The plasmon resonance-based optical trapping (PREBOT) method is used to achieve stable trapping of metallic nanoparticles of different shapes and composition, including Au bipyramids and Au/Ag core/shell nanorods. In all cases the longitudinal plasmon mode of these anisotropic particles is used to enhance the gradient force of an optical trap, thereby increasing the strength of the trap potential. Specifically, the trapping laser is slightly detuned to the long-wavelength side of the longitudinal plasmon resonance where the sign of the real component of the polarizability leads to an attractive gradient force. A second (femtosecond pulsed) laser is used to excite two-photon fluorescence for detection of the trapped nanoparticles. Two-photon fluorescence time trajectories are recorded for up to 20 minutes for single and multiple particles in the trap. In the latter case, a stepwise increase reflects sequential loading of single Au bipyramids. The nonlinearity of the amplitude and noise with step number are interpreted as arising from interactions or enhanced local fields amongst the trapped particles and fluctuations in the arrangements thereof.
©2007 Optical Society of America
The use of a tightly-focused laser beam to trap and manipulate nanometer to micron-sized particles has become widespread since its inception [1, 2, 3]. Indeed, the optical trapping or “tweezing” of such particles has resulted in new fundamental science and diverse applications in the biological and physical sciences [4, 5], including, the characterization of glassy and nonequilibrium phenomena in colloidal solutions [2, 6], measurement of the forces produced by RNA polymerase during DNA transcription , measuring the transition state dynamics of single molecule (RNA) folding , and the creation of arbitrary three-dimensional arrays of traps .
Optical trapping involves a competition between a “gradient force,” which is strongest at the location of maximum gradient of the intensity of a laser beam, and a “scattering force,” which results from the radiation pressure that tends to push an object along a beam’s propagation direction. Stable trapping occurs when the former is stronger than the latter, which is achieved when the trapping laser light is tightly focused with a high numerical aperture lens [2, 10]. The theory of operation is best described when operating in one of two regimes, either Rayleigh or ray optics. In the latter picture, wherein objects are large compared to the optical wavelength, refraction of the light leads to a momentum recoil that directs them toward the laser intensity maximum . Generally, this is true for objects whose refractive index is larger than that of the surrounding medium, whereas beam shaping has been used to obtain stable trapping when the object’s refractive index is smaller .
Alternatively, objects in the Rayleigh regime, viewed as induced dipoles upon interaction with an optical field, are small compared to the wavelength of the trapping laser and minimize energy by moving towards the maximum of the intensity. For a monochromatic field E, of angular frequency,ω, the time-averaged optical force acting on a particle in the Rayleigh regime is 
where <⋯> represents the time-average, α is the frequency-dependent electric polarizability of the particle (and is proportional to its volume), B is the magnetic component of the input optical field, nm is the real-valued index of refraction of the medium, c is the speed of light in vacuum, and σabs and σscatt are the absorption and scattering cross-sections, respectively. The first term is conventionally referred to as a gradient force and the second as a scattering force.
The ability to trap and steer nanoparticles of different shapes in three dimensions in solution could be useful in the creation of nanoparticle arrays with novel photonic applications. Furthermore, such a technique could be used to investigate optically-induced electromagnetic interactions between nanoparticles as a function of shape and/or composition [13, 14], analogous to the “optical binding” observed between micron-sized particles . However, Eq. (1) shows why trapping nanometer-sized particles has proved challenging; the gradient force scales as the particle volume and scattering force only as the area (i.e., cross-section). Therefore, the number of experiments that have demonstrated stable, three-dimensional trapping of objects with nanometer-scale dimensions have been limited. For example, objects such as carbon nanotubes [16, 17] and semiconductor nanowires  can be trapped because of their micron-scale extension in one dimension. In addition, gold spheres with diameters as small as 20 nm have been trapped by utilizing their large refractive index in the infrared [12, 19], but achieving stable, three-dimensional trapping is difficult , and is accompanied by significant heating .
Mitigating these difficulties is possible, enhancing trapping forces while avoiding unacceptable levels of heating, by exploiting optical resonances. As far back as 1977, it was observed that Mie resonances enhance the radiation pressure on optically levitated dielectric spheres , and the first experimental report of laser tweezers includes a proposal to use resonances to enhance trapping forces . Shortly afterwards, optical trapping of neutral atoms was demonstrated through careful tuning of laser frequency near atomic transitions . Although theoretical work has developed the idea of resonance enhancement for the room-temperature trapping of particles in solution , experimental success has been limited. Resonances have been shown to slow down the diffusion of single molecules in solution [25, 26], and it has been proposed that excitonic transitions in semiconductor nanocrystals could permit their optical trapping , but the trapping forces in these cases are limited by the broad linewidths and low saturation intensities of the transitions. Therefore, room-temperature trapping times are limited to a few milliseconds, and stable trapping would require operation at liquid-helium temperatures .
Noble metallic nanoparticles, such as gold and silver, are excellent candidates for trapping [29, 30]. They exhibit strong plasmon resonances, associated with the collective oscillation of conduction electrons in the particle, which have very high saturation intensities . In particular, metallic nanorods possess a strong longitudinal-plasmon resonance, corresponding to electron oscillation along the long axis of the rods, whose frequency can be tuned throughout the visible and near-infrared frequency range by controlling the aspect ratio and composition of the rods .
Recently, stable trapping of Au nanorods only 40–80 nm in length and 10–15 nm in diameter was accomplished using the near-resonance enhancement of the particle polarizability associated with the plasmon mode . In this paper we further develop this plasmon resonance-based optical trapping (PREBOT) method with independent control of both the trap and probe beams and extend the previous results for Au nanorods to the first demonstration of stable trapping of Au bipyramids and Au/Ag core/shell nanorods. Finally, we show that resonant optical traps can stably hold multiple nanoparticles for periods of minutes. Sequential loading of the trap will allow studying electromagnetic interactions between particles.
To further elucidate the operational principle of PREBOT we consider the case of metallic ellipsoids much smaller than the wavelength of the trapping laser. The dipole approximation is a sufficient model for the particle sizes used in our experiments [31, 34]. The large scattering and absorption cross-sections follow from the electric polarizability in the Rayleigh limit, where the polarizability for an ellipsoid with principal axis j parallel to the direction of polarization of the optical field is 
Here, V is the volume of the ellipsoid, ε o is the permittivity of free space, ε(ω) is the complex dielectric function of the ellipsoid, ε m is the dielectric constant of the medium (assumed to be real), and Lj is a geometrical factor. We model Au nanorods as prolate spheroids where only two of the three principal axes are independent. The frequency-dependent cross-sections for scattering and absorption are 
respectively, and k is the wavevector in the medium. Figure 1(a) shows the scattering and absorption cross-sections for the longitudinal plasmon mode of a Au nanorod with aspect ratio 4.6, and surrounding medium εm=1.77 (water). The resulting plasmon resonance occurs at 800 nm wavelength, and the cross-section for absorption is several times that of scattering. Figure 1(b) is a log-log plot of the scattering and absorption cross-sections for Au nanorods using the long axis length (and fixed aspect ratio of 4.6) up to 100 nm in length. The plasmon resonance remains centered at 800 nm. The magnitude of the gradient and scattering forces acting on a Au nanorod (15 nm diameter and 60 nm length) resulting from a 100 mW beam that is focused to a diffraction-limited spot is shown in Fig. 1(c). The gradient force is significantly increased when the trap laser wavelength is detuned to the long-wavelength side of the longitudinal plasmon resonance. An optimal detuning for Au nanorods is approximately 50 nm, where PREBOT achieves a trap potential depth of several kbT (kbT=4pN·nm).
Figure 2 is a schematic of the PREBOT setup. The trapping beam is obtained from a tunable, continuous-wave, linearly polarized Ti:Sapphire laser (Coherent). The beam is sent through a telescope (lenses L 1 and L 2), a tube lens, LT, and into an inverted microscope (Nikon). The telescope focal lengths are chosen such that the beam diameter at the back focal plane of the microscope’s water-immersion objective lens (Olympus UPLSAPO 60X 1.2 NA) is approximately 1 cm. The probe beam, derived from a pulsed and tunable Ti:Sapphire laser (Spectra-Physics Mai Tai) that produces sub-100 femtosecond-duration pulses at 80 MHz repetition rate, travels along a parallel path. The probe laser is tunable from 800 to 880 nm; the wavelength chosen depends on the ensemble spectra of the samples. The probe pulses are linearly polarized and spectrally centered at 800 nm for the experiments reported here. Dispersion to be encountered is pre-compensated by double passing a prism pair. The pulses travel through a 1:1 telescope (lenses L 3 and L 4) before being combined with the trap beam via a first surface reflection from a 1-inch diameter BK7 window. This configuration allows independent focus control of the trap and probe beams, which is essential for optimizing signal detection from trapped nanoparticles. The pulses are focused into the sample by the same microscope objective. A solution of nanoparticles is applied to a standard microscope coverslip (.17 mm thickness), and the microscope objective is adjusted to focus slightly above the coverslip surface. The emitted two-photon fluorescence from the sample is epi-detected, spectrally filtered and imaged onto either a thermoelectrically-cooled CCD (Andor, IXON 887) detector or a photon-counting avalanche photodiode (Perkin-Elmer, SPCM-AQR-15) via lens Li.
4. Results and Discussion
We used published methods to chemically synthesize the various types of nanoparticles [37, 38], whose aspect ratios were adjusted such that their longitudinal plasmon resonances are close to 800 nm. The nanoparticles are in aqueous solutions. The solutions contain CTAB (cetyltrimethylammonium bromide), which is a surfactant. For the Au/Ag core/shell nanorods, the solution also contains PVP (polyvinylpyrrolidone). The aspect ratio of the nanorods has a standard deviation of ~25% (volume has a standard deviation of ~30%). All sample concentrations were adjusted such that the arrival rate of the particles in the laser focus is very slow (approximately 1 particle every 10 sec), and the probability of more than one particle arriving at a time is vanishingly small . The Au/Ag core/shell nanorods are 60 nm long with 13 nm diameter and have a shell thickness of 2 nm and (ensemble) linewidth of 180 nm, while the Au bipyramids are 100 nm long and 30 nm diameter. TEM images and associated ensemble absorption spectra for the particle types are shown in Fig. 3. The rods and bipyramids have crystalline order, which is observed at higher magnification [37, 38]. Representative time traces, termed trapping trajectories, of Au bipyramids and Au/Ag core/shell nanorod samples are shown in Fig. 4. All the particle types studied here are aligned in the optical trap due to their anisotropy in shape and optical resonance . The CCD detector was used to record two-photon fluorescence time trajectories for 10,000 frames and 100 ms exposure time. In the case of the Au bipyramids, the trapping laser is tuned to 850 nm to optimize the trapping on the red side of the plasmon resonance shown in Fig. 1(c) , while the broader linewidth of the Au/Ag core/shell nanorods requires tuning the trapping laser to 880 nm.
The two-photon fluorescence trajectory for a trapped Au/Ag core/shell nanorod sample is shown in Fig. 4(a). During the first few hundred seconds nothing enters the trap and we only record detector readout noise. Shortly before 300 sec a nanorod enters and remains in the trap for the remainder of the time trace. The spikes in the trajectories are typically observed when a particle enters or escapes the trap or if another object briefly passes through the focus. Also, the signal emitted by a single Au/Ag core/shell nanorod is unpolarized as determined by rotating a polarization analyzer placed before the APD, as expected for two-photon fluorescence.
Figure 4(b) shows a two-photon fluorescence trajectory of trapped Au bipyramids. A single particle enters the trap at ~50 sec followed by a second particle at 125 sec. The fact that the signal intensity is not simply doubled by the addition of the second particle is probably because the latter is not identical to the first. Subsequently, one of the particles escapes the trap (most likely the first one that had originally entered) at ~150 sec, leaving a single particle in the trap until ~400 sec whereupon several particles enter the trap in quick succession and all but one leave at 425 sec. Other trapping conditions such as greater trap laser power or lower particle concentration give trajectories of single particles stably trapped for >10 min.
Assuming that a single step or plateau in a two-photon fluorescence time trajectory is equivalent to 1 particle in the trap, Fig. 5 shows a uniquely informative trajectory of sequential loading of Au bipyramids. We observe six distinct steps or plateaus (labeled 1–6), relative to the baseline (labeled 0), of sequential particle addition. The signal appears to increase nonlinearly with particle number, rather than the linear dependence expected for independent particles. The inset is a plot of the mean value of each step height versus the particle number. The error bars indicate the increasing standard deviation with particle number.
Figure 6 gives the distributions of signal intensity (fluctuations) associated with each step in Fig. 5. Specifically, the values are the instantaneous deviations divided by the mean values for steps 1–6 shown in Fig. 5. The histograms exhibit a positive skew, especially for steps 3–6. Furthermore, unlike the expected Poisson statistical behavior, the noise in the two-photon fluorescence signal is increasing faster than the mean values. Table 1 summarizes the normalized variance for each histogram. The variance for step 0 (i.e., the normalized detection noise) is included for reference.
Figure 7(a) is a series of (intensity normalized) frames of a movie showing the effect of single particles entering the trap, where each panel corresponds to a step (1–6) from Fig. 5. The movie is shown in Fig. 8. There are two interesting features to note. The first is that the diameter of the spot increases with each step. The second is that there is a slight “dip” in intensity near the center of the spot in the fifth panel. These features are more clearly seen in the radial profile plots of Fig. 7(b). The emergence of an apparent Airy disc has been previously observed for slightly defocused images of single emitters . We believe that these unusual features are due to specific geometrical configurations that the multiple particles take while in the trap. We are currently pursuing further studies to understand this behavior.
We assume that the nonlinear dependence of the total two-photon fluorescence signal on particle number in Fig.5 arises from inter-particle interactions [40, 41]. Finite difference time domain simulations (M.L., P.G.S. unpublished results) indicate that field enhancements extend to lateral distances as large as 50 nm from the center of the bipyramid. Therefore, proximity of particles would allow mutual field enhancements. Furthermore, since the minimum signal
is that of independent particles, position or configurational fluctuations that give local field enhancements would increase the total (instantaneous) signal, thereby giving a skew to positive signal values.
The stable trapping of multiple single particles also implies limits and constraints on possible plasmonic or photonic (i.e., scattering force) interactions . A strong (axial) inter-particle interaction would result in a large (red) shifted collective plasmon resonance  that could cause the particles to leave the trap. Conversely, a laterally disposed arrangement could result in a blue shift (i.e., parallel dipoles), which is not incompatible with the PREBOT effect [see Fig. 1(c)] if the interaction and spectral shift are small (e.g., <10 nm). The good stability of multiple particles in the trap (e.g., steps 4 and 6 are each 3 minutes long) suggests that the plasmonic interactions and spectral shifts are of order 10 nm or less. Measurements of spectral shifts to quantify these interactions are currently in progress.
Finally, although the power density of the cw trapping laser is ~108 W/cm2 we do not see evidence of particle melting or aggregation. Particle melting would result in a significant blue shift (~100 nm) of the longitudinal plasmon resonance  thereby weakening the PREBOT effect and ultimately resulting in the particle leaving the trap. Similarly the sequentially loaded trap does not form an aggregate of bipyramids since their very close proximity as an aggregate would lead to a significant shift of the plasmon resonance (particularly if they are touching or fused) and again would diminish the PREBOT effect causing the particles to rapidly leave the trap . We believe that the particles are kept apart due to both the “repulsive” interaction of aligned dipoles that would occur at close separations and the scattering of photons between particles, the latter being effective on ~100 nm length scales . Actually, repulsive interactions would be the critical element in allowing the multiple particles in a single trap to achieve some degree of spatial ordering that could explain the images in Fig. 7.
PREBOT holds promise for various applications including guided fabrication of ordered arrays of nanoparticles, manipulation of plasmonic nanoparticles in complex environments (e.g., cells), and provides a testbed for studying field-induced inter-particle interactions. The present paper used the PREBOT technique to demonstrate stable trapping of metallic particles of different shapes, such as Au bipyramids, and composition like Au/Ag core/shell nanorods. Stable trapping of multiple particles was achieved in the case of the bipyramids, which have narrow intrinsic plasmon linewidths and small inhomogeneous broadening. The associated two-photon fluorescence signal’s apparent nonlinear dependence with particle number for the case of sequential particle loading, affords qualitative insight into inter-particle interactions. Further studies are in progress to understand this nonlinear behavior as well as investigating trapping as a function of different polarization states of the laser. A cylindrically-symmetric trap potential is expected using circularly polarized light or a radially polarized vector beam . Both would result in tighter focusing and hence increased trapping stability compared to the conventional scalar (i.e., linearly polarized) beam and the associated asymmetric focusing .
The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.
We thank Professor Aaron Dinner for insightful conversations. This work was supported by the NSF (CHE-0616663). K. C. T. acknowledges support from the National Science Foundation (DBI-0511849). M.P. and N.F.S. acknowledge the Center for Nanoscale Materials, which is supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. N.F.S. acknowledges the John S. Guggenheim Foundation for a fellowship.
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