Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams Xiayin Liu and Daomu Zhao* Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received 25 February 2014; revised 12 May 2014; accepted 16 May 2014; posted 16 May 2014 (Doc. ID 207046); published 17 June 2014
We numerically investigate the radiation forces of multi-Gaussian Schell-model (MGSM) beams, in which the degree of coherence is modeled by the multi-Gaussian function, exerted on the Rayleigh dielectric sphere. By simulation of the forces calculation it is found that the steepness of the edge of the intensity profile (i.e., the summation index M) and the initial coherence width of the MGSM beams play important roles in the trapping range and stability. We can increase the trapping range at the focal plane by increasing the value of M or decreasing the initial coherence of the MGSM beams. It is also found that the trapping stability becomes lower due to the increase of the value of M or the decrease of coherence. Furthermore, the trapping stability under different conditions is explicitly analyzed. The results presented here are helpful for some possible applications. © 2014 Optical Society of America OCIS codes: (140.7010) Laser trapping; (030.1670) Coherent optical effects; (290.5870) Scattering, Rayleigh; (350.5500) Propagation. http://dx.doi.org/10.1364/AO.53.003976
1. Introduction
Recently, a new type of source named the multiGaussian Schell-model (MGSM) was introduced, whose degree of coherence is modeled by a multiGaussian distribution [1]. The MGSM beams form intensity plateaus in the far field, and the width of the flat center of its intensity profile can be controlled by the multi-Gaussian function. So it is clear that the MGSM source is different from the previously introduced sources [2,3], which have flattened intensity profile in the source plane but gradually become Gaussian on propagation. The propagation properties of the field generated by the MGSM source in free space and some linear random media were investigated [4]. In addition, Mei et al. extended the scalar MGSM beam to the electromagnetic domain and explored the polarization properties of such a beam [5]. Zhang and Zhao studied the scattering of the MGSM beam on a random medium and discussed 1559-128X/14/183976-06$15.00/0 © 2014 Optical Society of America 3976
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how the source parameters affect the far-zone properties [6]. All the studies show the novel far fields that the MGSM source generates have potential applications in beam shaping, free-space communication, and optical trapping. Optical trapping and manipulation of micrometersized particles has been the subject of many investigations since the pioneering work of Ashkin et al., who claimed that the radiation forces from a single laser beam could be used to trap the dielectric sphere (optical tweezers) [7]. It is known that the radiation forces consist of the scattering force and the gradient force, which arise from the momentum transfer caused by scattering of photons from the particles and the gradient of the electromagnetic field separately. The gradient force directs the particles to the region of maximum light intensity, while the scattering tends to push the particle out of the desired region. Therefore, in practice, a laser beam is always focused by a lens to achieve a refractionlimited optical spot (i.e., a larger gradient force) at the focal plane. To date, besides the fundamental Gaussian beams [8], a variety of beams have been
studied to trap particles. For example, GSM beams [9], anomalous hollow beams [10], and Lorentz– Gauss beams [11] were considered to trap the particles with refraction index higher than the ambient. Meanwhile, bottle beams [12] and elegant Hermitecosine-Gaussian beams [13] could be used to simultaneously trap two types of particles with refraction index higher or lower than the ambient. In addition, the trapping characteristics of other beams, such as radially polarized beams [14–16], vortex beams [17,18], Airy beams [19–21], and coherent and partially coherent beams [22–24], have also been explored. In all the studies, it is found that the radiation forces are closely related to the initial beam characteristics, such as the coherence, polarization, and spatial profile. In this paper, we will investigate the trapping effect of the MGSM beams. In Section 2, we first describe the intensity distributions of the MGSM beams through a focusing system, and then we analyze the radiation forces acting on a Rayleigh particle produced by the MGSM beams. In Section 3, we discuss the trapping stability in more detail. Finally, in Section 4, the conclusions are outlined. 2. Radiation Force Produced by Focused MGSM Beams
The cross-spectral density of the MGSM beam at two position vectors ρ10 and ρ20 of the source plane has a form of [1,6] M M ρ210 ρ220 1 X −1m−1 ρ10 ;ρ20 I 0 exp − m C0 m1 m 4σ 2 2 jρ − ρ j × exp − 20 10 ; (1) 2mδ2
W
0
in which I 0 is a constant given by I 0 P∕2πσ 2 with P being the input power and σ being the rms width of PM M is the northe source, C0 m1 −1m−1 ∕m m M malization factor with being for binomial coefm ficients, and δ is the rms width of the correlation. By employing the generalized Huygens–Fresnel principle, the cross-spectral density of the MGSM beam that passes through an ABCD optical system can be obtained as follows:
coefficients and the generalized phase radii of curvature, respectively. The intensity and the cross-spectral density of the MGSM beam at any point of the output plane are related by the formulas [25] I out ρ; z Wρ; ρ; z:
(3)
Consequently, the expression of the intensity of the MGSM beam through an ABCD optical system can be obtained by using Eqs. (2) and (3). Consider the propagation of the MGSM beam through an unapertured lens system (see Fig. 1). Let z be the separation between the input plane and the output plane, and f be the focal length. Then the ABCD matrix for the focusing system is
A B 1 C D 0
z 1
1 −1∕f
0 1 − z∕f 1 −1∕f
z : (4) 1
Upon substituting from Eq. (4) into Eq. (3), we can obtain the density distribution of the MGSM beams at the focal plane. In this paper, we choose P 100 mW, f 2 mm, λ 632.8 nm, σ 2 mm, and δ 2 mm. Figure 2 shows the transverse intensity distribution of the MGSM beams at the focal plane for different values of M. It can be found that intensity distribution depends on the value of M: the greater the value of M is, the smaller the magnitude of the intensity is and the flatter the profile becomes. It is well known that the particle whose radius is sufficiently smaller than the wavelength of light can be treated as a point dipole; in this case, the radiation force exerted on the dipole has two types: the scattering and gradient force, which come from the scattering by the dipole and the Lorenz force acting on the induced dipole, respectively. Assuming a homogeneous Rayleigh microsphere with refractive index np and radius a, the descriptions of the scattering force F sca and the gradient force F grad are given by [9,26] F⃗ sca ρ; z e⃗ z nm αI out ∕c;
(5)
2 M ρ1 ρ22 I0 X −1m−1 M 1 exp − Wρ1 ; ρ2 ; z C0 m1 mM 4Δσ 2 m Δ ikρ22 − ρ21 jρ2 − ρ1 j2 ; × exp − exp − 2R 2mΔδ2 (2) where Δ A2 B2 ∕4σ 4 k2 1 4σ 2 ∕mδ2 and R BΔ∕DΔ − A represent the generalized spreading
Fig. 1. Schematic of a MGSM beam through the focusing optical system. 20 June 2014 / Vol. 53, No. 18 / APPLIED OPTICS
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Fig. 2. (a) Intensity distribution for different values of M at the focal plane. (b) Two-dimensional intensity distribution for M 30.
F grad ⃗ρ; z 2πnm β∇I out ∕c;
(6)
where e⃗ z is a unity vector along the beam propagation, I out is the intensity of the focused beam at the output plane, and α 128π 5 a6 ∕3λ4 m20 − 1∕ m20 12 and β a3 m20 − 1∕m20 2 are the scattering cross section and the Polaris ability for the point dipole, respectively. m0 np ∕nm is the relative refraction index of the particles, with nm being the refractive index of the surrounding medium. According to Eqs. (2)–(6), we can obtain the radiation force at the output plane as follows: M ⃗e n αP X −1m−1 M 1 ρ2 ; exp − F⃗ sca ρ; z z m 2πσ 2 cC0 m1 m 2Δσ 2 m Δ (7) M −1m−1 e⃗ n βP X F⃗ grad;x ρ; z − x 2 m σ cC0 m1 m ρ2 ; × exp − 2Δσ 2
M m
x Δ σ
2 2
(8)
M X −1m−1 M 1 ⃗F grad;z ρ; z − e⃗ z nm βP σ 2 cC0 m1 m m Δ2 2B 4σ 2 ρ2 × 2AC 4 2 1 1− 2 mδ2 2σ 4σ k 2 ρ : (9) × exp − 2Δσ 2
Obviously, F⃗ sca is in the beam-propagation direction, while F⃗ grad is along the gradient of light intensity and acts as the restoring force directed toward the focusing center in the case of m0 > 1. In addition, it is evident that both F⃗ sca and F⃗ grad are affected by index M and the correlation width δ. Therefore, the distributions of the scattering force and the gradient force for different values of M and δ are plotted. The radius and refraction index of the particle and the refraction index of the ambient are chosen to be a 50 nm, np 1.59 (i.e., glass), and nm 1.33 (i.e., water), respectively. Figure 3 displays the distribution of the transverse gradient force F⃗ grad;x , the longitudinal gradient force F⃗ grad;z , and the scattering force F⃗ sca for different values of M at the focal plane. From Fig. 3 it can be found that a particle with m0 > 1 can be trapped by the MGSM beams at the focal point because there are stable equilibrium regions in Figs. 3(a) and 3(b) and the longitudinal gradient force is always larger than the forward-scattering force for different values of index M. The profile of scattering force contains a flat region being wider for larger values of M. In addition, with the value of summation index M increasing the transverse and longitudinal trapping ranges become larger; here, the trapping range is the distance from the equilibrium position to the position at which trapping starts to occur, but the trapping stability (i.e., the difference between the magnitudes of the force at the focal plane and at the boundary of the trapping region) reduces, which is similar to that of partially coherent flat-topped beams near the focal plane [22]. However, it is worthwhile to note that the
m m Fig. 3. Effect of index M on the radiation force of the MGSM beam at the focal plane: (a) F m grad;x ; (b) F grad;z ; and (c) F sca .
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m Fig. 4. Dependence of the radiation force of the MGSM beam at the focal plane on the coherence width δ, M 30: (a) F m grad;x ; (b) F grad;z ; and . (c) F m sca
trapping force and ranges of the MGSM beam at the focal plane are larger than those for the partially coherent flat-topped beams near the focal plane with the same trapping power and spot width. The affection of coherence on the gradient force and the scattering force of the MGSM beam with M 30 at the focal plane are indicated in Fig. 4. From Fig. 4 we can find that as the coherence of the MGSM increases, both the gradient force and the scattering force increase, but the stable trapping region becomes smaller and finally becomes a stable point. Although the trapping region of the MGSM beams with M > 1 decreases with the coherence increasing, it is always larger than that of the classic GSM beams with the same coherence. By comparison of the analyses of Figs. 3 and 4, it can be concluded that as the value of M increases, the trapping range becomes larger, but the stability decreases due to the reduction of radiation. On the contrary, as the coherence increases, the trapping range becomes smaller, but stability increases. Hence, it is important to choose suitable values of M and coherence to obtain a larger trapping range and higher stability. 3. Analysis of Trapping Stability
In this section, we will discuss the trapping stability in more detail. A necessary condition for stable trapping is that the backward axial gradient force must be great enough to overcome the forwardscattering force and the Brownian force at the position of the maximum intensity gradient [26]. According to Figs. 3(a) and 3(b), it can be found that the positions of the maximum intensity gradient (i.e., the effective focus plane) have a shift as the value of M increases. Through further calculation, we find the real focus plane is not at F 0; f but at F 0 0; ξ; here ξ z0
f 1
1 C0
PM
−1m−1 m1 m
; M 2 2 g f m 0
(10)
where g20 f ∕4σ 4 k2 1 4σ 2 ∕mδ2 ; obviously, the focal shift is related to the value of M. In the real focusing plane z ξ, the maximum transverse gradient force, the maximum longitudinal gradient
force, and the maximum scattering force are derived as follows: 0 1 g0 f σ F m @ q A ; ξ grad;x 1 g2 f 2 0
M M X
nm βP σ 2 cC0 m1 m
−1m−1 21 g20 f 2 3∕2 1 ; m e g0 f 3 σ
p 3g0 f F m grad;z 0; ξ 1 3 p M M −1m−1 3 31 g20 f 2 2 nm βP X 2 ; m σ cC0 m1 m 8g30 f 4 p 3g0 f F m 0; ξ 1 sca 3 M M n αP X −1m−1 31 g20 f 2 m2 : m 2πσ cC0 m1 m 4g20 f 2
(11)
(12)
(13)
The Brownian force due to the thermal fluctuation from the ambient (e.g., water) can be calculated by the formula [27] jF B j 12πκakB T1∕2 ;
(14)
where κ is the viscosity of the ambient, which is κ 7.977 × 10−4 Pas at T 300 K, a is the radius of the particle, and kB is the Boltzmann constant. x m In Fig. 5 the dependence of F m grad;x , F grad;z , and F sca on the summation index M, the coherence width δ, and the radius of particles a is calculated at the focal plane, and the maximum radiation forces are compared with jF B j. From Fig. 5(a) it can be seen that as the value of M increases the gradient force and the scattering force decrease and the gradient force is always larger than the scattering force and the Brownian force. So the MSGM beam with a larger value of M can be used to trap a particle, which offers some advantages over the trapping ranges of the classic GSM. But the values of M cannot be too large, 20 June 2014 / Vol. 53, No. 18 / APPLIED OPTICS
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m m Fig. 5. Comparison of F m grad;x , F grad;z , F sca , and F b (a) with different M, δ 2 mm; (b) with different δ, M 30; and (c) with different particle radius a, M 30.
because the radiation force reduces with M increasing, as discussed in Section 2. Figure 5(b) shows that with the coherence width increasing the gradient and the scattering force increase, which agrees with Fig. 4. It can be further found from Fig. 5(b) that when δ > 0.816 mm, the gradient force is greater than the scattering force and the Brownian force; in this case, the beam can trap a particle. So δ 0.816 mm is the low limit to stably trap particles for the case of M 30. The change of the forces versus radii of particles in the case of M 30 is plotted in Fig. 5(c). It can be seen from Fig. 5(c) that for the case in which a < 20 nm the Brownian force is greater than the gradient force and for a > 60 nm the scattering force is nearly equal to the gradient force. For both cases, the stability criterion is broken. Therefore, the particles with 20 nm < a < 60 nm can be stably trapped by the radiation forces produced by the MGSM beam. 4. Conclusion
We have studied the radiation forces on Rayleigh particles induced by focused MGSM beams. It is found that the radiation force produced by the MGSM beams can be used to trap particles with a highly refractive index at the focus plane. And the radiation forces are greatly affected by the summation index M and the coherence width of the beam. The transverse and longitudinal trapping range at the focal plane can be increased by increasing the value of M or decreasing the initial coherence of the MGSM beams. However, the trapping stability reduces due to the increase of the value of M and the decrease of coherence. So it is important to choose suitable values of M and the coherence of the MGSM beams to trap a particle. In addition, we explicitly discuss the trapping stability. Both the low limit of coherence and the range of the particle’s radius for stable trapping in the case of M 30 are determined. The results presented here are useful for some possible applications. This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11274273 and 11074219). 3980
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References 1. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012). 2. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). 3. R. Borghi, “Uniform approximation of paraxial flat-topped beams,” J. Opt. Soc. Am. A 30, 1099–1106 (2013). 4. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2163 (2012). 5. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15, 025705 (2013). 6. Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schellmodel beams on a random medium,” Opt. Express 21, 24781–24792 (2013). 7. 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, 288–290 (1986). 8. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970). 9. L. Wang, C. Zhao, L. Wang, X. Lu, and S. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32, 1393–1395 (2007). 10. Z. Liu and D. Zhao, “Optical trapping Rayleigh dielectric spheres with focused anomalous hollow beams,” Appl. Opt. 52, 1310–1316 (2013). 11. Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle,” Opt. Express 19, 9708–9713 (2011). 12. C.-H. Chen, P.-T. Tai, and W.-F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43, 6001–6006 (2004). 13. Z. Liu and D. Zhao, “Radiation forces acting on a Rayleigh dielectric sphere produced by highly focused elegant Hermitecosine-Gaussian beams,” Opt. Express 20, 2895–2904 (2012). 14. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). 15. S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007). 16. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010). 17. J. Shu, Z. Chen, and J. Pu, “Radiation forces on a Rayleigh particle by highly focused partially coherent and radially polarized vortex beams,” J. Opt. Soc. Am. A 30, 916–922 (2013). 18. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012). 19. Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). 20. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding
micoparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011). 21. Y. Yang, W. P. Zhang, Z. Y. Zhao, and J. G. Tian, “Optical forces on Mie particles in an Airy evanescent field,” Opt. Express 20, 25681–25692 (2012). 22. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboglu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009). 23. J. M. Auñón and M. Nieto-Vesperinas, “Optical forces on small particles from partially coherent light,” J. Opt. Soc. Am. A 29, 1389–1398 (2012).
24. J. M. Auñón and M. Nieto-Vesperinas, “Partially coherent fluctuating sources that produce the same optical forces as a laser beam,” Opt. Lett. 38, 2869–2872 (2013). 25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 26. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). 27. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83, 4534–4537 (1999).
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We numerically investigate the radiation forces of multi-Gaussian Schell-model (MGSM) beams, in which the degree of coherence is modeled by the multi-Gaussian function, exerted on the Rayleigh dielectric sphere. By simulation of the forces calculation it is found that the steepness of the edge of the intensity profile (i.e., the summation index M) and the initial coherence width of the MGSM beams play important roles in the trapping range and stability. We can increase the trapping range at the focal plane by increasing the value of M or decreasing the initial coherence of the MGSM beams. It is also found that the trapping stability becomes lower due to the increase of the value of M or the decrease of coherence. Furthermore, the trapping stability under different conditions is explicitly analyzed. The results presented here are helpful for some possible applications. © 2014 Optical Society of America OCIS codes: (140.7010) Laser trapping; (030.1670) Coherent optical effects; (290.5870) Scattering, Rayleigh; (350.5500) Propagation. http://dx.doi.org/10.1364/AO.53.003976
1. Introduction
Recently, a new type of source named the multiGaussian Schell-model (MGSM) was introduced, whose degree of coherence is modeled by a multiGaussian distribution [1]. The MGSM beams form intensity plateaus in the far field, and the width of the flat center of its intensity profile can be controlled by the multi-Gaussian function. So it is clear that the MGSM source is different from the previously introduced sources [2,3], which have flattened intensity profile in the source plane but gradually become Gaussian on propagation. The propagation properties of the field generated by the MGSM source in free space and some linear random media were investigated [4]. In addition, Mei et al. extended the scalar MGSM beam to the electromagnetic domain and explored the polarization properties of such a beam [5]. Zhang and Zhao studied the scattering of the MGSM beam on a random medium and discussed 1559-128X/14/183976-06$15.00/0 © 2014 Optical Society of America 3976
APPLIED OPTICS / Vol. 53, No. 18 / 20 June 2014
how the source parameters affect the far-zone properties [6]. All the studies show the novel far fields that the MGSM source generates have potential applications in beam shaping, free-space communication, and optical trapping. Optical trapping and manipulation of micrometersized particles has been the subject of many investigations since the pioneering work of Ashkin et al., who claimed that the radiation forces from a single laser beam could be used to trap the dielectric sphere (optical tweezers) [7]. It is known that the radiation forces consist of the scattering force and the gradient force, which arise from the momentum transfer caused by scattering of photons from the particles and the gradient of the electromagnetic field separately. The gradient force directs the particles to the region of maximum light intensity, while the scattering tends to push the particle out of the desired region. Therefore, in practice, a laser beam is always focused by a lens to achieve a refractionlimited optical spot (i.e., a larger gradient force) at the focal plane. To date, besides the fundamental Gaussian beams [8], a variety of beams have been
studied to trap particles. For example, GSM beams [9], anomalous hollow beams [10], and Lorentz– Gauss beams [11] were considered to trap the particles with refraction index higher than the ambient. Meanwhile, bottle beams [12] and elegant Hermitecosine-Gaussian beams [13] could be used to simultaneously trap two types of particles with refraction index higher or lower than the ambient. In addition, the trapping characteristics of other beams, such as radially polarized beams [14–16], vortex beams [17,18], Airy beams [19–21], and coherent and partially coherent beams [22–24], have also been explored. In all the studies, it is found that the radiation forces are closely related to the initial beam characteristics, such as the coherence, polarization, and spatial profile. In this paper, we will investigate the trapping effect of the MGSM beams. In Section 2, we first describe the intensity distributions of the MGSM beams through a focusing system, and then we analyze the radiation forces acting on a Rayleigh particle produced by the MGSM beams. In Section 3, we discuss the trapping stability in more detail. Finally, in Section 4, the conclusions are outlined. 2. Radiation Force Produced by Focused MGSM Beams
The cross-spectral density of the MGSM beam at two position vectors ρ10 and ρ20 of the source plane has a form of [1,6] M M ρ210 ρ220 1 X −1m−1 ρ10 ;ρ20 I 0 exp − m C0 m1 m 4σ 2 2 jρ − ρ j × exp − 20 10 ; (1) 2mδ2
W
0
in which I 0 is a constant given by I 0 P∕2πσ 2 with P being the input power and σ being the rms width of PM M is the northe source, C0 m1 −1m−1 ∕m m M malization factor with being for binomial coefm ficients, and δ is the rms width of the correlation. By employing the generalized Huygens–Fresnel principle, the cross-spectral density of the MGSM beam that passes through an ABCD optical system can be obtained as follows:
coefficients and the generalized phase radii of curvature, respectively. The intensity and the cross-spectral density of the MGSM beam at any point of the output plane are related by the formulas [25] I out ρ; z Wρ; ρ; z:
(3)
Consequently, the expression of the intensity of the MGSM beam through an ABCD optical system can be obtained by using Eqs. (2) and (3). Consider the propagation of the MGSM beam through an unapertured lens system (see Fig. 1). Let z be the separation between the input plane and the output plane, and f be the focal length. Then the ABCD matrix for the focusing system is
A B 1 C D 0
z 1
1 −1∕f
0 1 − z∕f 1 −1∕f
z : (4) 1
Upon substituting from Eq. (4) into Eq. (3), we can obtain the density distribution of the MGSM beams at the focal plane. In this paper, we choose P 100 mW, f 2 mm, λ 632.8 nm, σ 2 mm, and δ 2 mm. Figure 2 shows the transverse intensity distribution of the MGSM beams at the focal plane for different values of M. It can be found that intensity distribution depends on the value of M: the greater the value of M is, the smaller the magnitude of the intensity is and the flatter the profile becomes. It is well known that the particle whose radius is sufficiently smaller than the wavelength of light can be treated as a point dipole; in this case, the radiation force exerted on the dipole has two types: the scattering and gradient force, which come from the scattering by the dipole and the Lorenz force acting on the induced dipole, respectively. Assuming a homogeneous Rayleigh microsphere with refractive index np and radius a, the descriptions of the scattering force F sca and the gradient force F grad are given by [9,26] F⃗ sca ρ; z e⃗ z nm αI out ∕c;
(5)
2 M ρ1 ρ22 I0 X −1m−1 M 1 exp − Wρ1 ; ρ2 ; z C0 m1 mM 4Δσ 2 m Δ ikρ22 − ρ21 jρ2 − ρ1 j2 ; × exp − exp − 2R 2mΔδ2 (2) where Δ A2 B2 ∕4σ 4 k2 1 4σ 2 ∕mδ2 and R BΔ∕DΔ − A represent the generalized spreading
Fig. 1. Schematic of a MGSM beam through the focusing optical system. 20 June 2014 / Vol. 53, No. 18 / APPLIED OPTICS
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Fig. 2. (a) Intensity distribution for different values of M at the focal plane. (b) Two-dimensional intensity distribution for M 30.
F grad ⃗ρ; z 2πnm β∇I out ∕c;
(6)
where e⃗ z is a unity vector along the beam propagation, I out is the intensity of the focused beam at the output plane, and α 128π 5 a6 ∕3λ4 m20 − 1∕ m20 12 and β a3 m20 − 1∕m20 2 are the scattering cross section and the Polaris ability for the point dipole, respectively. m0 np ∕nm is the relative refraction index of the particles, with nm being the refractive index of the surrounding medium. According to Eqs. (2)–(6), we can obtain the radiation force at the output plane as follows: M ⃗e n αP X −1m−1 M 1 ρ2 ; exp − F⃗ sca ρ; z z m 2πσ 2 cC0 m1 m 2Δσ 2 m Δ (7) M −1m−1 e⃗ n βP X F⃗ grad;x ρ; z − x 2 m σ cC0 m1 m ρ2 ; × exp − 2Δσ 2
M m
x Δ σ
2 2
(8)
M X −1m−1 M 1 ⃗F grad;z ρ; z − e⃗ z nm βP σ 2 cC0 m1 m m Δ2 2B 4σ 2 ρ2 × 2AC 4 2 1 1− 2 mδ2 2σ 4σ k 2 ρ : (9) × exp − 2Δσ 2
Obviously, F⃗ sca is in the beam-propagation direction, while F⃗ grad is along the gradient of light intensity and acts as the restoring force directed toward the focusing center in the case of m0 > 1. In addition, it is evident that both F⃗ sca and F⃗ grad are affected by index M and the correlation width δ. Therefore, the distributions of the scattering force and the gradient force for different values of M and δ are plotted. The radius and refraction index of the particle and the refraction index of the ambient are chosen to be a 50 nm, np 1.59 (i.e., glass), and nm 1.33 (i.e., water), respectively. Figure 3 displays the distribution of the transverse gradient force F⃗ grad;x , the longitudinal gradient force F⃗ grad;z , and the scattering force F⃗ sca for different values of M at the focal plane. From Fig. 3 it can be found that a particle with m0 > 1 can be trapped by the MGSM beams at the focal point because there are stable equilibrium regions in Figs. 3(a) and 3(b) and the longitudinal gradient force is always larger than the forward-scattering force for different values of index M. The profile of scattering force contains a flat region being wider for larger values of M. In addition, with the value of summation index M increasing the transverse and longitudinal trapping ranges become larger; here, the trapping range is the distance from the equilibrium position to the position at which trapping starts to occur, but the trapping stability (i.e., the difference between the magnitudes of the force at the focal plane and at the boundary of the trapping region) reduces, which is similar to that of partially coherent flat-topped beams near the focal plane [22]. However, it is worthwhile to note that the
m m Fig. 3. Effect of index M on the radiation force of the MGSM beam at the focal plane: (a) F m grad;x ; (b) F grad;z ; and (c) F sca .
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m Fig. 4. Dependence of the radiation force of the MGSM beam at the focal plane on the coherence width δ, M 30: (a) F m grad;x ; (b) F grad;z ; and . (c) F m sca
trapping force and ranges of the MGSM beam at the focal plane are larger than those for the partially coherent flat-topped beams near the focal plane with the same trapping power and spot width. The affection of coherence on the gradient force and the scattering force of the MGSM beam with M 30 at the focal plane are indicated in Fig. 4. From Fig. 4 we can find that as the coherence of the MGSM increases, both the gradient force and the scattering force increase, but the stable trapping region becomes smaller and finally becomes a stable point. Although the trapping region of the MGSM beams with M > 1 decreases with the coherence increasing, it is always larger than that of the classic GSM beams with the same coherence. By comparison of the analyses of Figs. 3 and 4, it can be concluded that as the value of M increases, the trapping range becomes larger, but the stability decreases due to the reduction of radiation. On the contrary, as the coherence increases, the trapping range becomes smaller, but stability increases. Hence, it is important to choose suitable values of M and coherence to obtain a larger trapping range and higher stability. 3. Analysis of Trapping Stability
In this section, we will discuss the trapping stability in more detail. A necessary condition for stable trapping is that the backward axial gradient force must be great enough to overcome the forwardscattering force and the Brownian force at the position of the maximum intensity gradient [26]. According to Figs. 3(a) and 3(b), it can be found that the positions of the maximum intensity gradient (i.e., the effective focus plane) have a shift as the value of M increases. Through further calculation, we find the real focus plane is not at F 0; f but at F 0 0; ξ; here ξ z0
f 1
1 C0
PM
−1m−1 m1 m
; M 2 2 g f m 0
(10)
where g20 f ∕4σ 4 k2 1 4σ 2 ∕mδ2 ; obviously, the focal shift is related to the value of M. In the real focusing plane z ξ, the maximum transverse gradient force, the maximum longitudinal gradient
force, and the maximum scattering force are derived as follows: 0 1 g0 f σ F m @ q A ; ξ grad;x 1 g2 f 2 0
M M X
nm βP σ 2 cC0 m1 m
−1m−1 21 g20 f 2 3∕2 1 ; m e g0 f 3 σ
p 3g0 f F m grad;z 0; ξ 1 3 p M M −1m−1 3 31 g20 f 2 2 nm βP X 2 ; m σ cC0 m1 m 8g30 f 4 p 3g0 f F m 0; ξ 1 sca 3 M M n αP X −1m−1 31 g20 f 2 m2 : m 2πσ cC0 m1 m 4g20 f 2
(11)
(12)
(13)
The Brownian force due to the thermal fluctuation from the ambient (e.g., water) can be calculated by the formula [27] jF B j 12πκakB T1∕2 ;
(14)
where κ is the viscosity of the ambient, which is κ 7.977 × 10−4 Pas at T 300 K, a is the radius of the particle, and kB is the Boltzmann constant. x m In Fig. 5 the dependence of F m grad;x , F grad;z , and F sca on the summation index M, the coherence width δ, and the radius of particles a is calculated at the focal plane, and the maximum radiation forces are compared with jF B j. From Fig. 5(a) it can be seen that as the value of M increases the gradient force and the scattering force decrease and the gradient force is always larger than the scattering force and the Brownian force. So the MSGM beam with a larger value of M can be used to trap a particle, which offers some advantages over the trapping ranges of the classic GSM. But the values of M cannot be too large, 20 June 2014 / Vol. 53, No. 18 / APPLIED OPTICS
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m m Fig. 5. Comparison of F m grad;x , F grad;z , F sca , and F b (a) with different M, δ 2 mm; (b) with different δ, M 30; and (c) with different particle radius a, M 30.
because the radiation force reduces with M increasing, as discussed in Section 2. Figure 5(b) shows that with the coherence width increasing the gradient and the scattering force increase, which agrees with Fig. 4. It can be further found from Fig. 5(b) that when δ > 0.816 mm, the gradient force is greater than the scattering force and the Brownian force; in this case, the beam can trap a particle. So δ 0.816 mm is the low limit to stably trap particles for the case of M 30. The change of the forces versus radii of particles in the case of M 30 is plotted in Fig. 5(c). It can be seen from Fig. 5(c) that for the case in which a < 20 nm the Brownian force is greater than the gradient force and for a > 60 nm the scattering force is nearly equal to the gradient force. For both cases, the stability criterion is broken. Therefore, the particles with 20 nm < a < 60 nm can be stably trapped by the radiation forces produced by the MGSM beam. 4. Conclusion
We have studied the radiation forces on Rayleigh particles induced by focused MGSM beams. It is found that the radiation force produced by the MGSM beams can be used to trap particles with a highly refractive index at the focus plane. And the radiation forces are greatly affected by the summation index M and the coherence width of the beam. The transverse and longitudinal trapping range at the focal plane can be increased by increasing the value of M or decreasing the initial coherence of the MGSM beams. However, the trapping stability reduces due to the increase of the value of M and the decrease of coherence. So it is important to choose suitable values of M and the coherence of the MGSM beams to trap a particle. In addition, we explicitly discuss the trapping stability. Both the low limit of coherence and the range of the particle’s radius for stable trapping in the case of M 30 are determined. The results presented here are useful for some possible applications. This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11274273 and 11074219). 3980
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