## Abstract

The radiation forces and trajectories of Rayleigh dielectric particles induced by one-dimensional Airy beam were numerically analyzed. Results show that the Airy beam drags particles into the optical intensity peaks, and guides particles vertically along parabolic trajectories. Viscosity of surrounding medium significantly affects the trajectories. Random Brownian force affects the trajectories. Meanwhile, trapping potential depths and minimum trapping particle radii in different potential wells were also discussed. The trapping stability could be improved by increasing either the input peak intensity or the particle radius.

©2010 Optical Society of America

## 1. Introduction

Since Ashkin et al. proved that three-dimensional trapping of a dielectric particle is possible by use of a single, highly focused laser beam [1], optical tweezers have become an indispensable tool for manipulating small particles without any mechanical contact. The optical tweezers have been used to manipulate and trap micro-scale objects [1], liquid droplets [2], and even some submicron objects such as viruses [3] and silver nanoparticles [4]. Conventional tweezers usually utilize Gaussian light beams, which suffer from strong divergence off the focal plane, trapping particles with only a few micrometers apart in the axial direction. The “non-diffracting” beams, especially Bessel beams, do not spread while propagating, even if the beam diameter is reduced to the size of a tightly focused Gaussian laser beam. It has been used to trap atoms and microscopic particles in multiple planes [5], construct particle conveyor belts [6], sort microfluidic cells and transfect cells [7].

A second type of “non-diffracting” beam, the Airy beam, was observed in experiment [8,9]. Its key difference from the Bessel beam is that it additionally experiences a transverse acceleration and can self bend even in free space. Due to its unique properties, Airy beam has recently attracted a lot of attentions because of its potential applications in plasma guidance [10], vacuum electron acceleration [11], and generation of three dimensional optical bullets [12]. Jörg Baumgartl and colleagues [13,14] experimentally demonstrate the first use of the Airy beam in optical micromanipulation. As opposed to Bessel beams, it can transport microparticles along curved self-healed paths, and remove particles or cells from a section of a sample chamber. Its novelty is that the trapping potential landscape tends to freely self-bend during propagation. Moreover, the diffraction free distance and the bend degree of Airy beam can be controlled, and the acceleration direction can be switched by a nonlinear optical process [15]. These tunable properties make the Airy beam a versatile and powerful tool for optical manipulation.

Up to now, the optical micromanipulation by Airy light beam has been experimentally demonstrated, however, only qualitative theoretical analysis has been presented [13,14]. Further quantitative theoretical analysis is necessary to guide optical micromanipulation. Furthermore, the micromanipulation by Airy beam reported to date is related to Mie particles [13,14], whose radii are larger than the wavelength. To the best of our knowledge, manipulations of Rayleigh particles whose radii are smaller than λ/20 (where λ is the optical wavelength) by Airy beam have seldom been researched. Optical manipulation of particles in the Rayleigh range such as biomolecules, atoms and nanoparticles are of great significance, since conventional mechanical methods are more difficult to manipulate particles in the Rayleigh range with enough accuracy.

In this paper, the radiation forces and trajectories of Rayleigh particles in Airy beam is quantitatively analyzed. Trapping potential depths with different longitudinal positions and different particle radii are numerically calculated. Meanwhile, minimum trapping particle radii in different potential wells are also discussed. We believe that our numerical results would provide better guidance for further experimental investigation of optical manipulations.

## 2. Theory and description

In this section, we present the theoretical description of the trapping and propulsion of Rayleigh particles by the Airy beam. Rayleigh particles whose radii are much smaller than the wavelength in one-dimensional Airy beam with finite energy are considered in the whole paper. The expressions of optical gradient and scattering forces are introduced by the Rayleigh approximation, and the governed motion equations for particles moving in stagnant fluid are obtained. Finally, we discuss the trapping stability by introducing the trapping potential depths.

#### 2.1 Theory and description

The (1 + 1) dimensional optical Airy wave packet *ϕ* satisfies the normalized paraxial equation of diffraction [8]:

*ϕ*is the electric field envelope, $s=x/{x}_{0}$ represents a dimensionless transverse coordinate,

*x*

_{0}is an arbitrary transverse scale, $\xi =z/\left(k{x}_{0}^{2}/2\right)$ is a normalized propagation distance, and $k={2\pi n/\lambda}_{0}$ is the wave number of the optical field.

Assuming that the linear polarized electric field polarizes in *x* direction, the electromagnetic field of Airy beam can be analytically expressed as [11]:

*a*is the decay factor which is a positive quantity to ensure containment of the infinite Airy tail and can thus enable the physical realization of such beams.

An important and measurable physical quantity in evaluating radiation force of a light beam is the beam intensity, which is defined as time-averaged version of the Poynting vector and is given by [16,17]

Substituting Eqs. (2), (3), and (4) into (5), we can obtain the intensity as follows

The intensity can be rewritten into two components

where*c*is the speed of light in vacuum.

#### 2.2. Gradient force and Scattering force

The scattering force and gradient force are two kinds of main radiation force. Now we discuss the radiation force produced by an Airy beam on a Rayleigh dielectric sphere, whose radius is much smaller than the wavelength of laser beam (i.e., $a\le \lambda /20$). In this case, the particle is treated as a point dipole. There are two kinds of the radiation force: scattering force and gradient force. The gradient force produced by non-uniform electromagnetic fields is along the gradient of light intensity, which is expressed as [18,19]

*n*

_{1}and

*n*

_{2}are the refractive index of the particle and the surrounding medium respectively.

*R*is the radius of the particle.

The scattering force along the direction of light propagation is proportional to light intensity, which can be written as [18,19]

Thus, the gradient force and scattering force of *x* and *z* components can be written as follows

#### 2.3. Movement trajectories of particles

The viscous force of a particle in fluid or gas can be written as [20]

The Stokes drag coefficient of a particle moving in a stagnant fluid is ${C}_{Drag}=6\pi \eta R$, where *η*, *R* and *V* are the fluid viscosity, the particle radius and the velocity of particle, respectively.

Besides, a particle experiences Brownian force when it is in a fluid. As particle size decreases, the Brownian force becomes more dominant, resulting in strong thermal motion. The Brownian force can be expressed as [21]

where ${k}_{B}$ is the Boltzmann’s constant,*T*is the temperature of the medium surrounding a particle,

*ξ*is Gaussian white noise of unit strength [21].

When a light beam is acted on a particle, the particle experiences radiation forces such as optical gradient force and optical scattering force, which can be expressed in Eqs. (12)–(15). Thus the motion equations of a particle can be expressed as

*x*,

*z*) is the particle’s instant coordinate, ${F}_{x}={\left({F}_{grad}\right)}_{x}+{\left({F}_{scat}\right)}_{x}$, ${F}_{z}={\left({F}_{grad}\right)}_{z}+{\left({F}_{scat}\right)}_{z}$,

*G*and

_{p}*F*the gravitation and buoyancy of the particle respectively. The inertia (the left hand side of Eqs. (18) and (19)) can be ignored when particles are trapped in water [22,23]. However, the inertia term should be contained if the surrounding medium has small viscosity, such as gas.

_{b}#### 2.4. Trapping stability analysis

The Brownian motion is more dominant in the case of small particle, which results in strong thermal motion. Therefore, the trapping stability analysis is very important for Rayleigh particles. A necessary and sufficient condition for stable trapping is that the potential well of the gradient force trap is at least ten times larger than the kinetic energy of the Brownian particles. This can be expressed as [18]

*U* is the potential energy of the gradient force and can be expressed as

## 3. Results and discussion

In all our simulations, the incident beam is considered as a one-dimensional Airy beam, whose parameters are chosen as: *λ* = 1064 nm, *a* = 0.1 and *x*
_{0} = 2 μm, respectively. The peak intensity of input Airy beam is *I*
_{0} = 1.4518 × 10^{11} W/m^{2}. The *x* and *z* components of intensity distribution within a propagation distance of 150 μm are given in Fig. 1
. The Airy beam experiences constant transverse acceleration during propagation and can retain its intensity features over several diffraction lengths. Finally, it smears out while propagating due to the finite beam intensity. The maximum intensity of *z* component is located at around the incident plane *z* = 0, but the intensity of *x* component reaches the maximum after it propagates several diffraction lengths. Meanwhile, the maximum intensity of *z* component is one order larger than that of *x* component. Thus the micromanipulation on nanoparticles is mainly caused by the *z* component of the Airy beam.

Now we discuss the radiation forces of nanoparticles in the Airy beam. A 45 nm (radius) fused silica nanoparticle with refractive index *n _{p}* = 1.46 and density

*ρ*= 2.6 × 10

_{p}^{3}kg/m

^{3}is considered. The refractive index of the surrounding medium of water is 1.33. The gradient forces and scattering forces of the fused silica nanoparticle are shown in Fig. 2 . Results show that the gradient force alternates between positive and negative directions (see in Fig. 2(a) and (b)), while the scattering force exhibits only positive direction (see in Fig. 2(c) and (d)). The direction of gradient force depends upon the particle’s position, pointing to position of the closest optical intensity peak. These suggest that the Rayleigh particle in the Airy beam will be dragged into the closest intensity peak and transported along the direction of Poynting vector. As the multi-peak form of optical intensity, the Rayleigh particles will be trapped into different intensity peaks and move along multi-orbits. The maximum values of the scattering and gradient forces will shift towards the +

*x*direction as the longitudinal distance

*z*increases.

In addition, the *x* component of scattering force and the *z* component of gradient force reach maximum at around *z* = 80 μm. This can be explained by the propagation dynamics of the Airy beam of *x* component.

To analyze the dependence of radiation force on the size and refractive index of particle, we calculated the total radiation force *F _{x}* = (

*F*

_{scat})

*+ (*

_{x}*F*

_{grad})

*and*

_{x}*F*= (

_{z}*F*

_{scat})

*+ (*

_{z}*F*)

_{grad}*in different radii and refractive indices in Fig. 3 . For convenience, we choose the*

_{z}*z*= 0 plane for analysis. At the plane of

*z*= 0, the

*z*component of gradient force and the

*x*component of scattering force are both negligible compared to the

*z*component of scattering force and

*x*component of gradient force. Thus the total force at

*z*= 0 plane can be approximately expressed as

*F*= (

_{x}*F*)

_{grad}*, and*

_{x}*F*= (

_{z}*F*)

_{scat}*. As expected, the radiation force increases as the particle radius increases. The size sensitivity of the*

_{z}*z*component of the total force is much more pronounced than that of the

*x*component of the total force, because the scattering force yields a sixth order relationship with respect to particle radius, and the gradient force yields a third order relationship with respect to particle radius. The calculated radiation forces are stronger for larger refractive index particle, as shown in Figs. 3(b) and 3(d). This effect can be attributed to the relatively large refractive index contrast between particle and surrounding medium. This indicates that particles with larger radius and larger refractive index will be trapped and propulsed easier. When the radius of a particle decreases to a certain extent, it cannot be trapped by the beam. Then the Brownian force and random Brownian motion will be dominant.

Trajectories of fused silica nanoparticles with different radii and different initial *x* positions (*z* = 0) in an Airy beam are given in Fig. 4
. The surrounding medium is water with viscous coefficient *η* = 7.978 × 10^{−4} Pa·s and temperature *T* = 300 K. Results show that particles at different initial *x* positions move in different orbits. Particles are always dragged into the closest optical intensity peaks by the optical gradient force, and then move along parabolic shapes in the direction of the Poynting vector of the Airy beam. Particles at initial positions in the range of *x* = (−4.68 μm, 4 μm), (−8.18 μm, −4.68 μm) and (−11.04 μm, −8.18μm) (*z* = 0) are dragged into the primary, secondary and tertiary intensity peaks, respectively. Both 40 nm and 50 nm radius particles move along the same trajectory, which can be seen from Fig. 4(b).However, the moving details are quit different, which are given in the inset of Fig. 4(b). The Brownian force becomes more dominant as decreasing of particle sizes. Particles with smaller sizes move more irregularly. Results show that not all the intensity peaks can stably trap particles because of obvious Brownian motion. Therefore, it is necessary to analyze the stability of trapping particles when the Airy beam is used as the trapping beam.

It is well known that the optical gradient force and the optical scattering force of Airy beam act under an angle smaller than 90 degree. Thus, the scattering force may propel nanoparticles out of the curved Airy beam roads acting against the optical gradient force which confines particles to the rods. Figure 5
gives the trajectories of a 50 nm (radius) fused silica nanoparticle at the initial position *x* = −11 μm (*z* = 0) with different viscous coefficients of surrounding medium. The initial position of the nanoparticle is near the tertiary intensity peak of the Airy beam. When the viscous coefficient is 1.00 × 10^{−7} Pa∙s, the nanoparticle is dragged into the tertiary intensity peak, and moves along the parabolic line in the direction of the Poynting vector of the Airy beam. As the viscous coefficient decreases to 1.63 × 10^{−9} Pa∙s, the nanoparticle cannot be confined in the tertiary optical intensity peak any more. It hops into the secondary optical intensity peak, which oscillates along a parabolic line with a longitudinal distance of about 70 μm and then runs out of the orbit. When the viscous coefficient is 1.5 × 10^{−9} Pa∙s, the nanoparticle hops into the primary optical intensity peak and oscillates along the parabolic line with a longitudinal distance of about 30 μm. If the viscosity of the surrounding medium is large enough, such as water, particles will hardly run out of the intensity rods because of large viscous force. As the viscosity of the surrounding medium decreases, the nanoparticle will run out of the rods due to the scattering force.

Next, we discuss the trapping stability by analyzing the potential depths, which are shown in Fig. 6
. Figure 6(a) shows the potential depths of fused silicon nanoparticles with different x positions of the Airy beam. Results show that 50 nm radius particle can be trapped only in the primary maximum potential well when the peak intensity *I*
_{0} is 1.4518 × 10^{11} W/m^{2}. The potential depth decreases as the propagation distance increases. When propagation distance reaches 100 μm, the particle cannot be trapped any more. It will escape the optical potential well due to the Brownian force. Potential depths with different input peak intensities and particle radii are shown in Figs. 6(b) and (c), respectively. As the input optical peak intensity increases, the potential depth linearly increases. As a result, more secondary potential wells can also trap particle with radius of 50 nm. The potential depth has a cubic dependence on the radius of particle. When the input optical peak intensity *I*
_{0} is 1.4518 × 10^{11} W/m^{2}, particle with radius equal or less than 30 nm cannot be trapped by the Airy beam. Particle with radius of 40 nm can only be trapped in the main potential well of the Airy beam, and particle with radius of 50 nm can be trapped in both the primary and secondary potential wells.

The dependence of the minimum radius of trapped particles on input peak intensity is given in Fig. 7
. We only consider four main potential wells, which are named in turn from 1st to 4th potential wells. When the input peak intensity is *I*
_{0} = 1.4518 × 10^{11} W/m^{2}, particle with minimum radius of 37.7 nm can be trapped in the Airy beam. Larger input intensity is required to trap smaller particles. The 2nd, 3rd and 4th potential wells can trap particles with minimum radii of 52.2 nm, 62.4 nm and 71.5 nm, respectively. Trapping a particle with radius of 50 nm needs the minimum optical peak intensity 0.42*I*
_{0}. The stability of trapping particles can be improved by increasing either the input peak intensity or the particle radius. As the finite energy Airy beam diffracts out after propagating a distance, particles can only be trapped and propulsed in a finite distance. Finally, they will escape from the trapping potential well due to the Brownian random motion. In contrast, the particles can be trapped and propulsed into the potential wells by the Brownian random force again.

## 4. Conclusions

We have demonstrated through our analysis and simulation results that the Airy beam drags particles into the intensity peaks, and guides particles vertically along parabolic trajectories. Scattering and gradient forces of particles are quantitatively analyzed in an Airy beam, and are demonstrated sensitive to particle size and refractive index. Particles at different initial positions are dragged into different optical intensity peaks, and move along different obits. Particles in small viscous medium can hop between intensity rods and escape from the curved Airy beam roads. Viscosity of surrounding Random Brownian force affects the trajectory details. Moreover, trapping potential depths and minimum radii of particles trapped in different potential wells were also analysed. Potential depths increase as the particle size and optical peak intensity increase, and decrease as the beam propagation distance increases. As optical peak intensity increases, minimum radii of particles trapped in different potential wells of the Airy beam decrease. The trapping stability could be improved by increasing either the input peak intensity or the particle radius.

## Acknowledgments

We acknowledge financial supports from the National Natural Science Foundation of China (NNSFC) (grant 60678025), Chinese National Key Basic Research Special Fund (2006CB921703), Program for New Century Excellent Talents in University (NCET-05-0220), and 111 Project (B07013).

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