NATURE NANOTECHNOLOGY | VOL 8 | NOVEMBER 2013 | www.nature.com/naturenanotechnology 807
L
ight can exert a force on matter by means of momentum
exchange on scattering
1
. e existence of this force was rst
experimentally demonstrated by Lebedev
2
and Nichols and
Hull
3
in 1901 using thermal light sources (electric or arc lamps) and
a torsion balance. When the light was focused on a mirror attached
to the balance, the radiation pressure moved the balance from its
equilibrium position
2,3
. But the magnitude of these eects was con-
sidered insignicant for any practical use: to quote J.H. Poynting’s
presidential address to the British Physical Society in 1905 (reported
in ref.4), “A very short experience in attempting to measure these
forces is sucient to make one realize their extreme minuteness —
a minuteness which appears to put them beyond consideration in
terrestrial aairs.” It was not until 1970, and because of the advent
of lasers, that Arthur Ashkin showed that the use of optical forces to
alter the motion of micrometre-sized particles
5
and neutral atoms
6
could have applications in the manipulation of microscopic parti-
cles and of single atoms
4
.
ese pioneering works have developed into two very successful
research lines. On one hand, early techniques for laser cooling of
atoms
7–10
paved the way to modern ultracold atom technology
11
. On
the other hand, what is now commonly referred to as optical twee-
zers — that is, a tightly focused laser beam capable of conning par-
ticles in three dimensions
12
— has become a common tool for the
manipulation of micrometre-sized particles
13,14
and as a highly sen-
sitive force transducer
15
. But optical forces acting between ~1and
100nm, a range of primary interest for nanotechnology (Fig. 1),
have not been widely exploited because of the challenges in scal-
ing up the techniques optimized for atom cooling, or scaling down
those used for microparticle trapping. Indeed, ecient laser cooling
of atoms relies on light scattering close to a narrow spectral line,
without radiative losses, to reduce the atomic velocity distribution
11
.
Nanostructures lack these features, limiting both the cooling rate
and the minimum achievable temperature
11
. e techniques used
for manipulating microparticles rely on the electric dipole interac-
tion energy
16,17
. Because this scales down approximately with the
particle volume, thermal uctuations are large enough to over-
whelm the trapping forces at the nanoscale
18
.
New approaches were thus developed to stably trap and manip-
ulate nanoparticles. Over the past few years, these techniques
Optical trapping and manipulation of nanostructures
Onofrio M. Maragò
1
*, Philip H. Jones
2
, Pietro G. Gucciardi
1
, Giovanni Volpe
3
and Andrea C. Ferrari
4
*
Optical trapping and manipulation of micrometre-sized particles was first reported in 1970. Since then, it has been successfully
implemented in two size ranges: the subnanometre scale, where light–matter mechanical coupling enables cooling of atoms,
ions and molecules, and the micrometre scale, where the momentum transfer resulting from light scattering allows manipula-
tion of microscopic objects such as cells. But it has been dicult to apply these techniques to the intermediate— nanoscale—
range that includes structures such as quantum dots, nanowires, nanotubes, graphene and two-dimensional crystals, all of
crucial importance for nanomaterials-based applications. Recently, however, several new approaches have been developed and
demonstrated for trapping plasmonic nanoparticles, semiconductor nanowires and carbon nanostructures. Here we review the
state-of-the-art in optical trapping at the nanoscale, with an emphasis on some of the most promising advances, such as con-
trolled manipulation and assembly of individual and multiple nanostructures, force measurement with femtonewton resolution,
and biosensors.
have been successfully applied to a variety of objects, for exam-
ple metal nanoparticles (MNPs)
19–21
, plasmonic nanoparticles
(NPs)
22–30
, quantum dots
31,32
, carbon nanotubes (CNTs)
33–37
, gra-
phene akes
38,39
, nanodiamonds
40
, polymer nanobres
41
and semi-
conductor nanowires
42–49
. Typically these techniques rely either
on special properties of the trapped objects themselves — for
example force enhancement related to plasmonic resonances sup-
ported by the trapped particles
22–30
, or highly anisotropic geom-
etries, such as in CNTs and nanowires
35,38,43,44,46,47
— or on new
approaches to optical manipulation, such as exploiting the eld
enhancement due to plasmons supported by nanostructures on a
substrate
50–57
, or the feedback on the optical forces of the trapped
object
58
. Optical manipulation has been used to build compos-
ite nanoassemblies
32,42,43,59
. Optical tweezers have been developed
to measure forces with femtonewton resolution, enabling the
study of interactions between nanoobjects
34,59–64
. ey have also
been integrated with spectroscopic techniques, such as Raman
spectroscopy
33,36,38,65–71
and photoluminescence
40,44,45,48,49,72
, pav-
ing the way to the selection and manipulation of NPs aer their
individual characterization
36,38
. Optically levitated nanoparti-
cles have been laser-cooled towards their quantum-mechanical
ground state
73–76
.
Here, we review the state-of-the-art, open questions and future
directions in optical trapping and manipulation of nanostructures,
and show how the development of these techniques can aect nano-
science and nanotechnology.
Optical forces on nanostructures
In this section, we review how optical forces arise. We rst consider
the case of particles much smaller than the trapping wavelength
where one can make use of the Rayleigh approximation. We then
address the case of larger particles, where the full electromagnetic
scattering theory must be employed. We nally discuss plasmon-
enhanced forces and optical binding, particularly relevant for opti-
cal trapping and manipulation of nanostructures.
Forces in the dipole approximation. e optical response of
a nanostructure can be oen modelled as that of a dipole
16
or a
collection of dipoles
17
. e dipolar polarizability determines the
1
CNR-IPCF, Istituto per i Processi Chimico-Fisici, I-98158 Messina, Italy,
2
Department of Physics and Astronomy, University College London, London
WC1E 6BT, UK,
3
Department of Physics, Bilkent University, Cankaya, Ankara 06800, Turkey,
4
Cambridge Graphene Centre, University of Cambridge,
9JJ Thomson Avenue, Cambridge CB3 0FA, UK.
*e-mail: marago@me.cnr.it; acf26@hermes.cam.ac.uk
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PUBLISHED ONLINE: 7 NOVEMBER 2013 | DOI: 10.1038/NNANO.2013.208
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808 NATURE NANOTECHNOLOGY | VOL 8 | NOVEMBER 2013 | www.nature.com/naturenanotechnology
strength of interaction with an optical eld
16
. For a sphere of
radius a and relative permittivity ε, this can be written as
77
:
α
=
α
0
6πε
0
ik
3
α
0
1−
(1)
where α
0
is the point-like particle polarizability given by the
Clausius–Mossotti relation
77
α
0
= 4πε
0
a
3
(ε – 1)/(ε + 2); k is the
eld wavevector; and ε
0
is the vacuum dielectric permittivity. e
denominator in equation (1) acts as a correction to the Clausius–
Mossotti relation to account for the reaction of a nite-sized dipole
to the scattered eld at its own location
77
. e time-averaged force
acting on such a dipole is
16
:
=
F
j = x,y,z
αE
j
∇
E
j
Re
∑
∗
2
1
(2)
where E
j
are the electric eld components. Equation (2) can be
recast into the more explicit form
78
:
=
Re(α)
∇
Intensity gradientRadiation pressure Polarization gradient
2
+FE
Re(E × H )
+
∇
× E × E
2c
σ
4ωi
σcε
0
4
1
∗
∗
(3)
where σ is the extinction cross-section, E the electric eld, H the
magnetic eld, c the speed of light in vacuum, and ω the angular
frequency of the optical eld. e rst term in equation(3)is the
force due to the gradient of the electric eld intensity, which per-
mits three-dimensional connement in optical tweezers
12
as long as
it dominates the second and third terms. e second term, responsi-
ble for the radiation pressure, corresponds to a force in the propaga-
tion direction
5
. e third term is a force arising from the presence of
spatial polarization gradients
78
.
Forces beyond the dipole approximation. When a particle cannot
be approximated as a dipole, for example in the case of CNTs,
nanowires, graphene and other two-dimensional material akes,
the time-averaged radiation force F
rad
on the centre of mass due to
scattering of an electromagnetic eld is equal in magnitude, and
opposite in sign, to the rate of change of momentum of the electro-
magnetic eld itself
79–84
. erefore, F
rad
can be calculated by inte-
grating the optical momentum ux over a closed orientable surface
S surrounding the object
81,83
:
F
rad
= ∫
S
〈T
M
〉·dS (4)
where T
M
is the Maxwell stress tensor, accounting for the interaction
between electromagnetic forces and mechanical momentum
79,80
,
which can be calculated from the scattered elds, and dS is an out-
ward-directed element of surface area. e time-averaged radiation
torque Γ
rad
on the centre of mass can be calculated in an analogous
way as
85
:
Γ
rad
= −∫
S
〈T
M
〉 × r·dS (5)
where r is the position of the element of surface area.
e scattered electromagnetic elds in equations (4) and (5)can
be calculated using Maxwell’s equations. Oen, however, this turns
out to be a cumbersome procedure
79
. Various algorithms have
therefore been developed to handle this
79
. In the transition-matrix
(T-matrix) method
82–89
, the total electromagnetic eld — that is, the
sum of incident and scattered eld outside the particle and the eld
internal to the particle — is calculated by expanding all elds in a
common orthogonal basis set of functions and imposing boundary
conditions on the object surface
82,84,86,87
. Most oen, the T-matrix
method uses vector spherical wavefunctions
79
to take advantage of
the spherical symmetry of the scatterer, for example Au or poly-
mer NPs
84,88
. Because the T-matrix works best with objects highly
Figure 1 | The three size ranges of optical trapping. Objects of dierent sizes can be trapped within three main regimes (from left to right): atom trapping
(a few ångstrÖms to a few nanometres), nanotweezers (a few nanometres to a few hundred nanometres) and optical tweezers (from a fraction of a
micrometre up). The horizontal scale bar shows the average object size and the corresponding light wavelength. NV, nitrogen vacancy. Image of layered
material reproduced from ref. 169, © 2011 NPG.
Atom trapping Optical tweezers
Nanotweezers
nm10
−1
11010
2
10
3
10
4
Visible Infrared
Ultraviolet
Atoms
Molecules
Cells
Fullerenes
Nanowires and
nanotubes Graphene
Plasmonic nanoparticles
Synthetic colloids
Quantum dotsNV centres
Layered
materials
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symmetric in shape and composition, one can treat non-spherical
objects by modelling them as clusters of small spheres
29,41,89
.
Another method to calculate scattered electromagnetic elds
is the discrete dipole approximation (DDA)
90
, also referred to as
the coupled dipole model (CDM)
17
, where the particle is modelled
as a collection of dipoles. e force on each dipole is due to the
incident eld and the elds scattered by all other dipoles (equa-
tion (2)). e force acting on the particle is given by the sum of
the forces acting on each dipole. e torque on the particle can be
calculated in an analogous way. e DDA, although more compu-
tationally intensive than T-matrix
91
, can be directly applied to par-
ticles of any shape and composition. Hybrid methods
92,93
have also
been developed that make use of the T-matrix obtained by point-
matching the near-elds calculated with DDA to get the radiation
force andtorque.
Plasmon-enhanced forces. Two main approaches can be exploited
to use plasmons to enhance optical forces on nanoparticles. e rst,
discussed in this section, is to use the plasmons supported by trapped
MNPs to enhance their mechanical reaction to the elds
22–30
. e
second, covered in the section ‘Plasmonic tweezers’, uses plasmons
supported by nanostructures on a substrate, for example pads
52,53
,
nanoantennas
54,55
and nanoholes
56,58
, to generate enhanced elds, in
which nanoparticles can be more eectively trapped
50
.
e optical gradient forces (the rst term in equation(3)) expe-
rienced by nanoparticles are typically very weak (some femtonew-
tons or less), because the dipolar polarizability given by equation(1)
scales with the third power of the particle size
77
. e volume-scaling
of the maximum trapping force was evaluated explicitly in ref.94 for
polystyrene spheres, showing a decrease of three orders of magni-
tude in the maximum trapping force as the sphere radius decreased
from 100to 10nm. erefore, to conne nanoparticles against the
destabilizing eects of thermal uctuations, a signicantly higher
optical power is required: whereas a micrometre-sized polystyrene
sphere can be stably trapped with a fraction of a milliwatt in a stand-
ard optical tweezers set-up (Fig.2a,b), a 100-nm sphere requires
15mW (ref.12). is implies that for a 10-nm sphere ~1.5W would
be needed. e plasmonic nature
95
of MNPs can enhance the opti-
cal forces, so that stable trapping can be achieved at a much lower
power (~2–3mW; refs 26,29,71). On the one hand, far from plas-
mon resonances the optical response of small (<100-nm) spherical
MNPs is (mainly) the optical response of the free-electron plasma
95
yielding a large near-infrared (NIR) polarizability
95
. On the other
hand, MNPs are resonant systems
95
and their optical properties
(polarizability, cross-sections) are regulated by plasmon resonances
that can be tuned by changing size, shapes or aggregation
95
.
Svoboda and Block
19
compared 36.2-nm Au spheres with 38-nm
polystyrene ones, nding a maximum trapping force nearly seven
times as great for Au spheres, as a result of the (seven times) greater
polarizability at the 1,064-nm trapping wavelength
19
. Both Au nan-
oparticles (AuNPs, diameters 9.5–254nm)
20
and Ag nanoparticles
(AgNPs, diameters 20–275 nm)
21
have been optically trapped in
three dimensions. In both cases a maximum trapping force pro-
portional to the third power of the particle radius was observed for
diameters <100nm, with a crossover to a lower exponent for larger
radii
20,21
. is size-scaling behaviour was interpreted by accounting
for local heating
96,97
of the surroundings, and modelling the MNPs
as enclosed in a small steam bubble
88,29
.
Non-spherical MNPs, including Au nanorods (NRs)
22,24
(that
is, nanocylinders with an aspect ratio <10), Ag nanowires
98
and
aggregates of AuNPs
29
, can sustain plasmon resonances in a broad
spectral region in the visible/NIR. ese play a crucial role in
the enhancement of radiation forces and torques in optical twee-
zers
22–24,26–29
. More specically, elongated plasmonic nanostructures
(such as nanowires and NRs) are usually trapped with their axis par-
allel to the electric eld vector of the trapping laser, and orthogonal
to the propagation axis
22,24,84
. e strength of this aligning torque is
increased by tuning the laser close to the plasmon resonance
22
. is
provides a means to control their orientation by rotating the laser
polarization
26
. Plasmonic nanostructures with lengths from tens of
nanometres to several micrometres were aligned and rotated using
a single beam of linearly polarized NIR light
27
. Dienerowitz et al.
25
drew on elements of atom trapping
11
, changing the sign of the gradi-
ent force by blue-detuning the laser wavelength with respect to the
MNP plasmon resonance. us, particle connement was achieved
in the dark spot of an optical vortex beam. e frequency depend-
ence of the plasmon-enhanced radiation force was also used in a
system of two counterpropagating evanescent waves at dierent
wavelengths to selectively guide MNPs of dierent sizes in opposite
directions
30
. Cylindrical vector beams with radial polarization were
also suggested to trap plasmonic NPs, because for these structured
beams the second term in equation (3) (that is, the radiation pres-
sure that pushes particles out of the trap) is zero on the beam axis
99
.
Such structured beams trapped both dielectric microparticles
100,101
and single-walled nanotubes (SWNTs)
102
. Further analysis of the
optical forces
103
revealed, however, that in this case the polarization
gradient contribution to the optical force (the third term in equa-
tion (3)) can be signicant
103
and may eliminate the advantage of
such structured beams.
Resonant illumination of plasmonic NPs gives rise to strong
heating eects because of light absorption
104
. Temperature increases
of hundreds of kelvin were observed by trapping AuNPs adjacent
to uorophore-containing lipid vesicles with permeability sensitive
to temperature
97
. When heated above the gel-transition tempera-
ture, uorophores diused out of the vesicle
97
. Further experiments
made use of the diering longitudinal and transverse plasmon reso-
nances of AuNRs to control the local heating through the orienta-
tion of the AuNRs with respect to the electric eld vector of the
trapping laser
105
. It was suggested
105
that this would make AuNRs
sensitive and switchable remote-controlled heat transducers to
small-volumesamples
105
.
Optical binding forces. Optical binding forces emerge from
multiple scattering between several objects, and can result in the
formation of regular, ordered structures
106–108
. is oers a path
towards large-scale NP assembly and organization in one
109
, two
110
and three dimensions
111
. For example, one-dimensional chains of
MNPs were suggested as an ‘optical sail’
112
to achieve a high driv-
ing force on an attached nanoscopic object, taking advantage of
the huge extinction cross-section of the collective plasmon reso-
nance. Pairs of 200-nm AuNPs were optically bound perpendic-
ular to the direction of light propagation in an optical ‘line trap’
formed by reection of a line-shaped focused beam, with particle
separations multiples of the optical wavelength
109
, consistent with
predictions based on light scattering from Rayleigh (dipolar) parti-
cles
113
. Yan et al.
110
used 40-nm-diameter AgNPs, a size well within
the dipole approximation, with both a line trap and a cylindrically
symmetric Bessel beam trap, and observed dimers, chains and
‘photonicclusters’
110
.
Optical binding interactions can also trap and organize one-
dimensional carbon nanostructures. In Fig. 2c, we show SWNT
bundles illuminated in aqueous suspension
114
by counterpropagat-
ing evanescent elds formed by total internal reection at a glass/
water interface. We observe their self-organization into optically
bound chains, where the bundle axes align parallel to the chain axis
and to the direction of propagation of the incident beams. ese
chains break as soon as the evanescent eld is switched o.
Experimental designs and techniques
In this section, we review the most common experimental imple-
mentations relevant for optical tweezers, with an emphasis on those
used for trapping and manipulation of NPs and nanostructures.
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Optical tweezers. In the simplest conguration (Fig.2a), optical
tweezers can be generated by focusing a laser beam to a dirac-
tion-limited spot using a high numerical aperture (NA) objective
lens
4,115,116
. is serves the dual purpose of focusing the trapping
light and imaging the trapped object. Samples are oen placed in
small (few microlitres) microuidic chambers held on a motorized
or piezo-driven microscope stage with nanometre position resolu-
tion
116
. Generally, optical tweezers require little power (down to a
few milliwatts
115,116
): carbon and silicon nanostructures have been
trapped with as little as 1–2mW NIR light
34,35,38,46,117
. e optical
tweezer position can be controlled using two steerable mirrors
118,119
.
It is also possible to generate multiple optical tweezers by deecting
a single beam in various positions using, for example, an acousto-
optic deector — that is, a device where intensity and frequency of
an acoustic wave spatially controls the optical beam
115,120
.
Holographic optical tweezers. e range of optical tweezer
applications has been greatly expanded by the use of advanced
beam-shaping techniques, where the shape of a light beam is altered
by diractive optical elements (DOEs) to produce multiple optical
traps at denite positions
13,14
. Figure2d shows a schematic of a holo-
graphic optical tweezer (HOT) set-up, where the DOE is placed in a
plane conjugate to the objective focal plane so that the complex eld
distribution in the trapping plane is the Fourier transform of that in
the DOE plane
121,122
. Oen the DOE is a liquid-crystal spatial light
modulator (SLM) used to modulate the phase of the incoming beam,
because any modulation of the amplitude of the beam would entail a
loss of optical power
121,122
. erefore, various techniques have been
developed to determine the optimal phase modulation, for example
the Gerchberg–Saxton algorithm, based on iterative optimization of
the phase prole at the SLM in order to obtain the desired optical
L1
L2
DM
OBJ
Beam-
steering
mirror
Laser
beam
Illumination
DM
QPD
L3
C
To camera
Laser o
Free
Brownian motion
Laser on
Stable trap
5 μm
BM
−250
−250
−250
0
0
0
250
250
250
x (nm)
y (nm)
z (nm)
M
DM
Evanescent field on
Evanescent field o
Prism
Microscope
objective
SLM
Illumination
5 μm
5 μm
L2L1
DM
To camera
OBJ
C
a
a
m-
a
bc
d
Figure 2 | Basic experimental designs. a, Optical tweezers are obtained by focusing a laser beam to a diraction-limited spot, using a high-numerical-
aperture objective lens (OBJ). Additional optics is needed to steer the optical tweezer position (beam-steering mirror and telescope formed by lenses L1,
L2), to image the sample (illumination, dichroic mirrors DM, and camera) and to track it (condenser, lens L3 and quadrant photodiode QPD). The resulting
traces allow tracking of the Brownian motion (BM) and the calibration of the optical tweezer stiness. Inset: Bright-field image of (top) optically trapped
and (bottom) free SWNT bundle. b, Evanescent optical waves can be excited by total internal reflection at an interface between a high- and a low-refractive-
index medium, often a glass–water interface. The excited evanescent waves can be used to manipulate dielectric and metallic particles. A microscope
objective images the sample. c, The optical forces resulting from a standing evanescent wave created by the interference of two counterpropagating
evanescent fields (arrows) align SWNT bundles end-to-end (top). When the evanescent wave is switched o, the bundles are released from the locked
position and undergo thermal motion (bottom). Scale bar, 5 μm. d, HOTs rely on a programmable diractive optical element (SLM) for the creation, shaping
and control of multiple independent optical tweezers. Inset: (top) dark-field and (bottom) scanning electron microscope images of a 5×5 pattern of 80-nm
AuNPs deposited by HOTs on a glass substrate. Figure reproduced with permission from: a, ref. 35, © 2008 ACS: inset in d, ref.144, © 2011 ACS.
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tweezer conguration at the trapping plane
123–125
. Over the past few
years, HOTs have been used to manipulate and assemble nanostruc-
tures. For example, semiconducting nanowires have been translated,
rotated, cut and fused into complex structures (Fig.3)
42,126
.
Plasmonic tweezers. In the 1990s and early 2000s, various groups
theoretically suggested harnessing the enhancement associated with
a plasmonic resonance to realize nanoscopic optical tweezers, for
example by using the extremity of a sharp metallic tip
127,128
, the light
transmitted through a nanohole in a metallic lm
129
or metallic pat-
terns to create multiple trapping positions at the nanoscale
50
.
In 2006, Volpe et al.
51
showed that surface plasmon polaritons at
a glass/Au/water interface produce a 40-times increase in the optical
forces on micrometre-sized dielectric particles (Fig.4a), so that col-
lections of such particles could self-organize in large crystals
130
. But
a at metal lm features a homogeneous optical potential
51
, whereas
controlled trapping of single nano-objects requires patterning of the
surface to create three-dimensional conning optical potentials
50
.
is is achieved by using properly designed metal nanostructures
such as pads
52,53
, antennas
54,55
or nanoapertures
56,58
. A typical optical
set-up to excite plasmons is based on the Kretschmann congura-
tion shown in Fig.2b. With similar schemes it is also possible to
arrange microscopic particles in complex congurations corre-
sponding to the locations of metallic micropads, where a plasmonic
resonance can be excited (Fig.4d), and also to integrate such plas-
monic traps with a microuidic environment
131
. Other congura-
tions based on nanoantennas allow one to localize the eld intensity
in hotspots
54,55
(Fig.4c). Fractal plasmonic structures
132
can allow
tight foci below the diraction limit far away (hundreds of nanome-
tres) from the metallic structures (Fig.4d).
Plasmonic interactions can also be harnessed by using the active
feedback from the interaction between the optical tweezer beam
and the trapped particle. It is possible to overcome the scaling of the
optical forces with the third power of the object size, as well as the
increase in Brownian uctuation, by making use of an optical trap
realized with a nanoaperture in a metal lm (Fig.4e) in which the
particle itself has a strong inuence on the local electric eld. e
particle thus has an active role in the trapping mechanism, increas-
ing the stiness of the trap only when the particle tries to escape
58
.
Plasmonic double nanoapertures were also used for optical trapping
of single proteins, paving the way to direct optical manipulation of
smaller objects
56
. Note that whenever plasmonic nanostructures are
involved, the problem of heating must be faced. Wang et al.
57
have
described a method of reducing heat in a plasmonic trap by using a
heat sink integrated with the optical structure.
Photonic force microscopy. A photonic force microscope
(PFM)
133–135
is a scanning probe technique based on optical tweezers
(Fig.5). is concept was originally developed when scanning a die-
lectric particle trapped on a surface and observing how its Brownian
motion in the trap was modied by the probe–sample interaction
133
.
In this way, it was possible to measure extremely small forces down
to femtonewtons, as well as image surface features below the trap-
ping light diraction limit
133,134,136,137
.
e motion of a trapped particle subject to thermal uc-
tuations can be modelled in one dimension by the overdamped
Langevin equation
115
:
=
dx(t)
x(t)−
+
2DW(t)
γ
K
x
d(t)
(6)
where x(t) is the particle position, K
x
the stiness of the optical trap,
γ the friction coecient, D the Stokes–Einstein diusion coecient
and W(t) a white noise. When dealing with quantitative force meas-
urements, it is crucial to calibrate the optical trap stiness, K
x
. is
calibration can be obtained by measuring the Brownian trajectory
of the optically trapped particle using the deection of the trapping
beam onto a quadrant photodiode
136
, a device that allows one to map
the particle trajectory
135
. ese trajectories are typically analysed by
tting the autocorrelation function of x(t) to an exponential
138
: the
characteristic decay relaxation time of the autocorrelation function
is τ=γ/K
x
. Deriving the value of γ from hydrodynamics, it is then
possible to measure K
x
. Alternatively, it is possible to perform this
analysis in the frequency domain using the power spectral density
of x(t) (ref. 136), which can be tted to a Lorentzian lineshape
139
.
When dealing with force measurements in the presence of diusion
gradients on the probe, such as close to boundaries or objects, some
correction terms are necessary
18
, and these are more signicant for
a nanometre-sized probe
140
.
Using an optically trapped particle as a PFM probe may be
advantageous in imaging of so structures
135
, because the trap sti-
ness is low (10
–3
to 1pNnm
–1
)
15
compared with that of an atomic
force microscope cantilever (10 to 10
5
pNnm
–1
)
15
, and in volumet-
ric imaging
137
at high temporal resolution (tens of kilohertz sam-
pling rates
137
), which can be achieved by three-dimensional particle
b
5 µm
SnO
2
GaN
100 nm
10 µm
c
10 µm
10
µ
m
Position of focused laser beam
and trapped nanowire
a
Figure 3 | Optical manipulation and placement of nanowires. a, Semiconductor nanowires can be manipulated and assembled with optical forces.
The image shows a GaN nanowire laser-fused to a SnO
2
nanoribbon after manipulation and deposition with optical tweezers. The inset is a scanning
electron micrograph of the fused junction. b, Assembly of a rhombus constructed from semiconductor CdS-nanowires using HOTs. This entails nanowire
translation, cutting and fusion with the substrate. c, Optical tweezing of a In
2
O
3
nanowire (top) and placement by scanning optical tweezers, to connect
two branches of a circuit (bottom). Figure reproduced with permission from: a, ref.43, © 2006 NPG; b, ref.42, © 2005 OSA; c, ref.119, © 2009 OSA.
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NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2013.208
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