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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20384–20394
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Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam

Hua Cheng, Weiping Zang, Wenyuan Zhou, and Jianguo Tian  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20384-20394 (2010)
http://dx.doi.org/10.1364/OE.18.020384


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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 OSA

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

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

], 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

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

], liquid droplets [2

2. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006). [CrossRef]

], and even some submicron objects such as viruses [3

3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

] and silver nanoparticles [4

4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008). [CrossRef] [PubMed]

]. 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

5. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]

], construct particle conveyor belts [6

6. T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005). [CrossRef]

], sort microfluidic cells and transfect cells [7

7. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]

].

A second type of “non-diffracting” beam, the Airy beam, was observed in experiment [8

8. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

,9

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

]. 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

10. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]

], vacuum electron acceleration [11

11. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef] [PubMed]

], and generation of three dimensional optical bullets [12

12. A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]

]. Jörg Baumgartl and colleagues [13

13. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

,14

14. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009). [CrossRef] [PubMed]

] 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

15. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). [CrossRef] [PubMed]

]. These tunable properties make the Airy beam a versatile and powerful tool for optical manipulation.

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

8. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

]:
iϕξ+2ϕs2=0,
(1)
where ϕ is the electric field envelope, s=x/x0 represents a dimensionless transverse coordinate, x 0 is an arbitrary transverse scale, ξ=z/(kx02/2) is a normalized propagation distance, and k=2πn/λ0 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

11. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef] [PubMed]

]:
Ex=ikA0(2k2x02){exp(C)[(2a2+2iaξξ2/4+2s+2k2x02)Airy(D)     +(4a+iξ)Airy'(D)]},
(2)
Ez=ikA0(4k3x03){iexp(C)[2a3+9ia2ξ/2+a(3ξ2/2+6s4k2x02)i(2i+ξ3/8     3sξ/2+ξk2x02)Airy(D)+(6a2+4iaξ+2sξ2/24k2x02)Airy'(D)]},
(3)
By=kA0(4k2x02){iexp(C)[(2a2+2iaξξ2/4+2s4k2x02)Airy(D)     +(4a+iξ)Airy'(D)]},
(4)
where C=(asaξ2/8iξ3/96+ia2ξ/4+isξ/4),D=iaξ/2ξ2/16+s, and 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

16. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

,17

17. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

]

I=S(r,t)t=12Re[E×H*].
(5)

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

I=nε0c2|Ex|2ez+nε0c2|Ex||Ez|ex.
(6)

The intensity can be rewritten into two components
Iz=nε0c2|Ex|2,
(7)
Ix=nε0c2|Ex||Ez|
(8)
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., aλ/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

18. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]

,19

19. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005). [CrossRef] [PubMed]

]
Fgrad(x,z)=ε0πn22R3(m21m2+2)|EE*|                =ε0πn22R3(m21m2+2)(|Ex|2+|Ez|2),
(9)
where m=n1/n2 is the relative refractive index of the particle, 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

18. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]

,19

19. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005). [CrossRef] [PubMed]

]
Fscat(x,z)=(n2c)CprI(x,z),
(10)
where Cpris the cross section for the radiation pressure of particles. In the case of small dielectric particles under the Rayleigh regime, Cpr equals to the scattering cross section Cscat, which can be written as follows

Cpr=Cscat=83πk4R6(m21m2+2)2,
(11)

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

(Fgrad)x=ε0πn22R3(m21m2+2)x(|Ex|2+|Ez|2),
(12)
(Fgrad)z=ε0πn22R3(m21m2+2)z(|Ex|2+|Ez|2),
(13)
(Fscat)x=(n2c)CprIx(x,z),
(14)
(Fscat)z=(n2c)CprIz(x,z).
(15)

2.3. Movement trajectories of particles

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

20. A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009). [CrossRef] [PubMed]

]

FDrag=CDragV.
(16)

The Stokes drag coefficient of a particle moving in a stagnant fluid is CDrag=6πηR, 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

21. S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991). [CrossRef]

]
FBn=2CDragkBTξ.
(17)
where kB is the Boltzmann’s constant, T is the temperature of the medium surrounding a particle, ξ is Gaussian white noise of unit strength [21

21. S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991). [CrossRef]

].

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
mpd2xdt2=Fx(x,z)+CDragdxdt+FBn,
(18)
mpd2zdt2=Fz(x,z)+CDragdzdt+Gp+Fb+FBn,
(19)
where (x, z) is the particle’s instant coordinate, Fx=(Fgrad)x+(Fscat)x, Fz=(Fgrad)z+(Fscat)z, Gp and Fb 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

22. M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: a comparison between gas and liquid phase optical traps,” Opt. Express 16(11), 7739–7747 (2008). [CrossRef] [PubMed]

,23

23. A. Neild, T. W. Ng, and W. M. S. Yii, “Optical sorting of dielectric Rayleigh spherical particles with scattering and standing waves,” Opt. Express 17(7), 5321–5329 (2009). [CrossRef] [PubMed]

]. However, the inertia term should be contained if the surrounding medium has small viscosity, such as gas.

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

18. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]

]

U/(kBT)10,
(20)

U is the potential energy of the gradient force and can be expressed as
U=2πn2R3c(m21m2+2)[I(rpeak,z)I(rhollow,z)],
(21)
where I(rpeak,z) represents the intensity at the position where it reaches a peak, and I(rhollow,z) represents the intensity at the position where it reaches a hollow. Thus, the minimum radius of particle which can be trapped by optical forces can be derived as:

Rmin=[10kBTc2πn2(m2+2m21)]1/3×[I(rpeak,z)I(rhollow,z)]1/3.
(22)

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 × 1011 W/m2. The x and z components of intensity distribution within a propagation distance of 150 μm are given in Fig. 1
Fig. 1 Propagation dynamics of a finite energy Airy beam as a function of distance with (a) the z component and (b) the x component of the intensity, where λ = 1064 nm, a = 0.1, x 0 = 2 μm and I 0 = 1.4518 × 1011 W/m2, respectively.
. 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 np = 1.46 and density ρp = 2.6 × 103 kg/m3 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
Fig. 2 The radiation forces on a 45 nm (radius) fused silica nanoparticle at varying x positions for three different z planes. The gradient force of (a) x component and (b) z component; the scattering force of (c) x component and (d) z component.
. 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 Fx = (F scat)x + (F grad)x and Fz = (F scat)z + (Fgrad)z in different radii and refractive indices in Fig. 3
Fig. 3 The total radiation forces on fused silicon nanoparticles at varying x positions for z = 0 plane. Plots of total forces for particles with different radii and refractive indices of (a) and (b) x component; (c) and (d) z component. The refractive index is 1.46 in (a) and (c). The radius is 50 nm in (b) and (d).
. For convenience, we choose the 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 Fx = (Fgrad)x, and Fz = (Fscat)z. As expected, the radiation force increases as the particle radius increases. The size sensitivity of the 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
Fig. 4 The trajectories of fused silica nanoparticles in an Airy beam. Plots of trajectories for 50 nm (radius) particles at four different initial x positions (z = 0) in (a) and trajectories for two different radius particles both at the initial position of x = −3 μm (z = 0) in (b). The inset is the partial enlarged detail of the circled region.
. 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
Fig. 5 The trajectories of a 50 nm (radius) fused silica nanoparticle at the initial position of x = −11 μm (z = 0) with different viscosity of surrounding medium.
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
Fig. 6 Potential depths of fused silicon nanoparticles at varying x positions in an Airy beam. Plots of potential depths in (a) a 50 nm radius particle at different z planes when the input peak intensity of light is 1.4518 × 1011 W/m2, in (b) a 50 nm radius particle with different input peak intensity at z = 0 plane, in (c) three different radius particles at z = 0 plane with input peak intensity of 1.4518 × 1011 W/m2.
. 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 × 1011 W/m2. 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 × 1011 W/m2, 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
Fig. 7 Dependence of minimum radius of optical-field trapped particles on input optical peak intensity.
. 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 × 1011 W/m2, 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 0.42I 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).

References and links

1.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

2.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006). [CrossRef]

3.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

4.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008). [CrossRef] [PubMed]

5.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]

6.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005). [CrossRef]

7.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]

8.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

9.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

10.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]

11.

J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef] [PubMed]

12.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]

13.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

14.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009). [CrossRef] [PubMed]

15.

I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). [CrossRef] [PubMed]

16.

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

17.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

18.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]

19.

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005). [CrossRef] [PubMed]

20.

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009). [CrossRef] [PubMed]

21.

S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991). [CrossRef]

22.

M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: a comparison between gas and liquid phase optical traps,” Opt. Express 16(11), 7739–7747 (2008). [CrossRef] [PubMed]

23.

A. Neild, T. W. Ng, and W. M. S. Yii, “Optical sorting of dielectric Rayleigh spherical particles with scattering and standing waves,” Opt. Express 17(7), 5321–5329 (2009). [CrossRef] [PubMed]

OCIS Codes
(140.7010) Lasers and laser optics : Laser trapping
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: July 19, 2010
Revised Manuscript: August 12, 2010
Manuscript Accepted: August 28, 2010
Published: September 9, 2010

Virtual Issues
Vol. 5, Iss. 13 Virtual Journal for Biomedical Optics

Citation
Hua Cheng, Weiping Zang, Wenyuan Zhou, and Jianguo Tian, "Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam," Opt. Express 18, 20384-20394 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20384


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References

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]
  2. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006). [CrossRef]
  3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]
  4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008). [CrossRef] [PubMed]
  5. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]
  6. T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005). [CrossRef]
  7. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]
  8. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
  10. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
  11. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef] [PubMed]
  12. A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
  13. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
  14. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009). [CrossRef] [PubMed]
  15. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). [CrossRef] [PubMed]
  16. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]
  17. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]
  18. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]
  19. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005). [CrossRef] [PubMed]
  20. A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009). [CrossRef] [PubMed]
  21. S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991). [CrossRef]
  22. M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: a comparison between gas and liquid phase optical traps,” Opt. Express 16(11), 7739–7747 (2008). [CrossRef] [PubMed]
  23. A. Neild, T. W. Ng, and W. M. S. Yii, “Optical sorting of dielectric Rayleigh spherical particles with scattering and standing waves,” Opt. Express 17(7), 5321–5329 (2009). [CrossRef] [PubMed]

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