LETTERS
Subwavelength direct-write
nanopatterning using optically trapped
microspheres
EUAN McLEOD AND CRAIG B. ARNOLD*
Mechanical & Aerospace Engineering Department, Engineering Quadrangle, Princeton University, Princeton, New Jersey 08544, USA
*e-mail: cbarnold@princeton.edu
Published online: 8 June 2008; doi:10.1038/nnano.2008.150
A number of non-lithographic techniques are now available for
processing materials on the nanoscale, including optical
techniques1–7 capable of producing features that are much
smaller than the wavelength of light used. However, these
techniques can be limited in speed, ease of use, cost of
implementation, or the range of patterns they can write. Here
we report how Bessel beam laser trapping of microspheres
near surfaces can be used to enable near-field direct-write8,9
subwavelength nanopatterning. Using the microsphere as an
objective lens to focus the processing laser, we demonstrate
arbitrary patterns and individual features with minimum sizes
of 100 nm (which is less than one-third the processing
wavelength) and a positioning accuracy better than 40 nm in
aqueous and chemical environments. Submicron spacing is
maintained between the near-field objective and the substrate
without active feedback control. If implemented with an array
of optical traps, this approach could lead to a high-throughput
probe-based
method
for
patterning
surfaces
with
subwavelength features.
Various optical nanopatterning techniques can overcome the
theoretical diffraction limit of conventional focusing optics to
create feature sizes that are smaller than half the optical
wavelength, although aberrations and problems with alignment
can make it difficult to perform such subwavelength
nanopatterning in practice. Widely used strategies include
multiphoton absorption1,2 and near-field effects3–7. Feature sizes
on the order of tens of nanometres have been produced by
illuminating the tip of an atomic force microscope with a laser to
create near-field effects, but this approach—and similar
approaches involving near-field scanning optical microscopes—
require sophisticated feedback systems to maintain a precise
spacing between the probe and the substrate5,6. Near-field effects
can also be harnessed by coating a surface with a self-assembled
array of microspheres7 and subsequently illuminating this surface
with a laser pulse. This allows large areas to be patterned in a
single shot, but only random10 and hexagonally-close-packed
patterns7 have been demonstrated so far. Furthermore, the
spacing between features can be no smaller than the radius of
the microsphere.
Optical trap-assisted nanopatterning addresses the challenges
encountered with existing techniques. We combine the maskless
rapid-prototyping capabilities of direct writing with the
simplified alignment of Bessel beam optical traps and the high
resolution of near-field enhancement. A Bessel beam is used to
optically trap and manipulate a microsphere, which acts as a
near-field lens for nanopatterning, as shown in Fig. 1a,b. Once
trapped, the microsphere is illuminated with a pulsed gaussian
beam and concentrates this light directly below the sphere, locally
modifying the substrate. Despite the incident gaussian beam
having a width of several micrometres, the intensity is controlled
such that only locations with near-field enhancement are
modified, resulting in features on the order of 100 nm,
depending on the material response. In order to directly write
two-dimensional (2D) user-defined patterns, we translate the
substrate while keeping the trap fixed and fire laser pulses at
predefined intervals. The particle remains in the trap for the
duration of the experiment, except for isolated cases such as
the particle adhering to the substrate under certain conditions or
the stage moving exceedingly fast (see Supplementary
Information). The trapping laser should not damage the bead or
substrate, and should not be strongly attenuated by the fluid. The
pulsed laser should exhibit similar qualities regarding the bead and
fluid; however, strong absorption in the substrate is desirable.
Thus, trapping wavelengths in the visible/near-infrared and
processing wavelengths in the near-ultraviolet/visible generally
work well for many material systems. Here, we used a trapping
laser wavelength of 532 nm and a processing wavelength of 355 nm.
Bessel beam traps differ from conventional optical tweezers in
that they only exhibit significant gradients in the transverse
directions, leading to particle confinement in two dimensions
with a scattering force in the propagation direction11,12 (see
Supplementary Information, Fig. S1). Solvent-mediated
electrostatic double-layer repulsions between the charged
surfaces of the sphere and substrate balance this scattering force,
resulting in an equilibrium spacing on the order of 50 nm
determined by the trapping laser intensity and system
chemistry13,14 (see Supplementary Information, discussion and
Figs S2 and S3). The result of this force balance is that the
sphere self-positions in the propagation direction with accuracy
on the order of 10 nm, regardless of existing large-scale features
on the surface, thereby removing the need for feedback control
of the particle – surface spacing. In addition, if the sphere is
displaced from the central trap location, for example due to
ablated material leaving the substrate surface, the trap provides
a restoring force in all directions, returning the sphere to its
equilibrium position.
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413
LETTERS
0
White
light
14
Propagation direction (nm)
Axicon (for
Bessel beams)
532 nm CW
355 nm
pulsed
Glass coverslip
Spheres in solution
Substrate
Bandpass filter
x-y
stage
Near-field
focusing
50X
12
500
10
8
6
1,000
4
2
1,500
200 400 600 800 1,000
CCD
Polarization direction (nm)
Figure 1 Experimental setup and modelling. a, Schematic of the experimental setup, depicting the paths of the trapping laser (green) and the processing laser
(purple), and the location of the sample. The white light is used in conjunction with the 50 objective and CCD camera to monitor the process. b, Artist’s impression
showing a white microsphere that is optically trapped near the polyimide surface (orange) in the trapping laser beam (green). The processing laser covers a large
area, but only reaches a high enough fluence for material processing directly beneath the microsphere. The grey slabs represent glass coverslips, and the aqueous
environment is shown in pale blue. c, FDTD model of the intensity of light passing through a 0.76-mm polystyrene sphere (n ¼ 1.62) in water (n ¼ 1.34) above a
polyimide substrate (n ¼ 1.7, absorption depth ¼ 200 nm). The horizontal black line represents the top of the polyimide substrate. The plane shown is parallel to the
propagation and electric field polarization directions. Colour indicates the intensity enhancement relative to the incident wave.
1,400
FWHM (nm)
600
1,200
200
0
800
0
600
10
20
30
Laser fluence (mJ cm–2)
40
10
Number of spots
Spot FWHM (nm)
1,000
400
400
200
0
0
500
1,000
1,500
2,000
2,500
3,000
3,500
Bead diameter (nm)
8
6
4
2
0
100
150
200
250
FWHM value (nm)
Figure 2 Laser-written spot sizes. a, FWHM versus particle size for polystyrene spheres of several sizes and 0.52-mm silica spheres on a polyimide substrate.
Filled symbols indicate experimental material data, and open symbols indicate results of the FDTD-based laser/material interaction model. This model assumes a
threshold fluence of 48 mJ cm22 and a sphere – surface separation distance of 50 nm (see Supplementary Information for a discussion). The incident fluences are
colour-coded as follows: 5 mJ cm22 (blue circles), 7 mJ cm22 (light blue triangles), 12 mJ cm22 (green triangles), 13 mJ cm22 (yellow triangles) and 35 mJ cm22
(red squares). Error bars show one standard deviation of spot sizes on either side of the mean based on an average of 50 – 100 spots per data point. One data point
(star) is shown for silica spheres with 35 mJ cm22 incident laser fluence. b, Experimental (solid symbols) and simulated (open symbols) FWHM as a function of
incident laser fluence for 1.00-mm spheres (vertical black box in a). Vertical error bars represent one standard deviation in the measured spot size, and horizontal
error bars represent the shot-to-shot variation in laser pulses. c, Sample dataset illustrating the distribution in spot sizes for 0.76-mm polystyrene spheres with
12 mJ cm22 incident laser fluence.
Optical trap-assisted nanopatterning comes with a large
experimental parameter space that can be exploited to tailor feature
sizes to different applications. In this study, we vary the microsphere
material, substrate material, microsphere size, pulsed laser energy
and spacing between adjacent spots. A summary of these results is
presented in Fig. 2. Most of our data deals with polystyrene spheres,
because the high refractive index makes spheres easy to trap and
effective at concentrating light. We have also used silica spheres,
414
which exhibit lower-intensity enhancements and broadening of the
enhanced region. A variety of substrates including polyimide thin
films and sheets of polycarbonate, polyester and polyimide are
nanopatterned, but we primarily use polyimide thin films as our
model substrate because of its well understood response to laser
irradiation at 355 nm and relatively low damage threshold.
To understand the near-field effects of the pulsed laser, we refer
to previous models of near-field enhancement using polystyrene
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LETTERS
Figure 3 Nanopatterning a logo. Princeton University logo nanopatterned on
polyimide film using 0.76-mm beads. The 5-mm-wide shield was created using
overlapping subthreshold pulses (centres separated by 30 nm) at incident laser
fluences of 3 mJ cm22. At this fluence, trenches rather than bumps were
formed, as shown in the inset. Main scale bar, 2 mm; inset scale bar, 1 mm. The
elevations shown range from 230 nm to þ10 nm.
microspheres10, and generate finite-difference time-domain
(FDTD) simulations. Figure 1c shows the result of a sample
simulation of light being focused by a microsphere. Note that the
characteristic near-field polarization-induced two-lobe intensity
distribution only exists within a few nanometres of the sphere
surface, and does not extend to the substrate due to the finite
sphere-surface spacing. Therefore we do not expect to see this
effect expressed in the experimentally patterned substrate. In
order to develop our theoretical predictions of spot size, we first
assume a threshold fluence for material modification of
48 mJ cm22 based on reported values15,16. The theoretical spot
size is given by the maximum width of the region with fluence
greater than the threshold, at a plane 50 nm below the sphere, as
calculated by the FDTD simulation. This simple model for
laser/material interaction is helpful in theoretically estimating the
resulting feature size. A more accurate model would take into
account other material parameters and mechanisms, such as the
thermal diffusion length, chemical enhancement effects, pulse
duration effects, absorption in an ablating plume, long-term
incubation effects and so on. Such models may lead to nonlinear
absorption effects, which have the potential to further reduce
spot size17. See Supplementary Information, Figs S4 and S5, for
an illustration of the effect the substrate and spacing have on the
pulsed laser fluence enhancement.
One of the most critical parameters is the size of the microsphere.
As expected, smaller microspheres yield smaller spots, reflected in
Fig. 2; however, increased brownian motion also reduces positional
accuracy (see Supplementary Information). Experimentally, the
optimal results are achieved with 0.76-mm polystyrene spheres,
because they yield small spot sizes as predicted by our FDTD
calculations while still retaining relatively good positioning
accuracy and intensity enhancement. At low pulsed laser energies,
the 0.76-mm spheres produce spot sizes that are 130 nm wide on
average, with a standard deviation of 38 nm. We primarily use this
bead size for creating complex features, as presented in Fig. 3. For
larger spheres, the combination of longer focal lengths with larger
effective apertures leads to similar intensity enhancements
8
4
FWHM = 102 nm
100
50
0
0
2
100
0
(nm)
100
50
0
3,000
0
2,000
(nm) 1,000
0
500
1,000
1,500
(nm)
2,000
200
(nm) 400
0
(nm)
50
100
150
(nm)
200
0
–10
–20
–30
400
600
250
600
(nm)
200
(nm)
50
0 0
0
200
150
100
(nm)
(nm)
(nm)
(nm)
(nm)
6
6
4
2
0
200
0
–200
2,000
0
2,000
100
200
(nm) 200
(nm) 1,000
100
0
(nm)
0 0
1,000
(nm)
Figure 4 AFM characterization of various spot morphologies. a – f, Spot morphologies on a polyimide substrate (a – d,f) and a polycarbonate substrate (e).
In all figures, the maximum height is red and the minimum height is blue. a, An individual bump created using a 0.49-mm sphere with an incident laser fluence of
7 mJ cm22. b, Line scan through a, showing that the FWHM of this spot is 102 nm, well below the l /2 diffraction limit. c, Volcano-shaped spot created using a
1.00-mm sphere with an incident laser fluence of 13 mJ cm22. This shape is typical of higher energy pulses. d, A ridge line created using individual 7 mJ cm22
pulses and a 0.76-mm bead. The pulses are spaced 50 nm apart. e, A single hole created on bulk polycarbonate. Unlike for polyimide, low to medium energy pulses
form holes instead of bumps. f, A trench line created using overlapping pulses of 10 mJ cm22 and a 0.76-mm bead. The pulses are spaced 57 nm apart.
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LETTERS
(see Supplementary Information, Fig. S6), but the spot sizes rise in
this range.
Varying the pulsed laser energy induces several effects, such as
changes in feature size, morphology, modification mechanisms and
bead– surface adhesion. As expected, higher laser energies lead to
increased feature sizes (Fig. 2b). Morphologically, we see that as
the laser energy is increased, the features first take the form of
bumps (Fig. 4a,b) and transition into bumps with craters
(Fig. 4c). This effect has been well established in polyimide at
larger size scales16,18. Interestingly, near the transition between
these two regimes, spheres can be made to adhere to the surface.
A higher energy pulse can detach the bead, which can still be
used for further nanopatterning. We have successfully
nanopatterned polyimide using incident laser fluences in the
range of 2 –40 mJ cm22. At fluences above this, we approach the
damage threshold for unfocused laser pulses and the resulting
spot sizes become large.
The direct-write nature of optical trap-assisted nanopatterning
allows us to create both individual spots and continuous features
composed of overlapping spots (Figs 3 and 4d,f ). In both
geometries, a high degree of positional accuracy is necessary, and
error is most obvious when trying to build a continuous
structure. The continuous pattern shown in Fig. 3 is created
using overlapping pulses with sub-threshold fluence
(2.5 mJ cm22 compared to the 3 mJ cm22 threshold for 0.76-mm
beads). Although a single pulse at this laser energy is not
sufficient to form surface bumps, it can alter the polyimide
surface chemistry19, and multiple overlapping pulses lead to the
observed surface structuring. A statistical analysis of the central
horizontal line of the pattern shows that the standard deviation
of the central minimum is 38 nm, the mean full-width at halfmaximum (FWHM) is 263 nm, and the standard deviation in
FWHM is 49 nm. For comparison, theoretical analysis indicates
that the positional accuracy based on our experimental
conditions is approximately 30 nm (see Supplementary
Information). Accuracy can be improved by increasing the
trapping laser power; however, this will also decrease the sphere –
surface spacing.
Optical trap-assisted nanopatterning could be extended in a
number of ways. For instance, by using reactive fluids, mechanisms
such as etching20 or deposition21,22 can be harnessed. Adding
surfactants to modify the surface chemistry could also provide a
means to independently tune the sphere–surface spacing without
affecting the transverse positional accuracy. Moreover, the
throughput of many direct-write, probe-based processes is limited
by the use of a single probe; however, by using existing adaptive
optics devices (such as spatial light modulators23,24 or tuneable
acoustic gradient index lenses25,26) to generate addressable Bessel
traps in parallel, and then illuminating all these traps
simultaneously with a pulsed laser, it should be possible to
significantly increase throughput. Furthermore, in contrast to
other near-field techniques, the objective element can be easily
released from the trap and replaced without removal and
realignment of the substrate. Combining these ideas of being able
to simultaneously hold and exchange different sized particles
naturally leads to the concept of a complete nanopatterning
toolset in which tools are changed ‘on the fly’ to give researchers
great freedom over the size and morphology of the features
they create.
METHODS
SAMPLE PREPARATION
Polystyrene microspheres were obtained commercially from Bangs Labs
(0.49 mm and 0.76 mm, no. PS03 N), Ted Pella (1.00 mm, no. 610-38), and
416
Polysciences (3.0 mm, no. 17134). Silica spheres were purchased from
Bangs Labs (0.52 mm, no. SS03 N). These solutions are diluted with
deionized water until the sphere concentration is sufficiently low such that
multiple spheres do not enter the trapping region. In some solutions there
is a trace amount of Triton X-100 surfactant. Polyimide (HD MicroSystems,
no. PI2525) was spin-coated onto 150-mm-thick glass cover slips at
8,000 r.p.m. for 40 s. The resulting films were 3 mm thick. These films were
baked for 30 min at 150 8C, and then for 30 min at 350 8C. A 1 – 5 ml drop
of diluted microsphere solution was placed on the coated cover slip.
A sealed cell was created using a gasket-shaped piece of double-sided tape
(3M, no. 9589, 230 mm thick) in the middle and an uncoated cover slip on top.
This cell was mounted on a manual/piezo-controlled xyz translation stage
(ThorLabs, MAX301).
SAMPLE EXPOSURE
The sample cell was coarsely translated until a sphere was observed with white
light illumination using a 50 microscope objective at the back of the sample
and charge-coupled device (CCD) camera (Cohu 2622). The 400– 900 mW
linearly polarized 532-nm continuous-wave (CW) trapping laser beam was
directed through an axicon with cone angle 1788 and a 30 reducing telescope.
The actual power going through the sphere is expected to be on the order of
0.1 – 1 mW because of losses and the relatively large transverse extent of our
Bessel beam. Once a sphere was trapped, computer control directs the motion of
the translation stage and triggered a pulsed linearly polarized 355-nm laser
(Coherent Avia, 15-ns pulse length) according to user-defined programs. The
pulsed laser was focused as an Airy disk to a 10-mm-diameter spot centred on the
microsphere. This size was measured without a bead in the trap by increasing the
pulsed laser energy high enough for the rings of the Airy disk to modify the
polyimide. Pulsed laser energies in the range of 2– 30 nJ were used for
nanopatterning (measured immediately before the cell using an Ophir PD10).
The quoted fluences incident on the microsphere were calculated by dividing the
pulse energy by the area of the laser beam at the surface, resulting in values in the
range of 2– 40 mJ cm22. By increasing the laser energy beyond that used
for nanopatterning, we found the threshold for material modification
without a microsphere to be at 40 nJ, or 50 mJ cm22, in agreement with
the literature16,18.
SAMPLE ANALYSIS
The cells were disassembled by soaking the double-sided tape in ethanol.
Characterization of individual features written on the polyimide was performed
using tapping-mode atomic force microscopy (AFM). Statistical data on spot
sizes was obtained by capturing a single 2D AFM scan of 50– 100 spots and
subsequently using image processing routines. High-pass and low-pass median
filters were used to subtract the background and remove noise. For bumps, the
peaks of each individual spot were located, and then an average distance
calculated for each spot between its peak and nearby pixels that were 40 – 60% the
height of that peak. Twice this average yields the FWHM for a single peak. For
volcano-shaped structures, the centre of each spot was determined by finding the
peaks in the 2D correlation function between the original image and an annulus
with the approximate the dimensions of the spots. The FWHM was calculated
relative to the local maximum of the cratered structure.
FINITE-DIFFERENCE TIME-DOMAIN SIMULATION
A finite-difference routine was written using the Yee Algorithm27. The domain
was modelled using the dielectric constants for water, polystyrene and polyimide
at 355 nm. Absorption was only considered within the polyimide. At the input
face to the domain, a linearly polarized transverse electromagnetic wave
of unit intensity was applied, propagating normal to the substrate. All other faces
of the domain used a perfectly absorbing boundary condition28. The domain
dimensions were at least three times the sphere radius (five times in the
propagation direction) or twice the free-space wavelength. The number of cells in
the domain was chosen so that there were at least 8 cells per wavelength and 15
cells per sphere radius. The simulation was run for sufficient time that a wave was
able to propagate twice the length of the domain. Intensity and fluence
enhancements were calculated by averaging the magnitude of the Poynting
vectors over the last whole cycle. Feature widths were measured by calculating the
width of the beam where the fluence was greater than 48 mJ cm22, assuming a
sphere – surface spacing of 50 nm. The reasons for this spacing are discussed in
the Supplementary Information.
Received 21 February 2008; accepted 1 May 2008; published 8 June 2008.
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LETTERS
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Supplementary Information accompanies this paper at www.nature.com/naturenanotechnology.
Acknowledgements
We thank J. Fleischer and M. Haataja for providing helpful insight and critiques, and N. Kattamis for
providing helpful information regarding polyimide coating and thin film properties. Funding for this
work was provided by Princeton University and Air Force Office of Scientific Research (AFOSR)
(FA9550-08-1-0094).
Author contributions
E.M. performed the experiments. E.M. and C.B.A. conceived and designed the experiments, analysed the
data, and co-wrote the paper.
Author information
Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/.
Correspondence and requests for materials should be addressed to C.B.A.
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© 2008 Macmillan Publishers Limited. All rights reserved.
417
1
Supplementary Information
Subwavelength direct-write nanopatterning using optically trapped
microspheres:
Euan McLeod and Craig B. Arnold
Mechanical & Aerospace Engineering Department, Engineering Quadrangle, Princeton
University, Princeton, NJ 08544
Supplemental Discussion: Positional Uncertainties
The fundamental limit governing the positional accuracy of the technique is the
Brownian motion of the microsphere within the optical trap. In our cylindrical
geometry, the probability of finding a particle in a region between r and r + dr is given
by,
⎛ U (r ) ⎞
P(r )dr = 2πrA exp⎜ −
⎟dr ,
⎝ kT ⎠
(1)
where A is a normalization constant, U(r) is the potential energy of the radiation field, k
is Boltzmann’s constant, and T is the temperature. Although our 0.76 µm particles are
larger than the optical trapping wavelength, we will still approximate the optical
potential using the Rayleigh scattering model because it provides insight into the
physics of the trap stability without requiring Mie theory calculations. The potential
energy due to the gradient force on a Rayleigh particle is given in MKS units by1,
2πnb a 3
U (r ) = −
c
⎛ (n p / nb )2 − 1 ⎞
⎟ I (r ) ,
⎜
⎜ (n / n )2 + 2 ⎟
⎠
⎝ p b
(2)
where nb is the refractive index of the surrounding medium, np is the refractive index of
the particle, a is the radius of the particle, c is the speed of light, and I(r) is the intensity
of the trapping laser beam, in our case given by the profile shown in Fig. S1:
I (r ) =
P0
2 ⎛ 2.40r ⎞
J0 ⎜
⎟,
2
0.85a
⎝ a ⎠
(3)
assuming the Bessel beam is appropriately scaled so that the radius of the central lobe is
equal to the radius of the trapped particle, P0 is the power in the central lobe of the
Bessel beam, a is the particle radius, and 2.40 is first zero of J02(x).
Using a polystyrene particle of radius a = 380 nm and a Bessel beam with 100 µW of
power in the central lobe, we calculate the mean positional error to be r = 32 nm at
room temperature. This positional error can be reduced by increasing the trapping laser
© 2008 Nature Publishing Group
2
power, however there is an upper limit which depends on the electrostatic spheresurface repulsive force as discussed below.
As a first approximation, we can treat the thermal field in the liquid as being constant in
time. The continuous trapping laser will lead to an increased, but constant, temperature.
This constant thermal field in the liquid has a minimal effect on the positional
uncertainty. If instead of being at room temperature, the system were just below the
boiling point of water at 373 K, then the positional uncertainty would increase to 36 nm.
The processing laser will also have an effect on the thermal field within the liquid,
however this effect should occur at timescales much shorter than our procedure, because
we are using a nanosecond laser, but firing shots at rates less than 100 Hz. Hence, this
should not affect positional error.
Ablative material removal may temporarily displace the bead from the centre of the trap
until the optical force restores the particle to the centre position. We observed this in
the upper range of fluences used for nanopatterning, when features such as those in Fig.
4c were formed. In such situations writing rates need to be reduced somewhat to allow
the bead to return to the centre of the trap before the subsequent shot. We wrote
features at rates ranging from 1 to 100 Hz, depending on the distance between shots,
laser trapping power, and pulsed laser fluence.
Sample translation can also lead to positional error. If writing speeds are increased too
much, the particle may be lost from the trap when the Stokes drag exceeds the trapping
laser force on the particle. An external fluid flow would have the same effect. We
found that writing speeds on the order of microns per second still provided consistent
positioning accuracy.
Sharp, pre-existing asperities can also lead to positional error because the resulting
surface repulsion force is not normal to the trapping beam. However, a higher power
trapping laser can help to mitigate these uncertainties.
There are some factors which do not affect positional error that should be noted. First
the optical scattering force induced on the bead from the pulsed laser is negligible
compared to the continuous trapping laser force. At our fluences, the impulse from the
pulsed laser is on the order of FA/ c , where F is the fluence, A is the cross-sectional
area of the particle, and c is the speed of light2. This results in an impulse on the order
of 10-18 N s. Assuming a Stokes drag force of -6πηav, where η is the dynamic viscosity
of water and v is the particle velocity results in a maximum displacement on the order of
Angstroms.
Second, due to the low Reynolds number of the system, the motion of a microscopic
particle in water is always overdamped, and therefore there is no significant harmonic
motion of the particle within an optical trapping potential
In future experiments, we will experimentally characterize the above effects on the
positional accuracy of our technique.
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Fig. S1. Bessel beam profiles. (A) Plan view showing the calculated transverse
intensity variations of a Bessel beam. (B) Cross section of A illustrating the high
intensity of the central peak.
Supplementary Discussion: Sphere-Surface Spacing
The sphere-surface spacing is determined by the balance between the laser forward
radiation pressure force and a repulsive electrostatic double-layer force resulting in the
self-positioning depicted in Fig. S3.
As was done in the above Positional Uncertainties section, the probability of the spacing
between the particle and substrate being between z and z + dz is given by,
⎛ U ( z ) + U dl ( z ) ⎞
P( z )dz = B exp⎜ − l
⎟dz ,
kT
⎠
⎝
(4)
where B is a normalization constant, and Ul(z) is the potential induced by the laser
radiation pressure, and Udl(z) is the potential from the double layer. In order to keep the
equations simple, we will assume Rayleigh regime optical trapping as above, a Debye
length (κ-1) smaller than the microsphere radius, and equal surface potentials for the
sphere and substrate (ψ0). With such assumptions, the potentials will be given by3, 4,
128π 5 a 6 I 0 nb
U l ( z) =
3cλ4
⎛ (n p / nb )2 − 1 ⎞
⎜
⎟ z
⎜ (n / n )2 + 2 ⎟
⎝ p b
⎠
2
(
)
U dl ( z ) = εε 0 aψ 0 log 1 + e −κz ,
2
(5)
(6)
where I0 is the mean intensity incident on the microsphere, ε is the dielectric constant of
water, and ε0 is the free-space permittivity.
While the specific values of κ, ψ0, and Ι0 were not measured for our systems, we can
use some reasonable test values to arrive at approximate sphere-surface spacings4.
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Setting κ-1 = 100 nm and ψ0 = 90 mV and choosing the optical power passing through
the bead to be P0 = 100 µW, the resulting mean surface spacing is z = 50 nm, and the
positional error is given by
(z
2
− z
)
2 1/ 2
= 9 nm. The mean surface spacing can also
be determined by balancing the laser radiation pressure force with the electrostatic
double layer repulsion (Fig S2). If the temperature were raised to 373 K, then the mean
spacing would not change, and the spacing error would increase to 10 nm.
Several effects can alter the equilibrium sphere-surface spacing. Adding more
electrolyte to the trapping solution can decrease the Debye length, resulting in a closer
spacing. Modifying the surface chemistry of the sphere and substrate will alter the
surface potentials. Higher surface potentials increase sphere-surface spacing.
Increasing the laser power will decrease sphere-surface spacing. However, because the
electrostatic double-layer repulsion reaches a finite maximum at z = 0, there is an upper
limit on the trapping laser power so as to avoid contact and the resulting Van der Waals
adhesion between the sphere and the substrate (see Fig. S2).
The factors in the above Positional Uncertainties section that lead to transverse
positional uncertainties can also lead to longitudinal positional uncertainties.
For most of our theoretical calculations, we have assumed a separation distance of 50
nm based on the above reasoning. Using this separation distance, we find good
agreement between theoretical spot sizes and experimental predictions for small beads
in Fig. 2. However, as predicted above, the spacing should have some dependence on
sphere size, and this explains the discrepancies in Fig. 2 between the theory and
experiment for larger bead sizes.
In future experiments, we will incorporate an in situ apparatus for measuring spheresurface height, either based on total internal reflection microscopy5 or by comparing the
diffraction rings of a trapped bead and a bead adhered to the substrate6.
Fig. S2. Force balance between the laser and the electrostatic double layer. The forces
are calculated by taking the derivatives with respect to z of Eqs. 5 and 6. The
equilibrium sphere-surface separation will occur at z , where the absolute forces of the
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double-layer and laser pressure are equal. According to the above theory, κ −1 is the
1
2
Debye length and Fmax = εε 0 aψ 0 κ .
2
Fig. S3. Self-positioning of particles using Bessel traps. (A) Standard Gaussian traps
produce gradients in all three directions leading to a fixed position for the trapped
particle. Therefore they do not maintain sphere-surface spacing as the substrate changes
height, either due to roughness or existing surface features. (B) Bessel traps exhibit
gradients in the transverse direction. A scattering force in the propagation direction
causes the particle to be pushed towards the surface. Force balance due to repulsive
forces from the fluid and substrate maintain sphere-surface spacing regardless of the
absolute height of the substrate and enable the self-positioning effect.
A
B
Fig. S4. FDTD Simulations for a 0.76 µm sphere illustrating the effect of the substrate
on the pulsed laser fluence distribution. The sphere-substrate separation distance is 50
nm. (A) The width, in both the polarisation (dashed) and other transverse direction
(solid), of the region where the fluence surpasses the material threshold (48 mJ cm-2) for
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a 12 mJ cm-2 incident pulse. Thick black lines correspond to the model for an absorbing
polyimide substrate. Thin red lines correspond to the model without a substrate. Note
that at the substrate surface, the difference in spot size is less than 10% between the
models with and without the substrate. This small change indicates the lack of a
resonance between the sphere and substrate due to the highly absorbing nature of the
dielectric substrate. This is investigated further in Fig. S5. (B) The intensity
enhancement predicted as a function of the distance beyond the substrate. Colours are
the same as in (A).
Fig. S5. FDTD Simulations for a 0.76 µm sphere illustrating the effect of the spheresubstrate separation on the spot size at the surface. Hollow circles represent simulations
for the system that have explicitly included a polyimide substrate. Dashed lines indicate
estimates based on a model of a sphere surrounded entirely by water without a substrate.
Colour indicates the incident laser fluence on the sphere. As can be seen by the good
agreement between lines and circles, the substrate has a minimal effect on the spatial
size of the sphere-focused pulse at the surface because the polyimide substrate is an
absorbing dielectric. Spot sizes are calculated in the polarisation direction the same way
as in Fig. S4: the width of the region where the fluence surpasses the material threshold
(48 mJ cm-2). Note that for small spot sizes, maintaining accurate sphere-surface
spacing is necessary to achieve consistent spot sizes.
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Fig. S6. Simulated Intensity Enhancements. Calculations were performed with a
polyimide substrate 50 nm away from polystyrene microspheres. Note that intensity
enhancements fall rapidly for sphere sizes less than 0.76 µm, however they do not
dramatically increase for larger spheres. Higher enhancements for larger spheres would
be realized if the sphere-surface spacing were increased.
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