Journal of the Korean Physical Society, Vol. 60, No. 1, January 2012, pp. 155∼158

Beam Optics Approach to the Ray Optics Model for the Optical TrappingEfficiency of Optical Tweezers

SungHyun Kim, HyunIk Kim, HyeongJoon Jun, HyunJi Kim and Cha-Hwan Oh∗

Department of Physics, Hanyang University, Seoul 133-791, Korea

(Received 13 October 2011, in final form 2 November 2011)

The optical trapping efficiency of optical tweezers can be explained by the ray optics model whenthe particle size is much larger than the wavelength of laser light.The Optical trapping efficiency inthe ray optics model does not depend on the particle size. However, The most experimental resultsusually show that the optical trapping efficiency decreases as the particle size decreases. In thisstudy, we suggest a beam optics approach in order to take into account the particle size dependenceeffect in the ray optics model.

PACS numbers: 87.80.CcKeywords: Optical tweezers, Ray optics model, Trapping efficiency, Beam opticsDOI: 10.3938/jkps.60.155

I. INTRODUCTION

Optical trapping efficiency (or force) control of opti-cal tweezers is an important factor for applying themproperly. Especially, in the case of a biological sample,unnecessarily strong force can cause serious damage tothe sample. There are two representative models for de-scribing the optical trapping efficiency of optical tweezers[1,2]. Barton et al. suggested the electromagnetic model[1], in which the optical trapping force is induced approx-imately by the dipole interaction between the light andthe particle. This model is proper to explain the opticaltrapping efficiency when the particle size is much smallerthan the wavelength of light. Ashkin suggested the rayoptics model [2], in which the optical trapping efficiencyis calculated by using the momentum transfers from thelaser light to the micro-particle during successive reflec-tions and refractions at the boundary surface of the par-ticle. This model can be applied when the particle sizeis much larger than the wavelength of laser light. Theoptical trapping efficiency is independent of the particlesize in the ray optics model. In practice, however, mostexperimental data show a particle size dependence of theoptical trapping efficiency even in the region of particlesize where the ray optics model can be applied. Exper-imental results usually show that the optical trappingefficiency decreases as the particle size decreases [3-5].

In this study, we suggest a beam optics approach toexplain the particle size dependence of the optical trap-ping efficiency in the ray optics model. A change in the

∗E-mail: choh@hanyang.ac.kr; Tel: +82-2-2220-0926; Fax: +82-2-2295-6868

Fig. 1. Schematic diagram of ray propagation in opticaltweezers.

maximum incident angle of the laser beam into the par-ticle was considered through the equation of Gaussianbeam propagation. The Calculated efficiency was com-pared with the measured efficiency on polystyrene mi-crospheres as a function of the normalized diameter.

II. BEAM OPTICS APPROACH TO THERAY OPTICS MDDEL

When a particle is trapped in a chamber, the forcesacting on the trapped particle are the optical trappingforce, the buoyant force, and the gravitational force. Theoptical trapping force is taken to be the difference be-tween the buoyant and the gravitational forces. There-

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Fig. 2. Schematic diagram of the change in the maximumincidence and angle according to the particle size (a) smallparticle and (b) large particle.

fore, the axial trapping efficiency is measured by reduc-ing the laser power until the microsphere falls out of thetrap. The axial trapping efficiency can be determinedexperimentally by uesing the following relation [6];

Qax =(ρs − ρm)Vsgc

n1P, (1)

where ρs and ρm are the densities of the microsphereand the surrounding medium repectively, Vs the volumeof a microsphere, P the incident laser power, c the speedof light, n1 the refractive index of surrounding medium,and g the gravitational acceleration.

In the ray optics model, the optical trapping efficiencycan be described by using the gradient and the scatter-ing efficiencies of the rays in the laser beam. The axialtrapping efficiency is given by [2]

Qax = 4× 2πw2

0

∫ π2

0

dβ

×∫ w0

0

[exp

(−2r2

w20

)(qgcosϕ− qssinϕ)

]dr, (2)

where qg and qs are the gradient and the scattering effi-ciency for a single ray, respectively. w0 and ϕ are thebeam waist and the maximum incidence angle of thebeam, respectively. However, in the case of a Gaussianbeam, the practical maximum angle depends on size ofthe particles because the converging angle of a focusedGaussian beam is changed dramatically near the focalpoint. As shown in Fig. 2, the incidence angle varies withthe size of the particles. In the figure, incident angle isdetermined by using the coordinates in the intersectionbetween the particle and the light ray.

As can be seen in Fig. 2, the maximum incidence angledecreases as the particle size increases. The beam radiusand the radius of curvature of the Gaussian beam are

Fig. 3. Experimental set up of the optical tweezers system

expressed as follows:

w = w0

√1 +

(z

z0

)2

, R = z

[1 +

(z0

z

)2]

, (3)

where z0 is the Rayleigh range and w0 is the beam waist.As shown in Fig. 2, the radius of particle and the beamradius are related by following relation;

z2 + w2 = r2, (4)

From Eqs. (3) and (4), the beam radius and the radiusof curvature can be expressed by Rayleigh range, thebeam waist, and the radius of particle as follows;

w = w0

√z20 + r2

z20 + w2

0

,

R = z0r2 + z2

0√(z2

0 + w20)(r2 − w2

0). (5)

Finally, the maximum incidence angle of the light rayis given by

ϕ = sin−1[w

R

]= sin−1

w0

√z20 + r2

z20 + w2

0

z0r2 + z2

0√(z2

0 + w20)(r2 − w2

0)

.

(6)

III. EXPERIMENT

Figure 3 shows the experimental set up of a opticaltweezers system. A linearly polarized laser diode (SDL-5432H1, 200 mW at 834 nm) was used as the trappinglaser source. The elliptical beam shape of the laser wasreshaped into a circular beam shape by using an anamor-phic prism pair before the beam passed through thebeam expander. A Beam expender composed of, two

Beam Optics Approach to the Ray Optics Model for the Optical Trapping Efficiency · · · – SungHyun Kim et al. -157-

Fig. 4. Images of trapped polystyrene spheres.

double convex lenses was used to adjust the beam ra-dius. The laser beam, which was focused strongly bythe objective lens, made it possible to trap and manipu-late the polystyrene spheres (refractive index, n = 1.59).The objective lens was an oil-immersion-type lens witha magnificat of 100 (Zeiss, NA = 1.25), and the radiusof its back aperture was 3 mm. The polystyrene sphereswere dispersed at low concentrations only in distilled wa-ter and were put into chambers with height H of about 50µm. The polystyrene sphere stage was moved by usingtwo computer-controlled the motorized controllers (New-port, M-UTM24PP.1, resolution; 0.1 µm). The distancebetween the cover glass and the trap position, T waschanged by using the motorized controllers. The opticaltrapping efficiency was examined for the case in whichthe direction of the laser polarization was along the y-axis. The shold power at which the polystyrene spherejust started to fall out from the optical tweezers was mea-sured, and the optical trapping efficiency was determinedfrom Eq. (1).

IV. RESULTS

Figure 4 shows the images of polystyrene spherestrapped axially (a) and (b) and transversely (c) and (d).In Figs. 4(a) and (b), the trapped polystyrene spheres onthe right are shown to be distinct. As three untrappedpolystyrene spheres were still near the bottom of thechamber, they were shown as indistinct; that is, theywere out of focus (Fig. 4(b)). In Figs. 4(c) and (d), thetrapped microsphere, indicated by the arrow was manip-ulated to the right near the bottom of the chamber.

Figure 5 shows the measured and the calculated resultsfor the optical trapping efficiency with the normalizeddiameter of polystyrene sphere. The normalized diame-

Fig. 5. Normalized optical trapping efficiency as a functionof the normalized diameter of polystyrene sphere.

ter was defined as the ratio of the polystyrene diameterto the focused beam diameter. Since the optical trap-ping efficiency of optical tweezers sensitively depends onthe focused beam diameter, the normalized diameter ismore proper for comparing our optical trapping efficiencywith other experimental results. In our experiment, thefocused beam diameter was 1.5 µm. Therefore, for ex-ample, a normalized diameter of 2.0 corresponds to a thepolystyrene sphere diameter of 3 µm. In the experiment,we measured the optical trapping efficiency for sphereswith diametes of 1, 2, 3, 5, 8, 10, 12, and 20 µm, whichcorrespond to normalized diameter of 0.7, 1.3, 2.0, 3.3,5.3, 6.7, 8.0, and 13.3, respectively.

In most reported papers, the ray optics model hasbeen applied to explain the optical trapping efficiency(or force) when the particle size is larger than 10 µm.As expected, however the measured trapping efficiencydecreased as the size of a polystyrene sphere decreasedeven in the region of sphere sizs where the ray opticsmodel is valid. The solid line (Beam Optics) representsthe results calculated Eqs. (1) and (6). We assumed thatthe ray optics model was sufficient to explain the opticaltrapping efficiency at a polystyrene sphere diameter of 20µm, and we set the measured efficiency at 20 µm as thenormalized maximum efficiency expected in the ray op-tics model. As can be seen in the figure, the calculatedresults agreed well qualitatively with the experimentalresults in the range of normalized diameters above 5.3(polystyrene sphere diameter of 8 µm). This means thatthe particle size dependence of the optical trapping ef-ficiency can be explain by the beam optics approach inthe region of sphere diameters from 8 to 20 µm. How-ever as for the region of normalized diameters below 5,the beam optics approach could no longer be applied.Therefore, the validity of the electromagnetic model wasexamined in the region of normalized diameters below5. The dashed line (eletromagnetic) represents the re-

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sult calaulated by using the electromagnetic model [1].The in set shows the result calculated by using the elec-tromagnetic model comparing with experimental results.As can be seen in the inset, the electromagnetic modelcan explain the optical trapping efficiency up to a nor-malized diameter of 2. As a result, the optical trappingefficiency in the region of normalized diameters above5 and below 2 could be explained by using the beamoptics approach to the ray optics model and the electro-magnetic model, respectively. However, in the interme-diate region of normalized diameter from 2 to 5, neithermodel could be applied to explain the optical trapping ef-ficiency. Further study for the optical trapping efficiencyin the intermediate region is needed.

V. CONCLUSION

We suggested a beam optics approach in order to takeinto account the particle size dependence effect in the rayoptics model and we showed that the size dependence ofthe optical trapping efficiency in the region of normal-

ized diameters from 5.3 to 13.3 (sphere diameters from 8to 20 µm) could be explained by using the beam opticsapproach. Adjunctively, we confirmed that the electro-magnetic model could be applied up to the normalizeddiameter of 2. Further study for the optical trappingefficiency in the intermediate region of normalized diam-eters from 2 to 5 is needed.

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Lett. 63, 715 (1993).[4] H. Felgner, O. Mller and M. Schliwa, Appl. Opt. 34, 977

(1995).[5] K. B. Im et al., J. Korean Phys. Soc. 40, 930 (2002).[6] W. H. Wright, G. J. Sonek and M. W. Berns, Appl. Opt.

33, 1735 (1994).