thatisleftfromthebeamwaistandpositive
for whichisrightfromthebeamwaist.Forconsistencythecurvatureofthemirrorshastobedefinedtobenegativeifconcavewatchedfromrightorpositive -axis.Let'sput totheleftmirrorwithcurvature andthesecondmirrorwithcurvature at ,thenifwavefrontcurvatureistomatchthemirrorcurvature:
(*)
. (**)
(*):
(**):
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(*)-(**): (***)
From(*)follows
(**):
andsince hastobearealnumber,itis for and for .
With and(*)isthen
(*)-(**):(***)
whereyoumayfillin(***).Fig.29visualizesthebehaviorofbeamwaistpositionzwandbeamwaistradiusw0forthemirrorcombinationusedinTask2, =-1000mmand =1400mm.For <1000mmthebeamwaiststaysinsidethecavityinfrontoftherightmirrorandmovestotherightwithgrowingdandfor1400mm<d<2400mmthebeamwaistisneartheleftmirror.Thebeamwaistradiusisonlydefinedforthestabilityregions0mm< <1000mmandfor1400mm< <2400mm.Theangleofdivergenceisnotshownbutisproportionaltotheinversebeamwaistradius.
Fig.8:Calculatedbeamwaistpositionzwandbeamwaistradiusw0forr1=-1000mmandr2=1400mmwithdistancedbetweenthemirrorsfrom0to2800mm
Task4
InFig.32theleftscaleisforthemeasurementwithoutcoupling(OC)mirrorandtherightfortheoutputthroughhighreflective(HR)mirrors.Itcanbeseenthatthelaseroutputisabout25timeshigherwithanOCmirrorbuttheoutputdependenceonmirrorspacingismainlythesame.Perunittimethegasdischargesuppliesanactiveamplifyinggasvolumeby"pumping"withroughlythesameextractablelasingenergypervolumeofdischargeifthecurrentdensityisuniformthroughoutthevolume.Stilltheamplificationmaydependtosomeextentonradialdistancefromthecapillaryaxisbothduetoradialtemperatureanddensityvariationofthedischargegasandthedeexcitationeffectofthecapillarywallwhichisabottlenecktothepopulationinversion.Therearebothfrequencyandgeometriceffects:
Thelaseroutputpowerisdeterminedbythefractionoftheactivevolumewhereoscillatingmodeshavehighlightintensity:Onlywhereamodehashighintensity,itcaneffectivelyconversetheenergysuppliedbytheactivevolume.Theoutputpoweristhehigherthebettertheactivevolumeisfilledwithzonesofhighintensitybymodeswhichmayoscillate.Themorepossiblemodes,thebetterthevolumefitting.
Thelaseroutputpowerthroughthemirrorsisalsodeterminedbydiffractionlosses:Eachmodehasdiffractionlossesonthelasercapillaryandthecavitymirrorswhichallworkasmodeblinds.Iftheactuallyoscillatingmodeshavehighdiffractionlossestheygiveofflaserintensity(andpower)tothesidesalongthelasercavitybutnotthroughthemirrorsandtheselosseslessentheintracavityintensityandthustheeffectivityofenergyextractionfromtheactivevolume.
Forfrequencyeffects:
Perunittimethegasdischargeuniformlydistributesextractablelasingenergyamonggasatomswithdifferentamountofspeedindirectionofthecapillaryaxisandthusdifferentresonancefrequencies.Sincelongitudinalmodes(tothesametransversalmodenumber)differinfrequency,thehigherthenumberoflongitudinalmodesthehigheristhepossible
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outputpowerbecauseofbetterfillingofthefrequencyspace.
Withthecavitywithaflatmirrorononesideandthe1400mmmirrorontheotherforsmallmirrordistancesbothfundamentallongitudinalmodesandmodeswithnon-zerotransversalmodenumberandalsodoughnut-shapedmodesdooscillate.Theactivevolumeisoptimallyfilled,thediffractionlosseslowandtheoutputpowerishigh.Atamirrordistanceof600mmthecapillaryfiltersallnon-fundamentaltransversalmodes,onlysometransversalmodesmayoscillateandtheoutputpowerhasaminimum.Withrisingmirrordistancethenumberoflongitudinalmodesrisesandthustheoutputpower.
Fig.32:Outputpowerindependenceonmirrorspacingdwiththelasertubefixedinfrontoftheflatmirror.Thex-axiscorrespondstothemirrorspacingdinmm.
Formirrordistancesover1000mmtheactivevolumeisnolongeroptimallyfilledsincethemodediametercontractsontheflatmirrorandfordistancesover1200mmthediffractionlossesonthecurvedmirror,wherethediameterexpands,dominateandthelaserpowerdiminishesuntilthecavitybecomestotallyunstableat1400mm.Withthecavitywithaflatmirrorononesideandthe1000mmmirrorontheothermodeswithnon-zerotransversalmodenumberarefilteredalreadyattheminimalmirrordistance,thetubelength.Therisingofpossiblelongitudinalmodenumberscompensatesforthediminishingusefulfractionoftheactivevolumewithrisingmirrordistanceuntilthedistanceexceeds800mmandthediffractionlossesonthecurvedmirrordominate.
Fig.33showsthelaseroutputpowerdependenceontubepositionforacavitywithaflatmirrorononesideandthe1000mmconcavemirrorontheother.Theuppercurveappliesforamirrordistanceof700mmandthelowercurveforamirrordistanceof850mm.Astubepositionanarbitrarybutfixedpointonthetubeholderwasselected.Theopeningangleofthelaserbeamisgreaterforthegreatermirrordistanceof850mmsothediffractionlossesonthecapillaryopeningthatshowstothecurvedmirror,wherethebeamdiameteriswider,aregreaterandthelaserpowerdiminishesfasterwhenthetubeismovedtowardsthecurvedmirror.
InFig.34theleftscaleappliestotheoutputthroughtheOCmirrorandtherighttotheoutputthroughaHRmirrorofsamecurvaturewhichwassetasexchangefortheOCmirror.ThecavityconsistedofaHRflatmirrorononeanda1400mmconcavemirrorontheotherside.Theoutputdiminishesasthetubeismovedtowardstheendofthecavitywherethebeamdiameterisincreasingbecauseofthegreaterlossesofathickerbeamonthecapillary.BecauseoflessoutcouplinglossesthroughthemirrorstheHRconfigurationcantolerategreaterlossesonthecapillaryandthelowercurveislesssteepattheend.ThebendingofthecurveisduetothefactthatthetubeisstillintheRayleighrangeandthebeamdiameterdoesnotincreaselinearly.AlsotheintensityfractioncutbyacircularopeningofaGaussianbeamisnotlinearwithdiameter.
Fig.33:Laseroutputpowerindependenceonlasertubepositionfordifferentmirrordistances
Fig.34:LaseroutputpowerindependenceonlasertubepositionThex-axiscorrespondstothelasertubepositionzinmm.
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Task5
Fig.37showshowthelaseroutputpowervarieswiththedischargecurrentforacavityconfigurationwhereoptimaloutputcanbeextractedfromthelaserandaconfigurationthatisalmostunstable.InthefirstconfigurationwiththeOCmirrorsthelaserpowerisabout500timeshighercomparedtothelatteralmostunstableconfiguration.TheleftscaleappliestotheOCandtherightscaletotheHRconfiguration.IntheHRconfigurationcurveanexampleofcurvedistortionduetotemperaturevariationisshown(openingofawindowingermanfebruary).
Fig.37:Laseroutputpowerindependenceondischargecurrentinhighouputandinnearlyunstableconfiguration.Thex-axiscorrespondstothedischargecurrentinmA.
Fig.38:ComparisonofoutputdependencyoncurrentforsameresonatorgeometrybutwithHRorOCmirror.Thex-axiscorrespondstothedischargecurrentinmA.
IthastobekeptinmindthatthemultimeterdoesnotdetectpowerfluctuationsatfrequenciesexceedingsomeHertz.Inallcasesthelaserpowerfirstriseswithincreasingdischargecurrentasmightbeexpectedbecausethedischargesuppliesforthelaserpopulationinversion.Theintensityoftheemitteduncoherentlightfromthedischargeincreases,too.Butwhiletheuncoherentlightgetsalwaysmoreintensewithincreasingdischargecurrent,thecoherentlightintensitybeginstodecreaseatsomepoint–moreso,iftheintra-cavitylightintensityisnothigh.
Thisisbecausethedischargecurrentdensitygetsunstableatacertaincurrentdensityvalueforaspecifictubegeometryandgaspressure.Gasdischargesgenerallyhaveanegativeresistancesothecurrenthasthetendencytocontracttofilamentsiftherearenodampingprocessestoinhibitthis.Thereisacharacteristicformationtimeforthefilaments.Ifthetimethegasparticlesneedtopassthedischargespaceissmallerthanthisfilamentformationtime,thefilamentformationgetsdampedout.Thepassingtimeofthegasparticlesisdeterminedbythermalandturbulentdiffusiontimeconstants.Theonsetofdischargeinstabilitycanbedetectedasnoiseonthelaseroutputsignalandoccursaround7mAwiththetubesinuseinthisexperiment.
Therisinglinewidthofthelasertransitionwithrisingtemperatureandpressurecanenablemoremodestooscillate,ifthecavityconfigurationallowsthis.Sothedropofintensityforhighdischargecurrentsisless,ifmanymodesarepresent.Itisalsolessiftheintracavitylightintensityishigh,becausethensaturationforamodecanbereachedwithlowergain.Incaseoffeworweak,thatislow-gain,modesthelaserintensitygetsreducedforhighertemperaturesevenbeforeonsetofdischargeinstabilitybecausethedensityofatomswiththeadequatevelocityforaspecificmodefrequencysinksforbroadervelocitydistributionincaseofhighermediumenergyperatom.Thismaybesothoughthedensityofatomspervolumemaynotsinkincaseofconfinedgaswherethepressureriseswithrisingtemperature.
Task6
Asasimpledescriptionofthediffractiongratingthefollowingholds:Constructiveinterferenceofthelightdiffractedatadjacentgratinglinesoccurswhentheopticalpathlengthdifference ismultiplesofawholewavelength .Formanylinesthisleadstosharplydefinedanglesinwhichthelightofasinglefrequencyisdiffracted.Theopticalpathlengthdifference forlightemittedundertheangle fromtwolinesseparatedbydistance canbeevaluatedusingtrigonometryfromthefollowingsketch
tobe
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.
andwithconditionforconstructiveinterference
fornaturalnumber istheangle
or
since
.
Amoreprecisestatementaboutthediffractedlightofagratingmightbe:Ifthegratinghasindirection atransmissivityfunction andthetransmissivityisindependentondirection perpendicularto andthegratinggetsirradiatedwith
parallelmonochromaticlightofwavelength indirection perpendicularto and ,thediffractionpatternindirection istheFouriertransformofthetransmissionfunction scaledinsizewiththeinverseof ,thatisthewavenumber.
Withthegratingof600lines/mmis =1.67μm.
Table4:Measurementresults
[mm] red [mm] green [mm]
715 299 244
665 278.85 227
615 257 210
565 236 193
515 215 175.5
465 195 158
415 173 140.5
365 152 123.5
InFig.40istheequationforthelineofbestfitfortheredlight
=0.420 -1.048
andforthegreenlight
=0.345 -2.427
sotheanglesare =22.8°fortheredand =19.0°forthegreenlight.
Thisyieldswith =1,67μmawavelengthof0.645μmfortheHeNelaserandof0.544μmforthefrequencydoubledgreenalignmentlaser.Themeasurementerrorof2%comparedwiththeliteraturevaluesof0.633μmfortheHeNelaserand0.532μmforthegreenlaserisduetolengthmeasurementaccuracyoflessthan1mmwithmeasuredlengthssomehundredmm.
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Fig.40:Plotofthemeasurementresults