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Introduction: ThetechnologyofSpatialLightModulators(SLM)grewfromtheneedtoquicklyconvertdatainelectronicformintospatiallymodulatedcoherentopticalsignals.Thisallowsfortheabilitytointroduceinformationintoanopticalsystem,sincetheinformationcanbecarrieddirectlybytheopticalamplitude.Toachievethis,oneneedstomanipulatethecomplexopticalfieldstransmittedthroughtheopticalsystem.Originallyphotographicfilmswereusedforwavefrontmodulationbutmuchmorepowerfulopticalinformationprocessingsystemscanberealizedifthefilmisreplacedbyaspatiallightmodulatorcapableofchangingtransitedlightinrealtimeinresponsetoopticalorelectroniccontrolsignals(Goodman). Overthehistoryofopticalinformationprocessing,agreatmanydifferentSLMtechnologieshaveemergedalongwiththeproliferationofpracticalapplication.SuchSLMstechnologiesinclude:LiquidCrystalSLMs,magneto‐opticSLMs,deformablemirrorSLMs,andmultiple‐quantum‐well(MQW)SLMs.LiquidCrystalSLMsarethemostpresentineverydaytechnologyandwillbethetypeyouwillhavetheopportunitytoworkwith.(Goodman). InthislabyouwillbeinvestigatingthepropertiesandcapabilitiesofatwistednematicliquidcrystalSLM.TheSLMiscomposedofcarefullychosendimensionsofspatiallyseparatedliquidcrystalcells.LiquidCrystalsareconsideredaphaseofmatterinwhichthemoleculeorderisbetweenthecrystallinesolidstateandtheliquidstate.EachLCcelliscomposedoflong,cigar‐shapedmoleculessandwichedbetweentwoalignmentlayers,whichsettheangleofthemolecule’slongaxisalignmentattheinterface.Thetwolayers,however,donotsharethesameanglesothemoleculesforma“helix”structureastheytraversethecell,thusgivingrisetoatwistedappearance–Seefigure1(muchlikeputtingadeckofcardsbetweenyourtwohandsandfanningitaround–Hechtpg.372).Nematicliquidcrystalsareoneswherethemoleculestendtobeparallelbuttheirpositionisrandomlydistributedacrossthecell.AwonderfulsetofpicturesthatdoamuchgreaterjusticeatconveyingwhataTN‐LCDiscanbefoundinHechtonpages370‐373.

Thehelixstructureofthetwistednematiccrystalcanbeusedtochangethepolarizationstatusofincidentlight.Whenthepolarizationofthelightisparalleltothemoleculesofthecellattheentrancefacet,thepolarizationfollowsthetwistofthemoleculeaxis.Youcanthinkofthecellasbeingaseriesofthinwave‐plateseachwithaminutegradationoftheiropticalaxis.Thereforethelightleavesthecellwithapolarizationthatisperpendiculartotheincidentpolarization.Inordertorealizeadynamicopticalelement,avoltageisappliedtotheLCcell.Thisvoltagecauseschangesofthe

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Figure1:Twisted‐nematicliquidcrystalcell

molecularorientation,asisillustratedinfigure2forthreevoltagesVA,VB,Vc.Additionallytothetwistcausedbythealignmentlayers,themoleculesexperienceavoltage‐dependenttiltifthevoltageishigherthanacertainthresholdvoltage.Withincreasingvoltage,onlysomemoleculesclosetothecellsurfacearestillinfluencedbythealignmentlayers,butthemajorityofmoleculesinthecenterofthecellwillbetalignedparalleltotheelectricfielddirection.

Figure2.

Theliquidcrystalisbirefringentandthuscanbedefinedbytwoindicesofrefractionsimilartoawaveplate.TheuniquefeatureabouttheliquidcrystaldisplayintheSLMistheextraordinaryrefractiveindexisdependantontheanglethemoleculesmakewiththenormaloftheentranceandexitlayer,typicallydefinedasthez‐axisordirectionofpropagation.Theexplicitformulaisgivenby

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sin2(θ)neo2 Eq.1

Theangleθisafunctionoftheappliedvoltage,sothevalueofneoisalsodependantonvoltage.Asθ→0wecanseethevalueofneo→noandtheliquidcrystalbecomesisotropici.e.theopticalpropertiesofthematerialarethesameinalldirections.Hence,lightpassingthroughwillonlyexperienceonerefractiveindex.ThisideaisofimportancetounderstandinghowthepolarizationoflightischangedbytheSLM.

Thefollowingisashortlistofaccessibletexts,withgermanechaptersindicated,whichyoushouldreferencetoyourassimilationofsomeofthemoredifficultconcepts.FurtherReadings:DiffractionandFourierOptics:

1) HechtCh7.3‐7.4.4,Ch10‐11,Ch13.2‐13.2.32) FowlesCh53) IntroductiontoFourierOpticsCh2‐44) Pedrotti’sInro.ToOpticsCh11,13,21

SpatialLightModulators:1) Intro.ToFourierOpticsCh7.22) HechtCh8.123) Pedrotti’sIntro.ToOpticsCh17‐54) OptiXplorerManual

*Additionallyitwouldbebeneficialtoreviewpolarization,interferenceandsomegeometricalopticssuchasmicroscopes.Section1:AmplitudemodulationandprojectionObjectives:TounderstandthefollowingexperimentsandthefunctionalityoftheLCdisplay,thepolarizationofthelightplaysacrucialrole.ThereforethepolarizationcharacteristicsoftheLCdisplaywillbedeterminedfirst.Thenaprojectorsetupwillbeassembledusingthelightmodulatorastheimagesource.Duringtheexperimentcontrastandpixelsizewillbedetermined.

MeasurementofthepolarizationpropertiesoftheSLM.

Inthisexperiment,thepolarizationpropertiesofthelightmodulatorwillbedetermined.PlacetheunaddressedLCdisplaybetweentwopolarizers,oralternativelyalinearlypolarizedlightsourceandapolarizer.Thetwopolarizers

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eachhaveapolarizationaxis,whichare[notindicatedonthepolarizer.Usingapolarizerwithaknownfixedaxisandthepower‐meter,determinethepolarizationaxisofthetwovariablepolarizers.Markthepolarizationanglewithasmallpieceoftape.]indicatedbythesmallmarkontheoutsideedgeoftheinnermount.Alignthismarkwiththe0degreemark.NextensurethatthebeamincidentontotheLCdisplayisbothcollimatedandcompletelyilluminatingthedisplay.Thisisdoneusing,whatiscommonlyreferredtoas,acagesystem,whichconsistsoftwolensesoffocallengthsf=30mmandf=200mmattachedtoarailguide.Adjustthelensseparationinorderexpandandcollimatethebeam.Usingthesecondpolarizerasananalyzer,measuretheangulardistributionoftheintensitybysettingthepolarizeratzerodegreesandvaryingtheanalyzerangle;10‐degreeincrementswillsuffice.Recordthedataandgraphnormalizedintensityvs.analyzerangle.WithoutremovingtheSLM,commentontheintensitycurveproducediftheSLMwereremovedfromthesetup.

Q1)IsthelightleavingtheSLMlinear,ellipticalorcircularpolarized?Howcanyoutell?

ProjectorSetup

UsingaLED/Whitelightsource,placeanf=75mmlensinsuchawaythatthebeamiscollimatedandlargeenoughtoilluminatetheentireLCdisplay.Makesurethatnoscatteredlightbypassesthedisplayusingarectangularaperture,whichcanbeconstructedfromapieceofcardboard.Placethepolarizer,analyzerandSLMbehindthecollimatinglenssimilartofigure1.TheSLMispositionedasclosetothecollimatinglensaspossible,sothattheapertureofthedisplayisfullyilluminated.Placeanobjectivelens(f=100mm)behindthedisplayandadjustitinsuchawaythatafocusedandenlargedimageappearsonthescreen.Tryusinganirisdiaphragmwithdifferentapertures.Q2)Whatistheinfluenceoftheapertureregardingthedepthoffocus,brightness,andinfluenceoflensaberrations.

AlllensesusedarePlano‐convex/concave.Indicatewhichsideshouldface

thelightsourceinordertoreduceaberrationandthenimplementyourdecisioninordertooptimizeyourprojector.

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Figure3:Anexampleforaprojectorsetupwiththeopticalpathforillumination(black)and

theopticalpathforimaging(red)

OpentheOptiXplorersoftwarebygoingtoStart→AllPrograms→HoloeyeApp.Software2.8fortheOptiXplorer(x2).ThewindowthatwillopenshouldsayOptiXplorer2.8intheupper,leftcorner.Ifthisisnotthecaseortheprogramwillnotopen,re‐readtheinstructions,tryagainandifthereisstillaproblem,askforassistance.

UsetheOptiXplorersoftwaretoaddresstheSLMwithawhitescreenbyclickingonElementaryOpticalFunctions→ShowBlankScreen.Ahomogeneousgraylevelscreenshouldappear.Enlargetheblankscreensothatitfillstheentirecomputerdisplay.Positionthemousepointerontherightedgeofthewindowuntilatoolbarappears.Familiarizeyourselfwiththenameofeachbuttoninthetoolbaranditscorrespondingfunction;playaroundwiththesoftwareforafewminutes.

Onceyouarecomfortablewiththenamesandgeneralfunctionsofthe

toolbaradjustthegrayvalueoftheblankscreenuntilitisentirelywhite.Pressthe“Inverting”buttontotogglebetweenwhiteandblack.Next,rotatethepolarizers,whichareperpendiculartoeachother,inadequatestepsizes—10degreesworksfine—andmeasurethechangingcontrastquantitatively.Remembercontrastisgivenby,

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Imax − IminImax + Imin

Eq.2

whereImaxandImincorrespondtothewhiteandblackaddressedscreenrespectively.Findandrecordwhatpolarizer/analyzercombinationachievedthemaximumcontrast.Againbyclicking“ElementaryOpticalFunctions”scrolldownto“CircularAperture”andaddressacircleofreasonableradiusontotheSLM.Notewhathappenswhenoneorbothpolarizersareremoved.Keepthefirstpolarizersfixedattheoptimizedpositionandrotateonlythesecondone.Q3)Whatdoyouobserveintermsofcontrastwhenyourotatethesecondpolarizerby45/90/180degrees?

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PixelsizeoftheLCdisplay AddresstheLCdisplaywitharectangularobjectofknowndimensions.Usingtheprojectorsetupfromthepreviousexperiment,findthepixelsizeoftheLCdisplay;includeerror.Relationbetweenpixelvoltageandmodificationofthepolarizationstate

EverygraylevelcorrespondstoaspecificvoltageasingleLCDelementisaddressedwith.Thedifferentvoltageleadstoadifferenttiltintheliquidcrystalmoleculesandthereforeadifferentpolarizationstate.Determineforsixgraylevels(250,200,…0)therotationangleoftheanalyzerforthesmallestandlargestmeasuredpowervalues.

Thestateofpolarizationcanalwaysbedefinedasanellipse.Inparametricform,thesemi‐majoraxisacorrespondstomaximumpower,andaccordinglythesemi‐minoraxisbtotheminimumpower(measuredwiththeanalyzerrotated90degreestothemaximum).Theangleoftheanalyzerforthemaximumpowercorrespondstotheangle∆,whichdenotestherotationofthesemi‐majoraxistothex‐axis.

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Figure4:Rotatedpolarizationellipsesfordifferentgreylevels(GL’s)

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Section2:LinearandseparablebinarybeamslittergratingsObjectives:

Illuminatingaspatiallightmodulatorwithacoherentlightsourcegeneratesdiffractionpatternsbehindthedisplaysimilartothosethatappearbehindaconventionalopticalgrating.Onecanconsiderthenon‐addresseddisplayasanopticalgrating.Thereasonforthisliesinthestructureofasinglepixel.Itconsistsofthetransparentpartoftheliquidcrystalcellandthenon‐transparentpartofthecontrolelectronics.Assumingzerotransmissionforthispartofthecell,thedisplaycanbeseenasatwo‐dimensionalseparablegratingwithastructureasshowninFig.5.Withaso‐called“FourierLens”thediffractionpatternallowsconclusionstobedrawnaboutthecharacteristicsofthedisplay.

Figure5:Simplifiedmodelofapixelwithitstransmission

Generationofdiffractionpattern

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wavesfromacertainpartoftheilluminatedareaonthemodulatorcancontribute.Placingthemodulatordirectlyinfrontofthelensminimizesthiseffect.Ifthescreenisnotpositionedintherearfocalplaneofthelens,thesizeofthediffractionpatterndependsonthepositionofthelightmodulator.TryplacingthelensbehindtheSLM.ObservewhathappenswhentheSLM’spositionisvariedwithinthelens’srearfocallength.YoushouldnotethelensstillactsaFourierTransform,whichgeneratesafar‐fielddiffractionpatternatthelens’sfocalplane!

Figure6:AFourierlensgeneratesthefarfielddiffractionpatterninitsrearfocalplane

Adjustthesetupsoyoucanclearlyseethearrayofbrightspots.Youmayneedtoplaceadiverginglensbehindthesetupsothediffractionpatternisenlargedforbettermeasurements.Recallthepixelarraycanbethoughtofasatwo‐dimensionalgratingabletobeseparatedintotwoperpendicularN‐slitgratings.WiththisandFig.5inmind,determinetheslitwidthandseparationforaN‐slitgratinginthehorizontaldirection.Repeatthisprocedurealongtheverticaldirection.Q4)Fromyoumeasurements,whatisthepixelsizeofthedisplay?Compareyouanswerwithpreviousmeasurement.Also,whatisthefillfactor,orwhatpercentofeachpixelisthenontransparentpartoftheliquidcrystalcell?Includeerror.Section3:DiffractiveOpticsObjectives: DiffractiveOpticalElements(DOE’s)differfromtheclassicalopticalelementslikelensesandmirrorsbecausetheyarebasedontheirdiffractivepropertiesratherthanreflectionandrefraction.Whendealingwiththetraditionalopticalelements,diffractionphenomenaareconsideredasundesirablefeatures,whichinfluencestheperformanceofanopticalsystemandhenceshouldbeminimized.DOE’s,ontheotherhand,makeuseofdiffractiontomanipulatethewaveformofanincomingbeamoflight.BecauseofthenatureofdiffractionmostlyhighlymonochromaticandcoherentlightisusedwithDOE’s.Thefirstapplicationofwavefrontmanipulationwasholography.Holographyinspiredpeopletowavefrontprocessinginwhichthe

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Pg.9

surfaceofasubstratewasprocessedtochangeaninputwavefrontintoanotherform.Inprinciplelensesandotherclassicalopticalelementsareallwavefrontprocessers,buttheirfunctionalityislimitedtorelativesimpleactions.DOE’sallowformorecomplexwavefrontmanipulationsandresultedinmodernopticalapplicationsas‘holographicheadupdisplays’infighterjetsandrecentlyinautomobiles.

FresnelZoneLens(FZL)

TheopticalsetupforthisexperimentisilluminatingtheLCDwithaexpandedandcollimatedlaserbeam.OpenthesoftwareOptiXplorerandgotoElementaryOpticalFunctions→FresnelZoneLenses.Settheinnerradiusto35pixels.Measurethedistancebetweenthelensplaneandthenthfocus;thisisthefocaldistancef.Fromyourmeasurementsandtheformulafortheinnermostradiusofabinaryzonelens,

€

r = n ⋅ λ ⋅ f Eq.4

determineonceagainthepixelsizeoftheLCdisplay.Comparewithyourpreviousresults.Q5)Wheredoesequation3comefrom?Justabriefdescriptionisneeded.[Hint:Pedrotti136]

Focallengthofthediffractivelens

UsingtheOptiXplorersoftwareaddressablankscreenontheSLM.Withthetoolbarattherightwindowedgealensphasewillbeadded.FindthefocuscreatedbythisDOEandthecorrespondingfocallength.Forewarning,thefocallengthcangrowtoadistanceof3metersataphaseof25,soareasonablephasewouldbegreaterthan100.

CreatingandoptimizingaDOE

Addaf=250mmlensbehindtheSLMfollowedbyaf=30mmlens.Downloadanimage–lessthan200x200pixels–fromtheInternetoruploadonefromyourcamera/phoneintoafolder.IntheOptiXplorersoftwaregotoFile→OpenImageFileanduploadyourimage.Next,clickthe“ComputeDOE”buttonintheupperrightcorneroftheimage.Adjustyoursetupuntilyouseeyourimageasadiffractionpatternonthescreen. Usually,whenreconstructingaDOEthereisadisturbingbrightpointinthemiddle,thezeroorder.UsingadivergingrefractivelensandadiffractiveconverginglensaddedtotheDOE,onecouldspatiallyseparatethereconstructionplanesofthezeroandfirstorder.Beginbyremovingthef=250mmlensandbyreplacingitwith

Pg.10

af=150mmlensinfrontoftheSLM.CreateaDOEfromtheexamplesincludedwiththeOptiXplorersoftwarea.k.a.“grid.bmp”or“cords.bmp”.Fordifferentlensphases,thefocusplaneandtheFourierplaneisfocusedonthescreenbymovingthediverginglens.Forseveraldifferentlensphases,findthedifferencebetweenthefocusofthediffractedlightandthefocusoftheundiffractedlight.

Figure7.Anaddedlensphasecausesafocusingofthediffractedlight(red).Theundiffracted

lightisfocusedbehindthescreen(black).

ByaddingaprismphasetotheaddressedDOEisalsopossibletoseparatetheorders.Thediffractionpatternwillbeshiftedinthexandydirections(seefigure6).RemovethepositivelensinfrontoftheSLMandreplacethedivergingonewithaf=100mmlens.Adjustyousetupinsuchawayastoclearlyseetheimageontheprojectorscreen.Fordifferentprismphasevaluesrecordtheshiftofthediffractionpattern.

Figure8:Top:focusedzeroorder;Left:zeroorderdefocused;Right:diffractionpatternshifted

Section4:InterferometricmeasurementofthephasemodulationObjectives: Thephasemodulationthatcanbeachievedwithaspatiallightmodulatorforacoherentlightsourcecanbemeasuredwithatwo‐beaminterferencesetup.Two

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Pg.11

coherentandcollimatedlaserbeamscreatedbyadouble‐holemaskilluminatethedisplay.BothbeamsareseparatelyguidedtoanappropriatehalfoftheLCD.Theleftonewillbeaddressedwithaconstantgreylevelwhereastheotherhalfwillbeaddressedwithgreylevelsvaryingfrom0to255.AlensbehindthedisplayletsbothbeamsinterferewithoneanotherandamicroscopeobjectiveimagestheexpandedinterferencepatternontoaCCDcamera.Aphaseshiftasafunctionoftheaddressedgreylevelwillappearasashiftintheinterferencepatternperpendiculartotheopticalaxis.

Fig.9:Phasepatternsusedtodeterminethephasemodulation.

PhaseMeasurement

Theopticalsetupisshowninfigure7.Thelasermoduleemitsacollimatedbeam.Constructadouble‐holemaskwithholediameter~3mmandapertureseparation~7mm.Placethismaskintoyouropticalsystemsotwobeamsarecreatedandeachpassesthroughonehalfofthedisplay.AlinearpolarizerinfrontoftheSLMsetstheincomingpolarizationstate.ThelaseremitsellipticalpolarizedlightsotheintensityisdependentonboththeSLMandthefirstpolarizer’sstate.Toremovethisambiguitywemustsettheintensityofthelightleavingthefirstpolarizertobeindependentoftheangle.Q6)Howcanyoudothis?[Hint:Useapolarizerandawaveplate]

Usinganf=250mmlensandanobjectivelens(20x/0.4)imagetheinterferencepatternontotheCCDcamera.OpenthesoftwarePhaseCamandviewtheinterferencepattern.

Figure10:Twobeaminterferometertodetectthephaseshift

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ThePhaseCamsoftwareoffersanautomatedmeasurementoftheshiftintheinterferencepattern.Beginbyselectingthe“Preview”buttontomakesuretheinterferencepatternhasgoodcontrastandthedetectorisnotsaturated.Onceeverythingisproperlyadjustedpush“TestImage”.Astableimageoftheinterferencepatternwillbedisplayed.Byclickingtherightmousebuttoninsideoftheimageaparticularintensitylineisselected.Press“ReadoutLines”toshowasinusoidalintensityprofile.Iftheprofileseemsjitteryanduneven,increasethe“Averaging”numberuntiltheprofileissmooth.OpenthegraylevelwindowusingthecorrespondingbuttonandmakesurethatisoccupiesthefullscreenoftheLCD.Beforestartingthemeasurements,theincrementshavetobechosen.Thisoptionchangesthetimeandtheresolutionofonemeasurement.Thisisalwaysacompromisesincefastmeasurementshavealowresolutionandslowmeasurementshaveahighresolution. The“Start”buttonstartsthemeasurementandgraylevelsvaryingfrom0to255areaddressedontotheactivehalfofthegreylevelwindowinsuccession.Oncethesoftwareisdone,minimizethegreyvaluewindow.YoushouldseesomethingsimilartoFigure10below.Bypushing“ShowMeasurementPoints”aftertheimageisshown,themeasurementspointswillappearasreddots.Savethedataandopen

Figure11:PhaseCamSoftware

itinExcel.Somemeasurementpointswillhave“jumped”tothenextminimum.Thisissimplyfixedbyaddingorsubtractingtheperiodfromthesemeasurements.Findthevalueof∆yorthedistancetheminimumshiftedfromtheinitialminimum(grayvalueof0).Fromthismeasurementandtheperiod,plotthephaseshift,∆ϕ,asafunctionofgreylevel.Repeatthisprocedurefordifferentpolarizerandanalyzerconfigurations.Youwillneedthefollowingequationtodeterminephaseshift.

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Δϕ =2πg⋅ Δy Eq.5

Phase Cam

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6 Evaluation

6.1 Show image

If one presses this button when the measurement is finished an image will appear at the position of the former live image in which the selected measurement line for each addressed gray level is drawn one below the other. This image gives the first impression of the measurement and shows clearly if and how the interference pattern is shifted. A convenient function for the first rough determination of the total phase shift is implemented. If the user moves the cursor into the shown image and presses the right mouse button, a bar will appear at the position of the cursor. If the user holds the button and moves the cursor, a second bar appears and the distance in pixel will be shown just above the image (see figure 10). If the start and the end value are chosen for both bars one can see how far the interference pattern was shifted during the measurements. By taking the period of the interference pattern into account (see 4.2) the phase shift can easily be calculated.

FIG. 10: Metering bars

6.2 Show measurement points

If this button is pressed after the above explained image is shown, the measurement points (values with min intensity) will appear as red points. If the distribution of these points has no big jumps and irregularities (which can be introduced by vibrations or air flow) the measurement will be evaluable. It is

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Reference:Goodman,JosephW.IntroductiontoFourierOptics.NewYork,NewYork:TheMcGraw‐HillCompanies,1968Hecht,Eugene.Optics4thEdition.MenloPark,California:AddisonWesleyLongman,2002OptiXplorerEducationKitManual.HoloeyePhotonicsAG,2002