VOLUME 79, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 28 JULY 1997

eousic fieldh thesee tipThe

Theory of Nanometric Optical Tweezers

Lukas Novotny, Randy X. Bian, and X. Sunney XiePacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352

(Received 14 January 1997)

We propose a scheme for optical trapping and alignment of dielectric particles in aquenvironments at the nanometer scale. The scheme is based on the highly enhanced electrclose to a laser-illuminated metal tip and the strong mechanical forces and torque associated witfields. We obtain a rigorous solution of Maxwell’s equations for the electromagnetic fields near thand calculate the trapping potentials for a dielectric particle beyond the Rayleigh approximation.results indicate the feasibility of the scheme. [S0031-9007(97)03687-9]

PACS numbers: 42.50.Vk, 33.80.Ps, 41.20.Bt, 78.70.–g

a

ctoe-

i-iaoe

one

th

ticfgl

g3o

dslsli

eleecno

an-intsains.theby

on)e

ischants

nhenorty

o

thethe

Optical trapping by highly focused laser beams hbeen extensively used for the manipulation of submicrosize particles and biological structures [1]. Conventionoptical tweezers rely on the field gradients near the foof a laser beam which give rise to a trapping forcewards the focus. The trapping volume of these tweezis diffraction limited. Near-field optical microscopy enables the optical measurements at dimensions beyonddiffraction limit and makes it possible to optically montor dynamics of single biomolecules [2]. The potentapplication of optical near fields to manipulate atomsnanoparticles has been discussed in Ref. [3]. In this Lter, we present a new methodology for calculating rigously and self-consistently the trapping forces acting onanometric particle in the optical near field and proposnovel high-resolution trapping scheme.

The proposed nanometric optical tweezers rely onstrongly enhanced electric field at a sharply pointed metip under laser illumination. The near field close to themainly consists of evanescent components which derapidly with distance from the tip. The utilization othe metal tip for optical trapping offers the followinadvantages: (1) The highly confined evanescent fiesignificantly reduce the trapping volume; (2) the larfield gradients result in a larger trapping force; and (the field enhancement allows the reduction of illuminatipower and radiation damages to the sample.

High resolution surface modification based on the fieenhancement at laser-illuminated metal tips has beencently demonstrated [4]. It is essential to performrigorous electromagnetic analysis to understand the unlying mechanism for the field enhancement. Our analyis therefore relevant not only to optical tweezers, but ato other applications, such as surface modification, nonear spectroscopy and near-field optical imaging.

To solve Maxwell’s equations in the specific geomtry of the tip and its environment, we employ the multipmultipole method (MMP) which recently has been applito various near-field optical problems [5]. In MMP, eletromagnetic fields are represented by a series expansioknown analytical solutions of Maxwell’s equations. T

0031-9007y97y79(4)y645(4)$10.00

sn-alus-rs

the

lrt-

r-aa

etalpay

dse)n

ldre-aer-iso

n-

-

d-

of

determine the unknown coefficients in the series expsion, boundary conditions are imposed at discrete poon the interfaces between adjacent homogeneous domOnce the resulting system of equations is solved andcoefficients are determined, the solution is representeda self-consistent analytical expression.

Figure 1 shows our three dimensional MMP simulatiof the foremost part of a gold tip (5 nm tip radiusin water for two different monochromatic plane-wavexcitations. The wavelength of the illuminating lightl ­ 810 nm (Ti:sapphire laser), which does not matthe surface plasmon resonance. The dielectric constof tip and water were taken to be ­ 224.9 1 1.57iand ´ ­ 1.77, respectively [6]. In Fig. 1(a), a planewave is incident from the bottom with the polarizatioperpendicular to the tip axis, whereas in Fig. 1(b) ttip is illuminated from the side with the polarizatioparallel to the tip axis. A striking difference is seen fthe two different polarizations: in Fig. 1(b), the intensi

FIG. 1. Near field of a gold tip in water illuminated by twdifferent monochromatic waves atl ­ 810 nm. Direction andpolarization of the incident wave are indicated by thek andEvectors. The figures show contours ofE2 (factor of 2 betweensuccessive lines). The scaling is given by the numbers infigures (multiples of the exciting field). No enhancement attip in (a); enhancement ofø3000 in (b). The field in (b) isalmost rotationally symmetric in the vicinity of the tip.

© 1997 The American Physical Society 645

VOLUME 79, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 28 JULY 1997

nheinn

th

c(aie

tol

onthlleghtagancu

di

ohe

rma(

icle,toty

ly

,ce

l

ngel

aner

gee

ng

enhancement at the foremost part of the tip isø3000times stronger than the illuminating intensity, whereasenhancement beneath the tip exists in Fig. 1(a) [7]. Tresult suggests that it is crucial to have a large componof the excitation field along the axial direction to obtaa high field enhancement. Calculations of platinum atungsten tips show lower enhancements, whereasfield beneath a dielectric tip is reduced compared toexcitation field.

Figure 2 shows our calculation of the induced surfacharge density for the two situations shown in Figs. 1and 1(b). The incident light drives the free electronsthe metal along the direction of polarization. While thcharge density is zero inside the metal at any instantime (=== ? E ­ 0), charges accumulate on the surfacethe metal. When the incident polarization is perpendicuto the tip axis [Fig. 1(a)], diametrically opposed pointsthe tip surface have opposite charges. As a consequethe foremost end of the tip remains uncharged. Onother hand, when the incident polarization is parato the tip axis [Fig. 1(b)], the induced surface chardensity is almost rotationally symmetric and has thighest amplitude at the end of the tip. In both casessurface charges form an oscillating standing wave (surfplasmons) with wavelengths shorter than the wavelenof the illuminating light. While the field enhancement hbeen calculated in the electrostatic limit [8], the preseof surface plasmons indicates that it is essential to inclretardation in the analysis.

With the field distribution around the tip determinethe gradient force for a Rayleigh particle can be eascalculated as

F ­ say2d=E2, (1)

where a is the polarizability of the particle [9]. Theparticle tends to move to the higher intensity regiwhere its induced dipole has lower potential energy. Tassumptions inherent in Eq. (1) are that the external fi

FIG. 2. Induced surface charge density correspondingFig. 1(a) (left) and Fig. 1(b) (right). The surface charges foa standing wave in each case. In Fig. 1(a), the surface chwave has a node at the end of the tip, whereas in Fig. 1there is a large surface charge accumulation at the forempart, responsible for the field enhancement.

646

oisnt

dthee

e)

n

off

arnce,el

eehecethse

de

,ly

neld

to

rgeb)ost

is homogeneous across the particle and that the partdoes not alter the fieldE in Eq. (1). These assumptionshowever, do not hold for a nanometric particle closethe tip as shown in our calculation (Fig. 3). The intensicontours are distorted around a dielectric sphere (´ ­ 2.5,10 nm diameter) and the field inside the sphere is highinhomogeneous.

To overcome the limitation of Rayleigh approximationwe performed a rigorous treatment of the trapping forby applying the conservation law for momentum [10],

ddt

hGm 1 Gemj ­Z

≠VT ? n dS . (2)

≠V denotes a surface enclosing the particle andn isthe outwardly directed normal unit vector. The totamechanical and electromagnetic momenta inside≠V aredenoted byGm and Gem, respectively. T designatesMaxwell’s stress tensor given by

T ­ ´0´EE 1 m0mHH 212

s´0´E2 1 m0mH2d I .

(3)

Here, I denotes the unit dyad and, m are the dielectricconstant and magnetic permeability of the surroundimedium, respectively. For time-harmonic excitation, thtime average ofdGemydt is zero and the net mechanicaforce can be expressed as [11]

F ­

ødGm

dt

¿­

Z≠V

kT ? nl dS , (4)

wherek. . .l denotes the time average. We find that forparticle in the vicinity of the tip, the magnetic contributioto the force is approximately 2 orders of magnitude lowthan the electric one.

FIG. 3. Perturbation of the near field by a particle beintrapped ( ­ 2.5, 10 nm diameter). The field inside the particlis highly inhomogeneous, requiring rigorous calculation of thtrapping force. The arrow indicates the direction of the trappiforce. Same scaling as in Fig. 1(b).

VOLUME 79, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 28 JULY 1997

FIG. 4. Trapping potential of a particle (d ­ 10 nm, ´ ­ 2.5) in the vicinity of the tip. (a) Potential energy surface in the (x, z)plane (the tip is indicated by the shadow on the bottom plane). (b),(c) Normalized potential energy evaluated along thex and zdirections, respectively.

ten

)ththe

rely

ea.

thp

eue

s.s,aly

art

rtshisornng].d

ng

tsn

y

erpletip.e

or

n

To obtain the force field, the calculation was repeafor different center positions of the particle. The trappipotential (U) of a particle located atro was then deter-mined by

Usr0d ­ 2Z r0

`Fsrd dr . (5)

For a particle of diameterd ­ 10 nm, Fig. 4(a) showsthe potential energy surface in the (x, z) plane determinedfrom a grid of40 3 50 positions. Figures 4(b) and 4(cshow cross sections of the trapping potential alongforward and transverse directions of the tip. Sincetrapping potential is almost rotationally symmetric, wshow only results for the (x, z) plane. The potential isnormalized with the illuminating intensityI0 and withkT(k ­ Boltzmann constant,T ­ 300 K).

The potential in the (x, y) plane is quasiharmonic neathe potential minimum [Fig. 4(b)]. According to ththeory of Brownian motion of a particle in a harmonicalbound potential [12], the variance in thex direction nearthe potential minimum (x ­ 0) is given by

kx2l ­ kT

√≠2U≠x2

!21

. (6)

A similar equation holds for they direction. It followsthat for a trapping accuracy of 5 nm (kx2l1y2 or k y2l1y2),Eq. (6) necessitates thatI0 ­ 65 mWymm2. If the tip isplaced in the focus of a diffraction limited beam of ar0.1 mm2, the required laser power is 6.5 mW, which isnondestructive power level for nonresonant illumination

The trapping potential is sensitively dependent oncurvature and material of the tip, as well as the size, shaand dielectric constant of the particle being trappSmaller spheres require higher trapping power, becathe polarizability of the particle scales with its volum

dg

ee

a

ee,

d.se.

A sharper tip is preferred for trapping smaller particleAlthough it is generally easier to trap larger particlethere is also an upper limit for the particle size. Withparticle considerably larger than the tip radius, the rapiddecaying fields close to the tip affect only the nearest pof the particle’s surface.

The highly inhomogeneous field near the tip also exea mechanical torque on the particle being trapped. Ttorque can be calculated from a conservation law fangular momentum, similar to Eq. (2). Our calculatioshowed that the resulting torque tends to align the loaxis of an ellipsoidal particle towards the tip [Fig. 5(a)The final alignment of the particle, however, is dominateby the trapping potential (associated with the net trappiforce). The alignment of the particle in Fig. 5(c) isshown to be more stable than that in Fig. 5(b). With ilong axis perpendicular to the tip axis, a larger fractioof the particle’s volume is immersed into the highlinhomogeneous field near the tip.

A practical concern for the proposed tweezers is lasheating of the tip. High temperatures could cause samdamage and induce convection at the surface of theAccounting only for heat transport by conduction, thtemperature distribution is determined by

= ? skt=Td 2 cprsdTydtd ­ pabs , (7)

where kt , cp, and r are the thermal conductivity, theheat capacity, and the specific density, respectively. Fwater we used the valueskt ­ 0.61 3 1026 Wysmm Kd,cp ­ 4.18 JysgKd, r ­ 0.997 3 10212 gymm3, and forgold we applied kt ­ 3.17 3 1024 Wysmm Kd, cp ­0.129 JysgKd, r ­ 19.3 3 10212 gymm3. The sourceterm, pabs, denotes the absorbed power density giveby the solution for the electromagnetic fieldspabs ­s1y2d Rehjp ? Ej ­ sE2y2d Imh´j, wherej is the induced

647

VOLUME 79, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 28 JULY 1997

,ldai

gd

u

m

i

ten

e

beheer

orn

l.S.

all

at

l.,

,

tt.

,

eeR.l

by

allure

FIG. 5. Alignment of a prolate particle (long axis 10 nmshort axes 5 nm, ­ 2.5). (a) The field exerts a mechanicatorque on the particle and tends to align its long axis towarthe tip. Because the potential energy is lower for case (c) thfor case (b), the particle is more stable when trapped withaxis transverse to the tip. Same scaling as in Fig. 1(b).

current density. We solved Eq. (7) numerically, usinthe finite difference time domain method [13]. Tip anenvironment were divided into60 3 60 3 160 cubiccells with a grid size of 4 nm. A time step ofDt ­5 3 10217 s was used to ensure numerical stability.

Figure 6 shows the calculated steady-state temperatdistribution for I0 ­ 65 mWymm2. DT is the tempera-ture increase with respect to the original temperatu(300 K, for example). A maximum ofDT ­ 11.1 K isreached inside the tip at the location of the maximuabsorption [Fig. 1(b)]. The maximum value at the surfacof the tip, however, is onlyDT ­ 6.5 K. Note thatDTscales to the illuminating intensity. Our result indicatethat the temperature increase induced by laser heatis minimal for the intensity levels required for stabletrapping.

In the proposed trapping scheme, a sharp metalis brought to the focus of an illuminating beam whera particle has been trapped by conventional meaA polarization component along the tip axis enabletrapping of the particle to the near field of the tip. Thtrapped particle can be moved within the focal regio

FIG. 6. Distribution of the temperature increaseDT for anilluminating intensity ofI0 ­ 65 mWymm2.

648

sn

ts

re

re

e

sng

ip

s.s

n

of the illuminating light by translating the tip with apiezoceramic manipulator. The metal tip needs toinert and nonreactive to the trapped particle. Then, ttrapped particle can be released by turning off the lasillumination. The proposed scheme holds promise fnanometric manipulation of individual biomolecules itheir aqueous environment.

We thank Olivier Martin and Greg Schenter for helpfudiscussions. This research was supported by the UDepartment of Energy (DOE). Pacific Northwest NationLaboratory is operated for DOE by Battelle MemoriaInstitute under Contract No. DE-AC06-76RLO 1830.

Note added.—After submission of this Letter, an ar-ticle by Martin and Girard on the field enhancementpyramidal tungsten tips was published [14].

[1] A. Ashkin and J. M. Dziedzic, Science235, 1517 (1987),and references therein; S. Chu, Science253, 861 (1991).

[2] For reviews, see H. Heinzelmann and D. W. Pohl, AppPhys. A 59, 89 (1994); E. Betzig and J. K. TrautmanScience257, 189 (1992); X. S. Xie, Acc. Chem. Res.29,598 (1996).

[3] D. W. Pohl, in Forces in Scanning Probe Methods,editedby H.-J. Güntherodt, D. Anselmetti, and E. Meyer, NATOAdvanced Study Institutes, Ser. E, Vol. 286 (KluwerDordrecht, 1995), p. 235, and references therein.

[4] A. A. Gorbunov and W. Pompe, Phys. Status Solidi A145,333 (1994); J. Jersch and K. Dickmann, Appl. Phys. Le68, 868 (1996).

[5] Ch. Hafner, The Generalized Multiple Multipole Tech-nique for Computational Electromagnetics(Artech,Boston, 1990); L. Novotny, D. W. Pohl, and B. HechtOpt. Lett.20, 970 (1995).

[6] Since the tip size is still large compared to the mean-frpath of the electrons in gold [see, C. F. Bohren and D.Huffmann, Absorption and Scattering of Light by SmalParticles (Wiley, New York, 1983), p. 370], we believethat nonlocal effects are negligible.

[7] Different from a dielectric tip, we find that the intensityenhancement of a metal tip cannot be approximatedthat of a single polarizable sphere,4js´2 2 ´1dys´2 1

2´1dj2, ´2 and ´1 being the dielectric constants of tip andenvironment, respectively.

[8] W. Denk and D. W. Pohl, J. Vac. Sci. Technol. B9, 510(1991).

[9] The scattering force can be neglected because of the smparticle size and because there is no net radiation pressassociated with the evanescent fields close to the tip.

[10] J. A. Stratton, Electromagnetic Theory(McGraw-Hill,New York, 1941).

[11] J. P. Barton and D. R. Alexander, J. Appl. Phys.66, 2800(1989).

[12] S. Chandrasekhar, Rev. Mod. Phys.15, 1 (1943).[13] C. Temperton, J. Comput. Phys.34, 314 (1980).[14] O. J. F. Martin and C. Girard, Appl. Phys. Lett.70, 705

(1997).