Trapping two types of particles by modified circular Airy beams

Yunfeng Jiang; Zili Cao; Hehong Shao; Wanting Zheng; Bixin Zeng; Xuanhui Lu

doi:10.1364/OE.24.018072

1. Introduction

Nowadays, optical tweezers have become an important tool since they are first invented in 1986 by Ashkin [1]. 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 [28], radially polarized beam [21], partially coherent elegant Laguerre-Gaussian beam [16] 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 [29]. The first observed abrupt autofocusing beam is the circular Airy beam (CAB) [30]. Lately, various kinds of abruptly autofocusing beams have been proposed, such as CAB with optical vortex [31,32], modified CAB [33], non-paraxial abruptly autofocusing beams [34], 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 [17]. But it is difficult for CAB to trap low-refractive index particles as we have shown in our previous work [20].

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 MCAB

The MCAB is generated from the modification of the spectrum of CAB [33]. We assume the electric field of CAB at the initial plane is [29],

u (r) = C_{0} A i (\frac{r_{0} - r}{w}) \exp (a \frac{r_{0} - r}{w}),

where r₀ is related with the initial radius of CAB, w is a scaled parameter, a is the decaying parameter, C₀ is a constant related with the input power of CAB. So the Fourier spectrum of CAB can be expressed as [40],

U (k) = C_{0} w^{2} (\frac{r_{0}}{w} + k^{2} w^{2}) \exp (- a k^{2} w^{2}) \sqrt{\frac{3 k r_{0} + k^{3} w^{3}}{3 k r_{0} + 3 k^{3} w^{3}}} J_{0} (k r_{0} + \frac{k^{3} w^{3}}{3}),

where k is radial spatial frequency in the Fourier space. The Fourier spectrum of MCAB, U_m(k), can be generated by adding an apodization mask M to the spectrum of CAB [33],

U_{m} (k) = M (k) U (k) .

The mask M must be specially designed to reduce the low-frequency component,

M (k) = \frac{1}{1 + e^{- β (k - k_{c})}},

where β controls the steepness of apodization, and k_c is the cutoff position of the frequency.

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 [41]:

\overset{⇀}{E} (r, φ, z) = u_{m x} (r, z) \hat{x} + u_{m z} (r, φ, z) \hat{z} .

u_mx and u_mz can be calculated from the Hankel transform [20,42],

u_{m x} (r, z) = \int_{0}^{\infty} U_{m} (k) J_{0} (k r) e^{i z k_{z}} k d k,

u_{m z} (r, φ, z) = - i \int_{0}^{\infty} U_{m} (k) J_{1} (k r) e^{i k_{z} z} \frac{k^{2}}{k_{z}} d k \cdot \cos φ,

where

k_{z} = \sqrt{{(2 π / λ)}^{2} - k^{2}}

, λ is wavelength. So the intensity of MCAB can be expressed as,

I = I_{x} + I_{z} = {| u_{m x} |}^{2} + {| u_{m z} |}^{2} .

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 [33]. 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.

Fig. 1 Propagation characteristics of MCAB along the beam axis.

Download Full Size | PDF

Fig. 2 Transverse intensity profiles of MCAB with β = 3μm, k_c = 1μm⁻¹ at (a)-(c) focal position and (d)-(f) first valley position: (a)(d), I_x; (b)(e), I_z; (c)(f), I_x + I_z.

Download Full Size | PDF

Fig. 3 The intensity distributions of MCAB at peak positions and valley positions: (a) first peak position; (b) second peak position; (c) first valley position; (d) second valley position. The parameters of MCAB are the same as Fig. 1.

Download Full Size | PDF

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 k_c, but the rate of increase is very small when k_c 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 k_c. The first peak position, i.e., focal position, is z_f = 0.236mm; the first valley position is z = 0.250mm. Figure 2 shows the profiles of the two intensity components, I_x and I_z, at different planes. The intensity pattern of I_z 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 I_z is much less than I_x, and the total intensity is hardly affected by I_z. So we can neglect the influence of I_z 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 k_c 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, k_c = 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 k_c, the intensity at first valley position can be modulated to nearly zero. But the hollow region will disappear when k_c is too large. Figures 3(c) and 3(d) show that an intensity peak will appear at the center when k_c = 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 [43],

α = 4 π R^{3} \frac{ε_{p} - ε_{m}}{ε_{p} + 2 ε_{m}},

where R is the radius of the particle, ε_p is the dielectric function of the Rayleigh particle and ε_m is the dielectric function of the surrounding medium. So the gradient force F_g and the scattering force F_s can be calculated by the expressions [44]:

{\overset{⇀}{F}}_{g} = \frac{1}{4} ε_{0} ε_{m} Re (α) \nabla I,

F_{s} = \frac{ε_{0} ε_{m}^{3} k_{0}^{4}}{12 π} | α^{2} | I,

where ε₀ is the dielectric constant in vacuum, k₀ is the vacuum wave number. In our numerical calculation, we choose R = 40nm, the surrounding medium is water (ε_m = 1.33²). The radiation force exerted on two kinds of particles are analyzed below. We assume the refractive index of the high-refractive-index particle is n_h = 1.59 (i.e., glass microparticle); the refractive index of the low-refractive-index particle is n_l = 1.00 (i.e., air bubble), so

ε_{p} = n_{h}^{2}

or

ε_{p} = n_{l}^{2}

.

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, z_a = 0.238mm and z_b = 0.268mm. Note that the position of z_a is slightly shifted from the focal point (z_f = 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 z_a and z_b 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 k_c without affecting the trapping range. So the stiffness of the optical trap can be enhanced by choosing proper parameters of β and k_c.

Fig. 4 The distributions of the radiation force on the high-refractive-index particle with n_h = 1.59. (a) The longitudinal gradient force; (b) the scattering force; (c) the sum of the gradient force and the scattering force, z_a and z_b are the first two trapping positions; (d) the transverse gradient force at z_a; (e) the transverse gradient force at z_b.

Download Full Size | PDF

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 z_c = 0.250mm and z_d = 0.264mm. The trapping positions are exactly the valley positions in Fig. 1. These trapping positions do not vary with the parameters β and k_c of MCAB. From Figs. 5(d) and 5(e) we can see that the particle can also be transversely trapped at z_c and z_d 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 k_c is too large. As we can see in Figs. 5(d) and 5(e), for MCAB with β = 3μm, k_c = 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.

Fig. 5 The distributions of the radiation force on the low-refractive-index particle with n_l = 1.00. (a) The longitudinal gradient force; (b) the scattering force; (c) the sum of the gradient force and the scattering force, z_c and z_d are the first two trapping positions; (d) the transverse gradient force at z_c; (e) the transverse gradient force at z_d.

Download Full Size | PDF

We assume the maximum trapping force along the longitudinal direction and transverse direction as F_{z_max} and F_{gr_max}, respectively. Because of the asymmetric property of the distributions of the longitudinal radiation force [Figs. 4(c) and 5(c)]. The magnitude of F_{z_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 k_c. For the high-refractive-index particle, F_{z_max} and F_{gr_max} will attain maximum values for certain β and k_c [Figs. 6(a)-6(c)]. Take the optical trap at z_a as an example, for MCAB with k_c = 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 k_c are different for other optical traps. For the optical trap at z_b, the longitudinal trapping force become largest when β = 2μm, as k_c = 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, F_{z_max} and F_{gr_max} approach maximum values for certain β when k_c is constant [Figs. 7(a) and7(c)], F_{z_max} becomes maximum for certain β when k_c is constant [Fig. 7(b)]. However, the case of F_{gr_max} with k_c is much different. When k_c 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 k_c 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.

Fig. 6 Changes of the trapping force exerted on the high-refractive-index particle with β and k_c: (a) the change of longitudinal trapping force with β, when k_c = 1μm⁻¹; (b) the change of longitudinal trapping force with k_c, when β = 3μm; (c) the change of transverse trapping force with β, when k_c = 1μm⁻¹; (d) the change of transverse trapping force with k_c, when β = 3μm.

Download Full Size | PDF

Fig. 7 Changes of the trapping force exerted on the low-refractive-index particle with β and k_c: (a) the change of longitudinal trapping force with β, when kc = 1μm⁻¹; (b) the change of longitudinal trapping force with k_c, when β = 3μm; (c) the change of transverse trapping force with β, when k_c = 1μm⁻¹; (d) the change of transverse trapping force with k_c, when β = 3μm.

Download Full Size | PDF

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 [1]. 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 [45],

| F_{b} | = \sqrt{12 π η R k_{B} T},

where η = 8.0 × 10⁻⁴Pa∙s, is the viscosity of surrounding medium, R is the radius of the particle, k_B is the Boltzmann constant, T = 300K is the temperature. So the Brownian force F_b is about 2.2 × 10⁻³pN in this paper. The effective trapping forces exerted on different particles at different positions can be obtained from 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, k_c = 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 = z_a, and the low-refractive-index particle can be stably trapped at the plane z = z_c. 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 = z_b or z = z_d, 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.

Table 1. Trapping forces exerted on the trapped particle with n_h = 1.59

View Table | View all tables in this article

Table 2. Trapping forces exerted on the trapped particle with n_l = 1.00

View Table | View all tables in this article

5. Conclusions

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 k_c. The most appropriate values of β and k_c 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.

Funding

National Nature Science Foundation of China (NSFC) (11504274, 11474254); Zhejiang Provincial Natural Science Foundation (LQ16A040004).

References and links

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

2. S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990). [CrossRef] [PubMed]

3. L. Tskhovrebova, J. Trinick, J. A. Sleep, and R. M. Simmons, “Elasticity and unfolding of single molecules of the giant muscle protein titin,” Nature 387(6630), 308–312 (1997). [CrossRef] [PubMed]

4. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72(3), 1335–1346 (1997). [CrossRef] [PubMed]

5. P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature 436(7049), 370–372 (2005). [CrossRef] [PubMed]

6. Y. Pang, H. Song, J. H. Kim, X. Hou, and W. Cheng, “Optical trapping of individual human immunodeficiency viruses in culture fluid reveals heterogeneity with single-molecule resolution,” Nat. Nanotechnol. 9(8), 624–630 (2014). [CrossRef] [PubMed]

7. M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013). [CrossRef] [PubMed]

8. R. Liu, L. Zheng, D. L. Matthews, N. Satake, and J. W. Chan, “Power dependent oxygenation state transition of red blood cells in a single beam optical trap,” Appl. Phys. Lett. 99(4), 043702 (2011). [CrossRef]

9. R. Liu, Z. Mao, D. L. Matthews, C.-S. Li, J. W. Chan, and N. Satake, “Novel single-cell functional analysis of red blood cells using laser tweezers Raman spectroscopy: Application for sickle cell disease,” Exp. Hematol. 41(7), 656–661 (2013). [CrossRef] [PubMed]

10. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

11. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]

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

13. Q. Zhang, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003). [CrossRef]

14. 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]

15. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

16. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

17. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]

18. Z. Liu and D. Zhao, “Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermite-cosine-Gaussian beams,” Opt. Express 20(3), 2895–2904 (2012). [CrossRef] [PubMed]

19. Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle,” Opt. Express 19(10), 9708–9713 (2011). [CrossRef] [PubMed]

20. Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle,” Opt. Express 21(20), 24413–24421 (2013). [CrossRef] [PubMed]

21. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]

22. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

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

24. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]

25. A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011). [CrossRef] [PubMed]

26. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]

27. C. M. Herne, K. M. Capuzzi, E. Sobel, and R. T. Kropas, “Rotation of large asymmetrical absorbing objects by Laguerre-Gauss beams,” Opt. Lett. 40(17), 4026–4029 (2015). [CrossRef] [PubMed]

28. K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999). [CrossRef]

29. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

30. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]

31. J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express 20(12), 13302–13310 (2012). [CrossRef] [PubMed]

32. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef] [PubMed]

33. Y. Jiang, X. Zhu, W. Yu, H. Shao, W. Zheng, and X. Lu, “Propagation characteristics of the modified circular Airy beam,” Opt. Express 23(23), 29834–29841 (2015). [CrossRef] [PubMed]

34. R. S. Penciu, K. G. Makris, and N. K. Efremidis, “Nonparaxial abruptly autofocusing beams,” Opt. Lett. 41(5), 1042–1045 (2016). [CrossRef] [PubMed]

35. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011). [CrossRef] [PubMed]

36. I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85(2), 023828 (2012). [CrossRef]

37. P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. Martinez Matos, “Symmetric Airy beams,” Opt. Lett. 39(8), 2370–2373 (2014). [CrossRef] [PubMed]

38. P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014). [CrossRef] [PubMed]

39. S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. 38(14), 2416–2418 (2013). [CrossRef] [PubMed]

40. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]

41. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62(10), 1195–1201 (1972). [CrossRef]

42. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

43. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

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

45. K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]

Trapping force	MCAB (10⁻³pN)	CAB (10⁻³pN)
F_{z_max} (z = z_a)	4.5	1.5
F_{gr_max} (z = z_a)	43	14
F_{z_max} (z = z_b)	1.9	0.82
F_{gr_max} (z = z_b)	9.7	4.7

Trapping force	MCAB (10⁻³pN)	CAB (10⁻³pN)
F_{z_max} (z = z_c)	2.9	1.3
F_{gr_max} (z = z_c)	7.3	2.1
F_{z_max} (z = z_d)	1.5	0.76
F_{gr_max} (z = z_d)	3.5	1.5

Trapping force	MCAB (10⁻³pN)	CAB (10⁻³pN)
F_{z_max} (z = z_a)	4.5	1.5
F_{gr_max} (z = z_a)	43	14
F_{z_max} (z = z_b)	1.9	0.82
F_{gr_max} (z = z_b)	9.7	4.7

Trapping force	MCAB (10⁻³pN)	CAB (10⁻³pN)
F_{z_max} (z = z_c)	2.9	1.3
F_{gr_max} (z = z_c)	7.3	2.1
F_{z_max} (z = z_d)	1.5	0.76
F_{gr_max} (z = z_d)	3.5	1.5

Trapping two types of particles by modified circular Airy beams

Abstract

1. Introduction

2. Propagation characteristics of MCAB

3. Radiation forces of MCAB on two kinds of particles

4. Stable analysis

5. Conclusions

Funding

References and links

Cited By

Figures (7)

Tables (2)

Equations (12)

Optics Express