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IEEE TRANSACTIONS ON EDUCATION, VOL. 54, NO. 1, FEBRUARY 2011 133

Understanding Optical Trapping Phenomena:A Simulation for Undergraduates

Josep Mas, Arnau Farr, Jordi Cuadros, Ignasi Juvells, and Artur Carnicer

AbstractOptical trapping is an attractive and multidisci-plinary topic that has become the center of attention to a largenumber of researchers. Moreover, it is a suitable subject foradvanced students that requires a knowledge of a wide range oftopics. As a result, it has been incorporated into some syllabusesof both undergraduate and graduate programs. In this paper,basic concepts in laser trapping theory are reviewed. To providea better understanding of the underlying concepts for students,a Java application for simulating the behavior of a dielectricparticle trapped in a highly focused beam has been developed.The program illustrates a wide range of theoretical results andfeatures, such as the calculation of the force exerted by a beam

in the Mie and Rayleigh regimes or the calibration of the trapstiffness. Some examples that are ready to be used in the classroomor in the computer lab are also supplied.

Index TermsEducational technology, electromagnetic engi-neering education, geometrical optics, laser applications, physicaloptics.

I. INTRODUCTION

OPTICAL tweezers (OT) are highly focused light beamsthat are able to trap and manipulate small particles (from

micron to nanometer size) by taking advantage of the radiation

pressure effect. Nowadays, OT systems are a basic research toolin biophysics; for instance, single-molecule DNA experimentsor some cellular studies require the use of OT [1]. Particle ac-celeration induced by radiation pressure was first reported byAshkin [2], who was able to trap microscopic spheres using twocounterpropagating laser beams. In 1986, optical trapping witha single highly focused laser beam was achieved [3]. The trap-ping is achieved by momentum exchange between the beam andthe trapped particle via a scattering process. The force exertedon the particle (or bead) by the light can be decomposed intoa scattering component, which pushes the bead in the directionof propagation of the beam, and a gradient force that attractsthe bead toward regions of higher intensity. In order to create a

Manuscript received December 30, 2009; accepted March 16, 2010. Date ofpublicaton April 19, 2010; date of current version February 02, 2011. This workwas supported in part by the Agency for Management of University and Re-search Grants (Government of Catalonia) Grant MQD2007-00042 and the Insti-tute of Education Sciences (UB) grants REDICE06 B0601-02 and REDICE08B0801-01. J. Mas was supported by a FPU grant from the Spanish Ministry ofScience and Innovation. A. Farr was supported by a FI grant from the Govern-ment of Catalonia.

J. Mas, A. Farr, I. Juvells, and A. Carnicer are with the Applied Physics andOptics Department, Universitat de Barcelona, Barcelona 08028, Spain (e-mail:[email protected]).

J. Cuadros is with the Institut Qumic de Sarri, Universitat Ramon Llull,Barcelona 08022, Spain.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TE.2010.2047107

stable trap, the axial gradient force should match the scatteringforce. This can be achieved by focusing the beam using a highnumerical aperture (NA) objective.

In addition to their ability to manipulate small beads, OT canalso constitute a useful technique for measuring forces in the mi-croscopic world. When an external force is applied on a trappedparticle, it moves toward a new equilibrium position where theoptical force compensates the external force. As the trap poten-tial is assumed to be harmonic, it can be calibrated in a processthat involves obtaining the stiffness constant. Recently, several

experimental methods have been developed to measure the forceexerted by a light beam [4].Forces involved in optical trapping experiments can be pre-

dicted using the generalized LorentzMie theory (GLMT) [5].The GLMT is a general, rigorous, and rather complex approachthat provides accurate results. However, when the size of theparticle is much smaller than the wavelength of the beam, theso-called Rayleigh approximation can be used [6]. The Rayleighapproach is simpler than GLMT and supplies accurate enoughresults for most purposes within its range of applicability. Incontrast, the ray optics model (Mie regime) provides a simpleand intuitive description of optical trapping for particles whosediameter is larger than the wavelength of the laser used [7].

Despite the fact that OT are still a very active research area,optical trapping experiments are being introduced into someundergraduate and Masters programs in photonics, nanotech-nology, or biophysics. OT currently represent an attractive andmultidisciplinary area of study suitable for advanced studentswith a knowledge of a wide range of fields. For instance,building an OT setup involves optical engineering (opticalsystem design, microscopy), physics (electromagnetic theory,fluid mechanics), photonics (laser technology), electrical en-gineering (signal processing, hardware control), and so on.Related to this, several papers have been published pointing outthe usefulness of OT as an educational tool [8], [9].

Considerable knowledge of the underlying theories is neces-

sary to make the most of OT experiments. As the number ofpublished papers in this field is huge, students (and instructors)will have to access a large amount of reference work. In orderto provide basic and digestible information, the use of a numer-ical simulation as a complementary tool is suggested. Some OTsimulations can be found on the Internet. For instance, the readercan look up [10] (a simple animation of a trapped bead) and [11](a nice illustration of a DNA stretching experiment).

In this paper, a computer program designed to train studentsor nonspecialists in the basic Rayleigh and Mie theories is pre-sented; in addition, some illustrative results in OT theory thatcan be carried out using the same piece of software are dis-cussed. The article is organized as follows. In Section II, the

basic concepts involved in optical trapping theory, including

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134 IEEE TRANSACTIONS ON EDUCATION, VOL. 54, NO. 1, FEBRUARY 2011

force calibration are reviewed. How the application works is de-scribed in Section III, and several classroom activities are intro-duced in Section IV. Some results from laser trapping theoryusing the program are analyzed in Section V. A preliminary as-sessment of the innovation is included in Section VI, and finally,the conclusions are presented in Section VII.

II. OPTICAL TRAPPING REVIEW

Ashkin [2] demonstrated that a highly focused beam is able toaccelerate microscopic objects. Photons can interact with matterby transferring their momenta, which means that light is able toexert a force on a particle. OT setups require the use of a micro-scope objective to concentrate the photons coming from a laserbeam at a tiny spot. In this way, the target particle interacts withas many photons as possible since the momentum of each singlephoton is very small. A wide spectrum of accurate and sophisti-cated theories has gradually been developed to explain the trap-ping of particles of different sizes (see Section VI of [4] for a de-

tailed overview). None of them provides an exact description ofan optical trap, but predictions from the numerical solutions ofall the existing models span a significant range of experimentalsituations, leading to a fairly good understanding of the trappingproblem. However, for spherical particles (beads), the problemis usually described by approximations, which depend on theradius of the object used in the experiment and either the wave-length or the waist size of the beam.

Rayleigh regime : Forces acting on the bead aredescribed using electromagnetic theory [3].1

Mie regime : The interaction may be calculatedby means of ray optics [7].

A. Forces in the Rayleigh Regime

In the Rayleigh regime, spherical particles become dipolesthat scatter light. Let us suppose a bead is illuminated bya linearly polarized (1, 0) paraxial Gaussian beam

[12], propagating along the z-axis with beam waistradius and wavenumber ; is the wavelength inthe considered media. Let and be the refractive indicesof the medium and the particle, respectively; is the radiusof the particle; ; is the speed of light; is thepermittivity of free space; and is the wavelengthin the vacuum. Then, the beam intensity, , i.e., thetime-average of the Poynting vector, is written as

(1)

The polarization induced on the dipole is

, where is the polarizability of

the particle. The laser beam exerts a force on the dipole that

has two different components: gradient and scattering .The former is proportional to the gradient of the wave intensity,

1The dipole model in the Rayleigh approximation requires a homogeneouselectric field over the whole particle, which leads to a restriction on the size ofthe particle a < w . The Rayleigh regime is accurate enough to describe lateral

forces, which are what most experiments are concerned with. Nevertheless, asHarada et al. previously showed [6], a remarkable discrepancy appears in theaxial component, which is not accurately described above a = = 2 0 .

Fig. 1. Optical forces acting on a microsphere located near the focus of theobjective lens.

which means that points toward the focus of the microscope

objective (see Fig. 1). Following [6], is written as

(2)

The second component of the force is due to the scatteringand depends on the number of photons hitting the bead. It is anaxial force that pushes the sample along in the -direction.

(3)

The total force exerted on the particle is. The two contributions to the total

force, scattering and gradient, compete along the -direction(optical axis). The -component of the total force is zero

for

(4)

At this point, not far from the focus of the Gaussian beam, thetransverse components of the force are also zero, and thereforethe microsphere remains in an equilibrium position. In order toproduce a stable trap, the gradient force along the optical axishas to be larger than the scattering contribution. This is truewhen [3]

(5)

This relation is fulfilled only when the laser beam waistis small enough. For this reason, OT must be designed usingmicroscope objectives with high NA. If not, as the scattering

force increases, the gradient component does not compensatethe scattering force along the z-axis.

The main drawback of the Rayleigh regime comes from thelack of accuracy in the predictions for high NA because theunderlying theory has an intrinsically paraxial nature. In thesecases, the theory is restricted to providing a qualitative descrip-tion of the system [6], [13].

B. Forces in the Mie Regime

As explained above, if the diameter of the particle is muchlarger than the wavelength, the force exerted by the laser beamon a spherical particle is computed using the ray optics ap-proach. When the beam interacts with the particle, rays change

their direction according to the laws of geometrical optics. Theinteraction is described taking into account the reflections and

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MAS et al.: UNDERSTANDING OPTICAL TRAPPING PHENOMENA: A SIMULATION FOR UNDERGRADUATES 135

Fig. 2. Optical ray path within a spherical particle.

Fig. 3. Force component F ( x ) ( = 1 0 6 4 nm, P = 1 0 0 mW, w =0 : 6 m, n = 1 : 3 3 , n = 1 : 5 8 , a = 5 0 nm).

refractions inside the particle (see Fig. 2). The power of theincident ray is reduced to after reflections, whereand stand for the Fresnel coefficients [14]. Ashkin derivedanalytical expressions for the transverse (gradient) and longitu-dinal (scattering) components of the force exerted by a single

ray of power [7].

(6)

where and are the angles of incidence and refraction,respectively.

The ray optics approximation is particularly useful whendealing with large beads. The model is not restricted to low-con-vergence beams, so it not only explains the behavior of the trap,but also gives a reasonable estimate of its mechanical propertiesin real experiments [13].

C. Measurement of Transverse Forces

In both regimes, the transverse force components(and ) exerted on a particle vary linearly for small dis-placements from the center of the trap (Fig. 3). When the beadis moving close to the center, i.e., within about half of the beamwaist radius , the force is proportional to the position

, , and therefore the potential well is harmonic

. Similarly, the force in the -direction can alsobe described by .

Since the force exerted by the laser beam is proportional tothe displacement of the sample, OT are used to measure forcesin the microscopic world. As the spring constant can be ob-tained experimentally, a precise measurement of the position ofthe sample within the laser spot is required to obtain the forceexerted by the beam. The procedure for obtaining is basedon the analysis of the Brownian motion of the trapped sample.

One calibration method for is based on the reconstructionof the optical potential from the histogram of the position ofthe bead. The particle moves randomly within the trap, but thebead spends more time in those places where the optical poten-tial energy is lower. According to Florin [15], the probabilitydistribution of the position is related to the potential energy,thus

(7)

where is Boltzmanns constant and is the temperature. Fora large number of measurements of , the histogram provides

a good estimate of the spatial probability density (exceptfor a normalization constant), and the trap stiffness can beestimated from the variance of

(8)

The motion of a microparticle can be easily simulated usingNewtons second law (Langevin equation)

(9)

Here, is the drag force, is the drag coefficient,is the viscosity of the medium, and finally, is a

random force that accounts for the Brownian motion due to thewater molecules. For polystyrene or silica beads or cells, theReynolds number is very small, and therefore the inertial term

in (9) can be ignored [16].

III. DESCRIPTION OF THE APPLICATION

The applet simulates the behavior of a spherical dielectricparticle trapped by means of a laser beam focused by a mi-croscope objective. The program and the source code can bedownloaded from http://code.google.com/p/optical-tweezers/.The simulation, including the movement of the particle, isconfined to the plane. The calculation of the trapping force

is carried out using the geometrical optics approach (Mieapproximation) or in the Rayleigh approximation, according tothe size of the particle. The user can select the proper workingconditions via the selector located in the left-hand panel of thescreen.

In the left-hand panel (Fig. 4), a transverse section of the op-tical trap is shown. The z-axis corresponds to the beam propaga-tion direction, while the x-axis is the transverse coordinate. Notethat the light is coming from the bottom of the panel. Both theparticle studied and the trapping light beam are represented inthis panel. The optional buttons below the trap diagram allow thevisualization of three vectors representing the scattering, gra-dient, and net force exerted by the beam at the current position

of the particle. The current modulus of the net force is shownbelow the trap diagram.

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Fig. 4. Experimental settings tab.

The user can click on the particle and drag it to a new position.Starting from the position selected, a simulation of the particlestrajectory is calculated when the Animation button is clicked.For every frame of animation, the time is shown below the di-agram. The button Take bead to focus is used to place theparticle in the center of the panel. The data required to performthis simulation come from the dynamic solution of the Langevinequation (9). The time step used for the integration of the dif-ferential equation can be modified by activating the optionalTime step slider in the View menu. Special care must betaken when selecting large time-step values: If the time step is

much longer than the characteristic timescale of the movementof the bead within the optical trap (which depends on the exper-imental settings selected), the program could return a nonreal-istic animation.

In the Mie regime, the light beam is represented as a group ofseveral converging light rays in the plane, which are refractedthrough the dielectric particle. The momentum transferred to theparticle, i.e., the force exerted by the beam, is obtained by using(2) and adding the contribution of each ray. The power associ-ated with each ray is weighted according to a Gaussian intensityprofile. In the Rayleigh regime, the laser beam is represented bythe intensity map of the Gaussian beam.

In the right-hand panel, there are two different tabs for con-

figuring and analyzing the experiment. In the first tab (Exper-imental settings), the user can tune the values of the variables.The parameters available are the power and the wavelength ofthe laser, the NA of the microscope objective, the overfillingfactor, the radius and refractive index of the trapped particle, theviscosity and refractive index of the medium, and the tempera-ture. Notice that when using high NA objectives, the divergenceof the beam cannot be easily related to the beam waist. In thiscase, when the paraxial formula is not strictly valid, the valuesof NA in the Beam Geometry tab are displayed in red. This isnoticeable when working in the Rayleigh approximation.

The overfilling factor is the ratio of the full width at half-maximum (FWHM) of the beam intensity profile to the diameter

of the objective entrance pupil. Modification of the overfillingvalue changes the fraction of power that crosses the entrance

Fig. 5. Force analysis and calibration tab.

pupil, but also affects the shape of the force field and the stabilityof the trap, as analyzed in Section V. In the program, the over-filling factor setting is only available for the ray optics regime,where the relative power associated with each ray changes ac-cording to the overfilling factor. In the Rayleigh regime, theoverfilling setting is not available because the beam is no longerGaussian and the computational cost would be too high to per-form the calculations in real time during the animation.

By clicking on the Force analysis tab, the user can visualizethe force field created by the optical trap. In particular, the be-havior of the transverse and longitudinal compo-

nents of the force is shown. The scattering and gradient compo-nents and the net force can be selected independently by clickingon the optional buttons on the left panel. The transverse forcecurve is calculated at (but if does not exist, isobtained at ). By clicking the Plot details button, a newwindowpops up that indicates a set of parameters obtained fromthe force curves. If can be found, the equilibrium position,the maximum values of the force, and the slope of the curvesare calculated (see Fig. 5). Notice that defines the positionwhere the component of the total force is zero. Despite the factthat this is a necessary condition for trapping, the existence of

does not assure a stable trap.If the configured beam is able to trap particles, the net trans-

verse force component displays linear behavior in thecentral area of the trap. Its slope provides a measure of the stiff-ness of the trap, as explained in Section III. When the Cal-ibrate trap button is clicked, the transverse stiffness of the trapis obtained by means of the calibration method based on theanalysis of the Brownian motion. While the projection of thetrajectory of the bead on the x-axis is calculated, the -posi-tion histogram is updated, and the trap stiffness is continuouslycalculated from the -standard deviation and shown below thehistogram. This procedure emulates a continuous position mea-surement in the laboratory. When the number of points collectedis high enough, and if the stiffness value stabilizes, the his-togram displays a characteristic Gaussian shape. The stiffness

obtained with this method should match the slope of the trans-verse force curve.

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IV. SUGGESTED EXERCISES

In this section, some classroom exercises to be carried outusing the applet are presented.

A. Ray Optics Regime

1) Choose the Mie regime approach and select the visualiza-

tion of the scattering and gradient components and the netforce. Set the experimental variables with the following re-alistic values: Laser: A 50-mW, Nd:YVO4 doubled laser beam @

nm. Beam geometry: An oil-immersion objective 100x with

and . Medium: Water , temperature K,

viscosity mPa s. Particle: polystyrene spheres; radius m and

refractive index .2) Click on the optional force buttons (left panel). By pressing

the Animation button, the bead is displaced to the equi-

librium position if stable trapping is possible. Beam forcestry to compensate the effect of the temperature (Brownianmotion) to keep the particle in the trap. The Force anal-ysis tab shows the force fields. The equilibrium position

is slightly shifted to the upper side of the diagramfrom the focusing plane . This effect is corrobo-rated by the diagram, as the net force becomes zeroin . After stopping the simulation, the bead canbe centered in the focusing plane by clicking on the Takebead to focus button. Click on the Calibrate trap buttonto run the Brownian motion simulation. Verify that the stiff-ness reaches a stable value when the histogram shape be-comes Gaussian. When the simulation is stopped, the stiff-

ness value obtained provides an adjustment to the slope of.

3) When the simulation has finished, the user can click on thebead and drag it to an arbitrary position in the panel. Theuser can verify that if the bead is located near the focus,trapping is possible (notice the directions of force vectors).

4) Now set the refractive index of the particle to . Isthe beam able to trap the bead?

5) Repeat the analysis using different values for NA, power,overfilling, or particle size. Discuss under what conditionsis it easiest to trap the particle: high/low NA values? high/low laser power?

high/low overfilling values? large/small particle size?

B. Rayleigh Regime

Now, a partial repetition of the previous exercise is proposed.Initially, the variables are set to the following values:

Laser: A 50-mW, Nd:YVO4 doubled laser beam @nm.

Beam geometry: beam waist m. Medium: Water , temperature: K,

viscosity mPa s. Particle: polystyrene spheres; radius nm and re-

fractive index .

Is stable trapping possible in such conditions? Discuss whatmust be changed to improve the performance of the system.

Fig. 6. Lateral andaxialtrap stiffnesses fordifferent bead sizes in theray opticsregime. Valuesused in thissimulation: = 5 3 2 nm, P = 5 0 mW, N A = 1 : 2 , = 1

,n = 1 : 3 3

, = 0 : 6

mPa s,T = 3 2 0

K, andn = 1 : 5 7

.

C. Optical Tweezer Design

Now, the target is to design a system that can trap polystyrenespheres in water at K. The setup is illuminated with a

nm laser. Taking into account different sizes of beads,determine the following.

1) Discuss which available objective is more convenient, andwhat is the amount of overfilling and the minimum powerrequired to create a stable trap.

2) What are the stiffness and the trapping position ?

V. VERIFICATION OF SOME RESULTS FROM OPTICAL TRAPPINGUSING THE APPLET

The software enables a comprehensive and interactive anal-ysis of the interplay between the experimental parameters de-scribing the laser-sample interaction. Simple dependencies ofthe trap stiffness on, for instance, the laser power can beeasily simulated; nonetheless, it is the study of more compli-cated behavior, which cannot be readily inferred from a mathe-matical model, that turns the program into a useful tool. In thissection, a set of tasks to analyze the key parameters of the twomodels described in Section II are introduced. This should ulti-mately lead to a better and deeper understanding of the proper-ties of optical traps.

A. Ray Optics Regime

The use of numerical analysis becomes particularly importantin the ray optics limit. In this context, the vast majority of thequestions about the trap features can be uniquely solved througha computer simulation. For instance, the following calculationsare proposed.

Simulate the values of the lateral and axial trap stiffnessesfor different bead sizes. Notice that, as Fig. 6 shows, bothcurves should exhibit the same decrease as that foundin experiments [17]. Because the maximum force appliedby the beam occurs at the edge of the sphere, its value willnot depend on the size of the sample [7].

Analyze the relationship between the equilibrium positionof a trapped particle and the NA of the objective lens. As

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Fig. 7. The axial trapping position as a function of the objectives numericalaperture. Values used: = 1 0 6 4 nm, P = 5 0 mW, = 1 , n = 1 : 3 3 , = 0 : 6

mPa s,T = 3 2 0

K, andn = 1 : 5 7

, anda = 5

m.

Fig. 8. The normalized maximum force depending on the degree of overfillingalong the axial and transversal directions. The dashed line indicates the escapeforce as defined in the text. Values used: = 1 0 6 4 nm, P = 5 0 mW, N A =1 : 2 , n = 1 : 3 3 , = 0 : 6 mPa s, T = 3 2 0 K, and n = 1 : 5 7 , and a = 5 m.

NA decreases, the microsphere is displaced forward in the

-direction, as shown in the example (Fig. 7). The increasein the scattering force for low NA is the reason behindthis change. Photon scattering moves the equilibrium po-sition of the sample away from the focus until this forcecan no longer be balanced by the restoring gradient com-ponentat , in this case. This explains whythe use of such a high NA objective is required to createstable traps. A similar behavior is obtained in the Rayleighregime.

Study the maximum trapping force, as dependent on theoverfilling . As the overfilling increases, the axial forceshould also increase because of the diminishing scatteringeffect (Fig. 8). Beyond a certain optimum overfilling,

in the example, close to the values suggested by exper-iments ( and in [4] and [18]), the lateral

Fig. 9. Relation between k and the refractive index of the particle. Valuesused: = 1 0 6 4 nm, P = 5 0 mW, w = 0 : 5 2 m, n = 1 : 3 3 , = 0 : 6 mPa

s,T = 3 2 0

K, anda = 4 0

nm.

component will start limiting the escape force, that is, theforce needed to pull the sample out of the trap in any direc-tion. The value of at this intersection point is consideredto provide the best performance of the trap, correspondingto the case when the objective is slightly overfilled by thebeam.

B. Rayleigh Regime

In the Rayleigh approximation, the force is calculated througha couple of analytical expressions considering all the variables

involved except the overfilling [see (2) and (3)]. This makesthe analysis of this model more accessible. However, if one com-putes the force in the trapping plane , the trap stiffness nolonger has a trivial relation with the parameters because of thedependence of the beam width on the axial position . Takingthis effect into account in the experiments is important becauseit can yield a change in by a factor of 1.5 (with respect tothe stiffness at the plane) [4]. However, the correction isnegligible for particles with .

In this regime, the study of the relationship between andthe refractive index of the sample or the trap stiffness depen-dence on the radius of the particle is especially interesting.

Set the refractive index of the medium to that of water,

1.33, or similar, and choose a value for close to thisnumber. Fig. 9 shows the behavior of as a function of

.The result graphically explains why the trapping of par-ticles with a refractive index lower than that of the sur-rounding medium is unfeasible. Likewise, itexplains the higher lateral forces that can be achieved inthe laboratory using polystyrene beads com-pared to those achieved using silica particles .In practice, the refractive index of the microparticles usedin the experiments cannot be increased indefinitely becausethe trap becomes unstable [19].A similar variation in is found in the ray optics regime.

The reflection and transmission coefficients introduce thedependence on .

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Fig. 10. Trap stiffness for different values ofa in the Rayleigh regime. Valuesused: = 1 0 6 4 nm, P = 5 0 mW, w = 0 : 5 2 m, n = 1 : 3 3 , = 0 : 6 mPas, T = 3 2 0 K, and n = 1 : 5 7 .

Verify that the trap stiffness depends on followinga cubic function [see Fig. 10 and (2)]. The combinationof this result with the behavior for big particles givesrise to an absolute maximum of for diameters of

m [17], which are the bead sizes typically chosen in thelaboratory.

VI. ASSESSMENT OF THE INNOVATION

The applet presented in this paper was used by 18 students as

part of a Photonics course in the first semester of the 20092010academic year. Photonics is an elective course for senior under-graduates in Physics at the Universitat de Barcelona, Barcelona,Spain. This tool was the main object of a 2-h computer lab ses-sion on OT that used an adapted version of the first suggestedexercise (see Section IV-A). The topic was previously presentedin a lecture class. About a month after their having used the pro-gram, an assessment questionnaire was handed out to the stu-dents. This assessment instrument was designed to include twodifferent measurements: The first half of the questionnaire dealswith students perception of the applet; the second part is a shortlearning retention test. All 18 students turned in the question-naire, and all 18 responses were analyzed.

A. Students Perception

The perception part of the questionnaire included a four-itemLikert scale and two open-ended questions. The Likert scale wasmeant to measure the students perception on the usefulness ofthe applet for learning. The four four-level item levels of thescale are shown in Table I. The following two open-ended ques-tions asked for the best (or most noteworthy) and the worst (ormost in need of improvement) aspects of the applet. The resultsfor each item of the Likert scale are presented in Table I. Theyshow a highly positive opinion of the students on the applet;rescaling to a 0100 scale; the global average sums up to 83.

In the open-ended questions, students mention the followingideas.

TABLE IFREQUENCIES FOR THE RESPONSES TO THE LIKERT SCALE ITEMS OF THE

QUESTIONNAIRE. QUESTIONS AND ANSWERS WERE WRITTEN IN CATALAN INTHE QUESTIONNAIRE AND HAVE BEEN TRANSLATED FOR THIS PAPER. GRAYED

CELLS HIGHLIGHT THE MAXIMA IN THE ANSWERS TO EACH QUESTION:1)DISAGREE; 2) TEND TO DISAGREE; 3) TEND TO AGREE; 4) AGREE

As for the positive aspects, they remark that the programhelps learning (11 answers), especially improving visual-ization (9) and understanding (2) of the phenomena, andthat it is user-friendly and intuitive (5).

As for the negative side, or the characteristics that couldbe improved, users highlight that they would prefer to havemore time and explanations for using the applet in the labsession (6); some interface related issues (5), specificallythat graphical resolution of some charts is too low; and thatthe application is not intuitive enough and that it has a slowresponse. Five students did not report any negative aspect.

B. Learning Retention

The second part of the questionnaire included twoopen-ended questions to assess learning retention. Before

discussing the results in this part, it is important to highlightthat the assessment had not been previously announced and thatno further work on this topic was done between the academicuse of the applet and the assessment data collection (i.e., thelab reports had not yet been collected, and there had been noexams on this topic during this period).

The two open-ended questions were the following.1) Write down three variables that affect the stiffness of op-

tical tweezers in the Mie regime.2) Explain qualitatively how the trap stiffnesschanges the am-

plitude of the Brownian motion of a trapped particle.The first question was left blank by six students, eight stu-

dents gave two correct variables, and four wrote down a fully

correct answer. Thus, 67% of the students gave an at least par-tially correct answer. The second question was answered by 11students, and all of them gave a correct response. Thus, 61% ofthe students got it right. Both percentages show that significantlearning took place while using the applet and that part of thatlearning is retained a month after the learning activity. Both setsof data, students perception and learning retention, show con-sistently that the presented innovation is useful for learning thishighly complex topic.

VII. CONCLUDING REMARKS

Building from scratch an OT setup that is able to measureforces is a wonderful laboratory project for advanced under-

graduate students. The task requires considerable knowledge ofthe various physical theories involved and also some practical

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skills. Digital simulations can be powerful training tools if theyare designed to bridge the gap between theory and practice. Inthis paper, a comprehensive Java applet that covers relevant as-pects of optical trapping theory has been presented. In addition,some exercises related to real experimental situations have beenproposed. The use of the applet in the stage prior to actuallydesigning the system enables students to become more confi-dent in the lab since they acquire sound theoretical knowledgewith minimum effort. A preliminary assessment, analyzing stu-dents perception and learning retention data, suggests it can bea helpful tool to improve learning on this topic.

APPENDIX

About the Software: The complete source code and the ex-ecutable file can be downloaded from http://code.google.com/p/optical-tweezers/. The code for this software is written in Javaimplementing classes from the Open Source Physics Library(http://www.compadre.org/OSP/) and the Apache Commons

Mathematics Library (http://commons.apache.org/math/). Theprogram was compiled using the Sun Java SDK ver. 1.6.0_03.The software is licensed under a GNU General Public Licensev3.

ACKNOWLEDGMENT

The authors would like to thank Dr. M. Montes-Usategui andDr. X. de Pedro for their helpful comments and suggestions.

REFERENCES

[1] M. J. Lang and S. M. Block, Resource letter: Lbot-1: Laser-basedoptical tweezers, Amer. J. Phys., vol. 71, no. 3, pp. 201215, 2003.

[2] A. Ashkin, Acceleration and trapping of particles by radiation pres-sure, Phys. Rev. Lett., vol. 24, pp. 156159, 1970.

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Appl. Opt., vol. 47, pp. 31963202, 2008.

Josep Mas studied physics at the Universitat de Barcelona (UB), Barcelona,Spain, and obtained the M.Sc. degree in photonics from the Universitat Politc-nica de Catalunya, Barcelona, Spain, in 2008. Currently, he is a Ph.D. stu-dent with the Applied Physics and Optics Department, UB, and is working onholographic optical tweezers and video particle tracking for biophysical exper-

iments.

Arnau Farr received the M.Sc. degree in biophysics in September 2008 fromthe Universitat de Barcelona (UB), Barcelona, Spain, where he is currently aPh.D. student in the field of optical tweezers and biological applications.

He has been a member of the Applied Physics and Optics Department, UB,since 2006.

Mr. Farr is a Member of the Catalan Physical Society.

Jordi Cuadros received the B.Sc. and Ph.D. degrees in chemistry from the Uni-versitat Ramon Llull (URL), Barcelona, Spain, in 1997 and 2003, respectively.

He then spent two years as a Post-Doctoral Fellow with the Department of

Chemistry, Carnegie Mellon University, Pittsburgh, PA. Currently, he is an As-sociate Professor with the Applied Statistics Department, Institut Qumic deSarri, URL. Since completing his thesis, his research has mainly focused onthe use of simulations for teaching and learning physical sciences at the univer-sity level.

Prof. Cuadros is a Member of the Catalan Chemical Society, the AmericanChemical Society and its Division of Chemical Education, and the EuropeanScience Education Research Association.

Ignasi Juvells received the B.Sc. and Ph.D. degrees in physics from the Uni-versitat de Barcelona (UB), Barcelona, Spain, in 1972 and 1977, respectively.

He joinedthe faculty of theUB in 1972, where he is currentlya Full Professorand the Department Head of the Applied Physics and Optics Department. Heteaches courses on optics and image processing. He has worked on digital and

optical image processing, image quality analysis, holography, optical informa-tion processing, and biological and medical image applications.

Prof. Juvells is a Memberof theCatalan PhysicalSociety, theSpanishOpticalSociety, and the European Optical Society.

Artur Carnicer received the B.Sc. and Ph.D. degrees in physics from the Uni-versitat de Barcelona (UB), Barcelona, Spain, in 1989 and 1993, respectively.

Currently, he is an Associate Professor with the Applied Physics and Op-tics Department, UB, where he teaches undergraduate and graduate courses inphysical optics and photonics. His research focuses on the development of com-putational methods for the study of highly focused electromagnetic fields.

Prof. Carnicer is a Member of the Catalan Physical Society and the RoyalSpanish Physical Society

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