gaussian laser beams - university of colorado · pdf filegaussian laser beams week 1 ... b....

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 EnteringdataintoMathematicaorimportingdata.o Non-linearleast-squaresfitting.o Plottingdataandfitfunctiontogether.o 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,andanalysisshouldbetransferredtothelabnotebookbyprintingandtapingthepagesorkeepingtheminathreeringbinder.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.

MODELINGCHARACTERISTICSOFTHEPHOTODETECTOR

ThegoalofthispartofthelabistounderstandalotaboutthespecificationsgivenonthedatasheetfortheThorlabsPDA36ASwitchableGainAmplifiedPhotodetectors.Itisimportanttorealizethatdatasheets(alsocalledspecsheetsorspecificationsheets)provideamodelfortherealisticbehaviorofthedevice.Thismodelcanbetestedandimproved,aprocessmorecommonlycalled"calibration."

Question4

Basicfunctionoftheamplifiedphotodetector

a. Spendafewminutes(nomorethan15)towriteanexplanationusingwordsanddiagramstoexplainthephysicalmechanismforhowthephotodetectorconvertslightintovoltage.Youmayusethemanufacturer’sspecificationssheet,trustworthyonlineresources,abook,etc.Thespecificationissheetavailableatwww.colorado.edu/physics/phys4430/phys4430_sp16/datasheets/Thorlabs_PDA36A.pdf

b. 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?Wereyourmeasures

sufficientforanaccuratecalibration?c. Calibratingthegain

i. IsitpossibletomeasuretheV/Againforeachsetting,orcanyouonlymeasurethechangeingainasyouswitchthesettings?Why?Notethatthislabonlyrequiresrelativegain.

ii. Makeameasurementofthegainorrelativegainsettingsformostofthegainsettings.Ifyouneedtoadjustthelaserpower,tryblockingpartofthebeam.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.

Figure1Mountingassembliesforamirror(left)andalens(right).

Question7

SupposealaserbeamhasaGaussianintensityprofile𝐼 𝑥, 𝑦 = 𝐼*+,𝑒./ ,0120 30,andisincidentuponaphotodiode.Whatistheexpressionforthepowerhittingthephotodiodewhenaportionofthebeamisblockedbyarazorblade(seeFigure2:Razorblademountedonatranslationstage)?

a. Drawadiagramshowingthebeamandtherazor.b. Usingtheaboveexpressionfor𝐼(𝑥, 𝑦),writethemathematicalexpressionforthe

powerincidentonthephotodiodeasafunctionofrazorposition.Note,toaddressthisquestion,youwillneedtobecomefamiliarwiththeErrorFunction,erf(x).

Question8

Beforeyoutakedata:Createananalysisfunctiontofitatestsetofdata.

Note:NonlinearleastsquaresfittingiscoveredinMathematicaActivity2availableonthecoursewebsite.ThereisalsoaYoutubevideoavailableonleastsquaresfittingatwww.youtube.com/compphysatcu.

a. Whatisthefunctionalformforyourfitfunction?b. Isitalinearornonlinearfitfunction?Why?c. Whatarethefitparameters?Whydoyouneedthismany?d. Howdothefitparametersrelatetothebeamwidth?e. Downloadthedatasetfrom:

www.colorado.edu/physics/phys4430/phys4430_sp16/sample_data/Test_Profile_Data.csv.i. Makeaplotofthedata.ii. Makeafitandplotitwiththedata.iii. Checkthatthefitlooksgoodandyougetabeamwidthof𝑤 = 4.52×10.>m

Figure2:Razorblademountedonatranslationstage

Question9

Buildyoursetupformeasuringthebeamwidthofyourlaser.a. Drawadetailedschematicofthesetup(fromthelaserallthewaytothephotodetector,

includingthelensanditsfocalpoint).b. Afterassemblingyourexperiment,butpriortotakingalotofdata,howcanyouquickly

determineifthemeasurementisworking?c. Isitpreferabletouseadigitalmultimeteroroscilloscope?Why?d. Usethemeasurementschemetotakedataofpowervspositionoftherazor.Picka

positionwhereyourbeamhasameasurablewidth,andmeasureit.Justifyyourchoice.

Question10

Analysisoftherandomuncertaintysourcesa. Whatarepossiblesourcesofrandomuncertaintyinthephotodetectorvoltage?b. Howwouldyouestimatetheuncertaintyinthephotodetectorvoltagemeasurement?c. Whatisthelargestsourceofuncertainty?Why?

Question11

Analysisoftherealdata.a. UsetheanalysisproceduresverifiedinQuestion7tofindthebeamwidthsforeachdata

set.b. Plotyourfittogetherwithyourdatatomakesureitisgood.

WEEK2:DEVELOPINGAQUANTITATIVEMODELOFTHESPATIALPROPERTIESOFLIGHT

GOALS

Expandtwomodelsofthemostfrequentlyusedcomponentsintheopticsexperiments.Inweek1,wemeasuredtheprofileofthelaserandfoundittobeGaussiantoagoodapproximation.However,wedon'thaveanymodelforhowtheprofilechangesasthebeampropagates.

Also,wewillapplymeasurementandautomationtomorerapidlytakedata.Inparticular,youwillautomatetwothings:thedataacquisition,andthefittingandanalysisroutine.Thefullsetoflearninggoalsincludes:

1. Automateddataacquisition.a. LabVIEWb. USBDAQ(NIUSB-6009)

2. AutomatedfittingandanalysisofdatainMathematica3. UsingapredictivemodelofGaussianlaserbeams

a. ContrastGaussianbeamswithgeometricoptics4. MeasureprofilesofaGaussianbeam,andextracttheGaussianbeamparameters(typicallybeamwaist

radiusandposition).5. EffectofalensonGaussianbeams.

a. IsitstillGaussian?b. DoesthethinlensequationapplytoGaussianbeams?c. Whatlimitstheminimumachievablespotsize?

PRELAB:INTRODUCTION

Question12

Answerthesebeforereadingaheadinthelabguidebasedonyourexperiencefromlastweek'slab.

a. DoesthebeamalwaysstayaGaussianasitpropagates?b. DoesthebeamstayGaussianafteritgoesthroughalens?c. DoesthebeamstayGaussianafteritreflectsfromamirror?d. Howsmalldoesthebeamgetwhenitisfocusedbyalens?Doesitfocustoapoint?

Whyorwhynot?

Lightisapropagatingoscillationoftheelectromagneticfield.ThegeneralprincipleswhichgovernelectromagneticwavesareMaxwell'sequations.Fromthesegeneralrelations,avectorwaveequationcanbederived.

∇/𝑬 = 𝜇D𝜖D𝜕/𝑬𝜕𝑡/

(1)

Oneofthesimplestsolutionsisthatofaplanewavepropagatinginthe𝒛direction.

𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝐸,𝒙 cos 𝑘𝑧 − 𝜔𝑡 + 𝜙, + 𝐸2𝒚cos 𝑘𝑧 − 𝜔𝑡 + 𝜙2 (2)

Butasthemeasurementsfromlastweekshowed,thelaserbeamsarecommonlywellapproximatedbyabeamshapewithaGaussianintensityprofile.Apparently,sincetheseGaussianprofilebeamsexist,theymustbesolutionsofthewaveequation.ThenextsectionwilldiscusshowwederivetheGaussianbeamelectricfield,andgiveafewkeyresults.

PARAXIALWAVEEQUATION

Oneimportantthingtonoteaboutthebeamoutputfrommostlasersisthatthewidthofthebeamchangesveryslowlycomparedtothewavelengthoflight.Assumeacomplexsolution,wherethebeamispropagatinginthe𝒛-direction,withtheelectricfieldpolarizationinthe𝒙-direction.

𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝒙𝐴 𝑥, 𝑦, 𝑧 𝑒" VW.XY (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" VW.XY = 𝒙𝜇D𝜖D𝐴 −𝜔/ 𝑒" VW.XY (4)

Thiscanbesimplifiedrecognizingthat𝑘/ = 𝜔/ 𝑐/ = 𝜇D𝜖D𝜔/,wherethespeedoflightisrelatedtothepermeabilityandpermittivityoffreespaceby𝑐 = 𝜇D𝜖D .\ /.Also,the𝒙𝑒" VW.XY termiscommontobothsidesandcanbedropped,whichresultsin

𝜕/𝐴𝜕𝑥/

+𝜕/𝐴𝜕𝑦/

+𝜕/𝐴𝜕𝑧/

+ 2𝑖𝑘𝜕𝐴𝜕𝑧

= 0 (5)

Sofarwehavemadenoapproximationtothesolutionorthewaveequation,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 = 𝑬D3`3(W)

exp − ,0120

30 Wexp 𝑖𝑘 ,0120

/d W𝑒."e W 𝑒" VW.XY (8)

Where𝑬Disatime-independentvector(orthogonaltopropagationdirection𝒛)whosemagnitudedenotestheamplitudeofthelaser'selectricfieldandthedirectiondenotesthedirectionofpolarization.Thebeamradius𝑤(𝑧)isgivenby

𝑤 𝑧 = 𝑤D 1 + fWg3`0

/ (9)

𝑅(𝑧),theradiusofcurvatureofthewavefront,isgivenby

𝑅 𝑧 = 𝑧 1 + g3`0

fW

/ (10)

AndtheGuoyphaseisgivenby

𝜁 𝑧 = arctan g3`0

Wf (11)

Theremarkablethingaboutalltheseequationsisthatonlytwoparametersneedtobespecifiedtogivethewholebeamprofile:thewavelength𝜆andthebeamwaist𝑤D,whichisthenarrowestpointinthebeamprofile.ThereisamoregeneralsetofHermiteGaussianmodeswhichareshowninFigure3.Thelasercavitytypicallyproducesthe(0,0)modeshownintheupperleftcorner,butanopticalcavitycanalsobeusedtocreatetheseothermodesshapes–atopicthatcanbeexploredinthefinalprojects.

Figure3IntensitydistributionsforthelowestorderGauss-Hermitesolutionstotheparaxialwaveequation.Theaxesareinunitsofthebeamwidth,w.

MOREPRELAB:TRYINGOUTTHEGAUSSIANBEAMMODEL

Question13

Inweek1ofthelab,weassumedtheintensityprofileoftheGaussianbeamwasgivenby𝐼 𝑥, 𝑦 =𝐼*+,𝑒./ ,0120 30.TheequationfortheelectricfieldoftheGaussianBeaminEq.(8)lookssubstantiallymorecomplicated.Howaretheexpressionsforelectricfieldandintensityrelated?IsEq.(8)consistentwiththesimpleexpressionforintensity𝐼 𝑥, 𝑦 = 𝐼*+,𝑒./ ,0120 30?

Question14

TheGaussianbeamequationsgiveninEqs.(8)-(11)assumethebeamcomestoitsnarrowestwidth(calledthebeamwaist)at𝑧 = 0.

a. Howwouldyourewritethesefourequationsassumingthebeamwaistoccursatadifferentposition𝑧 = 𝑧3?

b. OnewaytocheckyouransweristomakesuretheequationssimplifytoEqs.(8)-(11)inthespecialcaseof𝑧 = 0.

Question15

a. Writeafunctiontofitthefollowingdatasetavailableat:www.colorado.edu/physics/phys4430/phys4430_sp16/sample_data/Test_beam_width_data.csv.Assumethewavelengthis𝜆 = 632.8nm.

i. Whatisthefunctionalformforyourfitfunction?ii. Whatarethedifferentfitparametersandwhatdotheymean?iii. Isitalinearornonlinearfitfunction?Why?

b. Youshouldgetthatabeamwaistof𝑤D = (93.9 ± 0.1)×10.smandoccursataposition𝑧3 = 0.3396 ± 0.0003m.

AUTOMATIONOFTHEMEASUREMENTANDANALYSIS

Inthislab,youwilluseLabVIEWandyourNIUSB-6009dataacquisitioncard.

Question16

a. Inweekonehowlongdidthetotalprocessofdatatakingthroughanalysistaketomakeameasurementofthebeamwidth𝑤?

b. Inthislabyoumayhavetotake20-30beamprofilesinordertomeasure𝑤Dand𝑧3.Howlongwouldthistakewithyourcurrentmethod?

c. Whatarethemosttimeconsumingportionsoftheprocess?Whichpartsoftheprocesswouldbenefitfromautomation?

Question17 YoushouldhavealreadycompletedthefirstLabVIEWlabskillactivityduringthelecturetime.Inordertodosetupyourmeasurementautomationyouwillneedtodotwothings:

a. Doquestions1and2oftheLabVIEWLabSkillActivity2.TheactivitygoesoverconnectingyourNIUSB-6009DataAcquisitiondevicetoyourcomputer.Theactivityisavailableonthe“Activities”pageonthecoursewebsite.

b. DownloadtheLabVIEWVIforacquiringdatafromthe“Hints”pageonthecoursewebsite.

Question18 a. ImplementtheautomationinLabVIEWandMathematicausingthebasicLabVIEWdataacquisitionVIprovidedtotheclass(seehintsbelow).

b. Beforeyougoon,makesuretheautomatedacquisitionandanalysisroutinegivesthesameresultasthemethodyouusedlastweek.

c. Howlongdoesyournewmeasurementmethodtake?(2-3minutesper𝑤measurementisverygood.)

HINTSONAUTOMATION

HerearesomehintsonhowtosetuptheThorlabdriverstoautomatemovingthetranslationstage.

1.Eachmotorhasadriverthatconnects(via)USBtoacomputerandhasapowercord.Theorderyouconnectthesematters(Ithinkit'sUSBtocomputerbeforeplugginginthepowerbuttrytheotherwayifneeded).

2.OpentheprogramAPTConfig.Fromthestagemenuselecttheserialnumberoftheactuator(w/otheletter'B'attheend).Thenselectthemotorserialnumberfromthedriverbox.Clickadd/changestageaccessories.

3.OpentheprogramAPTUser.Selectthestageserialnumberwhichshouldappear.Testthemotor.Ifitdoesn'tshowupordoesn'twork,gobacktostep1andtryagain.

4.OpenlabVIEW(restartitifopen).Followdirectionsat:www.thorlabs.us/images/TabImages/GuidetoLabVIEWandAPT.pdf

Eachmoduleshouldonlydoonething(e.g.setserialnumber-startcontroller-setjogsize-movejog-readposition-stopcontroller.

THEEXPERIMENT

TheGaussianbeammodeloflightisusefulbecauseitoftendescribesthebeamoflightcreatedbylasers.ThissectionwilltestthevalidityofthemodelforourHeNelaserbeam.Also,theeffectofalensonaGaussianbeamwillbetested,andtheGaussianbeammodelwillbecomparedwithpredictionsfromthesimplerraytheory.Lastly,theGaussianbeamtheorycanbeusedtodescribetheminimumpossiblefocussizeforabeamandalens.

Question19

MeasuringthebeamprofileofyourHeNelaser(removethelensfromyoursystem).Thereisastraight-forwardreasonthataHeNelasershouldproduceaGaussianbeam.Thelaserlightbuildsupbetweentwomirrors,andtheelectromagneticmodethatbestmatchestheshapeofthemirrorsistheGaussianbeam.

a. ConsideringEq.(8)-(11),whichaspectsoftheGaussianbeammodelcanyoutest?Arethereanypartsofthemodelyoucannottest?

b. Measurethebeamwidth𝑤versusdistancefromthelaser.Considercarefullywhatdistanceshouldbevarying.Isitthedistancefromlasertorazor,thedistancefromrazortophotodetector,orthedistancefromlasertophotodetector?Howdidyoudecidewhatpositions𝑧tomeasurethewidthat?(metersticksareavailable)

c. Fitthedatato𝑤(𝑧),thepredictedexpressionforaGaussianbeamgiveninEq.8.d. Whatisthevalueofthebeamwaist𝑤D?Wheredoesthebeamwaist𝑧3occurrelative

tothelaser?

Question20

HowdoesalenschangeaGaussianbeam?Pickanon-compoundlens(notthefancycameralenses)withfocallengthintherange100-200mm.Designandcarryoutanexperimenttoquantitativelyanswerthefollowing.Yourdataforthisquestioncanbeusedinthenextquestion.

a. DoesthebeamretainaGaussianprofileafterthelens?b. Whatisthenewbeamwaist𝑤Dandwheredoesitoccur?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