Optical tweezers computational toolbox

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Preprintof:

TimoA.Nieminen,VincentL.Y.Loke,AlexanderB.Stilgoe,GregorKn¨oner,AgataM.Bra´nczyk,NormanR.HeckenbergandHalinaRubinsztein-Dunlop“Opticaltweezerscomputationaltoolbox”

JournalofOpticsA9,S196-S203(2007)

Opticaltweezerscomputationaltoolbox

TimoANieminen,VincentLYLoke,AlexanderBStilgoe,

GregorKn¨oner,AgataMBra´nczyk,NormanRHeckenberg

andHalinaRubinsztein-Dunlop

CentreforBiophotonicsandLaserScience,SchoolofPhysicalSciences,The

UniversityofQueensland,BrisbaneQLD4072,Australia

Abstract.Wedescribeatoolbox,implementedinMatlab,forthecomputational

modellingofopticaltweezers.Thetoolboxisdesignedforthecalculationofoptical

forcesandtorques,andcanbeusedforbothsphericalandnonsphericalparticles,in

bothGaussianandotherbeams.Thetoolboxmightalsobeusefulforlightscattering

usingeitherLorenz–MietheoryortheT-matrixmethod.

1.Introduction

Computationalmodellingprovidesanimportantbridgebetweentheoryandexperiment—apartfromthesimplestcases,computationalmethodsmustbeusedtoobtainquantitativeresultsfromtheoryforcomparisonwithexperimentalresults.Thisisverymuchthecaseforopticaltrapping,wherethesizerangeoftypicalparticlestrappedandmanipulatedinopticaltweezersoccupiesthegapbetweenthegeometricopticsandRayleighscatteringregimes,necessitatingtheapplicationofelectromagnetictheory.Although,inprinciple,thesimplestcases—thetrappingandmanipulationofhomogeneousandisotropicmicrospheres—hasananalyticalsolution—generalisedLorenz–Mietheory—signi cantcomputationale ortisstillrequiredtoobtainquantitativeresults.Unfortunately,themathematicalcomplexityofLorenz–Mietheorypresentsasigni cantbarriertoentryforthenovice,andislikelytobeamajorcontributortothelaggingofrigorouscomputationalmodellingofopticaltweezerscomparedtoexperiment.

Ifwefurtherconsiderthecalculationofopticalforcesandtorquesonnon-sphericalparticles—forexample,ifwewishtoconsideropticaltorquesonandrotationalofnon-sphericalmicroparticles,themathematicaldi cultyisconsiderablygreater.onethee cientmethodsforcalculatingopticalforces

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andtorquesparticlesopticaltrapscloslyalliedtoMietheory—theT-matrixmethod(Waterman1971,Mishchenko1991,Nieminenetal.2003a).However,whiletheMiescatteringcoe cientshavearelativelysimpleanalyticalform,albeitinvolvingspecialfunctions,theT-matrixrequiresconsiderablenumericale ortforitscalculation.ItisnotsurprisingthatthecomprehensivebibliographicdatabaseoncomputationallightscatteringusingtheT-matrixmethodbyMishchenkoetal.(2004)listsonlyfourpapersapplyingthemethodtoopticaltweezers(Bayoudhetal.2003,Bishopetal.2003,Nieminen,Rubinsztein-Dunlop,Heckenberg&Bishop2001,Nieminen,Rubinsztein-Dunlop&Heckenberg2001).Sincethecompilationofthisbibliography,otherpapershaveappearedinwhichthisisdone(Nieminenetal.2004,Simpson&Hanna2007,Singeretal.2006),buttheyarefewinnumber.

Sincethepotentialbene tsofpreciseandaccuratecomputationalmodellingofopticaltrappingisclear,bothforsphericalandnon-sphericalparticles,webelievethatthereleaseofafreely-availablecomputationaltoolboxwillbevaluabletotheopticaltrappingcommunity.

Wedescribesuchatoolbox,implementedinMatlab.Weoutlinethetheoryunderlyingthecomputationalmethods,themathematicsandthealgorithms,thetoolboxitself,typicalusage,andpresentsomeexampleresults.Thetoolboxcanbeobtainedathttp://www.physics.uq.edu.au/people/nieminen/software.htmlatthetimeofpublication.Sincesuchsoftwareprojectstendtoevolveovertime,andwecertainlyintendthatthisonewilldoso,potentialusersareadvisedtochecktheaccompanyingdocumentation.Alongtheselines,wedescribeourplansforfuturedevelopment.Ofcourse,wewelcomeinput,feedback,andcontributionsfromtheopticaltrappingcommunity.

2.Fundamentals

Theopticalforcesandtorquesthatallowtrappingandmanipulationofmicroparticlesinbeamsoflightresultfromthetransferofmomentumandangularmomentumfromtheelectromagnetic eldtotheparticle—theparticlealtersthemomentumorangularmomentum uxofthebeamthroughscattering.Thus,theproblemofcalculatingopticalforcesandtorquesisessentiallyaproblemofcomputationallightscattering.Insomeways,itisasimpleproblem:theincident eldismonochromatic,thereisusuallyonlyasingletrappedparticle,whichis niteinextent,andspeedsaresomuchsmallerthanthespeedoflightthatwecanformostpurposesneglectDopplershiftsandassumewehaveasteady-statemonochromaticsingle-scatteringproblem.

Althoughtypicalparticlesinconvenientlyareofsizeslyingwithinthegapbetweentheregimesofapplicabilityofsmall-particleapproximations(Rayleighscattering)andlarge-particleapproximations(geometricoptics),theparticlesofchoiceareoftenhomogeneousisotropicspheres,forwhichananalyticalsolutiontothescatteringproblemisavailable—theLorenz–Miesolution(Lorenz1890,Mie1908).Whilethe

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applicationofLorenz–Mietheoryrequiressigni cantcomputationale ort,themethodsarewell-known.

Thegreatestdi cultyencounteredresultsfromtheincidentbeambeingatightlyfocussedbeam.Thewasdevelopedforscatteringofplanewaves,anditsextensiontonon-planeilluminationisusuallycalledgeneralisedLorenz–Mietheory(GLMT)(Gouesbet&Grehan1982)whichhasseensigni cantuseformodellingtheopticaltrappingofhomogeneousisotropicspheres(Renetal.1996,Wohlandetal.1996,MaiaNeto&Nussenzweig2000,Mazollietal.2003,Lock2004a,Lock2004b,Kn¨oneretal.2006,Nevesetal.2006).ThesamenameissometimesusedfortheextensionofLorenz–Mietheorytonon-spherical,butstillseparablegeometriessuchasspheroids(Han&Wu2001,Hanetal.2003).

Thesourceofthedi cultyliesintheusualparaxialrepresentationsoflaserbeamsbeingsolutionsofthescalarparaxialwaveequationratherthansolutionsofthevectorHelmholtzequation.Ourmethodofchoiceistousealeast-squares ttoproduceaHelmholtzbeamwithafar- eldmatchingthatexpectedfromtheincidentbeambeingfocussedbytheobjective(Nieminenetal.2003b).

Atthispoint,wecanwritetheincident eldintermsofadiscretebasissetof

(inc)functionsψn,wherenismodeindexlabellingthefunctions,eachofasolutionoftheHelmholtzequation,

Uinc=∞ n(inc)anψn,(1)

whereanaretheexpansioncoe cientsfortheincidentwave.Inpractice,thesummustbetruncatedatsome nitenmax,whichplacesrestrictionsontheconvergencebehaviourofusefulbasissets.Asimilarexpansionispossibleforthescatteredwave,andwecanwrite

Uscat=∞ kpkψk(scat),(2)

wherepkaretheexpansioncoe cientsforthescatteredwave.

Aslongastheresponseofthescatterer—thetrappedparticleinthiscase—islinear,therelationbetweentheincidentandscattered eldsmustbelinear,andcanbewrittenasasimplematrixequation

pk=∞ nTknan(3)

or,inmoreconcisenotation,

P=TA(4)

whereTknaretheelementsoftheT-matrix.ThisisthefoundationofbothGLMTandtheT-matrixmethod.InGLMT,theTTisdiagonal,whereasfornon-sphericalparticles,itisnot.

Whenthescattereris niteandcompact,themostusefulsetofbasisfunctionsisspherical(VSWFs)(Waterman1971,Mishchenko1991,Nieminen

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etal.2003a,Nieminenetal.2003b).SincetheVSWFsareadiscretebasis,thismethodlendsitselfwelltorepresentationofthe eldsonadigitalcomputer,especiallysincetheirconvergenceiswell-behavedandknown(Brock2001).

TheT-matrixdependsonlyonthepropertiesoftheparticle—itscomposition,size,shape,andorientation—andthewavelength,andisotherwiseindependentoftheincident eld.Thismeansthatforanyparticularparticle,theT-matrixonlyneedstobecalculatedonce,andcanthenbeusedforrepeatedcalculationsofopticalforceandtorque.Thisisthekeythatmakesthisahighlyattractivemethodformodellingopticaltrappingandmicromanipulation,sincewearetypicallyinterestedintheopticalforceandtorqueasafunctionofpositionwithinthetrap,evenifwearemerelytryingto ndtheequilibriumpositionandorientationwithinthetrap.Thus,calculationsmustbeperformedforvaryingincidentillumination,canbedonewiththeTThisprovidesasigni cantadvantageovermanyothermethodsofcalculatingscatteringwheretheentirecalculationneedstoberepeated.Thisisperhapsthethereasonthatwhileopticalforcesandtorqueshavebeensuccessfullymodelledusingmethodssuchasthe nite-di erencetime-domainmethod(FDTD),the niteelementmethod(FEM),orothermethods(White2000b,White2000a,Hoekstraetal.2001,Collettetal.2003,Gauthier2005,Chaumetetal.2005,Sunetal.2006,Wong&Ratner2006),thepracticalapplicationofsuchworkhasbeenlimited.

Since,asnotedabove,theopticalandtorquesresultfromdi erencesbetweentheincomingandoutgoing uxesofandangularcalculationoftheserequired.ThiscanbedonebyintegrationoftheMaxwellstresstensor,anditsmomentforthetorque,asurfacesurroundingtheparticle.Fortunately,intheT-matrixmethod,thebulkofthisintegralcanbeperformedanalytically,exploitingtheorthogonalitypropertiesoftheVSWFs.Inthisway,thecalculationcanbereducedtosumsofproductsoftheexpansioncoe cientsofthe elds.

Atthispoint,twocontroversiesinmacroscopicclassicalelectromagnetictheoryintrude.The rstoftheseistheAbraham–Minkowskicontroversy,concerningthemomentumofanelectromagneticwaveinamaterialmedium(Minkowski1908,Abraham1909,Abraham1910,Jackson1999,Pfeiferetal.2006).Thiscontroversyisresolvedforpracticalpurposesbytherealisationthatwhatisphysicallyobservableisnottheforceduetochangeintheelectromagneticmomentum,buttheforceduetotheThecontroversyisessentiallyoneofsemantics—whatportionofthetotalmomentumistobelabelled“electromagnetic”,andwhatportionistobelabelled“material”(Pfeiferetal.2006).approachcanbesummarisedascallingP/nctheelectromagneticmomentum ux,wherePisthepower,ntherefractiveindex,andcthespeedoflightinfreeThequantumequivalentisthemomentumofaphotoninamaterialmediumh¯k/n2=h¯0ontheotherhand,givesnP/castheelectromagnetic ux,orh¯k=nh¯k0photon.Thediscrepancyisresolvedbyrealisingthatthewaveofinducedpolarisationinthedielectriccarriesenergyandmomentum,equaltothedi erencebetweentheAbrahamandMinkowski

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pictures.ItissimplesttousetheMinkowskimomentum uxnP/c,sincethisisequaltothetotalmomentum ux.

Thesecondcontroversyistheangularmomentumdensityofcircularlypolarisedelectromagneticwaves(Humblet1943,Khrapko2001,Zambrini&Barnett2005,Stewart2005,Pfeiferetal.2006).Onthewecanbeginwiththeassumption

thattheangularmomentumdensityisthemomentofthemomentumdensity,r×(E/c,whichresultsinacircularlypolarisedplanewavecarryingzeroangularmomentuminthedirectionofpropagation.Ontheotherhand,wecanbeginwiththeLagrangianforanelectromagneticradiationandobtainthecanonicalstresstensorandanangularmomentumtensorthatcanbedividedintospinandorbitalcomponents(Jauch&Rohrlich1976).Foracircularlypolarisedplanewave,thecomponentoftheangularmomentum uxinthedirectionofpropagationwouldbeI/ω,whereIistheirradianceandωtheangularfrequency,indisagreementwiththe rstresult.Thedivisionoftheangularmomentumdensityresultingfromthisprocedureisnotgauge-invariant,anditiscommontotransformtheintegraloftheangularmomentumdensityintoagauge-invariantform,yieldingtheintegralofr×(E×H)/c.Jauch&Rohrlich(1976)carefullypointoutthatthistransformationrequiresthedroppingofsurfacetermsatin nity.Thereverseofthisprocedure,obtainingthespinandorbitaltermstartingfromr×(E×H)/c,involvingthesamesurfaceterms,hadalreadybeenshownbyHumblet(1943).Thecontroversythusconsistsofwhichofthetwopossibleintegrandstocalltheangulardensity.

However,theangularmomentumdensityassuchthatweareinterestedin,butthetotalangularmomentum uxthroughasphericalsurfacesurroundingtheparticle.Fortheelectromagnetic eldsusedinopticaltweezers,thisintegrated uxisthesameforbothchoicesofangularmomentumdensity.Crichton&Marston(2000)alsoshowthatformonochromaticradiation,thedivisionintospinandorbitalangularmomentaisgauge-invariant,andobservable,withitbeingpossibletoobtainthespinfrommeasurementoftheStokesparameters.Thetotalangularmomentum uxisthesameasthatresultingfromassumingadensityofr×(E×H)/c.Sincethetorqueduetospinisofpracticalinterest(Nieminen,Heckenberg&Rubinsztein-Dunlop2001,Bishopetal.2003,Bishopetal.2004),itisworthwhiletocalculatethisseparatelyfromthetotaltorque.

3.Incident eld

Thenaturalchoiceofcoordinatesystemforopticaltweezersissphericalcoordinatescenteredonthetrappedparticle.Thus,theincomingandoutgoing eldscanbeexpandedintermsofincomingandoutgoingvector(VSWFs):

Ein=

Eout=n=1m= nn∞ n∞ (2)anmM(2)nm(kr)+bnmNnm(kr),

(1)pnmM(1)nm(kr)+qnmNnm(kr).(5)(6)

n=1m= n

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Opticaltweezerscomputationaltoolbox

wheretheVSWFsare6

,2)(1,2)M(1(kr)Cnm(θ,φ)(7)nm(kr)=Nnhn ,2),2)nh(1(kr)h(1(kr)(1,2)nn(1,2)Bnm(θ,φ)Pnm(θ,φ)+Nnhn 1(kr) Nnm(kr)=krNnkr

,2)whereh(1(kr)arefunctionsofthe rstandsecondkind,Nn=nm(θ,φ),Cnm(θ,φ)=[n(n+1)] 1/2arenormalizationBnm(θ,φ)=r Ynmm(θ,φ)arethevectorsphericalharmonics(θ,φ)),andPnm(θ,φ)= rYn ×(rYn

(Waterman1971,Mishchenko1991,Nieminenetal.2003a,Nieminenetal.2003b),

mandYn(θ,φ)arenormalizedscalarsphericalharmonics.Theusualpolarsphericalcoordinatesareused,whereθistheco-latitudemeasuredfromthe+zaxis,andφistheazimuth,measuredfromthe+xaxistowardsthe+yaxis.

(1)(2)M(1)nmandNnmareoutward-propagatingTEandTMmultipole elds,whileMnm

SincetheseandN(2)nmarethecorrespondinginward-propagatingmultipole elds.

wavefunctionsarepurelyincomingandpurelyoutgoing,eachhasasingularityattheorigin.Since eldsthatarefreeofsingularitiesareofinterest,itisusefultode nethesingularity-freeregularvectorsphericalwavefunctions:

(2)RgMnm(kr)=1[M(1)nm(kr)+Mnm(kr)],(2)[N(1)RgNnm(kr)=nm(kr)+Nnm(kr)].2(8)(9)

Althoughitisusualtoexpandtheincident eldintermsoftheregularVSWFs,andthescattered eldintermsofoutgoingVSWFs,thisresultsinboththeincidentandscatteredwavescarryingmomentumandangularmomentumawayfromthesystem.Sinceweareprimarilyinterestedinthetransportofmomentumandangularmomentumbythe elds(andenergy,too,iftheparticleisabsorbing),weseparatethetotal eldintopurelyincomingandoutgoingportionsinordertocalculatethese.Theuserofthecodecanchoosewhethertheincident–scatteredorincoming–outgoingrepresentationisusedotherwise.

Weuseanpoint-matchingschemeto ndtheexpansioncoe cientsanmandbnmdescribingtheincidentbeametal.2003b),providingstableandrobustnumericalperformanceandconvergence.

Finally,oneneedstobeabletocalculatetheforceandtorqueforthesameparticleinthesametrappingbeam,butatdi erentpositionsororientations.ThetransformationsoftheVSWFsunderrotationofthecoordinatesystemortranslationoftheoriginofthecoordinatesystemareknown(Brock2001,Videen2000,Gumerov&Duraiswami2003,Choietal.1999).Itissu cientto ndtheVSWFexpansionoftheincidentbeamforasingleoriginandorientation,andthenusetranslationsandrotationsto ndthenewVSWFexpansionsaboutotherpoints(Nieminenetal.2003b,Doicu&Wriedt1997).Sincethetransformationmatricesforrotationandtranslationsalongthez-axisaresparse,whilethetransformationmatricesforarbitrarytranslationsarefull,themoste cientwaytocarryoutanarbitrarytranslationisbyacombinationofrotationandaxialtranslation.Thetransformationmatricesforbothrotationsandaxial

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translationscanbee cientlycomputedusingrecursivemethods(Videen2000,Gumerov&Duraiswami2003,Choietal.1999).

3.1.Implementation

Firstly,itisnecessarytoprovideroutinestocalculatethespecialfunctionsinvolved.Theseinclude:

(i)sbesselj.m,sbesselh.m,sbesselh1.m,andsbesselh2.mforthecalculationof

sphericalBesselandHankelfunctions.ThesemakeuseoftheMatlabfunctionsforcylindricalBesselfunctions.

(ii)spharm.mforscalarsphericalharmonicsandtheirangularpartialderivatives.(iii)vsh.mforvectorsphericalharmonics.

(iv)vswf.mforvectorsphericalwavefunctions.

Secondly,routinesmustbeprovidedto ndtheexpansioncoe cients,orbeamshapecoe cients,anmandbnmforthetrappingbeam.Theseare:

(i)bscpointmatchfarfield.mandbsc,describedin

(Nieminenetal.2003b),whichcancalculatetheexpansioncoe cientsforGaussianbeams,Laguerre–Gaussmodes,andbi-Gaussianbeams.Sincetheseroutinesaremuchfasterforrotationallysymmetricbeams,suchasLaguerre–Gaussbeams,aroutine,lgmodes.m,thatcanprovidetheLaguerre–Gaussdecompositionofanarbitraryparaxialbeamisalsoprovided.

(ii)bscplane.m,fortheexpansioncoe cientsofaplanewave.Thisisnotespecially

usefulforopticaltrapping,butmakesthetoolboxmoreversatile,improvingitsusabilityformoregenerallightscatteringcalculations.

Thirdly,thetransformationmatricesfortheexpansioncoe cientsunderrotationsandtranslationsmustbecalculated.Routinesinclude:

(i)wigner,implementingthealgorithmgivenbyChoietal.

(1999).

(ii)translate,implementingthealgorithmgivenbyVideen(2000).

4.T-matrix

Forsphericalparticles,theusualMiecoe cientscanberapidlycalculated.Fornon-sphericalparticles,amoreintensivenumericale ortisrequired.Weusealeast-squaresoverdeterminedpoint-matchingmethod(Nieminenetal.2003a).Foraxisymmetricparticles,themethodisrelativelyfast.However,asiscommonformanymethodsofcalculatingtheT-matrix,particlescannothaveextremeaspectratios,andmustbesimpleinshape.Typicalparticleshapesthatwehaveusedarespheroidsandcylinders,andaspectratiosofupto4givegoodresults.Althoughgeneralnon-axisymmetricparticlescantakealongtimetocalculatetheT-matrixfor,itispossibletomake

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useofsymmetriessuchasmirrorsymmetryanddiscreterotationalsymmetrytogreatlyspeedupthecalculation(Kahnert2005,Nieminenetal.2006).Weincludeasymmetry-optimisedT-matrixroutineforcubes.

ExpandingtherangeofparticlesforwhichwecancalculatetheT-matrixisoneofourcurrentactiveresearche orts,andweplantoincluderoutinesforanistropicandinhomogeneousparticles,andparticleswithhighlycomplexgeometries.

OncetheT-matrixiscalculated,thescattered eldcoe cientsaresimplyfoundbyamatrix–vectormultiplicationoftheT-matrixandavectoroftheincident eldcoe cients.

4.1.Implementation

OurT-matrixroutinesinclude:

(i)tmatrixmie.m,calculatingtheMiecoe cientsforhomogeneousisotropicspheres.(ii)tmatrixpm.m,ourgeneralpoint-matchingT-matrixroutine.

(iii)tmatrixpm,thesymmetry-optimisedcubecode.

5.Opticalforceandtorque

Asnotedearlier,theintegralsofthemomentumandangularmomentum uxesreducetosumsofproductsoftheexpansioncoe cients.Itissu cienttogivetheformulaeforthez-componentsofthe elds,asgiven,forexample,byCrichton&Marston(2000).Weusethesameformulaetocalculatethexandycomponentsoftheopticalforceandtorque,using90 rotationsofthecoordinatesystem(Choietal.1999).Itisalsopossibletodirectlycalculatethexandycomponentsusingsimilar,butmorecomplicated,formulae(Farsund&Felderhof1996).

The

axialtrappingQis

n∞ m2 Re(a Q=nmbnm pnmqnm)Pn=1m= nn(n+1)

n(n+2)(n m+1)(n+m+1)1 n+1(2n+1)(2n+3)

×Re(anma n+1,m+bnmbn+1,m

pnmp n+1,m qnmqn+1,m) (10)

inunitsofnh¯kperphoton,wherenistherefractiveindexofthemediuminwhichthetrappedparticlesaresuspended.ThiscanbeconvertedtoSIunitsbymultiplyingbynP/c,wherePisthebeampowerandcisthespeedoflightinfreespace.

Thetorquee ciency,ornormalizedtorque,aboutthez-axisactingonascattereris

τz=

n=1m= nn∞ m(|anm|2+|bnm|2 |pnm|2 |qnm|2)/P(11)

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Opticaltweezerscomputationaltoolbox

inunitsofh¯perphoton,where

P=

n=1m= n∞ n 9|anm|2+|bnm|2(12)

isproportionaltotheincidentpower(omittingaunitconversionfactorwhichwilldependonwhetherSI,Gaussian,orotherunitsareused).Thistorqueincludescontributionsfrombothspinandorbitalcomponents,whichcanbothbecalculatedbysimilarformulae(Crichton&Marston2000).Again,onecanconvertthesevaluestoSIunitsbymultiplyingbyP/ω,whereωistheopticalfrequency.

5.1.Implementation

Oneroutine,forcetorque.m,isprovidedforthecalculationoftheforce,torqueandspintransfer.Theorbitalangularmomentumtransferisthedi erencebetweenthetorqueandthespintransfer.Theincomingandoutgoingpower(thedi erencebeingtheabsorbedpower)canbereadilycalculateddirectlyfromtheexpansioncoe cients,ascanbeseenfrom(12).

6.Miscellaneousroutines

Anumberofotherroutinesthatdonotfallintotheabovecategoriesareincluded.Theseinclude:

(i)Examplesofuse.

(ii)RoutinesforconversionofcoordinatesandvectorsfromCartesiantosphericaland

sphericaltoCartesian.

(iii)Routinestoautomatecommontasks,suchas ndingtheequilibriumpositionofa

trappedparticle,springconstants,andforcemaps.

(iv)Functionsrequiredbyotherroutines.

7.Typicaluseofthetoolbox

Typically,foragiventrapandparticle,aT-matrixroutine(usuallytmatrix)willberunonce.Next,theexpansioncoe cientsforthebeamarefound.Dependingontheinterestsoftheuser,afunctionautomatingsomecommontask,suchas ndingtheequilibriumpositionwithinthetrap,mightbeused,ortheusermightdirectlyusetherotationandtranslationroutinestoenablecalculationoftheforceortorqueatdesiredpositionswithinthetrap.

Thespeedofcalculationdependsonthesizeofthebeam,thesizeoftheparticle,andthedistanceoftheparticlefromthefocalpointofthebeam.Evenforawidebeamandalargedistance,theforceandtorqueataparticularpositioncantypicallybecalculatedinmuchlessthanonesecond.

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(a)

(b)Figure1.Gaussiantrap(examplegaussian.m).ForceonasphereinaGaussian

beamtrap.Thehalf-angleofconvergenceofthe1/e2edgeofthebeamis50 ,

correspondingtoanumericalapertureof1.02.Theparticlehasarelativerefractive

indexequalton=1.59inwater,andhasaradiusof2.5λ,correspondingtoadiameter

of4.0µmiftrappedat1064nminwater.(a)showstheaxialtrappinge ciencyasa

functionofaxialdisplacementand(b)showsthetransversetrappinge ciencyasa

functionoftransversedisplacementfromtheequilibriumpoint.

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(a)

(b)

http://www.77cn.com.cnguerre–Gausstrap(example).ForceonasphereinaLaguerre–

Gaussbeamtrap.Thehalf-angleofconvergenceofthe1/e2outeredgeofthebeam

is50 ,asin gure1.Thesphereisidenticaltothatin gure1.(a)showstheaxial

trappinge ciencyasafunctionofaxialdisplacementand(b)showsthetransverse

trappinge ciencyasafunctionoftransversedisplacementfromtheequilibriumpoint.

ComparedwiththeGaussianbeamtrap,theradialforcebeginstodropo atsmaller

radialdisplacements,duethefarsideofthering-shapedbeamnolongerinteracting

withtheparticle.

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Figure3.Trappinglandscape(example).Themaximumaxialrestoring

forcefordisplacementinthedirectionofbeampropagationisshown,intermsofthe

trappinge ciencyasafunctionofrelativerefractiveindexandmicrospherediameter.

Thetrappingbeamisat1064nmandisfocussedbyanNA=1.2objective.Thistypeof

calculationisquiteslow,asthetrappingforceasafunctionofaxialdisplacementmust

befoundforagridofcombinationsofrelativerefractiveindexandspherediameter.

Attheleft-handside,wecanseethatthetrappingforcerapidlybecomesverysmallas

theparticlesizebecomessmall—thegradientforceisproportionaltothevolumeofthe

particleforsmallparticles.Intheupperportion,wecanseethatwhetherornotthe

particlecanbetrappedstronglydependsonthesize—forparticularsizes,re ectionis

minimised,andevenhighindexparticlescanbetrapped.

Morecomplextasksarepossible,suchas ndingtheopticalforceasafunctionofsomepropertyoftheparticle,whichcan,forexample,beusedtodeterminetherefractiveindexofamicrosphere(Kn¨oneretal.2006).

Figures1to4demonstratesomeofthecapabilitiesofthetoolbox.Figure1showsasimpleapplication—thedeterminationoftheforceasafunctionofaxialdisplacementfromtheequilibriumpositioninaGaussianbeamtrap.Figure2showsasimilarresult,butforaparticletrappedinaLaguerre–GaussLG03beam.Figure3showsamorecomplexapplication,withrepeatedcalculations(eachsimilartotheoneshownin gure1(a))beingusedtodeterminethee ectofthecombinationofrelativerefractiveindexandparticlesizeontrapping.Finally, gure4showsthetrappingofanon-spherical

光镊计算工具箱,由澳大利亚的著名研究人员开发。具有高超的matlab 编程技巧。看了准保你佩服。源程序另外下载

Figure4.Opticaltrappingofacube(examplecube.m).Asequenceshowingthe

opticaltrappingofacube.Thecubehasfacesof2λ/nmediumacross,andhasa

refractiveindexofn=1.59,andistrappedinwater.Sincetheforceandtorque

dependontheorientationaswellasposition,asimplewayto ndtheequilibrium

positionandorientationisto“release”thecubeandcalculatethechangeinposition

andorientationforappropriatetimesteps.Thecubecanbeassumedtoalwaysbe

movingatterminalvelocityandterminalangularvelocity(Nieminen,Rubinsztein-

Dunlop,Heckenberg&Bishop2001).Thecubebeginsface-up,centredonthefocal

planeofthebeam,andtooneside.Thecubeispulledintothetrapandassumes

acorner-uporientation.ThesymmetryoptimisationsallowthecalculationoftheT-

matrixin20minutes;otherwise,30hourswouldberequired.OncetheT-matrixis

found,successivecalculationsoftheforceandtorquerequirefarlesstime,ontheorder

ofasecondorso.

particle,acube.

Agreementwithprecisionexperimentalmeasurementssuggeststhaterrorsoflessthan1%areexpected.

光镊计算工具箱,由澳大利亚的著名研究人员开发。具有高超的matlab 编程技巧。看了准保你佩服。源程序另外下载

8.FuturedevelopmentWeareactivelyengagedinworktoextendtherangeofparticlesforwhichwecanmodeltrapping.Thiscurrentlyincludedbirefringentparticlesandparticlesofarbitrarygeometry.RoutinestocalculatetheT-matricesforsuchparticleswillbeincludedthemainwhenavailable.

Otherareasinwhichweaimtofurtherimprovethetoolboxarerobusthandlingofincorrectorsuspectinput,moreautomationoftasks,andGUItools.

Wealsoexpectfeedbackfromtheopticaltrappingandmicromanipulationcommunitytohelpusaddusefulroutinesandfeatures.

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