836 OPTICS LETTERS / Vol. 27, No. 10 / May 15, 2002

Fast position measurements with scanning lineoptical tweezers

Rajalakshmi Nambiar and Jens-Christian Meiners

Department of Physics and Biophysics Research Division, University of Michigan, Ann Arbor, Michigan 48109-1120

Received January 3, 2002

Scanning line optical tweezers are a powerful tool for the study of colloidal or biomolecular systems in thelow-force regime. We present a fast, high-resolution particle position measurement scheme that extends thecapabilities of these instruments into the realm of dynamic measurements. The technique is based on syn-chronous detection of forward-scattered laser light during a line scan. We demonstrate a position resolutionof better than 50 nm for bandwidths of as much as 40 kHz for pairs of microspheres trapped in a f lat linepotential at center-to-center separations of 1.7 6 mm. © 2002 Optical Society of America

OCIS codes: 170.4520, 170.6920.

With the advent of optical tweezers as a powerfulnew tool in biophysics and colloidal physics, a wealthof novel experiments has become feasible. In mostcases, optical tweezers are used to hold small particlessuch as a microspheres and bacteria with light forcesin the focus of a laser beam. In this mode, relativelylarge forces of as much as 100 pN can be applied toprobe single biomolecules, for instance, by stretchingthem.1 For most experiments the ability to manipu-late the system mechanically is not suff icient andneeds to be complemented by a method to measureforces and displacements directly. Therefore a num-ber of force- and position-measurement schemes havebeen deviced for use in conjunction with optical tweez-ers. Most of these schemes measure the displacementof the particle from the center of the focus of thelaser beam as a measure of the force acting upon theparticle. Commonly, this is accomplished by imagingof the forward-scattered or backscattered laser lightfrom the trapped particle onto a position-sensitivephotodetector.2 These force- and position-sensingschemes typically have a sensitivity of �1 pN or afew nanometers at a bandwidth of 1 kHz, limited bythermal motion of the microsphere in the trappingpotential.3 With correlation techniques this limit canbe circumvented, and even femtonewton force f luctua-tions on a millisecond time scale can be measured.4

All these techniques have in common that theparticle is trapped in a narrow and nearly harmonicoptical potential. Therefore the force acting on theparticle increases linearly with its displacement fromthe focus of the laser beam. Whereas this method isappropriate for force measurements at approximatelyconstant extensions of the molecule under study, it isoften desirable to work under constant-force or evenzero-force conditions without constraining the exten-sion of the molecule. This can be accomplished eitherby an active feedback mechanism or through the useof scanning line tweezers. In the latter scheme thelaser focus is rapidly scanned along a line, effectivelycreating an anisotropic optical potential that is f lator very shallow in the direction of the line scan butsteep in the other two dimensions. Such scanningline tweezers have been used in colloid and polymerphysics to study relatively small forces between par-

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ticles, such as entropic interactions in binary colloidmixtures.5 These experiments have been static innature or have looked at slow transport processes6

because they had to rely on video microscopy with itsinherently low time resolution with which to deter-mine the position of the microspheres in the opticalpotential.

In this Letter we present a fast position-measure-ment scheme for scanning line optical tweezers thatpermits dynamic measurements with bandwidths of asmuch as 40 kHz. We measured the positions of twomicrospheres trapped in a one-dimensionally f lat op-tical potential by monitoring the intensity of the for-ward-scattered laser light during the scan.

The experimental apparatus consists of a scanningline optical tweezer setup, which is integrated into acommercial inverted microscope, as shown in Fig. 1.The beam from a Nd:YAG laser �l � 1064 nm�is def lected with an acousto-optic modulator (In-traaction) and imaged through a beam-expandingtelescope of long-focal-length plano–convex lensesonto a beam-steering mirror, from where it is imagedthrough a second telescope onto the back apertureof an oil-immersion microscope objective (Zeiss Plan-Neof luar 1003 1.3 oil). At this point the laser beamtypically has an intensity of 300 mW. The micro-scope objective focuses the laser beam at a depth of10 mm into the sample cell. The fully transparentand hermetically sealed sample cell contains a dilutesuspension of yellow–green f luorescent latex micro-spheres at a typical volume fraction of f � 1027 ina TE NaCl buffer (10 mM Tris-HCl, 1 mM EDTA,10 mM NaCl; pH 8.0). Here the diameters of themicrospheres were 1.00 6 0.03 mm for one sample and0.50 6 0.04 mm for a second sample. A condenserlens on the other side of the sample cell collects thetransmitted laser light and images it onto a fastlarge-area photodiode, which has in conjunction withthe subsequent amplif ier a bandwidth of 10 MHz.The signal is recorded on a digital oscilloscope. Inaddition, epif luorescence microscopy yields an opticalimage of the trapped microspheres.

To realize a nearly f lat one-dimensional opticalpotential inside the sample cell, one scans the laserfocus at a constant speed along a line by applying

2002 Optical Society of America

May 15, 2002 / Vol. 27, No. 10 / OPTICS LETTERS 837

Fig. 1. Experimental setup of the scanning line opticaltweezers with forward-scattered light detection. A laserbeam is def lected with an acousto-optical modulator(AOM) and focused into a sample cell, where microspheresare trapped in the resultant optical potential. Theforward-scattered laser light is collected with a fastphotodiode in the image plane of the condenser lens.In addition, f luorescence microscopy provides an opticalimage of the trapped particles.

saw-toothed or triangular frequency modulation to theacousto-optical modulator. Typically, this line has alength of 25 mm at the focal plane of the microscopeobjective and is scanned at frequencies of 10–40 kHz.By applying different modulation functions one canrealize other potentials. In particular, a squarewave generates two separate, narrow traps with aseparation proportional to the amplitude of the wave-form.7 For the present study we used a combinationof the dual-trap mode and the f lat-potential modefirst to accurately position two microspheres withthe two traps and then to release them into a f latline potential. The initial center-to-center distancebetween the microspheres was varied from 5.99 to1.65 mm, and we measured it independently for eachdata set by determining the centroids of the imagesof the microspheres from a number of f luorescencevideo microscopy frames recorded immediately beforethe microspheres were released into the f lat potential.Immediately on release, the intensity of the trans-mitted laser light was recorded during the first linescan. The inset in Fig. 2 shows such an intensityprofile for a pair of microspheres of 1-mm diameter ata center-to-center separation of 4.4 mm.

We observed a diffraction pattern that, as Pralleet al.8 have shown, is generated by interferencebetween the laser light that is scattered from themicrosphere and the unscattered laser beam. Thepattern generally depends on the lateral and axialpositions of the microsphere with respect to the focusof the laser beam, but because the microsphere wasaxially confined in the trapping potential the observedinterference pattern corresponds to a static pictureof the positions of the spheres. Most notably, we

observed a pronounced minimum in the intensitywhen the laser beam was scanned through eachof the microspheres. This minimum was f lankedby much smaller secondary maxima, similar to theAiry-diffraction pattern of a circular particle, andits depth decreased with decreasing particle sizeto �25% modulation for the 0.5-mm spheres. Weused the time delay between the minima to measurethe relative separation between the microspheresin the optical potential. For this purpose we de-termined the locations of the minima throughleast-squares fits of a parabola to the lower portion ofthe intensity dips. This parabolic approximation isa suff iciently accurate, simple alternative to using amore complex, only partially theoretically known andspherically aberrated point-spread function. With aknown scan rate of the laser beam, the separationbetween the microspheres could immediately be calcu-lated. The width of the minima was, as expected fordiffraction from a small spherical particle, of the orderof the wavelength of the laser light. Therefore theminimum particle separation that we could measureis given by the Abbe limit; at smaller separations theminima are not resolved.

To determine the accuracy of this method we com-pared these measurements with the measurementsobtained from image analysis of the f luorescencemicroscopy picture. The results are shown in Fig. 2,where the microscopically measured separations are

Fig. 2. Separation of the microspheres in the optical linepotential as measured by video microscopy as a functionof the time difference between the interference minima.(a) Data for microspheres of 0.5-mm diameter at scan ratesof 0.99 �D�, 0.57 ���, and 0.29 ��� mm�ms. (b) Results forlarger microspheres with a diameter of 1.0 mm; inset, in-tensity of the forward-scattered light during a line scanthrough two microspheres at a center-to-center distance of4.4 mm and a scan rate of 0.57 mm�ms. We determinedthe positions of the minima in the interference patterns byfitting parabolas to the bottom halves of the intensity dips.These f itted parabolas are shown as bold curves.

838 OPTICS LETTERS / Vol. 27, No. 10 / May 15, 2002

Fig. 3. Residual deviations from a linear time–distancerelationship. The differences between the measurementof the separation between the microspheres by use of thetime delay between the interference minima and the linearregression from Fig. 3 and the microscopic measurementare shown for several laser scan rates and sphere sizes.Open symbols, small spheres of 0.5-mm diameter; filledsymbols, 1.0-mm spheres. The standard deviation fromthe expected linear behavior is shown as the one-s interval(solid and dashed lines, respectively).

plotted as a function of the time delay between theintensity minima for three scan rates, two sizes ofmicrospheres, and center-to-center separations thatrange from 1.65 to 5.99 mm. As expected, the graphsare highly linear, ref lecting the linear nature of thescans. Subtracting a linear regression line from thedata, we obtained the residuals shown in Fig. 3 as ameasure for the error of our position-measurementscheme. The data points scattered with a standarddeviation of typically 50 nm about the regression line,roughly independently of the size of the microspheresor the scan rate. The only notable exception is thedata for the small microsphere at the fastest scanrate, 40 kHz. Here, the standard error increased

to 110 nm. For the most part, the residuals do notpoint to systematic errors as a function of interpar-ticle separation. Instead, the scatter appears to befairly random, indicating a statistical error. Thisstatistical error has a number of sources, some ofwhich are not related to our position-measurementscheme with the scanning laser but result from un-certainties in the actual positions of the microspheresor their measurement with video microscopy: First,the positioning of the microspheres with the opticaltweezers in the dual-trap mode was imperfect, becausethe microspheres undergo Brownian motion in theirtraps. From a measurement of the stiffness of thesetraps we estimated the uncertainty in the particleposition to be �12 nm. Determination of the centerof the microspheres by f luorescence video microscopyis also fraught with uncertainties; we estimate thatthese measurements are good within 20 nm. Theremaining statistical errors related to the scanninglaser position measurement itself are due primarily tomechanical vibrations, electronic noise, and f luctua-tions in the laser intensity. Finally, minor systematicerrors, in particular at small interparticle separations,arise from an overlap of the interference patternsfrom the two particles. Systematic variations of thelaser intensity as a function of position during thescan and, in particular at higher scan rates and largerinterparticle separations, nonlinearities in the scanramp owing to the finite response of the acousto-optical modulator of �2-ms in our experiment mayalso contribute to the error.

Overall, the position-measurement scheme pre-sented here for particles trapped in a scanning lineoptical potential determines interparticle separationwith a typical resolution of �50 nm at bandwidthsof as much as 40 kHz, permitting new, fast dynamicmeasurements of biomolecular and colloidal systemsto be made in the low-force regime.

The authors acknowledge research support fromthe Alfred P. Sloan Foundation and the ResearchCorporation. J. C. Meiners’s e-mail address ismeiners@umich.edu.

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