Letter pubs.acs.org/NanoLett
Nano-Optical Conveyor Belt, Part I: Theory Paul Hansen,*,† Yuxin Zheng,‡ Jason Ryan,‡ and Lambertus Hesselink‡,† †
Department of Applied Physics and ‡Department of Electrical Engineering, Stanford University, Stanford, California 94305, United States ABSTRACT: We propose a method for peristaltic transport of nanoparticles using the optical force field over a nanostructured surface. Nanostructures may be designed to produce strong near-field hot spots when illuminated. The hot spots function as optical traps, separately addressable by their resonant wavelengths and polarizations. By activating closely packed traps sequentially, nanoparticles may be handed off between adjacent traps in a peristaltic fashion. A linear repeating structure of three separately addressable traps forms a “nano-optical conveyor belt”; a unit cell with four separately addressable traps permits controlled peristaltic transport in the plane. Using specifically designed activation sequences allows particle sorting. KEYWORDS: Plasmonic antenna, optical tweezer, nanoaperture, near-field, optical trapping
I
The independent control of traps required to mediate handoff may be achieved by designing neighboring resonators that respond to distinct wavelengths or polarizations. The resonant wavelengths of metallic nanoaperture traps depend on their shape and size. If the apertures are polarization-sensitive, two identical apertures in different orientations will work as well. Because tightly packed optical resonators will tend to couple and be “on” at the same time, crosstalk presents a major obstacle to independent control. Nanoapertures are superior to antennas in this regard: if their interior walls are separated from each other by one or two skin depths of light (on the order of 50 nm), they exhibit very little crosstalk and their trapping potential wells can easily overlap yet be independently controlled. We present a design for a nano-optical conveyor belt using the optical near field over closely packed C-shaped apertures of well-separated resonant frequencies. The power requirements for C-aperture optical traps will be shown to be modest compared to conventional focused-beam optical traps due to the stronger field gradients. To our knowledge, the C-aperture is the smallest resonant nanostructure to be employed for optical trapping, and it can be packed more tightly than any other demonstrated trap. Simulations show that apertures spaced 50 nm apart have low crosstalk even though their trapping wells overlap. Particles can be handed off down a row of apertures by modulating the illumination wavelength or polarization, forming a conveyor belt (this is demonstrated experimentally in Part II). A design for two-dimensional (2D) transport across a planar interface follows as a natural generalization of the conveyor belt idea. We first study the trapping forces exerted by a single resonant C-aperture cut through a gold film immersed in water.
n conventional optical trapping, polarizable particles are drawn by the dipole force into the brightest region of a focused laser beam, and they can be moved about by translating or refocusing the beam.1 Near-field trapping has been achieved as well using the evanescent fields near dielectric2 or metallic structures;3−6 the optical gradient force draws the particle toward the surface. It has been proposed as well to use strong propagating fields in hybrid plasmonic waveguides7 or fields enhanced by metallic apertures8 to trap particles with diameters of 5 nm and below. Although optical trapping is usually applied to dielectric particles, metal particles can be manipulated with light as well and have been trapped in the fields around plasmonic dipole antennas.9 A trapped particle may also be transported using optical forces. Evanescent waves can simultaneously trap and continuously propel particles, as has been shown near planar interfaces10,11 and waveguides.12,13 Optical forces near asymmetric nanoantenna arrays will bias Brownian diffusion in a chosen direction with a mean drift of several micrometers per second.14 Particles’ positions can also be controlled deterministically. One technique uses metal cylinders;15 under illumination by polarized light, optical traps are formed on two opposite sides of the cylinder. By rotating the polarization of the light, the optical traps are made to rotate around the cylinder, and they can drag a particle around its perimeter. This transport technique provides fine-grained control over particle position without the need for translation stages, spatial light modulators, or other complicated optical equipment. However, trapped particles could also be transferred from one near field trap to another. Two optical traps may be engineered so that their trapping force fields substantially overlap. If the traps are independently controlled, the particle may be moved at will from one trap to the other. We call this controlled transfer of a trapped particle between adjacent traps “handoff”. © XXXX American Chemical Society
Received: October 27, 2013 Revised: April 28, 2014
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produces the trap depth for each direction. The forces and trap depths are shown in Figure 2. To obtain the depth in the zdirection, we integrate Fz as the particle is moved from 20 nm above the aperture to 220 nm above the aperture. In a similar manner, we obtain the depth in the y-direction by integrating Fy as the particle moves from y = 0 nm to y = 300 nm, elevated 20 nm above the aperture. The aperture has no x-symmetry so we separately calculate the trap depth as the particle moves from x = 0 nm to x = −300 nm and from x = 0 nm to x = 300 nm. The direction of weakest trapping sets the power requirements for the trap. In simulation, the aperture is illuminated from below by a uniform plane wave with intensity of 1 mW/μm2. We measure force in units of pN and energy in units of kBT at T = 300 K. The C-aperture trap is deepest in the z-direction (2kBT) and weaker in x (1.56kBT) and y (1.74kBT). The force decays exponentially in the z-direction, indicative of the evanescent nature of the near-field. In the x- and y-directions, the force is strongest at a distance of 50 nm, which we call the trap radius; the transverse force falls to zero at the center of the trap. These forces exceed the forces exerted by a focused laser beam by a factor of 10 or more. The peak force exerted in a conventional optical trap is often quantified in terms of a dimensionless trapping efficiency Q:19
The C-aperture is a polarization- and wavelength-selective structure that may be designed to resonate at optical or nearinfrared wavelengths with a fwhm of roughly 100 nm.16,17 Capertures are named by their characteristic size α, the width of the two arm segments. A “C-50” aperture is shown in the simulation schematic Figure 1. This C-50 design with a 50 nm
Figure 1. Trapping simulation schematic with structural dimensions. A C-50 aperture (α = 50 nm) is shown. All force calculations to follow include the trapped particle in the optical calculations. The force is calculated with an integral around a flux box separated by at least 10 nm from the particle and gold surface.
characteristic size (arm height) resonates at 1080 nm in water. The particle under consideration is polystyrene (n = 1.58) with a diameter of 200 nm. “Trap depth” is the change in potential energy of the particle as it is moved from a position infinitely far away to a position close over the aperture. Because of an asymmetric optical scattering force, the trap depth depends on the direction along which the particle approaches the trap (alternatively, the particle is more likely to escape the trap along certain directions). Using the finite-difference time-domain (FDTD) method we calculate the electromagnetic fields around the aperture and particle, then integrate the flux of the Maxwell−Minkowski stress tensor18 around the particle to obtain the optical force for particles approaching the aperture along the +x, −x, −y, and −z directions. Integrating the force along the approach paths
Q=
Fc nP
(1)
where F is the peak force exerted, P is the incident power and c/n is the propagation speed of light through the background medium. The trapping efficiency varies with the size and polarizability of the trapped particle, and it has been tabulated before for the case of a tightly focused 1060 nm laser beam.19 We may perform the same calculation for the near-field trap, assuming a nominal total laser power of 1 mW. While the conventional focused beam trap obtains an axial (z) trapping efficiency of Q = 1.1 × 10−3 at d = 200 nm, the C-aperture trap has an efficiency of Q = 1.216 × 10−2 in x, Q = 1.506 × 10−2 in y, and Q = 6.745 × 10−2 in z.
Figure 2. Trapping force and potential over a C-aperture. Top row, left-to-right: force in x-, y-, and z-directions. Bottom row, left-to-right: potential energy in x-, y-, and z-directions. Incident optical power assumed to be 1 mW/μm2. B
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Optical forces grow weaker as the particle size diminishes. In the Rayleigh regime we expect the total force to scale as d3, but as the particle approaches the size of the trap the dependence becomes harder to predict. We calculate the force exerted on particles of varying size held 20 nm over the center of a C-50 aperture (Figure 3). The force exceeds 0.15 pN for diameters as
Figure 3. Optical force versus particle size. The particles are centered over a C-50 at a height of 20 nm above the surface, so the x- and yforces are close to zero. The incident optical intensity is 1 mW/μm2. Figure 4. (a) Intensity enhancement measured at z = 25 nm above an optical conveyor belt under illumination by 970, 1100, and 1230 nm xpolarized light. The apertures are C-40, C-50, and C-60 and their borders are separated by 50 nm. One unit cell of the conveyor belt (length Lx = 480 nm) is highlighted in the center. The entire simulation space (three unit cells, as shown) is periodic in the xdirection. The illumination intensity is 1 mW/μm2. (b) Conveyor belt force Fx, d = 200 nm, under uniform illumination of 1 mW/μm2. Vertical dashed lines mark locations of traps as determined by zero crossings of force. (c) Conveyor belt potential, d = 200 nm, under uniform illumination of 1 mW/μm2. Vertical dashed lines mark locations of traps as determined by zero crossings of force.
low as 100 nm, corresponding to a trapping efficiency of 0.04, exceeding the longitudinal efficiency of a focused laser beam trap (3 × 10−4) by 2 orders of magnitude. The shape and size of the trapping potential also change with particle diameter. We repeated the trap depth calculations for d = 400 nm (Figure 2). The trap size is larger for larger particles because the attractive force is mostly exerted on the particle− water interface closest to the aperture. The dependence of trap depth and radius on particle size provides a potential means of passively discriminating between particles of different sizes. Because C-apertures are resonant structures, apertures of nonoverlapping resonant wavelengths can be separately addressed by illumination wavelength. Even when separated by as little as 50 nm, a “blue” C-40 and “green” C-50 will respond nearly independently to their respective resonant wavelengths, and they can be used as independently controlled optical traps. By simultaneously turning off a blue light source and turning on a green light source, a particle trapped over the blue aperture can be transferred to the green aperture. This hand-off can succeed as long as the attractive force of each aperture extends over the adjacent aperture, a condition which our design meets as demonstrated below. A repeating chain (Figure 4) of three apertures (“blue” C-40, “green” C-50, and “red” C-60) separated by 50 nm can linearly transport a trapped particle over a long distance. Under broad illumination all the C-40s will transmit as one set, all the C-50s as one set and all the C-60s as one set, so all the traps can be controlled without moving any illumination beam. Consider a particle trapped over a blue aperture. At time t0, the blue illumination is turned off and green turned on. The particle is released from the blue trap, but it is drawn into the well of the adjacent green trap within several milliseconds. At time t1 ≈ 10 ms, the green traps are deactivated and the red traps activated, and the particle shifts again. On the next illumination cycle, it will move to the next blue aperture. This is the mechanism for a peristaltic optical conveyor belt. If necessary, particles can be prevented from entering the apertures by filling the apertures with dielectric. We show such a design in Part II. We demonstrate that handoff and transport are possible for a bead 200 nm in diameter by simulating a C-40-50-60 conveyor
belt, calculating the forces for bead positions along the x-axis at an elevation of 20 nm above the surface (Figure 4b,c). The simulation space is nine apertures wide in x (three unit cells) and uses periodic boundary conditions in the x-direction to simulate an infinitely long chain of apertures. Three illumination wavelengths (970, 1100, and 1230 nm) are considered in turn. For each case, the illumination is uniform (a plane wave) with intensity of 1 mW/μm2. The particle is included in the simulation and is also periodically repeated in x at a spacing of 1440 nm due to the boundary conditions, however the effect of duplicating the particle at such a distance is negligible. The sizes, resonant wavelengths (for isolated apertures and in the conveyor belt), resonance widths, and trap depths (along x) of the apertures are listed in Table 1. At its Table 1. Properties of C-Apertures in Conveyor Belt feature size α
label
resonant λ
fwhm
depth Ux
40 nm 50 nm 60 nm
blue green red
970 nm 1100 nm 1230 nm
71 nm 92 nm 128 nm
1.1 kBT 1.75 kBT 3 kBT
resonant wavelength, each aperture is shown to exert a positive force Fx > 0 on particles over the aperture to its left, and a negative force Fx < 0 on particles over the aperture to its right. Consequently, a particle over a dark aperture can be drawn into the well of an adjacent bright aperture. C
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Figure 5. (a) Simulated trajectories of 200 nm (red) and 400 nm (green) particles under 50 Hz forward motion protocol. Mean trajectories of 200 and 400 nm beads (nearly indistinguishable) are indicated with thick lines. Illumination protocol below. (b) Transport speed in forward motion protocol as a function of frequency. Inset: one cycle of the forward motion protocol. Full intensity is 18 mW/μm2 for blue, 11 mW/μm2 for green, and 7 mW/μm2 for red. (c) Simulated trajectories of 200 nm (red) and 400 nm (green) particles under 12.5 Hz sorting protocol. Mean trajectories of 200 and 400 nm beads are indicated with thick lines. Illumination protocol below. (d) Transport speed in sorting protocol as a function of frequency. Between 10 and 16 Hz, the two particle sizes will move in opposite directions, allowing sorting of particles. Inset: one cycle of the sorting protocol.
Although the conveyor belt exerts different forces on different-sized beads, 400 nm beads will also advance at 19 μm/s under this same illumination pattern. The staircase-like trajectories in Figure 5a indicate that when the illumination switches the particles move forward to the next trap immediately and then remain near the center of the trap until the next switch. As the repetition rate increases, the particles spend less and less time sitting still, and their mean velocity increases linearly with repetition rate until at some point the particles are unable to keep pace. This point marks the maximum velocity of the particle under the given illumination pattern. We find that 200 nm beads can move as fast as 33 μm/s and 400 nm beads at 22 μm/s. The dependence of velocity on repetition rate (Figure 5b) is linear for lower frequencies but drops off sharply after fcutoff = 110 Hz for 200 nm particles and fcutoff = 70 Hz for 400 nm particles. This suggests a simple way to separate particles of different sizes: modulate the illumination at a frequency above 60 Hz, and the larger particles will begin to lag behind the smaller ones. We exploit this speed limit to create a protocol to sort small and large particles. We concatenate two cycles of slow (50 Hz) B-R-G illumination with six cycles of fast (150 Hz) B-G-R illumination. The repetition rate of the combined illumination pattern is 12.5 Hz. In the slow phase, both particles move forward by two unit cells, but in the fast phase the large particles tend to sit still and the small particles move to the right by up to six unit cells. After two seconds of simulation, the 5000 small particle trajectories have all moved right and the 5000 large particle trajectories have all moved left with a mean separation distance of 19 μm (see Figure 6). Essentially complete separation of the two species can be achieved within
We also simulate trapped particles’ equations of motion to characterize the performance of the conveyor belt. The particle’s motion is modeled by the modified Langevin equation β , x(̇ t ) = F(x , t ) +
2kBTβ W (t )
(2)
where x(t) is the particle’s position along the x-axis, β is the viscous drag coefficient, F(x,t) is the optical force (we use Fx from the electromagnetic calculations), and W(t) is a noise term that models random collisions from fluid molecules. The drag coefficient depends on the sphere diameter, fluid viscosity, and distance to the gold surface.20 The equation of motion is solved numerically in the time domain with a stochastic Euler’s method21 xn + 1 = xn +
Δt F(tn , xn) + β
2kBT Δt Wn β
(3)
The values of the random variable Wn are normally distributed with unit variance and zero mean. We first study a forward transport protocol that sequentially uses only blue, then green, then red illumination (Figure 5). The intensities of these three lights will be referred to as R, G, and B for the red, green, and blue lights, respectively. Each light is on in turn for 6.67 ms, so the entire pattern repeats at 50 Hz. Because the trap depths per mW of incident power are not the same (see Figure 4c), the maximum intensity of B is scaled to 18 mW/μm2, of G to 11 mW/μm2, and of R to 7 mW/μm2; thus each trap set has a depth of 20 kBT for the motion calculations. A 200 nm particle is released at time t = 0 s from the point x = 0 nm and tracked until t = 0.25 s; this experiment is repeated 5000 times. We find that the mean forward velocity of the particle is 19 μm/s. D
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Figure 7. Intensity enhancement measured at z = 25 nm above a 2D conveyor belt. The unit cell of this x- and y-periodic design is shown under four illumination conditions: 979 nm x-polarized light (Bx), 1181 x-polarized light (Rx), 979 nm y-polarized light (By), and 1181 nm y-polarized light (Ry). Each illumination condition activates four traps, shown as bright spots in the figure. The mean center-to-center spacing of the apertures is 200 nm, and the unit cell is 693 × 1200 nm. All illumination intensities are 1 mW/μm2. N.B.: right- and left-facing apertures both respond to x-polarized light, and up- and down-facing apertures both respond to y-polarized light.
Figure 6. Distribution of particle positions under sorting protocol from 5000 trials for each particle size. By t = 2 s, 400 nm beads will be expected to have moved 10 μm to the left, and 200 nm beads 12 μm to the right.
one second over a belt length of little over 10 μm. Other illumination protocols for selectively moving particles will be presented in an upcoming paper. Because many nanostructures including C-apertures are polarization-sensitive, the polarization of light may be used to control optical traps as well. When x-polarized light is incident on a vertically oriented C-aperture, it gives rise to a very strong hotspot, but under illumination by y-polarized light there is very little field enhancement. A horizontally oriented “U” aperture (a 90° rotated C-aperture) will respond to y-polarized light but not to x-polarized light. A particle may thus be handed off between closely spaced horizontal and vertical apertures. Polarization can also be used to control a conveyor belt with traps of three different polarization states, as we present experimentally in Part II. Forward motion along a path requires a minimum of three sets of separately addressable traps. By adding a fourth addressable set of traps, it is possible to guide particles in various directions along a plane. One approach uses both polarization and wavelength as discriminants between traps. Two wavelengths of light, each with controllable x- or ypolarization, provide a total of four addressable sets. We will refer to these traps and their corresponding illumination intensities as Rx, Ry, Bx, and By (Table 2).
overlap adjacent apertures; in Part II, we present experimental data demonstrating that handoff of 200 nm beads is possible between apertures spaced 310 nm apart. Three simple transport protocols are shown in Figure 8. Other 2D trap arrangements and interesting illumination protocols will be presented in future publications.
Figure 8. Simple transport protocols on a 2D conveyor belt. Dimensions are as listed in Figure 7 and Table 2. Protocol 1 (Bx,Ry,By) creates clockwise transport. Protocol 2 (Rx,Ry,By) creates counterclockwise transport. Protocol 3 (Bx,By,Ry,Rx) creates linear motion northeast and southwest in alternating lanes.
Table 2. Apertures in 2D Conveyor Belt feature size α 40 40 60 60
nm nm nm nm
rotation angle
label
resonant λ
0°, 180° 90°, 270° 0°, 180° 90°, 270°
Bx By Rx Ry
979 nm 979 nm 1181 nm 1181 nm
The capture and transport properties of the nano-optical conveyor belt potentially lead to a number of applications. The peristaltic transport capability immediately suggests use as a platform technology for routing nanoparticles or biological cells, suitable for on-chip integration. Transport is inherently massively parallel, and many possible directions of motion can be determined by the pattern of traps on the chip. A spoke pattern can collect particles from a wide region and gather them to a central point. Closed rings can move particles in a circuit, which could be used to create holding patterns, recirculation, or even to impel the fluid medium in a channel. Using four or more addressable trap sets permits user-controllable routing, for instance, by selecting a direction at a junction of two conveyor belts.
There are many ways to arrange four trap types in a lattice for 2D transport. One such arrangement is shown in Figure 7. An aperture belonging to a given addressable set, for example, Rx, has three nearest neighbors on the lattice, one belonging to each of the three remaining addressable sets. Thus, by modulating the four illumination sources, a particle may be moved between any two apertures on the lattice. The mean center-to-center spacing of the apertures is 200 nm. Although this distance is slightly larger than the center-to-center spacing in the simulated 1D conveyor belt, the potential wells still E
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ACKNOWLEDGMENTS We acknowledge the United States National Science Foundation for partially funding this research, award no. 1028392.
Because the C-aperture itself has a different trapping radius and trap depth for different particle sizes, it has an inherent filtering capability that can be used for sorting particles. Widely spaced conveyor belts cannot hand off smaller particles; at the same time, the lateral force exerted on a large particle is very low, so there is also an upper limit to the size of particle that can be moved by a given design. We envision trap patterns and illumination protocols that can selectively route different types of particles along different paths. Selection could also be based on particles’ refractive indices (which determines the strength of the optical forces). These ideas will be explored in a future publication. Particle transport may be exploited for other purposes such as peristaltic nanoscale pumps, where trapped objects are used to drive fluid along a channel. A conceptual design for a pump is shown in Figure 9. Fluid will be driven by the motion of
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Figure 9. Rotary pump design. As particles are driven clockwise over unevenly spaced traps, fluid is drawn in on the left and expelled on the right.
particles over the conveyor belt. If the gaps between traps on the conveyor belt are larger on the left (inflow) side than on the right (outflow) side, the particles will also squeeze together on the outflow side, creating a pressure differential that draws fluid in on the left and expels it on the right. Similar designs could drive rotors between discrete displacement angles. The optical power to operate a conveyor belt increases with the number of simultaneously illuminated elements. The density of apertures in our 2D design is 19.24/μm2. Assuming a mean operating intensity of 15 mW/μm2, each aperture requires about 0.78 mW, and 1 W of average power can operate over 1200 apertures. Because the illumination need not be tightly focused, it can be provided by inexpensive diode lasers.22 We have described peristaltic hand-off of nanoparticles in an optical trapping system, a new method for control, routing, sorting and other manipulation of particles. This scheme can be implemented using simple optics and no moving parts. The feasibility of hand-off for 200 and 400 nm diameter particles is demonstrated explicitly. A long chain of separately controlled apertures may be used as a conveyor belt. Wavelength and polarization can both be used to discriminate between sets of traps.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. F
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