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Surface plasmon polariton assisted † optical pulling force 6 1 Mihail I. Petrov,‡ Sergey V. Sukhov,¶,‡,§ Andrey A. Bogdanov,‡,(cid:107),⊥ 0 2 Alexander S. Shalin,∗,‡,§,# and Aristide Dogariu¶ n a J 8 TheInternationalResearchCentreforNanophotonicsandMetamaterials,ITMOUniversity, ] s Birzhevajaline14,199034St. Petersburg,Russia,CREOL,TheCollegeofOpticsandPhotonics, c i t UniversityofCentralFlorida,4000CentralFloridaBlvd.,32816Orlando,Florida,USA, p o . Kotel’nikovInstituteofRadioEngineeringandElectronicsofRussianAcademyofSciences s c i (Ulyanovskbranch),Spasskayastr. 14,432011Ulyanovsk,Russia,IoffeInstitute, s y h Politekhnicheskayastr. 26,194021St. Petersburg,Russia,PetertheGreatSt.Petersburg p [ PolytechnicUniversity,Politekhnicheskayastr. 29,195251St. Petersburg,Russia,andUlyanovsk 1 v StateUniversity,LevTolstoystr. 42,432017Ulyanovsk,Russia 3 6 8 1 E-mail: [email protected] 0 . 1 0 6 1 †This is the pre-peer reviewed version of the following article: [M.I. Petrov, S.V. Sukhov, A.A. Bogdanov, : A.S. Shalin, A. Dogariu, Laser & Photonics Reviews. DOI:10.1002/lpor201500173 (2015)], which has been pub- v lished in final form at http: http://onlinelibrary.wiley.com/doi/10.1002/lpor.201500173/abstract. This article may be i X used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This article r maybeusedfornon-commercialpurposesinaccordancewithWileyTermsandConditionsforSelf-Archiving. a ∗Towhomcorrespondenceshouldbeaddressed ‡TheInternationalResearchCentreforNanophotonicsandMetamaterials, ITMOUniversity, Birzhevajaline14, 199034St. Petersburg,Russia ¶CREOL,TheCollegeofOpticsandPhotonics,UniversityofCentralFlorida,4000CentralFloridaBlvd.,32816 Orlando,Florida,USA §Kotel’nikovInstituteofRadioEngineeringandElectronicsofRussianAcademyofSciences(Ulyanovskbranch), Spasskayastr. 14,432011Ulyanovsk,Russia (cid:107)IoffeInstitute,Politekhnicheskayastr. 26,194021St. Petersburg,Russia ⊥PetertheGreatSt.PetersburgPolytechnicUniversity,Politekhnicheskayastr. 29,195251St. Petersburg,Russia #UlyanovskStateUniversity,LevTolstoystr. 42,432017Ulyanovsk,Russia 1 Abstract We demonstrate both analytically and numerically the existence of optical pulling forces acting on particles located near plasmonic interfaces. Two main factors contribute to the ap- pearance of this negative reaction force. The interference between the incident and reflected wavesinducesarotatingdipolewithanasymmetricscatteringpatternwhilethedirectionalex- citationofsurfaceplasmonpolaritons(SPP)enhancesthelinearmomentumofscatteredlight. The strongly asymmetric SPP excitation is determined by spin-orbit coupling of the rotating dipole and surface plasmon polariton. As a result of the total momentum conservation, the forceactingontheparticlepointsinadirectionoppositetotheincidentwavepropagation. We derive analytical expressions for the force acting on a dipolar particles placed in the proximity of plasmonic surfaces. Analytical expressions for this pulling force are derived within the dipoleapproximationandareinexcellentagreementwithresultsofelectromagneticnumerical calculations. Theforcesactingonlargerparticlesareanalyzednumerically,beyondthedipole approximation. Introduction In spite of wide popularity of volume optical manipulation (mainly, for biological research),1 it hasmajordrawbacksifappliedtolab-on-a-chipplatformsandmicro-fluidicdevices. Ontheother hand,manipulationwithevanescentfieldsandsurfacewavesiscompatiblewithco-planarintegra- tions needed for miniaturization of optical devices.2 The high spatial localization of evanescent field might enable to extend optical manipulation down to nanometer scale.3 The optical forces acting in a system can be significantly amplified if Surface Plasmon Polaritons (SPP) on metal- dielectricinterfacesareused.4,5 Inadditiontoenhancingthemagnitudeofopticalforces,SPPcan achievefurtherspatialconfinementdowntonanometerscale.4,6 Various aspects of SPP manipulation were already considered. Plasmonic amplification was used for trapping or acceleration of nano-sized particles using high gradients of local fields,4,7–9 andformodulationplasmonicsignalbyoscillatingnanoparticleinV-shapedwaveguide.10 Inother 2 situations, the propagating SPP waves were used for transportation of particles using localized scattering forces.11 Combination of gradient and scattering forces allows to construct robust SPP trap for mesoscopic metallic particles.12 Normal to the surface force was recently considered also in flat plasmonic and metamaterial structures.13,14 Here we discuss the unusual situation when nonconservative force from SPP wave creates attractive force on small particles acting along the interfacetowardsthesourceoflight(plasmonic”tractorbeam”). Recently,nonconservativepullingforces(”tractorbeams”)attractedconsiderableattentionbe- cause of an unusual behavior: they act in a direction opposite to the optical wave propagation, toward the source of light. Many types of tractor beams have already been suggested15 most of them being based on either structuring the incident field16–21 or on modifying the surroundings of the manipulated object.22–24 For instance, backward surface waves existing in certain structures can provide means for such manipulation.25 In other schemes, the negative optical forces appear because of the enhancement of optical linear momentum after interaction of light with an object. Attraction force can appear, for instance, when a particle is placed in the vicinity of a waveguide becauseofmodetransformationinsidethewaveguide.26 Another remarkable illustration of optical attraction is the case of micrometer-size particles floating on a liquid surface and driven towards the source of light due to the natural enhancement of light momentum in optically denser media.27,28 Of course, in terms of application in nano-photonic devices, this concept would be more interesting if it could be extended to solid substrates. The direct application of this method to solid surfaces is impossible because the manipulated objects reside entirely outside the substrate. However, such an object can still interact with the solid substrate by converting some of its evanescent field into waves that propagate into thesubstrate. Thisinteractioneffectivelychangesthescatteringpatternoftheobjectredistributing the nonconservative reaction forces. If the photons’ momentum in substrate increases, the object shouldexperienceabackwardforcetocompensatethissurgeofmomentum. Formetallicsurfaces, thisincreaseofmomentumcanoccurasaresultofSPPexcitation,whichhavelargewavenumbers and, therefore, large linear momenta. In this paper we explore this idea in detail and provide both 3 analytical and numerical results confirming the possibility of creating attractive optical forces for particlesplacedonsolidsurfaces. Particularattentionisdevotedtoplasmonicinterfaces. Results and discussion Forces acting on a point dipole near interface Westartbyexaminingasimplifiedsituationforwhichananalyticalsolutionispossible: thecaseof adipole locatedin thevicinity ofasolid substrate. We considera spherical dielectricnanoparticle withpermittivityε andradiusRlocatedinsidemediumwithpermittivityε . Theparticleisplaced m at the height z above a substrate with permittivity ε , as shown in Figure 1. The incident wave 0 s with wavelength λ impinges onto the surface at the angle θ. Assuming that the nanoparticle is small in comparison to the wavelength, R(cid:28)λ, one can consider it as a point electric dipole with polarizabilityα . 0 e v a w d e t c e fl e R Einc z Directional H inc k Air plasmon ε θ z m 0 R excitation ε Silver εs x Figure 1: Directional excitation of surface plasmon polaritons by a small particle placed in the vicinity of a metal surface. The recoil force created by plasmon polaritons on the particle can be directed opposite to the propagation of incident light. The insert depicts the geometry of the problem. 4 Thegeneralexpressionfortheopticalforceactingonapointdipolecanbewrittenas29,30 1 (cid:16) (cid:17) F= ∑Re p∗∇Eloc , (1) i i 2 i where p = α Eloc is the induced dipole moment, and Eloc is the local electrical field. The local 0 electricalfieldabovethesubstrate Eloc(r)=E0(r)+ED(r) (2) isthesuperpositionoftwocomponents: thefieldE0,whichincludesboththeincidentandreflected waves, and the scattered field ED, which includes field induced by the point dipole, and local fields generated in the substrate (see Figure 1). The second component can be found to be ED = ω2µ GˆR(r,r )p in terms of the dyadic Green’s function GˆR of the dipole-interface system31 (see 0 0 Supporting Information A for explicit expressions). Thus, the total induced dipole moment p = αÊ0 can be written using an effective polarizability tensor αˆ which also includes the nanoparticle interactionwiththesubstrate. Thiseffectivepolarizabilityisdefinedas αˆ =α (cid:2)I−α ω2µ GˆR(r ,r )(cid:3)−1, (3) 0 0 0 0 0 where I is the unity tensor, µ is the magnetic permittivity of vacuum, and ω is the frequency of 0 incidentwave. Thediagonalelementshavethefollowingform: α α 0 0 α =α = , α = . (4) xx yy 1−α ω2µ GR (r ,r ) zz 1−α ω2µ GR(r ,r ) 0 0 xx 0 0 0 0 zz 0 0 As a result of the field superposition in Eq. (2), the optical force acting along interface could also beseparatedintotwotermsF =F0+FD where x x x (cid:32) (cid:33) 1 ∂E0(D) F0(D) = Re p∗ . (5) x 2 ∂x 5 Term F0 corresponds to the x-component of the total radiation pressure generated by the incident x andthereflectedwaveswhileFD correspondstotheinteractionofthedipolewithitsownretarded x field. This separation is just a matter of convenience because, in fact, F0 also accounts for the x interaction with the substrate through the polarizability tensor in Eq. (4). It is important to note that, as long as the system depicted in Figure 1 remains invariant with respect to the motion of the dipole along its surface, F represents a purely nonconservative force. After some algebra, the x expressionfortheF0 becomes x Fx0 = 21kx(cid:104)Im(αxx)(cid:12)(cid:12)Ex0(cid:12)(cid:12)2+Im(αzz)(cid:12)(cid:12)Ez0(cid:12)(cid:12)2(cid:105) (6) 1/2 with k =ksinθ, and k=ε ω/c being the wavenumber in the upper half-space. The right hand x m side in Eq. (6) is always positive because Im(α ) > 0 as required by energy conservation for xx,zz exp(−iωt) time dependence. Thus, the force component F0 acts always along the direction of x incidentwavepropagation. It can be shown that the other force component, FD, corresponding to the dipole interaction x withitsretardedfield(seeSupportingInformationBfordetails)canbewrittenas k2 FD =− Im(p∗p )Im(cid:0)∂ GR(cid:1), (7) x x z x xz ε 0 whereε isthedielectricpermittivityofvacuum. 0 This formula is one of the main results of our paper. It indicates that, interestingly, this force component is nonzero when there is a phase difference between the x- and z-components of the dipolemoment. Dependingonthisphasedifference,theforceFDcanbeeitherpositiveornegative, x whichisofparticularinterestforourdiscussion. Wealsonotethat,becauseFDdependsonlyonthe x x- and z-components of the dipole moment, it is sufficient to analyze only the case of p-polarized incident waves. Remarkably, although FD acts on a dipolar particle located close to an interface, x thisforcecannotbepredictedbytheelectrostatictheoryofparticle-surfaceinteraction. Thisisdue tothefactthatFD isnonconservativeanditsdescriptionrequirestakingintoaccountphasesofthe x 6 fieldwhichareneglectedintheelectrostaticapproximation. Thephasedifferencebetweentwocomponentsofdipolemoment(leadingtodipole’srotation) isakeyrequirementfortheasymmetryofthedipole’sscatteringpatternforparticleslocatedinthe vicinity of interfaces. It also leads to the directional excitation of waves in a substrate due to the spin-orbitcouplingeffect. Thesituationistypicalforobjectsplacedclosetodielectricmedia,32,33 metal surfaces,34–36 or metasurfaces.37 This asymmetry produces a linear momentum imbalance thatshouldbecompensatedbyadditionalforceFDactingonthescatteringdipole. Whenthedipole x is located very close (kz (cid:28)1) to non-absorbing dielectric media with real reflection coefficient, 0 the p and p oscillatesynchronouslyand,consequently,FD=0. However,formetallicinterfaces, x z x the interference of incident and reflected waves can induce a phase shift of almost π/2 between the phases of the p and p generating, therefore, a rotating electric dipole. When the scattering x z fromthis rotatingdipoleexcitesSPP wavepropagatingpredominantlyin forwarddirection(along x-axis), one can anticipate an increase of the momentum of photons in the substrate. Then, due to linear momentum conservation law, the object should experience negative force F to compensate x thismomentumincrease. The analysis of Im(p∗p ) allows us to understand the dependence of FD on the angle of inci- x z x dence. Forexample,fordipole’slocationsclosetotheinterface(kz (cid:28)1)onegets(seeSupporting 0 InformationB) Im(p∗p )=|α |2|E0|2sin(2θ)Im(cid:2)r (θ)(cid:3), (8) x z 0 p where r is the Fresnel coefficient for p-polarized wave. It follows from this formula that, in p particular,FD =0forbothnormalandgrazinganglesofincidence. x As can be seen in Eq. (7), the magnitude and the direction of the force depend also on the derivative of the Green’s function, which can be expressed through the integral in Fourier space (seeSupportingInformationB):   ∞ Im(cid:0)∂xGˆRxz(cid:1)= 8π1k2Im(cid:90) k(cid:107)3rp(k(cid:107))exp(2ikzz0)dk(cid:107), (9) 0 7 wherek andk arethewavevectorcomponentslateralandnormaltotheinterface. Theintegration (cid:107) z is performed over the lateral component of the wave vector in the upper half-space. We note that the integration range 0 < k < k in Eq. (9) corresponds to the contribution from the propagating (cid:107) waves reflected from the surface, whereas the integration over k > k relates to the contribution (cid:107) from the modes evanescent in the upper space. This integral is evaluated for special cases in the SupportingInformationC. It is well-known31 that if the condition ε +ε < 0 is fulfilled, an evanescent surface plas- m s mon wave can propagate along the interface. In the integral in Eq. (9) it relates to a pole in the reflection coefficient at the SPP wavevector k =k[ε ε /(ε +ε )]1/2. Even though the reflec- SPP m s m s tion coefficient is purely real for k >k , the imaginary part of the integral is nonzero due to the (cid:107) regularization of the integrand divergence (see Supporting Information C). Besides, the Green’s function tensor components expressed in terms of similar integrals are incorporated into α and xx combined all-together define the force FD. When calculated near the surface plasmon (SP) res- x onance Re(ε +ε ) ≈ 0, in the case of low-absorbing substrate with Re(ε ) < 0, and far from m s s configurationalresonance,thisforcebecomes(seeSupportingInformationC): k4 k3 (cid:18) (cid:113) (cid:19) FD ≈− SPP SPP|α |2|E0|2sin(2θ)sin[2(k z +φ)]exp −2 k2 −k2z , (10) x 8ε k3 0 z 0 SPP 0 0 where φ is related to the phase of reflection coefficient, which in the case of non-absorbing media has the form r ≈ exp(2iφ). One can see from Eq. (10) that FD depends strongly on the SPP p x (cid:16) (cid:16) (cid:113) (cid:17)(cid:17) wavevector k ∝k7 exp −2 k2 −k2z , and right at the SP resonance, k → ∞, the SPP SPP SPP 0 SPP exponential factor zeroes the lateral force for fixed distance z . Thus, the maximal attractive force 0 is spectrally slightly red-shifted with respect to SP resonance. The same expression gives us the understanding of lateral force dependence on the elevation of the particle above the surface (z- coordinate). Because of the exponential factor the plasmonic force exponentially decays with height, and in point dipole approximation goes to zero exactly at the interface, assuming that φ ≈π in the metallic regime of the substrate. Generally, there exists an optimal height, at which 8 the SPP force reaches its maximal negative value for fixed wavelength and incidence angle. It can bederivedfromtheEq.(10)andforconsideredassumptionneartheSPresonanceisdescribedby (cid:16) (cid:113) (cid:17) simpleexpressionz ≈1/ 2 k2 −k2 ,whichisahalfofSPPpenetrationdepthintheupper max SPP medium. SPP enhanced optical pulling forces In this section we find the specific parameters corresponding to overall pulling force and com- pare analytical predictions to exact numerical calculations. First, as an example, we compute the longitudinal component of the optical force F for a dielectric nanoparticle with permittivity x ε = 3 (e.g., melamine) and radius R = 15 nm in air (ε = 1) placed in the proximity of a silver m substrate. The force on the dipole is evaluated within the model outlined before and the results are summarized in Figure 2. We have normalized the force F with the radiation pressure force x F = 1/2k|E0|2Im(α ) acting on the spherical nanoparticle in the absence of a substrate. This 0 0 force is nonzero as Im(α ) > 0 due to scattering of light, and it is proportional to nanoparticle 0 volume. Ascanbeseen,inthespectralregion330-400nm,theforceF resonantlybecomesnega- x tive exceeding the radiation pressure force by an order of magnitude. According to the dispersion curve shown in the inset, the localized SPP exists at wavelengths longer than 340 nm (dash-dot line), which fulfills the condition Re(ε +ε ) = 0 and explains the observed resonant behavior. s m Below340nmtheSPPisnotlocalizedandgivescontributiontothefar-fieldmodes. Totesttheanalyticalmodelandtofurtherextendthisforceconcepttolargerparticles,wehave evaluated the problem numerically with Comsol Multiphysics (see Methods section for details). The results are shown in Figure 2 with dotted line. In general, there is good correspondence between the numerical and analytical results; the minor discrepancy relates to the finite size of nanoparticleusedinthenumericalsimulations. Indeed,inthesimulationstheparticleissubjected tofieldsthatarestronglyinhomogeneousinthez-direction. Asaresult,thelocationoftheinduced dipole moment does not coincide with the geometrical centre of the particle and it is effectively located closer to the surface. In fact, the small discrepancy between analytical and numerical 9 Dipole model Dipole model without coupling to evanescent modes 5 Numerical simulations 0 −5 4.5 276 0 F m F/x V h, n e gt −10 Energy, 3.5 ght line S PP line 354Wavelen Li −15 2.5 496 0.01 0.02 0.03 Wavevector, 1/nm −20 300 310 320 330 340 350 360 370 380 390 400 Wavelength, nm Figure 2: The spectral dependence of the longitudinal force F acting on a dielectric particle with x radiusR=15nmanddielectricpermittivityε =3placedinairaboveasilversurfaceatz =27nm. 0 Dielectric permittivity of silver is taken from work by Johnson and Christy38 and the angle of incidence is θ =35◦. The solid line corresponds to the dipole model; dotted line corresponds to thenumericalcalculations. Thedashedlinecorrespondstodipolemodelwithexcludedevanescent modes contribution by integrating the Green’s function in the range 0 < k < k only. Inset: the (cid:107) dispersion curve of SPP at the silver/air interface. The dash-dot line corresponds to SP resonance wavelength. 10