Calibration of the Characteristic Frequency of an Optical Tweezer using a Recursive Least-Squares Approach

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1 Calibration of the Characteristic Freqenc of an Optical Tweezer sing a Recrsive Least-Sqares Approach Arna Ranaweera ranawera@engineering.csb.ed Bassam Bamieh bamieh@engineering.csb.ed Department of Mechanical and Environmental Engineering, Universit of California, Santa Barbara, CA Abstract We describe the se of a Recrsive Least- Sqares approach for calibration of the characteristic freqenc of an optical tweezer. Unlike commonl sed off-line calibration methods, the RLS algorithm does not reqire data averaging to redce the effects of Brownian motion. We se compter simlations to demonstrate that the RLS algorithm can be sed to efficientl calibrate the characteristic freqenc of a trapped, 1-micron diameter polstrene bead. However, experimental reslts sggest that appling the RLS method to an actal optical tweezer sstem is more difficlt becase of measrement errors and laborator noise. I. INTRODUCTION The optical tweezer is a device that ses a focsed laser beam to trap and maniplate individal dielectric particles in an aqeos medim. The laser beam is sent throgh a high nmerical apertre (highl converging) microscope objective that is sed for both trapping and viewing particles of interest. For small enogh displacements from the center of the trap, the optical tweezer behaves like a Hookeian spring, characterized b a fixed trap-stiffness. Fig. 1. Optical Tweezer. A single laser beam is focsed to a diffraction-limited spot sing a high nmerical apertre microscope objective. Dielectric particles are trapped near the laser focs. Several milliwatts of laser power at the focs can generate trapping forces on the order of piconewtons. While tin b conventional standards, this level of force is well sited for biomoleclar stdies. Althogh biological molecles are too small to be trapped at room temperatre, a molecle can be grasped once a trappable handle, sch as a polstrene bead, is (biochemicall) attached to that molecle [1. The optical tweezer is a controllable actator at the piconewtons scale. In experiments that investigate the forces that arise de to varios (sall biological) interactions, the trapping forces mst be accratel qantified. Since strongest trapping occrs when trapped particles are roghl the same size as the laser wavelength, most biological experiments are performed in this size regime [2. However, the absence of an accrate theoretical model for trapping behavior in this regime means that the capabilit of an optical tweezer is almost alwas qantified empiricall [3. The trapping capabilit of an optical tweezer can be qantified b its characteristic freqenc which is defined as the ratio of trap stiffness to viscos damping. Althogh calibration can be performed off-line sing either step response data or power spectrm data, sch methods are inherentl slow becase the reqire averaging to redce the effects of Brownian (thermal) motion [3, [4. Attempts to calibrate sing a Normalized Gradient (NG) algorithm are also severel hampered b thermal noise [3. In the remaining sections of this paper, we se a wellknown Recrsive Least Sqares (RLS) algorithm, which does not reqire averaging, to obtain significantl faster calibration of the characteristic freqenc. Compter simlations and experimental reslts are inclded. II. EQUATION OF MOTION The eqation of motion along the lateral x-axis for a trapped bead of mass m and lateral position x is given b mẍ = F T (x r ) + F D (ẋ) + F L (t) + F E (t), (1) where F T ( ) is the optical trapping force, F D ( ) is the viscos drag, F L ( ) is a Langevin (random thermal) force, and F E ( ) is an external driving force. The relative displacement x r is defined as x r := x x T, in which x T is trap position. Within the linear radis of x r < R l, the trap stiffness is approximatel constant and the trapping force is linear with respect to relative displacement: F T = αx r. (2) For a 1-µm diameter polsrene bead, R l.2 µm [5. The linear radis increases with bead size. The drag force can be expressed as F D = βẋ, (3)

2 where β > is the viscos damping factor from Stoke s eqation, β = 6πηr, in which r is the bead radis and η is the flid viscosit. For a 1-µm diameter polstrene bead in water at 2 C, m = mg and β.1 pns µm. The Langevin force has an average vale of zero, E{F L (t)} =, and constant power spectrm S L (f) (i.e., ideal white noise force) given b S L (f) = 4βk B T, (4) in which k B is Boltzmann s constant and T is the absolte temperatre [6. At biological temperatres, k B T is approximatel pnµm [6. Therefore, for a 1-µm diameter polstrene bead, S L (f) pn2 1 Hz. The natre of the external force F E(t) will depend on experimental conditions. For example, in biological experiments, the external force will arise de to the interaction between the trapped bead and biological particles. Assming the particle mass is negligible compared to the viscos drag, (1),(2), and (3) can be combined to obtain the noninertial eqation of motion for a trapped particle in the linear trapping region: = αx r βẋ + F L (t) + F E (t). (5) The characteristic freqenc of the trap (in radians per second) is defined as A. Transfer Fnctions := α β. (6) Defining trap position as the control inpt, := x T, and sing (6), (5) can be written in state space form as ẋ = x β F L + 1 β F E = x. (7) Defining Langevin distrbance d L := F L, and external distrbance d E := F E, and assming zero initial conditions, (7) can be expressed sing Laplace transforms as: X(s) = G (s)u(s) + G d (s) [D L (s) + D E (s), (8) in which the first order transfer fnctions are given b and G (s) = G d (s) = s + (9) 1 β s +. (1) A schematic block diagram of the linear plant P is shown in Figre 2. Fig. 2. Linear plant block diagram, in which := x T, d L := F L, d E := F E, and n is measrement noise. B. Zero-Order-Hold Representations A continos-time (CT) scalar sstem of the form ẋ = ax + b with corresponding transfer fnction = x (11) G(s) = b s + a (12) can be zero-order hold sampled with sampling time h to obtain the discrete-time () sstem x(kh + h) = a 1 x(kh) + (kh) (kh) = x(kh), (13) where k is a positive integer and a 1 = e ah = b ( 1 e ah ) a (14) [7. The sampled-sstem can also be represented b the difference eqation (kh) = H(q)(kh), where H(q) is the plse transfer operator given b H(q) = q + a 1, (15) in which q is the forward shift operator [7. From (14), if we se a identification algorithm to obtain parameter estimates â 1 and, the corresponding CT parameter estimates â and b are given b â = 1 h ln( â 1) ( ) b1 1 b = 1 + â 1 h ln( â 1). (16) From (14) and (15), the zero-order hold eqivalent of G (s) in (9) is given b H (q) = 1 e ωch. (17) q e ωch

3 For example, for h =.1 s, = 78 rad/s gives G (s) = 78 s+78 and therefore H (q) =.754 q.925, whereas = 65 rad/s gives G (s) = 65 s+65 and H (q) =.7653 q Note that the parameters a 1 and are relativel insensitive to variations in the continos parameters. In particlar, from (15) and (17), we obtain da 1 = d = he ωch, (18) d d which is jst nder.1 for the vales chosen above. III. RECURSIVE LEAST SQUARES ALGORITHM In this section, we will se the sampling time h as the nit of time. We wold like to compte the parameter estimate θ(k) := [â 1 b1 T at sample k that minimizes the weighted least-sqares criterion: θ(k) = min θ k w(k, n)[(n) φ T (n)θ 2, (19) n=1 where φ(k) := [ (k 1) (k 1) T is the regression vector that contains the inpt and otpt data, and w(k, n) is a weighting seqence with the propert, w(k, n) = λ(k)w(k 1, n), n k 1 w(k, k) = 1, (2) in which λ(k) is the forgetting factor [4. The RLS algorithm is given b where θ(k) = θ(k 1) + [ L(k) (k) φ T (k) θ(k 1) L(k) = P (k) = (21) P (k 1)φ(k) λ(k) + φ T (22) (k)p (k 1)φ(k) 1 [P (k 1) λ(k) P (k 1)φ(k)φ T (k)p (k 1) λ(k) + φ T, (23) (k)p (k 1)φ(k) [ k 1 P (k) := w(k, n)φ(n)φ (n) T (24) n=1 is the scaled covariance matrix of the parameters at sample k. The initial parameter vector is denoted as θ() = θ and the initial covariance matrix is denoted as P () = P. In other words, θ is what we gess the parameter vector to be before seeing the data, and P reflects or confidence in this gess [4. The initial regression vector is specified as φ() = [ T. The forgetting factor λ is constant for a slowl changing sstem and can be chosen according to λ = 1 1 K, (25) where K is the memor time constant. Data older than K samples are weighted b a factor e 1 36% compared to the most recent data [4. For an LTI sstem, it is natral to reqire that all data be given eqal weight, which implies that no data is disconted. B setting K in (25), we obtain λ = 1. IV. COMPUTER SIMULATIONS The CT sstem described b (8) was simlated sing Simlink with a simlation sampling time of T s =.1 ms, corresponding to 1 kilosamples per second (ks/s). The Langevin distrbance was modeled as bandlimited white noise with bandwidth 1 khz and constant power S L (f) = pn2 Hz. As mentioned in Section II, this represents the Langevin force that acts on a 1-µm diameter polstrene bead at biological temperatres. To facilitate comparison with experimental reslts in Section V, the actal characteristic freqenc of the simlated sstem is chosen as = 78 rad/s. According to Section II-B, for RLS sampling time h = 1 ms, the actal parameters are given b and θ = [ a1 a = b = 78 (26) = [ (27) To enable qantitative comparisons with the reslts from [3, we assme an initial characteristic freqenc gess of () = 65 rad/s, which corresponds to an error of 2% (13 rad/s). According to Section II-B, the initial parameter conditions are given b and θ = â() = b() = 65 (28) [ â1 () b1 () = [ (29) For invertibilit, we assme the initial covariance matrix is of the form [ p P =, p in which p >. In the simlations that follow, we will denote the inpt signal amplitde b A and inpt freqenc b f. Figre 3 shows reslts for an inpt sqare wave with A =.2 µm and f = 1 Hz. The top plot shows the inpt and otpt (I/O) signals, the middle plot shows the parameter estimates, and the bottom plot shows the CT parameter estimates. For comparison, the estimates are plotted as 1 + â 1 and, since, according to (14), we wold like these two qantities to converge to the same vale (= ). After flctating wildl at the onset, both the and CT parameter estimates eventall converge close to the correct vales of and, respectivel. The simlation is shown on an expanded time scale in Figre 4, in which the ± 5% and ± 2% limits are inclded in the bottom plot.

4 Fig. 3. Simlation of RLS for h = 1 ms, λ = 1, p = 1 4 ; sqare wave inpt, A =.2 µm and f = 1 Hz. Fig. 5. Hpothetical F L = simlation of RLS for h = 1 ms, λ = 1, p = 1 4 ; sqare wave inpt, A =.2 µm and f = 1 Hz Fig. 4. Simlation of RLS for h = 1 ms, λ = 1, p = 1 4 ; sqare wave inpt, A =.2 µm and f = 1 Hz. Dash-dotted lines on the bottom plot show ± 5% and ± 2% limits. Althogh the parameter estimates displa small flctations that do not disappear with time, the CT parameter estimates settle to within 5% of in nder.6 s and to within 2% in nder 8 seconds. The direct correspondence between the and CT estimates can be seen from the similar shape of their plots. As shown in (18), flctations in the estimates will amplif the CT estimates b a factor of over 1. The reason for the paramater flctations is the Langevin distrbance. This is demonstrated b Figre 5, which is a hpothetical simlation for zero Langevin distrbance. No oscillations are observed and the CT parameter estimates settle to within 5% of in nder 3 ms (3 iterations) and to within 2% in nder 4 ms (4 iterations). A. Effect of Inpt Signal Parameters 1) Inpt Freqenc: We fond that both f = 2 Hz and f = 2 Hz reslted in slower parameter convergence. The vale of f = 1 Hz was sed becase it gave the fastest convergence for the chosen vale of. This is consistent with the practical recommendation that inpt power be selected at freqenc bands in which a good model is particlarl important, or more formall, freqencies at which the Bode plot is sensitive to parameter variations [4. The optimal inpt freqenc for = 78 rad/s is f 12 Hz, which is close to 1 Hz. 2) Inpt Shape: We fond that a sqare wave provides faster convergence than both a sinsoidal inpt (f = 1 Hz) and a random inpt. Frthermore, we fond that random inpt signals ield erroneos paramater estimates. The speriorit of the sqare wave can be explained sing the crest factor C r, which shold be minimized to redce the covariance of the parameter estimates [4. For a discrete inpt seqence {(k)}, the crest factor is given b C 2 r = max k 2 (k) N k=1 2 (k), (3) lim N 1 N which is clearl at its theoretical lower bond for binar, smmetric signals sch as a sqare wave [4. B. Effect of RLS Algorithm Parameters 1) Forgetting Factor: According to (25), a forgetting factor of λ =.9999 corresponds to a memor time constant of K = 1 samples, which is eqivalent to t = 1 s for h = 1 ms. Figre 6 shows simlation reslts for λ = The CT parameter estimates settle to within 5% of in nder.6 s, bt the do not settle within the 2% limit. Clearl, the parameter estimates flctate more for the λ =.9999 case than for the λ = 1 case shown in Figre 4. Intitivel, since the past data is exponentiall disconted with time, there is less smoothing of the past data. Therefore, the identification algorithm is more sensitive to the Langevin distrbance and the convergence properties are diminished. 2) Initial Covariance Matrix: B decreasing initial covariance matrix gain from p = 1 4 to p = 1, we can specif greater confidence in or initial parameter gess. Figre 7 shows simlation reslts for p = 1.

5 Fig. 6. Simlation of RLS for h = 1 ms, λ =.9999, p = 1 4 ; sqare wave inpt, A =.2 µm and f = 1 Hz. Fig. 8. Schematic diagram of single-axis optical tweezer sstem. PBSC = Polarizing Beam Splitting Cbe, KM = Kinematic Mirror, AOD = Acosto-Optic Deflector, PSD = Position Sensing Detector, CCD = CCD Camera, DM = Dichroic Mirror Fig. 7. Simlation of RLS for h = 1 ms, λ = 1, p = 1; sqare wave inpt, A =.2 µm and f = 1 Hz. Comparing with Figre 3, we see that the smaller vale of p reslts in significantl smaller initial flctations in the parameter estimates. Frther simlations show that, for p = 1, the CT parameter estimates settle to within 5% of in nder.3 s and to within 2% in nder 8.5 s, which are similar to the settling times for p = 1 4. It shold be mentioned that, since the RLS algorithm is implemented sing a PC, the inpt and otpt position amplitdes can be scaled if necessar (within comptation limits), bt this does not affect convergence times for the RLS algorithm, so the data was not scaled. V. EXPERIMENTAL RESULTS A schematic diagram of or single-axis optical tweezer sstem is shown in Figre 8 [3. Since the RLS algorithm does not reqire real-time feedback control, we were able to investigate its performance b collecting inpt and otpt data from or sstem sing LabVIEW data acqisition software and hardware and then processing the data sing Matlab [3. Data was sampled at a rate of 2 ks/s with an analog lowpass filter at the Nqist freqenc. Figre 9 shows reslts for an inpt sqare wave with A =.2 µm and f = 1 Hz..5 Experimental Reslts (f = 1 Hz) Fig. 9. Implementation of RLS for experimental data with h = 1 ms, λ = 1, p = 1 4 ; sqare wave inpt, A =.2 µm and f = 1 Hz. After initial flctations, the parameter estimates appear to reach stead vales within 9 seconds. The vales of the CT estimates after 1 s are â = 88 rad/s and b = 86 rad/s. The slight disparit between these two estimates is likel de to attenation from the lowpass filter. The parameter convergence vales for different inpt freqencies are shown in Table I, inclding the two off-line methods. The RLS reslts are for 1 seconds of data, while the off-line reslts are for the average of 3 seconds of data [3. The RLS parameter estimates are not consistent with the reslts for off-line calibration methods sch as the power spectrm and step response [3. Frther implementation shows that for lower freqencies (f = 2 Hz and f = 5 Hz), the RLS algorithm does not converge to stead vales within 1 s. As shown in Section IV, this is almost certainl de to the mismatch in inpt freqenc and the characteristic freqenc of the sstem. The reslt is slower convergence for lower inpt freqencies. The inconsistencies can be de to several factors. For example, even slight misalignments in the position

6 Calibration method â b Convergence after 1 s RLS (f = 2 Hz) Unstead RLS (f = 5 Hz) Unstead RLS (f = 1 Hz) Stead Power Spectrm Step Response TABLE I PARAMETER ESTIMATES FOR EXPERIMENTAL DATA. detection sstem and flctations in the dnamic laser pointing sstem (the AOD) can case sstematic errors that distort the RLS calibration reslts. The power spectrm method, in particlar, is robst with respect to sch factors [6, [8. Additional sorces of measrement noise, sch as low-freqenc drift and other tpes of electronic bias, high freqenc amplifier noise, mechanical vibrations, and extraneos backgrond light can also contribte to erroneos estimates. Sch problems are inherent in an practical position detection sstem. The presence of an analog RC lowpass filter can also distort the RLS calibration reslts. However compter simlations show that the tpical effect of the lowpass filter is to redce the parameter estimates, not increase them. Therefore, it is nlikel that the lowpass filter cased the disparities in Table I. In light of the disparities between the reslts shown in Table I, more analsis and experimental work needs to be done before we can make definitive qantitative claims abot the convergence of the RLS algorithm when applied to tpical experimental data. VI. CONCLUSION This paper describes the calibration of the characteristic freqenc of an optical tweezer sing a wellknown recrsive least-sqares approach. For a sstem with characteristic freqenc of approximatel 12 Hz, we se compter simlations to show that a sqare wave inpt with freqenc 1 Hz can be sed to estimate the parameters to within 5% accrac in nder 1 s and to within 2% accrac in nder 1 s. We also demonstrate that initial parameter flctations can be significantl decreased b redcing the initial covariance matrix, bt this does not appear to improve stead state convergence times. We find that convergence properties are diminished for forgetting factors less than 1. For a sstem sch as an optical tweezer that is sbject to a large stochastic distrbance, the RLS method is, in theor, sperior to an on-line identification method sch as the NG algorithm, which is not designed to handle large distrbances. In fact, compter simlations show that the RLS algorithm is at least one order of magnitde faster than the NG algorithm [3. In practice, the characteristic freqenc of an optical trap can var de to laser flctations, local heating, and crosscontamination. Methods that depend on prel off-line data analsis do not accont for these effects and ma sggest an erroneos vale for the characteristic freqenc. Therefore, in a laborator environment in which experimental conditions are not entirel constant, the RLS algorithm cold potentiall provide a more reliable (p-to-date) measre of characteristic freqenc than the off-line methods that are widel in se. Frthermore, since the RLS algorithm is specificall designed for online identification, it is eas to implement sing a PC. Or experimental reslts are not consistent with previos, off-line calibration reslts. Althogh the RLS parameter estimates converge for an inpt freqenc of 1 Hz, the estimates are abot 12% greater than the offline vales. Frther analsis is needed before the RLS method can be recommended for accrate calibration of experimental reslts. In light of the effectiveness of the RLS method when applied to compter generated data, it is ver likel that the experimental difficlties can be overcome b redcing laborator noise and increasing the accrac of or position detection sstem. Also, the reslts of this paper pertain specificall to a 1-µm diameter polstrene bead trapped in water at biological temperatre. The characteristic freqenc for a 1-µm bead will be at least 1 times larger, whereas the Langevin noise power will be one-tenth. In ftre work, we will investigate the performance of the RLS algorithm for a 1-µm bead, which is widel sed in biophsics. It shold be pointed ot that this paper onl considers calibration of the characteristic freqenc. If specific information abot stiffness α and drag β are soght, the off-line power spectrm method is nrivalled in man aspects [6. However, de to its speed and ease of implementation, we believe that the RLS algorithm described here will prove to be a sefl tool for sers of optical tweezers. REFERENCES [1 S. Ch, Laser maniplation of atoms and particles, Science, vol. 253, pp , [2 A. D. Mehta, J. T. Finer, and J. A. Spdich, Reflections of a lcid dreamer: optical trap design considerations, in Methods in Cell Biolog (M. P. Sheetz, ed.), vol. 55, ch. 4, pp , Academic Press, [3 A. Ranaweera and B. 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