optical tweezers - university of florida · focused laser eld of an optical tweezers, the gradient...

Optical Tweezers

Experiment OT - sjh,rd

University of Florida — Department of PhysicsPHY4803L — Advanced Physics Laboratory

Objective

An optical tweezers apparatus uses a tightlyfocused laser to generate a trapping force thatcan capture and move small particles under amicroscope. Because it can precisely and non-destructively manipulate objects such as indi-vidual cells and their internal components, theoptical tweezers is extremely useful in biolog-ical physics research. In this experiment youwill use optical tweezers to trap small silicaspheres immersed in water. You will learn howto measure and analyze the frequency spec-trum of their Brownian motion and their re-sponse to hydrodynamic drag in order to char-acterize the physical parameters of the opticaltrap with high precision. The apparatus canthen be used to measure a microscopic bio-logical force, such as the force that propels aswimming bacterium or the force generated bya transport motor operating inside a plant cell.

References

D.C. Appleyard et al., Optical trapping forundergraduates, Am. J. Phys. 75 5-14(2007).

A. Ashkin, Acceleration and Trapping of Par-ticles by Radiation Pressure, Phys. Rev.Lett. 24 156-159 (1970).

K. Berg-Sorensen and H.Flyvbjerg, Power

spectrum analysis for optical tweezers,Rev. Sci. Instr. 75 594-612 (2004).

S. Chattopadhyay, R. Moldovan, C. Ye-ung, X.L. Wu, Swimming efficiency ofbacterium Escherichia coli, Proc. Nat.Acad. Sci. USA 103 13712-13717 (2006).

Daniel T. Gillespie The mathematics ofBrownian motion and Johnson noise,Amer. J. of Phys. 64 225 (1996).

Daniel T. Gillespie Fluctuation and dissipa-tion in Brownian motion, Amer. J. ofPhys. 61 1077 (1993).

S.F. Tolic-Norrelykke, et al., Calibration ofoptical tweezers with positional detectionin the back focal plane, Rev. Sci. Instr.77 103101 (2006).

University of California, Berkeley ad-vanced lab wiki site on optical trapping:www.advancedlab.org/mediawiki/

index.php/Optical_Trapping is one ofthe best references for this experiment.Be sure to check it out!

Introduction

The key idea of optical trapping is that a laserbeam brought to a sharp focus generates arestoring force that can pull particles into that

OT - sjh,rd 1

www.advancedlab.org/mediawiki/index.php/Optical_Trapping

www.advancedlab.org/mediawiki/index.php/Optical_Trapping

OT - sjh,rd 2 Advanced Physics Laboratory

focus. Arthur Ashkin demonstrated the prin-ciple in 1970 and reported on a working ap-paratus in 1986. The term optical trappingoften refers to laser-based methods for hold-ing neutral atoms in high vacuum, while theterm optical tweezers (or laser tweezers) typ-ically refers to the application studied in thisexperiment: A microscope is used to bring alaser beam to a sharp focus inside an aque-ous sample so that microscopic, non-absorbingparticles such as small beads or individual cellscan become trapped at the beam focus. Op-tical tweezers have had a dramatic impact onthe field of biological physics, as they allowexperimenters to measure non-destructivelyand with high precision the tiny forces gen-erated by individual cells and biomolecules.This includes propulsive forces generated byswimming bacteria, elastic forces generated bydeformation of biomolecules, and the forcesgenerated by processive enzyme motors op-erating within a cell. Experimenting withan apparatus capable of capturing, transport-ing, and manipulating individual cells and or-ganelles provides an intriguing introduction tothe world of biological physics.

A photon of wavelength λ and frequencyf = c/λ carries an energy E = hf and amomentum of magnitude p = h/λ in the di-rection of propagation (where h is Planck’sconstant and c is the speed of light). Notethat our laser power—up to 30 mW—focuseddown to a few square microns, implies laserintensities over 106 W/cm2 at the beam fo-cus. Particles that absorb more than a tinyfraction of the incident beam will absorb alarge amount of energy relative to their vol-ume rather quickly. In fact, light-absorbingparticles can be quite rapidly vaporized (op-ticuted) by the trapping laser. (Incidentally,your retina contains many such particles - seeLaser Safety below). While the scatterer andsurrounding fluid always absorbs some energy,

our infrared laser wavelength (λ = 975 nm) isspecifically chosen because it is where absorp-tion in water and most biological samples islowest. The absorption rate is also near a min-imum for the silica spheres you will study. Youshould keep an eye out for evidence of heatingin your samples, but because of the relativelylow absorption rate and because the particleshave good thermal conductance with the sur-rounding water, effects of heating should bemodest.

The theory and practice of laser tweezersare highly developed and numerous excellentreviews, tutorials, simulations, and other re-sources on the subject are easy to find online.

Physics of the trapped particle

The design, operation, and calibration of ourlaser tweezers draws on principles of optics,mechanics and statistical physics. We beginwith an overview of the physics relevant togenerating the trapping force and for calibrat-ing the restoring and viscous damping forcesassociated with its operation.

The laser force arises almost entirely fromthe elastic scattering of laser photons wherebythe particle alters the direction of the pho-ton momentum without absorbing any of itsenergy. It is typically decomposed into twocomponents: (1) a gradient force that every-where points toward higher laser intensitiesand (2) a weaker scattering force in the di-rection of the photon flow. For the sharplyfocused laser field of an optical tweezers, thegradient force points toward the focus and pro-vides the Hooke’s law restoring force respon-sible for trapping the particle. The scatter-ing force is in the direction of the laser beamand simply shifts the trap equilibrium positionslightly downstream of the laser focus.

The origin of both forces is similar: the par-ticle elastically scatters a photon and alters its

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Optical Tweezers OT - sjh,rd 3

ray

x

F

F1

F2

F

z

x

z

2 1

A B

z

Fz

ray 1 ray 2

CLaser beam

Fx

x

Figure 1: Ray model for the trapping force atthe focus of a laser beam. A particle displacedhorizontally (A) or vertically (B) from the focus(at x = y = z = 0) refracts the light away fromthe focus, leading to a reaction force that pulls theparticle toward the focus; (C) Schematic of therestoring forces Fx and Fz versus displacement xand z of the particle from the trap center. Nearthe beam focus, Fx ≈ −kx and Fz ≈ −k′z.

momentum. Momentum conservation impliesthat the scattered photon imparts an equaland opposite momentum change to the par-ticle. The net force on the particle is a vectorequal and opposite the net rate of change ofmomentum of all the scattered laser photons.

For particles with diameters d large com-pared to λ, the ray optics of reflection and re-fraction at the surface of the sphere provide agood model for the laser forces. The ray draw-ings in Figure 1 illustrate how laser beam re-fraction generates a trapping force. The laserbeam is directed in the positive z-directionand brought to a focus by a microscope ob-jective. Note that, owing to wave diffraction,the focal region has nonzero width in the xy

direction. Near the beam focus, a spherical di-electric particle alters the direction of a ray byrefracting it as shown in 1A. Momentum con-servation implies that the particle experiencesa force, indicated by F in the figure, that is di-rected toward the beam focus. If the particleis located below the focus, it refracts the con-verging rays (such as rays 1 and 2) as shownin 1B. The corresponding reaction forces F1

and F2 acting on the particle give a vectorsum F that is again directed toward the laserfocus. The net result of all the refractive scat-tering at any location in the vicinity of thefocus results in the gradient force that pullsthe particle into the beam focus. Reflectionat the boundaries between the sphere and themedium results in the scattering force in thedirection of the laser photons.

For smaller particles of diameter d � λ,Rayleigh scattering describes the interaction:The particle acts as a point dipole, scatter-ing the incident beam in a spatially dependentfashion that depends on the particle’s locationin the laser field. The result is a net force Fgiven by

F = α

(1

2∇E2 +

d

dt(E×B)

)(1)

where p = αE gives the particle’s induceddipole moment. The first term is in the direc-tion of the gradient of the field intensity, i.e.,the trapping force directed toward the laser fo-cus. The second term gives the weaker scatter-ing force—in the direction of the field’s Poynt-ing vector E×B.

The center of the trap will be taken asr = 0. For any small displacement (any di-rection) away from the trap center, the parti-cle is subject to a Hooke’s law restoring force,i.e., proportional to and opposite the displace-ment. Detailed calculations show that theforce constant is sensitive to the shape andintensity of the laser field, the size and shape

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of the trapped particle, and the optical prop-erties of the particle and surrounding fluid.Consequently, the Hooke’s law force is diffi-cult to predict. Furthermore, our apparatusoperates in an intermediate regime of particlesizes where neither the ray optics nor Rayleighmodels are truly appropriate. The diameter dof the silica spheres (SiO2) range in size from0.5-5 µm. Thus with the laser wavelength ofλ = 975 nm, we have d ∼ λ. Fortunately, wedo not need to calculate or predict the Hooke’slaw force constants based on these scatteringmodels. Instead, you will learn how to deter-mine them in situ—from measurements madewith the particle in the trap.

Consider the motion and forces in terms oftheir components. The laser beam in our ap-paratus is directed vertically upward, whichwill be taken as the +z direction so that the xand y coordinates then describe the horizon-tal plane. Because the laser beam and focus-ing optics are cylindrically symmetric aroundthe z-axis, the trap has the same properties inthe x-direction as in the y-direction. We needonly consider the equations for the x motion ofthe particle, and a similar set of equations willdescribe the motion in the y-direction. How-ever, the trapping force that acts along the zdirection is different than for x and y, as thelaser intensity in the focal region is clearly nota spherically symmetric pattern. The widthof the beam focus in its radial (xy) dimensionis very narrow. It is limited by wave diffrac-tion to roughly one wavelength (λ ∼ 1µm),whereas this is not the case in z. Hence therestoring force in z is not necessarily as strongas in xy.

If the focal “cone” has too shallow an angle(technically, a large f -number or small numer-ical aperture), particles may be trapped in thexy direction but not trapped along z. Thelaser beam will tend to pull small particles intoward the central optical axis and then push

them up and out of the trap. By employing alarge numerical aperture, our apparatus pro-vides excellent trapping in all three directions.

We will investigate the motions of the par-ticle in the xy directions only. Consequently,in the discussion that follows, when forces, im-pulses, velocities or other vector quantities arewritten without vector notation (e.g., F in-stead of F) and without explicit directionalsubscripts (e.g., Fz = −k′z), they representthe x-component of the corresponding vectorquantity. For example, the trapping force inthe x-direction is simply Ftrap = −kx.

What other forces act on the particle? Thelaser in our apparatus is directed verticallyupward—along the same axis as gravitationaland buoyant forces. Both silica spheres andbacteria are more dense than water and thusexperience a net downward force from thesesources. A constant force in the z-directionshifts the equilibrium position along the z-axisbut leaves the force constant unmodified. Forexample, the gravitational force on a mass mhanging from a mechanical spring of force con-stant k shifts the equilibrium by an amount−mg/k, but the net force F = −kz stillholds with z now the displacement from thenew equilibrium point. Thus, the Fx = −kx,Fy = −ky, Fz = −k′z “trapping force” canand will be taken as relative to the final equi-librium position and includes not only thetrue trapping force centered at the laser fo-cus, but also the laser scattering force andthe forces due to gravity and buoyancy. Keepin mind that these other forces are relativelyweak compared to the true trapping force andso the shift in the equilibrium position fromthe laser focus is rather small.

The fluid environment supplies two ad-ditional and significant forces to the parti-cle. The particles that we study with ourlaser tweezers are suspended in water wheremolecules are in constant thermal motion, i.e.,

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they are moving with a range of speeds in ran-dom directions. For still water with no bulkflow, the x-component of velocity (or the com-ponent along any axis) is equally likely to bepositive as negative and will have an expecta-tion value of zero: 〈vx〉 = 0. Its mean squaredvalue is nonzero, however, as the average ki-netic energy of the molecules is determinedby the temperature T . More precisely theequipartition theorem states that the meansquared value of any component of the veloc-ity, e.g. 〈v2

x〉, is related to the temperature Tby

1

2m⟨v2x

⟩=

1

2kBT (2)

where temperature is measured in Kelvin andkB = 1.38 × 10−23 J/K is Boltzmann’s con-stant. The value of vx for any given particle isa random variable whose probability distribu-tion is known as the Maxwell-Boltzmann dis-tribution:

P (vx)dvx =1√

2πσ2v

exp

(− v2

x

2σ2v

)dvx (3)

P (vx)dvx gives the probability that the veloc-ity component vx for a given particle lies in therange between vx and vx + dvx. The MaxwellBoltzmann distribution is a Gaussian distri-bution whose variance σ2

v = 〈v2x〉 = kBT/m

makes it satisfy the equipartition theorem.Likewise the other velocity components vy andvz obey the same distribution, (3), with thesame variance σ2

v .

Exercise 1 (a) Find the root-mean-square(rms) velocity in three dimension

√〈|v|2〉 =√⟨

v2x + v2

y + v2z

⟩=

√3kBT/m for water

molecules near room temperature (23 C).(b) Find the rms x-component of velocity,√〈v2〉 = σv and the number density of water

molecules (per unit volume). Use them to esti-mate the rate at which molecules cross through

(in either direction) a 1 µm diameter disk ori-ented with its normal along the x-direction.

Therefore, even if there is no bulk movementof the water, a small particle immersed in wa-ter is continuously subject to collisions frommoving water molecules. The collisional forceFi(t) exerted on the particle during the ithcollision delivers an impulse Ji =

∫Fi(t)dt to

the particle over the duration of the collision.By the impulse-momentum theorem, this im-pulse changes the particle momentum by thesame amount ∆pi = Ji; impulse is momentumchange and they can be used somewhat inter-changeably. For a one-micron particle in waterat room temperature, such collisions occur ata rate ∼ 1019 per second. Over some shorttime interval ∆t, the total impulse ∆p deliv-ered to the particle is the sum of the individualimpulses: ∆p =

∑i Ji, and the average colli-

sional force exerted on the particle over thisinterval is then Fc(t) = ∆p/∆t.

Theory cannot predict the individual im-pulses. A head-on collision with a high ve-locity water molecule delivers a large impulse,while a glancing collision with a low velocitymolecule delivers a smaller impulse. Depend-ing on the direction of the collision, Ji canpoint in any direction. Even when summedover an interval ∆t, ∆p will include a randomcomponent.

When no other forces act on the particle,the impulses push the particle slowly throughthe fluid along a random, irregular trajec-tory. This random motion is known as Brow-nian motion and is readily observed under amicroscope when any small (micron-sized orsmaller) particle is suspended in a fluid. Whenthe particle is trapped in an optical tweezers,the impulses act as a continuous perturbationthat pushes the particle in random directions.Because of the random component of the force,the particle motion is said to be stochastic(governed by probability distributions), and

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only probabilities or average behavior can bepredicted.

Exercise 2 You can estimate the averagespeed of Brownian motion from the fact thatthe speed of the microscopic particle at tem-perature T must also satisfy the equipartitiontheorem (2). For a silica sphere of diameter1 µm and a density of 2.65 g/cm3, what is itsrms velocity at room temperature? Is your re-sult still valid if the particle is in an opticaltrap?

Note that if a particle moves through thefluid at a velocity v, collisions are not equallylikely in all directions. More collisions will oc-cur on the side of the particle heading intothe fluid than on the trailing side and the to-tal impulse ∆p that is acquired by the particlewill acquire a non-zero mean. The direction ofthis impulse must tend to oppose the motionof the particle through the fluid. Macroscop-ically, we describe this effect by saying thatthe particle experiences a viscous drag forceFdrag that is proportional to (and in oppositedirection from) its velocity

Fdrag = −γv (4)

where γ is the drag coefficient. As they havethe same microscopic origin, there must bea connection between the magnitude of thesmall impulses ∆p and the strength of themacroscopic drag force. We can find this con-nection by noting that while the microscopiccollisions deliver momentum to the particleand drive its Brownian motion, the overalldrag force tends to slow the particle down. Onaverage these two effects must balance eachother exactly, so that the particle neither slowsto a halt nor accelerates indefinitely. Ratherthe particle maintains an average kinetic en-ergy in accord with the equipartition theorem(Eq. 2). In the following we investigate this

force balance in order to relate the magnitudeof the microscopic impulses to the drag coeffi-cient γ.

Therefore suppose that a micron-sized par-ticle is moving through a fluid. For claritywe consider only one component (say x) of itsmotion, although exactly the same argumentswill apply to its motion in y and z. Let ∆prepresent the x-component of the net vectorimpulse ∆p that is delivered to the particleduring an interval ∆t. Likewise v and p repre-sent the x-component of the velocity and mo-mentum, and Ji is the x-component of impulsefrom a single collision. Because of the high col-lision rate, ∆t can be assumed short enoughthat the particle velocity over this interval iseffectively constant, but still long enough toallow, say, a few thousand collisions or more—enough to apply the central limit theorem,which says that the sum ∆p =

∑i ∆pi will be

a random variable with a Gaussian probabilitydistribution no matter what probability dis-tribution governs Ji. Moreover, the Gaussiandistribution will have a mean µp = 〈∆p〉 equalto the number of collisions times the mean ofthe contributing Ji and it will have a varianceσ2p = 〈(∆p− µp)2〉 equal to the number of col-

lisions times the variance of the Ji. Becausethe number of collisions is proportional to ∆t,both the mean µp and the variance σ2

p shouldbe proportional to ∆t.

The total impulse ∆p is therefore a randomvariable that can be expressed as

∆p = µp + δp (5)

where µp is a constant (associated with themean of the impulse distribution) and δp is azero-mean Gaussian random variable: 〈δp〉 =0 with a non-zero variance 〈δp2〉 (associatedwith the variance of the impulse distribution).

In a still fluid with no bulk flow, a parti-cle at rest (v = 0) experiences collisions fromall directions equally. A collision delivering an

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impulse ∆pi, is exactly as likely as a collisiondelivering −∆pi. Consequently, ∆p is equallylikely to be positive as negative and its expec-tation value is zero: µp = 0. However if theparticle is moving through the fluid at veloc-ity v, the impulses tend to oppose the motion(as discussed above) and we expect the aver-age impulse µp will be proportional to v andopposite in sign. This tells us that the averagecollisional force µp/∆t is the x-component ofthe viscous drag force. Then from Eq. 4 wehave

µp∆t

= −γv (6)

How does a Brownian particle slow down orspeed up due to µp and δp? How does this pro-duce an average kinetic energy in agreementwith the equipartition theorem? The parti-cle’s kinetic energy changes because the finalmomentum pf = p + ∆p = p + µp + δp dif-fers from the initial momentum p = mv. Theenergy change is given by

∆E =p2f

2m− p2

i

2m

=1

2m

((p+ µp + δp)2 − p2

)=

1

2m

(µ2p + δp2 + 2pδp

+ 2µpp+ 2µpδp)

(7)

This energy change can be non-zero overany interval ∆t; the particle can gain or loseenergy in the short term. However, if theparticle is to remain in thermal equilibriumover the long term, the average energy changeshould be zero. Applying the equilibrium con-dition 〈∆E〉 = 0 will allow us to relate thevariance of δp to factors associated with µp.Therefore we need to evaluate the expectationvalue of the right side of this expression, whichis simply the sum of the expectation values of

each term:

〈∆E〉 =1

2m

( ⟨µ2p

⟩+⟨δp2⟩

+ 2 〈pδp〉

+ 2 〈µpp〉+ 2 〈µpδp〉)

(8)

Note first of all that the third term on theright side is 2 〈pδp〉 = 2m 〈vδp〉. Because theparticle velocity v and the random part ofthe collisional impulse δp are statistically inde-pendent, the expectation value of their prod-uct is the product of their expectation values:〈vδp〉 = 〈v〉〈δp〉. This term is zero because δpis a zero-mean random variable. The same ap-plies to the last term in the parentheses, whichcontains the product 2 〈µpδp〉. Because δp israndom and uncorrelated with µp, this termwill also be zero.

The first term on the right side involves µ2p,

where we have already noted that µp is propor-tional to the time interval ∆t. Therefore thisis the only term in the expression that variesas ∆t2, while every other term is proportionalto ∆t. Since we can choose ∆t as small as welike, we can make this term arbitrarily small incomparison to the other terms. We can safelydiscard this term as an insignificant contribu-tion to 〈∆E〉.

Now we can use Eq. 6 relating the viscousdrag behavior and µp. Making the substitu-tion µp = −γv∆t, we have the expectationvalue of the fourth term in Eq. 8: 2 〈µpp〉 =−2γ∆t 〈vp〉 = −2mγ∆t 〈v2〉. This term andthe remaining 〈δp2〉 term then give

〈∆E〉 = −γ⟨v2⟩

∆t+〈δp2〉2m

= −γ kBTm

∆t+〈δp2〉2m

(9)

where in the last line we used the equipartitiontheorem applied to the particle’s mean squarevelocity: 〈v2〉 = kBT/m.

Setting 〈∆E〉 = 0 and solving for 〈δp2〉 thengives ⟨

δp2⟩

= 2γkBT ∆t (10)

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Note this agrees with the prior assertion that〈δp2〉 should be proportional to ∆t. Moreoverit gives the proportionality constant, 2γkBT ,that is needed to keep the velocity distribu-tion in agreement with the equipartition the-orem. Although derived for a particle movingalong the x-axis, this same expression will ap-ply to each of the three dimensions, and so wefind the desired connection between the vis-cous drag coefficient, the mean squared ran-dom impulse (along any axis), and the tem-perature (i.e., the thermal equilibrium condi-tion).

Equation 10 leads directly to a versionof the fluctuation-dissipation theorem, whichsays that the variance of the fluctuating forcemust be proportional to the dissipative dragcoefficient γ and kBT . To see this, write thetotal collisional force as

Fc(t) =∆p

∆t

=µp∆t

+δp

∆t= Fdrag(t) + F (t) (11)

where F (t) = δp/∆t and, since δp is a zero-mean Gaussian random variable with a vari-ance given by Eq. 10, F (t) will be a zero-meanGaussian random variable with a variance⟨

F 2(t)⟩

=2γkBT

∆t(12)

F (t) is called the Brownian force. In equilib-rium, and on average, the energy lost by theparticle to the fluid via the drag force Fdrag(t)is balanced by the energy gained by the parti-cle from the fluid via the fluctuating Brownianforce F (t).

Note that different values of δp over anynon-overlapping time intervals arise from a dif-ferent set of collisions and thus will be statis-tically independent. For example, even for ad-jacent time intervals, the two δp values would

be equally likely plus as minus. This inde-pendence implies F (t) is uncorrelated in timewith 〈F (t)F (t′)〉 = 0, for t 6= t′ (or, at least,for |t− t′| > ∆t). Thus, F (t) is a very oddforce that fluctuates virtually instantaneouslyon all but the shortest time scales.

The local environment may produce otherforces on a small particle. The silica particlesin our experiment can adhere to a glass cov-erslip. A vesicle in a plant cell may be pulledthrough the cell by a molecular motor, while aswimming bacterium generates its own propul-sion force by spinning its flagella. These ad-ditional forces compete with the trapping andfluid forces. If these forces are known, mea-surements of the displacements they cause canbe used to determine the strength of the trap.If the trap strength is known, measured dis-placements can be used to determine these ad-ditional forces. Subsequent sections describehow to use the physics of Brownian motionand viscous drag to determine the strength ofthe trapping and drag forces.

We will need to know the position x of theparticle with respect to the trap. In principlewe could calculate x by analyzing microscopeimages collected with a camera. In practicethis does not work well because the displace-ments are very small and fluctuate rapidly.We can obtain higher precision and faster timeresolution if we detect the particle’s displace-ment indirectly by measuring the laser lightthat the particle deflects from the beam fo-cus. Light scattered by the particle travelsdownstream (along the laser beam axis) and—in our apparatus—is measured on a quadrantphotodiode detector (QPD). The QPD is dis-cussed in the experimental section. Here wemerely note that as the particle moves withinthe trap in either the +x or −x-direction, itdeflects some of the laser light in the same di-rection and the QPD reports this deflection bygenerating a positive or negative voltage V .

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For small displacements x of the particlefrom the beam focus, the QPD voltage is linearin the displacement (V ∝ x). Consequently,we can write

V = βx (13)

We will refer to β (units of volts/meter) asthe detector constant. Because the voltagegenerated by the QPD depends on the totalamount of scattered light, β depends on thelaser power as well as the shape and size ofthe particle and other optical properties of theparticle and liquid.

Analysis of Trapped Motion

How can we measure the strength of the trap?Suppose that a particle, suspended in water,is held in the optical trap. If we move themicroscope stage (that holds the sample slide)in the x direction at a velocity xdrive, the water(sealed in the slide) will move at that samevelocity. The water moves with the slide anddoes not slosh around because it is confined ina thin channel and experiences strong viscousforces with the channel walls. On the otherhand, the trap (whose position is determinedby the beam optics) will remain fixed so thatthe fluid and the trapped particle will then bein relative motion. The drag force is oppositethe relative velocity and thus given by −γ(x−xdrive). Like the Brownian force, the viscousforce is well-characterized and together theywill serve as calibration forces for the trap asdescribed next.

Together with the viscous force above, thetrapping force −kx, and the Brownian forceF (t), Newton’s 2nd law then takes the form

mx(t) = F (t)− kx− γ(x− xdrive) (14)

where m is the particle mass and x is its dis-placement with respect to the equilibrium po-sition of the trap.

Macroscopically the drag coefficient γ is re-lated to the viscosity of the fluid and the sizeand shape of the moving particle. For a sphereof radius a, γ is given by the Einstein-Stokesformula

γ = 6πηa (15)

where η is the dynamic viscosity of the fluid.While this equation is accurate for a spheri-cal particle in an idealized fluid flow environ-ment, the damping force is influenced by prox-imity to surfaces (the microscope slide) and issensitive to temperature and fluid compositionthrough the viscosity η. Thus it is appropriateto determine γ experimentally and compare itwith the Stokes Einstein prediction. A com-plete calibration includes a determination ofthe trap stiffness k, the detector constant β,and the drag coefficient γ.

We use the calibration method designed byTolic-Norrelykke, et al. The basic idea is todrive the stage back and forth sinusoidallywith a known amplitude and frequency andmeasure (via the QPD detector voltage V ) theparticle’s response to the three forces. Becausethe physics of heavily damped motion of a par-ticle in a fluid are well understood, the fre-quency characteristics of V (t) will reveal theparameters k, β, γ with good precision.

You are probably familiar with under-damped oscillators, for which the drag term−γx in Newton’s law is small in comparisonto the acceleration (“inertial”) term mx. Forsuch oscillators the acceleration is largely de-termined by the other (nonviscous) forces act-ing on the particle. However, the drag coeffi-cient γ in a fluid generally scales as the ra-dius a of the particle, whereas the mass mscales with the particle’s volume, m ∝ a3.Consequently, for sufficiently small particles(a ∼ µm), the inertial term is far smaller thanthe drag term, |mx| � |γx|. Under such con-ditions, the oscillator is strongly over-dampedand (to an excellent approximation) we may

September 25, 2015


drop the inertial term from Eq. 14. The parti-cle velocity is then determined by the balancebetween the viscous force and the other forcesacting on the particle. Physically this meansthat, if any force is applied to the particle,the particle “instantly” (see Exercise 3 below)accelerates to its terminal velocity in the di-rection of the applied force. When we dropthe mx term, the equation of motion becomesquite a bit easier to work with:

F (t) = kx+ γ(x− xdrive) (16)

Exercise 3 Suppose that the drag force −γxis the only force acting on the particle sothat the equation of motion becomes mx =−γx. Solve this equation for x(t) for a par-ticle with an initial velocity v0. Show thatthe velocity decays exponentially to zero andgive an expression for the time constant in-volved. (This would also be the time constantfor reaching terminal velocity when there areadditional forces acting on the particle.) Whatis the time constant for a 1µm diameter silicasphere moving through water (η ' 10−3 N-s/m2)? Integrate your solution for x(t) (as-suming x0 = 0) to determine x(t). If thesphere has an initial velocity v0 = 1 cm/s, ap-proximately how far does it travel before com-ing to rest? Give your answer in microns(µm).

Dropping the mx(t) term in Eq. 14 is equiv-alent to assuming that the time constant forreaching terminal velocity is negligible. Tosolve the resulting Eq. 16, first collect the xand x terms on the right side and multiplythroughout by ekt/γ

(F (t) + γxdrive) ekt/γ = (17)

γ((k/γ)xekt/γ + xekt/γ

)

Recognizing the right hand side as a deriva-tive, we find

x(t) = xT (t) + xresp(t) (18)

where

xT (t) =1

γ

∫ t

−∞F (t′)e2πfc(t′−t) dt′ (19)

xresp(t) =

∫ t

−∞xdrive (t′) e2πfc(t′−t) dt′ (20)

and

fc =k

2πγ(21)

has units of frequency (oscillations per unittime).

Equations 18-21 show that the motion x(t)has two components due to two sources. xT (t)is the response to the random Brownian forceF (t) and xresp(t) is the response to the motionof the surrounding fluid. They are integralsof the past values of the source terms withan exponentially decreasing weighting factorhaving a damping time, 1/2πfc, determinedby the ratio of the damping constant to thespring constant. This time constant is typi-cally in the millisecond range and thus onlyrecent past values contribute.

Applying a constant velocity flow (via a flowcell) so that xdrive = v0 creates a constant dragforce γv0 and causes a shift in the particle po-sition xresp = γv0/k. This is one common wayto get information about the trap parametersγ and k. Our apparatus uses an oscillatoryflow xdrive and looks for the predictable oscil-latory response in x(t) to provide the sameinformation.

Thus, the microscope stage (i.e., the fluid)will be driven back and forth sinusoidally witha known amplitude A and frequency fd. Thelocation of the stage xdrive (with respect to thetrap) is then given by

xdrive(t) = A cos(2πfdt) (22)

September 25, 2015


and the fluid has a velocity

xdrive(t) = −A2πfd sin(2πfdt). (23)

Exercise 4 Derive Eqs. 18-20 above. Evalu-ate the integral for xresp(t) given a constant ve-locity flow xdrive = v0 and show that it producesthe expected shift: xresp(t) = γv0/k. Also eval-uate the integral given the drive velocity ofEq. 23 and show that xresp(t) will be a sinu-soidal oscillation at the same frequency withan amplitude given by

A′ =A√

1 + f 2c /f

2d

(24)

Because F (t) is random, xT (t) is random—non-periodic and noisy. To characterize suchsignals, a statistical approach is typically usedin which the frequency components of x(t) areanalyzed. For that we need to return to Eq. 16and investigate the Fourier transform of themotion.

Consider the Fourier transforms of a trajec-tory x(t)

x(f) =

∫ ∞−∞

x(t) e−2πift dt (25)

and of the Brownian force F (t)

F (f) =

∫ ∞−∞

F (t) e−2πift dt (26)

The Fourier transform is evaluated for fre-quencies f covering both halves of the realaxis −∞ < f <∞ so that the inverse Fouriertransform properly returns the original func-tion. For example, x(t) is recovered from theinverse Fourier transform of x(f):

x(t) =

∫ ∞−∞

x(f) e2πift df (27)

Note that x has units of m/Hz and F has unitsof N/Hz.

A relationship between x and F is readilyobtained by taking the Fourier transform ofthe equation of motion, Eq. 16. That is, mul-tiply both sides by exp (−2πift) and integrateover dt. The result is

F = kx+ γ2πifx (28)

+2πγfdA

2i(δ(f + fd)− δ(f − fd))

To get Eq. 28, the Fourier transform of x(t)has been replaced by 2πif times the Fouriertransform of x(t)—as can be demonstrated byevaluating x(t) starting from Eq. 27. The ex-plicit form of xdrive as given by Eq. 23 has beenused and the Fourier transform of sin(2πfdt),which is given by (δ(f − fd) − δ(f + fd))/2i,has been applied. Solving for x then gives

x(f) =F

2πγ (fc + if)(29)

− fdA

2i (fc + if)(δ(f + fd)− δ(f − fd))

where we have replaced k by 2πγfc (Eq. 21).Equation 29 is a perfectly good description

of the particle response x—it just happens tobe Fourier transformed. We will use it to ex-tract information from measurements of x(t).

Discrete Fourier transforms

Although we have treated time t as a continu-ous variable that spans the range −∞→ +∞,in actual experiments we collect a finite num-ber of data values over a finite time intervalτ . A typical data set is a discrete samplingof the QPD voltage V (t) = βx(t) over a timeinterval τ ' 1− 2 sec, with measurements ac-quired at a uniform digitizing rate R around100,000 samples per second, i.e., with a timespacing between data points ∆t = 1/R. For

September 25, 2015


this discussion, we can consider β as given, sothat the data consists of values of x(tm) at aset of uniformly-spaced sampling times tm.

Let’s assume that measurements of x(t) aremade during the time interval from −τ/2 <t < τ/2. The integration in Eq. 25 needs tobe truncated so that t falls within this inter-val only. Of course, we expect to recover thepredicted results in the limit as τ →∞.

To analyze finite, discrete data sets, weneed to define the discrete Fourier transform(DFT). The DFT of x(t) is the version of theFourier transform that is comparable to Eq. 25but applies to a large (but finite) number L ofdiscretely sampled x(tm) values. If the mea-surement times tm are spaced ∆t = τ/L apartin time and the integration is over the range−τ/2 ≤ t ≤ τ/2, then we can write tm = m∆twith −L/2 ≤ m ≤ L/2. The finite integrationcorresponding to Eq. 25 is performed accord-ing to the rectangle rule and becomes

x(fj) =

L/2∑m=−L/2

x(tm)e−2πifjm∆t∆t (30)

The DFT is expected to accurately reproducethe true Fourier transform with some well un-derstood limitations discussed shortly.

The DFT is evaluated at fixed frequenciesfj = j∆f where

∆f =1

τ(31)

and −L/2 ≤ j ≤ L/2. That is, both x(t)and its DFT x(fj) contain the same numberof points, but each of the x(fj) has both areal and an imaginary part. However, the twoparts are not independent. If the x(t) are real(as is the case here), it is easy to demonstrate(from Eq. 26) that x(−f) = x∗(f). That is, foropposite frequencies, f and −f , the real partsare equal and the imaginary parts are nega-tives of one another. Thus x(tm) and x(fj)

both contain the same number of independentquantities. The two sets are just different waysof representing the same data.

The power spectrum

Another issue arises because the theory ofBrownian motion does not specify F (f).

At any frequency, F (f) is complex (sincee−2πift = cos 2πft− i sin 2πft). For any com-plex number z = x + iy = reiθ, x and y arethe real and imaginary parts of z, r is themodulus and θ = arctan(y/x) is the argu-ment or phase of z. The theory only predictsthe intensity given by the modulus squared:r2 = x2 + y2 = zz∗, where z∗ = x − iy is thecomplex conjugate of z. It does not predict thereal or imaginary parts of z individually or thephase. Moreover, the theory predicts that theFourier intensities F F ∗ obtained from a finiteFourier transform will be proportional to theintegration interval τ . The theory thus givesa result that is independent of τ only if theintensities are divided by τ . The traditionalcharacterization of the strength of a real, fluc-tuating function of time, such as the Brownianforce F (t) is its (two-sided) power spectrum orpower spectral density (PSD), defined as

PF (f) =F (f)F ∗(f)

τ(32)

defined for both positive and negative frequen-cies. As with x(t), F (t) is real and therefore

F (−f) = F ∗(f). This implies that Pf (−f) =P (f) and for this reason the power spectrumat f and −f is often added together to cre-ate the one-sided power spectrum. The powerspectrum at f = 0 is left unmodified. Itarises from any nonzero (DC offset) in thecorresponding quantity. For a Brownian forcePF (0) is expected to be zero as there is no longterm average force in any direction.

September 25, 2015


For f 6= 0, the power spectrum of theBrownian force is actually expected to be aconstant—independent of f . That PF (f) isflat and extends out to high frequencies is aresult of the collisional origin of the Brownianforce as described previously. Furthermore, inorder that the average speed of the particleobeys the equipartition theorem (Eq. 2), theone-sided PSD must depend directly on boththe temperature T and the viscous drag coef-ficient γ:

PF (f) = 4γkBT (33)

Equation 33 is another way of expressingthe fluctuation-dissipation theorem of Eq. 10.Here, it gives the relationship between γ andthe PSD for the fluctuating Brownian force.

For any frequency component f of a giventrajectory x(t), x(f) is also a complex randomvariable with a mean of zero. The square of itsFourier transform, x x∗, will have a non-zeromean and, as with F F ∗, is also proportionalto the integration time τ . Thus the powerspectral density for x is

P (f) =x(f)x∗(f)

τ(34)

and is also independent of τ .Again, because x(−f) = x∗(f), P (−f) =

P (f) and, as with the power spectrum PF (f),we add the components at f and−f (and leavethe component at f = 0 as is) to create theone-sided power spectrum defined for positivef only. This one-sided power spectrum, whichwe still call P (f), is then fit to the predictionsfor f > 0 given next. (We don’t fit at f = 0,as this component arises from any DC com-ponent in x(t) and is typically an artifact ofimperfectly positioning of the QPD.)

To derive the predicted relationship be-tween the one-sided power spectra for x(t) andF (t), consider the case where the stage oscilla-tions are turned off; A = 0 and the delta func-tions in Eq. 29 are gone. With only the Brow-

nian force contributing, multiply each side ofEq. 29 by its complex conjugate, divide by τ ,and add negative and positive frequency com-ponents to get

P (f) =PF (f)

4π2γ2(f 2c + f 2)

=kBT

π2γ(f 2c + f 2)

(35)

where Eq. 33 was used to eliminate PF (f).(From here on, all power spectra are the one-sided variety.)

Notice that PF (f) has units of N2/Hz andP (f) has units of m2/Hz. It makes sense toconsider these functions as a squared ampli-tude per unit frequency. For example, if weintegrate P (f) over a sufficiently small interval∆f centered around a frequency f0, we obtainP (f0)∆f . Using the one-sided PSD meansthis value would represent the mean squaredamplitude A2/2 of the oscillatory componentof x(t) at the frequency f0.

If the stage oscillations are turned back on,how do they affect the power spectrum? Wecan refer to Eq. 29 and see how the two deltafunction terms (resulting from the stage mo-tion of amplitude A at the drive frequency fd)contribute. The inverse Fourier transform ofthe delta function term in Eq. 29 shows thatit represents oscillations at the drive frequencyfd with an amplitude A′

A′ =A√

1 + f 2c /f

2d

(36)

(This result was derived in Exercise 4 fromthe response integral of Eq. 20. Here we seeit can be obtained using Fourier transformsas well.) The case fc � fd corresponds to aweak trap or high drive frequency and givesA′ = A; the amplitude of the particle oscilla-tion equals the amplitude of the stage oscil-lation. For stronger traps or lower drive fre-quencies, Eq. 36 shows how the trap attenu-ates the oscillation of the particle relative to

September 25, 2015


that of the stage; A′ is smaller than A by thefactor of

√1 + f 2

c /f2d .

Therefore the power spectrum of the parti-cle in the trap is the sum of two terms.

P (f) =kBT

π2γ (f 2c + f 2)

(37)

+A2

2 (1 + f 2c /f

2d )δ (f − fd)

= PT (f) + Presp(f) (38)

where PT (f) is the first term—the power spec-trum without stage oscillations (Eq. 35) andPresp(f) is the second term—the δ-functionterm. These two terms have the noteworthybehaviors discussed next.Presp is such that its integral over any fre-

quency interval that includes fd gives themean squared amplitude A′ 2/2 of the parti-cle’s sinusoidal response to the applied stageoscillations.

With the trap off (k = 0, fc = 0) and soPT (f) = kBT/π

2γf 2, i.e., it falls off as 1/f 2.With the trap on (fc 6= 0), fc plays the roleof a “cutoff frequency.” At high frequenciesf � fc, fc can be neglected compared to f andonce again, PT (f) = kBT/π

2γf 2—the sameas for the trap off; high-frequency oscillationsare unaffected by the trap. At low frequenciesf � fc, f can be neglected compared to fc andPT (f) = kBT/π

2γf 2c . The power spectrum

goes flat (becomes independent of f) and doesnot continue increasing as f decreases. More-over, this low-frequency amplitude decreasesas 1/f 2

c , i.e., the amplitude of the motion atlow frequencies decreases as the trap strengthincreases. Finally, PT (f) increases with tem-perature and decreases with γ; fluctuations inthe position of the particle are larger at highertemperatures and are suppressed by the vis-cous drag.

Equation 37 is for the particle’s position x,while we will actually measure the QPD volt-age V (t) = βx(t). Our experimentally deter-

mined power spectrum density will be that ofthe voltage V V ∗/τ , not the position xx∗/τ .As the Fourier transform is linear, the Fouriertransform of V (t) is related to that of x(t) bythe calibration factor β:

V = βx (39)

Accordingly, if we experimentally measureV (t) and then calculate PV (f), the PSD ofthe voltage data, then we expect

PV (f) = β2P (f) (40)

Keep in mind that the main prediction,Eq. 37, for P (f) was derived from continuousFourier transforms assuming an infinite mea-surement time, whereas our data are collectedin a discrete sampling over a finite interval τ .Because the discrete power spectrum PV (fj)is derived from a finite set of V (tm) collectedover a time interval τ spaced ∆t apart, it isnot expected to perfectly reproduce that pre-diction. However, the differences due to thefinite acquisition time and sampling rate arewell understood and predictable.

One aberration is aliasing. The highest fre-quency represented in PV (fj) is at j = L/2or fj = ∆fL/2, which is just half the sam-pling rate and called the Nyquist frequencyfNy. If the true power spectrum is zero forall frequencies above fNy, then PV (fj) shouldagree well with the true PV (f) at all fj. How-ever, if the true PV (f) has components abovefNy, these components show up as artifacts inPV (fj). Components in the true PV (f) at fre-quencies near f ′ = fNy + δf show up in thediscrete version PV (fj) at frequencies fj nearfNy − δf ; the true components are reflectedabout the Nyquist frequency. For example,for a 200 kHz sampling rate, the Nyquist fre-quency is 100 kHz and oscillations in V (t) at104 kHz, show up in PV (fj) near fj = 96 kHz.The effects of aliasing will be apparent in yourdata and can be dealt with easily.

September 25, 2015


The randomness of the trajectory over thefinite time interval leads to power spectra thathave random variations from the predictions—the PSDs will be noisy. The noise woulddecrease as we work toward the limit τ →∞. However, it is not practical to take everlonger measurements, with correspondinglylarger data sets. Data sets larger than a fewhundred thousand data points are tedious tomanage and analyze; the improvement in theresult does not justify the extra effort of han-dling and processing such large data arrays. Afar better way to approach the limit of τ →∞experimentally is just to collect a number ofdata sets of duration τ ∼ 1 sec and then aver-age the P (f) obtained from each set.

After each τ -sized V (t) is measured, its dis-

crete Fourier transform V (fj) is calculated andthen used to determine its power spectrumdensity PV (fj). After sufficient averaging ofsuch PV (fj), the predicted behavior will beginto appear—a continuous part from PT (f) anda sharp peak at fd due to Presp(f). This av-eraged PSD is fit to the prediction of Eq. 37(with Eq. 40) to determine the parameters ofthe optical trap: the trap constant β, the dragcoefficient γ and the force constant k = 2πγfc.

Exercise 5 In the optical trapping literature,typical reported values for the cutoff frequencyare in the range fc ' 102 − 103 Hz. Assumingthat these correspond to 1µm diameter spheri-cal particles in water at room temperature (295K), estimate the magnitude of the trap stiff-ness constant k. For fc = 100 Hz, what dis-placement would result if the full weight of a1µm diameter silica sphere hung from a springwith this force constant?

The equipartition theorem also applies to theaverage potential energy of a harmonic oscil-lator: ⟨

1

2kx2

⟩=

1

2kBT (41)

Use this relation to find the rms deviationof the particle from its equilibrium position:√〈x2〉. Compare this rms displacement and

the size of the shift in the equilibrium positiondue to gravity/buoyancy, with the particle di-ameter.

Exercise 6 Make two sketches of the PT (f)term in Eq. 37 for a particle in a trap with fc= 100 Hz. The first sketch should use linearscales (PT vs. f), while the second should usea log-log scale (logPT vs. log f) for 10−2fc ≤f ≤ 102fc.

Comparing the predicted PV (f) with oneactually determined from the measured QPDvoltage vs. time data is done in two steps: onefor the thermal component PT (f) and one forthe delta function response Presp(f). We willbegin with a discussion of the latter.

The main theoretical feature of a delta func-tion is that its integral over any region con-taining the delta function is one. Thus, thepredicted integral W of the PV (f) of Eq. 40associated with the delta function in Eq. 37 iseasily seen to be

W = β2

∫ ∞0

Presp(f) df

=β2A2

2 (1 + f 2c /f

2d )

(42)

In the experiment, the drive frequency fdwill be chosen so that there will be an exactinteger number of complete drive oscillationsover the measurement interval τ . This makesfd one of the frequencies at which the PV (fj) isevaluated and should produce one high pointin this PSD. You will determine the height ofthat point above the thermal background andmultiply by the spacing ∆f between points toget the experimental equivalent of integratingPV (f) over the delta function. In rare cases,you may see the experimental delta function

September 25, 2015


spread over several fj centered around fd. Inthese cases, the experimental integral is thesum of the amount these points exceed thethermal background times ∆f .

The experimental value of W obtained thisway is then used with Eq. 42 and the knownstage oscillation amplitude A, the drive fre-quency fd, and the value of fc (determined inthe next step) to determine the trap constantβ.

The force constant k and the drag coeffi-cient γ are found by fitting the non-δ-functionportion of the experimental PV (f) (f 6= fd) tothe prediction of Eq. 37 (with Eq. 40). That is,for all values of f except f = fd, the predictedPSD can be written

PV (f) =β2kBT

π2γ (f 2c + f 2)

(43)

For fitting purposes, this equation is more ap-propriately expressed

PV (f) =B

1 + f 2/f 2c

(44)

where B is predicted to be

B =β2kBT

π2γf 2c

(45)

The experimental PV (fj) is then fit to Eq. 44over a range of f (not including the point atf = fd) which then determines the fitting pa-rameters B and fc.

With fc determined directly from this fit,the experimental W is used with Eq. 42 todetermine β. Then, if we assume T is equalto the measured room temperature, the fittedB can be used in Eq. 45 with fc and β todetermine the value of γ. Finally, the forceconstant k = 2πγfc (Eq. 21) is determinedand the three trap parameters γ, β and k arethen known.

Apparatus

Overview

Our optical trap is based on the design of Ap-pleyard et al. The design uses an inverted mi-croscope to focus an infrared diode laser beamonto the sample and detects the deflection ofthat beam with a quadrant photodiode detec-tor (QPD). The design also illuminates thesample with white light and generates an im-age of the sample on a video camera. Thedetails are somewhat complex, as the sameoptical elements perform several functions si-multaneously. The layout is described below.Refer to Fig. 2 while considering the followingtwo optical paths:

The optical path for the infrared laser: Thediode laser is a semiconductor device thatoutputs its (λ = 975 nm) infrared beamto a single-mode optical fiber. A converg-ing lens (#1) receives the diverging lightexiting the fiber and collimates it to abeam with a diameter of ∼ 10 mm, or suf-ficient to fill the back aperture of the trap-ping objective (#3). A pair of mirrorsand the dichroic mirror (#2, infrared-reflecting) are used to steer the laser beamvertically upward, along the central axisof the objectives. The beam enters theback aperture of the lower microscope ob-jective (#3) (100× Nikon 1.25 NA, oil-immersion), which brings the beam to afocus at the sample, forming the opti-cal trap. The upper microscope objec-tive (#4) captures and re-collimates theinfrared light that has passed through thesample and directs this energy upward. Adichroic (infrared-reflecting) mirror (#5)then deflects the beam toward a converg-ing lens (#6), which focuses the beamonto the quadrant photodiode detector(#7, QPD).

September 25, 2015


(3) Objective

(100X)

A B

(4) Objective

(10X)

(5) Dichroic

mirror

(6) Lens

(7) QPD

(8) LED

stage

Camera

Trap

x

yz

(100X)

(2) Dichroic

mirror

Infrared laser

diode (1) Lens

C

Image: Ibidi Co.

Microchannel

optical fiber

D

100 µm

oil

100X Objective

(9) Lens

Figure 2: A and B show two views of the opti-cal system, illustrating the paths of the infraredtrapping rays (red arrows in A) and the visibleillumination rays (blue arrows in B) as they passthrough the same optical elements. The sampleis contained in a 100 µm-deep microchannel slideC, at the focus D of the 100× oil-immersion lens.

The optical path for visible light: An LED(#8) generates white light that passesthrough the dichroic mirror (#6) and isfocused by the upper objective (#4) ontothe sample. Transmitted light from thesample area near the trap is gathered bythe lower objective (#3) and with lens(#9) is brought to an image at the cam-era.

In this design the infrared laser serves tworoles. It traps the particle at the focus, and itis also used to detect the motion of the parti-cle within the trap. If there is no particle inthe trap, the infrared laser beam propagatesalong the optical axis of the instrument, i.e.,along the common cylindrical axis of the mi-

croscope objectives). The recollimated beamexiting the upper objective travels parallel tothe optical axis, and converging lens #6 bringsthis beam to a focus just a bit in front of thecenter of the QPD. However, if a small par-ticle is near the laser focus, the beam is re-fracted away from the optical axis. The colli-mated beam leaving the upper objective willthen propagate at an angle to the optical axis,and so it is focused by converging lens #6 toa spot that is displaced from the center of theQPD. The QPD reports this displacement asa voltage V , which is proportional to the par-ticle’s displacement x from the laser focus (seeEq. 39). The QPD actually detects deflectionsin the both the x and y directions, reportingtwo independent voltages Vx and Vy that youwill measure.

Hardware

Data acquisition board

The computer communicates with the tweez-ers apparatus via a USB connection orthrough a (National Instruments PCI-MIO-16E-4) multifunction data acquisition board(DAQ) located inside the computer. See Fig-ure 3.

The DAQ board supplies voltages thatmove the positioning stage in the xy planeand it reads voltages from the quadrantphotodiode—the raw data for analyzing par-ticle motion in the trap. Two components ofthe DAQ board are used to do these tasks—an analog to digital converter (ADC) and twodigital to analog converters (DACs).

The ADC and both DACs are 12-bit ver-sions, meaning they have a resolution of 1 partin 212 = 4096 of their full scale range. Forexample, on a ±10 V range setting, voltagesare read or written to the nearest 4.88 mV.An amplifier in the DAQ allows for full scaleranges on the ADC from ±10 V to ±50 mV.

September 25, 2015


Y

X

BNC2090

DAQ

PC

Z

quadrant photodetector

(QPD)

X diff

0-10 V VDAC

Y diffsum

piezocontrol

Signal cable(not used)

Figure 3: Schematic of electronic interface be-tween computer and tweezer apparatus. TheDAQ board in the PC has an analog-to-digitalconverter that reads data from the QPD, as wellas digital-to-analog converters that supply controlvoltages for the xy positioning of the microscopestage.

The DAC range is ±10 V. The ADC can readanalog voltages at speeds up to 500,000 read-ings per second, and the DACs can write out-put voltages at similar speeds. The ADC hasa high speed switch called a multiplexer thatallows it to read voltages on up to eight differ-ent inputs.

A cable connects the DAQ card in the PC toan interface box (National Instruments BNC-2090) that has convenient BNC jacks for con-necting coaxial cables between the various ap-paratus components and the DAQ input andoutput voltages.

Laser

The laser diode package (Thorlabs,PL980P330J) is premounted to a single-mode fiber which brings the laser light tothe apparatus. The package is mounted ona temperature stabilized mount (Thorlabs,LM14S2) kept at constant temperature bythe (Thorlabs, TED200C) temperature con-troller. An interlock requires the temperaturecontroller to be on before the laser currentcontroller will operate. The laser current is

adjusted and stabilized by a current controller(Thorlabs, LDC210C). The laser current canbe read off the controller. The laser turnson at a threshold current around 70 mA andthen the laser power increases approximatelylinearly with current over threshold.

Internal to the laser diode package, a small,constant fraction of the laser beam is made tofall on a photodiode which generates a currentproportional to the laser beam power. Thiscurrent is measured in the laser current con-troller and can be read if you set the frontpanel meter to display IPD. The supply al-lows you to scale IPD with any proportion-ality constant for display as PLD. By inde-pendently measuring the actual laser powerP out of the 100× objective as the laser cur-rent is varied, the proportionality between Pand IPD was confirmed and the proportion-ality constant has been adjusted so that PLDgives the laser power P out of the objective.Of course, P will not be PLD if the beam pathis blocked or if the alignment of the laser ischanged. The instructor should be involved ifa new calibration is deemed necessary.

Controller hub

There are six Thorlabs “T-Cube” electronicmodules mounted in the (Thorlabs, TCH-002)T-Cube controller hub. The modules, de-scribed below, are used to electronically con-trol the position of the microscope slide andto control and read the quadrant photodiodedetector. The hub supplies a signal path be-tween different modules and between all sixmodules and the computer’s USB bus.

Quadrant Photodiode Detector

A quadrant photodiode detector (Thorlabs,PDQ80A) is used to produce voltages that arelinearly related to the position of a particle inthe neighborhood of the laser focus. It has

September 25, 2015


V1V2

V3 V4

y

x

Figure 4: The QPD measures the intensity onfour separate quadrant photodiodes.

four photodiode plates arranged as in Fig. 4around the origin of the xy-plane. The platesare separated from one another by a fractionof a millimeter and extend out about 4 mmfrom the origin. The QPD receives the in-frared light from the laser and outputs a cur-rent from each quadrant proportional to thepower on that quadrant.

The Thorlabs TQD-001 module powers theQPD and processes the currents. It does notoutput the currents directly. Instead, it con-verts them to proportional voltages V1-V4 byadditional electronics to produce the followingthree output voltages. The x-diff voltage is thedifference voltage Vx = V1 +V4− (V2 +V3), they-diff voltage is Vy = V1 + V2 − (V3 + V4) andthe sum voltage is the sum of all four. Vx isthus proportional to the excess power on thetwo quadrants where x is positive compared tothe two quadrants where x is negative. Sim-ilarly for the y-diff voltage. The sum voltageis proportional to the total laser power on allfour photodiodes.

With no scattering, the light that is broughtto a focus by the 100× objective diverges from

there and is refocused by the 10× objectiveand lens #6 so that it again comes to a fo-cus just a bit in front of the QPD. The raysdiverge from this focus before impinging onthe photodiodes so that by the time they getthere, the spot is a millimeter or two in diam-eter and when properly centered will hit allfour quadrants equally.

With a particle in the trap, the scatteredand unscattered light interfere and produce aninterference pattern on the QPD that dependson the location of the scatterer. For small vari-ations of the particle’s position from equilib-rium, the QPD voltages Vx and Vy producedby these patterns are proportional to the par-ticle’s x and y positions. That is Vx = βx andVy = βy.

While the range of linearity between the Vand x is quite small—on the order of a fewmicrons, it is still large compared to the typ-ical motions of a particle in the trap (see Ex-ercise 5). Significantly, this voltage respondsvery quickly to the particle’s position so thathigh frequency motion (to 100 kHz or more)is accurately represented by V (t).

The QPD module has buttons for controlof its function and it has an array of LEDsthat show whether the beam intensity pat-tern is striking the QPD roughly in the center(center LED lit) or off-center (off-center LEDslit). The QPD is mounted on a manually-controlled, relatively coarse xy stage that willbe centered by hand during calibration.

Microscope stage and piezoelectric con-trol

The microscope stage is the component thatsupports the microscope slide between the100× and 10× objectives. It is built aroundthe Thorlabs MAX311D 3-axis flexure stage,which provides three means for positioning theslide in the trapping beam.

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First, and very crudely, you can manuallyslide the stage across the table for coarse po-sitioning in the x- and y-directions. You willneed to do this to put your slide into the beam,but you will find it difficult to position thesample to better than about ±1 mm using thismethod.

Second, the stage has a set of mi-crometers that can be turned manuallyto move the stage. Over a range ofabout 300µm, the micrometers operate asdifferential screws, in which two internalthreads with slightly different pitches turnsimultaneously—producing a very fine trans-lating motion around 50µm/revolution. Asyou continue turning the micrometer spindle,the differential operation runs out and the mo-tion switches to a coarser control in which thestage moves around 500µm/revolution. Thecoarse control can be obtained directly byturning the micrometers at the knurled ringup from the spindle, which then bypasses thedifferential screw.

Third, inside the stage there are three piezo-electric stacks that allow the computer tomove the stage along each of the x, y andz axes. Piezoelectrics (“piezos”) are crystalsthat expand or contract when voltages areapplied across two electrodes, which are de-posited on opposite sides of the crystal. TheThorlabs TPZ-001 piezo controller modulessupply these control voltages (up to about75 V). The piezos provide very fine and precisecontrol of stage motion, but only over shortdistance ranges: The full 75 V range generatesonly about 20 µm of stage motion. The piezovoltage can be read from an LED indicator onthe face of the module, which also has controlbuttons and a knob for the various modes ofcontrolling this voltage.

There are several ways to use the piezo con-troller. Manual mode, in which the voltage iscontrolled via the knob on the module, will be

disabled and is not used in our setup. (The dif-ferential micrometers are far more convenientmanual controls.) Only the following two elec-tronically controlled methods will be used.

One method is to use the DACs to supplyanalog voltages in the range of 0-10 V to theThorlabs piezo-control module. The Thorlabsmodule amplifies these voltages to the 0-75 Vscale and sends them to the piezo. This isthe fastest method and is the main one usedin our apparatus. Alternatively, the computercan communicate with the control module overthe USB bus to request a desired piezo voltage.

Unfortunately, piezos have strong hystere-sis effects. Their length, i.e., how far theywill move the stage, depends not only on thepresent electrode voltage, but also on the re-cent history of this voltage. One method todeal with piezo hysteresis is to obtain feed-back data from a strain gauge mounted along-side the piezo. The stage has one strain gaugefor each of the three axes. They are read bythe Thorlabs TSG-001 strain gauge modules,which are placed next to the matching piezomodule in the controller hub.

The strain gauge is a position transducerwith an output voltage that is very linear inthe displacement caused by the correspond-ing piezo. The output voltage from the straingauge module is internally wired to its corre-sponding piezo module through the controllerhub. The displacement of the strain gaugecaused by the piezo is indicated on a scale ontop of the strain gauge module in units of per-centage of the full scale: 0-100% for motion ofabout 20 µm.

Using strain gauge feedback, the controllerallows you to supply USB commands request-ing stage positions as a percentage (0-100) ofthe full scale motion (i.e., 0-100% of 20µm).This is the second mode of motion control usedin this experiment. Electronic feedback cir-cuitry adjusts the actual voltage sent to the

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piezo to achieve that percentage on the straingauge. Our setup has only two strain gaugecontrollers, which are used only on the x andy piezos. (We do not use the z-piezo.)

Another issue with the stage is cross talkbetween the x, y and z motions of the stagedue to its flexure design. The stage is capableof roughly 4 mm of travel in each direction,but the motions can couple to one another.Right around the middle position of the stage,changing x, y or z piezo or micrometer, shouldonly move the stage in the x, y or z direc-tion. However, as you move away from thiscentral position, changing the x-piezo or mi-crometer, for example, will not only changethe x-position of the stage; the flexure designcauses small changes in the y and z-positionsas well. In addition, the motion calibrationfactors—how much stage motion will corre-spond to a given micrometer or piezo changewill change. For example, when the stage isnear the limit in one or more of the three di-rections (±2 mm), changing the x piezo, say,will move the stage in the y- and z-directionsby as much as 30% of the amount moved in thex-direction. Consequently, it is worthwhile totry to operate the stage near the middle of itsx, y and z ranges.

Camera

A Thorlabs DCU-224C color video camera isused to observe and monitor the happeningsin the trap. It is also the means for transfer-ring a length scale from a calibration slide tothe motion caused by the piezo. The camerahas a rectangular CCD sensor with pixels ar-ranged in a 1280 × 1024 cartesian grid with5.3 µm spacing. Thus distances measured onthe image in pixels will scale the same way inx and y with real distances on the slide.

Software

UC480

The camera is controlled and read using theUC480 software program. This program hasfeatures for drawing or making measurementson the images, and for storing frames or videosequences. Select the Optimal Colors option atload time, then hit the Open camera button—the upper left item on the upper toolbar. Thedefault camera settings generally work fine,but if there are image problems, many cam-era settings can be adjusted to improve imagequality.

Note that there is a bad light path in ourapparatus that throws some non-image lightonto the camera sensor. This artifact can beeliminated by partially closing the adjustableaperture directly under the camera.

Become familiar with the measurementtools and the drawing tools on the utility tool-bar arrayed along the left edge of the screen.In particular, you will use the Draw circle,Draw line, and Measure tools. Other settingsand features can be found on the upper tool-bar or the menu system. Start with the con-trast and white balance set for automatic op-timization. Learn how to set and clear an AOIor area of interest (a rectangular area on thesensor) so that only the data from that areais sent from the camera to the program. Thisincreases the frame rate compared to using thefull sensor.

Initialize program

This program sets up all the T-Cube mod-ules to run in the appropriate modes used inother programs. It sets the piezo and straingauge feedback channel, zeros the piezos’ out-puts and then zeros the strain gauges. Fi-nally, it sets the x- and y-piezos near theirmidpoint voltages of 37.5 V, and sets the oper-

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ating mode to add this 37.5 V to the voltagesgenerated by signals applied to the externalinput. In this way, the piezo is near the mid-dle of its extension, and so both positive andnegative translation in x and y can be gener-ated by supplying positive or negative DACvoltages to the back of the piezo controller.

Oscillate Piezo program

This program creates sinusoidal waveformsfrom the two DACs for driving the x-and y-inputs of the piezo controllers. It is used inconjunction with the UC480 Camera programto calibrate the amplitude of the stage motionwhen driven by a waveform of a given ampli-tude and frequency.

Raster Scan program

This program is used to scan the x- and y-piezos in a slow scan mode using strain gaugefeedback while time averaging the signals fromthe QPD. The raster scan starts with a fixedvoltage applied to the x-piezo while the y-piezo is scanned back and forth over a user-defined range. Then the x-piezo is moveda small amount in one direction and the y-piezo is scanned again. This move-x-and-scan-y process is repeated until the x-piezo has alsoscanned over the user defined range. At eachxy value, the program digitizes the Vx and Vysignals from the QPD module and displays theresults for each of these signals.

This program is used to see how the QPDworks, give a sense for the intensity pattern onthe QPD, determine an approximate detectorconstant β, and see how β depends on boththe laser intensity and objective focusing.

Tweezers program

The main measurements are made from thisprogram. It has two tabbed pages along the

right. One is labeled Acquire and is for set-ting the data acquisition parameters, measur-ing the Vx and Vy signals from the QPD, andcomputing and averaging the PSD. The othertab is labeled Fit and is for fitting the PSD tothe predictions of Eq. 37.

The default parameters for data acquisitionshould work fine. The number of points ineach scan of Vx and Vy vs. t is forced to bea power of 2 (218 = 262144 is the default) sothat fast Fourier transforms can be used. Thesampling rate (number of readings per second)for the ADC is determined by dividing down a20 MHz clock on the DAQ board. The divisoris the number of 20 MHz clock pulses betweeneach digitization. The maximum speed of theADC is around 250 kHz when reading twochannels (Vx and Vy). The 105 default valuefor this divisor leads to a sampling rate around190 kHz. With 218 samples in each scan, eachscan lasts 218 ·105/(20×106 Hz) = 1.38 s. Theinverse of this time (0.73 Hz) is the frequencyspacing between points in the PSD.

The ADC has an instrument amplifier thatallows bipolar full scale (F.S.) voltages from±50 mV up to ±10 V. The F.S. range controlshould be set as small as possible without let-ting the Vx or Vy signals hit the range limits.

The two DACs used to drive the stage piezossend discretized sinusoidal waveforms with ad-justable amplitudes and with an adjustablephase between them. You can set the ampli-tude Ax or Ay to zero to get one-dimensionalback and forth stage motion. However, it isrecommended that the amplitudes be set equalwith a 90◦ phase difference so that the stagewill move with nearly circular motion. Thisway no matter what direction the QPD’s xand y responses are aligned to, the stage mo-tion will be sinusoidal with the chosen ampli-tude in those directions.

Recall that the drive frequency for the stagemust be made equal to one of the discrete

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points in the QPD power spectrum. For thisto happen there must be exact integer num-ber M of stage oscillations spanning the dataacquisition time. The default settings for thisnumber is 32 and, with a data acquisition timeof 1.38 s, gives a drive frequency fd = 23.2 Hz(point 32 of the PSD).

The output waveform is constructed with512 (= 29) points per period of the sinusoid.This is the maximum on-board buffer size foreach DAC and is not adjustable. The pro-gram must then calculate a separate (integer)divisor of the 20 MHz clock that determinesthe output rate for each point on this outputwaveform. In order for M periods of the out-put waveform to be exactly equal to the to-tal sampling time for the ADC, M must havecommon factors with the clock divisor for theADC. For example, if the default divisor (forthe ADC) of 3 ·5 ·7 = 105 is used, allowed val-ues for M would be any that can be made withsingle factors of 3, 5, and 7 and any number offactors of two. As a second example, an ADCdivisor of 5 ·5 ·4 = 100 gives a sampling rate of200 kHz and allowed values of M will be anythat can be made with one or two factors of5 and any number of factors of two. Selectingdisallowed values for M (those that would pro-duce a non-integer clock divisor) will disablethe continue button.

Once the data acquisition parameters havebeen accepted—by hitting an enabled con-tinue button—they cannot be changed withoutrestarting the program. One exception is theamplitude and phase of the drive waveforms.They can be adjusted by setting the new val-ues in the controls for them and hitting thechange amplitude button.

The fitting routine, accessed from the fittab, has several features designed for the datafrom this apparatus. First note the channelselector just above the graph. It is used toswitch between the two channels (0 or 1, i.e.,

the QPD x- or y-directions). The two cursorson the graph must be set to determine thepoints in between that will be used in the fitto Eq. 44. The PSD is normally displayed ona log-log scale, but this can be changed usingthe tools in the scale legend at the lower leftof the graph. Our PSDs show that many highfrequency and some low frequency noise com-ponents are being picked up in the Vx and Vysignals. They might originate from externallight sources, electrical interference, table andapparatus vibrations, etc. These unwantedsignals typically appear as spikes on top of thenormal Lorentzian shape of the PSD.

Spikes at the high frequency end of the PSDcan be eliminated from consideration by set-ting the second cursor below them. In fact,fits that include too many high frequency datapoints tend to take too long. Be sure to in-clude enough points above fc, but set the highcursor so there are less than 50,000 points inthe fit. Spikes between the cursors can beeliminated from the fit by setting their weight-ing factors to zero. This is done programmat-ically by telling the program how to distin-guish these spikes from the normal Lorentziandata. The criteria for eliminating the spikesthus requires an understanding of the normaland expected noise in the PSD.

Ordinary random variations in V (t) overany finite time interval lead to noise in theLorentzian PSD that becomes smaller as moredata is averaged. Watch PV (f) as you average50 scans and then stop the acquisition. Notethat the size of the noise (not the unwantedspikes) on the vertical log scale is nearly con-stant. While the band of noise may appeara bit wider at higher frequencies, this is atleast partially an artifact of the log f scalefor the horizontal axis; at higher frequenciesthe points are more closely spaced so that thenumber of 2-sigma and 3-sigma variations ap-pear more frequently per unit length along the

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f -axis.

Uniformly sized noise on a log scale im-plies the fractional uncertainty in PV (f) isconstant. Estimate the ±1-sigma fractionaluncertainty that would include about 68% ofthe data points in any small region of fre-quency. As you probably noticed above, thisfraction becomes smaller as more data is av-eraged. Check that it is roughly constant forall f even as PV (f) varies by one or more or-ders of magnitude. Enter this fraction in thecontrol for frac. unc. (fractional uncertainty).Then enter the rejection criterion in the re-ject control. For example, setting the frac.unc. control to 0.1 indicates that near any f ,68% of the PV (f) data points should be within±10 percent of the middle value. Setting thereject control to 3 would then exclude from thefit (set the weights to zero) any points morethan 30% “off.”

The program uses the fitted PSD at any f asthe central value for the rejection. For exam-ple, with the settings given above, any pointsmore than 30% from the current estimate ofthe fitted curve would be thrown out. Theinitial guess parameters define the current esti-mate of the fitted PV (f) according to Eq. 44,and these estimates must be set close enoughthat good points are not tossed. Click on theshow guess button to see the current estimateof the fitted PV (f) and the resulting rejectedpoints, which are shown with overlying ×’s.Clicking on the do fit button initiates a roundof nonlinear regression iterations excluding therejected points. After the fitting routine re-turns, click on the copy button to transfer theending parameter values from the fit to theinitial guess parameters and display the newpoints that would be rejected in another roundof fitting. Continue clicking the copy and thenthe do fit buttons until there are no furtherchanges in the fit.

PV (f) varies over several orders of magni-

tude and the fact that the fractional uncer-tainty is roughly constant over a wide range,indicates even the points out in the tails ofthe Lorentzian contain statistically significantinformation. If an equally weighted fit wereused, the points in the tails would not con-tribute to the fitting parameters as their con-tribution to the chi-square would be too smallcompared to the points at lower frequencieswhere PV (f) is much larger. Consequently,the fit should not be equally weighted. Be-cause the data point y-uncertainties σi areproportional to yi, the fit uses weights 1/σ2

i

proportional to 1/y2i . If the fitting function

accurately describes the data and the cor-rect fractional uncertainty is provided, thenormalized deviations between the data andthe fit (yi − yfit

i )/σy should be approximatelyGaussian-distributed with a mean of zero anda variance of one and the reduced chi-squarefor the fit should be about one.

Check the graph of normalized deviationsto verify the expected behaviors and check forsystematic deviations. (Click on the alternatetab control for the graphs to find this graph.)This graph also shows excluded points and canalso be used to make sure valid points are notbeing rejected. Even though the rejection cri-teria depends only on the product of the frac.unc. and reject controls, the frac. unc. controlshould be adjusted to get a reduced chi-squarearound one and the reject control should beadjusted so that only undesired points are re-jected from the fit. Correctly setting bothcontrols really only matters if you are inter-ested in determining the fitting parameter un-certainties. Recall that the fitting parametercovariance matrix scales with the assumed co-variance matrix for the input yi. With σi setproportional to yi, setting the fractional un-certainty to get a reduced chi-square of onedetermines the proportionality factor to usein order to get the best estimates for the true

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input and output covariance matrices.

Experimental overview

The basic tasks are to measure the trapstrength, calibration constant, and drag coeffi-cient for small particles of silica (SiO2) roughly0.5-1.5 µm in diameter. Having gained this ex-perience with the apparatus, you can then ex-periment with biological trapping by measur-ing, e.g., the force generated by a swimmingbacterium.

Note that it takes a couple of days to pre-pare the bacterial culture for this experiment,so you will need to plan ahead by notifyingyour instructor of the date when you plan toperform the bacterial study.

Laser Safety

Note that although this experiment is not dan-gerous, any eye exposure to the infrared laserbeam would be very dangerous. The beam isvery intense, with a power of several hundredmW, and it is invisible. Serious and perma-nent eye injury could result if the beam entersyour eye. Proper laser eye safety precau-tions must be used at any time that thelaser is running.

The apparatus is designed to keep the in-frared laser beam enclosed within its intendedoptical path and away from your eyes. The in-strument is safe to use as long as the laser re-mains enclosed. Therefore, laser safety meansthat you should not operate the laser whenthe beam enclosure is open or any portion ofthe optical pathway has been opened or disas-sembled. If you open or disassemble any com-ponents while the laser is powered you couldexpose yourself to the IR beam and suffer apotentially severe injury. Do not attempt toalign or adjust any part of the infrared laseroptical path.

The only point in the apparatus where thebeam leaves its confining path is at the sam-ple slide, between the two microscope objec-tives. In this region the beam is strongly con-verging/diverging and is not likely to presenta hazard to the user. However you should usecommon sense and avoid diverting the beamout of this region. Do not place shiny, metal-lic or reflective objects like mirrors or foil intothat region. Do not put your face close to theslide if the laser is on.

General concerns

In addition to laser safety issues, please takecare to observe the following precautions

• Alignment of the optical system: All op-tical elements have already been carefullyaligned and optimized. The only opti-cal adjustments you will need to makeinvolve the xyz positioning of the mi-croscope stage and xy positioning of theQPD. Do not attempt to move, dis-assemble or adjust the optical fiberor any of the mirrors and lenses andother optical components. If you dis-turb the laser alignment, the optical trapwill cease to function and it will requiretedious and time-consuming realignment.Any disassembly of the apparatus couldalso lead to accidental and very danger-ous eye exposure to the laser beam.

• The 100× objective: Please take care thatnothing (except immersion oil and lenspaper) ever touches the lens of the lowermicroscope objective. In focusing or ad-justing the stage you should not crash orscrape the slide against the lens.

• The laser optical fiber: Please do nottouch or handle the optical fiber. It isextremely delicate and costly to repair.

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• The laser settings: The laser beam poweris adjustable up to a maximum currentof ILD = 650 mA. The laser also has atemperature controller that has been pro-grammed to maintain the laser at its op-timum temperature. You can adjust thelaser current right up to the maximumlimit value, but please do not attempt tochange the limit or the laser control tem-perature.

Procedures

The following procedures should probably bedone in the order outlined below. They willtake more than one day. Be sure to follow theprocedures in the Cleaning Up section beforeleaving.

Initialization

Turn on the power supply for the controllerhub. Wait a few seconds for their firmware toinitialize and then run the Initialize program.Check that the LED light source is on.

Camera calibration

Find the Thorlabs R1L3S3P grid slide anddetermine which side has the grid patterns.Place a small drop of immersion oil over thesmallest (10µm) grid pattern and place theslide on the sample stage with the calibrationmarkings facing downward (oil side down).Then carefully slide the stage into positionover the objective, watching that you do notcrash the slide into it: the bottom of the slideshould be above the objective.

Start the UC480 camera program. Usingthe manual z micrometer, lower the slide downwhile watching the camera image for the gridto come into focus. You will need to get theslide quite close to the objective lens (less thana mm) to get into focus.

[Putting a drop of oil on the coverslip,putting the slide onto the stage (coverslipdown) and into the area just above the 100×objective will henceforth be referred to as “in-stalling” the slide. “Uninstalling” will meanraising the stage, sliding it away from the trap,and removing the slide.]

The 10µm grid is rather small and so coarseand fine adjustments in the x- and y-directionsmay also be needed just to get it into view. Ifyou are having trouble finding it, be sure theslide is correctly oriented with the grid sidedown. You may want to find the focus withone of the larger grid patterns first.

Now you can determine the pixel calibrationconstant: How many microns at the samplearea correspond to one pixel on the cameraimage? Note this is not the actual pixel size(5.3 µm/pixel), but rather that size divided bythe magnification, or roughly 0.05 µm/pixel.Use the camera software measuring tool todetermine the separation in pixels of knownlengths on the grid slide. (The grid squaresare 10×10µm.) Our camera pixels are squareand you should find the same values in the x-and y-directions. Determine the camera cali-bration constant in µm/pixel.

Next use the x- and y-micrometers on thestage to determine their sensitivity on the fine(differential) operation. The micrometer fine-control spindles are marked with 50 divisionsper rotation. Because the distance moved foreach division is somewhat variable, we will callthem m-units; 50 m-units per rotation. Usethe camera and grid markings to determinethese fine-control m-units per actual distancemoved. This calibration constant should benear 1m-unit per µm of real motion. However,remember that this calibration can change abit depending on how far the stage is from itscentral position.

Uninstall the grid slide, clean it with alco-hol, wiping it gently with sheet of lens paper,

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and place it back in its protective case.

Sample sphere preparation

You will need to prepare two solutions of1.2 µm diameter silica beads. Since you willneed to make measurements on a single sphere,getting their concentration correct is very im-portant. Too few spheres and it will be dif-ficult to find any. Too many and the sphereswill interfere with one another during the mea-surements.

Have the instructor show you proper use ofthe pipettors and vortex mixer. Be sure touse the vortex mixer just before sampling fromthe stock solution, any intermediate solutions,and just before loading your final solution intothe slide. The spheres tend to settle and thevortex mixer is needed to get them uniformlydistributed in the suspending liquid. If you donot mix, the density of spheres will be wrong.Moreover, if you don’t mix the main stock so-lution before taking a sample, you would bechanging the concentration of the remainingstock solution.

Prepare approximately 1.5 ml of a 150:1dilution of the stock solution of the 1.2µmspheres in deionized (DI) water. Even this di-luted solution is still much too dense for mea-surement and another 150:1 dilution is needed.For stuck spheres, this second dilution shouldbe into 1M NaCl water, which makes themstick to the slide. For free spheres, use DI wa-ter again. Only make the free sphere dilutionat this point. You may want to adjust theconcentration for the stuck spheres. Be sureto mark the vials with the sphere size, dilu-tion factor, date, and whether it is in water ora salt solution. At a dilution of 1502 = 22, 500,there should be an average of a few beads inthe camera image. If it turns out there aremore than 10 beads per image area, furtherreduce the concentration.

The Ibidi slide has wells on each side wherethe solution is introduced. The first sampleneeded will be the 1:22,500 dilution in DI wa-ter. Put about 100 µl in one well and use asyringe to suck it through the channel, takingcare not to suck air into the channel. (Addanother 50 µl as the well empties.) It is easierto see the liquid coming into the channel if theslide is placed on a dark background. Whenfilled, add or remove the solution to the wellsas necessary to get it about half-way high ineach well. If the heights are unequal, therewill be a pressure difference which will drivethe fluid from one well to the other until thepressure difference is eliminated. Even if youget the well heights equal by eye, small differ-ences can still drive the fluid and it can takeseveral minutes for the motion to cease. It canbe very difficult to see spheres if they are mov-ing with any but the smallest velocity flow.

Initial observation of a trapped sphere

Make sure the laser is off. Install the slideprepared above. As you bring the slide down,look for individual spheres undergoing Brow-nian motion. Spheres and small dirt parti-cles will often become stuck to the coverslipat the bottom of the channel or to the glassat the top of the channel. Find these surfacesand measure their separation in m-units to besure they are, in fact, the top and bottom ofthe 100µm channel. Being certain the focusis in the channel and just above the top of thecoverslip is often helpful in the hunt for freespheres. Spheres will be more dense at thebottom of the channel, but should be foundhigher up as well.

When you see spheres, turn on the laser toa current of approximately 400 mA and movethe stage around manually as you try to cap-ture a sphere into the trap. You will know thata particle is trapped because it will remain in

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the same location and same focus, even as youadjust the stage from side to side as well asup and down. Mark the trap position on thevideo image with the circle tool on the camerasoftware and save this drawing.

A particle trapped in the z-direction willnot change its focus (appearance on the im-age) when you adjust the stage up and downin the z-direction. While you are moving theslide, the trapped particle’s position is fixedbecause the laser focus is fixed relative to the100× objective. If you raise the slide enough,however, sooner or later the sphere will hit thebottom of the channel and then go out of focusif you continue raising the slide. Similarly, ifyou lower the slide enough, the sphere will hitthe top of the channel. If the trap is weak, youmay have problems keeping a sphere trappednear the top of the channel. Because of theoptical properties of the objective and sample,the trap force in the z-direction is expected toweaken as the sphere height increases.

Play around with this configuration a bit.Is the sphere density about right? Can youkeep a single sphere trapped for many minutesor do other spheres often wander in? Whileyou can compare measurements from differentspheres with nearly the same diameter, smallvariations in their trap constants will affectthe comparisons. Their dependence on laserpower, for example, is smoother when all mea-surements are from the exact same sphere. Toavoid spheres from wandering into an alreadyfilled trap, reduce the sample concentration.Setting it so there is about one sphere per cam-era image typically is about right. Working abit higher in the channel helps in this regardas well.

Check the top and bottom z-micrometer po-sitions as you demonstrate a trapped spherecan be moved from top to bottom of the chan-nel. Save a short video sequence of an isolated,trapped sphere. Be sure you have recorded the

trap position with a circle and have saved it asa drawing. It will be needed in later proceduresteps.

Always be sure to make measurements atleast 20 µm from the bottom of the channel.Viscosity effects cause the motion of the liquidaround the spheres to change when the spheresare close to the bottom or top surface of thechannel. Beyond 20 µm or so, the surfaces areeffectively infinitely far away as far as viscosityeffects go.

Uninstall the slide, empty it and refill itwith an appropriate dilution of spheres in saltwater to get, at most, a few per screen. This“stuck sphere” slide will be used in the nextprocedure.

Piezo calibration

You will next determine motion calibrationsinvolving the use of the piezo controls on thestage. To do a piezo calibration requires ob-serving a small object, such as a sphere, stuckto the slide. Install the slide prepared aboveand find a relatively isolated single spherestuck to the coverslip.

The direct DAC method of driving the piezois used in the main Tweezers program wherethe stage is set into sinusoidal oscillationsof known frequency and amplitude. Conse-quently, a calibration constant—from the am-plitude of the DAC drive voltage to the ampli-tude of the stage motion is needed. To performthis calibration, use the Oscillate Piezo pro-gram, which allows for convenient adjustmentsof the two DAC sinusoidal voltages. Theiramplitudes as well as their common frequencyand their phase difference are adjustable.

Because of the nonlinear piezo behavior, anapplied sinusoidal voltage of amplitude VDAC

will cause nearly sinusoidal oscillations of theposition with an amplitude A that dependsnonlinearly on VDAC. Run the Oscillate Piezo

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program while viewing a stuck sphere. Withthe VDAC at zero, the piezo doesn’t move andthe amplitude of the stage motion is zero. Asyou increase VDAC, the stage motion amplitudeincreases in a near-linear fashion with a smallquadratic component.

A = a1VDAC + a2V2

DAC (46)

If you apply equal amplitude oscillations toboth the x- and y-piezos and set them 90◦ outof phase with one another, the stage shouldmove in a circle with a radius given by Eq. 46.Or, you can set either the x- or y-oscillationamplitude to zero so that the stage movesback and forth in only one dimension. Useeither method. Measure the amplitude A ver-sus VDAC in the range from 1 to 3.5 V at a1 Hz frequency and fit that data to Eq. 46 todetermine a1 and a2. Be sure that there is noconstant term in the fit (as in Eq. 46) becausethe amplitude of the motion must be zero withno drive voltage. The peak-to-peak amplitude(2A) can be measured (in pixels) from cameraimages where you try to see and measure ei-ther the diameter of the circular motion or theextrema of linear oscillations. Be sure to mea-sure to the center of the spheres—a task moredifficult than it sounds as the extrema are of-ten faint and blurred. These measurementsare then converted to real stage motion by thepixel to stage distance factor determined fromthe previous grid pattern measurements. Set-ting a small AOI (area of interest) around thesphere will speed up the frame rate, which canbe quite useful in this step.

Next, measure the stage amplitude withVDAC = 1 V at several drive frequencies upto 40 Hz. Then try it at 3-V drive amplitude.At higher frequencies, the stage accelerationsfor a given amplitude are larger and the stageinertia can affect the motion.

When measuring the PSD for particles inthe trap, the stage oscillations will be in

the 10-30 Hz range, but will be at very lowamplitudes—a few tenths of a micron drivenby a VDAC of a few tenths of a volt. Theseoscillations are a bit too small to measure ac-curately with the camera. Instead, the cali-bration performed in this step should be ex-trapolated to these low amplitudes.

Leave the stuck-sphere slide mounted as itwill be used in the next procedure.

QPD calibration

For the in situ calibration described in the the-ory section, the detector constant β will be de-termined from the PSD of a trapped sphere onan oscillating stage. It is nonetheless worth-while to look at another method, describedhere, for determining β using a stuck sphere.This method shows why an in situ calibrationis so much better and demonstrates some ofthe limitations involved in either calibrationmethod.

Load the trap position drawing into theUC480 image and install the stuck-sphereslide. Move the slide so that there is a rel-atively isolated single sphere in the vicinity ofthe trap circle. Run the raster program. Itstarts in the calibration tab. Set the x and ystrain gauge percentages in various combina-tions from 10 to 90% and measure the parti-cle position on the camera for each x, y per-centage requested. Check for proportional be-havior between the motion and the percentageand determine the actual distance moved perstrain gauge percent.

Turn on the temperature controller andthen the laser current controller to about400 mA or about 15 mW. Move the slide in xand y so there are no beads within 10µm of thetrap circle, adjust the QPD xy-stage positionto roughly zero the Vx and Vy signals from theQPD controller. Start with the ADC full scalerange of ±10 V, but adjust it to the smallest

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range, once you know how big Vx and Vy get.(Don’t worry about keeping the sum voltagefrom saturating.) In the calibration tab, leavethe sample ADC rate at 10,000 readings persecond and averaging 500 such readings perpoint. This gives a 0.05 s averaging time orthree cycles of the 60 Hz power lines. Using amultiple of 1/60 second reduces noise at 60 Hz.

Try the default scan parameters first: scancentered at 50% for x and y, scan range ±15%in x and y. The Vx and Vy data at each pointin the scan are averaged according to the ac-quisition parameters above.

Manually move a relatively isolated, stucksphere to the center of the trap circle. Adjustthe z-focus so the sphere on the image appearsabout the way it did for a trapped sphere. Hitthe Start Scan button. The camera shouldshow how the raster scan proceeds and theprogram should go to the Acquire tab show-ing graphs of Vx and Vy vs. t as well as versuseach other, which update with each scan linemeasured.

When the scan is complete, the programgoes to the Analyze tab where you can see a2-D “intensity plot” of Vy versus x and y onthe left half and and Vx versus x and y on theright half. For an intensity plot, the values ofVx and Vy determine the plot color at each x, yvalue. Run the cursors through the center ofthe data in these plots to create the plots of Vxvs. x and Vy vs. y. These slices typically showoscillations as the sphere is scanned throughthe laser focal spot. You are trying to find themiddle of the pattern where Vx vs. x and Vyvs. y demonstrate nearly straight-line behav-ior. Find the slopes in this region in V/x-unitand V/y-unit. Also check the range of the lin-ear behavior in x- and y-units.

The x- and y-values on these graphs al-ways range from 0-1. (Because of the way thegraphs are stacked, x will range from 0-1 forthe left intensity plot of Vy and from 1-2 for

the right intensity plot of Vx. The range of 0-1(or 1-2) will correspond to the full scale motionrequested—30% for the default ±15% range.Divide your measured slope by the full scalerange to get the slopes in V/% and then divideby strain gauge calibration factor in µm/% toget the slope in V/µm. This slope is then theQPD detector calibration constant β. Simi-larly convert the x- and y-range for linear be-havior to a true distance and compare withthe expected rms displacements of a trappedparticle as from Exercise 5.

Because the QPD voltages should all be pro-portional to the power on the detector andthe power on the detector should be propor-tional to the overall laser power, the slopes inV/µm, say, should be proportional to the laserpower. Check this by performing and analyz-ing a raster scan at half the laser power usedabove.

Measured this way (with a stuck sphere),β is very sensitive to how the sphere is po-sitioned in the z-direction. When a floatingsphere is trapped by the laser, it will oscillatesomewhat in the z-direction because of theBrownian force, but its equilibrium position isat the center of the trap (z = 0) and cannot beadjusted. On the other hand, a stuck spherecan be placed anywhere in z by changing thez-position of the stage and its position will af-fect how well it will be focused on the camera.While your initial raster scan was at roughlythe same focus as a trapped sphere, do anotherwith the sphere moved slightly higher and/orlower (by changing the z-focusing) such thatthere is a modest change in the appearanceof the image. Note how much the stage wasmoved, run another raster scan and and checkhow this affects β. What does this say aboutthe assumption that β is a constant? Howwould a z-dependence to β affect the analy-sis?

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Full trap calibration

Adjusting the piezos with the strain gaugefeedback cannot always be done fast enough—particularly when applying an oscillatory mo-tion to the stage as for the in situ calibra-tion method. In this case, the computer’s twoDACs will be used to apply sinusoidal voltagesdirectly to the input of the x and y piezo mod-ules. The piezo module amplifies those volt-ages by about 7.5, adds them to the 37.5 Voffset and sends them on to the actual piezo-transducers in the stage.

Reuse or make a new slide with 1.2 µmspheres in DI water at an appropriate dilutionto get about 1 sphere per CCD image. In-stall it between the objectives, find a trappedsphere and adjust the stage’s z-position to getit about 30 µm above the coverslip.

Start the Tweezers program. Zero the x andy VDAC amplitudes so the stage does not os-cillate. Set the acquisition and timing param-eters. Begin acquiring the QPD signal andaveraging the calculated PSD PV (fj). Whenit is sufficiently smooth, stop the averaging,switch over to the Fit tab and do a fit of thePSD to Eq. 44.

Turn on the piezo oscillation of the stageand set the VDAC that would give a stage os-cillation amplitude A = 0.1 − 0.2 µm. Beginaveraging the PSD and perform a full analysisto determine β, γ and k. Repeat at differentlaser powers. Plot the trap strength k, the cal-ibration constant β, and the drag coefficient γas a function of laser power. Discuss the re-sults. Are k and β directly proportional tolaser power? Is γ constant? Can you see anysystematic behavior with power? Why mightthis be reasonable?

Possible additional studies

Repeat the calibration procedure forother sphere sizes. Our largest are 5.1µm

in diameter and present several difficulties as-sociated with their large size; they are about75 times heavier than 1.2µm spheres. We havespheres of diameter 0.5, 0.75, 1.0, 1.21, 1.5 and5.1µm. Most have not been studied. Exceptfor the 1µm spheres, the stock solutions areall 10% spheres by weight. Thus to get thesame concentrations in particles per unit vol-ume, the dilutions must scale in proportion tothe the sphere volume—twice the sphere di-ameter, 1/8 as much dilution. Scaling laws forthe parameters can be investigated.

Check out the Berkeley wiki for more in-formation on the following two investigations,which are only briefly described here.

Investigate flagella locomotion inE. coli bacteria. You will have to prepare adilution from a culture made one or two daysbefore. Don’t forget to request it well in ad-vance. Normal E. coli repeatedly swim a bitand then tumble changing their direction fromthe tumble. We use a strain that has beengenetically modified not to tumble—travelingmore or less continually in one direction.

Make and install a slide with an appropriatedilution. This might take a few tries as it isdifficult to predict the culture concentration.Capture a dead bacterium and use the Tweez-ers program to determine the trapping force.Then trap a swimming bacterium. Lower thelaser power until it swims free. Repeat forother E. coli on your slide. What does the dis-tribution (frequency histogram) of minimumlaser powers look like? Does the length of timeinside the trap affect the bacteria swimmingstrength? How well can you determine theswimming force generated by the bacteriumfrom this measurement?

With the laser off, watch a swimming bac-terium and determine their typical speeds. Es-timate the size of the bacteria. Its hydrody-namic radius is that value of a that gives theactual drag force Fd = −γv when the Einstein-

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Stokes formula (Eq. 15) is used with that a.Compare the two force determinations. Useyour measurements to estimate the power gen-erated by the flagellar motor of the swimmingbacterium.

Investigate vesicle transport in onioncells. You will have to bring in your ownonion. Be sure it is fresh—hard and tight,not mushy. Be sure to take any leftover onionhome. Do not dispose it in the lab trash. Itstinks up the room rather quickly.

Prepare a slide of onion epidermal cells in0.1 M salt solution. Look for vesicles (bags ofnutrients, waste or other cell material) floatingin the cytoplasm and others traveling alongspecialized filaments. Find one and trap it.Move the slide to see if it is freely floating orstuck to a filament. How much can a filamentstretch? What happens when you turn off thetrap?

Trap one on a filament and watch as othervesicles back up along that filament. Turn offthe trap and describe your observations. Trapan isolated vesicle on a filament and lowerthe laser power until it breaks free. Repeatfor others vesicles on filaments. Is the mini-mum laser power the same every time? Whatmight affect the distribution of minimum laserpower. Are there other quantitative measure-ments you can make?

Cleaning Up

When finished for the day, shut off the lasertemperature and current controllers. Closeall open LabVIEW programs and then turnoff power to the T-Cube hub. Most impor-tantly, this turns off all voltages to the piezos.Leaving a voltage on the piezos over long peri-ods can change their properties. Turn off thepower strip so the LED will turn off as well.(The computer and monitors are on a separatepower strip.)

Uninstall the slide, clean it with a Kim-Wipe and alcohol and use a syringe to run al-cohol through the channel two or three times.Then fill it with alcohol and leave it in a 200 mlcylinder also filled with alcohol. This storagetechnique will keep the slide from becomingfilled with bacteria or other living organismsand will help prevent breakage of the fragilecoverslip. (If it is left full, as the water evapo-rates, the coverslip will often crack.) The Ibidislides can be reused, but check the coverslipand dispose of the slide if it has more than afew small cracks.

Use a single sheet of lens paper (not aKim-Wipe, which is very abrasive) to wipe theoil from the 100× objective. Do not scrub.Wipe gently once in one direction.

Clean up the apparatus and sample prepa-ration area. Dispose of tissues in the trash canand dispose of glass or plastic slide material orpipettor tips in the disposal box by the sink.

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