The radiation force of modified circular Airy beams (MCAB) exerted on both a high-refractive-index particle and a low-refractive-index particle are analyzed in this paper. Our results show that the two kinds of particles can be simultaneously stably trapped by MCAB at different positions. Compared with the common circular Airy beams (CAB) with the same parameters, trapping forces on the two kinds of particles are greatly increased because of the enhanced abruptly autofocusing property and the appearance of hollow region in MCAB. The trapping forces can be modulated by varying parameters of MCAB, and it is important to choose appropriate parameters to trap particles in practice.
© 2016 Optical Society of America
Nowadays, optical tweezers have become an important tool since they are first invented in 1986 by Ashkin . They have been widely applied in the investigations of cells, viruses, atoms, colloids and other microparticles [2–9]. Usually, Gaussian beams are used to trap high-refractive-index (with respect to the surrounding medium) particles. But the application of spatial light modulator allows many different kinds of special laser beams to be used in optical tweezers [10,11]. Theoretical and experimental investigations demonstrate that, by using special laser beams, the performance of optical tweezers can be well improved [12–22], and some new optical micromanipulation techniques can be realized [23–27].
Usually, it is difficult to trap both high–refractive-index particles and low-refractive-index particles by a single beam. The highly focused optical vortex , radially polarized beam , partially coherent elegant Laguerre-Gaussian beam  have been proposed to trap both kinds of particles. However, for the radially polarized beam and the partially coherent elegant Laguerre-Gaussian beam, corresponding parameters of the beam must be changed for trapping different particles. In this paper, we propose a method of simultaneously trapping both kinds of particles by abruptly autofocusing beams.
The abruptly autofocusing beam is a kind of special laser beam, which can abruptly focus its energy at the focal point in the free space while maintaining a low intensity profile before . The first observed abrupt autofocusing beam is the circular Airy beam (CAB) . Lately, various kinds of abruptly autofocusing beams have been proposed, such as CAB with optical vortex [31,32], modified CAB , non-paraxial abruptly autofocusing beams , and so on [35–39]. Theoretical and experimental results show that CAB can be used to trap high-refractive-index particles [17,20]. Comparing with conventional optical tweezers under the same circumstance, the stiffness of optical trap will be increased and the numerical aperture can be decreased by using CAB . But it is difficult for CAB to trap low-refractive index particles as we have shown in our previous work .
In this paper, we have used the modified circular Airy beam (MCAB) to improve the performance of optical tweezers with common CAB in theory. The distributions of radiation forces generated by MCAB exerted on both high-refractive-index particles and low-refractive- index particles are analyzed. Our results show that the trapping force is great increased as compared to common CAB; and the two kinds of particles will be stably trapped at different positions by MCAB. Other useful results are also obtained in our investigations.
2. Propagation characteristics of MCAB40],33],
For simplicity, we assume MCAB at the initial plane is linear polarized in the x-direction. So the electric field of MCAB at arbitrary planes can be calculated by the Rayleigh- Sommerfield diffraction formula :20,42],
Because the low-frequency component of CAB contributes less to the focal intensity, the abruptly autofocusing property can be enhanced if the low-frequency component are blocked . The propagation characteristics of MCAB with different parameters are shown in Fig. 1, Fig. 2, and Fig. 3. In our numerical calculation, we choose r0 = 25μm, w = 2.5μm, a = 0.08, λ = 1064nm, and the incident light powers for both CAB and MCAB are 1W.
Figure 1 shows the abruptly autofocusing properties of MCAB and CAB along the beam axis. It is found that the focal intensity is greatly increased for MCAB. The focal intensity increases with kc, but the rate of increase is very small when kc is too large. Figure 1 also shows that peak positions and valley positions of the distributions of intensity along the beam axis do not change with β and kc. The first peak position, i.e., focal position, is zf = 0.236mm; the first valley position is z = 0.250mm. Figure 2 shows the profiles of the two intensity components, Ix and Iz, at different planes. The intensity pattern of Iz is not radially symmetric when the incident MCAB is linear polarized, as we can see in Figs. 2(b) and 2(e). From Fig. 2, we can find that Iz is much less than Ix, and the total intensity is hardly affected by Iz. So we can neglect the influence of Iz in our simulations.
From Figs. 3(a) and 3(b), we can see that several 3-D potential wells for the high-refractive-index particle are formed at peak positions. As kc increases, the intensity will increase, the transverse spot size will decrease. In Fig. 3(c), it is noteworthy that a nearly perfect hollow region appears at the first valley position for MCAB with β = 3μm, kc = 1μm−1, which indicates that a potential well for the low-refractive-index particles can be produced if we choose appropriate parameters. This hollow region will also appear at other valley positions behind [Fig. 3(d)]. By increasing kc, the intensity at first valley position can be modulated to nearly zero. But the hollow region will disappear when kc is too large. Figures 3(c) and 3(d) show that an intensity peak will appear at the center when kc = 1.5μm−1.
3. Radiation forces of MCAB on two kinds of particles
The Rayleigh dielectric particle can be considered as a point dipole in the light fields. Its polarisability α can be expressed as ,44]:
Figure 4 shows the distributions of the longitudinal and transverse radiation force of MCAB exerted on the high-refractive-index particle. The gradient force is directed to the equilibrium point in Fig. 4(a); the scattering force in Fig. 4(b) is directed to the propagation direction. The longitudinal trapping force can be defined as the sum of the longitudinal gradient force and the scattering force. Because of the enhancement of the abruptness of focusing, the longitudinal gradient force is increased for MCAB as we can see in Fig. 4(a). Although the scattering force of MCAB is larger [Fig. 4(b)], the effective longitudinal trapping force is increased due to the large gradient force of MCAB [Fig. 4(c)]. From Fig. 4(c), we can also see that there are several stable equilibrium points in the distribution of the longitudinal trapping force, such as, za = 0.238mm and zb = 0.268mm. Note that the position of za is slightly shifted from the focal point (zf = 0.236mm) because of the influence of the scattering force. From Figs. 4(d) and 4(e), we can see that the particle can be transversely trapped at za and zb by the transverse gradient force. The transverse gradient force of MCAB is greater than that of the common CAB. Figure 4 also shows that the magnitudes of longitudinal and transverse trapping force can be modulated by varying β and kc without affecting the trapping range. So the stiffness of the optical trap can be enhanced by choosing proper parameters of β and kc.
Figure 5 shows the distributions of the radiation force exerted on the low-refractive-index particle. As we can see in Figs. 5(a) and 5(b), comparing with CAB, the gradient force of MCAB is increased because of the appearance of the hollow region; but the scattering force is also increased for MCAB. Figure 5(c) shows that the effective longitudinal trapping force of MCAB is larger than that of CAB. The low-refractive-index particle can also be longitudinal trapped at several positions, such as zc = 0.250mm and zd = 0.264mm. The trapping positions are exactly the valley positions in Fig. 1. These trapping positions do not vary with the parameters β and kc of MCAB. From Figs. 5(d) and 5(e) we can see that the particle can also be transversely trapped at zc and zd by MCAB. The transverse gradient force of MCAB is greater than that of CAB, and the trapping range is slightly increased by using MCAB. However, MCAB is not able to trap low-refractive-index particles when kc is too large. As we can see in Figs. 5(d) and 5(e), for MCAB with β = 3μm, kc = 1.5μm−1, the center of the distribution of the gradient force is not an equilibrium point because of the appearance of the central peak. So appropriate parameters should be chosen to trap low-refractive-index particles by MCAB in practice.
We assume the maximum trapping force along the longitudinal direction and transverse direction as Fz_max and Fgr_max, respectively. Because of the asymmetric property of the distributions of the longitudinal radiation force [Figs. 4(c) and 5(c)]. The magnitude of Fz_max should be defined as the smaller one of the positive force and the negative force in the trapping region.
Figures 6 and 7 show the changes of the trapping forces exerted on the two kinds of particles with β and kc. For the high-refractive-index particle, Fz_max and Fgr_max will attain maximum values for certain β and kc [Figs. 6(a)-6(c)]. Take the optical trap at za as an example, for MCAB with kc = 1μm−1, the longitudinal trapping force become largest when β = 4μm, but the transverse trapping force become largest when β = 5.5μm. The most appropriate values of β and kc are different for other optical traps. For the optical trap at zb, the longitudinal trapping force become largest when β = 2μm, as kc = 1μm−1 is constant [Fig. 6(a)]. Similar conclusions can also be obtained for the low-refractive-index particle. As we can see in Fig. 7, Fz_max and Fgr_max approach maximum values for certain β when kc is constant [Figs. 7(a) and7(c)], Fz_max becomes maximum for certain β when kc is constant [Fig. 7(b)]. However, the case of Fgr_max with kc is much different. When kc is too large, the center of the beam is not an equilibrium position for the low-refractive-index particle as we have shown in Figs. 5(d) and 5(e). So the longitudinal trapping force do not exist when kc is too large as is shown in Fig. 7(d). In practice, we can modulate trapping forces at certain position for certain particle by choosing appropriate parameters of MCAB.
4. Stable analysis
For stable trapping, the trapping force of MCAB must be large enough to conquer the scattering force, the gravity of the particle, and the disturbance of the Brownian motion . As we can see in Figs. 4(c) and 5(c), the backward longitudinal gradient force is able to overcome the forward scattering force at the trapping position. For the two kinds of particles analyzed in this paper, the gravity forces are both about 10−6pN, which are much less than the trapping force.
The disturbance of the Brownian motion can be described as the Brownian force. For stable trapping, the trapping force must be larger than the Brownian force, which can be calculated by the equation ,Figs. 4 and 5. The values of these forces are listed in Table 1 and Table 2. For simplicity, we only analyze MCAB with β = 3μm, kc = 1μm−1 here. From Table 1 and Table 2, we can find that the trapping forces exerted on the two kinds of particles produced by MCAB are both larger than the Brownian force. So the high-refractive-index particle can be stably trapped at the plane z = za, and the low-refractive-index particle can be stably trapped at the plane z = zc. But for CAB with same parameters, the longitudinal trapping forces are not large enough to overcome the Brownian motion. So the two kinds of particles cannot be stably trapped by CAB. However, the two particles could not be stably trapped by MCAB at z = zb or z = zd, respectively. The longitudinal gradient forces are not large enough to overcome the disturbance of the Brownian motion. Increasing the incident power might be helpful to enhance the stability of the optical trap at these positions.
In summary, we have theoretically investigated the radiation force produced by MCAB exerted on the high-refractive-index particle and the low-refractive-index particle. It is found that the two kinds of particles can be simultaneously trapped by MCAB at different planes, and there are several trapping positions along the beam axis. Comparing with CAB, because of the great enhancement of the abruptly autofocusing property, the longitudinal trapping force and transverse trapping force exerted on the high-refractive-index particle are increased. Because of the appearance of the “perfect” hollow region, the trapping force exerted on the low-refractive-index particle produced by MCAB is much greater than that produced by CAB. Moreover, the optical trap formed by CAB will be greatly influenced by the Brown motion, especially for the low-refractive-index particle. But it is easier for MCAB with the same parameters to conquer the disturbance of the Brown motion. In practice, it is important to choose appropriate parameters of MCAB when applying this beam in optical tweezers. The longitudinal trapping force and the transverse trapping force exerted on the two kinds of particles can be controlled by parameters of β and kc. The most appropriate values of β and kc are related with the particle being trapped and the position of the optical trap. Because two types of particles could be trapped at different planes, MCAB might have great potential applications in optical micromanipulation, especially in optical sorting or other fields.
National Nature Science Foundation of China (NSFC) (11504274, 11474254); Zhejiang Provincial Natural Science Foundation (LQ16A040004).
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