Abstract

Motivated by the recent optical trapping experiments using ultra-short pulsed lasers [Opt. Express 18, 7554 (2010); Appl. Opt. 48, G33 (2009)], in this paper we have re-investigated the trapping effects of the pulsed radiation force (PRF), which is induced by a pulsed Gaussian beam acting on a Rayleigh dielectric sphere. Based on our previous model [Opt. Express 15, 10615 (2007)], we have considered the effects arisen from both the transverse and axial PRFs, which lead to the different behaviors of both velocities and displacements of a Rayleigh particle within a pulse duration. Our analysis shows that, for the small-sized Rayleigh particles, when the pulse has the large pulse duration, it might provide the three-dimensional optical trapping; and when the pulse has the short pulse duration, it only provides the two-dimensional optical trapping with the axial movement along the pulse propagation. When the particle is in the vacuum or in the situation with the very weak Brownian motion, the particle can always be trapped stably due to the particle’s cumulative momentum transferred from the pulse, and only in this case the trapping effect is independent of pulse duration. Finally, we have predicted that for the large-sized Rayleigh particles, the pulse beam can only provide the two-dimensional optical trap (optical guiding). Our results provide the important information about the trapping mechanism of pulsed tweezers.

© 2011 OSA

1. Introduction

Since Ashkin first demonstrated the optical trapping of particles using the radiation force produced by focused Gaussian beams [1], optical tweezers has been a powerful tool for trapping and manipulating of dielectric or biological micron-sized particles. Nowadays it has been applied to manipulate various tiny objects, such as dielectric particles [24], biological cells [58], neutral atoms [9, 10], molecule-level motors [11], colloid systems [12], and individual quantum dots [13,14], and recently it has been applied to the study of Brownian motion of particles [15,16]. Usually optical trapping or tweezers in many experimental and theoretical works are constructed by using the continuous-wave (CW) laser, such as Gaussian beams [1, 2], hollow-Gaussian beams [17], Bessel light beams [18, 19], and partially coherent beams [20].

As we know that the CW laser, with the power of a few milliwatts, can only produce the radiation force with an order of a few pN to manipulate micron-sized particles; while for pulsed laser, it can only produced the large peak radiation force with an order of 109N with the same averaged power. In 2004, a group in St Andrews made a comparison between the femtosecond and the CW optical tweezers, and have pointed out that femtosecond optical tweezers are as effective as CW optical tweezers [21]. Later, they had further pointed out that it is an average power effect for optical guiding of micron-sized particles by using the femtosecond pulsed Bessel light beams [19]. In 2005, Ambardekar et al. [22] used a pulsed laser to generate the large gradient force (up to 100pN) within a short duration (~45μs) for overcoming the adhesive interaction between the particles and the surface. Soon later, Deng et al. [23] theoretically pointed out that the axial pulsed radiation force (PRF) overcomes the adhesive interaction between the stuck particles and the surface, and results in optical levitation due to the axial displacement. In 2007, Pan et. al. realized the transverse two-dimensional (2D) optical trapping of CdTe quantum dots by a high repetition-rate picosecond pulsed laser with an input power as low as 100mW [13]. Most recently, De et. al. [24] demonstrated the stable optical trapping of latex nanoparticles (with diameter about 100nm) with ~120fs ultrashort pulsed laser at power levels where CW lasers cannot lead to a stable optical trap. Shane et. al. [25] reported the three-dimensional (3D) optical trapping by using the 12-femtosecond pulsed laser with 80 MHz repetition rate, and concluded that “the linear optical trapping using pulsed lasers is independent of the pulse’s duration or time profile” [26]. In our previous work [27], we have pointed out that for ultrashort pulses the PRF may result in different optical trapping effects, which are determined by the axial radiation force, since the profile of the axial radiation force is strongly affected by the pulse duration.

From the experimental data of optical trapping [13, 2426], there is a common point that all the experiments have only monitored the transverse trapping effects due to using the backward-scattering imaging (or fluorescence) technology, which only provides the information about the particle’s transverse motion. For a 2D trapping, using the backward-scattering imaging or fluorescence signals are sufficient enough; but for a real 3D trapping, besides its transverse motion, the particle’s axial motion along the light propagation must be monitored clearly, like in Ref [28]. by using 3D scanning technology. For a pulsed optical tweezers, as pointed out in the previous work [27] that the transverse trapping effects are similar for the pulses with the long or short pulse duration. Therefore, in order to show the effect of the pulse duration on the trapping effect, we suggest monitoring the particle’s axial motion.

For better understanding the trapping effect of ultrashort optical pulses with different durations, in this paper, we have analytically derived the expressions for both the particle’s velocity and displacement under the action of the pulsed radiation force (PRF) within the pulse duration. Based on these formulae, we have clear demonstrated the different trapping effects on the Rayleigh dielectric particles due to the change of the pulse duration. We hope our result can be helpful for the further investigations on the trapping effects of the pulse tweezers.

2. Formula for the PRF of a Gaussian-Shaped Pulse

For simplicity and without loss of generality, let us consider a Gaussian-shaped pulse radiating on a Rayleigh dielectric sphere, as shown in Fig. 1 . Followed with our previous work [27], the polarization of the pulse is assumed to be parallel with the x axis, and then the electric field of the paraxial narrow-spectral Gaussian-shape pulse can be written as [27]

E(ρ,z,t)=exE(ρ,z,t)=ex(iZRE0iZR+z)exp[iω0tikzkZRρ2+ikzρ22(z2+ZR2)]exp[(tz/c)2τ2],
where τ is the pulsed duration, ZR=kw02/2 is the Rayleigh distance, w0 is the transverse waist at the focusing plane of z=0, k=2π/λ0 is the wave number corresponding to wavelength λ0 (corresponding to the carrier frequency ω0), ex is the unit vector along the x direction, and E0 is determined by the relation: E02=42U/[π3/2n2ε0cw02τ], here n2 is the refractive index of the surrounding medium, c is the light speed in vacuum, ε0 and μ0 is the dielectric constant and permeability in vacuum, and U is the input energy of a single pulse. Equation (1) is applicable for the pulse whose pulse duration τ is much larger than one cycle of carrier wave, i.e., the so-called narrow-spectral condition1/τ<<ω0.

 

Fig. 1 Schematic for a Gaussian-shaped pulse passing through the focusing region.

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In Ref [27], we have obtained all components of the PRF, which includes the transverse PRF Fgrad,ρ, the longitudinal PRF Fgrad,z, the longitudinal temporal-effect PRF Ft, and the pulsed scattering force Fscat. All these forces act on the particle with radius a, which is much smaller than the wavelength of the pulse. In the following, we write these pulsed forces into two components: the transverse and axial components as follows:

Ftrans=Fgrad,ρ=ρ^2βI(ρ˜,z˜,t˜)ρ˜/[cn2ε0w0(1+z˜2)],
Faxial=Fgrad,z+Ft+Fscat=z^βI(ρ˜,z˜,t˜)n2ε0cZR[2ZR2z˜c2τ22ZRt˜cτ+z˜(1+z˜22ρ˜2)(1+z˜2)2]z^8μ0βI(ρ˜,z˜,t˜)t˜/τ+z^8z˜μ0βI(ρ˜,z˜,t˜)ZR/(cτ2)+z^(n2/c)CprI(ρ˜,z˜,t˜),
where(ρ˜,z˜,t˜)=(ρ/w0,z/ZR,t/τ) are dimensionless, β=4πn22ε0a3[(m21)/(m2+2)]and Cpr=(8π/3)(ka)4a2[(m21)/(m2+2)]2are, respectively, the polarizability and the radiation pressure’s cross section of a spherical particle in the Rayleigh regime, m=n1/n2 (here n1 is the particle’s refractive index), and ρ^is the unit vector in the radial direction and z^ is the unit vector along the light propagation. The function I(ρ˜,z˜,t˜) is the pulse intensity or irradiance, given by:
I(ρ˜,z˜,t˜)=P1+z˜2exp[2ρ˜21+z˜2]exp[2(t˜z˜ZRcτ)2],
where P=22U/[π3/2w02τ]. As pointed out in Ref [27]. (also in Ref [26].), the magnitudes of both the transverse and axial PRFs are greatly enhanced with decreasing of τ.

Figure 2 shows the typical changes of both Ftransand Faxial as a function of time under different τ. For Faxial, its dynamic property dramatically changes with decreasing of τ. In this example, the small Rayleigh particle with a=5nm is located at the position (ρ˜,z˜)=(0.2,0.5). In Fig. 2(a), for large τ (withτ=1ps), both Ftransand Faxial are negative, so that the particle suffers the transverse and axial restoring forces, and it is pulled back to the center of optical trapping during the pulse duration. For smaller τ, see Figs. 2(b) and 2(c) with τ=0.1 and 0.01ps, respectively, Ftrans is always negative, but in the axial direction Faxial is positive at the first half of the pulse, and then it becomes negative at the second half of the pulse. Thus, the particle can only be trapped in the transverse direction, while in axial direction it is first accelerated and then decelerated.

 

Fig. 2 Typical evolutions of transverse (dashed line) and axial (solid line) PRFs for pulses with different durations: (a) τ=1ps, (b) τ=0.1ps, and (c) τ=0.01ps. Other parameters areλ0=0.514μm, w0=1μm, a=5nm, U=0.1μJ, and m=n1/n2=1.592/1.332 (for example, the small glass bead and water). The particle is located at the position (ρ˜,z˜) = (0.2, 0.5).

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3. Velocity and Displacement of the Particle Due to the PRFs

In order to clear show the trapping effect on the Rayleigh particle, in this section, we will derive the particle’s velocity and displacement under the action of the PRF. It is easy to estimate that the particle’s displacement within a pulse duration [τ(0.01,10)ps] is much smaller than 0.1nm for a Rayleigh particle, therefore it is a good approximation that the change of the particle’s position does not affect on both the transverse and axial PRFs during a pulse duration, and the magnitudes of the radiation forces only change as the pulse propagates through the particle. Using the basic formulae of mechanics, v(t)=t[F(t1)/Mp]dt1 and s(t)=tv(t1)dt1, from Eq. (2a) and (2b), we can find the changes of the velocity v and displacement s of the particle in the transverse and axial directions as follows:

vtrans(ρ˜,z˜,t˜)=2βPρ˜cn2ε0w0(1+z˜2)2exp(2ρ˜21+z˜2)Φ1(z˜,t˜),
strans(ρ˜,z˜,t˜)=2βPρ˜cn2ε0w0(1+z˜2)2exp(2ρ˜21+z˜2)Φ2(z˜,t˜),
vaxial(ρ˜,z˜,t˜)=Y1(ρ˜,z˜)Φ1(z˜,t˜)+Y2(ρ˜,z˜)Φ3(z˜,t˜),
saxial(ρ˜,z˜,t˜)=Y1(ρ˜,z˜)Φ2(z˜,t˜)+Y2(ρ˜,z˜)Φ4(z˜,t˜),
where the subscripts “trans” and “axial” denote the transverse and axial components, respectively; and other functions are defined as follows

Y1(ρ˜,z˜)=P(1+z˜2)Mpexp[2ρ˜21+z˜2]{8z˜μ0βZRcτ2+n2CprcβMpn2ε0cZR[2z˜ZR2c2τ2+z˜(1+z˜22ρ˜2)(1+z˜2)2]},
Y2(ρ˜,z˜)=P(1+z˜2)Mpexp[2ρ˜21+z˜2](2βn2ε0c2τ8μ0βτ),
Φ1(z˜,t˜)=2πτ4{1+Erf[2(t˜z˜ZRcτ)]},
Φ2(z˜,t˜)=τ24exp[2(t˜z˜ZRcτ)2]+τ(t˜z˜ZRcτ)Φ1(z˜,t˜),
Φ3(z˜,t˜)=τ4exp[2(t˜z˜ZRcτ)2]+z˜ZRcτΦ1(z˜,t˜),
Φ4(z˜,t˜)=z˜ZRcτΦ2(z˜,t˜)τ4Φ1(z˜,t˜).

Here Mp is the particle’s mass. In the above calculations, we have assumed that the particle is initially stationary at position (ρ˜,z˜). Use Eqs. (4a)(4d), we can obtain the particle’s velocity and displacement, therefore we can analyze the motion status of the particle under the action of the PRFs.

4. Review Some Properties of the Particle’s Brownian Motion in Fluid

Before we discuss the action effect of the PRF, let us first see some characteristic properties of the particle in the surrounding fluid [2931]. Usually the particle’s motion in fluid can be characterized by the standard Langevin equation [29, 30]: MpX¨=Ffr+Fth+Fext, where Ffr is the friction force, Fth is the random fluctuating force due to the thermal fluctuation of Brownian motion, and Fext represents all external forces including the buoyant force, the gravitational force, and the PRFs considered here. The magnitude of Fth is given by [29] Fth=[2kBTγ/Δt]1/2, where kB is the Boltzmann’s constant, Tis the temperature, γ=6πηa is the stokes friction coefficient (ηis the viscosity of fluid), and Δt is the time slice over which Fth is used to average (i.e. cancel) itself out. This is not a real force but rather a noise density of Fth. Therefore the noise density of Fth increases as Δt decreases. For a free particle, i.e., Fext=0, at time t>>τp, the motion is diffusive with its displacement proportional to 2Dt. Here D=kBT/γ is the diffusion constant, τp=Mp/γ=2ρpa2/(9η) is the momentum relaxation time of the particle [31] and ρp is the density of the particle. For a short time, at t0 or t<τp, the motion of the free particle becomes ballistic with its displacement proportional to vrmst [15, 29], where vrms=kBT/M* is the root mean square (rms) velocity in the ballistic regime, and the effective mass M* is the sum of the mass of the particle and half of the mass of the displaced fluid [15].

In our cases, for a small glass bead with a=5nm andρp=2.4×103kg/m3, in water with η=7.977×104Pa·s at temperature T=300K, the quantities discussed in the above are as follows: the gravity of the particle is about 1.23×1020N, the buoyant force in water is about 0.51×1020N, the PRF is from 1012N to 109N, the relaxation time τp is about 16.7ps, the rms velocity of the particle is about 1.65m/s at short times t<τp, the diffusion constant is D=5.5×1011m2/s for t>>τp, Fth=7.88×1010N for Δt=1ps, Fth=1.93×1010N for Δt=τp, and Fth=7.88×1013N for Δt=1μs.

For a Rayleigh particle with a=50nm, its gravity is about 1.23×1017N and its buoyant force in water is about 0.51×1017N. The PRF on such a particle changes from 109N to 106N, the relaxation time τpin this case is about 1.67ns, the rms velocity of the particle is about 5.22cm/s at short times t<τp, the diffusion constant is D=5.5×1012m2/s for t>>τp, Fth=2.49×109N for Δt=1ps, Fth=6.10×1010N for Δt=τp, and Fth=2.49×1012N for Δt=1μs.

From these data, we can see that, for a smaller Rayleigh particle, the particle’s motion is dominated by Fth between two neighboring pulses, and Fth can totally erase the particle’s velocity (momentum) induced by the PRF; especially at time t<τp, the particle is the ballistic motion not diffusive motion, while at t>τp, the particle’s motion is diffusive. When the PRF is presented within the pulse duration, the particle’s motion is the combination effect of the ballistic motion and the displaced movement (due to the PRF). For a femtosecond pulsed laser with several MHz repletion rate, since Fth is dominated in the interval time between two pulses, within the short time region, t<τp, the particle’s rms velocity is very large. The average displacement of the particle due to the ballistic motion within 1ps is about 0.001nm, and the average displacement of the particle due to the diffusive motion for t>τp is not less than 0.1nm. These tiny displacements are still much larger than those induced by the single-pulse PRFs, although their directions are randomly changed due to the Brownian motion. Therefore it is expected that the femtosecond pulse laser may be used for trapping the particles but its trapping effect is much complex and is discussed in next section.

However, for a larger Rayleigh particle, Fth is only dominated between two neighboring pulses, and the particle’s motion is determined by PRF within the pulse duration. Meanwhile, the relaxation time τp for larger-sized particles is longer than that of smaller sized particles, therefore the particle’s velocity (or momentum) induced by the PRF can be better accumulated .

It should be emphasized that the dynamics of a particle in a fluid may causes a periodic compression and rarefaction of the fluid near it, thus may produce sound waves [30,32]. In our cases, we are focusing on the motion of the particle under action of the PRF, thus the other effects, such as the ultrasound generation associated with the motion of the particle, are beyond our discussion.

5. Discussion on the Status of Motion of the Particle

In this section, we will discuss the status of motion for the particle under the action of the PRF. In Fig. 2, we have pointed out that the PRF is greatly affected due to the change of pulse duration, especially for the axial PRF. In fact, for the transverse component, it always provides the restoring force only with its magnitude depending on the pulse duration. In our following discussions, it is also shown that the transverse trapping effect on the particle for the pulse with short duration is similar for that of the pulse with large pulse duration.

Figure 3 shows the changes of the velocity and displacement of the particle due to these components of the PRF. It is clear seen that when the particle is located at (ρ˜,z˜)=(0.2,0.5), which is displaced from the center of the trapping region, vtrans is always negative, so it leads to the negative transverse displacement in Fig. 3(c). Therefore the particle is transversely pulled back to the center (close to ρ˜=0) although the displacement within a single pulse duration is very tiny about several femtometer (fm). For a picosecond pulse laser with several MHz repletion rate or above, the value of strans within one second due to the transverse component of the PRF will be several hundred nanometers to a few micrometers, which could effectively overcome or counteract the diffusion effect of the particle in fluid. But for a ten-femtosecond pulse laser with the same repletion rate, the value of strans within one second due to the transverse PRF will be less than hundred nanometers, so it can but partially overcome the diffusion effect. Of course, by increasing the pulse power, the transverse trapping effect can be improved. From Figs. 3(a) and 3(c), under the condition of the same pulse power, we can qualitatively conclude that the transverse trapping effect on the particle by using the pulse with long pulse duration is better than that for using the pulse with short pulse duration. As τ decreases, the transverse trapping effect becomes worse and worse. This result could be examined by the experiment designed by Shane et al. [26].

 

Fig. 3 Time evolutions of the transverse and axial components for (a-b) the velocities vtrans and vaxial, and (c-d) the displacements strans andsaxial, of the particle under the action of the different pulses. Dashed lines are forτ = 1ps, solid lines for τ=0.1ps, dot-dashed lines forτ=0.03ps, and short-dashed lines for τ=0.01ps. In (b) and (d), the color arrows denote the ends of the pulses for τ=0.01ps and τ=0.03ps (i. e., the PRF nearly disappears). Other parameters are the same as in Fig. 2.

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However, in the axial direction, the velocity induced by the axial PRF is greatly affected by the pulse duration: For large τ, vaxial changes directly from zero to negative, which results from the negative axial force in Fig. 2(a). Thus it naturally leads to negative saxial, see the dashed line in Fig. 3(d). Combined with the transverse effect, for the pulse laser with large pulse duration (τ=1ps), it could realize the three-dimensional stable optical trapping. From Figs. 3(c) and 3(d), see the dashed lines, it is also found that the transverse trapping effect is nearly ten times that of the axial trapping effect for the pulse withτ=1ps.

For smaller τ, vaxial initially changes from zero to positive, and then decreases to negative, therefore saxial initially becomes positive and then slightly decreases. In fluid, the final value of vaxial induced by the PRF cannot be kept after the pulse leaves the particle, due to the effect of the Brownian motion and the damping process. Thus the particle is pushed along the light propagation since it may be still confined in the transverse plane, like optical guiding effect [19]. Therefore it is expected that the short-duration pulse only provides the 2D transverse trapping effect and the particle will axially move away from the trapping region after the limited time scale. In fact, it is understandable that for the short-duration pulse, it is like a light bullet to push the particle.

In Figs. 3(a) and 3(b), it is also found that the final values of both vtrans and vaxial are independent of the pulse duration. This property is the same as that in Ref [26]. As pointed out by Shane et. al. [25], regardless of the pulse duration the final momentum (velocity) transfer is a constant over a pulse duration. In our example, we find vtrans=1.06cm/s and vaxial=0.202cm/s for the particle located at the position (ρ˜,z˜)=(0.2,0.5) when a pulse completely passes through it. Here we would like to point out that the final values of vtrans and vaxial depend on the location of the particle in the trapping region. For the center of the trapping region, (ρ˜,z˜)=(0,0), both of them are exactly equal to zero, so that the particle will be axially pushed away from the center of the trapping region for the short-duration pulse, while for the long-duration pulse it stays at the center. We have calculated the final velocity distribution of the particle under action of the different pulses with τ=1,0 .1, and 0.01ps at time t˜=5. Our results show that the velocity distributions for the different pulses are exactly the same, as shown in Fig. 4 . Although the final velocity is dependent on the particle’s position, the final distribution (including their magnitude and direction of the velocities at different positions) is independent of pulse duration. From Fig. 4, it seems that the particle could be trapped stably regardless of the pulse duration. However, the dynamics of the velocity distribution induced by the PRF within the pulse duration is totally different for the pulses with different durations. In the next we will analyze the dynamic process of the trapping effect on the particle by the PRF.

 

Fig. 4 Final distribution of the particle’s velocity for the different pulses(τ=1,0 .1,and0 .01ps) at time t˜=5. Other parameters are the same as in Fig. 2.

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Now let us turn to discuss the dynamic process of trapping effect on a particle with different pulses. In the following cases, we still use the small Rayleigh particle with radius a=5nm, so that FScat is much smaller than Fgrad,z. Thus, for the pulse with large τ, the pulsed gradient force is dominated among the components of the radiation force, while for the pulse with shorter τ, Ft is dominated. It is expected that the dynamic process of trapping effect is different for different pulses.

Figures 5(a)5(c) show the distributions of the particle’s velocity near the focusing region at different times. In this case, the pulse duration is τ=1ps, so Ft is smaller than the gradient force due to large τ. As discussed in Section 2, if the particle moves away from the center, within the pulse duration the particle always suffers the opposite PRF which pulls it back to the center. From Figs. 5(a)5(c), it is seen that the velocity distributions are very similar to each other, but their magnitudes increase with the time accumulated. It tells us that the force field is stable. Correspondingly, there are similar behaviors for the displacements, see Figs. 5(d)5(f). From Fig. 5, it is obvious that the transverse displacement is nearly ten times of the axial displacement, and their directions are pointing toward the center.

 

Fig. 5 Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the pulse with τ=1ps at different times: t˜=1 for (a) and (d), t˜=0 for (b) and (e), and t˜=1 for (c) and (f). Other parameters are the same as in Fig. 2.

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For the pulse with several MHz repletion rate or above, we can estimate that the maximal value of saxial toward the center is less than 0.1μm within one second, which is smaller than the axial diffusion, so the pulse can only partially overcome the axial diffusive motion. Therefore, for the pulse with long pulse duration in our parameters, if the particle is trapped temporally in the trapping region, it can only be trapped within the limited time scale, but it will escape from the trapping region beyond a certain time scale, because the axial diffusion effect is larger than the axial trapping effect. Of course, if the pulse’s energy increases, the axial optical trapping effect can be improved better, therefore the stable 3D trapping effect can be obtained for the pulse with large duration.

However, Fig. 6 shows a different dynamic process for both the velocity and displacement of the particle under the action of the pulse with τ=0.01ps. In this case, it is clear seen that the particle’s motion status is very different from Fig. 5. When the PRF is presented, Ft is dominated. From Figs. 6(a) to 6(d), along the z direction, the particle is initially accelerated and then decelerated; in the transverse direction, the particle is confined near the focusing region. Unlike Fig. 5, here the velocity changes greatly as the pulse arrives. Until the pulse completely leaves the focus region, the velocity field just begins to form the centripetal distribution. But, at the end of the pulse, Fth begins to dominate the particle’s motion, so that the velocity field in Fig. 6(d) cannot be sustained and it is quickly wiped out due to Brownian motion and damping process. From Fig. 6(e) to 6(h), the particle’s displacement is transversely confined near the focus region, and it is also slightly pushed along the +z direction during a pulse duration. Therefore, for a short-duration pulse, it can only provide 2D transversal trapping, and the particle will be pushed along the pulse propagation. In experiment, the particle is optically guiding in the focusing region and moving along the direction of the light propagation.

 

Fig. 6 Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ=0.01ps at different times: t˜=1 for (a) and (e), t˜=0 for (b) and (f), t˜=1 for (c) and (g), and t˜=2 for (d) and (h). Other parameters are the same as in Fig. 2.

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Finally let us discuss the trapping effect on the large-sized Rayleigh particles. In our examples, the particle’s radius now becomes a=50nm, so that in these cases FScat is larger than Fgrad,z.

Figures 7 and 8 clearly show the dynamic changes of the particle’s velocity and displacement under the actions of the pulses with τ=1ps and τ=0.01ps, respectively. For the large-duration pulse (τ=1ps), as in Fig. 7, the pulsed scattering force dominates the particle’s axial motion, and the transverse PRF leads to the transverse trapping effect. For the short-duration pulse (τ=0.01ps), as in Fig. 8, the longitudinal component, Ft, dominates the particle’s axial motion, which also leads to the same effect, which is refer to the particle’s movement along the light propagation; and meanwhile, the transverse PRF still leads to the transverse trapping effect. Therefore, for the large-sized Rayleigh particles, when the pulsed scattering force is larger than the axial pulsed gradient force, the pulse always provides the 2D trapping effect (i.e., optical guiding), which is independent of the pulse duration τ. But the physical reasons for the particle’s axial movement are different as pointed out in the above.

 

Fig. 7 Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the PRF for the pulse with τ=1ps at different times: t˜=1 for (a) and (d), t˜=0 for (b) and (e), and t˜=1 for (c) and (f). Other parameters are the same as in Fig. 2 except for a=50nm.

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Fig. 8 Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ=0.01ps at different times: t˜=1 for (a) and (e), t˜=0 for (b) and (f), t˜=1 for (c) and (g), and t˜=2 for (d) and (h). Other parameters are the same as in Fig. 2 except for a=50nm.

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4. Conclusion and Remarks

In summary, we have investigated the optical trapping effect of ultra-short pulses acting on the Rayleigh dielectric particle. For the small-sized Rayleigh particles, when the pulsed scattering force is smaller than the axial pulsed gradient force, the pulse with longer pulse duration can provide the stable three-dimensional optical trapping under the suitable pulse power, but the pulse with shorter pulse duration can only provide 2D trapping effect (i.e., optical guiding). For the later case, it is revealed that the particle is only trapped transversely near the focusing region, and at the same time it is pushed along the direction of the pulse propagation due to the axial temporal-effect force. In fluid, the random fluctuation force dominates the particle’s motion in the interval between two neighboring pulses, so that the particle’s velocity (or momentum) induced by the previous pulse cannot be kept to the arrival of the next pulse. The trapping-time scale depends on the competition between the axial trapping effect (induced by the axial restoring force) and the axial displacement due to Brownian motion. Therefore, for a single particle, it is time-limited to observe the optical trapping effect on the small-sized Rayleigh particles in experiments. Even for the stable 3D optical trapping, if the axial trapping effect within the pulse duration is not strong enough, the particle may still be escaped due to the Brownian motion in the interval time of neighboring pulses. But, when the particle is in vacuum or in the surrounding with the very weak Brownian motion, the particle’s velocity (or momentum) resulted from the previous pulse can be accumulated to the next pulse, so that in these cases the particle can always be trapped stably due to the momentum transfer from the pulse.

Finally, we have pointed out that for the large-sized Rayleigh particles, when the pulsed scattering force is larger than the axial gradient force, the pulse can only provide the 2D optical trap, which is independent of the pulse duration τ.

As we know that, for a true 3D optical trapping by using the CW laser, usually it is very stable. Most recently, we have noted that there is an experimental demonstration of the axial movement of the microsphere driven by optical pulse [33]. In Ref [33], the microsphere starts up by the optical pulse, and then moves along the pulse propagation, finally it stops when the pulse disappears. Therefore, it is possible that in experiments one can monitor the Rayleigh particle’s axial movement for confirming the axial trapping effect for pulsed optical tweezers.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61078021), and Scientific Research Foundation of Returned Scholars, Zhejiang Province (G80611).

References and links

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]  

2. 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]  

3. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef]   [PubMed]  

4. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” in Methods in cell Biology, M. P. Sheetz, ed. (Academic Press, 1998), vol. 55, pp.1–27. [PubMed]  

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

6. A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994). [CrossRef]   [PubMed]  

7. J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995). [CrossRef]   [PubMed]  

8. M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998). [CrossRef]   [PubMed]  

9. A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978). [CrossRef]  

10. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986). [CrossRef]   [PubMed]  

11. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999). [CrossRef]   [PubMed]  

12. P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002). [CrossRef]   [PubMed]  

13. L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007). [CrossRef]  

14. L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008). [CrossRef]   [PubMed]  

15. T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010). [CrossRef]   [PubMed]  

16. Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007). [CrossRef]  

17. C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007). [CrossRef]  

18. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]  

19. H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004). [CrossRef]   [PubMed]  

20. L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007). [CrossRef]   [PubMed]  

21. B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Femtosecond optical tweezers for in-situ control of two-photon fluorescence,” Opt. Express 12(13), 3011–3017 (2004). [CrossRef]   [PubMed]  

22. A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. 30(14), 1797–1799 (2005). [CrossRef]   [PubMed]  

23. J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005). [CrossRef]   [PubMed]  

24. A. K. De, D. Roy, A. Dutta, and D. Goswami, “Stable optical trapping of latex nanoparticles with ultrashort pulsed illumination,” Appl. Opt. 48(31), G33–G37 (2009). [CrossRef]   [PubMed]  

25. J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008). [CrossRef]  

26. J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010). [CrossRef]   [PubMed]  

27. L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007). [CrossRef]   [PubMed]  

28. H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991). [CrossRef]  

29. E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975). [CrossRef]  

30. K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005). [CrossRef]  

31. B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005). [CrossRef]   [PubMed]  

32. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd English Edition, Revised, Translated from the Russian by J. B. Sykes and W. H. Reid, (Elsevier, 2009), pp. 281.

33. H. Li, Y. Zhang, J. Li, and L. Qiang, “Observation of microsphere movement driven by optical pulse,” Opt. Lett. 36(11), 1996–1998 (2011). [CrossRef]   [PubMed]  

References

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  • |

  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [CrossRef]
  2. 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]
  3. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
    [CrossRef] [PubMed]
  4. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” in Methods in cell Biology, M. P. Sheetz, ed. (Academic Press, 1998), vol. 55, pp.1–27.
    [PubMed]
  5. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987).
    [CrossRef] [PubMed]
  6. A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
    [CrossRef] [PubMed]
  7. J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
    [CrossRef] [PubMed]
  8. M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
    [CrossRef] [PubMed]
  9. A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
    [CrossRef]
  10. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
    [CrossRef] [PubMed]
  11. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
    [CrossRef] [PubMed]
  12. P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
    [CrossRef] [PubMed]
  13. L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
    [CrossRef]
  14. L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
    [CrossRef] [PubMed]
  15. T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
    [CrossRef] [PubMed]
  16. Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
    [CrossRef]
  17. C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
    [CrossRef]
  18. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
    [CrossRef]
  19. H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
    [CrossRef] [PubMed]
  20. L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
    [CrossRef] [PubMed]
  21. B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Femtosecond optical tweezers for in-situ control of two-photon fluorescence,” Opt. Express 12(13), 3011–3017 (2004).
    [CrossRef] [PubMed]
  22. A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. 30(14), 1797–1799 (2005).
    [CrossRef] [PubMed]
  23. J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005).
    [CrossRef] [PubMed]
  24. A. K. De, D. Roy, A. Dutta, and D. Goswami, “Stable optical trapping of latex nanoparticles with ultrashort pulsed illumination,” Appl. Opt. 48(31), G33–G37 (2009).
    [CrossRef] [PubMed]
  25. J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
    [CrossRef]
  26. J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
    [CrossRef] [PubMed]
  27. L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007).
    [CrossRef] [PubMed]
  28. H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
    [CrossRef]
  29. E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
    [CrossRef]
  30. K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
    [CrossRef]
  31. B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
    [CrossRef] [PubMed]
  32. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd English Edition, Revised, Translated from the Russian by J. B. Sykes and W. H. Reid, (Elsevier, 2009), pp. 281.
  33. H. Li, Y. Zhang, J. Li, and L. Qiang, “Observation of microsphere movement driven by optical pulse,” Opt. Lett. 36(11), 1996–1998 (2011).
    [CrossRef] [PubMed]

2011 (1)

2010 (2)

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

2007 (5)

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
[CrossRef] [PubMed]

L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007).
[CrossRef] [PubMed]

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
[CrossRef]

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

2005 (4)

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. 30(14), 1797–1799 (2005).
[CrossRef] [PubMed]

J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005).
[CrossRef] [PubMed]

2004 (3)

2002 (1)

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

2001 (1)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[CrossRef]

1999 (1)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

1998 (1)

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

1995 (1)

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

1994 (1)

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

1991 (1)

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

1987 (1)

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

1986 (2)

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]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

1978 (1)

A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

1975 (1)

E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Agate, B.

Ambardekar, A. A.

Arlt, J.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[CrossRef]

Ashkin, A.

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

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

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]

A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Bechhoefer, J.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

Bjorkholm, J. E.

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]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Brown, C. T. A.

Cable, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

Chu, S.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

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]

Dai, J.

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

De, A. K.

Deng, J. L.

Deng, Y.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Dholakia, K.

Dutta, A.

Dziedzic, J. M.

Florin, E.-L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

Forde, N. R.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Forró, L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Garcés-Chávez, V.

Gelles, J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Gittes, F.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Goswami, D.

Grier, D. G.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Hinch, E. J.

E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
[CrossRef]

Howard, J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Hunt, A. J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Ishikawa, A.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Jauffred, L.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Jeney, S.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Kheifets, S.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Kitamura, N.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Korda, P. T.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Koshioka, M.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Kulik, A. J.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Landick, R.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Lee, W. M.

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Li, H.

Li, J.

Li, T.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Li, Y. Q.

Little, H.

Lu, X. H.

C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
[CrossRef]

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
[CrossRef] [PubMed]

Lukic, B.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Masuhara, H.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Mazilu, M.

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Medellin, D.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Misawa, H.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Oddershede, L. B.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Pan, L.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Qiang, L.

Raizen, M. G.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Richardson, A. C.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Roy, D.

Sasaki, K.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Schnitzer, M. J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Shane, J.

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Shane, J. C.

Sheetz, M. P.

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

Sibbett, W.

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Tamai, N.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Taylor, M. B.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Tischer, C.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Wang, L. G.

Wang, L. Q.

Wang, M. D.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Wang, Y. Z.

Wei, Q.

Yin, H.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Zhang, Y.

Zhao, C. L.

Zhu, S. Y.

Appl. Opt. (1)

Biophys. J. (2)

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

J. Fluid Mech. (1)

E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

N. J. Phys. (1)

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

Nano Lett. (1)

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Lett. A (1)

C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
[CrossRef]

Phys. Rev. B (1)

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Phys. Rev. Lett. (5)

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Science (4)

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

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Other (2)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” in Methods in cell Biology, M. P. Sheetz, ed. (Academic Press, 1998), vol. 55, pp.1–27.
[PubMed]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd English Edition, Revised, Translated from the Russian by J. B. Sykes and W. H. Reid, (Elsevier, 2009), pp. 281.

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Figures (8)

Fig. 1
Fig. 1

Schematic for a Gaussian-shaped pulse passing through the focusing region.

Fig. 2
Fig. 2

Typical evolutions of transverse (dashed line) and axial (solid line) PRFs for pulses with different durations: (a) τ = 1ps, (b) τ = 0.1ps, and (c) τ = 0.01ps. Other parameters are λ 0 = 0.514 μm, w 0 = 1 μm, a = 5 nm, U = 0.1 μJ, and m = n 1 / n 2 = 1.592 / 1.332 (for example, the small glass bead and water). The particle is located at the position ( ρ ˜ , z ˜ ) = (0.2, 0.5).

Fig. 3
Fig. 3

Time evolutions of the transverse and axial components for (a-b) the velocities v t r a n s and v a x i a l , and (c-d) the displacements s t r a n s and s a x i a l , of the particle under the action of the different pulses. Dashed lines are forτ = 1ps, solid lines for τ = 0.1ps, dot-dashed lines for τ = 0.03ps, and short-dashed lines for τ = 0.01ps. In (b) and (d), the color arrows denote the ends of the pulses for τ = 0.01ps and τ = 0.03ps (i. e., the PRF nearly disappears). Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Final distribution of the particle’s velocity for the different pulses( τ = 1 , 0 .1 , a n d 0 .01 ps) at time t ˜ = 5 . Other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the pulse with τ = 1 ps at different times: t ˜ = 1 for (a) and (d), t ˜ = 0 for (b) and (e), and t ˜ = 1 for (c) and (f). Other parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ = 0.01ps at different times: t ˜ = 1 for (a) and (e), t ˜ = 0 for (b) and (f), t ˜ = 1 for (c) and (g), and t ˜ = 2 for (d) and (h). Other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the PRF for the pulse with τ = 1ps at different times: t ˜ = 1 for (a) and (d), t ˜ = 0 for (b) and (e), and t ˜ = 1 for (c) and (f). Other parameters are the same as in Fig. 2 except for a = 50nm.

Fig. 8
Fig. 8

Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ = 0.01ps at different times: t ˜ = 1 for (a) and (e), t ˜ = 0 for (b) and (f), t ˜ = 1 for (c) and (g), and t ˜ = 2 for (d) and (h). Other parameters are the same as in Fig. 2 except for a = 50nm.

Equations (14)

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E ( ρ , z , t ) = e x E ( ρ , z , t ) = e x ( i Z R E 0 i Z R + z ) exp [ i ω 0 t i k z k Z R ρ 2 + i k z ρ 2 2 ( z 2 + Z R 2 ) ] exp [ ( t z / c ) 2 τ 2 ] ,
F t r a n s = F g r a d , ρ = ρ ^ 2 β I ( ρ ˜ , z ˜ , t ˜ ) ρ ˜ / [ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) ] ,
F a x i a l = F g r a d , z + F t + F s c a t = z ^ β I ( ρ ˜ , z ˜ , t ˜ ) n 2 ε 0 c Z R [ 2 Z R 2 z ˜ c 2 τ 2 2 Z R t ˜ c τ + z ˜ ( 1 + z ˜ 2 2 ρ ˜ 2 ) ( 1 + z ˜ 2 ) 2 ] z ^ 8 μ 0 β I ( ρ ˜ , z ˜ , t ˜ ) t ˜ / τ + z ^ 8 z ˜ μ 0 β I ( ρ ˜ , z ˜ , t ˜ ) Z R / ( c τ 2 ) + z ^ ( n 2 / c ) C p r I ( ρ ˜ , z ˜ , t ˜ ) ,
I ( ρ ˜ , z ˜ , t ˜ ) = P 1 + z ˜ 2 exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] ,
v t r a n s ( ρ ˜ , z ˜ , t ˜ ) = 2 β P ρ ˜ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) 2 exp ( 2 ρ ˜ 2 1 + z ˜ 2 ) Φ 1 ( z ˜ , t ˜ ) ,
s t r a n s ( ρ ˜ , z ˜ , t ˜ ) = 2 β P ρ ˜ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) 2 exp ( 2 ρ ˜ 2 1 + z ˜ 2 ) Φ 2 ( z ˜ , t ˜ ) ,
v a x i a l ( ρ ˜ , z ˜ , t ˜ ) = Y 1 ( ρ ˜ , z ˜ ) Φ 1 ( z ˜ , t ˜ ) + Y 2 ( ρ ˜ , z ˜ ) Φ 3 ( z ˜ , t ˜ ) ,
s a x i a l ( ρ ˜ , z ˜ , t ˜ ) = Y 1 ( ρ ˜ , z ˜ ) Φ 2 ( z ˜ , t ˜ ) + Y 2 ( ρ ˜ , z ˜ ) Φ 4 ( z ˜ , t ˜ ) ,
Y 1 ( ρ ˜ , z ˜ ) = P ( 1 + z ˜ 2 ) M p exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] { 8 z ˜ μ 0 β Z R c τ 2 + n 2 C p r c β M p n 2 ε 0 c Z R [ 2 z ˜ Z R 2 c 2 τ 2 + z ˜ ( 1 + z ˜ 2 2 ρ ˜ 2 ) ( 1 + z ˜ 2 ) 2 ] } ,
Y 2 ( ρ ˜ , z ˜ ) = P ( 1 + z ˜ 2 ) M p exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] ( 2 β n 2 ε 0 c 2 τ 8 μ 0 β τ ) ,
Φ 1 ( z ˜ , t ˜ ) = 2 π τ 4 { 1 + E r f [ 2 ( t ˜ z ˜ Z R c τ ) ] } ,
Φ 2 ( z ˜ , t ˜ ) = τ 2 4 exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] + τ ( t ˜ z ˜ Z R c τ ) Φ 1 ( z ˜ , t ˜ ) ,
Φ 3 ( z ˜ , t ˜ ) = τ 4 exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] + z ˜ Z R c τ Φ 1 ( z ˜ , t ˜ ) ,
Φ 4 ( z ˜ , t ˜ ) = z ˜ Z R c τ Φ 2 ( z ˜ , t ˜ ) τ 4 Φ 1 ( z ˜ , t ˜ ) .

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