helium neon laser...helium neon laser (item no.: p2260701) curricular relevance additional...

Student'sSheetPrinted:12.12.201712:41:55|P2260701

Robert-Bosch-Breite10 Tel:+49551604-0 [email protected]

D-37079Göttingen Fax:+49551604-107 www.phywe.com

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




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.




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.




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:




.

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



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.




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.




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.




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




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.




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.




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.




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




Aghostbeammaybeseenasaresultofthewedgeformofthemirror:Multiplereflectionsoccurinsidethemirrorglassbecauseofimperfectantifreflectioncoatingofthemirror'sbackside.Thesereflectionsinterferewiththemainlaserbeam.Toreducethedisturbancecausedbythem,themirrorismadeinawedgeshape,thatisfrontandbacksidearenotparallel,sothereflectionsdifferindirectionfromthemainbeam.Theintensitydistributionisdistortedbytheghostbeam.Thisdistortionistobeneglected.

Intheidealcaseofinfiniteresistancethephotoelementvoltageisindependentonthelightintensityandequaltothebandgapoftheelement.Itdecreaseswithlowerresistancetogreaterextentthelessincidentlightisavailable.

Foraccurateproportionalityofphotoelementcurrenttoincidentphotonrateareversebiascouldbeappliedtothephotoelement.Thenanonlytemperaturedependentdarkcurrentfromintrinsicconductivityoftheelementmaybepresentandhastobesubtractedfromthereadings.Suchanaccuracyisnotneededinthisexperiment.

Thelaserpowercanbecalculatedwiththesensitivityofthephotoelementof0.48A/W.

Fig.24:Beamwidthmeasurementwithacaliper

Task4


SetthelaserupasinTask1,thatiswithHRflat/flatasleftmirrorandHRflat/1000mmasrightmirror.

Setthetubecurrentto5.5mA.

Movethelaserset-uptotheleftendoftheopticalbenchremovingtheadjustmentlaserandadjustlaseraxistoopticalbenchaxisbybeamwalking.

Movetherightmirrorto420mmdistance totheleftmirror,maximizelaseroutput.

Removetherightdiaphragmfromitsslidemountandinsertthesiliconphotoelementwithouttheblindthere.

Usethedigitalmultimetertodeterminetheactualresistancevalueofa1kΩresistor.

Besuretheinputresistanceofthemultimeterinuseisatleasttentimeshigherthantheusedresistorvalue.

Connectthephotoelementtothis1kΩresistorontheconnectionboxandalignitsuchthatitrecievesallthelaseroutput(s.Fig.31andcircuitdiagram).

Measurethevoltageontheresistor.

Ifthevoltageexceeds60mV,usealowerresistancevaluewhichyoucheckwiththedigitalmultimeter.

Connectthedigitalmultimetertomeasurethevoltagedropontheresistorwiththecurrentfromthephotoelementpassingtheresistor.


Registerthemaximumachievablephotoelementcurrentfordifferentmirrorspacingswiththelasertubeinfrontoftheleftmirrorandnotethevisiblemodeshape.

Ifthemodeshapeisnotround,themirrorsandbrewsterwindowsneedcleaning.

Alwaysnotedownthecurrentfrombackgroundlightwithnolaserbeam,thebeaminterruptedwithapieceofpaperbetweenleftmirrorandleftendofthelasertube,thetubelightandmaybeotherstraylightfromtheroomstillilluminatingthephotodiode;darkentheroomifexessivebackgroundispresent.




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




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.


Tocalculatethelaserpowerthesensitivityofthephotoelementisassumedtobe0.48A/W.

Notes:

Youalsomayanalyzethetime-dependenceofthelaseroutputpowerwithaLF-amplifyerandaloudspeaker(PhyweNr.13625.93and13765.00)orwithanoscilloscope(Phywe-Nr.11459.95).

ForlaseroutputpowermeasurementinthemWrangethephotoelementcurrentmaybedirectlymeasuredwiththedigitalmultimeter07123.00intheμAsettingwithoutconnectionboxandresistors.




Fig.35:Photoelementconnection

Fig.36:Experimentalset-up




Task6


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.




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




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




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




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




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.




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)




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.




Fig.26:Calculatedbeamradiusontherightmirrorwdandbeamwaistradiusw0accordingto(6)and(7)independenceonmirrorspacingdinmmwithmeasurementvaluesfromTable1.

Fig.27:Laserlightintensitymeasuredperpendiculartotheopticalaxis

Fig.28:Calculateddivergenceangleaaccordingto(8)independenceonmirrorspacingdinmmwiththemeasuredvaluesofTable2




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:




.

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:

(*)

. (**)

(*):

(**):




(*)-(**): (***)

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




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.




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




.

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.




Fig.40:Plotofthemeasurementresults

helium neon laser...helium neon laser (item no.: p2260701) curricular relevance additional...

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