REVIEW ARTICLE

REVIEWARTICLE

Informationprocessingwithnonlinearopticaltwo-dimensionalfeedbacksystems

MikhailAVorontsov†

ArmyResearchLaboratory,InformationScienceandTechnologyDirectorate,2800PowderMillRoad,Adelphi,MD20783,USA

Received29June1998,infinalform22October1998

Abstract.Herewediscussnewpotentialapplicationsinparallelimageprocessingand

adaptiveopticsfornonlinearspatio-temporalprocessesoccurringinoptical(opto-electronic)nonlineartwo-dimensionalfeedbacksystems.

Keywords:Parallelimageprocessing,adaptiveoptics,nonlineardynamics,nonlinearoptics

1.Introduction

Nonlinearopticalsystemswithatwo-dimensional(2D)feedbackloophaveappearedasanexcellenttoolforstudyingavarietyofcomplexnonlinearspatio-temporalphenomenaincludingpatterns,opticalsolitonsandlocalizedstates,driftingandrotatorywaves,andchaoticregimes[1–6].TheongoingmergerofclassicalFourieropticswithnonlinearopticsoccurringinnonlinearoptical2D-feedbacksystemsbringsadditional,effectivemethodsfornonlineardynamicsdesignandcontrol[7,8].

Despitethesenewprospectsthereareseveralproblemsrelatedtothepotentialapplicationofthesesystems,aswellasthemoregeneraldisciplineofnonlinearopticalspatio-temporaldynamics,totheareaofinformationprocessing.First,wearestillratherrestrictedinthechoiceofnatural(microscopic)opticalnonlinearitiesandcannotprovideanexactnonlinearitytypeasrequiredforaparticularapplication.Anotherissueimportantforinformationprocessingistheabilitytocontrolopticalnonlinearitytimeresponseandstrength.

Theseproblemscanbepartiallyovercomeusingartificiallydesignednonlinearitiesbasedonavarietyofopto-electronicdevices[9].Thefirstimplementationofthisapproachforthestudyofspatio-temporalphenomenawasmadeinthemid-1980swhenanopto-electronicdevice—theliquidcrystallightvalve(LCLV)—wasusedintheoptical2D-feedbacksystem[10].AtthepresenttimetheLCLV-based2D-feedbacksystemisoneofthemostextensivelyusednonlinearopticalmodelsintheanalysisofavarietyofnonlinearspatio-temporalphenomena[11–14].HereweconsiderbasicmodelsfornonlineardynamicsthatcanberealizedusingLCLV-based2D-feedbacksystems,

†E-mailaddress:vorontsov@iol.arl.mil

anddiscussproblemsrelatedtoinformationprocessingapplicationsusingthesesystems.

2.DynamicalmodelsofLCLV-based2D-feedbacksystems

2.1.Nonlinear-diffusionmodels

TheLCLV-based2D-feedbacksystemprovidesforopticalimplementationofthenonlinearreaction–diffusion-typeequation—apartialdifferentialequation(PDE)widelyusedinnonlineardynamics,synergetics,andinformationprocessing[15,16].ForthecaseofaLCLV-based2D-feedbacksystemthisequationreads[10,17]:

τ

∂u2

u(r,t)=D∇⊥u(r,t)+f[IFB(r,t)],∂t

IFB(r,t)=G[u(r,t),P(r),a],

(1)(2)

whereu(r,t)isphase-modulationintroducedbytheLCLV,2

istheLaplacianoperatorwithrespecttothespatial∇⊥

variables(r={x,y}),τistheLCLV’scharacteristictimeresponse,Disthediffusioncoefficient,andfisafunctiondescribingthemodulationcharacteristicsoftheLCLV;thatis,thedependenceofthephase-modulationu(r,t)onthefeedbackfieldintensitydistributionIFB(r,t)(intensitydistributionontheLCLVphotoconductivelayer).Equation(2)describestransformationofthewavefrontu(r,t)intothefeedbackintensitydistributionIFB(r,t)occurringinthesystem’sfeedbackloop.In(2)Gisanoperatorappliedtofunctionu(r,t),andvectoraandvector-functionP(r)describedependenceofthefeedbackintensityonexternalparametersandfunctionssuchasfeedbacklengthL,inputfieldintensityI0(r)andphaseϕ0(r)distributions.

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IntoaandP(r)wemayalsoincludeinputsignals,forexamplecontrolsignalsand/oraninputimageIim(r).

FormostLCLVtypesthemodulationfunctionin(1)canbeapproximatedbythefunctionf(IFB)=ptanh(bIFB+c),wherep,bandcareparametersdependentontypeofLCLVandtheoperationalmode[17].Inthevicinityoflowfeedbackintensitiesthemodulationfunctioncanberepresentedasthelineardependencef(IB)∼=KIFB.InthisfeedbackintensityrangetheLCLVactsasaKerr-typenonlinearelement[18].

Themathematicalmodel(1),(2)describingLCLV-based2D-feedbacksystemdynamicswasfirstintroducedin[10]basedonsimplified(phenomenological)considerationsoftheactuallyrathercomplicatedphysicalprocessesinvolvedinLCLVdynamics.Substituting(2)into(1)weobtainasinglereaction–diffusion-typePDE:

τ

∂∂t

u(r,t)=D∇2⊥u(r,t)+F[u(r,t),P(r),a],(3)

wherefunction

F[u(r,t),P(r),a]=f{G[u(r,t),P(r),a]}.2.2.Two-componentreaction–diffusionsystemsCombiningtwoLCLV-based2D-feedbacksystemswearriveatthetwo-componentreaction–diffusion-typemodels[19]:τ∂u

∂t

u(r,t)=Du∇2⊥u(r,t)+Fu[u(r,t),v(r,t)],(4)

τ∂v2

∂t

v(r,t)=Dv∇⊥

v(r,t)+Fv[v(r,t),u(r,t)],(5)whereu(r,t)andv(r,t)aredynamicalvariables(phase-modulationsintroducedbytheLCLVs),τu,τv,DuandDvaretheLCLV’scharacteristicresponsetimesanddiffusioncoefficients,andFuandFvarenonlinearfunctions.

Therichandintriguingnonlineardynamicsofthetwo-componentreaction–diffusion-typePDEhasbeentheinspirationanddeparturepointformanyinvestigators[15,20].Theattractivefeatureinopticalimplementationofnonlinearreaction–diffusionequationsistheparallelnatureofopticalprocessing,whichcanprovideextremelyhighprocessingspeedeveninthecaseofhigh-resolutioninputs.However,thereareseveralproblemsrelatedwithopticalmodelling(solution)ofreaction–diffusion-typeequationsusingLCLV-based2D-feedbacksystems.Oneofthemajorproblemsisrestrictioninthechoiceofnonlinearfunctions(Fin(3);Fu,Fvin(4),(5)).Inclassicalreaction–diffusion-typemodelsthenonlinearityisdescribedbyanN-(orS-)likenonlinearfunction.Moresophisticatednonlinearitiesareneeded,forexampleinimageprocessingbasedoncoupledreaction–diffusion(geometry-driven)equations[16].2.3.Choicesfornonlinearity

ThepossibilitiesforopticalimplementationofdifferentnonlinearitiesusingLCLV-based2D-feedbacksystemsareratherlimited.Hereweconsiderafewoftheavailablechoices.ThespecificN-liketypeofnonlinearfunctionintheformofcosorsinfunctionscanberealizedinaLCLV-basedinterferometerwith2Dfeedback[2,10].In

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thiscase,thefeedbackintensityisgivenbyIFB(r,t)=Iin(r){1+γcos[u(r,t)+ϕ(r)]}.InthelinearregionoftheLCLVmodulationcharacteristic,insteadof(1)and(2)weobtain[2,10]:τ∂∂t

u(r,t)=D∇2

⊥

u(r,t)+KIin(r)×{1+γcos[u(r,t)+ϕ(r)+󰀞]},(6)whereKisaparameterspecifyingnonlinearitystrength,γ

isthevisibilityoftheinterferencepatterninthefeedbackintensity,and󰀞isaphaseshift.

2.4.Nonlineardiffusion–diffraction-typemodelsMorecomplicatednonlinearitytypescanbeimplementedinasystemwithfielddiffractioninthefeedbackloop.TheLCLV-baseddiffractive2D-feedbacksystemcanbedescribedbyacouplednonlineardiffusion–diffraction-typemodel[21](seealso[12,13,18]):τ∂2

∂t

u(r,t)=D∇⊥

u(r,t)+f[IFB(r,t)],whereIFB(r,t)=|A(r,z=L,t)|2,

(7)−2ik

∂A(r,z,t)∂z

=∇2

⊥A(r,z,t),(8)and

A(r,z=0,t)=Iin(r)exp{i[u(r,t)+ϕ(r)]}.

(9)

HereA(r,z,t)isthefieldcomplexamplitudeinthediffractivepartofthefeedbackloopoflengthL,andk=2π/λisthewavenumber.

OpticalFourierfilteringinthefeedbackloopyieldsmodelsofnonlinearintegro-differentialequationsoftypeτ

∂∂t

u(r,t)=D∇2⊥u(r,t)+f[IFB(r,t)],󰀂󰀁󰀂2whereI󰀂FB(r,t)=󰀂󰀂A(r󰀌,L,t)h(r󰀌−r)d2r󰀌󰀂󰀂󰀂

(10)

−2ik

∂A(r,z,t)∂z

=∇2

⊥A(r,z,t),

(11)A(r,0,t)=Iin(r)exp{i[u(r,t)+ϕ(r)]}.

(12)

In(10)h(r)istheFourierfilter’sresponsefunction[7].FourierfilteringcanalsobeappliedtotheLCLV-basednonlinear2Dinterferometer,orcanbeusedinthetwo-componentreaction–diffusion-typesystem.

ThecouplingoftwoLCLVsthroughdiffractivefeedbackgaverisetothetwo-componentdiffusion–diffraction-typemodelanalysedin[22]:τ∂u∂t

u(r,t)=Du∇2⊥u(r,t)+fu[|Av(r,Lv,t)|2],(13)τ∂v

v(r,t)=Dv∇2⊥v(r,t)+fv[|Au(r,Lu,t)|2∂t

],(14)

−2ik

∂Au(r,z,t)∂z=∇2

⊥Au(r,z,t),∂Av(r,z,t)(15)

−2ik

∂z

=∇2

⊥Av(r,z,t),Au(r,0,t)=I(u)

in(r)exp{i[u(r,t)+ϕu(r)]},

(16)

Av(r,0,t)=

I(v)

in(r)exp{i[v(r,t)

+ϕv(r)]}.

Yetmorecomplexsystemdynamicscanbedesignedif

weapplygeometricalcoordinatetransformations(rotation,linearshift,scalechange,etc)inthefeedbackloop(seeforexample[2,10,11,14,21,23]).

3.Hybridopto-electronicnonlinear2D-feedbacksystems

3.1.Analoguesignalprocessingaspects

Allofthepossibilitiesintheopticalmodellingofnonlineardynamicsmentionedsofararequiteattractiveintheframeworkofnonlinearspatial–temporalcomplexitystudies,butveryoftentheyarenotexactlywhatweneedforpracticalapplicationofnonlinearspatio-temporaldynamics.Unfortunately,wemustconfessthatdespitenumerousattemptstodemonstratehowopticalsystemsbasedonnonlinearspatio-temporalphenomenaworkinsolvingpracticalproblems,westillhavenotprovidedavisibledemonstrationonasystemlevel(afewexamplesofspatio-temporaldynamicsapplicationsforatmosphericimagingmodellingaregivenin[24]).Itseemsthatoneofthereasonsforthisnon-successisthestrategythathasbeenpursued.Thisstrategyisbasedontheimplicitassumptionthatamongakaleidoscopeofnonlinearspatio-temporalprocesses,wewillbeabletofindsomethingpotentiallyusefulforapplication.Fromthispointofviewthewiderthepaletteofsystemsandmodelswestudythebetterthechanceforsuccess.Thismaybetrueingeneral,butnotinourparticularcase.Thenonlinearmodelswedealwitharesocomplicatedthatittakesmuchtimeandefforttoanalyse,understand,andrejecteachcandidateforpotentialapplication.

3.2.Imageprocessingbasedonnonlinear-diffusionPDEmodels

Asuccessfulexampleofadifferentstrategyhasbeendemonstratedrecentlyinthefieldofimageprocessingandcomputervision,wherenonlinear-diffusionequationswereappliedtoimageprocessing(noisesuppression,edge-enhancement,etc)[16].ThenonlinearPDEmodels(couplednonlinear-diffusionequations)wereobtainedbyminimizingapredefinedcost-functional.Thiscost-functionalaccommodatestheparticularrequirementsoftheprocessedimageu(r,t)(imagesmoothness,fidelitytotheinputimageg(r),preservationofimage-edges,etc).Thecontinuousgradientdescentprocedureappliedforcost-functionaloptimizationresultedinthefollowingtypeofcouplednonlinear-diffusionequations[25]:∂∂t

u(r,t)=∇2⊥u(r,t)−Ku[1−v(r,t)]2[u(r,t)−g(r)],

(17)∂∂t

v(r,t)=Dv∇2⊥v(r,t)−cv(r,t)+Kv[1−v(r,t)]|∇u(r,t)|,(18)whereDv,Ku,c,andKvareadjustableparameters

(cost-functionalweightingcoefficients).ThePDEsystem

Nonlinearopticaltwo-dimensionalfeedbacksystems

(17),(18)tendstoconvergetoastablesolution—theprocessedimage—providingfornoiseremovalandedge-enhancement.Thespecificformofthecouplednonlinear-diffusionequationsdependsonthechosencost-functional.Comparingequations(17),(18)withthepreviouslydiscussedmodelsindicatesanimportantdifference.TheRHSofthesystem(17),(18)containsbasicarithmeticaloperationsappliedtofunctionsu(r,t)andv(r,t)(subtractionandmultiplication),andalsoincludescalculationofthegradientmodules|∇u(r,t)|.Theseoperations,rathersimplefordigitalsignalprocessing,presentachallengingtaskforopticalimplementation.Theexamplepresentedhereillustratestosomeextentarathertypicalsituation.Ifwedepartfromaninformationprocessingproblemstatementandderiveacorrespondingnonlinearsystemofequationshavingadesirablesteadystatesolution(processedimage,etc),wealmostcertainlyobtainaPDEsystemthatcannotbeimplementedoptically.Theoppositeisalsotrue:ifourmajorconcernisthestudyofnonlinearspatio-temporaldynamicsbasedonopticalimplementation(LCLV-based2D-feedbacksystem,forexample)wehaveahardtimefindingapracticalproblemthatcanbesolvedusingmodelsavailableforopticalrealization.Ontopoftheopticalnonlinearprocessingdrawbacksalreadymentionedwecanaddafewmore:typically,conventionalopticalsystemsarebulky,heavy,vibration-sensitiveandratherexpensive.3.3.Opticalcomputations:newpotentials

Arationalquestionarisesfromtheabovediscussions:whydoweneednonlinearopticalprocessingifpowerfuldigitalcomputersandwell-developeddigitalsignalprocessing(DSP)toolsareavailable?Thereareseveralmotivationstocontinuesearchingforopticalprocessingtechniquesbasedonnonlinearspatio-temporaldynamics,andinsomesensethisistherighttimetodoso.DespitethesimplicityandflexibilityofDSP,solutionofnonlinearPDEsrequirestime-consumingcalculationsandcannotbedoneatreal-timeframerates.TosuccessfullycompetewithDSP,opticsshouldofferfast,inexpensive,flexible,compactandlowpowerarchitectures.Thereisnowaywecanmakethesedemandsrealifwemaintainopticalprocessingpurityasstatedbytheformula:everythingshouldbedoneoptically.Tochallengethedigitalworlditisimportanttohavetherightallies,andthesealliesareopticalmicro-electro-mechanicalsystems(MEMS)andmixed-mode(analogueanddigital)verylargescaleintegration(VLSI)system-basedtechnologyfornonlinearsignalprocessing.3.4.OpticalMEMS

Imagineamachinesosmallthatitisimperceptibletothehumaneye.Imagineworkingmachineswithgearsnobiggerthanagrainofpollen.Imaginethesemachinesbeingbatchfabricatedtensofthousandatatime,atacostofonlyafewpennieseach.Imaginearealmwheretheworldofdesignisturnedupsidedown,andtheseeminglyimpossiblesuddenlybecomeseasy.Aplacewheregravityandinertiaarenolongerimportant,buttheeffectsofatomicforcesandsurfacesciencedominate.

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Figure1.Actuatedmicro-electro-mechanicalmirrorwith

continuouspositionadjustment(scaleunitof100µmisshownbythewhiteline)[28].

Welcometothemicrodomain,aworldnowoccupiedbyanexplosivenewtechnologyknownasMEMS(MicroElectroMechanicalSystems)or,moresimplymicromachines...[26].

ThisquotedromanticintroductiontoMEMStechnologyreflectstheenthusiasmoftheparticipantsintheongoing‘secondsiliconrevolution.’TounderstandthesignificanceofMEMSforopticsweshouldjustreplacetheword‘machine’inthegivenquotewith‘opticalsystems’.MEMStechnologyallowsthefabricationofrathercomplicatedopticalsystemscomposedoflenses,mirrors,beamsplitters,lasers,andarraysofphoto-detectors,alongwithmicroactuatorsandXYZmicropositioners,onasinglesiliconchip[27].AsanexampleofopticalMEMS,anactuatedsilicon-micromachinedmicromirrorisshowninfigure1[28].Withelectrostaticcombactuatorsintegratedonthesamechip,continuousandhighlyaccurate(betterthan0.2µm)positionadjustmentsofthesemicromirrorsareobtained.OpticalMEMScanmakeopticalprocessingfast,inexpensive,compact,andlowpower.

3.5.Spatiallightmodulatorsonasiliconchip

Nonlinearoptical2Dinformation(image)processingusingopticalMEMSrequiresinformationtoolsfordatainputcompatiblewithsilicontechnology.Recentlydevelopedliquidcrystal(LC)-on-VLSIspatiallightmodulatorsexcellentlymatchopticalMEMSinprovidinghigh-resolutionandrelativelyfast2Dinformationinputtoacoherentopticalsystem[29].ThetypicalLC-on-VLSIspatiallightmodulator(SLM)issimilartoaDRAM(SRAM)memorychip,withathinlayerofLC(nematicorferroelectric)indirectcontactwiththechip’smemory.Memorycellsareelectricallyconnectedtometalmirrors.AlayerofLCissandwichedbetweenthearrayofmirrorsandapieceofglasswithaconductiveelectrode.ThevoltageappliedtoeachLCcellisdeterminedbytheinformationstoredinthechip’smemorycell.ThesevoltagesdrivetheoverlyingLCprovidingincidentlightphase(orpolarization)modulation[30].TheLC-on-VLSIspatiallightmodulator

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performsthesameroleastheLCLVdoesinourpreviouslydiscussedsystems,exceptthatwavefrontmodulationiscontrolledelectronicallyanddependsoninformationsenttothechip’smemory.CurrentlyavailableanalogueLC-on-VLSIphaseSLMscanprovideupto512×512resolutionwithafullframeloadingspeedofabout100µsandaphase-modulationdepthofabout0.6µm[31].Thearraysizeisabout8×8mmandthepixelpitchis15µm.

3.6.AnalogueVLSIsystemsforsensoryinformationprocessing

Rapidadvancesinthenumberoftransistorsincorporatedontoasinglesiliconchipandnewdesignandsynthesistechniqueshaveresultedintheappearanceofintegratedcircuitshavinghighsignalprocessingcapabilityonthechip.VLSIimagerswithover500000transistorsarenowcapableofperformingrathercomplicatedanalogueon-chipimageprocessingincludingamplification,imagesubtraction,spatialaveraging,ratiocalculation,edgeenhancement,andlocalgaincontrolallinreal-time[32].Inconjunctionwiththedynamicalmodelsdiscussedhere,considerthefollowingcontinuous-timepixel-dynamicsequation(node-equation)whichcanbeimplementedonanalogueVLSIsystems:τd

dt

u(rij,t)+u(rij,t)=F[u(rij,t),Iin(rij,t),v(rij,t)],(19)

whereu(rij,t)isthedynamicalstateofanindividualpixel(node)withcoordinaterij,(i,jarenodeindices),andτisVLSIsystemresponsetime.DependentonVLSIsystemimplementation,thedynamicalvariableu(rij,t)maybeacharge,voltageorcurrentatthenode.In(19)Iin(rij,t)areinputphotocurrentsfromtheimager’sphotoarray,v(rij,t)areexternalinputsandFisanonlinearfunction.UsingtranslinearMOS-basedcircuitsavarietyofnonlinearsignalprocessingfunctionscanbeimplementedatthepixellevelwithacharacteristicanaloguecalculationrateoftheorderofmicroseconds[33].Pixeldynamicscanalsoincludetheinteractionbetweenneighbouringpixels,providinganaloguelocalspatialaggregation-averaging(diffusiveorresistivenetworks)[32,34].Thusamoregeneralmodelfordynamicsofthepixelnetworkmaybedescribedbythenonlinearequationcontainingaconvolutionintegral:τd

dtu(r󰀁,t)+u(r,t)=

F[u(r󰀌,t),Iin(r󰀌,t),v(r󰀌,t)]ρ(r󰀌−r)d2r󰀌.

(20)

Tosimplifynotationwehaveusedthespace-continuousmodel.Thefunctionρ(r)in(20)describeslocalspatialaveraging.InanalogueVLSIsystemsρ(r)canbeapproximatedbyaGaussianfunctionofwidtha:ρ(r)=exp(−ρ2/a2).

3.7.Opto-electronicmodelsfornonlinear2D-feedbacksystems

Considertheopto-electronicsysteminfigure2thatservesasageneralmodelforthenonlinearimageprocessingsystemdiscussedbelow.Thesystemconsistsofahigh-resolutionphasespatiallightmodulator,forexamplethe

Figure2.Genericmodelforanonlinear2D-feedback

opto-electronicsystem.

LC-on-VLSIphaseSLM,andaphotoarray(VLSIimager)opticallymatchedwiththephasemodulatorinthesensethatthephotoarrayhasthesamesizeandpixelgeometry.TheSLMandphotoarrayarecoupledthroughcoherentwavediffractionovertheshortdistanceL.AnanalogueVLSIchipinterfaceswithboththeSLMandphotoarraytoprovideprogrammable2D-feedback.Dependingontheproblem’scomplexity,feedbackcomputationmaybeperformedontheimagerchip.InthiscasetheVLSIimageriscoupleddirectlywiththephaseSLM.UsingadvancedopticalMEMSandVLSItechnologies,thissystemcanpotentiallybeimplementedasanintegratedmicromachinedopto-electronicdevicethatincorporatesbothopticalandanalogueVLSIelectroniccounterparts.

Thesystem’smathematicalmodel(continuousform)includesthefree-spacepropagationequationfordiffractionofthecoherentwaveA(r,t):−2ik

∂A(r,z,t)2

∂z

=∇⊥A(r,z,t),

(0󰀁z󰀁L)

A(r,0,t)=I1/2

in(r)exp{i[u(r,t)+ϕ(r)]},

(21)

andtheequationdescribingfeedbacksignalprocessingperformedontheanalogueVLSIchip:τ

d

dtu(r󰀁,t)+u(r,t)=

F[u(r󰀌,t),IFR(r󰀌,t),v(r󰀌,t)]ρ(r󰀌−r)d2r󰀌,(22)

IFB(r,t)=|A(r,z=L,t)|2.

Hereu(r,t)isthewavefrontmodulationcomponentintroducedbythephaseSLM,andIin(r)andϕ(r)aretheinputwaveintensityandwavefrontmodulation,respectively.TheintensitydistributionIFB(r,t)isregisteredbythephotoarrayandusedasaninputfortheVLSIfeedbacksystem.Nextweconsiderspecificexamplesofopto-electronicsignalprocessingbasedonthedynamicalmodel(21),(22).

Nonlinearopticaltwo-dimensionalfeedbacksystems

4.Informationprocessingwithnonlinear2D-opto-electronicfeedbacksystems

4.1.Opto-electronicKerr-slice/feedback-mirrorsystem:adaptivephasedistortionsuppression

High-resolutionadaptive(real-time)wavefrontphasedistortioncompensationisanimportanttaskforanumberofapplications,forexampleinastronomy,lasercommunicationsandmedicine.In[7,18]itwasshownthatwavefrontphasedistortioncompensationcanbeobtainedusingthenonlinearspatio-temporaldynamicsofaKerr-slice/feedback-mirrortypemodel(7)–(9).Opto-electronicimplementationoftheKerr-slice/feedback-mirrortypemodelcanbeobtainedusingthebasicsystemarchitectureinfigure2.ByneglectingthediffusiontermfornodedynamicsintheKerr-slice/feedback-mirrormodel(7)–(9),insteadof(22)wehave:

τd

dt

u(r,t)+u(r,t)=KIFB(r,t),

(23)whereK<0isthefeedbackgaincoefficient.Accordingly,themodelrequiresrelativelysimplefeedbacksignalprocessingatthepixellevel.PhasedistortioncompensationspectralrangeiscontrolledthroughchoiceofthediffractionlengthL,ascompensationefficiencydependsonthefeedbackgaincoefficient.Thequalityofphasedistortioncompensationimproveswithgainincrease.However,spatio-temporalinstabilitiesareasignificantproblemandoccurwhen1

thecoefficient|K|exceedsthethresholdvalueKth=[1+sin(q2L/k)]−1,whereqiswavefrontphasespatial2distortionfrequency[7,18].Topreventinstabilities,morecomplicatedanalogueVLSIfeedbacksignalprocessingthatincludeslocalspatialaveragingbasedonlinearresistiveorMOStranslinearnetworkcircuitarchitecturecanbeused.Fornodedynamicswehavethefollowingmodel[7]:

τd

󰀁

dt

u(r,t)+αu(r,t)=KIFB(r󰀌,t)ρ(r󰀌−r)d2r󰀌,(24)

whereαisacoefficientcontrollingthecharacteristicsystemconvergencetime.Localspatialaveragingwithinthewindow-functionρ(r)=exp(−ρ2/a2)behavessimilartolow-passfeedbacksignalspatialfiltering.ThewidthoftheGaussian-typewindow-functionacanbechosenlargeenoughtosuppressinstabilities.Thenonlinearopto-electronicsystemdescribedbyequations(21),(24)providesefficientwavefrontphasedistortioncompensationforaninputwavehavingauniformintensitydistribution[Iin(r)=const].Non-uniformityintheinputintensitydistributioncanbetakenintoaccountbyusingmodifiedanaloguesignalprocessing.Thecorrespondingequationreads[35]τd

dtu(r󰀁,t)+αu(r,t)=K

[IFB(r󰀌,t)−Iin(r󰀌)]ρ(r󰀌−r)d2r󰀌.

(25)

Numericalresultsofhigh-resolutionphasedistortioncompensationbasedonparallelopto-electronicsignalprocessingasdescribedbythemodel(21),(25)areshowninfigure3[35].Dynamicalprocessconvergencetypicallyoccursduringatimeintervalof3–5τ.ExistingCMOStechnologyissufficienttobuildafeedbackVLSIimagerto

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Figure3.Realizationsofthedistortedinputwavefront(a),and

wavefrontcorrectedbythenonlinearopto-electronicsystemoutput(b).WavefrontdistortioncorrespondstotheKolmogorovspatialspectrum.

performpixeldynamicsinaccordancewithequation(25),providingaresolutionontheorderof200×200pixelsandanoperationalspeedofafewkHz.Notethatthedynamicalmodel(21),(25)isamodifiedversionoftheclassicalopticalKerr-slice/feedbackmirrormodelanditsLCLV-based2Ddiffractivefeedbacksystemimplementation.Despitethesesimilarities,itwouldbeachallengingtasktorealizethesemodifiednonlineardynamicsusingpurelyopticalmethods.4.2.Imagingprocessingforreal-timemotiondetectionThegeneralarchitectureshowninfigure2foranopto-electronic2D-feedbacksystemcanbeeasilymodifiedforthereal-timedetectionofmovingobjects[36].Thesystemschematicisshowninfigure4.AsetofimageframesfromacameraissenttothephaseSLM.TheSLMconvertstheinputimagesIim(r,t)intothecoherentwavephase-modulationϕ(r,t)=γIim(r,t),whereγisthephase-modulationdepthcoefficient.Thisphasemodulationisconsideredastheinputwavefrontdistortion,whichcanbecompensatedusingtheadaptivesystemdiscussedabove.Compensationoccursonlyforstationary-stateorrelativelyslowlychangingimages.Byelectronicallycontrollingthecharacteristicdynamicalprocessconvergencetimeαonecanadjustthesystemdynamicsinordertodetectnon-stationaryimageelementsintheinputimagestream.Asaresult,fort󰀄α−1thestationary-statefeedbacksignalδ(r,t)=ϕ(r,t)+u(r,t)approachesauniformintensitydistributionleadingtocontrastloss.Asshowninfigure5thelossofcontrastleastimpactsmovingcomponentsoftheinputimage,whichareclearlyseenontheblurredbackground.Theopto-electronicsystemwith2D-feedbackdescribedbythesystemofequations(21),(24)withϕ(r,t)=γIim(r,t)thusenablesdetectionofmovingobjects.

4.3.Opto-electronicmodelsforthenonlineardiffusionequation

Considerthepossibilitiesforopto-electronicimplementationofthenonlinear-diffusiontypemodels.Asshownintheappendix,diffractionofthepurelyphase-modulatedcoherentwaveA(r,0,t)=A0exp{iu(r,t)}overtheshortdistanceLresultsintheintensitydistribution

Id(r,t)∼=I0[1−L/(kb2)∇2

⊥

u(r,t)],(26)

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Figure4.Opto-electronic2D-feedbacksystemformovingobject

detection.ThesignalcontrollingthephaseSLMI󰀳(r,t)iscomposedofthefeedbacksignalIc(r,t)andtheinputimageIim(r,t)[u(r,t)=γIc(r,t)].

Figure5.DetectionofamovingcarasseenatthemonitorMinfigure4:(a)singleinputimageframeIim(r,t);(b)outputimageI󰀳(r,t=1.5τ)forasinglestationary-stateinputimageframeIim(r);(c)and(d)outputimagesforinputimagestreamcorrespondingtoamovingcar,(c)I󰀳(r,t=1.5τ)and(d)I󰀳(r,t=3.5τ).

whichisproportionaltotheLaplacianoperatorappliedtothephase-modulationfunctionu(r,t).In(26)I0=A2uniformintensitydistributionandthecoordinates{x,y0isa}arenormalizedbythepixelsizeb.ThisapproximationisvalidforrelativelyshortdiffractivedistanceswhereL/(kb2)󰀅1.FortypicalLC-on-VLSIspatiallightmodulatorswithapitchsizeof∼20µmtosatisfythisconditionthepropagationdistanceshouldbeontheorderof2–4mm.Thisopensthepotentialforintegrated(on-chip)implementationoftheLaplacianoperatorcalculation.BecauseoftheshortdiffractivedistanceboththephaseSLMandphotoarray(coupledwiththeSLMthroughwavediffraction)canbe

Figure6.Opto-electronicsystemforanaloguenonlinear-diffusionequationmodelling.

Figure7.Real-timeimageedge-detectionsystembasedon

phase-imagediffraction.

placedonthesamesiliconchipandtheentiresystemcanbedesignedasanopticalMEMS.

Thesystemcanalsobeusedasanelementarybuildingblockforparallelopto-electroniccalculations(modelling)ofthepreviouslydiscussednonlinear-diffusionPDEs.Considertheschematicfornonlinear-diffusionequationimplementationshowninfigure6.AcoherentinputwaveA0passesthroughaphaseSLMdrivenbyananalogueVLSIsystem.Assumealineardependencebetweenthecontrolimage(signal)appliedtotheSLMandthephase-modulationu(r,t)introducedbythephaseSLM.Theintensitydistributionintheplaneofthephotoarray(planez=L)Id(r,t)isusedasaninputforanalogueVLSIsignalprocessing.Thissignalprocessingincludesthefollowinganaloguecalculationsatpixellevel:(a)subtractionofthemeaninputintensityI0;(b)calculationofthenonlinearfunctionf[u(r,t)];and(c)integrationovertimetofthesumofthesignalsf[u(r,t)]+[Id(r,t)−I0].Node-dynamicscanbedescribedbytheequationgivenhereintermsofphase-modulation:

τ

d

dt

u(r,t)=K[Id(r,t)−I0]+f[u(r,t)],(27)

Nonlinearopticaltwo-dimensionalfeedbacksystems

Figure8.Infraredimages:inputimage(a);imageprocessedbyedgedetectionsysteminfigure7(b);imageprocessedbynonlinearinterferometerinfigure9(c).

whereτisthecharacteristictimeoftheVLSIsystemsignalprocessing,andKisacoefficient.Usingapproximation(26)weobtain

τ

ddt

u(r,t)=Dd∇2⊥u(r,t)+f[u(r,t)],(28)

whereDd=KI0L/(kb2)istheeffectivediffusioncoefficient.Thuswehavearrivedatthenonlinear-diffusion-typeequation.Initialconditionscanbecreatedusinganadditionalexternalsignal(image)appliedtotheSLM’sinputatthemomentt=0.Combiningtwophase-imagediffractionsystemsinfigure6wecanobtainanopto-electronicsystemforanalogueparallelcomputationofthecouplednonlineardiffusionequations.AnalogueVLSIsystemscanprovideawiderangeofnonlinearitytypesaswellasanumberofparallelarithmeticoperations

R7

MAVorontsov

Figure9.Schematicfornonlinear2D-feedbackinterferometerforreal-timeparallelimageprocessing.

appliedto2Dsignals.Bothoptionsareratherdifficulttoimplementusingpurelyopticaltools.Theoperationalspeedoftheentireopto-electronicsystemislimitedmostlybythespeedoftheSLMandcouldpotentiallybeontheorderofseveralhundredframespersecond.Themaindrawbackofparallelanaloguecalculationsistherelativelylowcalculationaccuracy—typicallylessthanonepercent.

4.4.Real-timeimageedgeenhancementbasedonphaseimagediffraction

OpticalcalculationoftheLaplacianoperatorcanbeusedforreal-timedetectionofimageedges—agenericoperationforcomputervisionandautomatictargetrecognitiontechniques.Thesimpleopto-electronicedge-detectionsystemiscomposedofacameraconnectedwithahigh-resolutionphaseSLMandaphotoarraylocatedatadistanceLfromtheSLM(figure7(b)).Inaccordancewithexpression(26)diffractionofthephase-modulatedwave(phaseimagediffraction)overtheshortdistanceLresultsintheintensitydistributionId(r,t)∼=I0[1−

22

L/(kb)∇⊥Iim(r,t)]proportionaltotheLaplacianoperatorappliedtotheinputimageframeIim(r,t).Signalprocessingincludessubtractionofthemeanimagecomponentandgaincontrol.Anexampleofopticaledge-imagedetectionfortheIRimageframeisshowninfigure8.Real-timedetectionofimageedgescanberealizedusinganumberofopticalimageprocessingsystems,forinstanceFourierfiltering,andholographicandnonlinearopticstechniques[29].Theopto-electronicsystemshowninfigure7hasperhapsthesimplestopticalimplementation—free-spacepropagationofaphasemodulatedwave.

4.5.Imagesegmentationusinganonlinear

interferometerwith2D-opto-electronicfeedbackNonlineardynamicsofthenonlinearinterferometer(equation(6))canbeimplementedusingtheopto-electronicalsystemshowninfigure9.Thesystem’sopticalpartrepresentsaMach–Zenderinterferometerhavingphase

R8

SLMsinbothlegs.Onephasespatiallightmodulator(SLM1)isdirectlyconnectedtoanimagingcameraandthesecondone(SLM2)isusedtointroducethecontrollablephase-modulationu(r,t).ThephotoarrayP1registerstheinterferencetermIout=Iin{1+γcos[u(r,t)+ϕ(r)+󰀞]},wherethephase-modulationcomponentsareϕ(r,t)=γIim(r,t)andu(r,t)=γIc(r,t),and󰀞isaconstant

2

phaseshift.TocreatethediffusiontermD∇⊥u(r,t)theapproachbasedonphaseimagediffractiondiscussedaboveisused.Forthis,photoarrayP2islocatedadistanceLfromSLM2.Thisphotoarrayregisterstheintensitydistribution

2

Id(r,t)proportionaltotheLaplacianoperatorD∇⊥U(r,t).Throughsignalprocessingthefollowingnode-dynamicsareperformed:

∂

u(r,t)=µId(r,t)+K{1+γcos[u(r,t)+ϕ(r)+󰀞]},∂t

(29)

whereµ,󰀞,τandKareconstantcoefficients.The2D-feedbackinterferometershowninfigure9canbeusedforanalogueimageprocessingandadaptiveopticsapplications.Withthesethepartofthesystemrelatedwithimplementationofthediffusionterm,aswellasthephotoarrayP2infigure9,canbeomitted.Experimentalresultsofadaptivephasedistortioncompensationusingthe2D-feedbackinterferometerarepresentedin[37].Inthereferencedcasethefunctionϕ(r,t)correspondedtophaseaberrationsintroducedintoonelegoftheinterferometer.

Nonlineardynamicsoftheopto-electronic2D-feedbackinterferometercanbeusedforreal-timeparallelimageprocessing:edge-detectionandimagesegmentation.Equation(29)wasanalysedthroughnumericalsimulationusingasaninput(functionϕ(r,t))theIRimageshowninfigure8(a).Theobtainedstationary-stateoutputimagepresentedinfigure8(c)illustratesimageedgeenhancementandsegmentationofdifferentimageparts.τ

Appendix

Considerdiffractionofthepurelyphase-modulatedcoherentwaveA(r,z=0)=A0exp{iu(r)}overthedistance

L.RepresentthefieldcomplexamplitudeA(r,z)intheformA(r,z)=a(r,z)exp{iφ(r,z)},wherea(r,z)isthemodulusofthefieldcomplexamplitudeandφ(r,z)isaphasemodulation.Substitutethisexpressionintothefree-spacepropagationequation(21).Afterthesetofderivationsweobtain

−2k∂a∂z

=2(∇a∇φ)+a∇2⊥φ,(30a)

2ka∂φ2∂z

=∇⊥a−a[(∂φ/∂x)2+(∂φ/∂y)2].(30b)

ForaninputwavewithauniformintensitydistributionandarelativelyshortpropagationdistanceLwecanassumethat

|∇a|󰀅|∇φ|and(∇a∇φ)󰀅a∇2

⊥φelsewherealongthepropagationpath.Inthiscaseinsteadof(30)weobtain

k∂I/∂z=−I∇2⊥

φ,(31a)2k

∂φ

∂z

=(∂φ/∂x)2+(∂φ/∂y)2,(31b)

whereI=|a|2isintensitymodulationalongthepropagationdistance.Assumingλ|∇φ|󰀅1wecanneglectthephase-modulationchangecausedbywavepropagation;thatis,φ(r,z)≈φ(r,z=0)=u(r).Inthiscaseequation(31a)

hastheanalyticalsolutionI(r,z)=A20exp{−kz∇2

⊥u}.FortheshortpropagationdistanceLthefollowingapproximation

isvalid:I(r,z=L)=Id(r)∼=I0[1−L/(kb2)∇2

u(r)],wherethetransversecoordinatesarenormalized⊥

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