Abstract

In this paper, we have considered the optical forces acting on submicron particles induced by arbitrary-order full Poincaré (FP) beams. Different from the traditional scalar beams, the optical forces of the FP beams include three contributions: the scattering, gradient, and curl forces. The last contribution is due to both the vectorial properties of the FP beams’ polarization and the rotating phase structure of the FP beams. We analytically derive all components of the optical forces of the FP beams acting on Rayleigh particles. The numerical results show that the optical curl force is very significant to the absorbing Rayleigh particles, and it has the same order with the scattering force. The total vortex force fields and their trapping effects of different order FP beams on the absorbing dielectric and metallic Rayleigh particles are discussed in detail. Our results may stimulate further investigations on the trapping effect of various vector-vortex beams on submicron or nanometer sized objects.

© 2012 OSA

1. Introduction

Optical trapping and manipulation have been widely used in various fields of application in physics, chemistry, and biology for the non-contact advantage, since Ashkin first demonstrated the optical trapping of particles using the radiation force produced by focused Gaussian beams [1], especially starting from the demonstration of single-beam optical trapping (also named as optical tweezers) in 1986 [2]. Traditionally, optical forces on small particles can be theoretically divided into two contributions: the gradient (or dipole) force and the scattering (or radiation pressure) force. The former is proportional to the gradient of the light intensity, while the latter is proportional to the Poynting vector. For many scalar (or linear-polarized) light fields without vortex phase structures, the usual gradient and scattering light forces provide the well-known descriptions. For the complex-vector or complex-vortex beams, such as radially polarized beams [35], cylindrical vector beams [6, 7], and vortex beams [8], currently most theoretical investigations are based on the ray optics [3] and the T-matrix method [4, 5, 8].

In the past ten years, one has known the exact expression of the timed-averaged total force of an arbitrary time-harmonic electromagnetic field on a small particle [9]. In this exact expression, except for the traditional gradient and scattering forces, there is an additional term proportional to Im[(E*)E], which is zero only when E is real (this is the case for a propagating or evanescent plane wave) [10]. One has found that this additional term plays an important role in determining the resultant optical forces on nanometer-sized absorbing particles [11]. In 2009, one clear understood the physical meaning of the additional term that is a so-called “curl force” associated to the nonuniform distribution of the spin density of the light field [12]. As pointed out in Ref [12], the curl force is actually associated to both the orbital and spin angular momentums. Later, one experimentally demonstrated the optical orbital angular momentum from the curl of polarization [13]. Very recently, the relevance of optical curl forces in highly focused Gaussian beams have been discussed for both linear [14] and radial [15, 16] polarized beams. The spin curl forces induced by Bessel beams have also been discussed in the content of optical tractor beams [17, 18]. In these previous examples, the curl forces along the propagation axis are impact to the total force field. In particular, in some Bessel beams the Poynting vector on the axis points against the beam propagation [19] while the total force, due to the curl contribution, push small particles along the propagation axis (see the supplementary information in Ref [17].). In fact, for light force effects of complex-vector and vector-vortex light fields, the method presented in Ref [12]. is very useful in analyzing the trapping and manipulation on the submicron or nanometer sized particles.

In this paper, we investigate the optical trapping effects of arbitrary-order full Poincaré (FP) beams acting on submicron or nanometer sized particles. The FP beams are very recently proposed by Beckley et. al. [20], and their main property is that the state of light polarization can span over the entire full Poincaré sphere with changing the azimuthal angle. Soon later, Han et. al. [21] further proposed and experimentally demonstrated the second-order FP beams with high-quality flattop profiles. To our best knowledge, the trapping effect of these FP beams has not yet been investigated. Here we focus on their light force effects on small particles, and derive all components of the optical forces of the FP beams acting on Rayleigh particles, and discuss the total vortex force fields and the corresponding trapping effects.

2. Formula of any order FP beams

According to the previous proposals [20, 21], we know that the first-order FP beam is the combination of the fundamental Gaussian and first-order Laguerre Gaussian (LG) beams of two orthogonal polarizations, and the second-order FP beam is the combination of the fundamental Gaussian and second-order LG beams of two orthogonal polarizations. Naturally, we may obtain the arbitrary-order FP beam consisting of the fundamental Gaussian and arbitrary-order LG beams of two orthogonal polarizations. Thus, the expression for any order FP beam can be written as

Em(r,φ,z)=cosγLG00(r,φ,z)x^+sinγLG0m(r,φ,z)y^
in the cylinder coordinates, where LG00(r,φ,z) and LG0m(r,φ,z) are the electric field distributions of the fundamental Gaussian and the m-th order LG beams, respectively, in the x and y directions. Here γ is a controllable factor which regulates the intensity profile of the beam [20], and r and φ are the radial coordinate and the azimuthal angle, respectively. The expressions of LG00(r,φ,z) and LG0m(r,φ,z) are, respectively, given by
LG00(r,φ,z)=2π1w(z)exp[ikr22Q(z)]exp[i(kzϕ)],
LG0m(r,φ,z)=2πm!1w(z)(2rw(z))mexp[ikr22Q(z)]×exp[i(kzϕ)]exp[im(φϕ)],
where w(z)=w0(1+z2/ZR2)1/2 is the beam’s radius at position z, Q(z)=ziZR, ϕ=tan1(z/ZR) is the Gouy phase, w0 is the beam waist of the Gaussian beam, and m=1,2,3, denotes the order number of the LG beam. Obviously, we can readily rewrite Eq. (1) into the following form:
Em(r,φ,z)=2π1w(z)exp[ikr22Q(z)]exp[i(kzϕ)]cosγ×{x^+y^tanγ1m!(2rw(z))mexp[im(φϕ)]}.
Equation (4) can also be written in the form of Jones vector, yielding
Em(r,φ,z)=E0(1Ωexp[iδ]),
where E0=2π1w(z)exp[ikr22Q(z)]exp[i(kzϕ)]cosγ, which denotes the common factor in the x and y components, Ω=tanγ1m!(2rw(z))m is the ratio between the amplitudes in the y and x components, and δ=m(φϕ) is the phase difference between the two components. From Eq. (5), it is clear that close to the optical axis (z axis), the polarization is mainly along the x direction since the value of Ω is close to zero due to small r, however the polarization of light gradually changes into y polarization for the large value of r (i.e., away from the axis), see Fig. 1(b) , when each ellipse, denoting the state of light polarization, becomes a vertical line. The phase delay is m times difference between the azimuthal angle and the Gouy phase shift, so that at any plane z the phase delay between two components will change from 0 to 2mπ. Thus, the state of polarization for such a kind of combined beams will span mtimes on the entire surface of Poincaré sphere. In fact, when m=1, Eq. (4) or (5) reduces to the case of the first-order FP beam, i.e., Eq. (6) in Ref [20]; when m=2, Eq. (4) or (5) reduces to the case of the second-order FP beam [21]. Therefore, we call such a kind of combined beams as the m-order FP beams. From the previous literatures, we know that the polarization of the FP beam rotates as it propagates. It is expected that such a rotation of polarization may lead to additional optical force acting on small objects, such as Rayleigh particles.

 

Fig. 1 (a) Schematic for a FP beam passing through the beam waist region (or the focusing region). (b) Distributions of intensities and polarizations of FP beams with different order m at the beam waist’s plane (z = 0). The states of light polarization are denoted by the different ellipses.

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3. Optical force of the m-th FP beams acting on Rayleigh particles

Usually the optical force, acting on a Rayleigh particle, is split into two parts: the gradient and scattering forces. However, since 2000 [9], one has gradually realized that there is an additional term contributed from the non-uniform distribution of polarization of light [12]. Now it is well known that for continuous-wave harmonic vector light fields, the timed-averaged optical force on a Rayleigh particle is given by [11, 12]

F=14Re(α)|E|2+k2Im(α)Re(E*×B)+12Im(α)Im[(E*)E],
where α is the complex polarizability, which can be given by [22]
α=α01iα0k3/(6πε0)
for a spherical particle with its radius amuch smaller than the light wavelength λ, and
α0=4πε0a3(εp/εm)1(εp/εm)+2
is the well known Clausius-Mossotti relation. In the above equations, k=ω/c, ε0 is the vacuum’s permittivity, c is the speed of light in vacuum, and εp and εm are the relative permittivities of the particle and the surrounding medium, respectively. In Eq. (6), the first term is known as the traditional gradient force, and the second term is the dissipative radiation-pressure force, and it can be simplified into the scattering force for non-absorption systems. The third term is recently explained as the optical curl force associated to the non-uniform distribution of the spin density of the light field [12]. For our cases of the FP beams, the optical curl force is associated with the distributions of both the azimuthal phase and rotating polarization.

In our case, we consider a paraxial FP beam radiating on a Rayleigh spherical particle, as shown in Fig. 1(a). For a FP beam, its polarization changes as the azimuthal angle φ varies and gradually rotates as the beam propagates along the zaxis. Substituting Eq. (4) into Eq. (6), after tedious calculations, we can analytically obtain the components of optical force due to different contributions as follows:

Fg,x=Re[α]I02Pε0cxw2(z){1+Ω2[1mw2(z)2(x2+y2)]},
Fg,y=Re[α]I02Pε0cyw2(z){1+Ω2[1mw2(z)2(x2+y2)]},
Fg,z=Re[α]I02P2ε0cz(z2+ZR2)[2(x2+y2)w2(z)(Ω2+1)Ω2(m+1)1],
Fs,x=Im[α]P2ε0cεmI0Ω{krz2+ZR2[zΩcosφzsinφcos(m(ϕφ))+ZRsinφsin(m(ϕφ))]mrΩsinφ},
Fs,y=Im[α]P2ε0cεmI0{krz2+ZR2[zsinφ+zΩcosφcos(m(ϕφ))ZRΩcosφsin(m(ϕφ))]+mrΩsin[m(ϕφ)+φ]},
Fs,z=Im[α]P2ε0cεmI0{k(1+Ω2)[1r2(z2ZR2)2(z2+ZR2)2]ZRz2+ZR2(mΩ2+Ω2+1)},
Fc,x=Im[α]PI0kr2ε0c(z2+ZR2){zcosφ+Ωsinφ(zcos[m(ϕφ)]ZRsin[m(ϕφ)])},
Fc,y=Im[α]PI0Ω2ε0c{krz(z2+ZR2)(Ωsinφ+cosφcos[m(ϕφ)])+krZR(z2+ZR2)cosφsin[m(ϕφ)]mrsin[m(ϕφ)+φ]+mrΩcosφ},
where I0=|E0|2, φ=tan1(y/x), and P denotes the input laser power. In the above, the subscripts “g” and “s” denote the components of the gradient and scattering forces, respectively; the subscript “c” denotes the contribution of the optical curl force due to the third term in Eq. (6), i.e., due to the non-uniform distribution of light polarization, which only affects the transverse optical force for the paraxial vector beams; and the subscripts “x”, “y” and “z” on the forces denote the directions of the force’s components. Therefore, using Eqs. (9a)-(9h), we can easily obtain the components of the resultant optical force along the x, y and z directions, as follows:
Fx=Fg,x+Fs,x+Fc,x,
Fy=Fg,y+Fs,y+Fc,y,
Fz=Fg,z+Fs,z.
Thus we can analyze the trapping effects of any order FP beams acting on the Rayleigh particles. It should be pointed out that, in our Eqs. (2)-(4) we have used the paraxial condition which will be violated for a tight-focusing condition when the beam spot size is smaller than the wavelength (i.e., 2w0<λ). In Ref [23], one use an important dimensional parameter, s=1/(kw0)=λ/(2πw0), to describe the accuracy of the paraxial approximation. When γ=π/3, the paraxial condition is very well. In our case, we will takew0=2λ, so s=0.08. Therefore the parameters used in our calculations still give out the sufficient accuracy with small uncertainty (close to the weak-focusing condition).

In the following section, we will show how the order number m affects on the trapping efficiency and how important the optical curl forces (Fc,x and Fc,y) are when the Rayleigh particles are the absorbing media.

4. Numerical results and discussions

Without loss of generality, in all our simulations we choose the following parameters: λ=500nm, m=2µm, P=1W, εm=1 (for air). Other parameters will be given out in the text.

First, let us discuss the magnitude and direction distributions of different force components due to the different physical contributions under two cases: non-absorbing and absorbing dielectric particles in Fig. 2 and Fig. 3 , respectively. For a non-absorbing particle, it is clear that the magnitude of the transverse gradient force in Fig. 2(a) is greatly larger than those of the curl and scattering forces, therefore the total transverse force is mainly manifesting the transverse restoring force, see Fig. 2(d). It is also shown that the magnitude of the curl force is the same order with that of the scattering force, see Figs. 2(b)-2(c) and Figs. 3(b)-3(c). Actually, in our case the optical curl force is also a kind of scattering force originated from the curl of the light angular momentum including the orbital and spin angular momentums, as pointed out in Ref [12]. For a strong-absorbing particle, the magnitudes of both the curl and scattering forces can be comparable to or larger than that of the gradient force by comparing Figs. 3(b)-3(c) with Fig. 3(a). Therefore the total transverse optical force may demonstrate the swirling force field. Comparing Fig. 2 with Fig. 3, it is shown that, the imaginary part of the particle’s permittivity has more impact on the magnitudes of the curl and scattering forces than that of the gradient force, which results in the vortex force field in Fig. 3(d) although the relative vector distributions of the curl and scattering forces are the same in both Figs. 2(b-c) and Figs. 3(b-c) except for their magnitudes.

 

Fig. 2 The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for the FP beam with m=1 and γ=π/4. Other parameters are a=50nm and εp=2.56 (no absorption).

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Fig. 3 The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for the FP beam with m=1 and γ=π/4. Other parameters are a=50nm and εm=2.56+i2.56 (with absorption).

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For better to understand the distinct property of the FP beams, Fig. 4 and Fig. 5 , respectively, plot the forces of a linear-polarized Gaussian beam and a linear-polarized LG01 beam for z=0 and γ=π/2 in Eq. (1). The other parameters are the same as in Fig. 3. From Fig. 4, we can see that both the transverse curl and scattering forces are zeros and the total transverse force is only contributed from the gradient force for a linear-polarized paraxial Gaussian beam, and these are very different from that of the FP beam (Fig. 2 and Fig. 3). The result obtained in Fig. 4 is actually the same as that in Ref [14] under the paraxial approximation. For a linear-polarized LG01 beam, see Fig. 5, the gradient force in the center region is pointing to outside due to the doughnut shape, and the curl and scattering forces are, respectively, non-zeros in the yand x directions with the same magnitudes. In Fig. 5, we still can see that the transverse curl and scattering forces provide the rotation effect. However, by comparing the results of the first-order FP beam with a linear polarized LG01 beam, see Figs. 3(a)-3(d) with Figs. 5(a)-5(d), it is found that, the curl and scattering forces of the FP beam are more complex than those of the linear-polarized beam, and the resultant force distribution of the FP beam at the plane of z=0 has a more stable rotation structure than that of the LG01 beam.

 

Fig. 4 The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for a linear-polarized Gaussian beam. Other parameters are the same as in Fig. 3.

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Fig. 5 The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for a linear-polarized LG01 beam. Other parameters are the same as in Fig. 3.

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In order to show how the light polarization affects on the resultant optical force, we plot Figs. 6 and 7 , as examples, to demonstrate the rotation of the optical force resulted from the rotation of the non-uniform polarization states as the beam propagates. In Fig. 6, for the first-order FP beam (m=1), as the beam propagates from z=0.5ZR to z=0.5ZR, the force field of the sum of the curl and scattering force gradually rotates and changes as the state of non-uniform polarization gradually rotates on the entire Poincaré sphere with changing the position z, see Figs. 6(d)-6(f). Combining with the centripetal distribution of the gradient force, the total transverse force demonstrates a bound rotating vortex field, see Figs. 6(g)-6(i), which provides a transverse stable but rotating trapping effect on the Rayleigh particle. There are similar result in the case of m=3, see Fig. 7, however the region of the vortex force field becomes larger and more complex in this case.

 

Fig. 6 Distributions of (a-c) the beam intensities and the states of polarization (different ellipses), and the magnitude and direction distributions of (d-f) the sum of the curl and scattering forces, and (g-i) the total transverse force at different planes: (a), (d), and (g) for z=0.5ZR, (b), (e), and (h) for z=0, and (c), (f), and (i) for z=0.5ZR. Other parameters are the same as in Fig. 3.

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Fig. 7 Distributions of (a-c) the beam intensities and the states of polarization (different ellipses), and the magnitude and direction distributions of (d-f) the sum of the curl and scattering forces, and (g-i) the total transverse force at the different planes: (a), (d), and (g) for z=0.5ZR, (b), (e), and (h) for z=0, and (c), (f), and (i) for z=0.5ZR, with m=3 and γ0.356π. Other parameters are the same as in Fig. 3.

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Figure 8 shows the effect of the particle’s absorption on the longitudinal optical force. As we know that for the large Rayleigh particles, the longitudinal scattering force is dominated and larger than the longitudinal gradient force [2, 23, 24]. Thus we consider a small Rayleigh particle with a=10nm. From Fig. 8, we clear see that that a slight absorption will destroy the stability of the longitudinal trapping effect. For example, when Im[εp]=0.01, the depth of the negative Fz becomes very weak, compared with the non-absorbing case; when Im[εp]=0.05, the value of Fz completely becomes positive, which provides the pushing force along the z direction. In fact, from Eq. (10c), it is seen that Fz is only related with the scattering and axial gradient forces, and it is not affected by the non-uniform polarization under the paraxial approximation. Therefore, in the following discussion, we do not pay attention on the longitudinal force Fz.

 

Fig. 8 Distributions of the z component of the total force acting on the small particle (a=10nm) under different absorptions (a) εm=2.56, (b) εm=2.56+0.01i, and (c) εm=2.56+0.05i, for the FP beam with m=1 and γ=0.25π.

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Now let us turn to discuss the effect of the particle’s absorption on the transverse optical force in detail. Since the transverse gradient force is always a centripetal force as a restoring force and it is also weakly affected by the particle’s absorption, also see Figs. 2 and 3, we use the maximal value of the transverse gradient force as a reference to compare the maximal curl and scattering forces with the corresponding gradient force. Figure 9 plots the dependences of the ratios of Fcmax/Fgmax, Fsmax/Fgmax, and |Fc+Fs|max/Fgmax on the imaginary part of the particle’s permittivity. Here all these forces are in the xy plane at z=0. It is found that with the increasing of Im[εp], the maximal optical curl and scattering forces increase and trend to be nearly the same order with the maximal gradient force. For the larger-sized particles, the ratios of the optical curl and scattering forces to the gradient force are much larger than the cases for the smaller-sized particles. This indicates that it is more efficiency to rotate the larger-sized particles with the large value of Im[εp].

 

Fig. 9 Dependence of (a) Fcmax/Fgmax, (b) Fsmax/Fgmax, and (c) |Fc+Fs|max/Fgmax on the absorption factor Im[εp] for the cases of the different particle’s sizes, under the radiation of the FP beam with m=1 and γ=0.25π.

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Figure 10 shows the effect of the order number m on the ratios of Fcmax/Fgmax, Fsmax/Fgmax, and |Fc+Fs|max/Fgmax, under the fixed parameters of Im[εp] and the particle’s size. It is clear seen that these ratios are increasing with the increase of m, which indicate that the higher-order FP beam may have powerful ability to rotate the particle in the transverse plane under the same input power. Here we would like to point out that the model we used [see Eqs. (1)-(3)] can be optimized by replacing a single w0with two independent adjustable parametersw0x and w0y in Eqs. (2) and (3). Then the result of the trapping effect can be further improved.

 

Fig. 10 Dependence of (a) Fcmax/Fgmax, (b) Fsmax/Fgmax, and (c) (Fc+Fs)max/Fgmax on the order value m with a=50nm and εp=2.56+2.56i. Other parameters are γ=0.25π for m=1, γ=π/3 for m=2, γ=0.356π for Re[εp], λ= for m=4, and γ=0.373π for m=5.

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In the above calculations, we have used the large value of Im[εp]. Actually, for the nano-scaled metallic particles, such large values of Im[εp] are possible. As examples, we have considered the distributions of total transverse optical forces of the second-order FP beam acting on the Al and Au nano-scaled particles with the size of a=50nm, see Fig. 11 . For an Al particle, its permittivity is εp=36.373+i9.412 at the wavelength λ=500nm [25]; while for an Au particle, its permittivity is εp=2.81+i3.18 at the same wavelength [10]. From Fig. 11, we can see that, for the metallic nano particles with the larger negative value of Re[εp] and the larger absorption, the vortex optical force field is very stable and it can rotate the particle along the dashed circle, see Figs. 11(a)-11(c). Of course, for the different order FP beam, the detailed distribution of the vortex force field may be different but the qualitative conclusion is similar. However, for the Au nano particle, at the operating wavelength, with the smaller value of Re[εp], the transverse stable rotating region [see the dashed circle in Figs. 11(d)-11(f)] will become larger and larger as the beam propagates along the z direction. From Fig. 11(d) to Fig. 11(f), we can see that, at z=0.5ZR, the resultant optical force field will drive the particle rotating into the center; while at z=0.5ZR, the particle may be spirally pushed out from the center to the outside region. The numerical results tell us that under certain parameters, the particle may be rotated out from the center. The condition for obtain the stable vortex force field should be further investigated.

 

Fig. 11 Distributions of the total transverse optical forces (in the unit of pN) of the second-order FP beam acting on (a-c) the Al nano-particle and (d-f) the Au nano-particle at different positions: (a) and (d) for z=0.5ZR, (b) and (e) for z=0, and (c) and (f) for z=0.5ZR. Other parameters are m=2 and γ=π/3.

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5. Conclusion

We have investigated the optical radiation force of the arbitrary order FP beam acting on the non-absorbing or absorbing dielectric Rayleigh particles and metallic Rayleigh particles. Due to the vector and rotation properties of the FP beams’ polarizations, it is found that the optical curl force due to the present of the non-uniform polarization becomes significant to the absorbing Rayleigh particles, and it has the same order with the scattering force. The result shows that for the strong-absorbing particles, both the transverse curl and scattering force of the FP beams can be comparable to or larger than the transverse gradient force, and the total transverse optical force will present the rotating-vortex force field. For the very-weak or non-absorbing Rayleigh particles, the longitudinal force may provide the stable trapping effect, mainly due to the longitudinal gradient force. As the order number m of the FP beams increases, the rotating effect becomes more and more powerful in the transverse plane under the same input laser power. For the metallic Rayleigh particles with large negative Re[εp] and positive Im[εp], the actual trapping effect is stably trapped and rotated in the transverse plane. Our results are useful for analyzing the trapping and rotating effects of different order FP beams acting on the absorbing Rayleigh particles.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61078021), and the National Basic Research Program of China (Grant No. 2012CB921602).

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24. L.-G. Wang and H.-S. Chai, “Revisit on dynamic radiation forces induced by pulsed Gaussian beams,” Opt. Express 19(15), 14389–14402 (2011). [CrossRef]   [PubMed]  

25. A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93(1), 139–143 (2008). [CrossRef]  

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. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett.32(13), 1839–1841 (2007).
    [CrossRef] [PubMed]
  4. S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A76(5), 053836 (2007).
    [CrossRef]
  5. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett.33(2), 122–124 (2008).
    [CrossRef] [PubMed]
  6. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009).
    [CrossRef]
  7. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18(10), 10828–10833 (2010).
    [CrossRef] [PubMed]
  8. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
    [CrossRef] [PubMed]
  9. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett.25(15), 1065–1067 (2000).
    [CrossRef] [PubMed]
  10. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A20(7), 1201–1209 (2003).
    [CrossRef] [PubMed]
  11. V. Wong and M. A. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B73(7), 075416 (2006).
    [CrossRef]
  12. S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
    [CrossRef] [PubMed]
  13. X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
    [CrossRef] [PubMed]
  14. I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun.284(10-11), 2430–2436 (2011).
    [CrossRef]
  15. I. Iglesias and J. J. Sáenz, “Light spin forces in optical traps: comment on “Trapping metallic Rayleigh particles with radial polarization”,” Opt. Express20(3), 2832–2834 (2012).
    [CrossRef] [PubMed]
  16. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization: reply to comment,” Opt. Express20(6), 6058–6059 (2012).
    [CrossRef] [PubMed]
  17. J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
    [CrossRef]
  18. J. J. Sáenz, “Optical forces: Laser tractor beams,” Nat. Photonics5(9), 514–515 (2011).
    [CrossRef]
  19. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A24(9), 2844–2849 (2007).
    [CrossRef] [PubMed]
  20. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18(10), 10777–10785 (2010).
    [CrossRef] [PubMed]
  21. W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett.36(9), 1605–1607 (2011).
    [CrossRef] [PubMed]
  22. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
    [CrossRef]
  23. 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]
  24. L.-G. Wang and H.-S. Chai, “Revisit on dynamic radiation forces induced by pulsed Gaussian beams,” Opt. Express19(15), 14389–14402 (2011).
    [CrossRef] [PubMed]
  25. A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B93(1), 139–143 (2008).
    [CrossRef]

2012

2011

W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett.36(9), 1605–1607 (2011).
[CrossRef] [PubMed]

L.-G. Wang and H.-S. Chai, “Revisit on dynamic radiation forces induced by pulsed Gaussian beams,” Opt. Express19(15), 14389–14402 (2011).
[CrossRef] [PubMed]

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun.284(10-11), 2430–2436 (2011).
[CrossRef]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

J. J. Sáenz, “Optical forces: Laser tractor beams,” Nat. Photonics5(9), 514–515 (2011).
[CrossRef]

2010

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
[CrossRef] [PubMed]

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18(10), 10777–10785 (2010).
[CrossRef] [PubMed]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18(10), 10828–10833 (2010).
[CrossRef] [PubMed]

2009

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009).
[CrossRef]

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

2008

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B93(1), 139–143 (2008).
[CrossRef]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett.33(2), 122–124 (2008).
[CrossRef] [PubMed]

2007

2006

V. Wong and M. A. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B73(7), 075416 (2006).
[CrossRef]

2003

2000

1996

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]

1988

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

1986

1970

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

Albaladejo, S.

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

Alonso, M. A.

Arias-González, J. R.

Asakura, T.

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]

Ashkin, A.

Beckley, A. M.

Bjorkholm, J. E.

Brown, T. G.

Chai, H.-S.

Chan, C. T.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
[CrossRef] [PubMed]

Chaumet, P. C.

Chen, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Cheng, W.

Chu, S.

Ding, J.

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Draine, B. T.

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

Dziedzic, J. M.

Guo, C.-S.

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Han, W.

Harada, Y.

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]

Heckenberg, N. R.

Iglesias, I.

I. Iglesias and J. J. Sáenz, “Light spin forces in optical traps: comment on “Trapping metallic Rayleigh particles with radial polarization”,” Opt. Express20(3), 2832–2834 (2012).
[CrossRef] [PubMed]

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun.284(10-11), 2430–2436 (2011).
[CrossRef]

Kawauchi, H.

Kozawa, Y.

Laroche, M.

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

Laroche, T.

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B93(1), 139–143 (2008).
[CrossRef]

Li, Y.

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Lin, Z.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
[CrossRef] [PubMed]

Marqués, M. I.

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

Ng, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
[CrossRef] [PubMed]

Nieminen, T. A.

Nieto-Vesperinas, M.

Novitsky, A. V.

Novitsky, D. V.

Ratner, M. A.

V. Wong and M. A. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B73(7), 075416 (2006).
[CrossRef]

Rubinsztein-Dunlop, H.

Sáenz, J. J.

I. Iglesias and J. J. Sáenz, “Light spin forces in optical traps: comment on “Trapping metallic Rayleigh particles with radial polarization”,” Opt. Express20(3), 2832–2834 (2012).
[CrossRef] [PubMed]

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun.284(10-11), 2430–2436 (2011).
[CrossRef]

J. J. Sáenz, “Optical forces: Laser tractor beams,” Nat. Photonics5(9), 514–515 (2011).
[CrossRef]

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

Sato, S.

Vial, A.

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B93(1), 139–143 (2008).
[CrossRef]

Wang, H.-T.

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Wang, L.-G.

Wang, X.-L.

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

Wong, V.

V. Wong and M. A. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B73(7), 075416 (2006).
[CrossRef]

Yan, S.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A76(5), 053836 (2007).
[CrossRef]

Yao, B.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A76(5), 053836 (2007).
[CrossRef]

Yonezawa, K.

Zhan, Q.

Adv. Opt. Photon.

Appl. Phys. B

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B93(1), 139–143 (2008).
[CrossRef]

Astrophys. J.

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

J. Opt. Soc. Am. A

Nat. Photonics

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5(9), 531–534 (2011).
[CrossRef]

J. J. Sáenz, “Optical forces: Laser tractor beams,” Nat. Photonics5(9), 514–515 (2011).
[CrossRef]

Opt. Commun.

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]

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun.284(10-11), 2430–2436 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A76(5), 053836 (2007).
[CrossRef]

Phys. Rev. B

V. Wong and M. A. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B73(7), 075416 (2006).
[CrossRef]

Phys. Rev. Lett.

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009).
[CrossRef] [PubMed]

X.-L. Wang, J. Chen, Y. Li, J. Ding, C.-S. Guo, and H.-T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010).
[CrossRef] [PubMed]

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

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett.104(10), 103601 (2010).
[CrossRef] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic for a FP beam passing through the beam waist region (or the focusing region). (b) Distributions of intensities and polarizations of FP beams with different order m at the beam waist’s plane (z = 0). The states of light polarization are denoted by the different ellipses.

Fig. 2
Fig. 2

The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for the FP beam with m=1 and γ=π/4 . Other parameters are a=50 nm and ε p =2.56 (no absorption).

Fig. 3
Fig. 3

The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for the FP beam with m=1 and γ=π/4 . Other parameters are a=50 nm and ε m =2.56+i2.56 (with absorption).

Fig. 4
Fig. 4

The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for a linear-polarized Gaussian beam. Other parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

The magnitude and direction distributions of (a) the gradient force, (b) the curl force, (c) the scattering force, and (d) total force in the plane of z=0 for a linear-polarized LG01 beam. Other parameters are the same as in Fig. 3.

Fig. 6
Fig. 6

Distributions of (a-c) the beam intensities and the states of polarization (different ellipses), and the magnitude and direction distributions of (d-f) the sum of the curl and scattering forces, and (g-i) the total transverse force at different planes: (a), (d), and (g) for z=0.5 Z R , (b), (e), and (h) for z=0 , and (c), (f), and (i) for z=0.5 Z R . Other parameters are the same as in Fig. 3.

Fig. 7
Fig. 7

Distributions of (a-c) the beam intensities and the states of polarization (different ellipses), and the magnitude and direction distributions of (d-f) the sum of the curl and scattering forces, and (g-i) the total transverse force at the different planes: (a), (d), and (g) for z=0.5 Z R , (b), (e), and (h) for z=0 , and (c), (f), and (i) for z=0.5 Z R , with m=3 and γ0.356π . Other parameters are the same as in Fig. 3.

Fig. 8
Fig. 8

Distributions of the z component of the total force acting on the small particle ( a=10 nm) under different absorptions (a) ε m =2.56 , (b) ε m =2.56+0.01i , and (c) ε m =2.56+0.05i , for the FP beam with m=1 and γ=0.25π .

Fig. 9
Fig. 9

Dependence of (a) F c max / F g max , (b) F s max / F g max , and (c) | F c + F s | max / F g max on the absorption factor Im[ ε p ] for the cases of the different particle’s sizes, under the radiation of the FP beam with m=1 and γ=0.25π .

Fig. 10
Fig. 10

Dependence of (a) F c max / F g max , (b) F s max / F g max , and (c) ( F c + F s ) max / F g max on the order value m with a=50 nm and ε p =2.56+2.56i . Other parameters are γ=0.25π for m=1 , γ=π/3 for m=2 , γ=0.356π for Re[ ε p ] , λ= for m=4 , and γ=0.373π for m=5 .

Fig. 11
Fig. 11

Distributions of the total transverse optical forces (in the unit of pN) of the second-order FP beam acting on (a-c) the Al nano-particle and (d-f) the Au nano-particle at different positions: (a) and (d) for z=0.5 Z R , (b) and (e) for z=0 , and (c) and (f) for z=0.5 Z R . Other parameters are m=2 and γ=π/3 .

Equations (19)

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

E m (r,φ,z)=cosγ LG 00 (r,φ,z) x ^ +sinγ LG 0m (r,φ,z) y ^
LG 00 (r,φ,z)= 2 π 1 w(z) exp[ ik r 2 2Q(z) ]exp[ i(kzϕ) ],
LG 0m (r,φ,z)= 2 πm! 1 w(z) ( 2 r w(z) ) m exp[ ik r 2 2Q(z) ] ×exp[ i(kzϕ) ]exp[ im(φϕ) ],
E m (r,φ,z)= 2 π 1 w(z) exp[ ik r 2 2Q(z) ]exp[ i(kzϕ) ]cosγ ×{ x ^ + y ^ tanγ 1 m! ( 2 r w(z) ) m exp[ im(φϕ) ] }.
E m (r,φ,z)= E 0 ( 1 Ωexp[ iδ ] ),
F = 1 4 Re(α) | E | 2 + k 2 Im( α )Re( E * × B )+ 1 2 Im(α)Im[( E * ) E ],
α= α 0 1i α 0 k 3 /(6π ε 0 )
α 0 =4π ε 0 a 3 ( ε p / ε m )1 ( ε p / ε m )+2
F g,x = Re[α] I 0 2 P ε 0 c x w 2 (z) { 1+ Ω 2 [ 1 m w 2 (z) 2( x 2 + y 2 ) ] },
F g,y = Re[α] I 0 2 P ε 0 c y w 2 (z) { 1+ Ω 2 [ 1 m w 2 (z) 2( x 2 + y 2 ) ] },
F g,z = Re[α] I 0 2 P 2 ε 0 c z ( z 2 + Z R 2 ) [ 2( x 2 + y 2 ) w 2 (z) ( Ω 2 +1 ) Ω 2 (m+1)1 ],
F s,x = Im[α]P 2 ε 0 c ε m I 0 Ω{ kr z 2 + Z R 2 [ zΩcosφzsinφcos(m(ϕφ)) + Z R sinφsin(m(ϕφ)) ] m r Ωsinφ },
F s,y = Im[α]P 2 ε 0 c ε m I 0 { kr z 2 + Z R 2 [ zsinφ+zΩcosφcos(m(ϕφ)) Z R Ωcosφsin(m(ϕφ)) ] + m r Ωsin[m(ϕφ)+φ] },
F s,z = Im[α]P 2 ε 0 c ε m I 0 { k(1+ Ω 2 )[ 1 r 2 ( z 2 Z R 2 ) 2 ( z 2 + Z R 2 ) 2 ] Z R z 2 + Z R 2 (m Ω 2 + Ω 2 +1) },
F c,x = Im[α]P I 0 kr 2 ε 0 c( z 2 + Z R 2 ) { zcosφ+Ωsinφ( zcos[m(ϕφ)] Z R sin[m(ϕφ)] ) },
F c,y = Im[α]P I 0 Ω 2 ε 0 c { krz ( z 2 + Z R 2 ) ( Ωsinφ+cosφcos[m(ϕφ)] ) + kr Z R ( z 2 + Z R 2 ) cosφsin[m(ϕφ)] m r sin[m(ϕφ)+φ]+ m r Ωcosφ },
F x = F g,x + F s,x + F c,x ,
F y = F g,y + F s,y + F c,y ,
F z = F g,z + F s,z .

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