INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J.Phys.B:At.Mol.Opt.Phys.39(2006)S637–S650doi:10.1088/0953-4075/39/15/S11

Opticallyinducedinter-particleforces:fromthebondingofdimerstoopticalelectrostrictioninmolecularsolids

DavidLAndrews1,RichardGCrispandDavidSBradshaw

NanostructuresandPhotomolecularSystems,SchoolofChemicalSciences,UniversityofEastAnglia,NorwichNR47TJ,UKE-mail:david.andrews@physics.org

Received15February2006Published24July2006

Onlineatstacks.iop.org/JPhysB/39/S637

Abstract

ThequantumelectrodynamicalformulationoftheCasimir–Polderinteractioninvitesaconsiderationofopticallyinducedinter-particleforces,whichariseinthesamelevelofperturbationtheory.Fortheobservationofsucheffects,quantitativeassessmentsofthecouplingmechanismsuggestlevelsofintensitythatarenowroutinelyavailablefrompulsedlasers.Inthispaper,atheoreticalanalysisoftheprincipalmechanismisfollowedbyadevelopmentfortwospecificsystems,chosentoillustratethebroadsignificanceandtherangeofsystemsinwhichsucheffectsmightbemanifest.ThefirstsystemisavanderWaalsdimer,alooselyboundandessentiallyisolatedmolecularpairinwhichasmallopticallyinducedshiftintheequilibriumbondlengthprovestobewellwithinthelimitsofmeasurementbymicrowaverotationalspectroscopy.Thesecondsystemillustratestheoppositeextreme,whereanopticallyinducedmodificationoftheforcesbetweendenselypackedmoleculesinthecondensedphaseisshowntoproduceanisotropicpatternsoflaser-inducedcompressionandexpansion,aneffecttermedopticalelectrostriction.Again,suchaneffectshouldbereadilymeasurable,althoughthenecessaryconditionsaresuchthatanumberofsecondaryeffectsmightalsobelikelytoarise.Theexperimentalchallengeofprecludingthosesecondaryeffectsinthecondensedphaseandunequivocallyidentifyingopticallyinducedintermolecularforcesisdiscussed.Possibleapplicationsarealsoentertained,includingopticalactuatorsfornanoscaleelectromechanicalsystems.

1.Introduction

Oneofthemostwidelycitedandbest-knownsuccessesofquantumelectrodynamics,QED,liesinitsapplicationtothedispersioninteraction,theelectromagneticcouplingbetween

1

Seniorauthor.

S637

0953-4075/06/150637+14$30.00©2006IOPPublishingLtdPrintedintheUK

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0000DLAndrewsetal

ssrr0000ABAB(a)(b)Figure1.Exampletime-ordereddiagramsforcalculationsof(a)theCasimir–Polderpotential,adaptedfrom[6]and(b)thelaser-inducedpotentialpresentedinthiswork.

polarizableparticles[1,2].Inthecaseofmolecules,thisinteractionoperatesuniversally,inadditiontoanycouplingassociatedwiththedirectinteractionofchargesordipoles(orindeedanyothermultipoles).TheCasimir–Polderformulaisanexampleparexcellenceshowcasingmanifestationsofretardation—anintrinsicfeatureofanytheorybasedonQED.Indeed,thestrikingandoriginallyunsuspectedmodificationtothedistancedependenceofthedispersioninteraction,whoseasymptoticformchangesfromR−6atshortrangetoR−7atlargerdistances[3,4],wasrigorouslyestablishedseveraldecadesbeforereceivingunequivocalexperimentalproof[5].Avarietyofinterpretationscanbeplacedonthedifferentmodelsassociatedwithalternativemethodsforcalculatingthefinalequation;inaformulationbasedonthePower–Zienau–WoolleyHamiltonian,thedispersioninteractionemergesfromtime-dependentperturbativecalculationsbasedonthepropagationofvirtualphotons[6].

Perturbativecalculationsofinter-particlecouplingaregenerallyperformedonabasisstateinwhichbothparticlesandtheradiationfieldareinthegroundstate.Thissystemstatecoupleswithothershort-livedstatesinwhichtheelectromagneticfieldhasanon-zerooccupationnumberforoneormoreradiationmodes.Equationsfordiagonalmatrixelementsinvokingasinglevirtualphotonleadtotheretardation-freeenergyofinteractionbetweenstaticmultipoles,theleadingcontributionbeingthefamiliardipole–dipoleinteraction.Necessarilyinvolvingbothaphotoncreationandaphotonannihilationevent,thiscalculationentailssecond-orderperturbationtheory.Thedispersioninteraction,traditionallyinterpretedasacouplingbetweenmutuallyinducedmoments,emergesfromthenextorder—afourth-orderperturbativecalculationbasedontheexchangeoftwovirtualphotons,eachcreatedatoneparticleandannihilatedattheother—asinthetime-orderedgraphoffigure1(a).Thetwovirtualquantamay,butneednot,overlapintimeastheypropagatebetweenthetwounits.

Althoughthecorrectrepresentationoflong-rangeretardationfeaturessignifiesthemostobviousvindicationofquantumelectrodynamicalmethods,thevirtualphotoninterpretationhasvalidityatalldistancesanditlendsafreshperspectivetothephysicsinvolvedintheshorterrangeR−6rangedependence(longknownasthevanderWaalsinteraction—theattractivepartoftheLennard–Jonespotential,whichislargelyresponsibleforthecohesionofcondensedphasematter).Aconsiderationofthephotonicbasisforthisinteractionsuggeststhatothereffectsmightbemanifestwhenintenselightispresent,i.e.whencalculationsareperformedonabasisstateforwhichtheoccupationnumberofatleastonephotonmodeisnon-zero.Itisimmediatelyencouragingtorecognizethatthisneednotinvokehigherordersofperturbationtheoryinordertogiveanon-vanishingresult;inprincipletheannihilationandcreationofonephotonfromtheoccupiedradiationmodecansubstituteforthepairedcreation

Opticallyinducedinter-particleforcesS639

andannihilationeventsofoneofthetwovirtualphotonsinvolvedintheCasimir–Poldercalculation.Thecorrespondingtime-orderedrepresentationisshowninfigure1(b).

Itisclearthattheresultofanysuchcalculationofanopticallyconferredpairenergywillexhibitalineardependenceonthephotonnumberoftheoccupiedmode.Whencastintermsofexperimentalquantities,thiswillbemanifestedasanenergyshiftwithacorrespondingproportionalitytotheirradianceofthroughputradiation.Beyondthisfeature,detailedcalculationisnecessarytogaugethesignificanceoftheeffectincomparisontothedispersioninteraction—especiallysincesingle-photonoccupationofallradiationmodesisinprincipleaccommodatedforeachvirtualphoton,whereasjustonemodewithamuchlargeroccupationnumbermightbeinvolvedinrepresentingideallaserthroughput.Thefirstdetailedcalculationsofalaser-inducedpairenergy[7–12]gaveresultsthatwereencouraging,butonlyforlevelsofintensity(typicallymegawattspersquarecentimetre)thatwereatthetimedifficulttoachieve.Withsuchintensitiesbecomingroutinelyavailable,interesthasrecentlybeguntorefocusonthephenomenoninanumberofrecentpublications[13–19].Moredetailonthepresentexperimentalcontextandpotentialsignificanceisgivenintheconcludingsection.Itshouldhoweverbemadeclearfromtheoutsetthattheseforcesarepairwiseinteractionsofmatterwiththeradiation,quitedistinctfromthemorefamiliaropticaltweezer[20,21]oropticalmolassesforces[22,23]whichoperateonindividualparticles1.

Inthenextsection,weoutlineabasisforthederivationofageneralresultforopticallyinducedinter-particleforces,associatedwithaperturbativeenergyshift.Attentionisdrawntothewayinwhicharesultapplicabletomoleculesshouldbemodifiedwhenappliedtotheinteractionofopticallysuspendednanoparticles.Insection3,theapplicationstomolecularsystemsareexploredindetailandexemplifiedbycalculationsontwoextremecases.First,attentionisfocusedonthelooselyboundcomplexesknownasvanderWaalsdimers,forwhichitisdemonstratedthatlaserlightshouldproduceasmallbutsignificantextensioninthebondlength.Inthespecificcaseofahydrogencyanidedimer,theresultshouldbemeasurablebyrotationalspectroscopy.Thesecondapplicationistomolecularsolids,forwhichitisshownthateachpairwiseinteractionbetweenneighbouringmoleculesissubjecttoopticalforcesthataffectalocalmechanicaldeformation—alinearcompressionalongthepolarizationdirectionaccompaniedbyanexpansionintheperpendicularplane.Thus,aneffectofopticalelectrostrictionisidentifiedanditsmolecularoriginisdetermined.Insection4,arangeofsecondaryeffectsisdescribed,theexperimentalchallengeofdeterminingthesedistinctiveoptomechanicaleffectsisgivenarealisticappraisalandsomepossibledeviceapplicationsareidentified.2.Perturbativecalculations

WebeginbydefiningthecomponentsofasimplesystemcomprisingapairofparticlesAandB,eachwithdistinctelectronicintegrityandelectricalneutrality,togetherwiththeradiationfield.Asisappropriateforthesubsequentapplicationsdetailedspecifically,theterm‘molecules’willbeusedinthefollowingasagenericdescriptoroftheseparticles.RepresentingthesysteminquantumelectrodynamicaltermsintheCoulombgaugeensuresthatthecouplingfieldsaredulyretardedandsatisfycausality[24].Inmultipolarform,thesystemHamiltonianmayberepresented,thus

󰀃ξ󰀃ξ

Hmol+Hint+Hrad.(2.1)H=

ξ=A,B

1

ξ=A,B

Thelattercommonlyresultfromresponsetoanintensitygradientoranexchangeofmomentumwiththeradiation,

respectively.

S0

ξ

DLAndrewsetal

ξ

Here,Hmolisthefield-freemultipolarHamiltonianformoleculeξ,operatorHintrepresentstheinteractionofξwiththeradiationfieldandHradistheradiationHamiltonian.Thetripartitesimplicityofequation(2.1)specificallyresultsfromadoptionofthemultipolarformoflight–matterinteraction,basedonawell-knowncanonicaltransformationfromtheminimal-couplinginteraction[25–27].ThisprocedureresultsinaprecisecancellationfromthesystemHamiltonianofallCoulombicterms,savethoseintrinsictotheinternalstructureoftheHamiltonianoperatorsforthecomponentmolecules.Intheelectricdipoleapproximation,ξ

isgivenbyHint

󰀃ξ−1

Hint=−ε0µ(ξ)·d⊥(Rξ),(2.2)

ξ

withµ(ξ)andRξ,respectively,denotingtheelectric-dipolemomentoperatorandtheposition

vectorofmoleculeξ.Theoperatord⊥(Rξ),representingthetransverseelectricdisplacementfieldatthatlocation,isexpressibleinthefollowingmodeexpansioninvolvingsummationsoveropticalwavevectors,p,andpolarizations,ε:

󰀎󰀃󰀍h󰀋¯cpε01/2󰀊(ε)⊥

¯(ε)(p)a†(ε)(p)e−i(p·Rξ).e(p)a(ε)(p)ei(p·Rξ)−e(2.3)d(Rξ)=i

2Vp,εHere,e(ε)(p)isthepolarizationunitvector(¯e(ε)(p)beingitscomplexconjugate,theadmission

ofcomplexpolarizationsallowingforcircularorellipticalaswellasplanepolarization);Visanarbitraryquantizationvolumeanda(ε)(p),a†(ε)(p)are,respectively,thephotonannihilationandcreationoperatorsforaradiationmode(p,ε).

Asdiscussedintheintroduction,tosecureageneralresultfortheopticallyinducedenergyshift,󰀅E,andhencetheassociatedforcebetweenAandB,requirestheimplementationoffourth-orderperturbationtheory—sincetheinteractionentailsfourmolecule-radiationfieldcouplingevents:

󰀂󰀁

󰀃󰀆i|Hint|t󰀇󰀆t|Hint|s󰀇󰀆s|Hint|r󰀇󰀆r|Hint|i󰀇

.(2.4)󰀅E=Re

(E−E)(E−E)(E−E)itisirt,s,rHere,allstatesarethoseofthesystem,i.e.thetwomoleculesplustheradiationfield;|i󰀇is

theunperturbedsystemstateinwhichbothmoleculesareintheirelectronicgroundstate,|r󰀇,|s󰀇and|t󰀇arevirtualstatesandEnistheenergyofstate|n󰀇.Thelattersignifiesoneofthebasisstatesfortheperturbativedevelopment,expressibleintheform

|n󰀇=|moln󰀇|radn󰀇≡|moln;radn󰀇,

(2.5)

with|moln󰀇and|radn󰀇definingstatesofthemolecularpairandtheradiation,respectively.Inequation(2.4),eachoperationofHintonthestatetoitsrightaffectstheannihilationorcreationofaphoton,asfollowsfrom(2.2)and(2.3).Thelaser-inducedinteractioninvolvestheannihilationofathroughputphotonatonemoleculeandthestimulatedemissionofanequivalent‘real’photonintotheradiationmode;thisismediatedbyintermolecularenergytransferthroughavirtualphotoncreatedatonesiteandannihilatedattheother.Themoleculesandthethroughputradiationsuffernooverallchangeinstate;theterm‘real’appliedtophotonsoftheinputmodedenotesquantaofelectromagneticradiationwithapropagationtimethatislongcomparedtotheopticalcycleandcorrespondingly‘real’characteristics[28].Inperformingenergyshiftcalculationsbasedon(2.4),detailedrepresentationsofallcontributorytermsareprovidedbyasetof48time-ordereddiagrams,oneofwhichisexhibitedinfigure1(b).Whensuchalargenumberoftime-orderingsisinvolved,arecentlydevisedalternativebasedonstate−sequencediagrams[29]provesadvantageous.Allthetime-orderingsareinfactaccommodatedinjusttwostate-sequencediagrams(oneforthecase

Opticallyinducedinter-particleforcesS1

Figure2.State-sequencediagramforallcontributionstothelaser-inducedpotentialassociatedwithrealphotonabsorptionatAandstimulatedemissionatB,correspondingto24distincttime-orderings.Eachsystemstateisrepresentedbyaboxandtheprogressionofsystemstates,|i󰀇,|r󰀇,|s󰀇,|t󰀇,|i󰀇inequation(2.4),isreadfromlefttoright.WithineachboxthestatesofmoleculesAandBaredesignatedbyadjacentcircles,AontheleftandBontheright.Blackcirclesdenoteanelectronicgroundstate;white,avirtualintermediatestate.Photonsfromthelaserandvirtualradiationarerepresentedbyωandφ,respectively.Theboldpathwaycorrespondstotheillustrativetime-orderingdepictedinfigure1(b).

wheretherealphotonabsorptionoccursatAandthestimulatedemissionatB,theotherwheretheoppositeapplies),oneofwhichisshowninfigure2.

Theexplicitresultfor󰀅Efollowsthesubstitutionofequations(2.2)and(2.3)into

|radn󰀇,respectively—(2.4),recognizingthatµ(ξ)andd⊥(Rξ)operateon|moln󰀇and√

(ε)

thelatterthroughthefollowingexpressions:a(p)|n(p,ε)󰀇=n|(n−1)(p,ε)󰀇and

√

a†(ε)(p)|n(p,ε)󰀇=n+1|(n+1)(p,ε)󰀇.Detailsofthecompletecalculationaregiveninarecentpaper[16].Thefollowingresulttherebyemerges,conciselyexpressibleusingtheimpliedsummationconventionforrepeatedCartesiantensorindices:

󰀎󰀍

󰀊󰀋nh¯ck±AB

¯(λ)Reei(λ)eα(k)V(k,R)α(k)exp(ik·R).(2.6)󰀅E(k,R)=ijklljk

ε0V

Here,kandh¯ckdenotetheinputwavevectorandphotonenergy,respectively,andRis

±

theinter-particledisplacementvector,R≡RB−RA.AlsoVjksignifiesthefullyretardedresonanceelectricdipole–electricdipoleinteractiontensoroftheform[30]

󰀆󰀇󰀆󰀇󰀋e∓ikR󰀊±ˆk−(kR)2δjk−Rˆk,ˆjRˆjR1±ikR)δ(2.7)Vjk(k,R)=−3R(jk

4πε0R3andαij(k)isthedynamicpolarizabilitytensorgivenas

󰀁󰀂issiissi󰀃µiµjµjµiξ

αij(k)=+,

˜si−h˜si+hE¯ckE¯ck

s

ξ

(2.8)

˜xy≡E˜x−E˜y,thetildedenotingtheinclusionofdampingfactorswhereµxy=󰀆x|µ|y󰀇andE

asappropriate.

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Figure3.Pairoflinearmoleculesbondedend-to-end.Here,φsignifiestheanglebetweenthepolarizationvectoroftheincidentlaserradiationandtheintermolecularseparationvector.

Beforeproceedingfurther,wenotethattheaboveresultinprincipleholdsfortwoparticlesinavacuum,eachwithpropertiesexpressibleintermsofwavefunctionsthatextendoverthewholeparticle—asisessentiallythecaseforthemoleculesofagas.Inthecondensedphase—thecaseofmolecularsolidsor,forexample,particlesopticallytrappedinahostliquid,dissipativeandrefractivecorrectionsduetotheelectronicpropertiesofthelocalenvironmentshouldbeappliedtoboththeinter-particlecoupling(whichistherebycastintermsofvirtualpolaritonsratherthanphotons)andalsotheinteractionswiththeopticalbeam.Theprocedureforintroducingsuchcorrectionsisintricatebuttheoutcomeisknown[31,32]andthenecessaryreformulationoftheaboveresultproducesthefollowingequation:

󰀎󰀍2󰀎󰀍

󰀋nk+2󰀊(λ)(λ)󰀂A¯vkknε0h±󰀂B

¯eeχ(k)V(nk,R)χ(k)expik·R,󰀅E(k,R)=Re()ijkliljkk

3n3Vk

(2.9)whereχ󰀂denotesalinearsusceptibilitytensorscaledbythecorrespondingparticlevolume,vkisthegroupvelocityforthemediumatopticalfrequencyckandnkisthecorrespondingcomplexrefractiveindex.Althoughwedonotpursueithere,itcouldbeamatterofconsiderableinteresttofurtherdevelopthisequationfornanoparticlessuchasquantumdotswithdistinctivedispersioncharacteristics.3.Applications

Theconditionsthatexpeditesignificantinter-particleforces,givensuitablyintenselaserlight,arethosewherepairsofhighlypolarizableparticlesarefoundinreasonablycloseproximity.Inpreviouspublicationswehavefocusedoninteractionsbetweennanoparticles,inonecasespecificallyaddressingcarbonnanotubes[15].Thelatterareparticularlywellsuitedascandidatesforobservingandexploitinglaser-inducedpairforces,becauseoftheirexceptionalelectronicproperties.Theprospectsforidentifyingsimilareffectsinsmallermoleculesmightatfirstappearinauspicious,giventheroughscalingofpolarizabilitywithmolecularsizeandthequadraticdependenceofthelaser-inducedforcesonmolecularpolarizability.Therearetwopossibleapproachestoaddresstheproblem.Oneistochooseasystemthatisamenabletoextremelyhighresolutionmechanicalorspectroscopicanalysis,sothatverysmallgeometricadaptationstoanopticalpairforcecanbedetermined.Theotherapproachistoprobeasysteminwhichsmallmechanicaleffectsareamplifiedbyscale.Inthefollowing,weentertaindetailedexamplesofthesetwodistinctcases.3.1.VanderWaalsmolecules

VanderWaalsmoleculesareweaklybound,usuallydimericmolecularstructures.Havingsignificantlylargermomentsofinertiathantheirmonomerparents—thisdifferenceenhancedbytheunusuallylong‘bond’holdingthecomponentunitstogether—suchdimersarereadilyidentifiablebyhighresolutionmicrowavespectroscopy.Themodelsystemtobeexaminedinmoredetailbelowconsistsoftwolinearmoleculeslyingend-to-end(figure3),with

Opticallyinducedinter-particleforcesS3

theintermolecularseparationvectorRidentifiedwiththeZ-axisandtheplane-polarizedthroughputradiationdefinedbyφ,i.e.theanglebetweentheeandRvectors.Thepolarization

ˆ.Fromequation(2.6),andvectorcanthusbewrittenincylindricalformase=sinφˆi+cosφk󰀌

acknowledgingthatI(k)=nh¯c2kVistheirradianceofthethroughputradiation,theenergyshiftis

󰀁󰀃󰀇󰀆A±BI±BABA

󰀅E(R)=sin2φ·αXJVJ±α+sinφcosφαVα+αVαReXJJKKZZJJKKXKKX

ε0cJ,K

󰀂

AB

+cos2φ·αZJVJ±KαKZcos(k·R)

(3.1)

wherethekandRdependencesarehenceforthsuppressed.Employingtheexplicitformofthe

VJ±Ktensorfrom(2.7),andwritingαXX=α⊥,αZZ=α󰀉foreachmolecule,equation(3.1)isexpressibleas

󰀎󰀍󰀏

󰀇󰀆2coskRksinkRIABAB

α⊥−2cos2φ·α󰀉α󰀉·+󰀅E(R)=sinφ·α⊥

2R3R24πε0c

󰀍2󰀎󰀐kcoskRAB

cos(k·R).(3.2)α⊥−sin2φ·α⊥

RIntheshort-rangeregion(kR󰀑1),theleadingtermofequation(3.2),󰀅E0,isfoundbytakingtheleadingtermsintheTaylorseriesexpansionsofsinkR,coskRandcos(k·R)togive

󰀆2󰀇IAB2ABsin(3.3)φ·αα−2cosφ·αα󰀅E0(R)=⊥⊥󰀉󰀉.2

4πε0cR3

Onisotropicallyaveragingthesystemwithrespecttotheincominglight,theenergyshiftofequation(3.3)iswrittenas

󰀆AB󰀇IAB

󰀅E0(R)=α(3.4)α−αα󰀉󰀉.26πε0cR3⊥⊥

ABAB

Theresult(3.4)vanishesifα⊥α⊥=α󰀉α󰀉(i.e.,ifthemoleculesaresphericallysymmetric);otherwiseitisnon-zeroanditssignsignifiesaforcethatiseitherattractiveorrepulsive,asdeterminedbytherelativemagnitudesofthepolarizabilitycomponents.

ThevanderWaalsdimer(HCN)2isonewidelyknownexampleofsuchamolecularsystem.TheintermolecularbondiswellmodelledbytheStockmayerpotential[33,34]:

󰀆󰀇󰀆󰀇

U(R,µA,µB)=4ε[(σ/R)12−(σ/R)6]+µA·µB/4πε0R3−3µA·RµB·R/4πε0R5,

(3.5)

whereRistheintermolecularseparation,σistheusualLennard–Jonesparameter,εisthewelldepthandtheµtermsaredipolemoments.Theeffectofintensethroughputlaserradiationistointroducealaser-inducedenergyshiftwhichcanbeincludedinaneffectivepotential,writing

(3.6)U(R)+󰀅E0(R)=U(R)+K/R3,

󰀇󰀆AB

AB2

where,fromequation(3.4),K=Iα⊥α⊥−α󰀉α󰀉/6πε0c.Themodificationtothepotentialenergysurfacechangestheequilibriumpositionofthebond,relatingtoacontractionorexpansion—dependingontheattractiveorrepulsivenature,respectively,ofthelaser-inducedenergyshift.Differentiatingequation(3.6)withrespecttoRgivesthefollowingatthenew

󰀂

=R0+δR0:equilibriumpositionR0

U󰀂(R0+δR0)−3K/(R0+δR0)4=0.

(3.7)

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˚belowtheappropriateFigure4.Diagramofthe(HCN)2dimer.BondlengthsaregiveninA

˚bonds.Thedimer’scentreofmassisalsoshown,0.605AfromthecentralHatom.

TakingtheleadingterminTaylorseriesexpansionsofbothtermsintheaboveequationleads

toanexpressionforδR0asfollows:

3K

δR0=4󰀂󰀂.(3.8)

R0U(R0)Differentiatingequation(3.5)twicewithrespecttoRgivesthefollowingexpressionforthechangeinequilibriumbondlength:

󰀆󰀂2󰀇󰀂2

πI(k)α⊥R/9c−α󰀉

δR0=,(3.9)

ε[26σ12/R9−7σ6/R3]−|µ|2/4πε0recognizingthatαA=αB=αandµA=µB=µ.Intheabove,α󰀂referstovolumepolarizabilities(αscaledby1/4πε0).

˚fromFigure4showsthestructureofthe(HCN)2dimer.Itscentreofmasslies0.605A

˘aboutthethecentralHatom,alongtheintermolecularbond.FromthemomentofinertiaI

centreofmass,therotationalconstantBofthedimercanbecalculatedfrom

˘(3.10)B=h/8π2Ic,withareportedvalue[35]of0.0584cm−1.Thechangeinequilibriumbondlengthinducedby

thelaserfieldwillmodifythemomentofinertiaandhencetherotationalconstantofthedimer.Substitutingdatafrompreviousstudies[34–36]intoequation(3.9),2itisreadilydeterminedthatarelativelymodestirradianceof1012Wcm−2willcausethedimerbondtoextendby1.72pm.Itisimportanttonotethat,despitethisbondlengthbeingmeasuredbetweenthecentresofeachmolecule,thelinearexpansionwillbealmostentirelyoperativethroughextensionoftheN–Hhydrogenbond.(Theprincipleoflaser-inducedbondextensioncouldbeappliedtoindividualbondsineitherHCNmolecule,butthepolarizabilityofeachHCNmoleculeismuchlargerthanthatofanyindividualatomiccomponent;alsotheN–Hintermolecularbondhasamuchlowerforceconstantthantheintramolecularbonds.)Thechangeinequilibriumbondlengthcanbeappliedtoequation(3.10),givinganewrotationalconstantof0.0579cm−1—adifferenceofabout1%.Thisisexperimentallyverysignificant,wellabovetheboundsoferrorinmicrowavespectroscopicmeasurements.Takingaccountofthedistributionofintensityacrossatypicallaserbeam,itisclearthattheeffectwouldbemanifestedinabroadeningaswellasashiftinspectrallines.3.2.Molecularsolids

Theeffectsofopticallymodifyingapairinteractionpotentialwillnowbeexploredinthecontextofmolecular(usuallyorganic)solids,wheremicroscopicmechanicaleffectscanbeamplifiedbyscale.Beforeproceedingwiththedetail,itisinterestingtofirstidentifyandinterpretaclassicalrepresentationoftheopticallyinducedpairforce.Tothisend,notethatequation(2.6)canverysimplybere-expressedas

󰀅󰀄

±1¯(3.11)󰀅E(R)=2ReVjk(k,R)Pj(k)Pk(k)exp(ik·R),

2

Weusestaticpolarizabilityfigures,asthecorrespondingdynamicvaluesarenotapparentlyavailable.

Opticallyinducedinter-particleforcesS5

eφR(a)

θe(b)

Figure5.Parallelcylindricalmoleculesbondedside-to-side:(a)intheXZ-planeand(b)intheXY-plane.Thepolarizationvectoroftheincidentlaserradiationisdefinedbyitsangulardispositionagainsttheintermolecularseparationvectorandthemolecularaxis(φandθ,respectively).

nowcastinducedelectricalpolarizationsP(andnotingtheclassicalcorrespondence󰀌󰀌intermsofnh¯ckε0V→E22,whereEistheelectricfieldoftheradiation).Forneighbouringmolecules,theshort-rangelimitoftheexponentialisappropriateandwehave

󰀄󰀅±1¯¯󰀅E(R)=σjk(k)Pj(k)Pk(k)≡2ReVjk(k,R)Pj(k)Pk(k),(3.12)whichcaststhecouplingtensorσjkasaneffectivelocalstresstensor.Takingthesecond

derivativewithrespecttovectorcomponentsofR(andforconcisenesssuppressingthek-˜signifiesthemicroscopicresponsedependence),theensuingtensorx

∂2󰀅E∂2

¯k.˜il≡=XijklPjPk≡σjkPjP(3.13)x

∂Ri∂Rl∂Ri∂Rl

Inasolidwithperfectelasticity,thesecond-orderstraintensorwouldbedirectlyproportionalto˜;theaboveequationthusdesignatesthefourthranktensorXijklasaneffectiveelectrostrictivex

coefficient.Althoughtheterm‘electrostrictive’3isusuallyemployedinconnectionwithstaticelectricfields,weadopttheterm‘opticallyelectrostrictive’tosignifytheeffecttobedetailedbelow—thestemofthelatteradjectiveservingasareminderthattheassociatedeffectisaresultofelectricalinteractionsbetweenthematterandtheradiation.Incontrastto‘piezoelectric’,theterm‘electrostrictive’alsosignifiesconsistencywithaquadraticdependenceontheinducedpolarization—andacorrespondingindependenceofthelatter’ssign[38,39].

Toproceed,itisappropriatetoenvisageanarrangementconsistingofparallel,linearmoleculesasshowninfigure5.Suchastructurecanbefound,forexample,insolidscomprisedofpoledpolymers(averagingouttheeffectsofmisregistrationofchainends)orsmectic/nematicliquidcrystals4.Focusingononeneighbouringpairofmoleculesindetail,wespecificallyconsiderapairofparallel,cylindricallysymmetricmoleculeswithamutualseparationvectorRorthogonaltotheir‘long’molecularaxes.IdentifyingRwiththeZ-axisandthemolecularaxiswiththeX-direction,andassumingthesystemisirradiatedwithplane-polarizedlight,thepolarizationvectorofincidentradiationcanbedefinedas

ˆincylindricalform,whereφandθaretheanglese=sinφcosθˆi+sinφsinθˆj+cosφk

madebyewithRandthemolecularaxis,respectively.Fromequation(2.6),thelaser-inducedenergyshiftexperiencedbythispairis

󰀊IABAB

VXXαXX+sin2φsin2θαYResin2φcos2θαXX󰀅E(R)=YVYYαYY

ε0c

󰀋AB

cos(k·R),(3.14)+cos2φαZZVZZαZZ

Thereisanotherquitedistinctbutnowlesscommonusageoftheterm‘electrostrictive’inconnectionwithDrude

andNernst’stheoryofionsolvation—see,forexample[37].

4Inthecaseofliquidcrystalsanappliedelectricfieldisgenerallyrequiredtomaintainthestructureindomainsofexperimentallyamenablevolume.

3

S6DLAndrewsetal

sincealltheothertermsareequaltozero.EmployingtheexplicitformoftheVJ±Ktensorand,foreachmoleculewritingαYY=αZZ=α⊥,αXX=α󰀉,equation(3.1)isdifferentiated󰀌togivethefollowingexpressionfortheforceinducedbetweentheparticlesFind=−∂󰀅Eind∂R:F0ind(R)=

󰀈AB󰀆2󰀆󰀇󰀇󰀉3I2AB22

αsinαφ2+sinθ−2+ααsinφcosθ,󰀉󰀉24πε0cR4⊥⊥

(3.15)

aftertakingtheleadingtermsintheTaylorseriesexpansionsofcoskR,sinkRandcos(k·R)

(effectingacorrectiontotheresultgivenin[16]).Thisiseffectivelytheonlyoperativepairforce,sinceotherinter-particleforcesinthesolidarebalancedatequilibrium.Itisinstructivetoconsiderthreespecialcases,inwhichthepolarizationvectoroftheincidentlightisparalleltoeitherthemolecularaxisortheseparationvector,oritisorthogonaltoboth.Inthefirstcase,whereφ=90◦andθ=0◦,theforceisequalto

F0ind(R)

=

AB

α󰀉3Iα󰀉24πε0cR4

,(3.16)

signifyingarepulsiontendingtoincreasetheintermolecularseparation.Inthesecondcase

φ=0◦,giving

F0ind(R)

AB

α⊥3Iα⊥=−,22πε0cR4

(3.17)

correspondingtoadecreaseintheintermolecularseparation.Finally,whenφ=θ=90◦,

equation(3.15)gives

F0ind(R)

AB

α⊥3Iα⊥=,24πε0cR4

(3.18)

againsignifyingarepulsion.(Itmaybenotedthatforsphericallysymmetricspecies,the

firstandthirdcasesareidentical.Inanyothersystem,therelativestrengthsoftherepulsive

ξξξ

forcesdependonthemagnitudesofα󰀉andα⊥.)Forelectronicallyprolatemolecules,α󰀉is

ξ

greaterthanα⊥,soequation(3.16)willsignifyalargerrepulsionthantheattractiongivenbyequation(3.17);thereverseholdstrueforoblatemolecules.

Theabovemodelaccountsforonlytwodimensionsofmolecularpackingintheseanisotropicsolids.Inordertoaccountforthethird,wemustreturntotheend-to-endmodelexploredinanotherconnectionintheprevioussubsection.Fromequation(3.3),whenthepolarizationvectoroftheincidentradiationisparalleltotheintermolecularseparationvector,theensuingforceisequalto

F0ind(R)

=−

AB

3Iα󰀉α󰀉22πε0cR4

,(3.19)

onceagainsignifyinganattraction.Ifthepolarizationvectorisorthogonaltotheseparation

vector,theenergyshiftemergesas

F0ind(R)

AB

α⊥3Iα⊥=,2

4πε0cR4

(3.20)

signifyingrepulsion.Thus,irradiatingsuchasolidwithplane-polarizedlightwillaffecta

contractionparalleltothepolarizationofthelaserbeamandanexpansionintheothertwodimensions,bothlinearlydependentontheirradianceofthelaser.

Again,therespectiveattractiveandrepulsiveforcesarereadilyexplicableinclassicalelectrodynamicterms.Theoscillatingelectricfieldoftheradiationinducesexactlyin-phase

Opticallyinducedinter-particleforcesS7

e(i)Re(ii)e(iii)(a)e(v)e(iv)R(b)Figure6.Diagramsdepictingkeyorientationsoftheincidentlaserradiationwithrespecttothemolecularsystem.In(a),themoleculesarebondedside-to-sideandthepolarizationoftheincidentradiationis(i)paralleltothemolecularaxis;(ii)paralleltoRand(iii)orthogonaltoboth.In(b),themoleculesareend-to-endandthepolarizationis(iv)paralleltoand(v)orthogonaltotheintermolecularseparationvector.

(and,ifthemoleculesaresimilarlyaligned,parallel)oscillatingdipolesinbothmolecules.Asshowninfigure6,wheneandRareorthogonal,thesedipoleslineup‘head-to-head’—accountingfortherepulsiveforceexhibited—whereaswhentheyareparallel,thedipoleslineup‘head-to-tail’—hencetheattraction.Itisalsoworthremarkingthatacompressionparalleltoetogetherwithanexpansionindirectionsperpendiculartoeproducesasoliddeformationcharacterizedbyanincreaseinvolume5thatscaleslinearlywiththeirradianceI.Thissignifiesachangeinthelocaldensity—andhencealsoalocalrefractiveindex—thatscaleslinearlywithI.So,thiseffectleadstoanoptomechanicallyinducedintensity-dependentrefractiveindex[40].Thatistosay,itcontributestoaneffectwhich,insolids,isdominatedbymuchmorewidelyknownelectronicmechanismssuchastheopticalKerreffect—and,inabsorbingregions,photothermaleffectsduetospatiallyinhomogeneousheating.Intriguingly,electrostrictionwasindeedconsideredinconnectionwithanintensity-dependentrefractiveindexintheearlydaysoflaserphysics[41],butitsubsequentlyemergedthatothermechanismsplayamoredirectandsignificantrole.3.3.ComparisonwithCasimir–Polderforces

ItisinstructivetogaugetherelativesignificanceoftheforcesdescribedabovewiththoseassociatedwiththeCasimir–Polderinteraction.Inarigidsystem,theforceinducedbythispotentialcanbeexpressed,fornear-zoneinteractions,as[6]

󰀃|µr0(A)|2|µs0(B)|227

.(3.21)FCP(R)=−27Er0+Es012π2ε0Rr,s

Inthespecialcaseofsphericalmolecules,thenetexpansiontoleadingordersfrom(3.19)and(3.20)cancels,as

thenumericaldifferencebyafactorof2iscompensatedbythefactthattherearethetwodirectionsperpendiculartothepolarizationvector.

5

S8DLAndrewsetal

Forsimplicity,thiscanbecomparedtothelaser-inducedforceinonedirectionforidentical,

ξξ

sphericalparticles(i.e.,α⊥=α󰀉=α)onafixed,cubiclattice;themagnitudeanddirectionalityoftheforcebothdependontheorientationofthesystemwithrespecttothepolarizationvectoroftheincidentradiation.ForthosepairswithRparalleltoe,theforcecanbewritten,fromequation(3.19),asfollows:

F0ind

3Iα2

.(R)=−22πε0cR4

(3.22)

Fororderofmagnitudeestimations,thetransitiondipolemomentscanbeapproximatedas

productsofanelectronchargeandtheBohrradius,i.e.asea0;likewisetheenergydifferencesapproximatetoe2/4πε0a0.Itthusfollowsthatthepolarizability,fromequation(1.8),isequal

3

.Substitutingtheseexpressionsintoequations(3.21)and(3.22),theratioofthetwoto4πε0a0

forcescanbecalculatedas

F016π2Iε0a0R3ind(R)=,FCP(R)3ce2

(3.23)

which,atanappliedirradianceof1012Wcm−2,isequalto2.8×10−3.SincetheCasimir–

Polderinteractionisalwayspresent,thelaser-inducedinteractioncanbeviewedinenergytermsasacorrectioneitherenhancingordiminishingthepaircoupling,asdeterminedbythelaserpolarization.Withtheincreasinguseofhigh-intensitylightinlaserexperimentstoday,itappearsthattheopticaleffectwillexertasignificantinfluenceonthemolecularmechanics.4.Conclusion

Intheintroduction,themainpurposewastorehearsealineofreasoning,basedonquantumelectrodynamicalconcepts,thatbothsuggeststhepossibilityandoffersthemeansofcalculatinglaser-inducedpairforces.Thedetailedtheorythatensueshasbeendevelopedinsection2,itsuniversalityofapplicabilitydemonstratedanditskeyfeaturesdelineated.Recognizingtheexperimentalchallengeslikelytobeinvolvedincharacterizingsuchphenomena,physicallyidentifiableresultshavebeendeterminedfortwophysicallyverydifferentsystems,thecorrespondingestimatessuggestingentirelyrealisticprospectsforexperimentaldetermination.Inparticular,anopticalelectrostrictionphenomenonhasbeenidentified,itselectrodynamicinterpretationexploredanditsrelationtoCasimir–Polderforcesdiscussed.Throughthefullquantumelectrodynamicalanalysispresentedabove,keyparametersthatdeterminethesizeandcharacterofopticalelectrostrictionhavebeenidentifiedanditssignificanceassessed.Inthefollowing,itisouraimtoputtheseresultsfullyintocontext.

First,itneedstobereaffirmedthatopticallyinducedpairwiseforcesbetweenparticlesareclearlydistinguishablefromthevariousformsofopticalforcethatintenselaserlightcanexertonindividualparticles.Thelattercategoryofforcesismuchbetterknown—andwidelyapplied,inmethodsasdisparateasopticaltweezersandopticalmolasses.Someofthemostobviousdifferencesthatariseinthecaseofopticallyinducedinter-particleforcesarethatthelatterdependneitheronintensitygradientsnoronabsorptionprocesses.Thus,insystemswhere,forexample,opticalmolassesareemployedfortrappingatoms,opticalpairforceswilloperateinadditiontotheforcesonindividualatoms,andmaytherebyinfluencetheensemblebehaviour.Morespecifically,whenanyparticlesareheldatequilibriuminanopticaltraportweezersset-up,thesmallopticallyinducedforcesthatalsooperatemaybesufficienttoproduceobservabledeparturesfromthebehaviourexpectedofindependentparticles.Insomeearlierwork,ithasalreadybeenshownthataweakbindingeffectcaninsomecasesleadtoopticalclusteringandinotherstopatternformation[42].

Opticallyinducedinter-particleforcesS9

Insuitablecases,ithasbeenshownthatopticallyinducedpairforcesshouldproveaccessibletoexperimentaldeterminationingasesandinsolids,usingwidelydifferentexperimentalmethods.Wenotethatthelevelsofintensitychosenforourmodelcalculationsarereadilyattainable,forexample,fromthefocusedirradianceofacurrentgenerationfemtosecondmode-lockedtitanium:sapphirelaser.ForvanderWaalsmolecules,measurementbyamicrowavespectroscopictechnique,whileundoubtedlyhighlydemanding,hasthevirtuethatnootherprocessesobviouslycompete.Amoredirectalternativemightbetoattemptdetectionusingatomicforcemicroscopy,withoneofthemoleculestetheredonasubstrate.Thesameexperimentalobjectivitywillbemuchhardertoachieveinthecaseofopticalelectrostriction.Thoughclearlydependentonthechoiceoflaserwavelength,thetypicallybroadopticalabsorptionbandsassociatedwiththesolidstatewillmakeitdifficulttoensureadegreeoftransparencysufficienttoobviatepotentiallymuchlargerthermaleffects—suchasthewell-knownchangeinrefractiveindexinvolvedinthermallensing.Itisironicthatthisneedtoensureoperationwelloutsideregionsofabsorptionmilitatesagainstexploitingthedispersionbehaviourofthemolecularpolarizability,whichmightotherwiseofferanobviousresonanceroutetoenhancement.

Inseekingthebestexperimentalmethodology,thehugerecentadvancesinthetechnologyofultra-preciseforcemeasurementmightoffersomeencouragement.Nonetheless,sincedirectlyoperativemechanicaleffectsarenecessarilylimitedbythespatialbeamwidthofthelaserlight,itmaybemorepracticaltolookforsecondaryeffects,suchasthechangeinrefractiveindexnotedinsection3,whicharebothconfinedtoandmonitoredbytheradiationitself.ArelatedpossibilitymightbetoseekforadetectionofchangesintheRamansignatureoftheemergentlight,adaptingsomeofthehighlysensitivespectroscopicmethodsthatareemployedtodetectmechanicalstressinsemiconductors.Oneotherverydifferentpossibilitywouldbetomeasurenanoscaleforcesbetweenwaveguidesasafunctionofthroughputintensity,ashasbeenproposedinarecentcollaborationbetweengroupsatMITandHarvard[43].Thetheoryisofcoursecastverydifferently,butatafundamentallevelthephysicsisundoubtedlyrelated.

Althoughwehavefocusedontheintrinsicinterestofthesubject,wewouldendbypointingoutthepossibilitiesfordeviceapplications.Inthecontextofdramaticallyacceleratingdevelopmentsinthefieldofnanoelectromechanicalsystems,anymechanismthatcanreproduciblydeliverareversibleandultrafastmechanicalresponse,actuatedbylight,appearstobeofconsiderablemerit.Again,inthefieldofmicrofluidicsthereisscopetoconsiderhowlaser-inducedmolecularforcesmightbeusedtocontrolthemixingorpumpingofextremelysmallvolumesinlab-on-a-chipandassociateddevices;intheseandinmanyotherareasthereisplentyofscopeforspeculation.Itisourhopethat,withthefundamentalprinciplesfullyestablished,thedevelopmentofsomeoftheseapplicationscannowbegin.Acknowledgments

ThispaperisofferedasatributetoEdwinPower,whosememorablylucidteachingwasaprivilegetoexperience(DLA),andwhoseworkcontinuestoinspiremembersoftheQEDgroupatUniversityofEastAnglia.ThegroupgratefullyacknowledgesfinancialsupportfromtheEngineeringandPhysicalSciencesResearchCouncil.References

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