type: phd summary section: image processing and imaging...

ÓPTICA PURA Y APLICADA. www.sedoptica.es

Opt. Pura Apl. 50 (1) 75-91 (2017) © Sociedad Española de Óptica

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Type:PhDsummarySection:ImageProcessingandImagingTechniques

Participant in the 5th Ed. Justiniano Casas Prize (2016)

ProgrammableDiffractiveOpticalElementswithApplicabilityinOphthalmicOptics

Elementosópticosdifractivosprogramablesconaplicabilidaden

ópticaoftálmica

LennyA.Romero1P*,MaríaS.Millán2S.1.Facultad de Ciencias Basicas, Universidad Tecnológica de Bolivar, Km 1 vía Turbaco,

Cartagena, Colombia2.Dep. Optics and Optometry, Universitat Politècnica de Catalunya BARCELONATECH, C/ Violinista Vellsolà,

37, 08222 Terrassa (Barcelona), Spain(*)E-mail:[email protected] S:SEDOPTICAmember/P:SRCOmember

DOI:10.7149/OPA.50.1.49509

ABSTRACT:

ThispaperpresentsresultsoftheresearchrelatedwiththedevelopmentofthedoctoralthesisofLennyAlexandraRomeroPérez.Thisworkcoversthedesign,thecharacterizationandtheanalysisofseveralprogrammable diffractive optical elements (PDOEs) such as: Fresnel lenses, multifocal lenses,integratedcombinationsofphase-maskswith lenses,andDOEswithextendeddepthof focus (EDOF)like theLight Swordoptical element and thePeacockEye.Thishasbeenpursued to address severalproblems of human vision, like myopia, hyperopia, astigmatism, and presbyopia, by means of theimplementationofsuchDOEsonaHoloeyeLiquidCrystalonSiliconHEO1080Pspatiallightmodulator.Wehavedevelopedandimplementedseveralalgorithmsforgeneratingthenecessaryopticalelementstocompensatethedifferentametropies.Simulationandexperimentalresultsdemonstratethatseveralof the considered DOEs have the sufficient imaging performance and thus the potential forcompensatingametropiesandpresbyopia.

Key words: extended depth of focus, diffractive optical elements; ophthalmic optics, visual optics,spatiallightmodulator.

RESUMEN:

Este trabajopresenta resultadosde la investigación enmarcada en la tesis doctoral realizadapor ladoctoraLennyAlexandraRomeroPérez,lacualfueenfocadaeneldiseño,caracterizaciónyanálisisdediversoselementosópticosdifractivosprogramables (PDOEsporsussiglasen inglés)programablestalescomo:lentesdeFresnel,lentesmultifocales,combinacionesintegradasdemáscarasdefaseconlentes y elementos con capacidad de extender la profundidad de foco (EDOF del inglés, extendeddepthoffocus),comoloselementosdifractivosllamadosespadadeluzyojodepavo.EstatesisbuscasolventarpormediodelusodeDOEsenunmoduladorespacialdeluzdiversosproblemasdelavisiónhumana, como lamiopía, hipermetropía, astigmatismo y la presbicia. La implementación se llevó acabo en un modulador Holoeye LCoS (del inglés Liquid Crystal on Silicon) de 1080P. Se handesarrollado e implementado distintos algoritmos para la generación de los elementos ópticosnecesarios para la compensación de las ametropías. Los resultados obtenidosmediante pruebas desimulaciónyexperimentaciónpermitencontrastarlaspropiedadesdelosDOEsanalizadosydestacaraquellos que por su funcionalidad en la compensación de ametropías pueden alcanzar una mayoraplicabilidad. En este artículo presentamos los resultados más relevantes obtenidos en estainvestigación.

Palabras clave: elementos ópticos difractivos programables, profundidad de foco extendido, ópticaoftálmica,ópticavisual,moduladorespacialdeluz.



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REFERENCESANDLINKS/REFERENCIASYENLACES[1] V.Laude,“Twisted-nematicliquid-crystalpixelatedactivelens,”Opt.Commun.153,134–152,(1998).

https://doi.org/10.1016/S0030-4018(98)00143-6[2] M. S. Millan, J. Otón, and E. Perez-Cabre, “Dynamic compensation of chromatic aberration in a

programmablediffractivelens,”OptExpress,14,9103–9112,(2006). https://doi.org/10.1364/OE.14.009103

[3] L.A.Romero,M.S.Millan,andE.Perez-Cabre,“Lenteprogramablemultifocal:Combinacióncoaxialymultieje,”Opt.PuraApl.,43,101–112,(2010).

[4] L.A.Romero,M.S.Millan,Z.Jaroszewicz,andA.Kolodziejczyk,“Doublepeacockeyeopticalelementforextendedfocaldepthimagingwithophthalmicapplications,”JBiomedOpt,17,046013–046013,(2012). https://doi.org/10.1117/1.JBO.17.4.046013

[5] L.N.ThibosandA.Bradley,“Useof liquid-crystaladaptive-opticstoaltertherefractivestateoftheeye.,”OptomVisSci,74,7,581–587,(1997). https://doi.org/10.1097/00006324-199707000-00028

[6] G.Li,D.Mathine,P.Valley,P.Äyräs,J.Haddock,M.Giridhar,G.Williby,J.Schwiegerling,G.Meredith,and B. Kippelen, “Switchable electro-optic diffractive lens with high efficiency for ophthalmicapplications,”Proc.Natl.Acad.Sci.U.S.A.103,6100-6104(2006). https://doi.org/10.1073/pnas.0600850103

[7] L.A.Romero,M.S.Millan,andE.Perez-Cabre,“Opticalimplementationofmultifocalprogrammablelenswithsingleandmultipleaxes,”JournalofPhysics:ConferenceSeries,274,012050(2011).https://doi.org/10.1088/1742-6596/274/1/012050

[8] J. Otón, P. Ambs, M. S. Millan, and E. Perez-Cabre, “Multipoint phase calibration for improvedcompensationofinherentwavefrontdistortioninparallelalignedliquidcrystalonsilicondisplays,”Appl.Opt.46,5667–5679,(2007). https://doi.org/10.1364/AO.46.005667

[9] N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focaldepth,”Opt.Lett.16,523–525,(1991). https://doi.org/10.1364/OL.16.000523

[10] J.Sochacki,S.Bara,Z.Jaroszewicz,andA.Kołodziejczyk,“Phaseretardationoftheuniform-intensityaxilens,”Opt.Lett.17,7–9,(1992). https://doi.org/10.1364/OL.17.000007

[11]A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, and M. Sypek, “The light sword optical element—a newdiffractionstructurewithextendeddepthoffocus,”J.Mod.Opt.37,8,1283–1286,(1990). https://doi.org/10.1080/09500349014551431

[12]Z. Jaroszewicz,A.Kołodziejczyk,D.Mouriz, and J. Sochacki, “Generalized zoneplates focusing lightintoarbitrarylinesegments,”J.Mod.Opt.40,601–612,(1993). https://doi.org/10.1080/09500349314550661

[13] J.Otón,M.S.Millán,andE.PérezCabré,“Programmablelensdesigninapixelatedscreenoftwisted-nematicliquidcrystaldisplay,”Opt.PuraApl.38,47–56,(2005).

[14] J. Otón, M. S. Millan, E. Perez-Cabre, P. García-Martínez, G. Cristóbal, B. Javidi, and S. Vallmitjana,“Imaging characteristics of programmable lenses generated by SLM,” AIP Conference Proceedings,860,471–480(2006). https://doi.org/10.1063/1.2361252

[15]R.R.Bennett,“Rabbetts'ClinicalVisualOptics,”(1998).[16]Z. Jaroszewicz,A.Kołodziejczyk,D.Mouriz, and J. Sochacki, “Generalized zoneplates focusing light

intoarbitrarylinesegments,”J.Mod.Opt.40,601–612,(1993). https://doi.org/10.1080/09500349314550661

1.IntroductionProgrammable diffractive lenses, displayed on liquid-crystal pixelated devices are of common use inopticalprocessorsandinformationoptics[1].Veryoftentheyaremultiplexedtootherelementssuchasfiltersorhologramssoastoimplementacomplexdiffractiveopticalelement[2-4].Despitetheirattractiveproperties of refreshment and flexible design, the application of programmable lenses to ophthalmicoptics for ametropia compensation is, however, very limited. Some liquid crystal devices have been



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alreadyproposedfortheiruseasophthalmiclensesandsomeprototypescanbefoundintheliterature[5,6].Theyusuallyhaveadiffractivedesign,eitherdisplayedonapixelateddeviceoronadeviceconsistingof other arrays of cells (e.g.,with circular symmetry in Ref. [6]). Some practical difficulties arisewhenconsidering theophthalmicapplicationofprogrammable lensesdisplayedon liquid-crystaldevices: lowefficiency,chromaticaberrations(emphasizedwithdiffractivedesigns),polarizedlightrequirements,etc.

The attractive properties of refreshment and flexible design can be achieved when displaying theprogrammablediffractive lens(alsocalledPhaseFresnelhologram,kinoformlens,andactive lens)onapixelated liquid crystal screen that acts as a phase-only spatial light modulator. Modern LCoS (liquidcrystal on silicon) devices reach 2π phase modulation with almost inexistent amplitude coupledmodulation,highefficiency,andsmallpixelpitch(about8microns)sothatitispossibletogeneratelensesofpowerupto9D(Dstandsfordiopter)forawavelengthof633nm.Ontheotherhand,thesedevicesaretypicallyreflectiveandhavearelativelysmallaperture(e.g.0.9cmx1.5cm),whicharecleardrawbacksinophthalmic applications. Other practical limitations are related to the mechanics and electronicsrequirementsofthecurrentstateoftheartofspatiallightmodulators(SLMs).But,apartfromspectacles,thereareotherophthalmicapplications thatmaystillbeconsidered,as shown inRef. [5].For instance,programmable lenses could be inserted in the eyepieces of optical instruments or in phoropters foroptometricassessmentofvisualacuity.Inthesecases,theuseofdiffractiveprogrammablelenseswouldpotentiallycompensateforthepossiblerefractiveerror(i.e.ametropia)oftheobserver(orpatient)withsimilarorevenmoreaccuracythanconventionalcomponents.

Inthiswork,weexplorethepotentialcapabilityofdiffractivelensesandprogrammablediffractiveopticselements(PDOEs)withextendeddepthof focus(EDOF)displayedonapixelatedLCoSdevicetoquicklydetermineandcompensatefortherefractiveerroroftheobservereye.Weconsidercommonametropies,suchasmyopia,hyperopia,astigmatismandpresbyopia.Inafirststage,wedemonstratethefeasibilityofthe proposal on an artificial eye in optical bench. The artificial eye, which is initially emmetropic, isconverted into ametropic by introducing other lenseswith the appropriate adding power. The LCoS isthenprogrammedtodisplayaseriesoflensestocompensateforthedisorderinducedintheartificialeye.Afine-tuningoftheopticalpower(or,equivalently,thefocallength)oftheprogrammablediffractivelensallowsthecompensationoftheametropia.Wecomparethecompensationachievedbytheprogrammablelens displayed on the LCoSwith the compensation achieved by introducing a conventional ophthalmiclens from a trial lens case. To this end, we compute the modulation transfer function (MTF) for thecompensation.

Inasecondstage,weusedtheSLMlikeamultifocallens.Wehaveexaminedthecombinationofdiffractivelenses in one device [3, 7]. In this stage we introduce four coding methods for merging the phasedistributionsofthreeFresnellenses,(f1,f2,f3),ofdifferentfocallengthintoasingle-phasedistribution.Theresulting phase is implemented experimentally via an LCoS SLM (HEO 1080P Holoeye) previouslycalibratedandcharacterized[8].Thecombinationoflensesisarrangedfortwoconfigurations:coaxialandmultiaxis. In the first one, all lenses have a common optical axis,whereas in the latter the lenses haveparallel separatedaxes.Wedescribe themain featuresof such lensesby taking into account the actualtechnicalconstraintsgivenbytheSLMspecifications.Wedescribedifferentmethodsforspatialencodingthe three combined lenses, and we show numerical simulations and experimental results with anextendedtest-objectforeachcodingmethod.

Finally,we study a several diffractive optical elements (DOEs)with EDOF. Themost promising opticalelementsforimagingwithEDOFinreal-timeseemtobeopticalelementsfocusinganincidentplanewaveintoafocallinesegment.Theseelementscanberegardedasmodifiedlenseswithcontrolledaberrations.Themodificationshouldleadtooutputimagescharacterizedbythepossiblehighestcontrast,brightnessand sharpness. Among the most important of these elements we present the Axicon [9], the Axilens(AXL)[10], the Light Sword Optical Element (LSOE) [11], and the Peacock Eye (PE) [4, 12]. Moreover,thereareworksthathaveshowntheadvantagesoftheseopticalelementsasanalternativesolutiontothecompensationofophthalmicaberrations inagedhumanvision[4].Theseelements illustrateapotentialapplicabilityofEDOFimagingcomponentstopresbyopiacompensation.ThePDOEswillbedisplayedonaparallel-aligned liquid crystalon silicon spatial lightmodulator (LCoSSLM),whichworks inphaseonlymodulationregime[8].TheresultsobtainedwithallPDOEswillbecomparedwithamultifocallens.Thelatter consists of threephasediffractive lenseswith the sameaxis that are spatiallymultiplexed. It hasthreefocalpointscoincidingwiththeextremesandthecenteroftherequireddepthoffocussegment[7].To test the optical performance of all the elements, we obtain the point-spread function (PSF), themodulation transfer function (MTF), and their evolution along the optical axis. To better visualize the



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imagingperformanceofeachPDOEweacquireimagesofanextendedobjectwithincoherentillumination.The MTF is to be computed using a slanted border extracted from the image of the extended object.Additional results concerning incoherent imaging of extended objects placed at different distances areincluded.Theresultshavebeenobtainedexperimentally.

2.RepresentationofthePDOEsbymeansapixelateddevice Let us denote the quadratic phase pattern of a spherical lens, in spatial coordinate (x0) andwith focallengthf,byl(x0,f),whichisthecontinuousfunction

l(x0, f ) = exp − j πλ f

x02⎛

⎝⎜⎞⎠⎟, (1)

represented in one spatial dimension for the sake of simplicity. The wavelength of the design isrepresentedbyλ.ThepixelatedstructureoftheSLMisresponsibleforaspatialdiscretizationofthephasedistributiondisplayedonthemodulator.Thisdiscretedistributioncanbewrittenas

l m, f( ) = exp − jπλ f

L2

Nl2 m

2⎛⎝⎜

⎞⎠⎟

∀ m∈ − Nl

2+1, Nl

2⎡⎣⎢

⎤⎦⎥, (2)

wherem is the pixel position and L is the diameter of the lens aperture that corresponds to a givennumberofpixelsNl. Ingeneral,L≤L0,whereL0 is the totalSLMaperture.AsEq. (2) reveals, thephasevaluevariesmorerapidlyintheperipherythaninthecenterofthelensaperture(Fig.1).AccordingtotheNyquistcriterion,thephaseshiftbetweentwoneighborpixelsofEq.(2)attheborderofthelensaperturemustbelowerthanorequaltoπ[13,14].Thisconditionresultsinaminimumvalueforthefocallength,namedthereferencefocallength,oftheprogrammablelens.ForagivenSLM,withaparticularpixelpitch,therequirementestablishesthatthesamplingfrequencyfitsfortheNyquistcriterionatthebordersofthelens aperture. Assuming that L = Nlδx0 , where δx0 is the pixel pitch, the reference focal length fref, orequivalently,thereferencepower,canbedeterminedfromtheexpression[13,14],

fref =Nl

λδx0

2 , (3)

Fig.1.Spatialdiscretizationandphaseprofileofa(modulo-2π)programmablelensaddressedtoanSLM.Reddotsrepresentthephasevaluesaddressedtotheelementsofthepixelatedscreenafterthespatialdiscretizationandphasequantizationofthe

continuousphaseprofile(insolidblueline).VerticalredlinesindicatetheNyquistlimitbeyondwhichtheeffectsofundersamplingmayappear.

Forinstance,takingintoaccountthespecificationsofHoloeyeLCos-SLMusedinthiswordNl=1080,𝛿𝑥#=8μm)andλ=633nm,weobtainafref,=0.109mor,equivalently,anopticalpowerof9.16D.WeassumethattheSLMworksinphase-onlyregime,withaminimalamplitudemodulationandalinearphaseresponse.We also assume that the phase modulation range is restricted to 2π radians (modulo-2π), whichcorrespondstothephasemodulationdepthavailabletoreproducetheprogrammedphaselenswith,forinstance, a focal length of f =fref. Furthermore, the phasemodulation range is usually discretized into a



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number of levels addressable through the video graphic array card. In our experiments, 256 discretephaselevelswereavailable.

3. Compensation of refractive errors induced in an artificial eye throughmonofocalsprogrammablelenses.Letusrecalltheprinciplesforthecompensationofrefractiveerrorsofametropichumaneyes.Wefirstlyconsider an artificial emmetropic eye (Fig. 2(a)) consisting of a camera lens, namely a photographicobjective(LobinFig.2),andaCCDsensor.Fortheproofofconceptdescribedinthiswork,weconsideranoversimplifiedversionoftheeye,hereafternamedartificialeye.ThephotographicobjectivesimulatestheopticalsystemoftheeyewhereastheCCDsensorplaystheroleoftheretina.Therelativepositionofthesetwoelementsisfixed,sothatanobjectplacedatafardistance,morespecificallyinfinity,isfocusedontheCCDsensorplacedatthebackfocalplaneoftheobjective,F'Lobinthesamewaythatanemmetropiceyewouldfocustheimageofsuchanobjectontheretina.Differentrefractiveerrorscanbesimulatedusingthis artificial eye in an optical bench. Let us briefly describe the myopia and the hyperopia, theastigmatismandthepresbyopia.

3.a.RefractiveErrorsMyopiaandhyperopia:Amyopiceyehashigheropticalpowerthananemmetropiceye.Consequently,theimageofa farobject throughamyopiceye focuses in frontof theretina.Asdepicted inFigure2(b),anadditionalconverginglens,LM,beforetheemmetropicartificialeyepermitstoinducemyopiatoit.Inthisartificialeyewithinducedmyopia,theCCDsensorwouldcaptureablurredimage.Inordertocompensatethe artificial eye for the induced refractive error, a third lens can be placed in front of the simulatedmyopiceye.Adiverginglens,LC,wouldcompensateforthemyopiceffectintroducedbyLM,providedthatthevirtualbackfocalplaneofLC(F'LC)coincideswiththefrontfocalplaneofLM(FLM),thatis,F'LC≡FLM,asshowninFig.2(c).TakingintoaccountthedistancedbetweenLCandLMandthepowerofLM,wecomputethepowerofLC,orequivalently, itsfocal length,toobtainafocusedfinal imageattheCCDsensorplane(Fig.2(c)).Thus,

fLC' = fLM + d, (4)

wherethecoordinateoriginofeachdistanceis indicatedinFig.2(c).Thosedistancesinthedirectionoflightpropagationarepositive,andnegativeotherwise.

Ahyperopiceyehasloweropticalpowerthananemmetropiceye.Consequently,theimageofafarobjectthroughahyperopiceyefocusesbehindtheretina.Figure3showsthiscase.Hyperopiaisinducedintheartificial emmetropic eye by the use of a diverging lens,LH. A converging lens,LC, compensates for therefractiveerrorprovidedthatplanesF'LCandFLHcoincideatthesamepositionoftheopticalaxis,thatis,F'LC≡FLH.Ifthisconditionismet,thefinalin-focusimagewillbeobtainedintheCCDsensorplane(Fig.3(c)).

Fig.2.Simulationofanartificialmyopiceyeonanopticalbenchanditsophthalmiccompensation.(a)Emmetropiceye;(b)myopiceye(myopiainducedbylensLM);(c)myopiacompensationbylensLC;(d)longitudinaldisplacement,Δ,introducedbytheplano

parallelplateeffectofthebeamsplitter(BS).

Astigmatism:TheintroductionofacylindricallensLAinfrontoftheartificialemmetropiceyepermitstoinduceastigmatism.Withtheuseofsphero-cylindricalorplano-cylindricallensesinthepositionofLM,a



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varietyof cases canbe simulatedanddescribed in termsofLM (Fig. 2) andLH (Fig. 3) for theprincipalmeridians: from compound myopic astigmatism up to compound hypermetropic astigmatism, passingthroughsimplemyopicastigmatism,mixedastigmatism,andsimplehypermetropicastigmatism.Applyingin-planerotationstotheastigmaticlensLA,wechangetheaxisdirection,orequivalently,theweakerandstrongerprincipalmeridiansoftheartificialastigmaticeye.Asaconsequenceoftheinducedastigmatism,theCCDsensorcapturesblurredimages.Althoughtheyappeartobedifferentfromthedefocusedimagescausedbysimplesphericalametropia (i.e.,myopiaorhypermetropia), thecompensationof the inducedastigmatism does not involve, from the conceptual point of view, a rather different solution. Thecompensationwith the lensLC canbe achievedby taking into separate consideration the twoprincipalmeridiansoftheartificialastigmaticeye.Apartfromtheblurringeffectsintheuncorrectedastigmaticeye,there is some distortion in the image because of the different magnifications in the two principalmeridians.Aftercorrection,asharpimageisrestored,but,dependingontheseverityofastigmatism(i.e.,thedifferenceofpowerbetweentheprincipalmeridians)andthepowerandpositionofLC,someresidualdistortionmaystillaffectthefinalimage.

Fig.3.Simulationofanartificialhypermetropiceyeonanopticalbenchanditsophthalmiccompensation.(a)Emmetropiceye;(b)

hypermetropiceye(hypermetropiainducedbylensLH);(c)hypermetropiacompensationbylensLC;and(d)longitudinaldisplacement,Δ,introducedbytheplano-parallelplateeffectoftheBS.

Presbyopia:Theyoungeyeisabletoincreaseitspower(accommodation)bymodifyingthecurvatureofthe crystalline lens, and consequently, it is able of focusing on objects placed at different distances.Presbyopia isadecline inthis focusingabilitythatappearsasthecrystalline lensages.Therefore,whentheemmetropiceyebecomesadditionallypresbyopic, itneedsanextra focusingpower in the formofa“nearaddition”ofpositivepower. If thepresbyopiceyeiseithermyopicorhypermetropic, itneedstwokinds of compensations: to focus distant objects (distance correction) and to focus close objects (nearaddition).Thenearadditionofpositivepowerisaddedtothedistancecorrection[15].Theneedforanearadditiondependsonboth the available amplitudeof accommodationand the applicationdemands (i.e.,theworkingdistanceforthenearvisualtask).Itispossibletosimulatethissituationinanopticalbenchbyplacingtheobjecttestatarelativelyclosedistancefromtheartificialeye(Fig.4),forinstance,about10timesitsfocallength.A“young”emmetropicartificialeyewouldaccommodate,thatis,wouldincreaseitspower (represented by the converging lens LQ in Fig. 4(a)) to focus correctly the object on the sensor.However, an “old” emmetropic eye that suffers from presbyopia has insufficient amplitude ofaccommodation,andthepowerincrease(representedbythelensLPinFig.4(b))isbelowdemand.Insuchasituation,thepresbyopicartificialeyeformstheimagebehindtheCCDsensor,thuscapturingablurredimage of the object. This loss in the available amplitude of accommodation can be compensated by anadditional converging lens,LC, placed in frontof thepresbyopicartificial eye.This lens images thenearobjecttest,locatedatadistancea,ontothefirstfocalplaneoflensLP,thusobtainingasharpfinalimageinthesensorplane[Fig.4(c)].TherefractivepowerofthecompensatinglensLCmustbeadjustedsothatthelensequation

− 1a+ 1a '

= 1fC' , (5)

isfulfilled,andtheconditionforimagedistance a ' = fP + d, issatisfied(Fig4(c)).



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Fig.4.Simulationofpresbyopiaonanopticalbenchanditsophthalmiccompensation.(a)Accommodatedeyewiththeobjecttestplacedatneardistance.LQprovidesthenecessaryadditionalopticalpowertotheemmetropiceye(b)Presbyopiceye.TheadditionalopticalpowerofLPisunderdemand(c)PresbyopiceyecompensatedwiththeadditionalopticalpoweroflensLC.

3.b.OpticalExperimentsExperimental results are provided to compare the compensationachieved by programmable lensesdisplayedonanLCoSSLMwith thecompensationachievedbyophthalmic lenses froma trial-lenscase.Figure5showsthesetupusedintheexperiment.Theobjectisilluminatedwithspatiallyincoherentlight.Thewavelengthused is λ=632.8nm (He-Ne laser).The collimating lensL1 permits to simulate that theobject test is at a far distance (infinity) provided O is located in the front focal plane of lens L1. Thissituation simulates the observation of a distance object by an emmetropic eye.We use the 1951USAFresolutiontestchartastheobject(Fig.6).TheSLMusedtodisplaytheprogrammablecompensatinglensinourexperiments,LC,isanLCoSdevicethatworksinreflectivemode.AfterreflectionintheSLM,theBSreflectspartofthemodulatedbeamtowardtheartificialeye.TheemmetropicartificialeyeisrepresentedinFig.5by thecombinationofa lens (aphotographicobjective,Lob)andaCCDsensor.Throughout theexperiments, this artificial eye is going to be affected by myopia, hypermetropia, astigmatism, orpresbyopia.ThesetupshowninFig.5hasbeenusedto test theperformanceof theprogrammable lensdisplayedontheLCoSphaseonlypanelasacompensatinglensLCforrefractiveerrors.

Fig.5.Opticalsetupusedintheexperiment:A-attenuator,λ∕2-retarderhalf-waveplate,SF-microscopeobjectivecombinedwithaspatialfilter,D-rotatingdiffuser,O-objecttest,L1-collimatinglens,BS-beamsplitter,SLM-spatiallightmodulatoractingeitherasaprogrammablecompensatinglens(LC)oraflatmirror(M),LX-lensinducingtherefractiveerror(beingX=M,H,A,orPforamyopic,hypermetropic,astigmatic,orpresbyopiceye,respectively),Lob-photographicobjectiveandCCDsensor.Whenthetriallenswastobeusedforcompensatingtherefractiveerror,L*CwasinsertedcloselybeforeLXand,atthesametime,theSLMwasprogrammedto

actasaflatmirror(M).

Fig.6.Experimentalimageofthestandard1951USAFtestchartcapturedbytheCCDsensoroftheartificialemmetropiceye.Onlythecentralareaoftheimageisdisplayed.Whitezonesdonotappearentirelyuniformasaneffectcausedbytherotatingdiffuserplaced

againstthetest.

Experimentallywemust consider that the compensating lenses LC and L*C would not have exactly the



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samepowerbecauseofthedifferentdistancestheyarefromtheartificialeye.Besides,thepresenceofaBSintheraypath,betweentheSLMandtheartificialeye, introducesa longitudinaldisplacementintheposition of the second focal point of the compensating programmable lens LC. The plane parallel plateequivalent to the right angle reflectionprismof theBSproduces such a displacement,whichhas to betakenintoaccount intherefinedcalculationofthecompensatingpower.The longitudinalshiftΔcanbecomputedbytheexpression

Δ = n −1n

b, (6)

wherenandbdenotetherefractiveindexandthecubedimensionoftheBS.Forbothcases,myopiaandhypermetropia[Figs.2(d)and3(d)],thisshiftwillmodifythefocallengthoftheprogrammedlensf'Cby

fC' = fx + d − Δ, (7)

where fx will be fM or fH in case of myopia or hyperopia, respectively. Equation (7) is valid forcompensatingtheastigmatismaswellbecauseitcanbedescribedintermsofLMandLH intheprincipalmeridiansseparately.

Cylindricalophthalmiclensesweresuccessivelyplacedinfrontoftheemmetropiceyetosimulatesimpleastigmatismindifferentdirections(LX=LAinFig.5).Simplemyopicastigmatisminthehorizontaldirectionwas induced by inserting a +1.5D cylindrical lens (first focal length fA=−666.6 mm), whereas simplehypermetropic astigmatism in the vertical directionwas induced by inserting a −1.5 D cylindrical lens(first focal length fA=+666.6 mm). Fig. 7(a) show the corresponding defocused images as the artificialastigmatic eye in each situation captured them. Ametropy compensation was first achieved by aprogrammed cylindrical lens displayed on the SLM with the same orientation as the ophthalmiccylindrical lens used asLA. According to Eq. (7), and keepingd =90mm, the calculated focal lengths tocompensate for thehorizontalandverticalametropiawere f 'C=-594mmand f 'C=740mm,respectively.Thefocal lengthoftheprogrammedlensdisplayedontheSLMwasadjustedtoexperimentallyoptimizethecompensation.Theexperimentalfocallengthsthatbestperformedweref'C=−578mmandf'C=+813mm,respectively,whichare ingoodagreementwiththepredictedvalues.Sharply focused imageswereobtainedbythecompensatedartificialeye,andtheycanbeseeninFig.7(b).

Fig.7.Experimentalresultsforthecompensationoftheinducedsimplemyopicastigmatisminthehorizontaldirection(left),simplehypermetropicastigmatismintheverticaldirection(right).(a)Defocusedimageduetotheinducedastigmatism;focusedimageobtainedintheCCDsensorwhenthecompensationwasdoneby(b)displayinaprogrammablelensontheSLMor,(c)usingan

ophthalmiccylindricallens;(d)MTFfunctionscomputedfromthefocusedimages(b)and(c),alongwiththeMTFoftheemmetropiceye.

ConventionalcompensationbyophthalmictriallensesplacedinfrontoftheartificialeyewhileaddressingaflatmirrortotheSLMwasalsocarriedout.Theavailablecylindricaltrial lensof−1.5Dopticalpowerwas used to compensate for the simple myopic astigmatism for a distance d=10 mm. For the



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compensationofthesimplehypermetropicastigmatism,acylindricallensof+1.5Dfromthetrialsetwasused.Figure7(c)containsthecapturedimagesafterthecompensationforthesimplemyopicastigmatismin the horizontal direction (left) and the simple hypermetropic astigmatism in the vertical direction(right)by these conventionalophthalmic trial lenses.Thequalityof these images is comparedwith theimages obtained after the compensation done by the programmed lenses on the SLM, through thecomputation of the MTF curves [Fig. 7(d)]. In both cases, the best performance corresponds to theprogrammablecompensatinglensesdisplayedontheSLM,whoseMTFcurveshaveaslowerdecreaseinthe frequencydomain than the conventional trial lenses and are closer to the emmetropic behavior. Infact,thecompensationofsimplehypermetropicastigmatism(vertical)withatrial lenspresentsanMTFcurve that reveals a pseudoresolution effect (side lobe centered at ≈0.65 cycles∕pixel) with reversalcontrast,thusindicatingthemediocrequalityofthiscompensation.

To simulate presbyopia in the optical bench, a near objectwas obtained by approaching lensL1 to theUSAF test (O in Fig. 5) so that its conjugated image,which corresponds to testA in Fig. 4,was locatedapproximately at 500 mm from the artificial eye. In such a situation, the acquired image by theemmetropiceyewasdefocused,unlesssufficientaccommodation(about+2D)wasavailable.Weassumedtheaccommodationcapabilitywaslimitedtojust+1D,thatis,theartificialeyesufferedfrompresbyopia.To simulate thispresbyopia, a lensLP of +1Dwasplacedat the locationofLX (Fig. 5) and adefocusedimagewascapturedasshowninFig.8(b).FromtheopticalpowerofLP(orequivalently,itsfocallength

f'P=1000mm)andtakingintoaccountthatthedistancebetweenthecompensatinglensandtheartificialpresbyopiceyewasd=110mm,thenecessaryadditionalpowerofthecompensatinglensLCwas+1.44D.Compensation forpresbyopiawas firstattemptedwith theprogrammable lensdisplayedon theSLM.Arangeoffocallengthsfrom+600to+720mminstepsof5mmweresequentiallyaddressedtotheLCoSdevice, obtaining the best compensation for f 'C =+700mm,which corresponds to an optical power of+1.43 D. The captured image after this compensation is shown in Fig. 8(c). After that, we applied thealternativecompensationwithaconventionaltriallens.IfaplanemirrorwasdisplayedontheSLM,andanophthalmic trial lensof+1.5Dwasused, thecompensationwasalsoachievedasshown inFig.8(d).ComparingbothresultsintermsofMTFvalues[graphsdepictedinFig.8(e)],abetterperformanceoftheprogrammablelensisobtainedincomparisontotheophthalmictriallens.

Fig.8.Experimentalresultsofthecompensationforpresbyopia.(a)Reference:capturedimageofadistantobjectbytheartificialemmetropiceye.(b)Defocusedimageofanearobject(placedat500mmfromtheeyeapproximately)obtainedwhenthepresbyopicartificialeyehasjust+1Dofaccommodation.(c)Refocusedimageaftercompensationwiththeprogrammablelensdisplayedonthe

SLM,whichprovidedthenecessaryadditionalpower(focallengthfinelytunedto+700mm).(d)Refocusedimageaftercompensationbyanophthalmictriallenswithadditionalopticalpowerof+1.5D.(e)MTFcurvescorrespondingtofigures(a),(c),

and(d).

4.Opticalimplementationofmultifocalprogrammablelens.4.a.SpatialencodingthemultifocallensTocombinemultiple lenses ina single-phasedistribution, twodifferentencodingmethodsof thephasedistribution are used: by rows or randomly. In the codification by rows, each row of pixels of themodulator is assigned to one sublens, and the process is repeated sequentially until filling in the SLMaperture. In the random codification, each pixel (x, y) of the SLM’s aperture is assigned a value of kbetween0and1,randomly.Ifk≤(1/3),thepixelrepresentsthephaseofl1(x,y).If(1/3)<k≤(2/3),thepixelrepresentsthephaseofl2(x,y)andfinally,if(2/3)<kthepixelrepresentsthephaseofl3(x,y).

Two configurations of multifocal lens are considered: the lenses share a common optical axis (coaxialconfiguration)orhavedifferentparallelaxes(multi-axisconfiguration).Inallcases,threesublensesl1,l2,l3withrespectivefocallengthsf1,f2,f3,arecombined.Allthefocallengthsusedintheseexperimentsaregreaterthanfref.Thefocallengthsforl1,l2,l3sublensesaref1=35cm,f2=25cmandf3=30cm,respectively,



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wheref2istheclosesttothefrefoftheEq.3

Coaxialconfiguration.Theopticalaxesofthethreesublensesl1,l2,l3coincide.Figure9showsagreylevelrepresentationofthephasedistributionofthemultifocal lenswithcoaxialconfiguration:(a)withphasecodificationbyrows,(c)withrandomcodificationofphase.

Fig.9.Phasedistributionsforthreelenses(l1,l2,l3)ofdifferentfocallengthincoaxialandmultiaxisconfiguration.(a)and(b)Multifocallenscodedbyrows,(c)and(d)multifocallensrandomlycoded.

Multi-axis configuration. In this configuration the optical axes of the three sublenses are separate andparallel between them. Each lens can be described as follows:

l1 x0 + xd , f1( ) , l2 x0 , f2( ) , l3 x0 − xd , f3( ) .

Figure9showsagreylevelrepresentationofthephasedistributionofthemultifocallenswithmulti-axisconfiguration:(b)withphasecodificationbyrows,(d)withrandomcodificationofphase.

4.b.SimulatedandExperimentalresultsofthemultifocallensInordertoobtaintheimagesformedbyprogrammablemultifocallensdesignedaccordingtothedifferentconfiguration shown in section 4.a, we present the numerical simulation results obtained bymeans ofFresnelpropagation.Theexperimentalresultswereobtainedusingthesetupofthefigure5,butforthisexperiment,theCCDcameraoperatedwithoutphotographicobjective,thatis,thelensesLXandLobwereremoved. In this way, the multifocal lens formed images at different distances in the image space.DisplacingconvenientlytheCCDsensortheimageswerecaptured.

InFig.10weshowthesimulatedandexperimentalresultsobtainedintheimageplaneforthemulti-focallenscodedbyrowsandwithcoaxialconfiguration.Inthecoaxialconfigurationthethreeimagesoverlapdue to the fact that the optical axes of the three sublenses coincide. When the three sublenses areregularly encoded by rows on the SLM, vertical repetitions of the image appear due to the multiplediffraction orders introduced by the coding procedure. We verified the correspondence between thenumerical and experimental results, although, only the central zero-order image is shown because theimagesofthehigherorderswereoutsidethefieldoftheCCD.

InFig.11weshowthesimulated(above)andexperimental(below)resultsobtainedintheimageplane,for themulti-focal lens with coaxial configuration and random phase encoding. In each image plane acoaxial superposition of three images is observed,while one of them is in focus the other two appeardefocused.

Figure 12 shows the results for the multifocal lens with multi-axis configuration and random phaseencoding. In thiscase, themulti-axisconfigurationdoesnot formamixtureof in-focusandout-of-focussuperposedimages.Themulti-axisconfigurationpermitstospatiallyseparatetheseimages.



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Fig.10.Coaxialconfigurationcodedbyrows.Theimages(a),(c),(e)representsimulatedimageplanesand(b),(d),y(f)experimentalresults.(a)-(b)l1(f1=35cm),(c)–(d)l2(f2=25cm),(e)-(f),l3(f3=30cm).

Fig.11.Multifocallenswithcoaxialconfigurationandrandomphaseencoding.Simulated(top)andexperimental(bottom)results.

l1(f1=35cm),(c)–(d)l2(f2=25cm),(e)-(f),l3(f3=30cm)

Fig.12.Multifocallenswithmulti-axisconfigurationandrandomphaseencoding.Simulated(left)andexperimental(right)results.

l1(f1=35cm),(c)–(d)l2(f2=25cm),(e)-(f),l3(f3=30cm).

We foundtheexperimental results tobeconsistentwith thenumericalsimulationscarriedout foreachconfiguration. The experimental results bring out the fact that the multi-axis configuration is



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characterizedbyprovidingagoodcontrastintheimagesplanesthatareinfocus.Ontheotherhand,thecoaxialconfigurationimageswithpoorcontrastduetotheimageoverlappingonthesameaxis.Regardingthe encoding methods, we have verified that the row encoding method generates multiorder imagereplicas given by the repeating pattern introduced by the codification, whereas the random encodingmethodhasnotthisdisadvantageouseffect.

5.ProgrammableDiffractiveOpticalElementsforExtendingtheDepthofFocusInthissectionweevaluateanopticalperformanceofavarietyofProframmableDOEswithEDOFsuchas:the Forward Logarithmic Axicon (FLAX) [9], the Axilens (AXL) [10], the Light Sword Optical Element(LSOE) [11], thePeacockEyeOpticalElement (PE)[16] andDoublePeacockEyeOpticalElement (DPE)[4]. These elements illustrate a potential applicability as EDOF imaging components for presbyopiacompensation.ThePDOEswillbedisplayedonaLCoSSLM.TheresultsobtainedwithallPDOEswillbecomparedwithamultifocal lens.The latterconsistsof threephasediffractive lenseswiththesameaxis(coaxial configuration) that are spatially multiplexed. It has three focal points coinciding with theextremesandthecenteroftherequireddepthoffocussegment.Totesttheopticalperformanceofalltheelements,weobtain thepoint-spread function (PSF), themodulation transfer function (MTF), and theirthrough-focusevolutionalongtheopticalaxis.TobettervisualizetheimagingperformanceofeachPDOEwe show images of an extended object with incoherent illumination. The MTF was computed using aslantedborderextracted fromthe imageof theextendedobject.Theresults shownhavebeenobtainedexperimentally.

5.a.ProgrammableDOEsForward logarithmic axicon optical element (FLAX):An axicon is an optical element that transforms anincidentplanewaveintoanarrowfocalsegmentwithuniformintensity.Itischaracterizedbyarefractivepowerthatdecreaseswiththeradialdistance(r)toitscenter,thustheperipheralraysarefocusedtoanaxial point located farther than the focus for the central rays. Figure 13(a) shows the geometricalparameters and distribution of rays for an axicon illuminated by a plane wave of wavelength l. Thetransmittancefunctionforthiselementhasthefollowingform:

t r( ) = −ik 2a ln 1+ ar2 f1( )⎡⎣ ⎤⎦ ; (8)

where𝑘 = 2𝜋 𝜆and𝑎 = 𝑓, − 𝑓. 𝑅. = Δ𝑓 𝑅.andr=(x2+y2)1/2.Figure14(a)showstheresultingphasedistributionofthisDOErepresentedingreylevels.Thisiswhatisultimatelysenttothemodulator.

Axilens optical elements (AXL):This element also concentrates the incoming energy in a segmentof theopticalaxis.Ithasafocallengththatvarieswiththeradialcoordinatebuttheassociatedphaseretardationfunctiondiffers fromtheconventionalquadraticphase.Thiselement iscomposedofconcentricringsofinfinitesimalwidth.R is theradiusof the lensaperture, fandDfrepresentsthefocalof the lensandthelengthofthefocalsegmentrespectively.Figure13(b)showsthegeometricalparametersanddistributionofraysforanaxilensilluminatedbyaplanewaveofwavelengthlandFigure14(b)showstheresultingphasedistributionofthisDOErepresentedingreylevels.Thetransmittancefunctionoftheaxilenscanbewritten[10]as

t r( ) = −ikr2 2 f1 + Δf r2 R2( )⎡⎣ ⎤⎦{ } ; (9)

Light Sword Optical Element (LSOE) This element is a counterpart of the axilens where the radialmodulation of the focal has been replaced by an angular one [11]. The transmittance function has thefollowingforminthepolarcoordinatesystem:

t r,θ( ) = −ikr2 2 f1 + Δfθ 2π( )⎡⎣ ⎤⎦{ } ; (10)

where r,q are the radial and angular coordinates respectively. Analyzing Eq. (10),we can see that thephasefunctionoftheLSOEisanunconventionalFresnellensthathasafocallength𝑓 + Δ𝑓 2𝜋.Therefore,theLSOE focusesan incidentplanewave intoa focal segmentDf .When𝜃𝜖 0, 2𝜋 then thesegment isstretched from a distance f1 to a distance f1+Df in front of the LSOE plane. This segment is orientedperpendicularly to the sector, like is shown in Fig.13(c). Figure 14(c) shows the resulting phasedistributionofthisDOErepresentedingreylevels.Thisiswhatisultimatelysenttothemodulator



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Peacockeyeopticalelement(PE) [4]:Thegeometricalparametersanddistributionofrays forapeacockeyeand illuminatedbyaplanewaveofwavelengthl areshown in figure13(d). In thePEelement, thefocal segment is aligned with the optical axis. The transmittance function of this PDOE by which anincidentplanewaveisfocusedontoafocalsegmentoftheopticalaxiswithuniformintensitydistributionisgivenby

t x, y( ) = −ik y2

2 Lx A + d( ) −ALx + dA

2

L2ln LAx + d

⎡

⎣⎢

⎤

⎦⎥ ; (11)

whereAisasquareapertureuniformlyilluminatedbyaplanewaveofwavelengthλ.Listhelengthofthefocalsegmentofextremes[f1,f2].ThecentralpointofthissegmentisatadistancedfromdePEaperture.Figure14(d)showstheresultingphasedistributionofthisDOErepresentedingreylevels.

DoublePeacockeye(DPE):TheDoublepeacockeyeelementismadebyspatiallymultiplexingtwosinglePEelements[4]ThesesinglePEselementsweredesignedsothattheircorrespondingfocalsegmentsarearrangedinsuchawaythatonefocalsegmentislocatedaftertheotheralongtheopticalaxis,withsomepartialoverlapping.The total lengthcovers therequireddepthof focus.WehaveconsideredaRandomdistributionconfigurationforthedesignofthedoublemultiplexedpeacockeye.Inthisconfigurationthedevice aperture is segmented into small windows of 3x3 pixels. The phase of either one or the otherpeacockeye(onlyoneofthem)isdisplayedoneachwindowaccordingtoamosaicrandomdistribution(Figure14(e)).

Fig.13.GeometricalparametersandschematicdistributionofrayswithaninputplanewavefocusedbyallDOEs.(a)FLAX,(b)AXL,

(c)LSOE,theinfinitesimalangularsectoroftheelementfocusesanincidentplanewaveontoasegmentPP1orientedperpendicularlytotheopticalaxis,(d)PE.

Fig.14.PhasedistributionsforthePDOEsinrepresentedingreylevels.(a)FLAX,(b)AXL,(c)LSOE,(d)PE,(e)DPE.

5.b.OpticalexperimentandprogrammableDOEsdesignconditionsAllprogrammableDOEsweredesignedsothattheDOFlieswithinaspecificrangeoffocalsgivingrisetoafocalsegmentwithinthe interval [f1, f2]with fixedextremesat theaxialdistancesof f1=30cm(power indiopters of 3.33D) and f2=80cm (1.25D). In the case of the single PE had a focal segment [f1, f2] thatcoincidedwith the requested focal segment, that is f1=30cmand f2=80cm.ThedoublePEhad two focalsegments[f1,f2],and[f3,f4],thatcoveredtherequestedtotalfocalsegment,withsomeoverlap(ofabout5cm)inthecenter,thatis,f1=30cm,f2=58cm,f3=53cmandf4=80cm.Intheopticalexperiment(Figure13),each phase diffractive elementwas displayed on the SLM controlled by computer. AHe-Ne laser beam(element1inFig.(15))wasusedforillumination.Usingasmallpinhole(element4)intheopticalaxis,thebeam was spatially filtered and afterwards collimated to obtain the PSFs produced by each opticalelement displayed on the SLM (element 9) in the first series of experiments. In the second series ofexperiments, a groundglass rotatingdiffuser (element5)was locatedagainst theobject (element6) toobtainincoherentillumination.Theextendedobjectwasplacedatthefrontfocaldistanceofanauxiliarylensof faux=200mm,thustheimageis locatedat infinity.Theextendedobjectusedisthefigure#2from



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thetestUSAF(1.5mmlateralsize)anditcoveredanangularfield≈0.07’≈4.2”.TheobjectwasimagedbytheDOEdisplayedon theSLM.ACCDsensor (element11)wasdisplaced throughseveralplanesof thefocal segment for capturing the image.We fixed the capturing parameters of the CCD camera so as toavoidthesaturationofthecamera.

Fig.15.ExperimentalsetuptoevaluatetheextendeddepthoffocusofPDOEs.Thissetupisusedtoimageadiffusingextendedobject(element6)inthesecondseriesofexperiments.ToobtaindthePSFs(firstseriesofexperiments)theelement5and6areremoved

andtheelement4isshifttotheirposition.

Figure16.ExperimentalPSFsandimagesofanextendedobjectalongdifferentpositionsofthefocalsegmentobtainedwiththephaseofthePDOEsrepresentedingraylevelsontheleftcolumn.ThepositionsfromthePDOE(top)areexpressedincmand

diopters(D).



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Figure 16 shows the experimental results for the PSFs and the images of the extended object alongdifferentpositionsofthefocalsegmentobtainedforthedifferentPDOEs.Figure17showsexperimentalMTFscalculatedfromthePSFsofthePDOEs(Fig17).ThespatialfrequencyoftheMTFsisnormalizedfortheallcases.Asfortheconstantofnormalization,wehaveconsideredthediffraction-limitedcutoffspatialfrequencyintheobjectspace,inourexperimentfcutoff=238cycles/degree.ForthesomeDOEs,wepresenttwocross-sectionsoftheMTFscorrespondingtoperpendiculardirections,horizontal(H)andvertical(V).

An overview of the image quality produced by all DOEs along the focal segment reveals that with theexception of the multifocal lens, all remaining elements extend the DOF. However, not all elementspreserve imagequality significantlyalong the focal segment. Inboth theAxiconandAxilens, theirPSFscollect most of the incident energy in the central maximum. However, the energy collected aroundexternalringsisnotnegligible,whichleadstoadecreaseinimagequality.Becausethecentralmaximumconcentratesmostoftheenergyalongseveralplanesofthefocalsegment,theDOFisextended.InthecaseoftheFLAXitperformssufficientlywellinthefirsthalfofthefocusrangeandtheperformancedecreasesforthefollowingplanes,althoughit isnotaconsiderabledecline.Thebest imagequalityappears intheplanesof the focal segment, roughly from30cm to50cm. In the caseof theAXL thebest imagequalityappearsintheplanesofthefocalsegmentfrom40cmto80cm.AnalyzingtheMTFoftheaxiconwecanseethat fordistancesgreater thanz=50cmtheMTFsdecrease rapidly from thezeroth frequencyonward.Therearecontrastinversionsthatproduceblurryimages.IncontrasttotheeffectobservedintheMTFsofthe axicon, theMTFs of the axilens do not decrease as rapidly and have greater resemblance betweenthem.Whichmeansthatitismoreinsensitivetomisfocusthantheaxicon.Thisisevidencedbytheimagesoftheextendedobject,whichisbetterdefinedthroughoutmostofthefocalrange∆z.

TheLSOEcollectahighamountofincidentlightatthemostintensepartofthemaximum.Theseelementshavethepropertyof focusingwithina focalsegment,howeverthemainmaximumisnot locatedontheopticalaxis.Fromtheexperimentalresults,wehavelocatedandplottedaseriesofmaximaalongthefocalsegmentfordifferentimageplanesforbothelements.InthecaseoftheLSOEitperformssufficientlywellintheintermediateimageplanesandtheperformancedecreasesattheextremes(f1=30cmandf2=80cm),althoughitisnotaconsiderabledecline.Thebestimagequalityappearsintheplanesnearthecentralpartof the focal segment, roughly from 50cm to 70cm. In the focal planes outside this range, the imagesdegraderelativelyquickly.TheseresultsareconsistentwiththeMTFsshowninFigure17.

Figure17.ExperimentalMTFscomputedfromthePSFsofFigure16

The performances of the peacock-based elements show a real focal segment, somewhat shorter thanexpected, where defocus is remarkably reduced. In the case of the single peacock eye, the best imagequalityappearsinthecentralpartofthefocalsegment,letussay,from50cmupto60cm.Imagesdegraderatherquicklyoutsidethiscentralparttowardtheextremesofthedesignedfocalsegment(from30cmto80cm). In case of the double peacock eyes, however, the image quality benefits from two separatesegmentsofgoodperformance(thefirst,from35cmupto45cmandthesecond,from65cmto75cm),yetmaintaining an acceptable performance in the central part (from 50cm up to 60cm) of the total focalsegment,wherebothfocalcomponentsoverlap.Evenintheextremesofthetotalfocal.Theseresultsshedsomelightontheproblemofdesigningopticalsystemswithdefocusinvariance.Moreover,thereexistsapotentialapplicabilityforophthalmicapplications,likeinpresbyopiacompensation.



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5.c.DoublepeacockeyeopticalelementforextenddepthimagingforpresbyopiacompensationWeillustratethepotentialapplicabilityofthephasepeacock-baseddiffractiveelementsasEDOFimagingcomponents for presbyopia compensation. The object was axially shifted from the infinite (objectvergenceindioptersequalto0D)towardtheLCoSSLM.Tocoverlongobjectdistances(frominfinite(0D)to1m(1D),approximately),weshiftedtherealobjectwithinthefrontfocaldistanceoftheauxiliarylens(faux=20cm). For shorter object distances,we removed the auxiliary lens anddirectly shifted the objectalongthebenchtowardtheLCoSSLM.Figure18showstheexperimentalimagescapturedbythecamerafor the two elements: the single peacock eye and the double peacock eye. The latter wasmultiplexedbased on the addition of transmittances. At short and intermediate object distances (up to 165cm orobjectvergence0.61D),Figure18clearlydemonstratesbetterresultsfortheDPEthanforthesinglePE.Forfarobjectdistances(from165cmtoinfinite),however,thesinglepeacockeyeassuresmoreaccurateresults.

Inthecaseofthedoublepeacockeye,weshouldsaythattheimageobtainedfortheobjectplacedat90cmisstillacceptableincomparisonwiththeothers;consequently,thispositionconstitutesthe“nearobjectpoint” fortheEDOFimagingelement.Sincethe imageplaneof thenearobjectpoint is locatedat65cm(imagevergence1.5D)behindtheLCoSSLM, it impliesthatthedoublepeacockeye isoperatingwithafocal length of 38 cm (power of 2.6D) approximately according to a simple calculation in the paraxialopticsapproach.This result is consistentwith thevaluesconsidered in thedesignof thisPDOE(a totalfocalsegmentfromf1=30cmtof2=80cm).The“nearobjectpoint”forthesinglepeacockeyewouldbeat142cm(objectvergence0.70D)approximately.The“remoteobjectpoint”wouldbeinfiniteforthesinglepeacockeye,whereasforthedoublepeacockeyeitwouldbeatabout2m(0.5D).Forthedoublepeacockeye working with objects located at this distance or further, there is an effect that reminds thesimultaneousdoubleimage,withoneofthembetterfocusedthantheother.

Figure18.Experimentaldepthoffield(objectspace)fortwopeacockeyebaseelements.

6.ConclusionsAmetropiescanbeencompensatedbymeansofprogrammedmonofocallensesdisplayedontheSLM.Theuse of programmable lenses allowed a fine-tuningprocedure to obtain thenecessary optical power forcompensationinsitu.Theprocedureconsistedinsequentiallyreprogrammingandrefreshingthelenseson the modulator with a range of optical power values close to the initial calculated value. This wascarriedouttooptimizethesharpnessoftheimageontheretinaoftheartificialeye(theCCDsensor).Aswe have shown in the experimental results, this in situ fine tuning procedure of the programmablecompensation lens has clear advantages over the conventionalmethod of placing ophthalmic lenses ofdifferent optical power (with a ordinary precision not higher than of 0,25D) on a set of glasses or aphoropter.Certainly,withtheuseofanSLMitispossibletoobtainamoreprecise,rapid,andcomfortablecompensationduetothefactthatitdoesnotrequirethechangingofphysicalcompensationlenses.

In the second stage,we have studied the generation of a PDOE that consists of a combination of threesublenseswithdifferentfocallengthsandwithopticalaxesineithercoaxialormulti-axisconfiguration.Inaddition, for both configurations, we have used two methods for the spatial encoding of the phasedistributionof the composite system:by rows, and randomly.We found the experimental results tobeconsistentwith the numerical simulations carried out for each configuration. The experimental resultsbring out the fact that the multi-axis configuration (with separate parallel axis) is characterized byproviding a good contrast in the images planes that are in focus. On the other hand, the coaxial



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configuration imageswithpoor contrast due to the imageoverlappingon the sameaxis.Regarding theencodingmethods,wehaveverifiedthat therowencodingmethodgeneratesmultiorder imagereplicasgivenbytherepeatingpatternintroducedbythecodification,whereastherandomencodingmethodhasnotthisdisadvantageouseffect.

Finally, the experimental results presented in last stage prove that the PDOEs designed to focus anincident plane wave into a segment of the optical axis, satisfactorily perform as EDOF imagingcomponents.Wehave implementedseveralPDOEswithEDOFsuchas: theForwardLogarithmicAxicon(FLAX),theAxilens(AXL),theLightSwordopticalelement(LSOE),thePeacockEye(PE),andthedoublePeacockEye(DPE)whosephasefunctionisequivalenttotheso-calledprogressiveophthalmiclens.EachPDOEhas a focal segment that extends between the design focals [f1, f2] and its performance has beencomparedtoaprogrammabletrifocalFresnellens.WehaveevaluatedthequalityoftheimageformedbyeachPDOEbymeansofthePSFandtheMTF.

The imagequalityproducedbyallDOEsalong the focal segment reveals that,with theexceptionof thetrifocallens,allelementsarecapableofextendingsmoothlythedepthoffocus.However,notallelementspreserve the image quality in a significant way along the focal segment. The FLAX and AXL are theelementsthatperformworstwiththeimagequalityrapidlydecreasingfromthecentralpartofthefocalsegmenttotheextremeoflessopticalpower.OutofalltheremainingDOEs,theLSOEandtheDPEsaretheones thatproduce thebest imagequalityalong the focal segment.Nonetheless, theseelementsalsodisplay a loss of resolution and contrast. Therefore, the results suggest that a trade-off between theextendeddepthoffocusandimagesharpnessshouldbeachieveddependingontheapplication.Moreover,these results opennewavenues for thedesign of optical systems invariant to defocus oriented towardophthalmicapplications,likethecompensationofpresbyopia.

AcknowledgementsThisstudywaspartiallysupportedbyprojectDPI2013-43220-RfromtheSpanishMinisteriodeEconomíayCompetitividadandFEDER.

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