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
Since Ashkin first demonstrated the optical trapping of particles using the radiation force produced by focused Gaussian beams , 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 [2–4], biological cells [5–8], neutral atoms [9, 10], molecule-level motors , colloid systems , 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 , Bessel light beams [18, 19], and partially coherent beams .
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 N 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 . 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 . In 2005, Ambardekar et al.  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.  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 . Most recently, De et. al.  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.  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” . In our previous work , 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, 24–26], 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 . by using 3D scanning technology. For a pulsed optical tweezers, as pointed out in the previous work  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 , 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 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 condition.
In Ref , we have obtained all components of the PRF, which includes the transverse PRF , the longitudinal PRF , the longitudinal temporal-effect PRF , and the pulsed scattering force . 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:27]. (also in Ref .), the magnitudes of both the transverse and axial PRFs are greatly enhanced with decreasing of τ.
Figure 2 shows the typical changes of both and as a function of time under different τ. For , its dynamic property dramatically changes with decreasing of τ. In this example, the small Rayleigh particle with is located at the position . In Fig. 2(a), for large τ (withps), both and 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 0.1 and 0.01ps, respectively, is always negative, but in the axial direction 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.
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 [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, and , from Eq. (2a) and (2b), we can find the changes of the velocity and displacement of the particle in the transverse and axial directions as follows:
Here is the particle’s mass. In the above calculations, we have assumed that the particle is initially stationary at position . 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 [29–31]. Usually the particle’s motion in fluid can be characterized by the standard Langevin equation [29, 30]: , where is the friction force, is the random fluctuating force due to the thermal fluctuation of Brownian motion, and represents all external forces including the buoyant force, the gravitational force, and the PRFs considered here. The magnitude of is given by  , where is the Boltzmann’s constant, Tis the temperature, is the stokes friction coefficient (ηis the viscosity of fluid), and is the time slice over which is used to average (i.e. cancel) itself out. This is not a real force but rather a noise density of . Therefore the noise density of increases as decreases. For a free particle, i.e., , at time , the motion is diffusive with its displacement proportional to . Here is the diffusion constant, is the momentum relaxation time of the particle  and is the density of the particle. For a short time, at or , the motion of the free particle becomes ballistic with its displacement proportional to [15, 29], where is the root mean square (rms) velocity in the ballistic regime, and the effective mass is the sum of the mass of the particle and half of the mass of the displaced fluid .
In our cases, for a small glass bead with nm and, in water with Pa·s at temperature K, the quantities discussed in the above are as follows: the gravity of the particle is about N, the buoyant force in water is about N, the PRF is from N to N, the relaxation time is about ps, the rms velocity of the particle is about at short times , the diffusion constant is m2/s for , N for ps, N for , and N for μs.
For a Rayleigh particle with nm, its gravity is about N and its buoyant force in water is about N. The PRF on such a particle changes from N to N, the relaxation time in this case is about ns, the rms velocity of the particle is about at short times , the diffusion constant is m2/s for , N for ps, N for , and N for μs.
From these data, we can see that, for a smaller Rayleigh particle, the particle’s motion is dominated by between two neighboring pulses, and can totally erase the particle’s velocity (momentum) induced by the PRF; especially at time , the particle is the ballistic motion not diffusive motion, while at , 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 is dominated in the interval time between two pulses, within the short time region, , 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 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, is only dominated between two neighboring pulses, and the particle’s motion is determined by PRF within the pulse duration. Meanwhile, the relaxation time 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 , which is displaced from the center of the trapping region, 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 ) 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 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 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. .
However, in the axial direction, the velocity induced by the axial PRF is greatly affected by the pulse duration: For large τ, changes directly from zero to negative, which results from the negative axial force in Fig. 2(a). Thus it naturally leads to negative , see the dashed line in Fig. 3(d). Combined with the transverse effect, for the pulse laser with large pulse duration (ps), 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 withps.
For smaller τ, initially changes from zero to positive, and then decreases to negative, therefore initially becomes positive and then slightly decreases. In fluid, the final value of 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 . 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 and are independent of the pulse duration. This property is the same as that in Ref . As pointed out by Shane et. al. , regardless of the pulse duration the final momentum (velocity) transfer is a constant over a pulse duration. In our example, we find cm/s and cm/s for the particle located at the position when a pulse completely passes through it. Here we would like to point out that the final values of and depend on the location of the particle in the trapping region. For the center of the trapping region, , 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 and ps at time . 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.
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 nm, so that is much smaller than . 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, 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 ps, so 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.
For the pulse with several MHz repletion rate or above, we can estimate that the maximal value of 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 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, 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, 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.
Finally let us discuss the trapping effect on the large-sized Rayleigh particles. In our examples, the particle’s radius now becomes 50nm, so that in these cases is larger than .
Figures 7 and 8 clearly show the dynamic changes of the particle’s velocity and displacement under the actions of the pulses with ps and ps, respectively. For the large-duration pulse (ps), 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 (ps), as in Fig. 8, the longitudinal component, , 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.
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 . In Ref , 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.
This work was supported by National Natural Science Foundation of China (No. 61078021), and Scientific Research Foundation of Returned Scholars, Zhejiang Province (G80611).
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