## Abstract

An all optical method for dielectric Rayleigh particle sorting possesses significant advantages. Here, we describe an approach that applies optical scattering forces to translate varied sized particles differentially from a surface followed by the introduction of an optical standing wave to maintain and tighten the positional tolerance of the differentiated particles in the medium. Numerical simulation demonstrates the workability of this scheme; which is highly dependent on Brownian forces typically dominant at this length scale. It also shows the significant impact of temperature and medium viscosity on the operation of this technique.

©2009 Optical Society of America

## 1. Introduction

Nanospheres offer exciting vistas in burgeoning areas such as drug delivery [1] and nanoscale lithography [2]. For the latter field in particular, the availability of precise nanosphere sizes is crucial. This can be accomplished by developing improved methods of manufacture which is challenging to achieve and maintain [3]. Another approach will be to create nanospheres with a looser size tolerance and then introduce a sorting method to obtain the right sizes. Contacting sorting approaches are not feasible at the nanoscale due to the inherent propensity for handling damage. The use of momentum transfer between laser beam and particle offers an exciting avenue for nanosphere sorting without destruction or contact. The majority of optical methods reported rely on using the flowrate of particles in tandem with a landscape of radiation forces to operate somewhat as either sieves or deflecting elements to carry out sorting [4–11]. It is well accepted, however, that a good control of the fluidic flow necessitates careful design and skilled tuning. A recent report of using displaced and flashing optical landscapes offers a flow independent approach to sorting micron scale particles [12]. At the sub-micron scale, particle motion and thus sorting, is strongly influenced not only by fluid flow but by Brownian diffusion. The coupling of photophoresis with asymmetric potential cycling presents a possible non-flow approach for sorting at this regime [13]. Plasmon excitation of particles using laser light is yet another method amenable for sub-micron scale sorting [14]; albeit it is limited to differentiating metallic particles.

The use of an optical standing wave for particle trapping was first reported by Zemanek et.al. [15]. As the scattering force is strongly suppressed with a laser beam that is reflected strongly from a mirror, trapping in the axial direction is greatly enhanced and occurs at multiple axial locations with this method. Nevertheless, single particle manipulation using standing wave traps remains limited by two practical issues. Firstly, moving the laser beam together with a mirror is cumbersome, and secondly an approach of applying radiation from two opposing coaxial optical fibers can achieve a trapping effect as well [16]. Optical standing waves, however, manifest themselves in a periodic field akin to interference fringe patterns and have been demonstrated to be effective in the context of multiple particle transport and sorting [17]. Similarly, ultrasonic standing waves are predominately used for the manipulation of large numbers of micro particles [18, 19].

In this work, we propose a method of sorting Rayleigh spherical particles that uses optical scattering forces to move particles differentially according to size followed by an optical standing wave to trap the sorted particles in layers at the opportune moment. The key to this method is the relationship of the scattering forces to the particle radius to the sixth power. Numerical simulation is performed to verify the workability of this scheme using established optical particle force relationships and Brownian dynamics.

## 2. Theory and description

A key feature in the sorting scheme lies in the devise an optical setup that will permit both optical scattering and standing wave trapping to occur. The proposed setup (Fig. 1) has two laser beams at different wavelengths illuminating the sample at opposite directions, aligned such that they are coaxial. A step response optical filter, with transmittance characteristics shown in the inset A, is placed between the two laser beams. As the beam from below (laser 1) operates at λ > λ_{o}, the filter will allow almost all light to pass upwards (in a diagrammatic sense). There are two components to the resulting force acting on the particles; the scattering and gradient force. These need be considered in two directions, axial and radial (assuming the system is axially symmetric). The scattering and gradient force acting on *a* Rayleigh particle of radius a at any radial position *r* is given by [20]

$${\mathrm{Fz}}_{\mathrm{grad}\_\mathrm{tw}}=-\hat{z}\frac{32{n}_{2}{a}^{3}}{c}\mathrm{Pm}\prime z\prime \frac{A}{{k}_{\mathrm{tw}}^{2}{w}_{\mathrm{tw}}^{4}+{z\prime}^{2}}\left(1-2{r}^{2}A\right)\text{exp}\left[-2{r}^{2}A\right]$$

$$A=\frac{{k}_{\mathrm{tw}}^{2}{w}_{\mathrm{tw}}^{2}}{{k}_{\mathrm{tw}}^{2}{w}_{\mathrm{tw}}^{4}+4{z\prime}^{2}}$$

Here *ẑ* represents the unit vector in the *z* or axial direction, with *z′* being the distance in the *z* direction from the beam waist, *w _{tw}* is the waist radius,

*r*is the radial distance,

*k*= 2π/λ, λ the wavelength of light,

_{tw}*m*′ = (

*m*

^{2}-1)/(

*m*

^{2}+2), and

*m*=

*n*/

_{1}*n*the relative refractive index of the particle,

_{2}*n*the refractive index of the particle,

_{1}*n*the refractive index of the surrounding medium,

_{2}*c*the speed of light in vacuum, and

*P*the power of the beam. We refer to the forces generated with laser 1 as being traveling wave forces – denoted with the subscript

*tw*– to avoid repetition that the axial force consists of both a scattering and gradient component. Note, however, that the scattering force is dominant in this configuration. Based on the assumption that the radial scattering force is zero, the gradient force is given by [20]

As a second laser beam from above (laser 2) operates at λ < λ_{o}, the filter will almost totally reflect the irradiation back upwards. This will then create an optical standing wave in which trapping can occur. The trapping forces acting on the Rayleigh particle, if the waist of the beam is assumed to be close to the mirror surface and the mirror surface is perfectly reflecting (thus making the axial scattering force equal to zero), is given by [15]

$${\mathrm{Fr}}_{\mathrm{grad}\_\mathrm{sw}}=-\hat{r}\frac{16{n}_{2}{a}^{3}}{c}\mathrm{Pm}\prime r{B}^{2}\left(2+2\mathrm{cos}\varphi +\frac{4z}{{k}_{\mathrm{sw}}{w}_{\mathrm{sw}}^{2}}\mathrm{sin}\varphi \right)\mathrm{exp}\left[-2{r}^{2}B\right]$$

$$\varphi =2{k}_{\mathrm{sw}}z+\frac{4z{k}_{\mathrm{sw}}{r}^{2}}{{k}_{\mathrm{sw}}^{2}{w}_{\mathrm{sw}}^{4}+4{z}^{2}}-2{\mathrm{tan}}^{-1}\left(\frac{2z}{{k}_{\mathrm{sw}}{w}_{\mathrm{sw}}^{2}}\right)+\psi $$$$B=\frac{{k}_{\mathrm{sw}}^{2}{w}_{\mathrm{sw}}^{2}}{{k}_{\mathrm{sw}}^{2}{w}_{\mathrm{sw}}^{4}+4{z}^{2}}$$

where *Ψ* denotes the phase shift upon reflection, *z* is the axial distance (if the focal points of both lasers align, *z*′ = *z*) and all other terms are as previously described. It should be noted that the radial scattering force is also zero in this case.

Let us consider the principle in the axial (*z*) direction. Particles of various sizes are initially located at the surface of the filter. When laser 1 is turned on, this causes the particles to move upwards from the surface under the influence of the travelling wave force. The larger particles will experience a greater motive force upwards than the smaller ones as the dominant scattering force is proportional to radius to the sixth power. In addition Brownian motion will cause a Gaussian spread of the particles’ end positions. After a specific period, chosen for optimum sorting efficiency, laser 1 is switched off and laser 2 switched on. This fixes the location of the separated particles and prevents further Brownian lead motion from washing out the sorting achieved by the axial traveling wave forces.

It can be seen that while Brownian forces are present in this approach, it is not utilized in the sorting. Brownian forces can generally be used in the scheme of sorting by providing motion to particles. Even when the Brownian type particle movement can be made to dominate in one axis, it is random in nature [13]. The use of traveling wave forces dominated by scattering forces provides a high level of size separation sensitivity (radius to the sixth power). Brownian motion will continue to exist in the system but will represent an artifact to be surmounted. Whilst the optical forces are expressed above in terms of radial and axial components, it is much more convenient to break the radial force into *x* and *y* components when dealing with Brownian motion. The Langevian equations for motion of a single particle can be resolved in the respective axis as

where *M* is the particle mass, *β* = 6*πμa* is the drag coefficient (*μ* being the viscosity), *k* the Boltzmann’s constant, *T* the temperature, *i* is the axis under consideration (whether x, y or z) *F _{opt}*(

*i*) the optical force, and ξ is Gaussian white noise of unit strength [21]. The dynamics can be taken for which inertia is ignored (

*M*= 0), this is an acceptable approximation even in air for particles of these radii [22].

## 3. Modeling and results

The Langevian relationships in Eq. (4) can be solved in x, y and z (axial) directions via numerical time steps. We consider particles of radii 24, 27 and 30 nm in air, and a refractive index of 1.6. The sizes of the particles are restricted to the Rayleigh regime by the equations. The traveling wave laser has a wavelength of 650 nm, a waist of 2 μm positioned at the upper surface of the filter, power is 10 W, and is used only as a single pulse. In the case of the standing wave the waist (2 μm) is again located at the top filter surface. However, the beam has a wavelength of 600 nm and power of 1 W. The time step used in the simulation is 50 ns and we consider operation at temperatures of 100, 200 and 293K. Figures 2(a) to 2(d) depict the axial and radial forces by the traveling and standing waves respectively, calculated in the case for a 30nm radius polystyrene spherical particle. The Brownian fluctuations occur on top of this force and depending on particle size, viscosity and temperature can be the dominant effect. Note that as we assume the viscous forces to be highly dominant at this size-scale, the first term in Eq. (4) relating to inertia can be neglected. This allows inference that at each moment in time the particles instantaneously reach a terminal velocity dictated by local optical conditions and the random variable governing the Brownian forces.

Considering the case of using a traveling wave alone, we see that separation occurs fleetingly by examination of the results in Fig. 3. In this depiction, each size particle is assigned a color, and three lines are plotted to represent the probability (the 10^{th} percentile, the median, and the 90^{th} percentile) in which the each sized particle will likely reside in the axial location sense. The number of particles used was 500 and the time step applied was 50 ns; these values arrived at through a procedure seeking convergence in the simulation results. Even with a high power laser beam (10W) at 200K, the Brownian forces causes wide spreading of the particles within the 0.3 s considered. The simulation is performed for nitrogen, the viscosity of which at 200 K is 12.79 μPa s. One point to note is the particles showing a tendency to translate directionally and quickly from the surface initially before succumbing to random Gaussian movement further away. This is consistent with the optical axial force distribution of the traveling wave. Secondly the Gaussian spread of the Brownian forces is not reflected by a strict Gaussian distribution of the end location of the particles (the 10^{th} percentile, lowest, is further from median line, than the 90^{th} percentile, highest). This is because the Brownian forces may induce off-axis trajectories as the particles move with each time step. When the particles move off-axis, axial forces drop; which reduces the restoring capability of particles moving towards an on-axis trajectory. This skews the distribution of the particles. This is evident in the scatter plot of the particle end locations furnished in Fig. 3(b). This result illustrates the fleetingly observable separation caused by the traveling wave forces alone.

The next set of simulations is similar; however with laser 1 turned off and laser 2 on after 0.015 s. From this point in time, particle motion is dictated by a standing wave regime. The simulation demonstrates (Fig. 4(a)) that the standing wave locks the location of the particles somewhat to their sorted locations after 0.015 s. It can be seen that the degree of separation is improved between all three sizes – with clear separation between the largest and smallest particle sizes. It should be noted that when held by the standing wave, the spread of particles is such that they are located in a number of standing wave equilibrium positions, the key point being that they are held spatially separate in time. Figure 4(b) gives a scatter plot of the particles end location (axial, radial) which depict the extent of distribution.

To investigate the effect of temperature (which also affects medium viscosity), we perform further simulations at 293 K (17.7 μPa s) and 100 K (6.66 μPa s). The results are shown in Figs. 5(a) and 5(b) respectively. It can be seen that separation improves with lowered temperatures. When particles are subject to Brownian forces alone, their trajectory will spread over time Δ*t* along one axis by

This corresponds to a Gaussian distribution with a standard deviation of the squared-root term on the right where *β* is the drag coefficient. Both temperature and viscosity terms are present which indicate an interrelation [23]. For the three cases examined, a standard deviation of approximately 3.3 μm can be expected after 0.015 s for the 30 nm radius particles, this converts to an expected 4.3 μm spread between the location of the median line, and that of the 10^{th} and 90^{th} percentile (we term these the lower and upper spread respectively). In a scenario in which only Brownian forces are present, the effect of increased temperature can be expected to be almost entirely cancelled by that of increased viscosity.

The spreads that occurs under the optical traveling wave and Brownian force influences are presented in Table 1. One obvious trend is that the minimum lower spread is obtained for the lowest temperature and the lower spread is greater than the upper spread. Firstly, we examine the 100K case, wherein the total spread is considerably lower than intuitively perceived if we consider on-axis movement alone. Essentially, if a particle moves forward at a faster rate due to Brownian motion, there is 50% chance that it will find itself in a region of lower optical forces (as the optical force drops with axial distance). Hence its progress is slowed, providing a tendency to keep the particles bunched together. There is naturally also Brownian motion affecting radial movement of the particles as well. This causes the particles to enter regions of lowered axial optical forces at radial displacements from the optical axis. This hinders progress along the axial direction and causes a particle distribution skew; as seen by the trend of a larger lower spread than upper spread across all temperatures examined. Associated with the radial Brownian forces that tend to move the particles off-axis are the optical radial forces which try to return the particles back on axis where the optical progression of the particles is highest. These radial forces are most efficient when they act against low drag forces; which mean that the particles should be held more effectively on axis at lower viscosity regimes. This accounts for the lower spreading observed in the radial location scatter plots. A second natural observation is the faster axial progression at lowered temperatures in which viscosity is also lowered accordingly.

Clearly, lowered temperatures are best for sorting as the associated lower viscosity causes the optical forces to move particles more efficiently as well as reducing the influence of Brownian motion. This is confirmed in Table 2, which gives the percentage of each particle type in different axial zones. The distribution curves of the particles at 0.03 s are shown in Fig. 6; which have been produced by counting the number of particles in 0.1 μm axial steps. This distance is lower than the distance between adjacent standing wave potentials; so the curves have a sharply changing profile within a bell (non-Gaussian) envelope. The axial zones were selected as being separated at the points at which these distribution profiles intersect. Thus when considering the sorting of 3 particle sizes at 200 K, the intersection between the 24 μm and 27 μm profiles is used for the lower boundary, and the intersection between the 27 μm and 30 μm profiles for the upper boundary. Hence, in sorting two particles sizes (e.g. the 24 μm and 30 μm particles) at any specific temperature, the boundary to be used is essentially the intersection of these two distribution profiles.

As has been seen, the Brownian forces alone are largely unaffected by temperature due to the almost linear relationship between viscosity and temperature. However, the drop in viscosity helps the optical forces become more dominant. Other options to achieve the same result would be to increase the laser power. However, this has been limited to 10W in this study, as a value achievable with commercially available pulsed (for operation at 0.015 s duration) lasers.

The second stage of the manipulation – which is the actuation of the standing wave using laser 2 – is also affected by the relationship between optical and Brownian force magnitudes. As such, the optical forces must be able to dominate the artifact Brownian forces, ensuring successfully trapped particles, as discussed by Kramers [24]. A medium with higher viscosity supports longer times for the standing wave forces to bring the particles back to the potential well as a consequence of Brownian induced fluctuations. There is a clear trend that the higher the temperature, the less well the clouds of particles are held in a constant distribution. This is demonstrated by the slightly diverging lines in the second phase of the 293 K simulation. Once again, raising the laser power used would aid this; but practicable issues will have to be considered as this laser would be need to operate continuously once actuated.

In both cases, increased particle radii would both increase optical force effects with both lasers; the scattering force in the traveling regime being related to the sixth power (the laser beam has been set such that the dominant traveling wave effect is the scattering), and the standing wave having a cubic relationship. In the case of larger particles in a standing wave, the affinity to the bright fringes will change periodically with increased size, however, as the collection of particles is over many wavelengths, this will not play a significant role, provided the amplitudes remain high enough [25]. The values of the radii used here are limited by the Rayleigh regime criteria which is one twentieth of the optical wavelength. This is a restriction imposed due to the equations used, rather than a physical restriction. As the radius is increased [20] the sixth power relationship for scattering is diminished to a lower value; however there is still a strong force dependency on radius adequate for effective sorting, and the resultant forces should be strong enough to allow room temperature operation as the Brownian forces are also reduced with increasing radius (to the power ½).

The essential feature in this scheme lies in the use of the high power relationship that allows for sorting of relatively similar sized nanoparticles, where clear sorting between particles differing in radius by 20% has been observed in the simulation results even at room temperature. A realization of this method will rely on successful trapping at low temperatures, and a method for subsequent particle extraction or fixing.

## 4. Conclusions

A scheme has been presented, whereby, the highly dependant force amplitude relationship on particle size in a traveling wave can be utilized for sorting Rayleigh dielectric particles operating in a regime where Brownian forces are dominant. The method uses a two phase scheme, starting with the traveling wave before alternating to a standing wave at an opportune moment to restrict movement of particles in time and space to resist the sorting from being washed out by subsequent Brownian fluctuations. A high degree of sorting has been demonstrated between nanosized particles which vary by just 10% in their radii. The nature of the competing forces has been examined by varying the temperature at which the simulations are performed.

## Acknowledgments

The authors acknowledge support from ARC Discovery project grant DP0878454.

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