High trapping forces for high-refractive index particlestrapped in dynamic arrays of counterpropagating

optical tweezers

Astrid van der Horst,1,2,3,* Peter D. J. van Oostrum,2 Alexander Moroz,2,4

Alfons van Blaaderen,2 and Marileen Dogterom1

1Fundamenteel Onderzoek der Materie (FOM) Institute for Atomic and Molecular Physics (AMOLF),Kruislaan 407, 1098 SJ Amsterdam, The Netherlands

2Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University,Princetonplein 5, 3584 CC Utrecht, The Netherlands

3Currently at Department of Physics, Simon Fraser University, 8888 University Drive,Burnaby, British Columbia V5A 1S6, Canada

4Currently at www.wave-scattering.com

*Corresponding author: astrid.van.der.horst@sfu.ca

Received 6 November 2007; revised 1 May 2008; accepted 2 May 2008;posted 2 May 2008 (Doc. ID 89374); published 5 June 2008

We demonstrate the simultaneous trapping of multiple high-refractive index (n > 2) particles in a dy-namic array of counterpropagating optical tweezers in which the destabilizing scattering forces are can-celed. These particles cannot be trapped in single-beam optical tweezers. The combined use of twoopposing high-numerical aperture objectives and micrometer-sized high-index titania particles yieldsan at least threefold increase in both axial and radial trap stiffness compared to silica particles underthe same conditions. The stiffness in the radial direction is obtained from measured power spectra; cal-culations are given for both the radial and the axial force components, taking spherical aberrations intoaccount. A pair of acousto-optic deflectors allows for fast, computer-controlled manipulation of the indi-vidual trapping positions in a plane, while the method used to create the patterns ensures the possibilityof arbitrarily chosen configurations. Themanipulation of high-index particles finds its application in, e.g.,creating defects in colloidal photonic crystals and in exerting high forces with low laser power in, forexample, biophysical experiments. © 2008 Optical Society of America

OCIS codes: 140.7010, 170.4520, 230.1040.

1. Introduction

In 1986, Ashkin et al. introduced the optical trappingof dielectric particles using a single-beam gradienttrap [1]. In such a configuration, known as opticaltweezers, a large gradient in light intensity is createdby tightly focusing a laser beam using a high-numerical aperture (NA) objective. For particles witha refractive index np higher than the index nm of the

surrounding medium, this gradient provides the ne-cessary force Fgrad to balance the destabilizing scat-tering force Fsc. In general, an increase in np or inthe radius r of the particle yields an increase in theseforces, but the dependence on r and on m ¼ np=nm isstronger forFsc than forFgrad. This can clearly be seenin the Rayleigh regime (r ≪ λ, the wavelength of thetrapping laser), where these dependencies are givenby Fsc ∼ r6ðm2 − 1Þ2=ðm2 þ 2Þ2 and Fgrad ∼ r3ðm2 −

1Þ=ðm2 þ 2Þ [1]. For our particles, with r≃ λ in theMie regime, interactions are not described correctlyby Rayleigh scattering and the force relations are

0003-6935/08/173196-07$15.00/0© 2008 Optical Society of America

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more complicated. Also, although the physically in-sightful decomposition of the interaction of light withparticles of this size into Fsc and Fgrad becomes lessmeaningful, formally it can still be done, and onefinds that also for these particles Fsc depends morestrongly on r and m than does Fgrad. This limits thesize and refractive index of particles that can betrapped in a single-beam gradient trap and, with this,a limit is set to the trapping force of a single-beamgradient trap.By using a second, counterpropagating trapping

beam, the radiation pressure in the propagation di-rection can be canceled. Such counterpropagating ordual-beam traps have been used before in severalconfigurations [2–6]. In these cases, however, the fo-cusing of the laser beam was mild, either due to theuse of low-NA objectives or because the diameter ofthe beam was kept small in comparison with theaperture of the objectives; the possibility of a hightrapping force for a counterpropagating trap was,therefore, not fully exploited.Here we demonstrate the trapping of high-

refractive index titania particles in an array of coun-terpropagating traps of which the positions can bedynamically changed. We combine this counterpro-pagating trapping with overfilling of the twohigh-NA objectives, thereby canceling the radiationpressure, while a large intensity gradient providesa high trap stiffness. The array of counterpropagat-ing traps was created by time-sharing the laser beamover the positions of the array using a pair of acousto-optic deflectors (AODs), ensuring fast, computer-controlled dynamics of the individual traps in aplane. The possibility of trapping an array of high-index particles at once, and being able to manipulateindividual particles within such an array indepen-dently, finds its application in, for example, manipu-lating high-refractive index rodlike particles in threedimensions (3D), e.g., ZnO nanorods [7]; patterningsurfaces for colloidal epitaxy [8]; and creating defectsin colloidal photonic crystals. For titania and silicaparticles, the radial stiffness was derived from mea-sured power spectral density (PSD) curves [9]. In ad-dition, calculations were done to obtain both theaxial stiffness and the radial stiffness [10,11], includ-ing taking spherical aberrations due to the use of anoil immersion objective to trap in ethanol into account[12]. Both the measurements and the calculationsshow higher trap stiffnesses for titania particles com-pared to silica particles under the same conditions.Also, the effects of the use of high-NA objectives, par-ticle radius r, and refractive index n on the trap stiff-ness are investigated using calculations.

2. Experimental Methods

To allow for counterpropagating trapping in our set-up [Fig. 1(a)], the condenser of an inverted micro-scope (Leica DM IRB) is replaced by a high-NAobjective [7,13]. The distance between the two objec-tives (both: Leica 100×, 1:4NA, oil immersion) is0:52mm when overlapping their focal planes. An in-

frared laser beam (Spectra Physics Millennia,1064nm, 10W cw) is split at a polarizing beam split-ter cube, while the rotation of the wave plate (WP)determines the ratio between the power sent tothe inverted objective and that sent to the uprightobjective. In both beam paths a pair of lenses(L3i;u and L4i;u, all f ¼ 80mm) forms a telescope toprovide manual displacement of the laser focus.The use of dichroic mirrors DMi;u allows for illumina-tion and imaging in the visible, including confocalmicroscopy. Before splitting, the laser beam passesa pair of AODs (IntraAction DTD-276HB6), posi-tioned at a plane conjugate to the back focal planesof the objectives. The signal to the AODs is suppliedby direct digital synthesizers (Novatech DDS 8m)controlled by a LabVIEW (National Instruments)program. By fast repositioning of the laser focus,multiple time-shared traps are created [14]. The po-sition of the traps can be preprogrammed or changedinteractively. The beam is expanded in two steps: be-fore the AODs by a 6× beam expander and after theAODs ∼2× by the lenses L1 (f ¼ 120mm) and L2(f ¼ 250mm). For position detection of a particle ina nonshared trap, the fraction of the infrared beamthat leaks through dichroic mirror DMi is imagedonto a quadrant photodiode (QPD) that is placedat the front camera port of the microscope.

The sample cell, consisting of two coverslips (Men-zel No. 1) sealed together with candle wax, was filledwith a dilute mixture of SiO2 (n ¼ 1:45, diameter1:4 μm) and TiO2 (n ¼ 2:4, diameter 1:1 μm) particles.Ethanol (n ¼ 1:36) was the initial dispersion and, be-cause the refractive index of ethanol is close to,though slightly higher than that of water (n ¼1:33, commonly used in biophysical experiments),we kept the particles dispersed in ethanol.

Fig. 1. Schematics of (a) the optical tweezers setup and (b) thecounterpropagating traps scanned with one pair of AODs. A pat-tern of inverted beam traps (∘) is a mirror image of the pattern ofupright beam traps (×) with the mirror plane top-to-bottom in theimage. In addition, the magnification between the inverted andthe upright beam path might differ. By adding a mirrored and ap-propriately scaled pattern to the original traps, counterpropagat-ing traps (⊗) can be created for which the position of each trap canbe chosen arbitrarily.

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Apattern scannedby theAODs is imaged inside thesample by both objectives. Theupright objective, how-ever, will give amirror image of the pattern imaged bythe inverted objective [see Fig. 1(b)]. To not be limitedto symmetric patterns using a single pair of AODs, wescan both the desired pattern and its mirror image.Then, inside the sample, we place the pattern fromthe inverted beam opposite the mirror image fromthe upright beam, creating an array of counterpropa-gating traps. By scaling the added mirror image, wecan also compensate for differences in magnificationbetween the inverted and the upright beam paths in-troduced by the telescopes or by the use of objectiveswith different magnifications. We aligned the coun-terpropagating traps by visually checking on the cam-era image that a trapped silica particle did not changeposition when the inverted beam and the uprightbeam were alternately blocked.

3. Experimental Results

The method of creating the counterpropagating trappatterns by also scanning themirror image allows forrapid manipulation of multiple high-index particles,as can be seen in Fig. 2. Here, stills from a movieshow how patterns of eight traps were combined toform four counterpropagating traps (plus eightsingle-beam traps that were not used). The arrayof counterpropagating traps, positioned at a distanceof 12 μm from either wall, was filled with one silicaand three titania particles. The total power insidethe sample was 44mW, corresponding to 5:5mWfor each individual counterpropagating trap. Thepattern was scanned at 4:5kHz and changed in 34steps over a total period of 1:2 s, yielding an averagespeed of the particles of ∼20 μm=s. In bright-field mi-croscopy, the apparent size of the imaged particles islarger than the actual size [15]. For particles with ahigher refractive index this effect is more pro-nounced, causing the 1:1 μm diameter TiO2 particles

to appear larger than the 1:4 μm diameter SiO2particle.

Three-dimensional trapping of the TiO2 particlesin single-beam traps was not possible for any ofthe dozens of particles we tried to trap; they werepushed along the beam axis. This is in accordancewith the absence of a stable trapping position forthese particles in the single-beam calculations (Pre-sented in Section 4).

We measured the radial trap stiffness κx usingQPD measurements. By fitting a Lorentzian to thenormalized PSD curve of the QPD signal (Fig. 3)we obtained the roll-off frequency f 0 and usedκx ¼ 12π2ηrf 0, with η as the viscosity of the medium[16]. The high-frequency parts of the spectra for tita-nia deviate from the −2 slope typical for Brownianmotion. This is possibly caused by not capturingall the light with the high-NA objective for thesestrongly scattering titania particles. However, wedo not expect this to influence f 0 significantly. Forthese measurements, one stationary counterpropa-gating trap was used, with powers inside the trapof 22, 44, 88, and, only for the titania particle,176mW (measured using the two-objective method[17]). For a power of 44mW in the sample we foundf 0 to be 499Hz for the 1:1 μm diameter TiO2 particleand, with η ¼ 1:2 × 10−3 Ns=m2 for ethanol, thisyields a stiffness of κx ¼ 39pN=μm. The trap stiffnessincreased linearly with increasing laser power. Be-cause of the mechanical noise in the curves for the1:4 μm diameter SiO2 particle, we fitted these Lor-entzians manually. Assuming for the three curvesan increase in stiffness linear with laser power, wefound at 44mW an f 0 of 115� 10Hz, correspondingto κx ¼ 11:4pN=μm, thus, a factor of 3.4 lower stiff-ness than for the smaller titania particle.

4. Calculations

For our calculations we used the Mie–Debye repre-sentation first given by Maia Neto and Nussenzveig

Fig. 2. (Multimedia online; ao.osa.org) Stills from a movie inbright-field microscopy of four counterpropagating traps filledwith one 1:4 μm diameter SiO2 (arrow) and three 1:1 μm diameterTiO2 particles. The particles are positioned 12 μm away fromeither wall, and the pattern is changed in 34 steps in 1:2 s. Theshort version of the movie (1:8MB) shows this complete changeof the pattern four times, while the longer version of the movie(12:3MB) shows it 28 times. Scale bar is 2 μm.

Fig. 3. PSD curves of the normalized signal for a SiO2 (gray solidcurve, diameter 1:4 μm) and a TiO2 (black dotted curves, 1:1 μm)particle in ethanol for given laser powers inside the counterpropa-gating trap. For clarity only one curve is shown for SiO2. Lorent-zian fits (dashed curves) are plotted for the 44mW curves, with f 0at 115Hz (SiO2) and 499Hz (TiO2). Highest laser powers result inlowest plateau values.

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for the axial direction [10], expanded to 3D by Mazol-li and co-workers [11], and complemented by Vianaet al.[12] to include spherical aberrations (i.e., thefocal depth of a light ray depends on the distanceof this ray to the optical axis). In these calculations,the parameters describing the laser trap configura-tion are the beam opening angle θ and γ, the ratioof the objective focal length to the beam waist ω0.We used θ ¼ 64:245°, γ ¼ 1:21, and laser wavelengthλ ¼ 1064nm. The spherical aberrations, due to therefractive index mismatch between the coverslipand the ethanol, depend on the distance from the geo-metric focus to the glass surface. In our calculationsthis distance was 12 μm.In Fig. 4(a), the trapping force F per laser power P

(at the back focal plane) is plotted (F=P ¼ Qnm=c,with Q the trapping efficiency and c the speed oflight). Stable trapping occurs at the position whereFz ¼ 0 and dFz=dz < 0. In a single-beam gradienttrap a 1:1 μmdiameter SiO2 particle (solid curve) willbe trapped at a position in front of the focal plane ofthe objective (negative z) due to the spherical

aberrations [18,19]. For a TiO2 particle of the samesize, however, no stable trapping occurs (dashed–dotted curve).

To calculate the force on a particle in a counterpro-pagating (cp) trap, we averaged the curves for the up-right and the inverted beam, which in ourexperiment have perpendicular polarization. (Thecurve for the upright trapping beam is the negativeof the inverted one, with the z axis mirrored at theposition of stable trapping for the 1:4 μm diameterSiO2 particle, corresponding to the experimental con-figuration.) For the 1:1 μm diameter SiO2 particle wefind that the curve for counterpropagating trapping(dashed curve) does not differ much from the curvefor single-beam gradient trapping, which is in accor-dance with the small role played by the scatteringforce due to the small difference in refractive indexbetween silica and ethanol. For the TiO2 particlein the counterpropagating trap, however, we nowfind a stable trapping position (dashed curve). Wefind the axial trap stiffness κz (the absolute valueof dFz=dz at the trapping position) to be 877 and217pN=ðμm ×WÞ for 1:1 μm diameter TiO2 andSiO2 in ethanol, respectively. Thus, κz is 4× higherfor the TiO2 particles than for the SiO2 particle underthe same conditions. For 1:4 μm diameter silica par-ticles as used in our experiments κz ¼ 192pN=ðμm ×WÞ, yielding a factor of 4.6 (data not shown).

Figure 4(b) shows the radial forces at z ¼−0:628 μm (the calculated position of stable trappingfor 1:4 μm SiO2 with z ¼ 0 as the geometric focus ofthe objective). Because of the symmetric configura-tion, these radial forces are the same for both thesingle-beam and the counterpropagating configura-tion. We find stiffnesses of 3754, 1284, and655pN=ðμm ×WÞ for the 1:1 μm TiO2, 1:1 μm SiO2,and 1:4 μm SiO2, respectively, giving for the titaniaparticle an increase in κxy of 2:9× (compared to1:1μm SiO2) and 5:7× (compared to 1:4 μm SiO2).Also, in the axial as well as the radial direction,we see a higher maximum force for the trapped tita-nia particle as compared to silica.

Spherical aberrations,which spread out the focus ofthe laser beam and increase the opening angle of theouter light rays, affect the trapping forces [19] andcan, depending on the trapping configuration, eitherincrease or decrease trap stiffnesses. We thereforealso calculated the axial and radial forces while dis-regarding these aberrations [10,11]. The absence ofspherical aberrationswould, for example, be obtainedby using water immersion objectives to trap in water.In our calculations, however,weused ethanol to inves-tigate the effects only of the spherical aberrations andnot have an added effect of a change in ratio of the re-fractive indicesm. Theaxial trap stiffness κz [Fig. 5(a)]is 1043, 409, and 272pN=ðμm ×WÞ for 1:1 μmdiameter TiO2 and 1:1 μmand 1:4 μmSiO2 in ethanol,respectively. This corresponds to a κz that is 2:6×high-er for the TiO2 particle than for the SiO2 particle un-der the same conditions,while compared to the 1:4 μmdiameter silica particles (datanot shown) the increase

Fig. 4. Calculations of the force per 1W laser power for 1:1 μmdiameter TiO2 (n ¼ 2:4) and SiO2 (n ¼ 1:45) particles in ethanol(n ¼ 1:36) in Gaussian single and cp traps using 100×, 1:4NA,oil immersion objectives. Spherical aberrations are accountedfor. (a) Axial force, with z as the offset of the particle from the focalplane of one objective and (b) radial force at the trapping plane for1:4 μm SiO2 (z ¼ −0:628 μm ) as a function of the distance to theoptical axis. In addition, the curve for 1:4 μm SiO2 is shown.

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is 3:8×. The code to calculate these axial forces with-out spherical aberrations, based on [10], is madeavailable online [20]. For the radial stiffness κxy [Fig. 5(b)] we find 4921, 1399, and 732pN=ðμm×WÞ for the1:1 μm diameter TiO2, 1:1 μm SiO2 (increase is 3:5×),and 1:4 μm SiO2 (6:7×), respectively. Also here themaximum forces are higher for titania than for silica.The shown radial results are for z ¼ 0:028 μm, the cal-culated position of stable trapping for 1:4 μm SiO2 inethanol, chosenas theplane of symmetry for the coun-terpropagating beams when spherical aberrationsare not included.The trapping forces are not only influenced by the

refractive indices, but also by the distance betweenthe focal planes of the objectives, the beam diameterwith respect to their backapertures, and the particle’ssize parameter β ¼ r=λ. The dependencies are non-linear and these parameters have not been optimizedin our experiments nor in our calculations. To illus-trate this, we calculated for a varying refractive indexn, trap stiffnesses κz and κxy for a 1:4 μmdiameter par-ticle in ethanol. The curves in Fig. 6, with andwithout

spherical aberrations taken into account, show thatour experimentally used titania (n ¼ 2:4) does notgive the largest increase in trap stiffness, neither ax-ial nor radial. In addition, we investigated the oscilla-tions in Q, due to Mie resonances, for increasing sizeparameter β, seen both experimentally [21,22] as wellas in calculations [10,22]. We plotted κz and κxy asfunctions of the particle radius r for titania [Fig. 7(a)] and silica [Fig. 7(b)] in ethanol. The calculationstook spherical aberrations into account andweredonefor a single position—on the optical axis at z ¼−0:628 μmour chosen plane of symmetry for the coun-terpropagatingbeamconfiguration, as dictated by thesilica particles used. For the titania particles this po-sition is not for every r the position forwhichFz ¼ 0 (i.e., the axial trapping position).However, these resultsdo give an indication of the large variations in trapstiffness for changes in particle size, especially fortitania.

To investigate our choice for using high-NA (100×,1:4NA) objectives, we compare the calculations withthose for low-NA (60×, 0:85NA) objectives as used inseveral other counterpropagating tweezer configura-tions [4,5]. Figure 8 shows the results for the counter-propagating trapping of a 1:1 μm diameter TiO2particle in ethanol. In the radial direction the stiff-nesses are 4921 and 3085pN=ðμm×WÞ for 1:4NAand 0:85NA, respectively (data not shown), whilethe axial stiffnesses are 1043 and 103pN=ðμm ×WÞ, respectively. The use of high-NA objectivesyields a large increase in trap stiffness, especially inthe axial direction. (These results are in accordancewith the results of Rodrigo et al. [23] for their coun-terpropagating generalized phase contrast method;their calculated trapping efficiencies using low-NAobjectives are an order of magnitude lower than whatwe find for our high-NA objective configuration.)

Fig. 5. Calculations of the force per 1Wlaser power for 1:1 μmdia-meter TiO2 (n ¼ 2:4) and SiO2 (n ¼ 1:45) particles in ethanol(n ¼ 1:36) in Gaussian single and cp traps using 100×, 1:4NA,oil immersion objectives. Spherical aberrations are neglected.(a) Axial force, with z as the offset of the particle from the focalplane of one objective and (b) radial force at the trapping planefor 1:4 μm SiO2 (z ¼ 0:028 μm) as a function of the distance tothe optical axis. In addition, the curve for 1:4 μm SiO2 is shown.

Fig. 6. Calculated trap stiffness κ for varying indices of refractionn of a 1:4 μm diameter particle in ethanol (n ¼ 1:36) with and with-out spherical aberrations. Single-beam trapping (solid curves) islimited to n < 1:87; for higher n the particle will be pushed alongthe beam axis. Because of symmetry, where single-beam trappingis possible, the radial stiffness κxy is the same as κxy for cp trapping.

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5. Conclusions and Discussion

We demonstrated the trapping of high-refractive in-dex particles in counterpropagating optical tweezers

and found that, as expected from our calculations,these particles could not be trapped in single-beamoptical tweezers. In our calculations we comparedcounterpropagating trapping with high-NA objec-tives, as used in our experiments, with that of low-NA objectives, as used by several others [4,5].Although it should be noted that the axial positionof the beams was not optimized in either of the cases,the use of high-NA objectives yields a large increasein obtainable axial trap stiffness.

From our calculations between the optimal trap-ping size for a silica particle and a titania particleit follows that for counterpropagating traps 0:86 μmdiameter tiania particles can be trapped 9× strongeraxially than optimally sized silica particles (1:16 μmin diameter). These calculations do not take polariza-tion of the laser light, beam misalignment, transmit-tance of the objectives, or radial dependency of thetransmittance [24] into account. Also, any interfer-ence of the focused high-NA laser beams [25] has notbeen included. Therefore, our results give an upperbound of the increase in trap stiffness that can be ex-pected for particles with an index of n ¼ 2:4 inethanol.

In addition, the measured threefold increase in ra-dial trap stiffness for a 1:1 μm diameter TiO2 particlecompared to a 1:4 μm diameter SiO2 particle in etha-nol indicates that the use of high-index particles com-bined with counterpropagating optical tweezers isindeed a promising method to exert high forces atlow laser powers. Our calculations for this experi-mental configuration show an even higher obtainableincrease (6:7×). These results are relevant also foraqueous solutions (H2O, n ¼ 1:33) used in most bio-logical applications, as the difference in refractive in-dex as compared to ethanol is small. The discrepancybetween the absolute values of calculated and mea-sured forces might be due to many factors, includingdifferences in the particle diameters, refractive in-dices, beam alignment, effects of polarization inthe focus, and beam diameter. All these parametersplay a role in the increase of trapping forces, withnonlinear dependencies, such as the Mie resonancesfor increasing particle size. However, calculationscan be useful in sampling the expansive parameterspace to optimize trap stiffness or, e.g., maximumtrapping force.

The method of scanning both the pattern and itsmirror image provides flexibility in manipulatingmultiple high-index particles by supplying the possi-bility of arbitrarily chosen dynamic configurations ofthe counterpropagating traps while avoiding the costof a second pair of AODs.

We thank A. I. Campbell and J. S. Schütz-Widoniak for the TiO2 particles and J. P. Hoogen-boom for the SiO2 particles. This work is partof the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM), whichis financially supported by the Nederlandse Organi-satie voor Wetenschappelijk Onderzoek (NWO).

Fig. 7. Calculated trap stiffness κz and κxy for varying radius r of(a) titania (n ¼ 2:4) and (b) silica (n ¼ 1:45) particles in cp traps inethanol (n ¼ 1:36). The stiffness is calculated at fixed depthz ¼ −0:628 μm, which is not for every radius the position for whichFz ¼ 0. Spherical aberrations were taken into account.

Fig. 8. Calculated axial force (assuming no spherical aberrations)for a 1:1 μm diameter TiO2 particle (n ¼ 2:4) in ethanol (n ¼ 1:36)for high-NA objectives (100×, 1:4NA; solid curves) and low-NA ob-jectives (60×, 0:85NA; dashed curves). There is no stable trappingposition for the single-beam traps. In the counterpropagating (cp)beams, the axial stiffness κz at the stable trapping position is1043pN=ðμm×WÞ (1:4NA) and 103pN=ðμm×WÞ (0:85NA).

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