Gaussian Beams Lab - Physics · 2020. 1. 13. · GAUSSIAN LASER BEAMS WEEK 1 INTRO: MEASURING A...
Transcript of Gaussian Beams Lab - Physics · 2020. 1. 13. · GAUSSIAN LASER BEAMS WEEK 1 INTRO: MEASURING A...
GAUSSIANLASERBEAMSWEEK1INTRO:MEASURINGAGAUSSIANBEAM;CALIBRATINGYOURPHOTODETECTOR
GOALS
Inthislab,youlearnaboutmountingopticsandphotodetectorsandtryoutsometechniquesthataregenerallyusefulinopticslabsandelsewhere.Inparticular,youwillsetupasimpleopticssystemformeasuringthewidthofyourlaserbeamandintheprocesswillhavetomountandalignthelaserandoptics.
• Proficiencywithnewequipmento Laser:mountingittotable,pluggingitin,turningiton.o Mountingoptics:
§ Mirrors§ Lenses§ Post,postholders,bases
o Aligningoptics§ Mirrors(usingtwomirrorstoadjustabeamtoanydesiredpositionandangle)§ Lenses
o Translationstage§ Mountingittotheopticstable§ Mountingopticsonit§ Readingthemicrometerposition§ Measuringmicron-scaledisplacements
o Amplifiedphotodetector§ Usingit§ Understandinghowitworks§ Modelingitsbehavior§ Readingthespecification/datasheet.
• NewskillstoapplyfromLabSkillActivitieso EnteringdataintoMathematicaorimportingdatao Non-linearleast-squaresfittingo Plottingdataandfitfunctiontogethero Extractingbasicfitparameterswithstandarduncertainties
• Experimentaldesigno Calibrationofthephotodetectoro Modelingthephotodetector
LABNOTEBOOKGUIDELINES
Thelabnotebookwillplayanimportantroleinthiscourse.Youwilluseyournotebookforkeepingrecordsofmanythingsincluding
• Answeringpre-labquestionsfromthelabguide.• Answeringin-labquestions• Recordingdata• Includingplotsofdata• Analysisandresults
• Diagramsandpictures• Proceduresofexperimentsthatyoudesign
Thelabnotebookwillbeanimportantpartofyourgradebecauselearningtokeepagoodlabnotebookisanimportantpartofyourprofessionaldevelopment.Youmayfindithelpfultowriteupmanyofyournotesonthecomputer,forexample,withinMathematicaoranotherprogram.Thisisfine.However,beforeyournotebookisturnedin,thenotes,plots,andanalysisshouldbetransferredtothelabnotebookbyprintingandtapingthepagesorkeepingtheminathree-ringbinder.Therewillalsobeformallabreportsandoralpresentations,butthesewillberestrictedtoalimitedportionoftheexperimentalworkyouhaveconductedinthelab.
DEFINITIONS
Optic–Anyopticalcomponentthatmanipulatesthelightinsomeway.Examplesincludelenses,mirrors,polarizingfilters,beamsplitters,etc.
Optomechanics–Thiscategoryincludesopticsmountsandthecomponentstoalignthem.Examplesinthelabincludepost,postholders,bases,lensmounts,adjustablemirrormounts,rotationmounts,andtranslationstages.
SETTINGUPYOURLASERANDMOUNTINGOPTICS
Whenyoustartworkinginthelabyoushouldhaveanemptyopticsbreadboard.Theshelfabovethebreadboardshouldhave
• Anoscilloscope• Awaveformgenerator• TripleoutputDCPowerSupply• Setofballdrivers• Opticscaddytoholdopticsalreadymountedon0.5"posts.• Setof1/4-20and8-32screws,setscrews,washers,andnuts.
Question1 a. Getalaserfromthecabinetsandmountitonyourworktable.Youshoulduse2"postsandpostholdersforthelaser,whichwillsetthelaserataconvenientheightformostoftheopticslabs.
b. EachpersoninyourgroupisresponsibleforassemblingamountedlensormirrorasshowninFigure1.Intheend,youwillneedatleast2mirrorstocompletethenexttask.
Asyouaremountingtheoptics,choosetheheightssothatthelaserhitsthecenterofeachopticandthebeamhorizontal.
Question2
"Walkingabeam"Mountanarrowtubeatarandompositionwitharandomorientationonyouropticsbreadboard.
a. Useonlytwomirrorstogetthebeamtopassthroughthecenterofyourtube.Thistechniqueiscommonlycalled"walkingabeam."Havingtwomirrorsallowsyoutoindependentlyadjusttheangleandpositionofyourbeam.
b. Drawadiagramoftheconfigurationofyourlaser,mirrors,andtube.
Question3 Lensalignment.UsingyoursetupfromQuestion2,addalensafterthemirrors.a. UsingyoursetupfromQuestion2,insertalenstochangethedivergence/convergence
ofthebeambutkeepitspropagationdirectionthesame.b. Whenthiscondition(thebeampropagationisunchanged)ismet,wheredoesthebeam
intersectthelens?
Note:Thisisthepreferredmethodofaddingalenstoanopticalsetup.
Figure1Mountingassembliesforamirror(left)andalens(right).
MODELINGCHARACTERISTICSOFTHEPHOTODETECTOR
ThegoalofthispartofthelabistounderstandalotaboutthespecificationsgivenonthedatasheetfortheThorlabsPDA36ASwitchableGainAmplifiedPhotodetectors.Itisimportanttorealizethatdatasheets(alsocalledspecsheetsorspecificationsheets)provideamodelfortherealisticbehaviorofthedevice.Thismodelcanbetestedandimproved,aprocessmorecommonlycalled"calibration."Notethattherearetwopowerswitches,oneonthepowersupplyandoneonthephotodetector.Thephotodetectorwillrespondtolightwiththepoweroffbutitwon'tworkwellandchangingthegainwillhavelittleeffect.
Question4
Basicfunctionoftheamplifiedphotodetector
Spendafewminutes(nomorethan10)towriteanexplanationusingwordsanddiagramstoexplainthephysicalmechanismforhowthephotodetectorconvertslightintovoltage.Youmayusethemanufacturer’sspecificationssheet,trustworthyonlineresources,abook,etc.Thespecificationsheetavailableathttps://physicscourses.colorado.edu/phys4430/phys4430_sp20/UsefulDocs/PDA36A-Manual.pdf
a. UsethedatasheettoestimatetheconversionofWattsoflightintoAmpsofcurrentforHeliumNeonredwavelength(632.8nm)andfortheFrequencydoubledNd:YAGlaser(Greenlaserpointerwavelength,532nm)?i. Howwouldyouconvert“AmpsperWatt”into“electronsperphoton”?ii. Whatistheelectron/photonconversionefficiencyfortheredHeNeandgreen
doubledNd:YAGlasers?iii. Isthisnumberlessthan,equalto,orgreaterthanone?Whatdoesthisnumbertell
youabouthowthephotodiodeworks?
Question5
CalibratingtheThorlabsPDA36Aphotodetectoroffsetandgain.Calibratingthephotodetectorisespeciallyimportantwhenyoutakeadatasetthatusesmultiplegainsettings.Havinganaccuratecalibrationofthegainandoffsetwillletyoustitchthedatatogetheraccurately.
a. HereyouwillencountergainvaluesthatarepresentedonalogarithmicdB(decibel)scale,whichisobtainedbytaking20×log(Vout/Vin).Forexample,20dBofgaincorrespondstoelectronicvoltageamplificationbyafactorof10.AdBscalecouldalsobedefinedas10×log(Pout/Pin),wherePisthepower.Explaintheconversionbetweenthesetwoscalesandwhythismakessense.
b. Calibratingtheoffsetvoltageistheoutputofthephotodetectorwhennolightisincidentuponthedevice.i. Calibratetheoffsetofthephotodetectorasafunctionofgainsetting.ii. Quantitativelycompareittothespecificationsgiveninthetable.Isyourmeasured
valuewithinthespecifiedrangegivenonthePDA36Aphotodetectordatasheet?iii. Whatmeasuresdidyoutaketoeliminatestraylight?Wereyourmeasuressufficient
foranaccuratecalibration?c. Calibratingthegain
i. IsitpossibletomeasuretheV/Againforeachsetting,orcanyouonlymeasurethechangeingainasyouswitchthesettings?Why?Notethatthislabonlyrequiresrelativegain.
ii. Makeameasurementofthegainorrelativegainsettingsformostofthegainsettings.Ifyouneedtoadjustthelaserpower,tryblockingpartofthebeam.Note,youwillneedtomaketwomeasurementsatonegainsettingwhenyoublockthebeam.Whatsystematicerrorsourcesareofmostconcern?
iii. Quantitativelycompareyourresultswiththerangeofvaluesgivenonthedatasheet.Doyoubelieveyourresultsprovideamoreaccurateestimateofthephotodetectorgainthanthedatasheet?Whyorwhynot?
iv. UsingthePDA36Aspecsheetandyourmeasurements,whatisthepowerofyourlaser?Doesthisagreewiththelaserpowershownonthelaser?
v. Hypothetically,howwouldyoumeasuretheabsolutegain?
Question6
Followup:Writemathematicalexpressionsthatconvertstheincidentpower(thelight)𝑃!"tothephotodetectorvoltage𝑉andthephotodetectorvoltage𝑉toinputpower𝑃!".Takeintoaccountallrelevantparameterssuchasthephotodetectorgainsetting(indB)andoffsets.
MEASURINGTHEBEAMWIDTH
Note:ManyofthedataanalysistechniquesinthissectionwilluseskillsfromtheclassActivities.
Thegoalofthissectionistodevelopameasurementtechniqueandanalysisschemetomeasurethewidthofabeam.Theschemewillletyoumeasurethewidthinonedirection.ThetechniqueismostusefulforbeamsthatareapproximatelyGaussianprofileinintensity.InthesecondweekofthelabyouwillusethistechniquetoexperimentallyanswerquestionsaboutGaussianbeams.
ThebasicschemeinvolvesmeasuringthepowerinthelaserbeamasthebeamisgraduallyblockedbyarazorbladeusingasetupsimilartoFigure2
Figure2:Razorblademountedonatranslationstage
Question7
SupposealaserbeamhasaGaussianintensityprofile𝐼 𝑥, 𝑦 = 𝐼!"#𝑒!! !!!!! !!,andisincidentuponaphotodiode.Whatistheexpressionforthepowerhittingthephotodiodewhenaportionofthebeamisblockedbyarazorblade(seeFigure2:Razorblademountedonatranslationstage)?
a. Drawadiagramshowingthebeamandtherazor.b. Usingtheaboveexpressionfor𝐼(𝑥, 𝑦),writethemathematicalexpressionforthe
powerincidentonthephotodiodeasafunctionofrazorposition.Note,toaddressthisquestion,youwillneedtobecomefamiliarwiththeErrorFunction,erf(x).Whatassumptions,ifany,didyouneedtomakeinevaluatingtheintegral?Hint:ifyouaremovinginthexdirection,whatisgoingonintheydirection?
Question8
Beforeyoutakedata:Createananalysisfunctiontofitatestsetofdata.
Note:NonlinearleastsquaresfittingiscoveredinMathematicaActivity2availableonthecoursewebsite.ThereisalsoaYouTubevideoavailableonleastsquaresfittingatwww.youtube.com/compphysatcu.
a. Whatisthefunctionalformforyourfitfunction?b. Isitalinearornonlinearfitfunction?Why?c. Whatarethefitparameters?Whydoyouneedthismany?d. Howdothefitparametersrelatetothebeamwidth?
Downloadthedatasetfrom:https://physicscourses.colorado.edu/phys4430/phys4430_fa19/UsefulDocs/Test_Profile_Data.csv
i. Makeaplotofthedata.ii. Makeafitandplotitwiththedata.iii. Checkthatthefitlooksgoodandyougetabeamwidthof𝑤 = 4.52×10!! m.If
yougetadifferentvalue,checkwithyourinstructortounderstandtheproblem.Whatistheuncertaintyonyourmeasurement?
Question9
Buildyoursetupformeasuringthebeamwidthofyourlaser.a. Drawadetailedschematicofthesetup(fromthelaserallthewaytothe
photodetector).Donotusealensforthismeasurement.b. Afterassemblingyourexperiment,butpriortotakingalotofdata,howcanyouquickly
determineifthemeasurementisworking?c. Isitpreferabletouseadigitalmultimeteroroscilloscope?Why?d. Usethemeasurementschemetotakedataofpowervspositionoftherazor.Pay
attentiontotheunitsofthetranslationstage.Pickapositionwhereyourbeamhasameasurablewidth,andmeasureit.Justifyyourchoice.
Question10
Analysisoftherandomuncertaintysourcesa. Whatarepossiblesourcesofrandomuncertaintyinthephotodetectorvoltage?b. Howwouldyouestimatetheuncertaintyinthephotodetectorvoltagemeasurement?c. Whatisthelargestsourceofuncertainty?Why?
Question11
Analysisoftherealdata.a. UsetheanalysisproceduresverifiedinQuestion8tofindthebeamwidthforyourdata.
Besuretoincludetheuncertainty.b. Plotyourfittogetherwithyourdatatomakesureitisgood.
WEEK2:DEVELOPINGAQUANTITATIVEMODELOFTHESPATIALPROPERTIESOFLIGHT
GOALS
Inweek1,wemeasuredtheprofileofthelaserandfoundittobeGaussiantoagoodapproximation.However,wedon'thaveanymodelforhowtheprofilechangesasthebeampropagates.Also,wewillapplyautomationtomorerapidlytakedata.Thefullsetoflearninggoalsincludes:
1. Automateddataacquisition.a. LabVIEWb. USBDAQ(NIUSB-6009)
2. FittingandanalysisofdatainMathematica3. UsingapredictivemodelofGaussianlaserbeams
a. ContrastGaussianbeamswithgeometricoptics4. MeasureprofilesofaGaussianbeam,andextracttheGaussianbeamparameters5. EffectofalensonGaussianbeams.
a. IsitstillGaussian?b. DoesthethinlensequationapplytoGaussianbeams?c. Whatlimitstheminimumachievablespotsize?
PRELAB:INTRODUCTION
Question12
Answerthesebeforereadingaheadinthelabguidebasedonyourexperiencefromlastweeka. DoesthebeamalwaysstayaGaussianasitpropagates?b. DoesthebeamstayGaussianafteritgoesthroughalens?c. DoesthebeamstayGaussianafteritreflectsfromamirror?d. Howsmalldoesthebeamgetwhenitisfocusedbyalens?Doesitfocustoapoint?
Whyorwhynot?
Lightisapropagatingoscillationoftheelectromagneticfield.ThegeneralprincipleswhichgovernelectromagneticwavesareMaxwell'sequations.Fromthesegeneralrelations,avectorwaveequationcanbederived.
∇!𝑬 = 𝜇!𝜖!𝜕!𝑬𝜕𝑡!
(1)
Oneofthesimplestsolutionsisthatofaplanewavepropagatinginthe𝒛direction.
𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝐸!𝒙 cos 𝑘𝑧 − 𝜔𝑡 + 𝜙! + 𝐸!𝒚 cos 𝑘𝑧 − 𝜔𝑡 + 𝜙! (2)
Butasthemeasurementsfromlastweekshowed,thelaserbeamsarecommonlywellapproximatedbyabeamshapewithaGaussianintensityprofile.Apparently,sincetheseGaussianprofilebeamsexist,theymustbesolutionsofthewaveequation.ThenextsectionwilldiscusshowwederivetheGaussianbeamelectricfield,andgiveafewkeyresults.
PARAXIALWAVEEQUATION
Oneimportantthingtonoteaboutthebeamoutputfrommostlasersisthatthewidthofthebeamchangesveryslowlycomparedtothewavelengthoflight.Assumeacomplexsolution,wherethebeamispropagatinginthe𝒛-direction,withtheelectricfieldpolarizationinthe𝒙-direction.
𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝒙 𝐴 𝑥, 𝑦, 𝑧 𝑒! !"!!" (3)
Thebasicideaisthatthespatialpatternofthebeam,describedbythefunction𝐴(𝑥, 𝑦, 𝑧),doesnotchangemuchoverawavelength.InthecaseoftheHeNelaseroutput,thefunction𝐴 𝑥, 𝑦, 𝑧 isaGaussianprofilethatchangesitswidthasafunctionof𝑧.IfwesubstitutethetrialsolutioninEq.(3)intothewaveequationinEq.(1)weget
𝒙𝜕!𝐴𝜕𝑥!
+𝜕!𝐴𝜕𝑦!
+𝜕!𝐴𝜕𝑧!
+ 2𝑖𝑘𝜕𝐴𝜕𝑧
− 𝑘!𝐴 𝑒! !"!!" = 𝒙𝜇!𝜖!𝐴 −𝜔! 𝑒! !"!!" (4)
Thiscanbesimplifiedrecognizingthat𝑘! = 𝜔! 𝑐! = 𝜇!𝜖!𝜔!,wherethespeedoflightisrelatedtothepermeabilityandpermittivityoffreespaceby𝑐 = 𝜇!𝜖! !! !.Also,the𝒙𝑒! !"!!" termiscommontobothsidesandcanbedropped,whichresultsin
𝜕!𝐴𝜕𝑥!
+𝜕!𝐴𝜕𝑦!
+𝜕!𝐴𝜕𝑧!
+ 2𝑖𝑘𝜕𝐴𝜕𝑧
= 0 (5)
Sofar,wehavemadenoapproximationtothesolutionorthewaveequation,butnowweapplytheassumptionthat𝜕𝐴 𝑥, 𝑦, 𝑧 𝜕𝑧changesslowlyoverawavelength𝜆 = 2𝜋 𝑘,soweneglecttheterm
𝜕!𝐴𝜕𝑧!
≪ 2𝑘𝜕𝐴𝜕𝑧
(6)
Andfinally,wegettheparaxialwaveequation
𝜕!𝐴𝜕𝑥!
+𝜕!𝐴𝜕𝑦!
+ 2𝑖𝑘𝜕𝐴𝜕𝑧
= 0 (7)
OnesetofsolutionstotheparaxialwaveequationareGauss-Hermitebeams,whichhaveanintensityprofileslikethoseshowninFig.3.Thesearethesamesolutionsasforthequantumsimpleharmonicoscillator,atopicthatcouldbefurtherexploredasafinalproject.
ThesimplestofthesesolutionsistheGaussianbeam,whichhasanelectricfieldgivenby
𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝑬!!!!(!)
exp − !!!!!
!! !exp 𝑖𝑘 !!!!!
!! !𝑒!!" ! 𝑒! !"!!" (8)
Where𝑬!isatime-independentvector(orthogonaltopropagationdirection𝒛)whosemagnitudedenotestheamplitudeofthelaser'selectricfieldandthedirectiondenotesthedirectionofpolarization.Thebeamradius𝑤(𝑧)isgivenby
𝑤 𝑧 = 𝑤! 1 + !"!!!!
! (9)
𝑅(𝑧),theradiusofcurvatureofthewavefront,isgivenby
𝑅 𝑧 = 𝑧 1 + !!!!
!"
! (10)
AndtheGuoyphaseisgivenby
𝜁 𝑧 = arctan !!!!
!" (11)
Theremarkablethingaboutalltheseequationsisthatonlytwoparametersneedtobespecifiedtogivethewholebeamprofile:thewavelength𝜆andthebeamwaist𝑤!,whichisthenarrowestpointinthebeamprofile.ThereisamoregeneralsetofHermiteGaussianmodeswhichareshowninFigure3.Thelasercavitytypicallyproducesthe(0,0)modeshownintheupperleftcorner,butanopticalcavitycanalsobeusedtocreatetheseothermodes–atopicthatcanbeexploredinthefinalprojects.
Figure3IntensitydistributionsforthelowestorderGauss-Hermitesolutionstotheparaxialwaveequation.Theaxesareinunitsofthebeamwidth,w.
MOREPRELAB:TRYINGOUTTHEGAUSSIANBEAMMODEL
Question13
Inweek1ofthelab,weassumedtheintensityprofileoftheGaussianbeamwasgivenby𝐼 𝑥, 𝑦 =𝐼!"#𝑒!! !!!!! !!.TheequationfortheelectricfieldoftheGaussianBeaminEq.(8)lookssubstantiallymorecomplicated.Howaretheexpressionsforelectricfieldandintensityrelated?IsEq.(8)consistentwiththesimpleexpressionforintensity𝐼 𝑥, 𝑦 = 𝐼!"#𝑒!! !!!!! !!?
Question14
TheGaussianbeamequationsgiveninEqs.(8)-(11)assumethebeamcomestoitsnarrowestwidth(calledthebeamwaist)at𝑧 = 0.
a. Howwouldyourewritethesefourequationsassumingthebeamwaistoccursatadifferentposition𝑧 = 𝑧!?
b. OnewaytocheckyouransweristomakesuretheequationssimplifytoEqs.(8)-(11)inthespecialcaseof𝑧! = 0.
Question15
Writeafunctiontofitthefollowingdatasetavailableat:https://physicscourses.colorado.edu/phys4430/phys4430_fa19/UsefulDocs/Test_beam_width_data.csvAssumethewavelengthis𝜆 = 632.8nm.
i. Whatisthefunctionalformforyourfitfunction?ii. Whatarethedifferentfitparametersandwhatdotheymean?iii. Isitalinearornonlinearfitfunction?Why?
a. Youshouldgetthatabeamwaistof𝑤! = (93.9 ± 0.1)×10!!mandoccursataposition𝑧! = 0.3396 ± 0.0003m.
AUTOMATIONOFTHEMEASUREMENTANDANALYSIS
Inthenextstep,youwilluseLabVIEWandyourNIUSB-6009dataacquisitioncardtoautomatetheprocedurefor
measuringthewidthofthelaserbeam.Youcandothiswithyourownlaptoporwiththelaptopsinlab.Iffor
Question16 YoushouldhavealreadycompletedatleastthefirstLabVIEWlabskillactivityduringthelecturetime.Inordertosetupyourmeasurementautomationyouwillneedtodotwothings:
a. Doquestions1and2oftheLabVIEWLabSkillActivity2.TheactivitygoesoverconnectingyourNIUSB-6009DataAcquisitiondevicetoyourcomputer.Theactivityisavailableonthe“Activities”pageonthecoursewebsite.
b. CreateaLabVIEWVIthatperformstheautomateddatataking(movingthetranslationstage,recordingtheposition,recordingthevoltagefromthephotodiode,repeat).InstructionscanbefoundintheAppendix.
Question17 a. TestandruntheautomatedLabVIEWprogramandevaluatetheresultusingthesameMathematicaanalysisasbefore.
b. Beforeyougoon,makesuretheautomatedacquisitionandanalysisroutinegivesthesameresultasthemethodyouusedlastweek.
c. Howlongdoesyournewmeasurementmethodtake?(2-3minutesper𝑤measurementisverygood.)
Question18
a. Inweekone,howlongdidthetotalprocessofdatatakingthroughanalysistaketomakeameasurementofthebeamwidth𝑤?
b. Inthislab,youmayhavetotake20-30beamprofilesinordertomeasure𝑤!and𝑧!.Howlongwouldthistakewithyourcurrentmethod?
c. Whatarethemosttimeconsumingportionsoftheprocess?Whichpartsoftheprocesswouldbenefitfromautomation?
somereasonyoucannotgetthistowork,youcanstillcompletethelabinamuchlessefficientfashionbytakingdatamanually.
THEEXPERIMENT
TheGaussianbeammodeloflightisusefulbecauseitoftendescribesthebeamoflightcreatedbylasers.ThissectionwilltestthevalidityofthemodelforourHeNelaserbeam.Also,theeffectofalensonaGaussianbeamwillbetested,andtheGaussianbeammodelwillbecomparedwithpredictionsfromthesimplerraytheory.Lastly,theGaussianbeamtheorycanbeusedtodescribetheminimumpossiblefocussizeforabeamandalens.
Question19
MeasuringthebeamprofileofyourHeNelaserwithoutanylenses.Thereisastraight-forwardreasonthataHeNelasershouldproduceaGaussianbeam.Thelaserlightbuildsupbetweentwomirrors,andtheelectromagneticmodethatbestmatchestheshapeofthemirrorsistheGaussianbeam.
a. ConsideringEq.(8)-(11),whichaspectsoftheGaussianbeammodelcanyoutest?Arethereanypartsofthemodelyoucannottest?
b. Measurethebeamwidth𝑤versusdistancefromthelaser.Considercarefullywhatdistanceshouldbevarying.Isitthedistancefromlasertorazor,thedistancefromrazortophotodetector,orthedistancefromlasertophotodetector?Howdidyoudecidewhatpositions𝑧tomeasurethewidthat?(metersticksareavailable)
c. Fitthedatato𝑤(𝑧),thepredictedexpressionforaGaussianbeamgiveninEq.9.d. Whatisthevalueofthebeamwaist𝑤!(includinguncertainty)?Wheredoesthebeam
waist𝑧!occurrelativetothelaser?
Question20
HowdoesalenschangeaGaussianbeam?Pickanon-compoundlens(notthefancycameralenses)withfocallengthintherange100-200mm.Designandcarryoutanexperimenttoquantitativelyanswerthequestionsbelow.Considercarefullywheretoputthelens.Yourdataforthisquestioncanbeusedinthenextquestion.
a. DoesthebeamretainaGaussianprofileafterthelens?b. Whatisthenewbeamwaist𝑤!andwheredoesitoccur?c. Whatfactorsaffectthebeamprofileafterthelens?d. Doesthemeasured𝑤(𝑧)matchtheGaussianbeampredictiongiveninEq.(9)?
Question21 Quantitativelymodelingtheeffectofalens.Oneofthesimplestwaystomodeltheeffectofalensisthethinlensequation,whichisbasedonaraymodeloflight.(seeFigure4)
1𝑆!+1𝑆!=1𝑓
a. RedrawFigure4toshowhowitwouldchangewhenthelightismodeledasaGaussianbeam,ratherthanrays.Inparticular,whereshouldthebeamwaistsoccur?Whatdeterminestherelativewidthofthebeamwaist?
b. ExperimentallytesttheaccuracyofthethinlensequationfortheimagingofGaussianbeams.Yourdatafromthepreviousquestioncanprobablybeused.Istheagreementwithintheestimateduncertainties?
c. Systematicerrors:Underwhatconditionsshouldthethinlensequationbemostvalid?Howdotheseconditionscomparetoconditionsofyouractualmeasurements?Canyougetbetteragreement?
Figure4Diagramshowingthefocusingoflightbyathinlensintherayapproximation.Thediagramidentifiesthequantitiesinthethinlensequation:imagedistance,objectdistance,andfocallength.
PROJECTIDEAS
1. PredictingthebehaviorofcomplexopticalsystemsusingABCDmatricestotransformGaussianBeams.2. Buildanopticalcavity.Studythecouplingoflightintothecavity,andspatialfilteringintodifferentTEM
modes.Replicatetheawesomepictures.3. Analogybetweenparaxialwaveequationinfreespaceand2DSchrodingerwaveequation.Solvingthe
Schrodingerequationoptically.Addingapotential.Tunneling.Etc.4. Usingatranslatable,rotatableslittomapoutthebeamprofileofafunkypatternusingtheRadon
transform,whichisusedinreconstructingCTscans.Perhapsthereissomebetterapplicationoftomographyalso.
REFERENCES
1. http://people.seas.harvard.edu/~jones/ap216/lectures/ls_1/ls1_u3/ls1_unit_3.html(GaussianBeamtheory)
2. http://en.wikipedia.org/wiki/Gaussian_beam
APPENDIX:LABVIEWAUTOMATIONGUIDE
SETTINGUPTHEMOTOR
YouwilllikelyneedtodownloadadditionaldevicedriversforLabVIEW.Youshoulddownloadthe32-biton64-bitWindowsAPTsoftwarefromwww.thorlabs.com/software_pages/ViewSoftwarePage.cfm?Code=Motion_ControlinordertogettheMGMotorlibraryusedbelow.Onceyouhavedonethis,followthenextsteps:
1. StartwithLabVIEWclosedonyourcomputerandnocordsconnectedtothemotorcube.2. ConnecttheUSBtotheKST101controllercube,THENturnonthepower.3. OpenAPTConfigonthecomputer.Yourmotorshouldbeseenbythesoftware.Clickthemotordrop
downmenuandselectyourmotor.Now,inthedropdownmenuforStage,findZST225(B)andselectit.Withtheseoptionsinplace,selecttheAdd/ChangeStageAssociationbutton.
4. CloseAPTConfig
Thisshouldconnectthemotorcorrectly,buttoverify,openAPTUser.ThiswillopenaLabVIEWVIwithamotorcontrolonthefrontpanel.Thismotorcontrolshouldhaveyourserialnumberinthetoprightcornerandshouldalsohave“STAGE:ZST225(B)”.Clickingthearrowpointingupandarrowpointingdownshouldmovethemotorineitherdirection.IfnoneofthisisseenwhenyouopenAPTUser,themotorwasnotsetupcorrectlyandyoushoulddisconnecttheUSBandpowersupplyfromthemotorandstartagainstep1.
LABVIEWVIIMPLEMENTATION
TounderstandhowtoautomatethedatatakingprocessusingLabVIEW,considerwhatyoudowhiletakingdatabyhand.Theprocessincludes:movingtherazor(stage),readingtheposition,readingthephotodiodevoltage,repeat.Inordertoautomatethisprocess,twopiecesofequipmentareused:themotortomovethestageandtheNI-DAQtoreadthevoltage.
I. Thefirststepindatatakingismovingtherazor.Toautomatethis,weneedtousethemotorcube.ToconnectthecubetoLabVIEW,followthesesteps:A. Onthefrontpanel,placeanActiveX Container(Underthe“.NET&ActiveX”tab)B. RightclicktheActiveXContainer→ Insert ActiveX Object... → MGMotor ControlC. RightclickMG17Motorintheblockdiagram→ Create → Property for MG17MotorLib…
→ HWSerialNumD. RightclickHWSerialNum → Change to Write
1. WiretheMG17MotorreferenceouttothereferenceinofHWSerialNum.ThisallowsustotelltheLabVIEWtolookforthemotorcubewithyourserialnumber(i.e.yourcube)
E. AddaNumericalControltothefrontpanel1. RenametoHWSerialNum2. Intheblockdiagram,wireittotheinputofHWSerialNum.Thiscontrolallowsyoutoinputyour
motor’sserialnumbereasilyonthefrontpanelincaseyouchangecubes.F. RightclickMG17Motorinblockdiagram→ Create → Method for MG17MotorLib… →
StartCtrl1. WirethereferenceoutofHWSerialNumnodetoreferenceinofStartCtrl.2. Atthispoint,ourLabVIEWisbasicallysaying“Lookforamotorwiththisserialnumber(inputin
thefrontpanelnumericalcontrol)andstartcontrollingit.”Nowwehavetomakeitmove.G. RightclickMG17Motorinblockdiagram→ Create → Method for MG17MotorLib… →
SetJogStepSize1. Thisnodeallowsustotellthemotorhowmuchtomoveanditusesunitsofmillimeters.2. WirethereferenceoutofStartCtrltoreferenceinofSetJogStepSize.
H. AddaNumericalControlonfrontpanel1. RenameitSetJogStepSize(mm)2. Intheblockdiagram,wirethiscontroltofStepSizeinputofSetJogStepSize.3. ConnectreferenceoutofStartCtrltoreferenceinofSetJogStepSize.
I. AddaNumericalConstantof0intheblockdiagram1. WiretotheIChanIDinputofSetJogStepSize2. ThisconcludestheinitializationportionoftheLabVIEW.
J. RightclickMG17Motoriconinblockdiagram→ Create → Method for MG17MotorLib… → SetJogVelParams
1. WirethereferenceoutofHWSerialNumtoreferenceinofthisblock.K. AddthreeNumericalControlstothefrontpanel
1. RenameonetoMinVelocity(mm/s)andwireitintheblockdiagramtothefminVelinputofSetJogVelParam
2. RenameonetoMaxVelocity(mm/s)andwireitintheblockdiagramtothefmaxVelinputofSetJogVelParam
3. RenamethelastnumericalconstantAcceleration(mm/s/s)andwireitintheblockdiagramtothefaccninputofSetJogVelParam.
4. Thesecontrolhowthemotormoves.Theyshouldbesettoaminimumvelocityof0mm/s,amaximumvelocityof1mm/s,andanaccelerationof1mm/s/s.
II. Thenextstepinvolvestheactionthatwewantthemotortoexecute,aswellasthedatatakingprocessA. Inordertoseparatethe‘action’partoftheLabVIEWfromthe‘initialization’,firstaddaFlat
Sequence StructureintheblockdiagramundertheStructurestab1. DragtheFlat Sequence Structuretoincludeeverythingintheblockdiagramsofar(the
initializationpart)2. RightclicktheFlat Sequence StructureandselectAdd Frame After.Thesecond
frameinthissequenceloopwillcontainalltheactionofmovingthemotorandtakingthedata.B. Thedatatakingprocesshasonekeyfeaturethatshouldbeaddressedatthispoint,the“repeat”part
(Moverazor,takedata,repeat).InordertomakeourLabVIEWrepeatthisprocess:1. AddaWhile Loop(fromtheStructurestab)insidethesecondframeoftheSequence2. AddaStopbuttontothefrontpaneland,intheblockdiagram,wirethestopbuttontothered
stopcircleinthecorneroftheWhile Loop.ThisloopwillexecuteitscontentsrepeatedlywhiletheloophasaBoolean“true”value.OncetheLabVIEWisrunning,pressingthisstopbuttononthefrontpanelwillstoptheWhileLoopfromcontinuing.
C. WeneedtoputourotheractionsinsidethisWhile Loop.Wewanttheseactionstobeperformedinasequence,soinsidetheWhile Loop,addanotherFlat Sequence Structure(wewillrefertothisloopassequence#2)
1. Add3framestosequence#2foratotalof4framesD. Insidethefirstframeofsequence#2,wewanttherazortomoveoneJogStep.
1. RightclickMG17Motor → Create → Method for MG17MotorLib… → MoveJogandplaceinside1stframeofsequence#2
2. WirereferenceoutfromSetJogStepnodetoreferenceinofMoveJog3. AddaNumericalConstantof0totheblockdiagramandwireittotheIChanIDinput4. AddoneNumericalControltothefrontpanelandrenameitJogDirection5. WireittotheIJogDirinputandenteritsvalueaseither1or2(thevaluescorrespondtoextending
orretracting)E. Thenextstepinthedatatakingprocesstowaitsomeamountoftimetoensurewedonotreadthe
photodiodevoltagewhiletherazorismoving.1. AddtheWaitfunction(undertheTimingtab)tothe2ndframeofsequence#22. AddaNumericalControltothefrontpanel,renameitWait(ms)3. WirethiscontroltotheWaitfunction.Thisallowsyoutochangethewaittimefromthefront
panel.F. AtthispointweneedtointroducetheNI-DAQ.First,findaworkingDAQ.Plugitintothecomputervia
theUSB.WiretheBNCoutputfromthephotodiodeintothe+/-analoginputsoftheDAQlabeledAI01. Inthe3rdframeofsequence#2addMeasurement I/O → NI-DAQmx → DAQ Assist2. AwindowshouldpopandyoushouldclickAcquire Signal → Analog Input →
Voltage → Channelsai03. AnewwindowshouldpopupforadditionalsettingoptionscalledDAQAssistProperties.Under
TimingSettings,changetheAcquisitionModeto1Sample(OnDemand)4. ClickOKtoclosethewindowandsaveyoursettings.
G. Inthe4thframeofsequence#2selectFile I/O → Write to Measurement FileH. Awindowshouldopenforpropertiesofthisnode,ifitdoesnotthenrightclicktheWriteto
MeasurementFileiconandselect“Properties”1. UndertheFilenamesection,selectAskUsertoChooseandchecktheboxAskOnlyOnce.This
allowsyoutoenterthefilenamewhenLabVIEWisrun.2. UnderIfafilealreadyexistsselectAppendtofile.Thismakesitsoeachdatapointisaddedtothe
samefile,insteadofoverwritingordeletingolddatapoints3. Thefileformatshouldbeleftastextformatbecauseit’seasiesttouploadintoMathematica4. Undertheheaderoption,selectNoHeaders.Atthispointallthesettingsshouldbeaswewantso
closethiswindow
I. Nowweneedtofeedourdataintothisiconsoitknowswhattowritetofile.Thedatathatwewantshouldbeintheform(position,voltage)
1. Insequence#2frame4addExpress → Sig. Manip. → Merge SignalsJ. ThismergedsignalwilloutputtotheWritetoFilenodeandwewantthepositionvalueintheupper
inputofMergeSignalsandthevoltagevalueinlowerinput1. WiretheDataoutputfromtheDAQ-Assisticoninframe3tothelowerinputoftheMerge
Signalsoperator2. AddaNumericalIndicatoronthefrontpanelandrenameitVoltage(V)andwirethedataoutput
oftheDAQ-Assisttoitinthe3rdframeofsequence#2intheblockdiagram.Thisprovidesareal-timecheckofthephotodiodevoltagefromthefrontpanel
K. Wewanttoknowthepositionofthestage.RightclickMG17Motor → Create → Method for MG17MotorLib… → GetPositionandplacetheiconinthe3rdframeofsequence#2
1. WirethereferenceoutfromMoveJogtothereferenceinofGetPosition2. WireaNumericalConstanttotheIChanIDinputandenteritsvalueas03. Right-clickonpfPositionandselectCreate → Indicatortogiveusareal-timeindicatorof
theposition.4. Right-clickthenewindicatorandselectCreate → Local variable5. Right-clickonthenewlocalvariableandselectChange to readandwireittotheinputof
pfPosition6. WiretheoutputofpfPositiontotheupperinputoftheMerge Signalsoperator
L. Nowthedataisintheform(position,voltage)andthemergedsignalshouldbewiredintotheSignalsinputoftheWrite to Filenode.
Andwiththat,theLabVIEWiscomplete!
Atthispoint,takeastepbackandtakealookatthebigpictureoftheLabVIEWandhowitoperates.Therearemanydifferentapproachestoautomatingthisspecificprocessandthereisalwaysmorethanonewaytoapproachanyproblem.InthiscasewhatwedidwaswebrokeuptheLabVIEWinto2majorparts;theinitializationandtheaction.Toinitialize,wesaid“lookforamotorwiththisserialnumber,startcontrollingitandsetitsjogstepsizetobethissize.”Withthatdone,wemovetotheactionpart.TheWhileLoopsays“continuedoingthisuntilIpressthestopbutton”andwhatwearetellingittodois;movetherazer,waitforittostopmoving,readthephotodiodevoltageandcombineitwiththepositionmeasurement,andfinallywritethedatatoafile.OnceitexecutesthatprocesstheWhileLoopstartsitalloveragain.
SometipsonhowtoactuallyusetheLabVIEW.
1. Makesurethemovevelocityonthemotorcontrolcubeismaxedoutat1mm/s.Thisischangedonthemotorcubeitselfthroughthemenu.
2. TheJogStepSizecanbeverysmall(0.05mmforexample).3. Todetermineagoodamountoftimetowait,youcanstartwiththekinematicequation:x=½at2.Fora
stepsizeof0.05mm,thisindicatesatimeof316ms.Youmayalsoneedtoworryaboutthemaximumvelocityifyouhavealargestepsize.Ingeneral,youprobablywanttoaddanother200msforsafety.Thiswillensurethatvibrationsfromthemovementhavedampedout.
4. Checkwhereagoodstartingpositionisonthecontrolcubedisplay.Then,onceadatasetistakenyoucanquicklymovethemotortothestartpositionusingthecontrolwheelonthecube.Youdefinitelywanttoincludedatabefore,during,andaftertherazorstartstoblockthelight.
ThefinalLabVIEWrequiresyoutoinputtheserialnumber,stepsize,andwaittimeonthefrontpanel.Withthosevaluesinplace,runningtheLabVIEWandselectingthefilesavelocationareallthatneedstobedonetostarttheprocess.Tostop,pressthestopbuttonanditwillcutofftheWhileLoopandendtheprocess.
FormoredetailsonhowLabVIEWandThorlabsproductsinteract,youcanlookat:www.thorlabs.us/images/TabImages/GuidetoLabVIEWandAPT.pdf