PHYSICAL REVIEW E 93, 022137 (2016)

Nonconservative dynamics of optically trapped high-aspect-ratio nanowires

Wen Jun Toe,1 Ignacio Ortega-Piwonka,2 Christopher N. Angstmann,2 Qiang Gao,3 Hark Hoe Tan,3

Chennupati Jagadish,3 Bruce I. Henry,2 and Peter J. Reece1,*1School of Physics, The University of New South Wales, Sydney NSW 2052, Australia

2School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia3Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia

(Received 13 August 2015; published 24 February 2016)

We investigate the dynamics of high-aspect-ratio nanowires trapped axially in a single gradient force opticaltweezers. A power spectrum analysis of the dynamics reveals a broad spectral resonance of the order of kHz withpeak properties that are strongly dependent on the input trapping power. A dynamical model incorporating linearrestoring optical forces, a nonconservative asymmetric coupling between translational and rotational degreesof freedom, viscous drag, and white noise provides an excellent fit to experimental observations. A persistentlow-frequency cyclical motion around the equilibrium trapping position, with a frequency distinct from thespectral resonance, is observed from the time series data.

DOI: 10.1103/PhysRevE.93.022137

I. INTRODUCTION

Optical trapping studies of low-symmetry objects, such asrods, ellipsoids, and oblate spheroids, provide new dimensionsto the study of optical trapping dynamics. In addition to morecomplex hydrodynamic interactions, nonspherical objects aresubject to preferential trapping orientations and associatedrestoring torques when displaced away from equilibrium.When the trapping point is positioned arbitrarily with respectto the geometric center, perturbative forces can lead to complexcross-coupling between translational and rotational motion [1].Shape mediated effects have been discussed in the contextof tailored optical force profiles, including the generationof optical lift [2] and negative radiation pressure [3], andhave consequently been implemented in a practical settingfor passive force clamping in photonic force microscopy [4].

One interesting peculiarity that has seen growing attentionis the tendency of low-symmetry particles to undergo persistentoscillatory motion in thermal equilibrium. This type of motionwas first articulated for optically trapped rods by Simpson andHanna, where it was related to the influence of nonconservativeoptical scattering forces, inducing coupling between rotationand translation [5]. The concept has its genesis in an earlierstudy by Roichman et al. who explored the influence ofthe nonconservative scattering force in producing circulatingcurrents in the stochastic trajectories of spherical particles inan optical tweezers [6]. These concepts have been extendedby Saberi and Gittes who consider general nonconservativeforcing effects in optical tweezers [7].

Our interests lie in experimental investigations of weaklytapered, high refractive index, indium phosphide (InP)nanowires of uniform composition, held in a linearly polarized,gradient force optical tweezers [8]. Nanowires are a particularsubclass of nonspherical objects with nanoscale radial dimen-sions and lengths that extend to several micrometers, givingaspect ratios in excess of 100:1. For trapping experiments, theextended axial dimension can exceed the depth of focus of thetrapping beams by several times the Rayleigh range, thereby

*p.reece@unsw.edu.au

creating a trapping force that is localized to a particular pointon the axis. Earlier studies have shown that the equilibriumtrapping position of these nanowires has been determined tobe near one end, with the major segment of the length sittingbelow the trap far from the geometric center [9], which isqualitatively consistent with modeling of tapered rods [10].The optically trapped nanowires are aligned closely to theoptical axis of the tweezers and experience small perturbationsfrom the equilibrium point due to stochastic motion. Notethat these trapping conditions exclude the possibility of majororientation changes that are observed in other studies [11,12].

Here we present experimental evidence for the presenceof resonant fluctuations in the stochastic motion of opticallytrapped nanowires. A power spectral density analysis of thenanowire motion reveals a broad resonance peak of the ofthe order of kHz with characteristics dependent on the inputtrapping power and the trapping height above the coverslip. Adynamical model is constructed with the following features:linear restoring optical forces acting at an equilibrium trappingpoint on the rod; an additional nonconservative force providingasymmetric coupling between translational and rotationaldegrees of freedom; a viscous drag applied at a point on therod; and white noise accounting for the random fluctuations ofthe fluid. This model provides an excellent fit to experimentalobservations of the power spectra. Further, a winding numberanalysis of the positional time series reveals clear cyclicalmotion around the trapping point.

II. METHOD

Our experimental apparatus consists of a single beamgradient force optical trap generated at the focus of a highnumerical aperture oil immersion microscope objective lens(Nikon E Plan Achromat 100x 1.25NA) using a Nd:YAG laseremitting at 1064 nm (Laser Quantum IR Ventus). Dynamicsteering of the laser focus is provided by a two-axis acousto-optic deflector (AOD) (Gooch and Housego 45035 AOBD)and spatial light modulator (SLM) (Hamamatsu LCOS-SLMx10468-03) positioned within the optical train at planesconjugate with the back aperture of the objective. SLM-basedaberration correction and overfilling of the back aperture are

2470-0045/2016/93(2)/022137(7) 022137-1 ©2016 American Physical Society

http://dx.doi.org/10.1103/PhysRevE.93.022137

WEN JUN TOE et al. PHYSICAL REVIEW E 93, 022137 (2016)

x

zθyZ

2D-AOD

1064 nmLASER

λ/2

λ/2 CCD

OBJ

COND

PSD RELAY

SAMPLE

SLM

LAMP

y

FIG. 1. Schematic of our experimental setup and geometrywithin the optical trap with relevant coordinates and angles. In theexperimental arrangement a diode pumped solid state laser is usedwith adaptive optics to create a diffraction limited spot within thefocal plane of the objective (OBJ). The nanowires are trapped nearthe end of the wire and aligned along the laser axis. Light scatteringfrom the nanowire is collected by the condenser (COND) and usedto perform back focal plane interferometry with a position sensitivedetector (PSD).

used to produce a diffraction limited spot optimized for opticaltrapping. The power of the trapping laser at the imaging planeis controlled by the AOD and set between 10 and 90 mW. Aschematic diagram of the trapping arrangement is shown inFig. 1.

The indium phosphide (InP) nanowires used in the study areepitaxially grown on InP (111)B substrates by metal-organicchemical vapor deposition (MOCVD) using gold nanoparticlecatalysts. The nanowires are between 5 and 10 μm havea tapered aspect of approximately 0.3◦ as characterizedby electron microscopy. Further information on the growthprocess can be found elsewhere [13].

For tweezers measurements, the nanowires are dispersedinto an aqueous solution by sonication and transferred toa hermetically sealed chamber at a dilution where singlenanowires can be isolated. Visualizing the nanowires in thetrap is achieved using a charge-coupled device (CCD) camera(AVT Stingray) together with diascopic illumination suppliedby the condenser. Back focal plane interferometry is performedby collecting the forward scattered light from the trappednanowires with a high numerical aperture condenser lens(Olympus S Plan Fluor 40x 0.6NA) and projected onto aposition sensitive diode (PSD) (Pacific Sensor, DL16-7PCBA)located at a conjugate plane to the condenser back aperture.The PSD is connected to a field-programmable gate array(FPGA) multifunction acquisition card (National InstrumentsPCIe-7852R) and time series measurements of the nanowireposition fluctuation with respect to the trapping center are

recorded at a sampling rate of 10 kHz for a period of 10 s.The linearity of the PSD detector response with respect tothe nanowire position in this orientation has been previouslyverified [8].

III. THEORY

We treat the nanowire as a rigid body in the shape of aconical frustum. The stochastic motion of the nanowire withinan optical trap can be described by a set of Langevin equationsin terms of translational (x,y,z) and angular coordinates(θx,θy,θz). These co-ordinates are shown schematically inFig. 2.

We assume that the optical tweezers give rise to linearrestoring forces that operate through the point (x,y,z). We referto this point as the trapping point on the nanowire. We alsoassume that the optical tweezers give rise to linear restoringtorques about the trapping point in the angles (θx,θy). Thetrapping point is typically near the large radius end of thetapered nanowire, above the position of the center of mass [9].We also consider the possibility of a nonconservative linearforce in the x and y directions proportional to θy , and θx ,respectively. The total linearized optical force at the trappingpoint is given by

FO =

⎛⎜⎝

Fx

Fy

Fz

⎞⎟⎠ = −

⎛⎜⎝

kxx + cyθykyy + cxθx

kzz

⎞⎟⎠, (1)

where kx,ky,kz are positive but there is no restriction on thesign of cx or cy . The restoring torque about the trapping pointis taken to be

TO =

⎛⎜⎝TθxTθyTθz

⎞⎟⎠ = −

⎛⎜⎝

kθx θx

kθy θy

0

⎞⎟⎠. (2)

The motion of the nanowire is opposed by a viscousdrag that depends on nanowire’s shape and surface area. Weassume that there is a center of drag, also called a centerof hydrodynamics stress [14], and the viscous drag operatesthrough this point. We further assume that the center ofdrag is distinct from the trapping point. This is a reasonable

FIG. 2. Schematic illustration of the coordinate system used inthe model for the dynamics of the nanowire.

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NONCONSERVATIVE DYNAMICS OF OPTICALLY TRAPPED . . . PHYSICAL REVIEW E 93, 022137 (2016)

assumption given that the trapping point is typically above thecenter of mass, and the surface area is balanced at a pointbelow the center of mass for a conical frustum. With the aboveassumptions, neglecting inertial effects in the overdampedlimit, the optical forces acting through the trapping positionwould induce a torque about the center of drag. We label thecenter of drag by coordinates (u,v,w) and in the small angleapproximation we have

x = u + �θy,y = v − �θx, (3)z = w + �,

where � is the distance between the center of drag and thetrapping position. The induced torque is given by

TD =

⎛⎜⎝TuTvTw

⎞⎟⎠ = r × FO,

where r = (x − u,y − v,z − w)T . Explicitly, retaining termsthat are linear in the coordinates x,y,z,θx,θy , and then usingthe transformations in Eq. (3) we have

TD =

⎛⎜⎝

�kyy + �cxθx−�kxx − �cyθy

0

⎞⎟⎠ =

⎛⎜⎝

�kyv − �2kyθx + �cxθx−�kxu − �2kxθy − �cyθy

0

⎞⎟⎠.

(4)

Using the transformations in Eq. (3) we can express the opticalforce in the center of drag coordinates as

FO =

⎛⎜⎝

Fu

Fv

Fw

⎞⎟⎠ = −

⎛⎜⎝

kxu + kx�θy + cyθykyv − ky�θx + cxθx

kzw + kz�

⎞⎟⎠. (5)

We can combine the above optical forces and torques in asingle vector as follows:

GO =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

Gu

Gθy

Gv

Gθx

Gw

Gθz

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

Fu

τv + τθyFv

τu + τθxFw

τw + τθz

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (6)

Given that Gθz is zero we ignore motion in the θz coordinatein the following.

Finally we include the drag proportional to the velocity ofthe center of drag and we include independent white noiseterms in each of the coordinates with zero mean, and variancegiven by the Einstein relation for kinetic theory. This resultsin the dynamical equation

−� �̇δq + �ζ (t) − K �δq = �0, (7)where δ�q = (δu,δθy,δv,δθx,δw)T is a generalized coordinatevector, representing small displacements around the equilib-rium values u∗ = 0, θ∗y = 0, v∗ = 0, θ∗x = 0, and w∗ = −�.The drag matrix is assumed to be diagonal in the center of

drag coordinates, so that

� =

⎛⎜⎜⎜⎜⎜⎝

γu 0 0 0 0

0 γθy 0 0 0

0 0 γv 0 0

0 0 0 γθx 0

0 0 0 0 γw

⎞⎟⎟⎟⎟⎟⎠. (8)

The white noise vector �ζ (t) = (ζu,ζθy ,ζv,ζθx ,ζw)T has theexpected value 〈ζi(t)〉 = 0 and correlations 〈ζi(t)ζj (t +τ )〉 = 2kbT γiδij δ(τ ). The matrix K is the optical stiff-ness matrix defined by −Kδ�q = GO with GO evaluatedat u = u∗ + δu, θy = θ∗y + δθy, v = v∗ + δv, θx = θ∗x + δθx ,and w = w∗ + δw. Substituting Eqs. (2), (4), and (5) intoEq. (6) evaluated at �q = �q∗ + δ�q, we have

K =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

kx �kx + cy 0 0 0�kx �

2kx + �cy + kθy 0 0 00 0 ky −�ky + cx 00 0 −�ky �2ky − �cx + kθx 00 0 0 0 kz

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(9)

The experimental observations on the motion of thenanowire are the position co-ordinates, x̂,ŷ,ẑ near the up-permost tip, a distance �̂ above the center of drag. We let δ̂�q =(δ̂x,δθy,δ̂y,δθx,δ̂z)

Tdenote the generalized coordinates of the

observation point. The dynamical equation in the observedcoordinates can be found using the linear transformationδ̂�q = Jδ�q with

J =

⎛⎜⎜⎜⎜⎜⎜⎝

1 �̂ 0 0 0

0 1 0 0 0

0 0 1 −�̂ 00 0 0 1 0

0 0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎠

. (10)

Substituting the linear transformation into Eq. (7) we obtainthe equations of motion

dδ̂�qdt

= −Mδ̂�q + �ξ (t), (11)

where

M = J�−1KJ−1 (12)and

�ξ (t) = J�−1�ζ (t). (13)It is easy to verify that M has the block structure

M =

⎛⎜⎜⎜⎜⎜⎝

Mx,x Mx,θy 0 0 0

Mθy,x Mθy,θy 0 0 0

0 0 My,y My,θx 0

0 0 Mθx,y Mθx,θx 0

0 0 0 0 Mz,z

⎞⎟⎟⎟⎟⎟⎠, (14)

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WEN JUN TOE et al. PHYSICAL REVIEW E 93, 022137 (2016)

where

Mx,x = kxγu

+ �̂Mθy,x,

Mx,θy =−�̂kx + kx� + cy

γu+ �̂Mθy,θy ,

Mθy,x =�kx

γθy,

Mθy,θy =−�̂kx� + kx�2 + cy� + kθy

γθy,

My,y = kyγv

− �̂Mθx,y,

My,θx =�̂ky − ky� + cx

γv− �̂Mθx,θx

Mθx,y = −�ky

γθx,

Mθx,θx =−�̂ky� + ky�2 − cx� + kθx

γθx,

Mz,z = kzγw

.

The above approximations thus lead to a description ofthe dynamics in three sets of coupled equations of motionthat describe the dynamics over the XZ plane (x,θy), theYZ plane (y,θx), and the z axis. These subsystems retain thenonconservative effects and can be treated independently. Forexample, the two equations that describe the dynamics in theYZ plane can be written explicitly as

dδ̂y

dt= −My,y δ̂y − My,θx δθx +

1

γvζv(t) − �̂

γθxζθx (t), (15)

dδθx

dt= −Mθx,y δ̂y − Mθx,θx δθx +

1

γθxζθx (t). (16)

This system can be written in vector form as

d �ψdt

= A �ψ + L�ζ , (17)

where

�ψ =(

δ̂y

δθx

), (18)

A =(−My,y −My,θx

−Mθx,y −Mθx,θx

), (19)

and

L =(

1γv

− �̂γθx

0 1γθx

). (20)

The power spectrum for this system can be written as [15]

S(ω) = (A − iωI)−1LQLT (A + iωI)−T , (21)

where I is the 2 × 2 identity matrix and

Q =(

ηv 0

0 ηθx

), (22)

with ηi = 2kBT γi .This leads to a power spectral density for each coupled

coordinate that has the form

Si(ω) = αiω2 + βi

(ω2 − �)2 + (μω)2 , (23)

where

μ = −ky�2

γθx+ cx�

γθx− ky

γv− kθx

γθx(24)

is the trace of the matrix A,

� = kykθxγθx γv

(25)

is the determinant of the matrix A, and αi and βi are positiveparameters arising from the white noise at the center of drag.For example

αy = �̂2ηθx

γ 2θx

+ ηvγ 2v

, (26)

and

βy = [kθx − (cx + �̂ky)� + ky�2]2ηv + [cx + ky(�̂ − �)]2ηθx

γ 2v γ2θx

.

(27)We note that the equilibrium solution of Eq. (17) is unstableif μ > 0 and this places a constraint on the size of thenonconservative coupling in this model.

To allow for detector noise in the experimental measure-ments of the power spectrum we carry out fits below based onthe functional form

Si(ω) = αiω2 + βi

(ω2 − �)2 + (μω)2 + σi, (28)

This functional form has a maximum at a nonzero frequency

ω∗ =

√√√√√(� + βi

αi

)2− βi

αiμ2 − βi

αi(29)

provided that

� >2βiαi

(√1 + αi

βiμ2 − 1

). (30)

If � � |μ| then

ω∗ ∼ √� − 14

(βi

αi

)μ2

(1

�

)3/2. (31)

IV. RESULTS

In Fig. 3 we present experimental measurements of thepower spectrum from a typical nanowire trapped at differenttrapping heights above the chamber. The figure also shows fitsbased on Eq. (28) with parameters as shown in Table I.

The physical dimensions of the nanowire are approximately50 nm by 5 μm in length, Fig. 3(c) inset. The power spectrum

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NONCONSERVATIVE DYNAMICS OF OPTICALLY TRAPPED . . . PHYSICAL REVIEW E 93, 022137 (2016)

Frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (

arb.

uni

ts)

0 0.002 0.004 0.006 0.008 0.01

0

10

20

Time (s)

Aut

ocor

rela

tion

(arb

. uni

ts)

5μm

0 1000 2000 3000 4000 50000

1

2

3(b)

0 1000 2000 3000 4000 50000

1

2

3(a)

(c)

FIG. 3. (a) Experimental power spectrum of a nanowire trappedapproximately 10 μm from the cover slip (blue dots). The dashed(red) line is a Lorentzian fit which is characteristic of Browniandynamics in an optical trap; and the solid (black) is the presentedmodel. (b) Experimental power spectrum of the same nanowiretrapped approximately 50 μm from the cover slip (red dots). Thesolid (black) line is the fitted spectrum from the presented model.(c) Autocorrelation function of the nanowire motion approximately10 μm from the cover slip (dashed) and approximately 50 μm fromthe cover slip (solid). The inset is a bright field image of the untrappednanowire.

in Fig. 3(a) was recorded when the nanowire was trappedat a height of approximately 10 μm and is plotted togetherwith two fits; a Lorentzian (dashed), which is consistent withuncoupled dynamics [16], and our model power spectrum,Eq. (28). We do not observe a second Lorentzian componentin the power spectrum corresponding to the rotation dynamics,however from previous work analyzing autocorrelation datait is noted that the characteristic frequency for rotation issomewhat smaller compared with the translation motion [17].When the trapping height above the cover slip is increasedto approximately 50 μm a broad resonant peak is apparentin the power spectrum (red), shown in Fig. 3(b). The fit ofthe power spectrum using Eq. (28) shows excellent agreementwith the experimental data. The resonance is also observed inthe autocorrelation function of the nanowire trajectory, wherea fluctuation between a positive and negative correlation can beseen, as shown in Fig. 3(c). A fit of the data taken at the lowertrapping height Fig. 3(a) also reveals a strongly suppressedpeak at a similar frequency. The full height of the chamber isestimated to be in excess of 100 μm and so we exclude the

TABLE I. Fitting parameters for power spectrum at differenttrapping heights.

Height (μm) � (Hz2) |μ|(Hz) β/α α σ∼10 1.76 × 105 843 7.06 × 104 4.08 × 105 0.048∼50 1.53 × 105 350 2.43 × 105 8.94 × 104 0.044

possibility of direct interaction with the chamber walls. Wenote that changing the height of the trap will induce sphericalaberrations and accentuate other aberrations already presentwithin the trap. Increased aberrations will change the relativecontributions from gradient and scattering forces for a givenfixed input power, and this may be implicated in the heightdependence of the resonance. Given that the theoretical modelprovides a good fit to the power spectrum at both trappingheights it is reasonable to assume that the height dependencemay be incorporated into the model through different valuesof system parameters. The lack of an explicit dependence inthe model prohibits predictions of the resonant peak in thepower spectrum with the variation in trapping height so thatthis remains an empirical observation.

From the experimental measurements and theoretical fitswe can make the following observations: (i) The model reducesto a Lorentzian when cx = 0,cy = 0,� = 0, i.e., there is nononconservative optical force and there is no induced torquefrom the drag, independent of the experimental observationpoint �̂. (ii) A nonzero peak can occur when there is anonconservative optical force and induced drag provided thatEq. (30) is satisfied. (iii) The increase in the position of theresonance peak ω∗ is correlated with a decrease in μ2, Eq. (29).This may correspond to a decrease in the restoring forces andtorques, or a decrease in the distance between the center of dragand the trapping position, Eq. (24). Both of these possibilitiesare consistent with increased optical aberrations with depth.(iv) The height of the peak increases with increasing α.

To further investigate the dynamics of the resonance, thepower spectrum of a trapped nanowire of length 4.6 μm was

101 102 103

10−3

10−2

10−1

100

Frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (

arb.

uni

ts)

10.4 mW24.0 mW43.0 mW54.9 mW62.3 mW69.1 mW77.6 mW84.5 mW

IncreasingPower

FIG. 4. Power spectral density of a single nanowires held at afixed trapping power measured for different trapping powers. Thepeak frequency shifts given by the

√� increases linearly with trapping

power, while the Brownian fluctuations are strongly suppressed. Thesolid lines represent the fitted model which includes nonconservativecross-coupling between translation and rotation. The arrow indicateshow the power spectral density changes with increasing trappingpower.

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WEN JUN TOE et al. PHYSICAL REVIEW E 93, 022137 (2016)

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

x 106

Trapping Power (mW)

ε (H

z2)

0 10 20 30 40 50 60 70 80 90 1000

5k

10k

15k

Trapping Power (mW)

α

0

1

2

3x 10

6

β/α

0

0.5k

1.0k

1.5k

2k

|μ| (H

z)

β/αα

|μ|ε

(a)

(b)

FIG. 5. Fitting parameters of the power spectral density datapresented in Fig 4. (a) � increase quadratically and μ linearlywith trapping power, which corresponds to linear increase in boththe peak frequency and width. (b) The noise strength α showsno clear dependence on trapping power indicating a constanttemperature, while β/α increases quadratically with trapping power.All dependencies are consistent with a trap stiffness that is linearlyproportional to the trapping intensity in the presence of asymmetriccoupling.

recorded as a function of input trapping laser powers rangingfrom 10 to 90 mW. The power dependent results are presentedin Fig. 4 along with the associated fits using Eq. (28); againwe note exceptional agreements between experimental resultsand the model. Qualitatively we see that the peak positionof the resonance is blueshifted from 200 to 1800 Hz withincreasing trapping power together with a reduction of theamplitude of the fluctuations. The parameters associated withthe fits are given in Fig. 5. We observe that � increasesquadratically with trapping power, while |μ| increases linearly.This is consistent with the model equations, Eqs. (25) and (24),under the assumption that the trap stiffness parameters kiscale linearly with power. The parameter α is observed to beroughly constant across all trapping powers, which indicatesthat there is no obvious heating across the different trappingpowers. Note too that α is independent of the trap stiffness inEq. (26). Finally, we note that while the trapping power dictatesthe relative strengths of the trap stiffness (conservative) andradiation pressure (nonconservative), it should not modify theratio of the two competing forces under conditions where theharmonic approximation for the trap stiffness is valid.

The fitting parameters are shown in Table II.In addition to the characteristic power spectrum described

above, another important property observed in this system isits tendency to produce a continuous cycling of the positionand orientation under steady state conditions. This cyclicmotion was detected using a winding number analysis in whichwinding events were denoted when the nanowire trajectorymoved around the center of the trap in either a clockwise(negative) or anticlockwise (positive) motion. In an unbiasedsystem the rates of cycling in either direction should be

TABLE II. Fitting parameters for power spectrum at differenttrapping powers.

P (mW) � (Hz2) |μ|(Hz) β/α α σ10.4 3.819 × 104 213 2.035 × 104 1.379 × 104 0.0001624.0 3.155 × 105 552 3.000 × 105 1.179 × 104 0.000743.0 8.467 × 105 782 5.193 × 105 1.915 × 105 0.00954.9 1.499 × 106 969 8.977 × 105 4.816 × 105 0.0362.3 1.871 × 106 1111 1.094 × 106 5.76 × 105 0.0369.1 2.516 × 106 1375 1.535 × 106 7.798 × 105 0.04577.6 3.143 × 106 1518 1.877 × 106 5.593 × 105 0.04484.5 3.935 × 106 1841 2.209 × 106 1.31 × 106 0.096

balanced, leading to an average of zero winding. The resultspresented for different powers in Fig. 6(a) indicate that thenanowires have a tendency to persistently cycle around acentral point within the trap in a particular sense; shorttime-scale fluctuations correspond to Brownian fluctuationsin the trend. The winding behavior is persistent over manyseconds and is reproducible for each individually trapped wire.The cycling in a particular sense suggests an underlying biaswithin the trapping geometry, however cycling in the oppositesense is also observed for some trapping events. We findthat for different powers the rate of winding increases beforeplateauing at higher powers; this is clearly observed in a plotof the gradient of the winding number with trapping power,shown in Fig. 6(b). The characteristic frequency of the windingis much less than the frequency of the broad resonance peakand it is not obviously detectable in the power spectra in Fig. 4.

0 1 2 3 4 5 6−2000

−1500

−1000

−500

0

500

Time (s)

Win

ding

Num

ber

(tur

ns)

10.4 mW24.0 mW43.0 mW54.9 mW62.3 mW

0 10 20 30 40 50 60 70 80 90

−300

−200

−100

0

Trapping Power (mW)Win

ding

Rat

e (t

urns

/s)

(a)

(b)

FIG. 6. (a) A winding number plot of the nanowire motion fordifferent trapping powers. Small time-scale fluctuations in the wind-ing decorate a clear linear increase with time, indicating persistentcycling in one direction. (b) The rate of winding, given by the gradientof the winding plots, is observed to increase approximately trappingpower for lower powers and plateau at higher powers.

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NONCONSERVATIVE DYNAMICS OF OPTICALLY TRAPPED . . . PHYSICAL REVIEW E 93, 022137 (2016)

V. SUMMARY

In this work we have described experimental results andtheoretical modeling on the dynamics of high-aspect-rationanowires trapped axially in a single gradient force opticaltweezers. Experimental measurements of the power spectraldensity reveal a broad peak of the order of kHz. The positionof this peak is blueshifted with increasing input trappingpower. A separate analysis of time series data reveals apersistent low frequency cyclical motion, less than one kHz,around the equilibrium trapping position. The frequency of thismotion is not readily identifiable in the power spectral density.We introduced a theoretical model for the motion of thenanowire, which incorporates linear restoring optical forces,a nonconservative asymmetric coupling between translationaland rotational degrees of freedom, viscous drag, and white

noise. This model provides an excellent fit to the experimentalpower spectral density observations. Our experimental andtheoretical results provide clear evidence for either a noncon-servative coupling between translational and rotational modesin optical trapped nanowires, or an induced torque arising fromlow Reynolds viscous drag, or a combination of both. Furtherexperimental and theoretical studies on this model systemmay be able to elucidate further on these and other complexdynamic behaviours, and this may provide new insights intoanalogous biophysical systems.

ACKNOWLEDGMENTS

The Australian National Fabrication Facility (ANFF) andAustralian Research Council (ARC) are acknowledged forsupporting the nanowire growth facilities utilized in this work.

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