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