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Dynamic motion analysis of optically trappednonspherical particles with off-axis positionand arbitrary orientation

Jun-Sik Kim and Seung-Woo Kim

We present a general computational method of determining radiation pressure forces and torques exertedon small particles by a converging beam of light. This method, based on a ray optics model of opticaltrapping, allows time-series dynamic motion analysis to be performed on nonspherical objects that areinitially positioned off the optical axis with arbitrary orientation. Comparison tests of computer simu-lation with experimental results prove that the proposed model can be used to predict complicatedtrapping behavior of microfabricated objects. © 2000 Optical Society of America

OCIS codes: 140.0140, 140.3300, 140.7010.

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1. Introduction

The study of optical trapping of micrometer-sizedparticles began with spheres,1,2 and extended to non-pherical objects.3–6 Nonspherical examples of in-erest are microfabricated functional objects such asadiation pressure micromotors,7,8 ring cylinders,9

and scanning force microscopes.10 When the objectsize is sufficiently larger than the wavelength of light,the ray optics approach for determination of radiationpressure forces and torques acting on sphere particlesis applicable even to nonspherical objects.11–18 How-ever, nonspherical shapes generally lose the perfectgeometric symmetry of spheres, requiring that notonly the position but also the angular orientation ofthe object be taken into account for complete analysisof trapping behaviors.15–19 Here we propose a com-putational method of ray optics, which determinesthe radiation pressure forces and torques acting onarbitrary-shaped objects that are positioned off theoptical axis with arbitrary orientation. This modelis then extended to dynamic analysis to provide time-series kinetic trajectories of an object in the process oftrapping.

The authors are with the Department of Mechanical Engineer-ing, Korea Advanced Institute of Science and Technology, ScienceTown, Taejon 305-701, Korea. The e-mail address for S.-W. Kimis swk@kaist.ac.kr.

Received 12 November 1999; revised manuscript received 15May 2000.

0003-6935y00y244327-06$15.00y0© 2000 Optical Society of America

2. Ray Optics Model of Optical Trapping

Figure 1 shows the representative geometry of theoptical trapping that we investigated. A collimatedlaser beam is convergent in the upward directionagainst gravity by a high numerical aperture objec-tive with a focal length of f. The beam that entersthe aperture of the objective is assumed to maintaina Gaussian intensity distribution of I~r! 5 I0exp~22r2yw0

2! with a beam radius of w0, where r isthe radial distance measured from the optical axiswithin the aperture plane of the objective. The raypath follows the principles of ray optics for a simpleanalysis.11 A more elaborate hybrid approach canbe adopted to take into account features such as thewavelength and the nonzero focal spot radius.17 Butthe ray optics model reduces the subsequent burdenof numerical computation without significant dis-agreement with experimental observation if the par-ticle dimensions are an order of magnitude largerthan the wavelength and the focal spot radius of thelaser beam.20 When trapping starts, the object isinitially located off the optical axis with an arbitraryorientation in three dimensions.

To obtain necessary expressions for force andtorque equilibrium, the Cartesian coordinates frame$A% is defined with its z axis being in line with theptical axis and its origin being fixed at the focal pointf the upward converging beam. For the object tondergo a kinetic change, two moving Cartesian co-rdinate frames $A9% and $B% are additionally intro-uced to be embedded within the object with theirrigins located at the mass center of the object.

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Frame $A9% has the same orientation as frame $A%,whereas $B% is tilted so that its three orthogonal unitvectors always coincide with the principal axes of themomentum of inertia of the object. The position vec-tor of APBorg directs the origin of frame $B% in terms of$A% coordinates, whereas the angle vector of AQB 5A@ux uy uz#

T indicates the orientation of $B% with re-spect to $A%.

The coordinates ~x9, y9, z9! given in $B% are con-verted into their corresponding coordinates ~x, y, z! in$A%, following the transformation rule of

AP 5 A@x, y, z#T 5 BAR BP 1 APBorg,

where

BP 5 B@x9, y9, z9#T. (1)

In the expression in Eq. ~1!, the rotation matrix BAR

epends on the instantaneous values of AQB, which isexplicitly expressed as21

BAR 5 F ux

2X 1 C uxuy X 2 uz S uxuz X 1 uy Suyuy X 1 uz S uy

2X 1 C uyuz X 2 ux Suxuz X 2 uy S uyuz X 1 ux S uz

2X 1 CG , (2)

where u [ @ux2 1 uy

2 1 uz2#1y2, C [ cos u, S [ sin u,

X [ 1 2 cos u, ux [ uxyu, uy [ uyyu, and uz [ uzyu.The radiation pressure force is generated by the

interaction of the laser beam with the object followingthe rules of reflection and refraction. For analysis, arepresentative ray passed through point ~r, w! in theaperture plane of the objective with the direction co-sine vector of Adi 5 A@sin f cos w sin f sin w cos f#T.Cone angle f was obtained from the geometric rela-tionship of rmaxyr 5 tan fmaxytan f as depicted inFig. 1. Consequently, the trajectory line equation ofthe ray in frame $A% is described as

AP 5 A@x, y, z#T 5 Adi 3 s, (3)

where s is the scalar quantity of distance measuredfrom the origin of frame $A%. The surface that sur-rounds the object is conveniently described in frame

Fig. 1. Ray optics model for optical trapping of the upward beam.A cylindrical micro-object is shown to represent nonsphericalmicro-objects.

328 APPLIED OPTICS y Vol. 39, No. 24 y 20 August 2000

$B% by a set of implicit functions of F1~x9, y9, z9! 5 0,F2~x9, y9, z9! 5 0, . . . , Fm~x9, y9, z9! 5 0 such asBO 5 $~x9, y9, z9!u~x9, y9, z9! [ ~F1 5 0 ø · · · ø Fm 5 0!%.

(4)

To determine the radiation pressure force exerted bythe ray, the intersecting point of AP with BO needs tobe identified first. We performed this computationin frame $B% simply by transforming the ray equationof Eq. ~3! into frame $B% such as

BP 5 B@x9, y9, z9#T 5 Bdi 3 s 1 BPo, (5)

where ABR 5 B

AR21 5 BART, Bdi 5 B

ARTAdi, and BPo 5

BART~2APBorg!. We then determined the intersect-ing point by solving parameter s to equate Eqs. ~4!nd ~5!. If there are more than two possible inter-ept points, the intercept point generated by themallest s parameter represents the first surface en-ountered by the ray. Next, we computed the radi-tion pressure forces and torques generated by thendividual ray by considering the rules of ray reflec-ion and refraction following the method proposed byauthier.17 The resulting forces and torques are po-

larization dependent; for a linearly polarized beam,reflectance R is given as R 5 Rp cos2 a 1 Rs sin2 a,where a is the angle between the electric field vectorand the plane of incidence. For circularly polarizedand unpolarized cases, the reflectance becomes R 5~Rp 1 Rs!y2. Integration of the contributions of in-dividual rays over the full aperture of the objectivefinally gave the force components ~Fx9, Fy9, and Fz9!and torque components ~Tx9, Ty9, and Tz9! that affectthe whole object.

Kinetic behavior of the object in response to theresulting forces and torques is described by the well-known translation and rotation laws of dynamic mo-tion22:

Fx 5 mvx 1 bx vx, Fy 5 mvy 1 by vy,

Fz 5 mvz 1 bz vz 1 ~m 2 mm!g, (6)

Tx9 5 Ix9vx9 2 ~Iy9 2 Iz9!vy9vz9 1 btx9vx9,

Ty9 5 Iy9vy9 2 ~Iz9 2 Ix9!vz9vx9 1 bty9vy9,

Tz9 5 Iz9vz9 2 ~Ix9 2 Iy9!vx9vy9 1 btz9vz9, (7)

where m represents the mass of the object, v is thevelocity; b is the damping coefficient, mmg is thebuoyancy force, I is the moment of inertia, v is the an-gular velocity, and bt is the rotational damping coef-ficient. Note that the translational motions of Eqs.~6! are written in frame $A%, which conveniently takesinto account the gravity and buoyancy force withoutcoordinate transformation. On the other hand, therotational angular motions of Eqs. ~7! are described inframe $B%, where the principal moments of inertiaremain constant regardless of the relative orientationof $B% with respect to $A%. The reference point of thetorques in Eqs. ~7! coincides with the origin of rotat-ing frame $B%. Frame $B% is time variant as itchanges with the position and orientation of the ob-

~mi$

w

tg

vwc

jAmctmi

u

ject after trapping has started. Equations ~6! and7! are therefore numerically solved in a recursive

anner with a time step of Dt. With index k denot-ng the time duration such as t 5 kDt, moving frameB% is discretized as $B~k!% for k 5 0, 1, . . . . During

each computation, after the incremental motions oftranslational position and angular orientation werecomputed during time interval Dt, $B~k!% was updatedto $B~k 1 1!% with the new position vector APB~k11!organd the transformation matrix B~k11!

A R. The latteras modified by simple multiplication in the form of

B~k 1 1!A R 5 B~k!

A RB~k 1 1!B~k! R, (8)

where B~k11!B~k! R is the incremental rotational matrix

hat we obtained by substituting the incremental an-ular motions B~k!DQB~k11! 5 B~k!@Dux Duy Duz#

T deter-mined from Eqs. ~7! into the rotational matrixformula of Eq. ~2!. The computation of motion anal-ysis begins with the initial position and orientation ofthe object and continues until equilibrium is reachedwith no significant kinetic change or the object fallsout of the beam. Computation can be terminated byoperator intervention if optical trapping results incyclic repetitive motion or chaotic nonrepetitive mo-tion.

For accurate dynamic analysis, damping needs tobe identified for the nonspherical shape under con-sideration. The Stokes law provides an exact ana-lytical solution for spheres such as b 5 6pmR, whereR is the radius of the sphere and m is the dynamicviscosity of the medium. For nonspherical shapes,however, rigorous numerical computation of fluid dy-namics is generally required if the actual complexgeometry is to be considered in detail.23 In our in-estigation, to facilitate the computational burden,e approximated the translational damping coeffi-

ient to be b 5 Cf 3 6pmR, where Cf is the compen-sation factor that matches experimental results on

Fig. 2. Schematic of the experimental apparatus.

the basis that the object is assumed to be a spherewith effective radius R. The rotational damping co-efficient bt is also estimated as bt 5 br#2, where r# is theradius of gyration that lies in the r# 5 Ry2 ; R range.

3. Simulations and Experiments

Now the dynamic computation method proposed sofar is verified through computer simulation, and itsvalidity is examined by comparison with experimen-tal results. Figure 2 shows the schematic of the ex-perimental apparatus that we used for simulationand experiments. The light source that we used fortrapping is a TEM00 mode laser beam generated froman 826-nm-wavelength laser diode and subsequentlycollimated in a 4-mm-diameter beam. A 1003 ob-ective with 0.7 numerical aperture ~Mitutoyo M Planpo! focuses the beam onto the micro-objects sub-erged in water. A CCD camera observes the

hange in motion of the objects during trappinghrough an infrared block filter. Figure 3 shows theicro-objects to be tested, made of fluorinated poly-

mide ~6FDAyTFDB! with a 1.525 refractive index

Fig. 3. Scanning electron microscope photograph of the fabricatedmicro-objects.

Fig. 4. Computed z-axis force in frame $A% versus x-rotation anglex and y-rotation angle uy at position ~1,1,1! mm.

20 August 2000 y Vol. 39, No. 24 y APPLIED OPTICS 4329

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and fabricated by optical lithography with subse-quent oxygen-reactive ion etching.24,25

A series of simulations were performed on a ringcylindrical object with a 3.5-mm inner radius, a 7-mmouter radius, and a 4-mm height. The ring cylinder

as a refractive index of 1.525 and the surroundingater has a refractive index of 1.33. The variation

f the force with the orientation of the object is de-cribed in frame $A%, and the torque is presented in

frame $B%. The laser beam has 30-mW power in thetrap region. Figures 4–6 show the instantaneousresults when the object was stationary at x 5 1 mm,

5 1 mm, and z 5 1 mm off the optical axis, and itsrientation varied in the 0° # ux # 180°, 0° # uy #

180°, and uz 5 0° ranges. Figure 4 reveals that the-axis force that determines whether the object fallsr rises at the particular position depends on therientation of the object. Figure 5 shows the x-axisorce Fx that was negative for 0° # uy # 30° and 0° #

x # 180°, which implies that with the particularrientation the object experiences a pulling force to-

Fig. 6. Computed x-axis torque in frame $B% versus x-rotationangle ux and y-rotation angle uy at position ~1,1,1! mm.

330 APPLIED OPTICS y Vol. 39, No. 24 y 20 August 2000

ward the laser beam axis. Otherwise, Fx becomespositive for the object to be pushed away from thebeam. Similarly, Fig. 6 indicates that the x-axistorque Tx becomes positive for 0° # ux # 90° andnegative for 90° # ux # 180°. If Tx is positive thex-axis angle of the cylinder increases and vice versa.This observation implies that the equilibrium point ofthe x-axis torque exists at ux 5 90°. But the object athis position with a stable torque angle of 90 deg isot in force equilibrium. Figures 7–9 show otheresults when the object is positioned at x 5 4 mm, y 5

0 mm, and z 5 2 mm off the optical axis, and itsorientation varies in the 0° # ux # 180°, 0° # uy #180°, and uz 5 0° ranges. Figure 7 reveals that, forhe z-axis force, Fz highly depends on the orientation

of the object. Figure 8 shows that Fx is negative for30° # uy # 120°, which implies that with the partic-ular orientation the object experiences a pulling forcetoward the laser beam axis. Otherwise, Fx becomespositive for the object to be pushed away from thebeam. Similarly, Fig. 9 indicates that the x-axistorque Tx becomes positive for 0° # ux # 90° andnegative for 90° # ux # 180°.

Fig. 8. Computed x-axis force in frame $A% versus x-rotation anglex and y-rotation angle uy at position ~4,0,2! mm.

Fig. 5. Computed x-axis force in frame $A% versus x-rotation anglex and y-rotation angle uy at position ~1,1,1! mm.

Fig. 7. Computed z-axis force in frame $A% versus x-rotation anglex and y-rotation angle uy at position ~4,0,2! mm.

1

rFt

sfiTmrm

mt

mt

m

Now Figs. 10 and 11 show two results of dynamicsimulation for two initial conditions: x 5 1 mm, y 5

mm, z 5 1 mm, ux 5 0, uy 5 0, and uz 5 0; and x 54 mm, y 5 0 mm, z 5 2 mm, ux 5 0, uy 5 0, and uz 5 0.The time-series kinetic trajectory of the object differsdepending on the initial conditions, but, as trappingproceeds, the object is pulled into an identical equi-librium position. The time interval used for this nu-merical computation was 1.0 ms. The dynamicsimulation result of Fig. 11 is animated in series inFigs. 12~a!–12~d! in direct comparison with the cor-esponding experimental results as illustrated inigs. 12~e!–12~h!. The 30-mW laser beam in the

rap region is irradiated perpendicular to the plane of

Fig. 9. Computed x-axis torque in frame $B% versus x-rotationangle ux and y-rotation angle uy at position ~4,0,2! mm.

Fig. 10. Dynamic simulation results for initial conditions of x 5 1m, y 5 1 mm, z 5 1 mm, ux 5 0, uy 5 0, and uz 5 0: ~a! position

rajectories and ~b! orientation trajectories.

Fig. 11. Dynamic simulation results for initial conditions of x 5 4m, y 5 0 mm, z 5 2 mm, ux 5 0, uy 5 0, and uz 5 0: ~a! position

rajectories and ~b! orientation trajectories.

the photograph with its focal point indicated by thecross mark in Figs. 12. This dynamic simulationreveals that the final position of the ring at equilib-rium is 4.2-mm offset from the optical axis. The finaltable orientation is 90° rotated from its initial con-guration, being in parallel with the optical axis.hese simulation results are generally in good agree-ent with the experimental data. We estimated the

eal damping coefficients in the simulation by deter-ining Cf to match the animated time-series images

with the experimental data of linear and rotationalmotion. The results are b 5 4.0 3 1027 Nsym, bt 51.2 3 10217 N ms, Cf 5 3, and r# 5 0.8R, where R isthe outer radius of the cylindrical ring.

In the experiments, the bottom wall of the samplechamber on which the cylindrical ring is initiallyplaced prevents the ring from rotating perpendicularto its initial configuration and retards the time ittakes to reach equilibrium. To eliminate this walleffect for more accurate comparison with simulationdata, we trapped the object on the bottom of the sam-ple chamber and intentionally dragged it upward by

Fig. 12. Comparison of animated simulation results with exper-imental results for initial conditions of x 5 4 mm, y 5 0 mm, z 5 2

m, ux 5 0, uy 5 0, and uz 5 0 performed in water. A 30-mW laserbeam was irradiated perpendicular to the plane of the photographwith the focal points indicated by the crosses. Simulation resultsof the time that elapsed after trapping began is ~a! 0.0 s, ~b! 0.6 s,~c! 1.0 s, ~d! 1.2 s. The ring cylinder is represented by a solidmodel for animation. Experimental results ~e! before trappingand after ~f ! 0.6 s, ~g! 1.0 s, ~h! 1.2 s time has elapsed.

20 August 2000 y Vol. 39, No. 24 y APPLIED OPTICS 4331

ticles by use of Nd:YAG and Ti:Al O lasers,” Opt. Lett. 19,

4

moving the laser focus. Then we freed the ring toallow it to fall downward by blocking the laser beamuntil the orientation of the ring was close to the in-tended initial orientation of ~ux 5 0°, uy 5 0°, uz 5 0°!in simulation. Then before the falling ring cameinto contact with the bottom, we retrapped the objectby unblocking the laser and its time-series kinetictrajectory is observed. Both experiment and simu-lation reveal that the different initial position andany slight deviation in the initial orientation fromzero angles results in different final orientations atequilibrium. However, it is noted that the final po-sition is invariantly found at 4.2-mm offset from theoptical axis as depicted in Fig. 12~d!. In addition,the final orientation is always rotated 90 deg parallelto the optical axis, whatever its actual angles are atequilibrium. When there is no damping, the objectcan rotate freely about the optical axis under slightdisturbance without any additional energy input.This phenomenon is in good agreement with thatreported previously in Ref. 8.

4. Conclusions

We have presented a general computational methodof determining radiation pressure forces and torquesexerted on small particles by a converging beam oflight. This method, based on a ray optics model ofoptical trapping, allows time-series kinetic analysisto be performed on nonspherical objects that are ini-tially positioned off the optical axis with arbitraryorientation. Comparison of computer simulationwith experimental results proved that the proposedcomputational method provides a good quantitativeanalysis of actual dynamic behavior of micro-objects,providing a useful means for the design of nonspheri-cal micro-objects such as a radiation pressure micro-motor and optical probes for scanning forcemicroscopes.

References1. A. Ashkin, “Acceleration and trapping of particles by radiation

pressure,” Phys. Rev. Lett. 24, 156–159 ~1970!.2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Ob-

servation of a single-beam gradient force optical trap for di-electric particles,” Opt. Lett. 11, 288–290 ~1986!.

3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipu-lation of viruses and bacteria,” Science 235, 1517–1520 ~1987!.

4. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trappingand manipulation of single cells using infrared laser beams,”Nature ~London! 330, 769–771 ~1987!.

5. S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bac-terial flagella measured with optical tweezers,” Nature ~Lon-don! 338, 514–518 ~1989!.

6. S. Sato and H. Inaba, “Second-harmonic and sum-frequencygeneration from optically trapped KTiOPO4 microscopic par-

332 APPLIED OPTICS y Vol. 39, No. 24 y 20 August 2000

2 3

927–929 ~1994!.7. E. Higurashi, H. Ukita, H. Tanaka, and O. Ohguchi, “Opti-

cally induced rotation of anisotropic micro-objects fabricatedby surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 ~1994!.

8. E. Higurashi, R. Sawada, and T. Ito, “Optically induced rota-tion of a trapped micro-object about an axis perpendicular tothe laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 ~1998!.

9. E. Higurashi, O. Ohguchi, and H. Ukita, “Optical trapping oflow-refractive-index microfabricated objects using radiationpressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 ~1995!.

10. L. P. Ghislain and W. W. Webb, “Scanning-force microscopebased on an optical trap,” Opt. Lett. 18, 1678–1680 ~1993!.

11. A. Ashkin, “Forces of a single-beam gradient laser trap on adielectric sphere in the ray optics regime,” Biophys. J. 61,569–582 ~1992!.

12. G. Roosen and C. Imbert, “Optical levitation by means of twohorizontal laser beams: a theoretical and experimentalstudy,” Phys. Lett. A 59, 6–8 ~1976!.

13. W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, “Lasertrapping in cell biology,” IEEE J. Quantum Electron. 26,2148–2157 ~1990!.

14. R. C. Gauthier and S. Wallace, “Optical levitation of spheres:analytical development and numerical computations of theforce equation,” J. Opt. Soc. Am. B 12, 1680–1686 ~1995!.

15. R. C. Gauthier, “Ray optics model and numerical computationsfor the radiation pressure micro-motor,” Appl. Phys. Lett. 67,2269–2271 ~1995!.

16. R. C. Gauthier, “Theoretical model for an improved radiationpressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 ~1996!.

17. R. C. Gauthier, “Trapping model for the low-index ring-shapedmicro-object in a focused lowest-order Gaussian laser-beamprofile,” J. Opt. Soc. Am. B 14, 782–789 ~1997!.

18. R. C. Gauthier, “Theoretical investigation of the optical trap-ping force and torque on cylindrical micro-objects,” J. Opt. Soc.Am. B 14, 3323–3333 ~1997!.

19. R. C. Gauthier and M. Ashman, “Simulated dynamic behaviorof single and multiple spheres in the trap region of focusedlaser beams,” Appl. Opt. 37, 6421–6431 ~1998!.

20. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametricstudy of the forces on microspheres held by optical tweezers,”Appl. Opt. 33, 1735–1748 ~1994!.

21. J. J. Craig, Introduction to Robotics ~Addison-Wesley, Reading,Mass., 1989!.

22. T. R. Kane and D. A. Levinson, Dynamics: Theory and Ap-plications ~McGraw-Hill, New York, 1985!.

23. M. P. Omar, M. Mehregany, and R. L. Mullen, “Electric andfluid field analysis of side-drive micromotors,” J. Microelectro-mech. Syst. 1, 130–140 ~1992!.

24. E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, and K. Itao,“Fabrication and optical rotation characteristics of anisotropi-cally shaped micro-objects made of fluorinated polyimide,” J.Jpn. Soc. Prec. Eng. 61, 1021–1025 ~1995!.

25. E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, and R.Sawada, “Optically induced rotation of dissymmetricallyshaped fluorinated polyimide micro-objects in optical traps,”J. Appl. Phys. 82, 2773–2779 ~1997!.