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