Abstract

We consider a class of fields generated by passing an isotropic Gaussian Schell-model beam through a wavefront-folding interferometer. The output field has various intensity profiles for different phase differences, including the central peak and doughnut shapes. The radiation force on a Rayleigh dielectric particle produced by the highly focused fields is investigated. Numerical results demonstrate that the new fields can be used to trap high-index particles at the focus for the specular case and nearby the focus for the anti-specular case. It is further revealed that the position, the range of particle sizes and the low limit of correlation length for stable trapping could be modulated by adjusting the phase difference.

© 2016 Optical Society of America

1. Introduction

Since Ashikin et al. [1,2] first demonstrated that the radiation pressure of laser light can be used to trap small particles, many novel applications have been developed in various domains of science, especially in biophysical science to trap living cells and organelles [3,4], and in atomic physics to manipulate neutral atoms [5,6]. The radiation force exerted on a particle originates from the total exchange of momentum and energy between the incident photons and particles. Conventionally, the optical traps or optical tweezers are mainly constructed by highly focused laser beam to achieve a stable single-beam axial gradient force trap. Among the single-beam gradient force traps (SGFTs), the focused Gaussian beams are used the most, which trap the high-index particles (with the refractive index higher than the ambient) in the beam while expel the low-index particles (with the refractive index lower than the ambient) out of the beam. The trapping characteristics for other SGFTs [7–15] have also been explored, such as bottle beams [7], zero-order Bessel beams [8], self-focused laser beams [9], and so on.

The concept of specular cross-spectral density function (CSD) was introduced by Gori and associates in 1988 [16]. By definition, a cross-spectral density function is to be specular if it is even with respect to x1 (in one dimension), which implies that it is even also with respect to x2. Some related properties of such optical fields have been studied, among which the field is spatially completely coherent between the points located symmetrically with respect to the origin of the coordinate axis. Besides, the specularity property is conserved during the field propagation. It is also proposed that specular fields can be generated starting from any partially coherent field by use of Porro-prism interferometer. In spite of the peculiar properties, little work has been done since its introduction except the study of Ponomarenko and Agrawal [17]. Not long ago, it has been experimentally demonstrated that specular and antispecular light beams [18] are generated by passing a Gaussian Schell-model (GSM) beam through a wavefront-folding interferometer (WFI).

In this paper, we consider the field generated by passing an isotropic GSM beam through a WFI. The output field of the WFI has various intensity distributions for different phase difference, including the central peak shape and doughnut shape. Then, the field is focused onto a particle by a lens and the radiation force produced by the highly focused field is investigated. It is found that the new fields can be used to trap high-index particles and the position for stable trapping can be modulated by adjusting the phase difference. Finally, we analyze the condition for stable trapping.

2. Fields through a lens

The wavefront-folding interferometer [19] illustrated in Fig. 1 is mainly used to measure the spatial coherence properties of partially coherent fields. Key elements of the interferometer are the two perpendicularly oriented right-angle prisms PRx and PRy, which retroreflect the incident field in the x and y direction, respectively. The prisms are slightly tilted with respect to the optical axis to observe interference fringes. However, we assume the device perfectly aligned to generate a field with specular and antispecular cross-spectral density function.

 

Fig. 1 The wavefront-folding interferometer. S is the source, BS is a non-polarizing beam splitter, PRx and PRy are right-angle prisms.

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Suppose that a single partially coherent beam with the scalar optical field E0(x,y) is incident on the interferometer. This beam is split into two light beams by the beam splitter. Due to the folding effect of the right-angle prisms, the field E(x,y) at the output of the interferometer corresponds to the superposition of the two beams [18]

E(x,y)=12[E0(x,y)+E0(x,y)exp(iϕ)],
where ϕ is the phase difference between the two optical paths. As a result, the corresponding cross-spectral density functions W(x1,y1,x2,y2) and W0(x1,y1,x2,y2) are associated by the following relation
W(x1,y1,x2,y2)=12[W0(x1,y1,x2,y2)+W0(x1,y1,x2,y2)]+12[W0(x1,y1,x2,y2)exp(iϕ)+W0(x1,y1,x2,y2)exp(iϕ)].
As can be seen from Eq. (2), whatever the partially coherent field of the input is, the CSD of the field at the output is specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) if ϕ=2lπ and anti-specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) if ϕ=2lπ+π, where l is an integer.

We consider the output field of the interferometer as a secondary source, which propagates through a lens system as shown in Fig. 2. The transfer matrix of this system is

[ABCD]=[1z01][101/f1]=[1z/fz1/f1].
Here z is the axial distance from the lens to the output plane, Δz=zf is the distance between the focus and the output plane, and f is the focal length of the thin lens.

 

Fig. 2 Schematics of the output field of a WFI focused onto a particle. F is the geometrical focus.

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Assume that the field entering the interferometer is an isotropic GSM beam [20], whose cross-spectral density function is

W0(x1,y1,x2,y2)=I0exp(x12+x22w02)exp(y12+y22w02)exp[(x1x2)22σ02]exp[(y1y2)22σ02].
Here w0 denotes the rms width of the beam and σ0 represents the rms width of correlation. I0 is a constant given by I0=4P/(πnmε0cw02) with P being the beam power, nm the refractive index of the ambient, ε0 the dielectric constant and c the speed of light in vacuum. In our calculations, we choose P=100mW, nm=1.332, λ=632.8nm, f=2mm, w0=2mm and σ0=2mm, unless otherwise stated in the text.

On substituting from Eq. (4) into Eq. (2), we can obtain the CSD of the secondary source

W(x1,y1,x1,y2)=I0exp(x12+x22w02)exp(y12+y12w02){exp[(x1x2)22σ02]exp[(y1y1)2σ022]+cosϕexp[(x1x2)22σ02]exp[(y1y1)2σ022]}.
It is obvious that this field is indeed specular when ϕ=2lπ and anti-specular when ϕ=2lπ+π. The spectral density of this field is given by
S(x,y)=I0exp(2x2w02)exp(2y2w02)[1+cosϕexp(2x2σ02)exp(2y2σ02)].
Equations (5) and (6) show that the new secondary source is no longer a GSM source, but constituted by the superposition of two sources, only one of which is the GSM source. However, in the particular case that ϕ=2lπ+π/2, the new source is exactly the GSM source again, i.e., W(x1,y1,x2,y2)=W0(x1,y1,x2,y2). The corresponding intensity profiles are demonstrated in Fig. 3 for the three values of the parameter ϕ=0,π/2,π. It is seen from Fig. 3(a) that in the specular case (ϕ=0) the field has a sharper central peak than the GSM beam while in the antispecular case (ϕ=π) it has a dark notch in the center. As illustrated in Figs. 3(c) and 3(e), one can find that the width of the peak or notch could be adjusted by changing the correlation width of the incident field, which differs from the GSM beam shown in Fig. 3(d).

 

Fig. 3 Intensity distributions of the GSM, specular and anti-specular fields at the (a), (c)-(e) input plane and (b) focal plane. Figures (c)-(e) correspond to ϕ=0, ϕ=π/2 and ϕ=π for different values of σ0, respectively. (c)-(e) black solid curve, σ0/ω0=0.5; red short dashed curve, σ0/ω0=1; blue short dashed-dotted curve, σ0/ω0=2.

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By employing the generalized Huygens-Fresnel principle, the CSD of this field in any transverse plane z>0 is given by

W(x1,y1,x2,y2;z)=4Pπnmε0cw2(z)×exp(x12+x22w2(z))exp(y12+y22w2(z))exp[ik2R(z)(x12x22)]exp[ik2R(z)(y12y22)]×{exp[(x1x2)22σ2(z)]exp[(y1y2)22σ2(z)]+cosϕexp[(x1+x2)22σ2(z)]exp[(y1+y2)22σ2(z)]},
and the spectral density has the form
S(r)=4Pπnmε0cw2(z)exp(2x2w2(z))exp(2y2w2(z)){1+cosϕexp[2x2σ2(z)]exp[2y2σ2(z)]}.
Here k=2π/λ represents the wave number, λ being the wavelength, the position r=(x,y,z). The parameters
w(z)=w0(1z/f)2+z2/zR2,
σ(z)=σ0(1z/f)2+z2/zR2,
R(z)=z+[zzR2(1z/f)1f]1,
where zR=kw02β/2 and β=(1+w02/σ02)-1/2 denote the Rayleigh range and the state of coherence of the beam. It is obvious from Eq. (7) that the specularity and anti-specularity properties of the CSD are always conserved during the field propagation. The contrast of Figs. 3(a) and 3(b) indicates that the central peak distribution for the case ϕ=0,π/2 and the dark-centered distribution for the case ϕ=π always hold. The beams are shape-invariant during propagation because w(z) and σ(z) grow at the same rate due to the isotropy of the incident beam.

3. Radiation force produced by the highly focused fields

In the limit that the radius a of the particle is much smaller than the wavelength of the light, i.e., aλ/20, the Rayleigh approximation is applicable and the particle could be treated as a simple point dipole. Under this condition the radiation force acting on the particle can be described by two components: the scattering force and the gradient force which arise from momentum changes of the light due to the scattering by the particle and the Lorenz force acting on the particle, respectively. The scattering force is proportional to the intensity of the beam and is along the beam-propagation direction, which is given by

FScat(r)=nmcCprI(r).
Here Cpr represents the cross section of the radiation pressure of the particle. For a small dielectric particle in the Rayleigh regime where the particle scatters the light isotropically, Cpr equals the scattering cross section Cscat and is given by
Cpr=Cscat=83π(ka)4a2(m21m2+2)2,
where m=np/nm is the relative refractive index of the particle with np being the refractive index of the particle. I(r) is defined as a time-averaged version of the Poynting vector and is expressed as
I(r)=eznmε0c2S(r)=ezI(r),
where ez is the unit vector in the beam-propagation direction. The gradient force arises from the inhomogeneous field distribution, which is given by

FGrad(r)=2πnma3c(m21m2+2)I(r).

Substitute Eqs. (8) and (14) into Eq. (15), we can obtain the transverse component FGrad,x(r) and the longitudinal component FGrad,z(r) of the gradient force at the output plane

FGrad,x(r)=4Pnma3c(m21m2+2)(4x)w2(z)exp[2(x2+y2)w2(z)]×{1w2(z)+cosϕ[1w2(z)+1σ2(z)]exp[2(x2+y2)σ2(z)]},
FGrad,z(r)=4Pnma3c(m21m2+2)1w2(z)exp[2(x2+y2)w2(z)]{2w02α(z)w2(z)[2(x2+y2)w2(z)1]+2cosϕ[2(x2+y2)α(z)(σ02σ4(z)+w02w4(z))w02α(z)w2(z)]exp[2(x2+y2)σ2(z)]},
where α(z)=(1/zR2+1/f2)z1/f. Without loss of generality, in the following simulations we choose the radius of the particle a=25nm, and the refractive indices of two kinds of particles: np=1.592 (glass) or np=1 (air bubble).

In Fig. 4, we demonstrate the radiation forces produced by highly focused specular (ϕ=0) and GSM (ϕ=π/2) beams on high-index particles (np=1.592). For the transverse gradient force FGrad,x, the positive value means the force is along the +x direction while the negative means it is along the x direction. Similarly, the positive (negative) FGrad,z means that the longitudinal gradient force is along the +z (z) direction. It can be seen from Figs. 4(a)-4(c) that there is a stable equilibrium at the focus for the high-index particles illuminated by highly focused specular or GSM beam. The radiation forces produced by the specular beam are analogous to that produced by the GSM beam due to their similar intensity profiles and evolutions. However, the transverse gradient force, the longitudinal gradient force and the scattering force produced by the specular beam are larger than that produced by the GSM beam.

 

Fig. 4 Radiation forces produced by highly focused specular (ϕ=0) and GSM (ϕ=π/2) beams on high-index particles (np=1.592). Black solid curve, the specular beam; red dashed curve, the GSM beam.

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Figure 5 demonstrates the radiation forces produced by highly focused anti-specular beams (ϕ=π) on high-index particles (np=1.592) and low-index particles (np=1). For the particles with m<1, the longitudinal gradient force FGrad,z maintains zero because the doughnut shape of the anti-specular beam holds during the beam propagation. Consequently, the low-index particles accelerate toward the beam propagating direction which are drawn by the scattering force. However, for the particles with m>1, there are two stable equilibrium points at the positions x=±0.168μmin the focal plane. Therefore, one can use the highly focused anti-specular beam to trap or manipulate high-index particles nearby the focus.

 

Fig. 5 Radiation forces produced by highly focused anti-specular beams (ϕ=π) on high-index particles (m>1) and low-index particles (m<1). Black solid curve for the particles with np=1.592, red dashed curve for the particles with np=1.

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In our case, the analysis of Eq. (6) reveals that in the interval 0ϕ2π/3 the field has a central peak, but as ϕ increases in the interval 2π/3<ϕπ, a central dip gradually appears. Consequently, the radiation force exerted on the particles changes as we adjust the phase difference. In Fig. 6, we illustrate the radiation forces on the particles with m>1 for the three values of ϕ=2π/3,3π/4,5π/6. It can be seen from Figs. 6(a) and 6(d) that for the case ϕ=2π/3, the particles with m>1 are stably trapped at the focus. However, as ϕ increases, the trapping becomes unstable at first [see Figs. 6(b) and 6(e)] and then a new stable equilibrium is generated near the focus [see Figs. 6(c) and 6(f)]. For the case ϕ=5π/6, the particles could be trapped at the positions x=±0.15μmin the focal plane. Thus, it can be concluded that the position for stable trapping shifts from the focus to the position nearby the focus with increasing the phase difference ϕ.

 

Fig. 6 Radiation forces exerted on the particles with m>1 for different values of ϕ. (a), (d), ϕ=2π/3; (b), (e), ϕ=3π/4; (c), (f), ϕ=5π/6.

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4. Analysis of trapping stability

In Figs. 7-8, we plot the maximum transverse gradient force FGrad,x at two points A,B, the maximum longitudinal gradient force FGrad,z at other two points C,D, and the scattering force FScat at points C,D. In order to trap particles stably, two necessary conditions must be satisfied for a single-beam trap. One necessary criterion for stable trapping is that the backward longitudinal gradient force must be greatly larger than the forward scattering force, i.e., R=|FGrad,z|C,Dmax/|FScat|C,D>1. It is obvious from Figs. 7-8 that the magnitude of the scattering force is much smaller than the axial gradient force, which is also revealed in Figs. 4-6.

 

Fig. 7 Dependence of |FGrad,x|A,Bmax (black solid curve), |FGrad,z|C,Dmax (red dashed curve), |FScat|C,D (blue dotted-dashed curve), and |Fb| (green dotted curve) on the radius a for different ϕ. (a) ϕ=0; (b) ϕ=π/2; (c) ϕ=2π/3; (c) ϕ=π.

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Fig. 8 Dependence of |FGrad,x|A,Bmax (black solid curve), |FGrad,z|C,Dmax (red dashed curve), and |FScat|C,D (green dotted-dashed curve) on σ0 for different ϕ; the horizontal dotted lines denote |Fb|. (a) ϕ=0; (b) ϕ=π/2.

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On the other hand, the Brownian motion of the particles will strongly affect the stability of trapping when the particles are very small. The magnitude of the Brownian force is given by [21] |Fb|=12πηakBT, where η=7.977×104Pas is the viscosity for water at the room temperature T=300K and kB being the Boltzmann constant. To judge whether the trapping is stable or not, we compare the radiation forces with |Fb|. Figure 7 demonstrates the dependence of |FGrad,x|A,Bmax, |FGrad,z|C,Dmax, |FScat|C,D and |Fb| on the radius a of the particle for different ϕ. It can be found that the low limit of a to have a stable trapping for ϕ=0,π/2,2π/3,π is 3.8,5.0,6.6,8.1nm, respectively. On the other hand, the up limit of a is 31.6nm because the Rayleigh approximation is no more valid if the particle size is too large. Thus, we can conclude that the range of particle sizes to have a stable trapping can be modulated by changing the phase difference.

Figure 8 illustrates the dependence of |FGrad,x|A,Bmax, |FGrad,z|C,Dmax, and |FScat|C,D on σ0 for different ϕ. It can be seen that the magnitude of the radiation forces, including FGrad,x, FGrad,z and FScat, increases quickly as the correlation length σ0 increases, also indicated in [22]. The cross point 1 denotes the low limit of σ0 for stable trapping. It is found that the low limit of σ0/w0 to have a stable trapping for ϕ=0 and π/2 is 0.15 and 0.19, respectively. When σ0 is smaller than the cross point 1, the beam is unable to trap the particles.

The second condition for stable trapping can be given by using the Boltzmann factor [2]:

Rthermal=exp[Umax/(kBT)]1,
where the maximum depth of the potential well Umax is expressed as
Umax=πε0nm2a3|m21m2+2|Smax.
Equation (18) means that the potential well generated by the gradient force must be much larger than the kinetic energy of the particle. At the room temperature of T=300K for the particles with np=1.592, Rthermal1.079×109 for the specular beam, Rthermal3.284×105 for the GSM beam. It is obvious that in our case the Brownian motion can be overcome and the particles can be stably trapped by the highly focused fields.

5. Conclusions

In this paper we have considered the field generated by passing an isotropic GSM beam through a WFI. It was found that the output field of the WFI has a variety of CSD functions for different phase difference, including the specular, anti-specular and GSM forms. Then, we have investigated the radiation forces on Rayleigh dielectric particle produced by such highly focused fields. The numerical results demonstrate that the new fields can be used to trap high-index particles at the focus for the specular case and nearby the focus for the anti-specular case. It is further revealed that the position, the range of particle sizes and the low limit of correlation length for stable trapping could be modulated by adjusting the phase difference. Our results may have applications in particle trapping and manipulation.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253).

References and links

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

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

3. Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996). [CrossRef]   [PubMed]  

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

5. A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978). [CrossRef]  

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

7. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43(32), 6001–6006 (2004). [CrossRef]   [PubMed]  

8. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]  

9. J. Y. Ye, G. Chang, T. B. Norris, C. Tse, M. J. Zohdy, K. W. Hollman, M. O’Donnell, and J. R. Baker Jr., “Trapping cavitation bubbles with a self-focused laser beam,” Opt. Lett. 29(18), 2136–2138 (2004). [CrossRef]   [PubMed]  

10. 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]  

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

12. X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015). [CrossRef]  

13. M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014). [CrossRef]  

14. X. Liu and D. Zhao, “Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams,” Appl. Opt. 53(18), 3976–3981 (2014). [CrossRef]   [PubMed]  

15. X. Peng, C. Chen, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Optically trapping Rayleigh particles by using focused partially coherent multi-rotating elliptical Gaussian beams,” Chin. Opt. Lett. 14(1), 011405 (2016). [CrossRef]  

16. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988). [CrossRef]  

17. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004). [CrossRef]   [PubMed]  

18. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015). [CrossRef]   [PubMed]  

19. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988). [CrossRef]  

20. P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986). [CrossRef]  

21. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]  

22. 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]  

References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [Crossref]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
    [Crossref] [PubMed]
  3. Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
    [Crossref] [PubMed]
  4. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987).
    [Crossref] [PubMed]
  5. A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
    [Crossref]
  6. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
    [Crossref] [PubMed]
  7. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43(32), 6001–6006 (2004).
    [Crossref] [PubMed]
  8. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
    [Crossref]
  9. J. Y. Ye, G. Chang, T. B. Norris, C. Tse, M. J. Zohdy, K. W. Hollman, M. O’Donnell, and J. R. Baker., “Trapping cavitation bubbles with a self-focused laser beam,” Opt. Lett. 29(18), 2136–2138 (2004).
    [Crossref] [PubMed]
  10. 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]
  11. C. Zhao, L. Wang, and X. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
    [Crossref]
  12. X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
    [Crossref]
  13. M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014).
    [Crossref]
  14. X. Liu and D. Zhao, “Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams,” Appl. Opt. 53(18), 3976–3981 (2014).
    [Crossref] [PubMed]
  15. X. Peng, C. Chen, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Optically trapping Rayleigh particles by using focused partially coherent multi-rotating elliptical Gaussian beams,” Chin. Opt. Lett. 14(1), 011405 (2016).
    [Crossref]
  16. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
    [Crossref]
  17. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
    [Crossref] [PubMed]
  18. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
    [Crossref] [PubMed]
  19. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
    [Crossref]
  20. P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
    [Crossref]
  21. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
    [Crossref]
  22. 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]

2016 (1)

2015 (2)

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
[Crossref] [PubMed]

2014 (2)

M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014).
[Crossref]

X. Liu and D. Zhao, “Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams,” Appl. Opt. 53(18), 3976–3981 (2014).
[Crossref] [PubMed]

2012 (1)

2007 (2)

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

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

2004 (4)

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43(32), 6001–6006 (2004).
[Crossref] [PubMed]

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

J. Y. Ye, G. Chang, T. B. Norris, C. Tse, M. J. Zohdy, K. W. Hollman, M. O’Donnell, and J. R. Baker., “Trapping cavitation bubbles with a self-focused laser beam,” Opt. Lett. 29(18), 2136–2138 (2004).
[Crossref] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

1999 (1)

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

1996 (1)

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

1988 (2)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

1987 (1)

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

1986 (3)

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]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

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

1978 (1)

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[Crossref]

1970 (1)

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

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

Ashkin, A.

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

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

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

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[Crossref]

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

Baker, J. R.

Berns, M. W.

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

Bjorkholm, J. E.

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

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

Cable, A.

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

Chang, G.

Chen, B.

Chen, C.

Chen, C. H.

Chu, S.

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

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

Deng, D.

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

Dholakia, K.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Dziedzic, J. M.

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Hollman, K. W.

Hsieh, W. F.

Kawata, S.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Liu, X.

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

X. Liu and D. Zhao, “Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams,” Appl. Opt. 53(18), 3976–3981 (2014).
[Crossref] [PubMed]

Liu, Y.

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

Liu, Z.

Lu, X.

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

Lu, X. H.

Luo, M.

M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Melville, H.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Norris, T. B.

O’Donnell, M.

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

Partanen, H.

Peng, X.

Peng, Y.

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

Roskey, D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Sharmin, N.

Sonek, G. J.

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

Summers, M. D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Tai, P. T.

Tervo, J.

Tromberg, B. J.

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

Tse, C.

Turunen, J.

H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
[Crossref] [PubMed]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Wang, L.

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

Wang, L. G.

Wang, L. Q.

Wright, E. M.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Yang, X.

Ye, J. Y.

Zhao, C.

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

Zhao, C. L.

Zhao, D.

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014).
[Crossref]

X. Liu and D. Zhao, “Optical trapping Rayleigh particles by using focused multi-Gaussian Schell-model beams,” Appl. Opt. 53(18), 3976–3981 (2014).
[Crossref] [PubMed]

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]

Zhou, M.

Zhu, S. Y.

Zohdy, M. J.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[Crossref]

Biophys. J. (1)

Y. Liu, G. J. Sonek, M. W. Berns, and B. J. Tromberg, “Physiological monitoring of optically trapped cells: assessing the effects of confinement by 1064-nm laser tweezers using microfluorometry,” Biophys. J. 71(4), 2158–2167 (1996).
[Crossref] [PubMed]

Chin. Opt. Lett. (1)

Laser Phys. (1)

M. Luo and D. Zhao, “Simultaneous trapping of two types of particles by using a focused partially coherent cosine-Gaussian-correlated Schell-model beam,” Laser Phys. 24(8), 086001 (2014).
[Crossref]

Opt. Acta (Lond.) (1)

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta (Lond.) 33(3), 315–326 (1986).
[Crossref]

Opt. Commun. (3)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (1)

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

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[Crossref]

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

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

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Science (1)

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

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Figures (8)

Fig. 1
Fig. 1 The wavefront-folding interferometer. S is the source, BS is a non-polarizing beam splitter, PRx and PRy are right-angle prisms.
Fig. 2
Fig. 2 Schematics of the output field of a WFI focused onto a particle. F is the geometrical focus.
Fig. 3
Fig. 3 Intensity distributions of the GSM, specular and anti-specular fields at the (a), (c)-(e) input plane and (b) focal plane. Figures (c)-(e) correspond to ϕ=0 , ϕ=π/2 and ϕ=π for different values of σ 0 , respectively. (c)-(e) black solid curve, σ 0 / ω 0 =0.5 ; red short dashed curve, σ 0 / ω 0 =1 ; blue short dashed-dotted curve, σ 0 / ω 0 =2 .
Fig. 4
Fig. 4 Radiation forces produced by highly focused specular ( ϕ=0 ) and GSM ( ϕ=π/2 ) beams on high-index particles ( n p =1.592 ). Black solid curve, the specular beam; red dashed curve, the GSM beam.
Fig. 5
Fig. 5 Radiation forces produced by highly focused anti-specular beams ( ϕ=π ) on high-index particles ( m>1 ) and low-index particles ( m<1 ). Black solid curve for the particles with n p =1.592 , red dashed curve for the particles with n p =1 .
Fig. 6
Fig. 6 Radiation forces exerted on the particles with m>1 for different values of ϕ. (a), (d), ϕ= 2π /3 ; (b), (e), ϕ= 3π /4 ; (c), (f), ϕ= 5π /6 .
Fig. 7
Fig. 7 Dependence of | F Grad,x | A,B max (black solid curve), | F Grad,z | C,D max (red dashed curve), | F Scat | C,D (blue dotted-dashed curve), and | F b | (green dotted curve) on the radius a for different ϕ. (a) ϕ=0 ; (b) ϕ=π/2 ; (c) ϕ= 2π /3 ; (c) ϕ=π .
Fig. 8
Fig. 8 Dependence of | F Grad,x | A,B max (black solid curve), | F Grad,z | C,D max (red dashed curve), and | F Scat | C,D (green dotted-dashed curve) on σ 0 for different ϕ; the horizontal dotted lines denote | F b | . (a) ϕ=0 ; (b) ϕ=π/2 .

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E( x , y )= 1 2 [ E 0 ( x , y )+ E 0 ( x , y )exp( iϕ ) ],
W( x 1 , y 1 , x 2 , y 2 )= 1 2 [ W 0 ( x 1 , y 1 , x 2 , y 2 )+ W 0 ( x 1 , y 1 , x 2 , y 2 ) ] + 1 2 [ W 0 ( x 1 , y 1 , x 2 , y 2 )exp(iϕ)+ W 0 ( x 1 , y 1 , x 2 , y 2 )exp(iϕ) ].
[ A B C D ]=[ 1 z 0 1 ][ 1 0 1/f 1 ]=[ 1z/f z 1/f 1 ].
W 0 ( x 1 , y 1 , x 2 , y 2 )= I 0 exp( x 1 2 + x 2 2 w 0 2 )exp( y 1 2 + y 2 2 w 0 2 )exp[ ( x 1 x 2 ) 2 2 σ 0 2 ]exp[ ( y 1 y 2 ) 2 2 σ 0 2 ].
W( x 1 , y 1 , x 1 , y 2 )= I 0 exp( x 1 2 + x 2 2 w 0 2 )exp( y 1 2 + y 1 2 w 0 2 ){ exp[ ( x 1 x 2 ) 2 2 σ 0 2 ] exp[ ( y 1 y 1 ) 2 σ 0 2 2 ] +cosϕexp[ ( x 1 x 2 ) 2 2 σ 0 2 ]exp [ ( y 1 y 1 ) 2 σ 0 2 2 ] }.
S( x , y )= I 0 exp( 2 x 2 w 0 2 )exp( 2 y 2 w 0 2 )[ 1+cosϕexp( 2 x 2 σ 0 2 )exp( 2 y 2 σ 0 2 ) ].
W( x 1 , y 1 , x 2 , y 2 ;z)= 4P π n m ε 0 c w 2 ( z ) ×exp( x 1 2 + x 2 2 w 2 ( z ) )exp( y 1 2 + y 2 2 w 2 ( z ) )exp[ ik 2R(z) ( x 1 2 x 2 2 ) ]exp[ ik 2R(z) ( y 1 2 y 2 2 ) ] ×{ exp[ ( x 1 x 2 ) 2 2 σ 2 ( z ) ]exp[ ( y 1 y 2 ) 2 2 σ 2 ( z ) ] +cosϕexp[ ( x 1 + x 2 ) 2 2 σ 2 ( z ) ]exp[ ( y 1 + y 2 ) 2 2 σ 2 ( z ) ] },
S(r)= 4P π n m ε 0 c w 2 ( z ) exp( 2 x 2 w 2 ( z ) )exp( 2 y 2 w 2 ( z ) ){ 1+cosϕexp[ 2 x 2 σ 2 ( z ) ]exp[ 2 y 2 σ 2 ( z ) ] }.
w( z )= w 0 ( 1z/f ) 2 + z 2 / z R 2 ,
σ( z )= σ 0 ( 1z/f ) 2 + z 2 / z R 2 ,
R( z )=z+ [ z z R 2 ( 1z/f ) 1 f ] 1 ,
F Scat ( r )= n m c C pr I ( r ).
C pr = C scat = 8 3 π ( ka ) 4 a 2 ( m 2 1 m 2 +2 ) 2 ,
I ( r )= e z n m ε 0 c 2 S( r )= e z I(r),
F Grad ( r )= 2π n m a 3 c ( m 2 1 m 2 +2 )I( r ).
F Grad,x ( r )= 4P n m a 3 c ( m 2 1 m 2 +2 ) ( 4x ) w 2 ( z ) exp[ 2( x 2 + y 2 ) w 2 ( z ) ] ×{ 1 w 2 ( z ) +cosϕ[ 1 w 2 ( z ) + 1 σ 2 ( z ) ]exp[ 2( x 2 + y 2 ) σ 2 ( z ) ] },
F Grad,z ( r )= 4P n m a 3 c ( m 2 1 m 2 +2 ) 1 w 2 ( z ) exp[ 2( x 2 + y 2 ) w 2 ( z ) ]{ 2 w 0 2 α( z ) w 2 ( z ) [ 2( x 2 + y 2 ) w 2 ( z ) 1 ] +2cosϕ[ 2( x 2 + y 2 )α( z )( σ 0 2 σ 4 ( z ) + w 0 2 w 4 ( z ) ) w 0 2 α( z ) w 2 ( z ) ] exp[ 2( x 2 + y 2 ) σ 2 ( z ) ] },
R thermal =exp[ U max / ( k B T ) ]1,
U max =π ε 0 n m 2 a 3 | m 2 1 m 2 +2 | S max .

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