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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2004; 00:1–19

Modeling, Identification, and Control of a Spherical ParticleTrapped in an Optical Tweezer

A. Ranaweera∗, B. Bamieh

Department of Mechanical and Environmental Engineering,University of California, Santa Barbara, CA 93106-5070.

SUMMARY

We provide an introduction to modeling, identification, and control of a spherical particle trappedin an optical tweezer. The main purpose is to analyze the properties of an optical tweezer from acontrol systems point of view. By representing the noninertial dynamics of a trapped particle as astochastic differential equation, we discuss probability distributions and numerically compute firstmean exit times. Experimentally measured mean passage times for a 9.6-micron diameter polystyrenebead within the linear trapping region show close agreement with theoretical calculations. We applya recursive least squares method to a trapped 9.6-micron diameter polystyrene bead to study thepossibility of obtaining faster calibrations of characteristic frequency. We also compare the performanceof proportional control, LQG control, and nonlinear control to reduce fluctuations in particle positiondue to thermal noise. Assuming a cubic trapping force, we use computer simulations to demonstratethat the nonlinear controller can reduce position variance by a factor of 65 for a 1-micron diameterpolystyrene bead under typical conditions.

key words: Optical Tweezers, Modeling, Identification, Control.

1. INTRODUCTION

The optical tweezer is a device that uses a focused laser beam to trap and manipulate individualdielectric particles in an aqueous medium. The laser beam is sent through a high numericalaperture (highly converging) microscope objective that is used for both trapping and viewingparticles of interest. Arthur Ashkin and his colleagues—Joseph Dziedzic, John Bjorkholm, andSteven Chu—at AT&T (Bell) Laboratories demonstrated the first working optical tweezer in1986. Since then, optical tweezers have been used to trap dielectric particles with diametersin the range of tens of nanometers to tens of microns. For small enough displacements fromthe center of the trap (up to, maximally, 100-300 nm) [1], the optical tweezer behaves like aHookeian spring, characterized by a fixed trap-stiffness. Several milliwatts of laser power atthe focus can generate trapping forces on the order of piconewtons, typically 1-100 pN [2].

∗Correspondence to: Department of Mechanical and Environmental Engineering, University of California, SantaBarbara, CA 93106-5070. Email: [email protected]; Tel: 805-893-5134; Fax: 805-893-8651.

Received October 27, 2004

2 A. RANAWEERA, B. BAMIEH

While tiny by conventional standards, this level of force is well suited for biomolecular studies.For example, a force of 10 pN is sufficient to pull a bacterium such as Escherichia coli throughwater at ten times the speed at which it can swim [3].

Another reason for the popularity of optical tweezers in the field of cell biology is becausethey can be used to trap cells or organelles without damage [4]. By choosing a trapping laserwith a wavelength in the near-infrared range, optical damage to biological specimens can bereduced by one or two orders of magnitude [5]. According to Mehta et al., “a general goalin molecular biophysics is to characterize mechanistically the behavior of single molecules”[6]. Before the advent of optical tweezers, biophysicists did not possess a noninvasive toolcapable of manipulating individual molecules. Instead, they would examine the behavior ofa large clump of molecules and use various averaging techniques to infer the behavior of asingle molecule. The optical tweezer has made such inference methods obsolete by enablingthe direct study of individual molecules. Although biological molecules are too small to betrapped at room temperature, a molecule can be grasped once a trappable ‘handle’, such as apolystyrene bead, is (biochemically) attached to that molecule as shown in Figure 2 [7]. Opticaltweezers have been used “to trap and manipulate dielectric spheres, viruses, bacteria, livingcells, organelles, colloidal gold, and even DNA. Such traps . . . have measured elasticity, force,torsion, position, surface structure, and the interaction between particles” [8]. For trappedparticles with a diameter of 1 µm, the range of forces that can be measured using an opticaltweezer is 0.2–200 pN, which partially overlaps the 2–400 pN range in which a wide varietyof cellular processes occur [9]. For studies that require larger forces, atomic force microscope(AFM) cantilevers or glass microneedles are more suitable choices because they are stronger(less compliant) than optical tweezers [10].

The main purpose of this paper is to analyze the properties of an optical tweezer from acontrol engineering point of view. By doing so, we hope to achieve two objectives:

1. Enhance the arsenal of tools available to users of optical tweezers, especially in biophysicsand microfluidics.

2. Provide a framework that encourages future contributions from the control systemscommunity.

As shown in Figure 2, polystyrene beads are widely used in optical tweezer experiments;hence, in this paper, we will specifically study the behavior of trapped polystyrene beads.In Section 2, we describe the dynamics of a trapped particle. After discussing differentmathematical models of the optical trapping force in the lateral plane, we derive equationsof motion for a trapped particle. Descriptions of the system are developed for both a cubictrapping force model and a linear force model. We convert the deterministic descriptions intostochastic differential equations, which can be manipulated using stochastic control theory. InSection 3, we discuss the probability distribution of a trapped particle and numerically computeits mean first exit time. In Section 4, we briefly describe popular off-line identification methodssuch as the equipartition method, the power spectrum method, and the step response method.We then discuss the application of on-line calibration methods to achieve faster calibrations.We represent the optical tweezer system as a sampled-data system which allows the useof discrete time (DT) parameter estimation using Recursive Least Squares (RLS); we usecomputer simulations and experimental results to demonstrate the performance of the RLSmethod. In Section 5, we discuss feedback control. The performance, including limitations, ofstandard linear controllers are studied using computer simulations. We also discuss a nonlinear

Int. J. Robust Nonlinear Control 2004; 00:1–19

MODEL., ID., CONTROL OF SPHERICAL PARTICLE TRAPPED IN OPTICAL TWEEZER 3

controller that achieves Global Asymptotic Stability (GAS) of the origin. In Section 6, wesummarize the conclusions that can be drawn from the material in this paper.

2. MODELING

Although the physics behind optical tweezers is not trivial, its behavior can be explainedtheoretically using two separate models. For a trapped particle with diameter d much largerthan the wavelength λ of the trapping laser (d� λ), a ray optics model shows good agreementwith measured results, whereas for a particle with diameter much smaller than the trappingwavelength (d � λ), an electromagnetic field model provides best agreement [4]. In theintermediate size regime (d ∼ λ), electromagnetic theory has yielded better results than rayoptics, but neither model has been satisfactory [4]. Because strongest trapping occurs whentrapped particles are roughly the same size as the laser wavelength, most biological experimentsare performed in the intermediate regime. In the absence of an accurate theoretical model, theperformance of optical tweezers in this regime is determined empirically.

2.1. Trap Force Model

For relative displacements within the trapping radius R, we can model the trap as a cubicrestoring spring:

FT =

{α3x

3r − α1xr for |xr| < R =

√α1α3

0 otherwise,(1)

where the relative position xr is defined as xr := x− xT , in which xT is the trap (laser focus)position. Figure 3 shows a typical cubic trapping force model in which α3 = 22 pN/µm3,α1 = 10 pN/µm, and R = 674 nm. The effective trap stiffness, which is defined asαe(xr) := −FT

xr, is greatest near the trap position (laser focus). We obtained the nonlinear

spring constants α1 and α3 by fitting a cubic polynomial to experimental results published bySimmons et al. in [2]. According to their force model, which is also shown in Figure 3, the trapexerts a linear restoring force for relative displacements xr of up to 200 nm, which we denoteas the linear trapping radius Rl [2]. The trap has no effect on beads that are more beyond thetrapping radius of R = 675 nm (from the trap center) [2]. According to the cubic force model,the maximum restoring force of 2.60 pN occurs at |xr| = RF = 389 nm. Both curves areconsistent with Ashkin’s (theoretical) ray-optics calculations that predict that the maximumtrapping force occurs at about one bead radius [11]. Although the magnitude of the force shownin Figure 3 will vary depending on parameters such as laser power and numerical aperture, thecurve accurately depicts the qualitative trapping behavior of a well-aligned optical tweezer, asdiscussed in [11].

Within the linear region of |xr| ≤ Rl < R, the effective trap stiffness is approximatelyconstant (αe = α = 10 pN/µm in Figure 3), and the trapping force is linear with respect torelative displacement, FT = −αxr, for |xr| < Rl. Clearly, the linear force model overestimatesthe experimental force model outside of the linear region. On the other hand, the cubic forcemodel underestimates the experimental force model everywhere (except at the origin), buthas a similar profile (shape) within the trapping radius. Hence, we can view the cubic forcemodel as a useful, but conservative estimate of the experimental force model. When considering



motion that is restricted to the linear region, it is appropriate to use the simple linear forcemodel, keeping in mind that the cubic model’s departure from linearity is under 9% withinthe linear region [12]. For motion outside of the linear force region, the cubic model can beused as a convenient approximation of the experimental force model.

It should be mentioned that an optical tweezer traps particles in not only one, but threedimensions (Figure 1). For a well-aligned trap, the trapping profile is cylindrically symmetricabout the axial z direction in which laser light propagates. As a result, ignoring polarizationeffects, the optical trapping force along the lateral y axis is identical to that along the lateralx axis. Therefore, when considering motion in a lateral plane, it is convenient to use polarcoordinates instead of Cartesian coordinates [12]. In the remainder of this paper, we willinterpret the spatial coordinate x as representing radial position in the lateral plane from thecenter of the trap.

The trapping force of an optical tweezer is proportional to the laser power [8]. We denotethe laser power by the factor ρ > 0, which is defined relative to the power level used in [2];that is, ρ = 1 corresponds to approximately 100 mW at the focus. The trapping force alsoincreases with numerical aperture, so the force models in Figure 3, and therefore, the materialin this paper, pertain to a 1.25 NA (numerical aperture) microscope objective, which was usedin [2]. However, as shown in [11], the trapping force for a spherical particle always has a profilethat qualitatively matches (1). Therefore, our results can be extended qualitatively to highernumerical apertures (for example, NA = 1.3 or 1.4), if necessary.

2.2. Equation of Motion

The equation of motion along the lateral x-axis for a trapped bead of mass m is given by

mx = FT (xr) + FD(x) + FL(t) + FE(t), (2)

where FT (·) is the optical trapping force, FD(·) is the viscous drag, FL(t) is a Langevin (randomthermal) disturbance force, and FE(t) represents other external forces. The drag force can beexpressed as FD = −βx, where β > 0 is the viscous damping factor given by Stoke’s equation,β = 6πηrb, in which rb is the bead radius and η is the fluid viscosity. For a 1-µm bead trappedin water at room temperature, β ≈ 0.01 pNs/µm. It should be noted that the drag coefficientgiven by Stoke’s equation requires adjustment for particles located in close proximity to the wallof the fluid cell; furthermore, the viscosity of water decreases dramatically with temperature[13].

Due to the scaling effect, for a microscopic particle bound in a harmonic potential at lowReynolds’ number (i.e., “small particles moving not too fast in a viscous medium”), viscousdrag dominates inertia [14]. Consequently, in practice, the mass of the trapped particle issmall enough that it can be ignored. For example, for a 1-µm diameter polystyrene bead,m ≈ 5.5× 10−10 mg, and the effective bandwidth of the trap (with the force profile shown inFigure 3) is not more than α1

β = 1 krad/s. The power fraction, drag force, (2), and (1) can becombined to obtain the noninertial equation of motion for a trapped particle:

0 = ρψ(xr)(α3x3r − α1xr)− βx+ FL(t) + FE(t), (3)

in which

ψ(xr) :={

1 for |xr| < R0 otherwise. (4)



Along one axis, the Langevin force has an average value of zero and a constant, one-sided power spectrum (i.e., ideal white noise force) given by S+

L (f) = 4βkBT , where kB

is Boltzmann’s constant and T is absolute temperature [14]. Correspondingly, denoting theDirac delta function by δ(t), the covariance rL(t) of the radial Langevin force is given byrL(t) = RLδ(t) = 2βkBTδ(t), since RL = 1

2S+L , in magnitude [15]. At biological temperatures,

kBT = 4×10−3 pNµm [14]; hence, for a 1-µm polystyrene bead trapped at room temperature,RL ≈ 8 × 10−5 µm2. The nature of the external force FE(t) will depend on experimentalconditions. For example, in biological experiments, the external force may arise due to theinteraction between the trapped particle and a biological particles.

2.2.1. State Model By defining trap position as the control input, u := xT , and measurementnoise as n, we can express (3) using a first-order state model:

x = ρψ(x− u)

β

[α3(x− u)3 − α1(x− u)

]+

1β

(FL + FE) (5)

y = x+ n. (6)

Because it does not account for angular position θ, the state form (5) is not a complete statespace description. For the purposes of this paper, however, we are not concerned with angularposition. Ignoring the external force FE and comparing open loop (5) with the standardstate form from [15], x = f(x, t) + σ(x, t)e(t), in which e(t) is white noise with covariancere(τ) = δ(τ), we see that f(x, t) = f(x) and σ(x, t) = σ are given by

f(x) = ρψ(x)β

[α3x

3 − α1x]

(7)

σ2 =2kBT

β. (8)

For a 1-µm bead trapped in water at room temperature, σ2 ≈ 0.8 µm2.

2.2.2. Stochastic Differential Equation We can express (5) as a stochastic differentialequation (SDE):

dx = ρψ(x− u)

β

[α3(x− u)3 − α1(x− u)

]dt+

FL

βdt. (9)

In the open loop case, by comparing the above SDE with the standard state form from [15],dx = f(x, t)dt+ σ(x, t)dw, in which w is a Wiener process with incremental covariance dt, wecan verify that f(x) and σ2 are given by (7) and (8).

2.2.3. Linear Trapping Region Clearly, the Langevin force and the external force can beinterpreted as disturbances. Within the linear trapping region, for zero initial conditions, (3)can be expressed using Laplace transforms as:

X(s) = Gyu(s)U(s) +Gyd(s) [FL(s) + FE(s)] . (10)

For the noninertial case, the first order transfer functions are given by

Gyu(s) = ωc

s+ωcGyd(s) =

1β

s+ωc, (11)

where the characteristic frequency (bandwidth) ωc of the trapped particle is defined as ωc := αβ ,

in which frequency is measured in radians per second. These transfer functions will be usedfor recursive system identification in Section 4.



3. STOCHASTIC ANALYSIS

As described in Section 2.2, a trapped particle experiences random position fluctuations dueto the Langevin force. As a result, the optical tweezer is a nanometer-scale stochastic system.

3.1. Probability Distribution

Defining p = p(x, t;x0, t0) as the probability of being in state x at time t given that the particlewas (initially) in state x0 at time t0, the conditional probability distribution p satisfies theFokker-Planck equation (also known as the Kolmogorov forward equation) given by

∂p

∂t= − ∂

∂x(pf) +

12∂2

∂x2(σ2p), (12)

where f and σ are defined according to (7) and (8), for our system [15, 16]. The initial conditionis specified as p(x, t0;x0, t0) = δ(x − x0). The time-dependent Fokker-Planck equation (12)can be solved numerically, but we have not included the solution in this paper. (Solutionprocedures can be found in most texts on partial differential equations.) To classify the natureof boundaries for a trapped particle, we consider the probability distribution in steady-state.Using (7), (8), and (12), we can obtain the following analytical solution for the steady-stateprobability distribution:

p(x) =

p(0)eρ x24kBT (α3x2−2α1) for |x| < R

p(0)e−ρα21

4kBT α3 for |x| ≥ R,(13)

which is shown graphically in Figure 4 [16]. Because there is no restoring force outside of thetrapping radius R, the probability density p(x) has a nonzero value for all positions outsideof the trapping radius. In other words, in the absence of finite absorbing boundaries, theparticle has a finite probability of being anywhere in the radial x-direction (i.e., in the lateralplane). In terms of classification of boundary conditions, this implies that a trapped particlehas accessible boundaries at all locations in the lateral plane [17]. In practice, the fluid cellwhich contains trapped particles has lateral dimension of approximately 20 mm and the fieldof view is typically about 100 µm. Therefore, for practical purposes, we can impose (virtual)absorbing boundaries at x = ±50 µm, which has been done for the distributions shown inFigure 4.

3.2. Mean First Exit Time

In the presence of accessible boundaries, we can define the first exit time T1 as the randomvariable,

T1 = T1(x0,−R,R) := sup{t|X(τ) ∈ (−R,R), 0 ≤ τ ≤ t}, (14)

where X(τ) is the random variable corresponding to particle position x with initial conditionX(0) = x0 [17]. We can use results from [17] to calculate the probability density function ofthe first exit time [16], but for the purposes of this paper, we will investigate the mean firstexit time, which can be obtained using much simpler calculations.

For our system, the mean first exit time m1 := E{T1} in the radial x direction is given by



the linear second order ordinary differential equation

12σ2 d

2m1

dx20

+ f(x0)dm1

dx0= −1, (15)

with two-point boundary conditions, m1(−R) = m1(R) = 0 [18]. Substituting f and σ from(7) and (8), we obtain:

kBT

β

d2m1

dx20

+ ρ(α3x

30 − α1x0)β

dm1

dx0+ 1 = 0, (16)

which can be solved numerically. The mean exit time for ρ = 1 (100 mW) is bounded, butextremely large, with a maximum in the vicinity of over 10100 trillion years for x0 = 0. Thislength of time is unbounded for all practical purposes! As shown in Figure 5, reducing thepower to ρ = 0.01 (1 mW) drastically reduces the mean exit time such that its maximum isapproximately 2.51 s at x0 = 0. For comparison, the mean exit time for ρ = 0 is 0.57 s, whichcorresponds to free diffusion of an untrapped particle due to Brownian motion.

Figure 6 shows the maximum mean exit time m1(0) as a function of the laser power factorρ for ρ ≤ 0.1 (10 mW). The solid line pertains to a 1-µm diameter polystyrene bead in waterat biological temperature (σ2 = 0.8 µm2; β = 0.01 pNs/µm). For comparison, three othercombinations of σ2 and β have also been plotted. The maximum mean exit time (solid line)for ρ = 0.05 (5 mW) is 3.63× 104 s, or about 10 hours, which is more than sufficient for mostoptical tweezer experiments.

3.2.1. Experimental Results Verifying the theoretical mean exit time results from Section 3.2is difficult for a number of reasons, including limited position detector range, excessively longexperiment duration, and the possibility of axial escape [16]. However, we can compute themean passage time for particles within the linear region quite easily by detecting zero crossingsand subsequent excursions outside of the radius r of interest. The mean passage time isanalogous to the mean exit time, but with R in (14) replaced by r < R [17]. Figure 7 showsthe calculated (experimental) mean passage times for a 9.6-µm diameter bead in a Phosphate-Buffered Saline (PBS) solution, which is used to prevent beads from clumping together. Forcomparison, theoretical values according to (16) are shown for the linear case. Clearly, thetheoretical and experimental values are in close agreement. The slight discrepancies for lowand high values of r are most likely due to unmodeled nonlinearities in the position detectorresponse; furthermore, experimental mean passage times for low r are artificially inflated dueto quantization errors. Although the experimental results in this section pertain to a bead thatis larger than the 1-µm bead studied in the previous sections, the linear force model appliesto beads of any size, as long as they remain within the linear region.

4. IDENTIFICATION

For quantitative measurements of (biological) force using an optical tweezer, trappingparameters such as characteristic frequency, stiffness, and drag must be calibrated with a highdegree of accuracy. In particular, we require system identification methods that are effectiveat the nanoscale.



4.1. Off-Line Identification

Most optical tweezer users, such as physicists and biophysicists, use off-line calibrationmethods. Three of the most common off-line methods are the Equipartition method, the powerspectrum method, and the step response method.

In the equipartition method, the trap stiffness α is obtained by measuring the thermalfluctuations of trapped particle position x. According to the Equipartition theorem, for aparticle bound in a harmonic potential, kBT = ασ2

x, where σ2x is the variance [19, 14]. Although

knowledge of the viscous drag coefficient is not needed, this method requires both a well-calibrated position detector with a high analog bandwidth and an environment with minimalnoise [19].

Much information can be obtained from the power spectrum of a trapped particle. Thepositive (single-sided) power spectrum S+

x (f) is given by S+x (f) = kBT

βπ2(f2c +f2) , where fc := ωc

2π

[14, 12]. If one fits the power spectrum S+x (f) to a Lorentzian shape, the roll-off frequency

should be equal to the characteristic frequency of the trap in Hertz, fc [14]. Therefore, withprior knowledge of viscous drag β, trap stiffness α can be obtained from the roll-off frequency.Furthermore, for low frequencies f � fc, the power spectrum is approximately constant,S+

x (f) ≈ S0, given by S0 = kBTβπ2f2

c, which shows that β and α can be calculated once fc and S0

have been measured [14]. Although the calculated power spectrum is asymptotically unbiased,it is an extremely erratic function: the standard deviation of each point is typically equal to itsmean [14, 20]. Fortunately, if we calculate the power spectra for many different data sets, themagnitude of the power spectrum for a chosen frequency will be uncorrelated [20]. Therefore,to obtain a smooth curve, we can calculate power spectra for many data sets and then averagethem. The true spectral characteristics of the system can be obtained from the smooth, averagepower spectrum of an infinite number of data sets [14, 20].

The step response of a trapped particle is often used to calibrate characteristic frequency.Since the Langevin force FL(t) has an average value of zero, for a small trap step sizexT (0+) (i.e., within the linear force region Rl), the average step response is given byx(t) = xT (0+) [1− e−ωct], which shows that the characteristic frequency ωc can be obtainedfrom the step response data. Furthermore, trap stiffness α can be obtained from knowledgeof the viscous damping factor β [19]. A calibrated detector is not required, but the timeconstant for trap movement must be faster than the characteristic damping time of the particle,τc := 1/ωc = β

α , which requires a fast actuator such as an acousto-optic deflector (AOD) [19].A more robust method for obtaining ωc is to re-arrange the expression for the average stepresponse and take the natural logarithm, to obtain ln

[1− x(t)

xT (0+)

]= −ωct. If necessary, initial

particle velocity data can be used to calculate (model) the entire nonlinear trapping force of anoptical trap. For example, Simmons et al. used initial velocity data and knowledge of viscousdrag to compute the experimental force model shown in Figure 3 [2].

4.2. On-Line Identification

In practice, the characteristic frequency of an optical trap can vary due to laser fluctuations,local heating, and cross-contamination. Methods that depend on purely off-line (batch)data analysis do not account for these effects and may suggest an erroneous value forthe characteristic frequency. Therefore, in a laboratory environment in which experimentalconditions are not entirely constant, on-line (recursive) parameter estimation methods could



potentially provide a more reliable (up-to-date) measure of characteristic frequency than theoff-line methods from Section 4.1. This is especially true for experiments in which the effectivetrap stiffness changes due to a force interaction between a trapped bead and an externalentity such as a biological molecule. Furthermore, recursive identification methods enables theimplementation of adaptive control.

4.2.1. Sampled Data System For implementation of recursive identification using a computer,a continuous-time (CT) SISO model of the form Y (s) = G(s)U(s), with first order transferfunction (11), can be zero-order hold sampled with sampling time h to obtain the discrete-time(DT) difference equation y(kh) = H(q)u(kh), where k is a positive integer and H(q) is the firstorder pulse transfer operator given by H(q) = b1

q+a1, in which q is the forward shift operator

and a1 = −e−ωch, b1 = 1 − e−ωch. Hence, the causal, zero-order hold equivalent of Gyu(s) in(11) is given by:

Hyu(q) =(1− e−ωch)q−1

1− e−ωchq−1, (17)

which implies a causal difference equation of the form y(kh)+a1y(kh−h) = b1u(kh−h). Notethat, b1 = 1 + a1. The zero-order hold equivalent of Gyd(s) in (11) is given by

Hyd(q) =1α (1− e−ωch)1− e−ωchq−1

, (18)

which is equivalent to a causal difference equation of the form y(kh)+ a1y(kh−h) = b1α d(kh).

The numerator contains a direct term that implies that the white noise is assumed to gothrough the denominator dynamics of the system before being added to the output [20]. Thecombined difference equation corresponding to the models (17) and (18) is given by

y(kh) + a1y(kh− h) = b1u(kh− h) +b1αd(kh), (19)

in which d(kh) is white noise with variance σ2d. If we normalize the white noise such that

its coefficient in (19) is 1, we obtain the standard ARX form, y(kh) + a1y(kh − h) =b1u(kh− h) + e(kh), in which {e(kh)} is white noise with variance σ2

e = ( b1α )2σ2

d.

4.2.2. Recursive Least Squares Algorithm In this section, we normalize the sampling time hto unity. We would like to compute the DT parameter estimate θ(k) := [a1 b1]T at sample kthat minimizes the weighted least-squares criterion:

θ(k) = arg minθ

k∑n=1

w(k, n)[y(n)− φT (n)θ]2, (20)

where φ(k) := [−y(k − 1) u(k − 1)]T is the regression vector that contains the input andoutput data, and w(k, n) is a weighting sequence in which λ(k) is the forgetting factor [20].The standard RLS algorithm is outlined in [20]. The initial parameter vector is denoted as

θ(0) = θ0 and the initial covariance matrix of the paremeters is denoted as P0 =[p 00 p

], in

which p > 0. The initial regression vector is specified as φ(0) = [0 0]T . The forgetting factoris set to λ = 1 for an LTI system.



4.2.3. Computer Simulations The CT system described by (10) was simulated using Simulinkwith a simulation sampling time of Ts = 0.1 ms, corresponding to 10 kilosamples per second(kS/s). The Langevin disturbance was modeled as band-limited white noise with bandwidth10 kHz and constant power SL(f) = 1.6× 10−3 pN2

Hz . This represents the Langevin force thatacts on a 10-µm diameter polystyrene bead at biological temperatures. To facilitate comparisonwith experimental results, the actual characteristic frequency of the simulated system is chosenas ωc = 100 rad/s. According to Section 4.2.1, for RLS sampling time h = 1 ms, the actualparameters are given by a = b = 100 and θ = [a1 b1]T = [−0.9048 0.09516]T . We assumean initial characteristic frequency guess of ωc(0) = 80 rad/s, which corresponds to an errorof 20% (13 rad/s). According to Section 4.2.1, the initial parameter conditions are given bya(0) = b(0) = 80 and θ0 = [a1(0) b1(0)]T = [−0.9231 0.0769]T . In the simulations that follow,we denote the input signal amplitude by A and input frequency by f .

Figure 8 shows simulation results for an input square wave with A = 200 nm and f = 10 Hz.For comparison, the DT estimates are plotted as 1 + a1 and b1, since we expect these twoquantities to converge to the same value. After fluctuating wildly at the onset, both the DTand CT parameter estimates eventually converge close to their correct values of b1 and ωc,respectively.

The parameter estimates display fluctuations that diminish with time. In particular, the CTparameter estimates settle to within 5% of ωc in under 0.6 s and to within 2% in under 1 s.The direct correspondence between the DT and CT estimates can be seen from the similarshape of their plots. The reason for the paramater fluctuations is the Langevin disturbance.

The value of f = 10 Hz was used because it gave the fastest convergence for the chosenvalue of ωc. This is consistent with the practical recommendation that input power be selectedat frequency bands in which a “good model is particularly important”, or more formally,frequencies at which the “Bode plot is sensitive to parameter variations” [20]. In fact, wefound that both f = 2 Hz and f = 20 Hz resulted in slower parameter convergence. We alsofound that a square wave provides faster convergence than a sinusoidal input. The superiorityof the square wave can be explained using the crest factor Cr, which should be minimized toreduce the covariance of the parameter estimates [20]. For a discrete input sequence {u(k)},the crest factor is given by

C2r =

maxk u2(k)

limN→∞1N

∑Nk=1 u

2(k), (21)

which is clearly at its theoretical lower bound for binary, symmetric signals such as a squarewave [20]. Computer simulations for an input Pseudo-Random Binary Sequence (PRBS) withA = 200 nm and cutoff at 40 Hz yield convergence results that are comparable to a squarewave with f = 10 Hz. A PRBS is particularly useful in situations in which we have poorknowledge of the characteristic frequency. Simulation results for binary filtered white noisewas not promising: parameter convergence was slow compared to the square wave and thePRBS.

In general, filtering will not distort the input-output relationships provided both the inputand output are subject to exactly the same filters [20]. Effects of filtering can also be neglectedby sampling fast enough that the Nyquist frequency is significantly greater than the bandwidth.Computer simulations show that an analog RC lowpass filter with a 1 kHz bandwidth willreduce the parameter estimates by about 5%, while a bandwidth of 5 kHz will reduce theestimates by about 1%.



4.2.4. Experimental Results Since the RLS algorithm does not require real-time feedbackcontrol, we were able to investigate its performance by collecting input and output data fromour system using LabVIEW data acquisition software and hardware and then processing thedata using Matlab. Data was sampled at a rate of 2 kS/s with an analog lowpass filter at theNyquist frequency. Figure 9 shows results for an input square wave with A = 150 nm andf = 10 Hz. The lowpass filter has a higher cutoff than the Nyquist frequency to a avoidundesirable distortion of the output signal; the price we pay is the signal contains somemeasurement noise, but this is negligible. Also, alignment errors in the position detectionsystem can distort the estimation data. In the case of a square wave, such offsets can beremoved by off-line averaging.

After initial fluctuations, the parameter estimates appear to reach steady values within 9seconds. The values of the CT estimates after 30 s are a = 112 rad/s and b = 113 rad/s.Experimental results for a PRBS were not encouraging. In theory, the PRBS should providefast convergence, comparable to a square wave input. Offsets or misalignments in the positiondetection system can severely distort the estimation. In the case of square wave, we were ableuse averaging to calculate and remove the offset from our data prior to identification. For thePRBS, this is much more difficult to do because the signal has large periods. In practice,ofcourse, we should not have to adjust offsets when implementing an on-line calibrationbecause that would defeat the purpose of on-line calibrations. The solution is to implementan automatic tracking system that can automatically align the laser beam onto the detector;the tracking criterion can be implemented in an RLS manner. Alternatively, a second (diffuse)laser beam can be used purely for position detection [19].

5. CONTROL

A well-tuned position feedback system can be used to convert the optical tweezer into eitheran isometric position clamp (used to keep the position of trapped particles constant) or anisotonic force clamp (used to keep the force acting on trapped particles constant) [19, 21, 2].

5.1. Proportional Control

For a first order plant, PI control is one of the most straightforward and commonly used controlstrategies. For a system with purely zero-mean white noise disturbances, integral control iscounter-productive [12]. Therefore, we consider proportional control u = −kpy, which is inwide use in the biophysics community.

The practical tradeoffs that can be expected for different proportional control gains areshown in Figure 10 †. The left figure shows position standard deviation and the right figureshows maximum absolute position (infinite norm). Results are shown for both the linear forceand for the cubic force. In the cubic case, position variance decreases until about kp = 35,gradually increases for 35 < kp < 40, and dramatically increases for kp ≥ 40. Up to kp ≈ 40,the net effect of the controller is to increase the effective stiffness of the trap. For kp = 35,the variance is 1.36× 10−5 µm2, which is approximately 29 times smaller than the open loop

†All variance values from the simulations have been normalized to conform with the Equipartition theorem.



variance of 4× 10−4 µm2. For the minimum variance near kp = 35, the particle has a maximumexcursion that is slightly beyond the trapping radius, according to the right figure. For valuesof kp greater than approximately 30, the particle has excursions outside of the trapping radiusof 674 nm, which results in an occasional absence of trapping force and a correspondingincrease in variance. As kp is further increased, the relative position develops larger excursionsoutside of the trapping radius until it eventually escapes from the trap, resulting in verylarge position variance. Such excursions are a result of the proportional controller being tooaggressive because it is not designed to handle the nonlinear trapping force. For the lineartrapping force, both variance and maximum absolute position display minima near kp ≈ 100,which agrees with theoretical analysis in [12]. The drop in absolute position for very highgains is due to the limited bandwidth of the nonlinear system. As the gain is increased, thecontroller attempts to move the particle at very high velocities (frequencies), which are beyondthe system’s bandwidth; the system behaves like a lowpass filter for very high gains.

5.2. LQG Control

LQG control is an obvious strategy for position regulation, assuming noise inputs are zero-meanwhite noise processes and the system is linear [22]. Computer simulations show that the LQGcontroller significantly reduces position fluctuations due to thermal noise. The (open loop)variance of 4 × 10−4 µm2 is decreased by a factor of approximately 27, which is comparableto the performance of the proportional controller [12].

5.3. Nonlinear Control

As shown in Figure 3, the nonlinear trapping region of an optical tweezer is much largerthan the linear trapping region. In this section, we describe the performance of a nonlinearcontrol law that achieves global asymptotic stabilization (GAS) of the origin by utilizing ourknowledge that the trapping force is, in fact, nonlinear for large relative displacements. From[23], the globally asymptotically stabilizing nonlinear feedback control law u is of the form

u = y − ω tanh (kty) , (22)

in which, kt > 0 and 0 < ω < R = 674 nm [23]. By choosing ω within this range, GAS isguaranteed because xr will always exist within the region in which the nonlinear restoring forceFT (xr) is not zero (except at the origin) [23]. Furthermore, by picking ω = RF = 389 nm, inaddition to achieving GAS, the particle will also be driven into (restricted to) the region inwhich the nonlinear restoring force FT (xr) is maximized [23].

The tradeoffs associated with different nonlinear control gains kt are shown in Figure 11. Forthe cubic trapping force, position variance decreases until about kt = 2 × 106 and graduallyincreases for larger kt. For kt = 2 × 106, the variance is 6.18× 10−6 µm2, which is over 64times smaller than the open loop variance of 4× 10−4 µm2.

6. CONCLUSION

In this paper, we provide a basic introduction to the dynamic behavior of optical traps. Bycharacterizing their properties using terminology from control engineering, we have providedmathematical descriptions that should be accessible to anyone interested in studying optical



traps in greater depth. For example, the use of a cubic trapping force model enables thederivation of analytic expressions.

We developed stochastic differential equations that enable computation of probabilitydensity functions and first exit times. For a given optical tweezer configuration, the first exittime is an extremely useful measure of trapping capability because it quantifies the time horizonduring which experiments can be conducted before trapped particles are lost. It is especiallyimportant to use lower power levels when studying biological samples to avoid damaging themwith heat; furthermore, in applications in which a single laser beam is time-shared to trap manyparticles, it is important to quantify low power trapping capabilities. Published informationabout minimum power levels are based on experimental observations; for example, Smith et al.observe that “polystyrene spheres could be trapped with powers at the back of the objectiveas small as 5 mW” [8]. The theoretical framework we have developed can be used to verifysuch statements and also quantify the trapping capabilities for various power levels. This willsave both time and money.

We show that the mean first exit time for a given trapping force model can be computednumerically. We calculated values for a 1-µm diameter polystyrene bead trapped in water atbiological temperature; in particular, for laser powers of greater than approximately 5 mW atthe focus, the mean first escape time is extremely large, and unbounded for most practicalpurposes. We show that the maximum mean exit time increases exponentially with laser power.Using experimental data for a trapped, 9.6-µm diameter polystyrene bead, we calculated themean first passage time and its standard deviation within the linear trapping region. Theexperimental value shows close agreement with theoretical calculations. Since the mean passagetime is very sensitive to parameter values, it can (potentially) be used to verify the results ofother calibration methods.

We briefly described off-line (batch) calibration methods that are in wide use in thebiophysics community. We propose the use of on-line calibration to achieve faster and moreup-to-date calibrations. By describing the sampled-data system using an ARX model, we wereable to implement recursive least squares (RLS) estimation to calibrate the characteristicfrequency of an optical tweezer. Computer simulation show that a well-aligned system canbe calibrated within 2% in under 1 second using an RLS approach. Computer simulationsshows that further improvements can be achieved by identification in closed loop. However,in practice, the RLS method requires a very well-calibrated position detector and an inputsignal. Slight offsets in the position detection system can cause significant calibration errors.For this reason, calibration using a PRBS was not successful for our system; for a square waveinput, the alignment offset can be subtracted quite accurately. For practical implementationof the RLS method, an automatic tracking system is suggested for the position detector.This will minimize laser misalignments due to human error. Alternatively, a second (diffuse)laser beam can be used purely for position detection [19]. It should be noted that calibrationinconsistencies can arise due to factors such as slight misalignments in the position detectionsystem and fluctuations in the dynamic laser pointing system. The power spectrum method, inparticular, is robust with respect to such factors [14, 19]. Additional sources of measurementnoise, such as low-frequency drift and other types of electronic bias, high frequency amplifiernoise, mechanical vibrations, and extraneous background light can also contribute to erroneousestimates. Such problems are inherent in any practical position detection system.

Although not analyzed in this paper, it should be mentioned that optical tweezers can beused to detect force jumps (discontinuities) in biological experiments. For example, such force



jumps occur in experiments involving unzipping DNA [24] and interactions between proteinssuch as actin and myosin [21]. In the future, we propose the use of on-line identification methodssuch as RLS for the fast detection of force interactions. In particular, the use of innovations(prediction errors)–which has been used succesfully in AFM experiments [25]–shows muchpromise for optical tweezers.

We discussed the performance of linear controllers that are designed for a linear trappingforce and a nonlinear controller that is designed for a cubic trapping force. Computersimulations show that the linear controllers are effective for low gains because the particledoes not leave the trapping radius. However, for high gains, the linear controllers are counter-productive because they drive the particle beyond the trapping radius. The nonlinear controlleris effective for all gains because it is specifically designed to prevent the particle from leavingthe trapping radius. In fact, we can choose the nonlinear controller parameters in such away as to maximize the restoring force at all times. The performance of each controller issummarized in Table I. All three controllers are capable of reducing the position variancesignificantly (compared to open loop). The nonlinear hyperbolic tangent controller is superiorto both the proportional controller and the LQG controller. For a linear trapping force, weexpect the LQG controller to provide position variance properties that are superior to theproportional controller because the former uses a Kalman filter to obtain position estimatesthat are optimal according to a variance criterion; however, for the cubic trapping force model,the Kalman filter is no longer optimal and the controller performance is degraded. In practice,if proportional control works well-enough, there is usually no need to replace it because itis easiest to implement. Although the variance reduction factor for the nonlinear controller isimpressive (factor of∼ 65) compared to the open loop case, the variance reduction factor is onlyabout 2.2 compared to the optimal proportional controller. In fact, a proportional controllerwith saturation provides equally effective, if not better, performance than the hyperbolictangent controller. In practice, the most likely control option is proportional control withsaturation, for example, u = −ωsat(kpy). The saturation function limits the over-reaction ofthe linear controllers; for analog control, saturation can be implemented quite easily using aZiener diode. For best performance, the saturation level ω can be set to ω ≈ RF = 389 nm,which roughly limits the relative position to the maximum force radius RF .

We hope the material in this paper will both enhance the arsenal of tools available tousers of optical tweezers (especially in biophysics and microfluidics) and also encourage futurecontributions from the control engineering community.

7. APPENDIX

A schematic diagram of our optical tweezer setup is shown in Figure 12, which is describedin detail in [26, 12]. The various optical components are positioned according to a schemesuggested by Fallman and Axner in which the trapping force is kept constant regardless oflaser beam position [27]. A general view of the optical components in our system is shown inFigure 13.

REFERENCES



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2. R. M. Simmons, J. T. Finer, S. Chu, and J. Spudich, “Quantitative measurements of force and displacementusing an optical trap,” Biophys. J., vol. 70, pp. 1813–1822, 1996.

3. S. M. Block, “Making light work with optical tweezers,” Nature, vol. 360, pp. 493–495, 1992.4. H. Felgner, O. Muller, and M. Schliwa, “Calibration of light forces in optical tweezers,” Appl. Optics,

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tweezers for undergraduate laboratories,” Am. J. Physics, vol. 67, no. 1, pp. 26–35, 1999.9. M. P. Sheetz, “Preface,” in Methods in Cell Biology (M. P. Sheetz, ed.), vol. 55 (Laser Tweezers in Cell

Biology), pp. xi–xii, Academic Press, 1998.10. A. D. Mehta, J. T. Finer, and J. A. Spudich, “Reflections of a lucid dreamer: optical trap design

considerations,” in Methods in Cell Biology (M. P. Sheetz, ed.), vol. 55 (Laser Tweezers in Cell Biology),ch. 4, pp. 47–69, Academic Press, 1998.

11. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,”in Methods in Cell Biology (M. P. Sheetz, ed.), vol. 55 (Laser Tweezers in Cell Biology), ch. 1, pp. 1–27,Academic Press, 1998.

12. A. Ranaweera, Investigations with Optical Tweezers: Construction, Identification, and Control. PhDthesis, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara,September 2004.

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Biology (M. P. Sheetz, ed.), vol. 55 (Laser Tweezers in Cell Biology), ch. 8, pp. 129–156, Academic Press,1998.

15. K. J. Astrom, Introduction to Stochastic Control Theory. Academic Press, 1970.16. A. Ranaweera, K. J. Astrom, and B. Bamieh, “Lateral mean exit time of a spherical particle trapped in an

optical tweezer,” in Proceedings of the 43nd IEEE Conference on Decision and Control, (Paradise Island,Bahamas), December 2004. To appear.

17. A. T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill,1960.

18. D. R. Cox and H. D. Miller, The Theory of Stochastic Processes. John Wiley, 1965.19. K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-

resolution position sensing,” IEEE J. Select. Topics Quantum Electronics, vol. 2, no. 4, pp. 1066–1076,1996.

20. L. Ljung, System Identification. Prentice Hall PTR, 2nd ed., 1999.21. J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White, “Movement and force

produced by a single myosin head,” Nature, vol. 378, pp. 209–212, 1995.22. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control. Wiley, 1996.23. A. Ranaweera, A. R. Teel, and B. Bamieh, “Nonlinear stabilization of a spherical particle trapped in

an optical tweezer,” in Proceedings of the 42nd IEEE Conference on Decision and Control, (Maui, HI),pp. 3431–3436., December 2003.

24. U. Bocklemann, P. Thomen, B. Essevaz-Roulet, V. Viasnoff, and F. Heslop, “Unzipping DNA with opticaltweezers: High sequency sensitivity and force flips,” Biophysical Journal, vol. 82, pp. 1537–1553, March2002.

25. A. Sebastian, D. R. Sahoo, and M. V. Salapaka, “An observer based sample detection scheme for atomicforce microscopy,” in Proceedings of the 42nd IEEE Conference on Decision and Control, (Maui, HI),pp. 2132–2137., December 2003.

26. A. Ranaweera and B. Bamieh, “Calibration of the characteristic frequency of an optical tweezer usingan adaptive normalized gradient approach,” in Proceedings of the 2003 American Control Conference,(Denver, CO), pp. 3738–3743, IEEE, June 2003.

27. E. Fallman and O. Axner, “Design for fully-steerable dual-trap optical tweezers,” Appl. Optics, vol. 36,no. 10, pp. 2107–2113, 1997.



Figure 1. Basic optical tweezer. A single laser beam is focused to a diffraction-limited spot using ahigh numerical aperture microscope objective. Dielectric particles are trapped near the laser focus.

Figure 2. Typical biomechanics experiment. The ends of the DNA molecule are attached to polystyrenebeads which are trapped and moved using optical tweezers.

Table I. Summary of controller performance according to computer simulations, assuming a cubictrapping force. Listed properties are best performance values relative to open loop.

Controller Variance reduction factor

LQG 27Proportional 29Hyperbolic tangent 65



−1 −0.5 0 0.5 1

−2

0

2F

T (

pN)

Experimental modelCubic model

−1 −0.5 0 0.5 1

0

5

10

xr (µm)

α e (pN

/µm

)

Experimental modelCubic model

Figure 3. Cubic and experimental optical force models for a 1-µm diameter polystyrene bead. Topfigure shows trapping force; bottom figure shows effective stiffness. Experimental model is from [2], inwhich a 1-µm diameter bead was trapped in water using laser power of approximately 100 mW (at

the focus) using a 1.25 NA microscope objective.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−150

−100

−50

0

I(x)

(µm

2 /s)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

5

10

15

20

x (µm)

p(x)

ρ = 1 ρ = 0.1

Figure 4. Normalized steady state probability distribution calculations for ρ = 1 and ρ = 0.1, assumingfinite absorbing boundaries at x = ±50 µm. The nonzero tails of the probability distributions are too

small to be seen.



−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

m1 (

s)

x0 (µm)

ρ = 0.01

Figure 5. Mean exit time for ρ = 0.01 (1 mW). Maximum is m1(0) = 2.51 s.

0 0.02 0.04 0.06 0.08 0.1

100

102

104

106

m1(0

) (s

)

ρ (100 mW)

Maximum Mean Exit Time

β = 0.010; σ2=1.6β = 0.010; σ2=0.8β = 0.020; σ2=0.8β = 0.005; σ2=1.6

Figure 6. Maximum mean exit time as a function of laser power factor. The mean exit timeincreases exponentially with laser power. Solid line corresponds to a 1-µm polystyrene bead at room

temperature.



0.02 0.04 0.06 0.08 0.1 0.12 0.1410

−4

10−2

100

r (µm)

m1 (

s)

Mean Passage Time

Theoretical Experimental

0.02 0.04 0.06 0.08 0.1 0.12 0.14

102

104

r (µm)

Num

ber

of p

asse

s

Experimental

Figure 7. Maximum mean exit time within the linear region for a 9.6-µm polystyrene bead. Top plotshows measured mean exit time, including measured standard deviation bounds (dashed lines); bottom

plot shows number of pertinent crossings (passes) outside of the radius of interest.

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

I/O

(µ

m)

t (s)

uy

100

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

Su (

V2 /H

z)

f (Hz)

0 2 4 6 8 100.09

0.095

0.1

0.105

0.11

0.115

0.12

DT

t (s)

1+a∧1

b∧1

b1

0 2 4 6 8 1080

90

100

110

120

t (s)

CT

(ra

d/s)

a∧

b∧

ωc

Figure 8. Simulation of RLS for h = 1 ms, λ = 1, p = 104; square wave input, A = 200 nm andf = 10 Hz. The top left plot shows the input and output signals; the top right plot shows the powerspectrum of the input signal; the bottom left plot shows the DT parameter estimates, and the bottomright plot shows the CT parameter estimates. Dash-dotted lines on the bottom right plot show ωc±5%

and ωc ± 2% limits.



0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

I/O

(µ

m)

Experimental Results (f = 10 Hz)

uy

0 2 4 6 8 10

0.07

0.08 D

T1+a∧

1b∧

1

0 2 4 6 8 1070

80

90

t (s)

CT

(ra

d/s) a∧

b∧

Figure 9. Implementation of RLS for experimental data with h = 1 ms, λ = 1, p = 104; square waveinput, A = 150 nm and f = 10 Hz.

100

101

102

103

0

0.005

0.01

0.015

0.02

0.025

0.03

σ x (µm

)

kp

Standard Deviation

Open Cubic F

T

Linear FT

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|x| ∞

(µm

)

kp

Maximum Absolute Position

x=xr (Open)

x (Cubic FT)

x (Linear FT)

xr (Cubic F

T)

xr (Linear F

T)

Figure 10. Computer simulation of proportional controller performance. Left figure shows positionstandard deviation and right figure shows maximum absolute position; solid lines correspond tocubic trapping force and dashed lines correspond to linear trapping force. Simulations assumeα1 = 10 pN/µm; α3 = 22 pN/µm3 (cubic force) and α3 = 0 (linear force); β = 0.01 pNs/µm;measurement noise power S+

n = 2×10−10 µm2/Hz, and sampling time 0.01 ms. For the cubic trappingforce, variance has a minimum near kp = 35, at which the particle has a maximum excursion that is

slightly beyond the trapping radius R = 674 nm.



105

0

0.005

0.01

0.015

0.02

0.025

σ x (µm

)

kt

Standard Deviation

Open Cubic F

T

Linear FT

105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

|x| ∞

(µm

)

kt

Maximum Absolute Position

x=xr (Open)

x (Cubic FT)

x (Linear FT)

xr (Cubic F

T)

xr (Linear F

T)

Figure 11. Computer simulation of nonlinear controller performance. Left figure shows positionstandard deviation and right figure shows maximum absolute position; solid lines correspond tocubic trapping force; and dashed lines correspond to linear trapping force. Simulations assumeα1 = 10 pN/µm, α3 = 22 pN/µm3 (cubic force) and α3 = 0 (linear force), β = 0.01 pNs/µm, andsampling time 0.01 ms. For the nonlinear trapping force, variance has a minimum near kt = 2× 106.

Figure 12. Schematic diagram of single-axis optical tweezer system. PBSC = Polarizing Beam SplittingCube, KM = Kinematic Mirror, AOD = Acousto-Optic Deflector, PSD = Position Sensing Detector,

CCD = CCD Camera, DM = Dichroic Mirror.



Figure 13. Optical tweezer system. Left image shows position detection system; right image showssteering optics.


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