Sub-micron scale Brownian swimmer or surfer in onedimensional standing wave optical traps

Martin Siler, Tomas Cizmar, Petr Jakl and Pavel Zemanek

Institute of Scientific Instruments, Academy of Sciences of the Czech Republic,Kralovopolska 147, 612 64 Brno, Czech Republic

ABSTRACT

An optical conveyor belt created by a standing wave can be used to deliver Brownian particles in a controlledway. The dependence of the particle speed on the speed of traveling standing wave is not simple but two basicmodes can be distinguished. For very slow standing wave motion the particle is tightly coupled to the potential,i.e. it “surfs along with the potential wave” and therefore it is called the “Brownian surfer”. For bigger standingwave velocities the particle behaves like a swimmer afloat on the surface of the ocean and thus it is called the“Brownian swimmer”. The mutual speed of the particle and the standing wave is studied experimentally andtheoretically.

Keywords: Brownian motion, optical ratchet, optical force, colloidal particle, interference optical trap, opticalconveyor belt

1. INTRODUCTION

An indivisible part of all microscopic phenomena are stochastic processes generally called as the Brownianmotion. In most cases it is regarded as a noise that has to be reduced. Nevertheless the presence of a randommovement might sometimes evoke greater understanding of the studied systems or bring some new unexpectedphenomena. Random movement can be used e.g. to calibrate stiffness of optical traps1–3 or it participates in theeffect of stochastic resonance.4–7 The “gedanken” experiment of ratchet and pawl presented by Smoluchowskiand later extended by Feynman8 shows a possibility to rectify a random motion of particles together with someother preconditions, such as asymmetric potential and time dependency of yet unspecified quantity, to createmacroscopic movement of the particles, so called Brownian motors or ratchets.9

The interference of two counterpropagating beams creates a periodic structure of standing wave nodes andantinodes. If a dielectric object having refractive index higher than the surrounding medium is placed into thisinterference field, it feels the same periodic structure as an array of potential wells - optical traps. It is alreadyknown that bigger gradients of electric field intensity implies better localization and stronger confinement ofobjects in this structure in comparison to single beam optical trap.10,11 Different types of beams have been usedto create one dimensional array of optical trap - for example Gaussian beams,10 Bessel beams12,13 or evanescentwaves.14,15 The change of phase in one of the interfering beams results in the movement of the whole structureof the optical traps together with the confined objects. Therefore the beads can be delivered over microscopicaldistances in a way similar to a conveyor belt and so the whole device is called optical conveyor belt (OCB).12

However the object stays in one trap only if it is sufficienly deep. In swallower traps the particle is confinedweakly and therefore it starts to jump to the neighboring traps more frequently. For the practical realization ofthe OCB it is desirable to know the dependence of the average bead velocity on the velocity of the OCB. Thejump rate between neighboring optical traps would increase for the high velocities of OCB and therefore theaverage bead velocity would decrease. There has to exist some optimal velocity of the OCB that would deliverconfined beads under maximal average speed.9 To find this optimal speed one have to use the Fokker-Planckequation16 because the system consisting of a bead and moving OCB (periodic potential profile) is far fromthe thermal equilibrium state. In this paper we study this dependence theoretically, by means of Monte-Carlosimulation and experimentally by the evanescent conveyor belt.14

Further author information: (Send correspondence to P.Z.)P.Z.: E-mail: pavlik@isibrno.cz, Telephone: +420 541 514 202, Fax: +420 541 514 402

Optical Trapping and Optical Micromanipulation III, edited by Kishan Dholakia, Gabriel C. Spalding,Proc. of SPIE Vol. 6326, 63262K, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.680597

Proc. of SPIE Vol. 6326 63262K-1

2. THEORY

The behavior of a particle localized in the traveling potential is described by the Langevin equation16,17

γx(t) = −U ′(x(t) − f(f)) + ξ(t), (1)

where γ is the Stokes coefficient, x(t) the particle coordinate, U(x) is periodic potential with period L and f(t)is function describing movement of the potential landscape. In this work we consider an overdamped motion ofa particle and therefore the mx term is omitted. The thermal fluctuations are modeled by an unbiased Gaussianwhite noise ξ(t) with zero mean 〈ξ〉 = 0 and correlation

〈ξ(t)ξ(t′)〉 = 2γkBTδ(t − t′), (2)

where δ(t) is the Dirac delta function, kB is Boltzmann constant and T is temperature. Upon introducingauxiliary variable

y(t) ≡ x(t) − f(t) (3)

into (1) we obtainγy(t) = −U ′(y(t)) − γf(t) + ξ(t). (4)

Further we will consider so-called genuine traveling potential ratchet9 with function f(t) in the following form:

f(t) = ut. (5)

In the other words, the potential U(x − f(t)) in (1) is given by a periodic array of traps traveling at constantvelocity u along x axis. After inserting (5) into (4) we obtain

γx(t) = −U ′(x(t)) − γu + ξ(t), (6)

the auxiliary variable y was renamed back to x. Formally, the Eq. (6) describes movement in tilted periodicpotential, i. e. the particle moves in periodic potential U(x) and is affected by constant force F = −γu. We mayincorporate both the potential and the force into one effective potential

Ueff(x) ≡ U(x) + xγu. (7)

Note, that although the effective potential is not periodic, its derivation along x, dU/dx, is still periodic functionwith period L. The next step is to consider a statistical ensemble of the stochastic processes belonging toindependent realizations of random fluctuations ξ(t). The corresponding probability density P (x, t) in space xand time t describes the distribution of Brownian particles. The time evolution of P (x, t) is given by Fokker-Planck equation as follows9,16

∂

∂tP (x, t) =

∂

∂x

{U ′

eff(x)η

P (x, t)}

+kBT

η

∂2

∂x2P (x, t). (8)

The first term on the right-hand side is called drift term and the second diffusion term.

The quantity of the foremost interest in the context of transport in the periodic systems is the particle current〈x〉, defined as time-dependent ensemble average over the velocities. Obviously, the probability density P (x, t)contains the entire information about the system. The simplest way to establish a connection between P (x, t)and 〈x〉 follows by averaging (6) and taking into account 〈ξ〉 = 0, i. e. 〈x〉 = −〈U ′

eff(x(t))/γ〉. Since the ensembleaverage means by definition an average with respect to probability density P (x, t) we obtain

〈x〉 = −∞∫

−∞dx

U ′eff(x)γ

P (x, t). (9)

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To solve Fokker-Planck equation (8) it is useful to introduce so-called reduced probability density:

P (x, t) ≡∞∑

n=−∞P (x + nL, t). (10)

For this quantity it follows that

P (x, t) = P (x + L, t),

L∫0

dxP (x, t) = 1. (11)

With P (x, t) being a solution of the Fokker-Planck equation (8) it follows with the periodicity of U ′eff that also

P (x + nL, t) is a solution of the Fokker-Planck equation for any integer n. Since the Fokker-Planck equation islinear, it is also satisfied by reduced density (10). In other words, as far as the particle 〈x〉 current is concerned,it suffices to solve the Fokker-Planck equation with periodic boundary conditions.

For physical reasons we expect that the reduced probability density P (x, t) indeed approaches a steady stateP st(x) in the long time limit t → ∞. The reduced probability is then

P st = N γ

kBTe−Ueff (x)/kBT

x+L∫x

dy eUeff(y)/kBT , (12)

where N is normalization constant is given by (11) as follows:

N =kBT

γ

⎡⎣

L∫0

dx

x+L∫x

dy e(Ueff (y)−Ueff(x))/kBT

⎤⎦−1

. (13)

Eqs. (12) and (13) can be used to calculate probability current and particle current

〈x〉 = LN [1 − e(Ueff(L)−Ueff (0))/kBT ]. (14)

For the specific form (7) of effective potential we can further simplify (14) by exploiting that Ueff(L)−Ueff(0) =Lγu. The average velocity 〈x〉 of the original problem (1) is

〈x〉 = 〈f〉 + 〈y〉, (15)

where 〈y〉 is given by (14) and we obtain

〈x〉 = u − LkBT

γ

[exp(γuL/kBT ) − 1]∫ L

0dx

∫ x+L

xdy exp

{U(y)−U(x)+(y−x)γu

kBT

} . (16)

The particle current 〈x〉 is always smaller than the original velocity of potential movement u and it has alwaysthe same sign as u. The first statement comes from the fact that the term exp(γuL/kBT ) − 1 has the samesign as u and the denominator of the fraction in (16) is always positive. Therefore the absolute value of 〈x〉 issmaller than the absolute value of u. The proof of the second statement can be found in.9 In case of weak andfast moving potential, i. e. U(x) � γux, we obtain 〈x〉 as 〈x〉 = 0.

One can say that for the high velocities the particle does not feel the presence of traveling potential andbehaves like a swimmer afloat on the surface of the ocean and may thus be called Brownian swimmer. For smallvelocities u we have a Brownian surfer that is riding in tightly coupled way on the potential. The current 〈x〉tends to zero for both very small and very large speeds u and it has maximum at the largest u which still supportsthe surfing mode. Figure 1 shows an example of the mean velocity of bead 520 nm in diameter calculated fordifferent traveling speeds u and different heights of potential barriers ∆U .

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0100

200300

400500

02

46

810

−20

0

20

40

60

80

u [µ m s−1]∆ U [k

B T]

⟨ x . ⟩ ≡

vB

ead [µ

m s

−1 ]

Brownianswimmer

Browniansurfer

Figure 1. The mean velocity of a polystyrene bead of 520 nm in diameter placed to traveling periodic potential. Thepotential has spatial profile U = −∆U/2 cos(4πnextx/λ), where ∆U is the potential well depth, next is the refractive indexof the surrounding medium (water) and λ is the vacuum wavelength of interfering light waves, λ = 532nm. One can seethe regions where the bead behaves as the Brownian surfer(on the left) and as a Brownian swimmer (on the right).

3. RESULTS

Further, we will concentrate on the potential profile coming from the interference of two counterpropagatingplane waves. The potential is:

U(x) = −∆U

2cos 2kx, (17)

where ∆U is the trap depth (depth of the potential well) and k = 2πnwater/λ is wave vector in the water.

3.1. Monte Carlo Simulations

Before the experimental realization we used Monte Carlo (MC) simulations to study behavior of a bead in thetraveling potential. In each simulation at least 106 steps were performed and a time difference between the stepswas 2.5µs. This choice of time step fulfills the the condition

√2Ddt � L, where D is the diffusion constant,

D = kBT/γ. This condition states that the average displacement of a bead in time interval dt is much lowerthan distance between neighboring potential minima. Figure 2 shows an example of the bead behaviour. Therecord of the positions is showed in a stationary coordinate as well as in coordinate system traveling with thepotential landscape. In both coordinate systems the jumps between traps are visible. In the figure, the velocityof potential (further referred as velocity of optical conveyor belt vOCB) is vOCB = 50µms−1. The average velocityof the bead obtained from the MC simulation in this interval is 28.83 µms−1 while velocity given by Eq. (16) is28.77 µms−1. The coincidence is very good.

The simulation of the bead movement can be used to make a histogram of the bead position with respect tothe center of optical trap. The histogram can be compared to the reduced density probability (12). Figure 3shows three examples of such histograms and reduced density probability. The simulations were performed for

Proc. of SPIE Vol. 6326 63262K-4

0 10 20 30 40

0

200

400

600

800

1000

1200

t [ms]

x [n

m]

(a)

0 10 20 30 40−900

−800

−700

−600

−500

−400

−300

−200

−100

0

100

t [ms]

x −

vO

CB t

[nm

]

(b)

Figure 2. Monte-Carlo simulation of the bead positions in the traveling standing wave with respect to fixed coordinateorigin (a – left) and with respect to coordinate origin traveling with standing wave (b – right). The speed of the OCBis vOCB = 50 µms−1, trap depth ∆U = 5 kBT , bead diameter d = 520 nm and the wavelength in water λwater = 400 nm.The dashed lines shows borders between neighboring optical traps.

−100 −50 0 50 1000

0.5

1

1.5

2x 10

7

x [nm]

Pst

[arb

. uni

ts]

vOCB

= 10 µ m s−1

vOCB

= 50 µ m s−1

vOCB

= −150 µ m s−1

vOCB

= 0

Figure 3. Reduced probability density or bead positions constructed as a histogram from the MC simulated data forstationary OCB an for different velocities of OCB (10, 50, -150 µms−1) is shown by noisy lines. The trap depth is∆U = 5 kBT and bead diameter is d = 520 nm. The smooth curves are density probabilities calculated by Eq. (12)for the same values. The interval of x positions shown here covers one trap, i. e. the histograms are plotted in interval〈−L/2; L/2〉.

Proc. of SPIE Vol. 6326 63262K-5

OCB velocities 10, 50 and -150 µms−1 and for stationary OCB. The trap depth was ∆U = 5 kBT . The noisylines shows histograms of bead positions and smooth lines were calculated for the same values of vOCB and ∆U .Note that the positions of bead are taken relative to the center of trap where bead is located at the given time.The height of histogram is changed in order to assure the unity area under the histogram. One can see that withincreasing velocity the central peak is broadening and moving away from the center of the trap. The maximumof the histogram (and the reduced density probability) is deviated in opposite direction than the direction of theOCB.

Further simulations showed very good coincidence between values obtained from MC simulations and depen-dences calculated form Eqs. (12) and (16).

3.2. Experimental results

We used an experimental set-up based on the interference of two counterpropagating evanescent waves.14,15 Up tothese days, the interference of counter-propagating evanescent waves was generated using a retroreflecting mirrorand it was used for generation of periodic light pattern for near-field microscopy experiments.18 In contrast tothe previous methods of the generation of a periodic near-field interference pattern by a retroreflecting mirror18

we used a setup with two independent counter-propagating Gaussian beams that were focused on the top surfaceof the prism. Tuning the incident angle may result in either a surface patterned evanescent fields or a propagatinglight field just above the surface. This set-up enables us to alter the relative phase between the two beams in acontrolled fashion by a movable mirror. Its movement results not only in the motion of the standing wave butalso in motion of objects confined in this standing wave. We have also used the OCB to deliver beads of differentsizes and to sort diluted heterogeneous mixture of beads according to their sizes.15

We wanted to find out how the velocity of the standing wave influences the velocity of the particle movement.At first we focused on the properties of a bead confined in the stationary OCB. The movement of the polystyreneparticle (radius a = 260 nm) was recorded by fast CCD camera (IDT X-Stream XS3, 4GB) recording 6120 fpsduring a period of 1 s. The position of the bead center was evaluated with subpixel accuracy by our newmethod of particle tracking15 and we obtained resolution in nanometers. Figure 4a shows the histogram ofbead positions along x axis. Since the OCB is at the rest, i. e. vOCB = 0, the reduced probability density (12)gives the Boltzmann distribution of particle positions.1 We used the Boltzmann distribution to fit the measuredhistogram with unknown parameter ∆U and fixed values of temperature T , Boltzmann constant kB and knownpositions −L/2 ≤ x ≤ L/2 with respect to the bottom of the trap. These positions were shifted to the interval〈−L/2, L/2〉. Coincidence between the fit and the data is shown in Fig. 4a. It gave the depth of the optical trap∆U = (9.0 ± 0.3) kBT .

Further, the sawtooth voltage was applied to the piezoelectric actuator (PIFOC – Physik Instrumente) andthe movable mirror moved at a speed vOCB. The bead movement was again recorded by the fast camera withthe sampling frequency 6120 fps. Figure 5 shows the position of the particle (full line) and displacement ofits trap site at t = 0 (dotted line) if the OCB moved with two velocities. The tracking of the same particlein two motional and one stationary measurements was performed with same particle within 20 minutes. Fromthe values shown in Fig. 5 we calculated average speeds of confined particles and also the speed of the OCBmovement. The velocities were calculated as a mean value from a linear fit to each forward and backward partsof the particle movement. The first and the second column in Table 1 show the velocity of the OCB and thevelocity of captured beads obtained by the fit. For the slower velocity of the OCB the bead can be considered as“Brownian surfer” because its speed is only slightly smaller than the speed of the OCB. For the faster motion ofthe OCB the bead is between “surfing” and “swimming” states because it still moves with the OCB but with alow relative speed. One can see from Fig. 5, that for bigger vOCB a particle does not follow the OCB motion andjumps between neighboring optical traps. Since our detection method enables precise determination of the beadposition with respect to the standing wave nodes and antinodes (even in motion)15 we analyzed this particlepositions especially in motional standing waves. Figures 4b and 4c show histograms of the bead position if theOCB moved. We used (12) to find trap depths ∆U of the motional waves while we assumed constant velocitiesof OCB vOCB. The trap depths are shown in the third column of Table 1.

Small velocities vOCB cause only small deviations from the equilibrium position while bigger velocities of theOCB cause more frequent jumps between neighboring traps accompanied by wider histograms. The decrease

Proc. of SPIE Vol. 6326 63262K-6

0

0.2

0.4

0.6

0.8

1

arb.

uni

ts

vOCB = 0 µm s-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9ar

b. u

nits

vOCB = -47 µm s-1vOCB = 47 µm s-1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-100 -50 0 50 100

arb.

uni

ts

x [nm]

(a)

(b)

(c) vOCB = -98 µm s-1vOCB = 98 µm s-1

Figure 4. Histogram of positions of the bead captured in the standing wave trap. Noisy line shows experimental data ofx coordinate and smooth line shows the fit by the reduced density probability (12). The radius of the polystyrene beadis 260 nm. The OCB is at the rest (top – a), traveling with velocity 47 µms−1 (middle – b) or 98 µms−1 (bottom – c).(b) and (c) show histograms for the OCB moving from right to left with solid lines and histograms for the OCB movingfrom left to right with dashed lines.

Table 1. The experimentally and theoretically obtained values of the bead velocities in the stationary and motional OCBare shown here. The columns 1 and 2 show velocity of the OCB and the bead obtained from Fig. 5. In the column 3there are shown the trap depths obtained by fitting histograms in Fig. 4 by Eq. (12). The columns 4 and 5 shows thebead velocity calculated by Eq. (16) for the trap depth of stationary OCB (column 4) and for the trap depths given incolumn 3 (column 5). For further details see the text.

vOCB [µms−1] vexpbead µms−1 ∆U [kBT ] vteor.,1

bead [µms−1] vteor.,2bead [µms−1]

0 – 9.0 ± 0.3-47 -43 ± 5 7.6 ± 0.3 -45.4 -42.747 44 ± 5 7.7 ± 0.4 45.4 43.0-98 -34 ± 5 5.9 ± 0.4 -67.4 -30.798 36 ± 7 5.8 ± 0.3 67.4 29.6

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-3

-2

-1

0

1

2

3

x [µ

m]

-4

-3

-2

-1

0

1

2

0 0.2 0.4 0.6 0.8 1

x [µ

m]

t [s]

Figure 5. Record of bead positions while moving standing wave in different speeds. The solid noisy lines show positionsof the bead during the movement and the dotted sawtooth curves show the displacement of the original bead positionat t = 0 corresponding to the motion of the standing wave (OCB). The top plot corresponds to slower OCB motion (47µms−1) and the lower plot to 98 µms−1.

of the trap depth with increasing velocity of the OCB is unexpected because the trap depth does not changeif the OCB moved. But the theoretical model does not take into account the presence of the boundary as wellas the exponential decay of electric field and potential above the prism. These effects lead to the change of thebehavior of the bead in the trap and demonstrate themselves as a change of the histogram shape and thereforeas a change of the trap depth.

Further, we used Eq. (16) to calculate velocities of the bead in the motional OCB. The values are shown inthe fourth column of Table 1. These values were calculated for velocities of OCB shown in the first column inthis Table and for the trap depth of the stationary OCB, i. e. ∆U = 9 kBT . We obtained values that are slightlybigger than the experimental values for slowly moving OCB but for the OCB moving fast the values are verydifferent. We suppose this is due the presence of the prism surface as discussed above. Therefore we used thevalues of the trap depths obtained by the histogram fitting (in the column 3 of Table 1). The bead velocitiesobtained by this way are shown in the last column of Table 1 and they coincide better with the experimentalvalues.

4. CONCLUSIONS

The random motion of a bead in the traveling periodic potential was studied in this work. We used the onedimensional Fokker-Planck equation to predict the mean velocity of a particle in this motional potential. TheMonte Carlo simulations were used to confirm the theoretical results. It was shown that one can use the reduceddensity probability P to find the parameters of the motional system. The P is used in a similar way to aBoltzmann distribution in the stationary systems.1

An array of the optical traps in the optical conveyor belt (OCB) can be used for delivery of sub-micrometersized beads. Experimentally, the OCB was obtained by an interference of two counterpropagating evanescentwaves. The obtained OCB potencial landscape was periodic and we moved it with different velocities. Therefore

Proc. of SPIE Vol. 6326 63262K-8

the above described theoretical results can be applied to the Brownian motion of a bead in the OCB (eitherstationary or moving). In this work we studied stationary OCB and OCB moving with two different velocities.For the lower velocity the confined bead behaved like a “Brownian swimmer” while for faster velocity the beadwas between “surfing” and “swimming” states because it moved with the OCB but with a low relative speed.

We compared experimental values of the bead mean velocity to the theoretical predictions. It was found thatthe presence of the surface influences the particle behavior. The exponential decay of the evanescent field wasnot taken into account. In the further studies more velocities of the OCB will be studied and the influence ofthe prism surface will be considered in greater details.

ACKNOWLEDGMENTS

This work was partially supported by the Institutional Research Plan of the Institute of Scientific Instruments(AV0Z20650511), Grant Agency of the Academy of Sciences (IAA1065203), European Commission via 6FPNEST ADVENTURE Activity (ATOM3D, project No. 508952), and Ministry of Education, Youth, and Sportsof the Czech Republic (project No. LC06007).

REFERENCES1. E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by

thermal noise analysis.,” Appl. Phys. A 66, pp. 75–78, 1998.2. I. M. Tolic-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “Matlab program for precision calibration of

optical tweezers,” Comp. Phys. Commun. 159, pp. 225–240, 2004.3. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75,

pp. 594–612, 2004.4. L. Gamaitoni, P. Hanggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, pp. 223–

287, 1998.5. T. Wellens, V. Shatokhin, and A. Buchleitner, “Stochastic resonance,” Rep. Prog. Phys. 67, pp. 45–105,

2004.6. A. Simon and A. Libchaber, “Escape and synchronization of a Brownian particle,” Phys. Rev. Lett. 68,

pp. 3375–3378, 1992.7. D. Babic, C. Schmitt, I. Poberaj, and C. Bechinger, “Stochastic resonance in colloidal systems,” Europhys.

Lett. 67, pp. 158–164, 2004.8. R. P. Feynmann, R. B. Leighton, and M. Sands, Feynmann Lectures on Physics, vol. 1, Addison-Wesley,

1977.9. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Physics Reports 361, pp. 57–265,

2002.10. P. Zemanek, A. Jonas, L. Sramek, and M. Liska, “Optical trapping of nanoparticles and microparticles using

Gaussian standing wave.,” Opt. Lett. 24, pp. 1448–1450, 1999.11. P. Zemanek, A. Jonas, P. Jakl, M. Sery, J. Jezek, and M. Liska, “Theoretical comparison of optical traps

created by standing wave and single beam,” Opt. Commun. 220, pp. 401–412, 2003.12. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron

objects,” Appl. Phys. Lett. 86, pp. 174101–1–174101–3, 2005.13. T. Cizmar, V. Kollarova, Z. Bouchal, and P. Zemanek, “Sub-micron particle organization by self-imaging

of non-diffracting beams,” New. J. Phys. 8, p. 43, 2006.14. M. Siler, T. Cizmar, M. Sery, and P. Zemanek, “Optical forces generated by evanescent standing waves and

their usage for sub-micron particle delivery,” Appl. Phys. B 84, pp. 157–165, 2006.15. T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and

detection of sub-micron objects in a motional standing wave,” Phys. Rev. B , to be published.16. H. Risken, The Fokker-Planck Equation, Springer-Verlag, Berlin, 1996.17. C. W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Berlin, 2004.18. M. Gu and P. C. Ke, “Effect of depolarization of scattered evanescent waves on particle-trapped near-field

scanning optical microscopy,” Appl. Phys. Lett 75, pp. 175–177, 1999.

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