Abstract

We theoretically study the three-dimensional behavior of nanoparticles in an active optical conveyor. To do this, we solved the Langevin equation when the forces are generated by a focusing system at the near field. Analytical expressions for the optical forces generated by the optical conveyor were obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is illuminated by a radially polarized Hermite-Gauss beam. Trajectories, in both the transverse plane and the longitudinal direction, are analyzed showing that the behavior of the optical conveyor can be optimized by conveniently choosing the configuration of the mask of the two annular pupils (inner and outer radius of the two rings) in order to trap and transport all particles at the focal plane.

© 2014 Optical Society of America

1. Introduction

In the 1970’s, Ashkin [1] demonstrated the optical trapping of particles using the radiation force generated by a focusing Gaussian beam. Since its first demonstration, optical manipulation by using focusing beams has been converted into a powerful tool for microscopic manipulation in different research fields like physics, biology, colloid science, or microfluidics. Among the different optical systems developed, the non diffracting Bessel beams have been one of the most used in optical nanotrap technology. Theoretical and experimental studies of non-paraxial Bessel beams and the resulting optical forces acting on a nanoparticle have been reported, such as the cases of a single Bessel beam or a standing Bessel beam obtained by illumination of an axicon with a linearly polarized beam [2, 3]. Using the same type of electric and magnetic field of Bessel beams given in [3], different computational models have been employed for analyzing the dynamic of particles produced by the optical forces generated by Bessel beams comparing the scattering Mie theory and geometrical ray optics [4]. Recently, Bessel beams, when the axicon is illuminated by a linearly polarized plane wave with different topological charge, have been used to study the behavior of microparticles near the center of an optical vortex beam [5]. Moreover, the influence of the orbital angular-momentum, using linear stability analysis on a spherical particle, has also been studied, [6] showing that a particle cannot be stably confined at the region of negative longitudinal optical force originated by Bessel beams with topological charge in the absence of ambient damping.

Recently, Ruffner and Grier [7] experimentally demonstrated and analyzed the properties of a class of tractor beam obtained by the interference of two coaxial Bessel Beams that differ in their axial wave numbers. For this, it was employed linearly polarized light illuminating a computer designed phase profile which was focused using a high numerical aperture objective. In this paper, we theoretically examine in detail the particle dynamics for different configurations of the tractor beam type described in [7] when radially polarized Bessel beams are used in order to improve the trapping of spherical particles [8, 9]. These radially polarized Bessel beams will be obtained by focusing a radially polarized beam using a high-aperture system that illuminates a two-ringed phase-only transmission function. The choice of this polarization state, together with two annular pupils, gives a sharp focal spot [10, 11]. Analytical expressions for the electric field and optical forces generated by the optical conveyor will be obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form. Our theoretical study will be carried out in the near field and for high aperture system using a vectorial diffraction analysis which differs to the scalar diffraction approximation used by Ruffner and Grier [7].

2. Theoretical background

The electric field components (using cylindrical coordinates) in the vicinity of the focus of a radially polarized beam can be obtained by using vectorial diffraction theory as follows [12, 13]:

er=A0αhr(θ)l0(θ)T(θ)J1(krSin(θ))Exp(ikzCos(θ))dθeψ=0ez=iA0αhz(θ)l0(θ)T(θ)J0(krSin(θ))Exp(ikzCos(θ))dθ
where k is the wavenumber, A is an amplitude constant, and l0(θ) is the apodization function that we have assumed that is an order one Hermite-Gauss mode:
l0(θ)=Exp(β2Sin(θ)2Sin(α)2)βSin(θ)Sin(α)
where hr(θ)=Cos(θ)Sin(2θ), hz(θ)=2Cos(θ)Sin2(θ), α is the angular semi-aperture of the focusing system given by α = sin−1(NA/n). NA is the numerical aperture and β is the ratio of the pupil radius and the beam waist, n is the refractive index between the high numerical optical system and the sample. Following the definitions given in reference [13], the main parameters used in Eqs. (1)(3) are shown in Fig. 1. The apodization function is modified by a mask complex function T (θ) given by:
T(θ)={g1θ1δ12θθ1+δ12g2θ2δ22θθ2+δ220otherwise
where we have assumed that g1 = 1 and g2 = Exp(iξt), so emergent fields from the rings described in transmission Eq. (3) differ in their relative phase. This linear relative phase ξt difference makes the conveyor work [7]. The inset in Fig. 1, shows the transmittance of the mask illuminated by a radially polarized beam considered (continuous line) compared to the same mask illuminated by a linearly Gaussian beam as apodization function (dashed lines). Values of the employed parameters are given in Table 1, the blue lines correspond to θ2 = 0.6, the red lines to θ2 = 0.7 and finally green lines correspond to θ2 = 0.8. As can be observed, the first ring is the same in all cases.

 

Fig. 1 Main parameters used in Eqs. (1)(3). The inset, shows |lo(θ)T (θ)| (continuous line) compared to linearly polarized Gaussian beam as apodization function (dashed lines). Values of the used parameters are given in Table 1

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Tables Icon

Table 1. Values of the parameters used in the numerical simulations and theoretical axial range

If δ121 and δ221, then, analytical solutions to integrals 1 can be obtained if we also assume that:

  1. The dependence on θ of the amplitude components of integrals 1 can be approximated to their constant value evaluated at the middle point (θl; l = 1, 2) of each annular ring described by Eq. (3).
  2. The phase of integrals 1 can be linearized, so using a Taylor expansion at θl point, we obtain:
    Exp(ikzCos(θ))Exp(ikzCos(θl)Sin(θl)(θθl)))
Using these approximations, the integrals in θ variable of Eq. (1) can be analytically solved, so Eq. (1) can be written as:
er(r,z)=Al=12glh^rlJ1(kslr)Exp(ikclz)Sinc(kslzδl2)δleψ=0ez(r,z)=iAl=12glh^zlJ0(kslr)Exp(ikclz)Sinc(kslzδl2)δl
where ksl = kSin(θl), kcl = kCos(θl), and ĥul = l0(θl)hu(θl) being u = (r, z). Equation (4) shows that the focusing of a radially polarized beam by using a high-aperture system with mask function T (θ) composed by two annular rings, generates a superposition of coaxial Bessel beams that produces a sharp focal spot [10]. This analytical electric field is similar to that proposed in reference [7] except the Sinc functions that limit the efficiency of the electric field in z-propagation direction. Moreover, another difference is given by the use of vectorial diffraction theory of a radially polarized incident field, which implies that the emergent radial field is a superposition of Bessel J0 beams and the axial field is a superposition of Bessel J1 beams radially symmetric. It is important to note, that using this methodology with field integrals described in reference [12] the resulting electric fields for linear polarization lack of rotational symmetry and the particles dynamic at different axial planes will depend on the initial angular position of the particle.

Equation (4) can be written as:

er(r,z)=ir(r,z)Exp(iφr(r,z))eψ=0ez(r,z)=iz(r,z)Exp(iφz(r,z)),
where we have introduced ir = |er|2, and iz = |ez|2 as the radial and axial intensity of the electric field, respectively. In the same way, we have introduced φr, and φz as the radial and axial phases of the electric field, respectively.

The intensity at the focal region is given by:

I=|er|2+|ez|2=ir+iz
The electric fields generated when the transmittance given by Eq. (3) is illuminated by a radially polarized beam and focused by a high-aperture system, can be obtained from Eqs. (4) and (5), respectively. Introducing Eq. (4) into Eq. (5) we obtain:
ir=(δ12J1(ks1r)2h^r12Sinc(δ1ks1z2)2+δ22J1(ks2r)2h^r22Sinc(δ2ks2z2)2)+2δ1δ2J1(ks1r)J1(ks2r)Cos(ξt+(ks2ks1)z)h^r1h^r2Sinc(δ1ks1z2)Sinc(δ2ks2z2)
iz=(δ12J0(ks1r)2h^z12Sinc(δ1ks1z2)2+δ22J0(ks2r)2h^z22Sinc(δ2ks2z2)2)+2δ1δ2J0(ks1r)J0(ks2r)Cos(ξt+(ks2ks1)z)h^z1h^z2Sinc(δ1ks1z2)Sinc(δ2ks2z2)
φr=Tan1(δ1J1(ks1r)h^r1Sin(kc1z)Sinc(δ1ks1z2)+δ2J1(ks2r)h^r2Sin(ξt+kc2z)Sinc(δ2ks2z2)δ1J1(ks1r)h^r1Cos(kc1z)Sinc(δ1ks1z2)+δ2J1(ks2r)h^r2Cos(ξt+kc2z)Sinc(δ2ks2z2))
φz=Tan1(δ1J0(ks1r)h^z1Cos(kc1z)Sinc(δ1ks1z2)+δ2J0(ks2r)h^z2Cos(ξt+kc2z)Sinc(δ2ks2z2)δ1J0(ks1r)h^z1Sin(kc1z)Sinc(δ1ks1z2)+δ2J0(ks2r)h^z2Sin(ξt+kc2z)Sinc(δ2ks2z2))

2.1. Optical forces acting on a nanoparticle

We focus here on the optical forces acting on a particle in the Rayleigh regime (radius rp << λ/20) for which it is accomplished that the scattering is so weak. Therefore, according to [14], the time average optical forces acting on the particle (assuming it in a region where an electric field (er, eψ, ez) exists) can be expressed as:

Fr=12[α^(erer*r+eψeψ*r+ezez*r)]Fψ=12[α^(erer*rψ+eψeψ*rψ+ezez*rψ)]Fz=12[α^(erer*z+eψeψ*z+ezez*z)]
where ℜ denotes the real value of the expression, * is a complex conjugate value and α̂ = αR + I is the complex value of the polarizability particle, which for the dielectric particles considered can be obtained by [15]:
αR=9a3(εpε)ε(εp+2ε)(9+4a6k06)εp2+4(92a6k06)εεp+4(9+a6k06)ε2αI=6a6k03(εpε)2ε(9+4a6k06)εp2+4(92a6k06)εεp+4(9+a6k06)ε2
being k0 the wavenumber in vacuum, a the particle radius and εp the particle dielectric permittivity and ε the dielectric permittivity of the medium where the dipolar particle is embedded. Introducing Eq. (5) into Eq. (11), we obtain that:
Fr=14αRIr+αI2(φrrir+φψriψ+φzriz)Fψ=0Fz=14αRIz+αI2(φrzir+φψziψ+φzziz)
In last equation, the terms proportional to αR are the gradient force and the terms proportional to αI are the scattering force, so Eq. (13) can be written in a compact form as:
F=αR4I+αI2uiuφu
where it is shown that the gradient force is proportional to the gradient of total intensity of the electric field and the scattering force depends on the gradient of phases of each component of the field. It is interesting to note that, recently, it has been demonstrated that gradientless optical fields can act as tractor beams along a properly chosen interface of two materials with different refractive indices [16]. In order to solve the particle dynamics, we must solve the Langevin equation [17]:
md2Rdt2=F(R(t))γdRdt+(t)
where R⃗(t) is the position vector of the particle at time t, m is the particle mass, γdRdt is the frictional force of a particle, and ℱ⃗(t) is a random function force with time. ℱ⃗(t) has a Gaussian probability distribution with correlation function <i(t), j(t′)>= 2γKBi,jδ(tt′), where kB is Boltzmann’s constant and T is the temperature. Coefficient γ = 6πηa, where η is the viscosity of the media.

3. Design of an optimal active tractor beams

By introducing Eqs. (7)(10) into Eq. (14) we can obtain analytical expressions of the generated gradient and scattering optical forces components. As can be deduced from Eqs. (7) and (8), contributions to the axial conveyor’s intensity (r → 0) are given by iz, because ir is null at r = 0. Thus, the axial intensity is described by:

I(z)=(δ12h^z12Sinc(δ1ks1z2)2+δ22h^z22Sinc(δ2ks2z2)2)+2Cos(ξt+(ks2ks1)z)h^z1h^z2Sinc(δ1ks1z2)Sinc(δ2ks2z2)
In order to obtain the same intensity in the coordinate origin as was realized in reference [7], we take the width of the second ring as δ2 = δ2e according to:
δ2e=h^z1δ1h^z2=δ1l0(θ1)Cos(θ1)Sin(θ1)2l0(θ2)Cos(θ2)Sin(θ2)2
As can be observed, the width of the second ring depends on the width of the first ring, the ring’s positions and the apodization function evaluated at each ring. Introducing δ2e in Eq. (16), the axial intensity when the beam ratio is 1:1 at t = 0 is given by:
I(z)=δ12h^z12(Sinc(δ1ks1z2)2+Sinc(δ2eks2z2)2)+δ12h^z12(2Cos(ξt+(ks2ks1)z)Sinc(δ1ks1z2)Sinc(δ2eks2z2))

Equation (18) is similar to the result obtained at reference [7] but the Sinc functions (that are the beam’s amplitude variable for each z plane) limit the axial range of the optical conveyor and also contributes to the axial force according to Eq. (14). The maximum theoretical axial range Δz of the conveyor described by the axial intensity 18, can be deduced from the arguments of the Sinc functions (the ones that multiplies the cosine function) by using:

Δz=λMax(δ2eSin(θ2),δ1Sin(θ1))
For optimizing the energetic response of the optical conveyor, we locate the first ring centered at the position where incident field l(θ) shows the maximum value:
θ1=Sin1(Sin(α)2β)
Moreover, we choose:
β=Sin(α)2Sin(αδ12)
so that the first ring gives us the maximum aperture angle of the system. By taking these parameter values, it is accomplished that θ1 > θ2 and as consequence δ2eCos(θ2) > δ1Cos(θ1). Then, according to Eq. (19), the maximum theoretical axial range of the optical conveyor will be given by:
Δz=λδ2eSin(θ2)
Since, in order to optimize the tractor beam, we have two degrees of freedom, δ1 and θ2.

3.1. Numerical results

Taking into account previous points, we analyze three configurations of the same tractor beam (see Table 1), taking in all cases δ1 = 0.06. For this, the NA system was 1.1, and the refractive index between the lens and the sample is n=ε=1.33 (water). Using these numerical values, according to Eqs. (20) and (21), we have that θ1 = 0.942.

Then, δ2e values for each conveyor configuration can be obtained by introducing in Eq. (17) Eq. (2) together with the numerical values of δ1 and θ1. In Table 1, the obtained results for θ2 values between 0.6 to 0.8 are shown. This analysis is limited to this θ2-range because at lower values than 0.6, δ2e increases and condition δ2 << 1 is not fulfilled. On the other hand, higher θ2-values as 0.8 originate only one ring for the used numerical values.

Figure 2 shows the normalized intensity distribution at the initial time for the three analyzed conveyors configurations shown in Table 1. As can be seen, according to condition 17, the maximum intensity value is the same for all of them; moreover, the theoretical axial range increases as the θ2 value raises (Fig. 3). However, at higher θ2 values, the intensity maxima at axial positions (different to z = 0) decreases (Fig. 3). The maximum transport efficiency will be obtained when particles are axially confined. Then, we are going to analyze the best trapping configuration at the focal plane.

 

Fig. 2 Normalized Intensity I = ir + iz at time t = 0 as a function of coordinates r and z for the three conveyors configurations (T1,T2 and T3) described in Table 1

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Fig. 3 Normalized Axial Irradiance I(r = 0) = iz at time t = 0s versus z-coordinate for the three conveyors configurations (T1 (blue),T2 (red) and T3 (green)) described in Table 1.

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For this purpose, in order to study the dynamical behavior of particles at three configurations, we have numerically solved Eq. (15) for particle sizes of a = λ/30 and a = λ/60, initially located at the focal plane using parameters values ξ = 10Hz and λ = 633nm. The particle dielectric constant was εp = 2.223 (PMMA), the constants γ = 6πηa and m=43ρπa3 have been obtained using the water viscosity coefficient η = 8.9 × 10−4 (Pa s) and the material density (PMMA) ρ = 1.19 × 103 (Kg m−3). The corresponding dynamics for the conveyors previously described are shown in Figs. 48. These figures have been obtained averaging more than one hundred individual paths for each initial conditions. As can be observed, when the intensity is low (dotted and dashed lines in Figs. 4(a) and 4(b)), the particles do not reach the theoretical axial range due to Brownian and frictional effects. For higher intensity values the particles nearest to the origin (blue and green colors continuous lines in Figs. 4(a) and 4(b)) reach the theoretical axial range, while those farthest do not, especially the smaller ones. It can be explained by taking into account the intensity side lobes showed in Fig. 2(T1). The particles will be stably trapped in this potential wells out of optical axes. The axis can only reached by random process of scape from the potential well. Moreover this behavior can be deduced from Figs. 4(c) and 4(d), that shows the time and intensity dependence of the particle radial positions for conveyor configuration T1. As may be seen, the T1 conveyor configuration traps only particles located near the origin (r < λ) at the focusing plane. When particles are axially trapped (green and blue lines in Figs. 4(c) and 4(d)), the axial dynamics are practically the same as Figs. 4(a) and 4(b) show. It is important to remark that the trap dynamical times at the focal plane are not related to axial dynamical times; the first one is on the order of 0.1 s and the second is about 3 s, so trapping process is faster than axial transport. Figure 5 shows the time and intensity dependence of the particle’s axial and radial positions for conveyor configuration T2. As can be observed, when the intensity is low (Figs. 5(a) and 5(b) dotted lines), the particles do not reach the theoretical axial range as occurs for configuration T1. For the case of higher intensity values (see Figs. 5(a) and 5(b) dashed and continuous lines), the particles reach nearly the same maximum Δz value, at similar times, irrespective of its initial radial position value. In fact, larger particles attain the maximum theoretical value of axial range as can be seen in Fig. 5(a) if the intensity is high enough. For a lower intensity value (Fig. 5(a) dashed) and smaller particles (see Fig. 5(b) dashed), the axial range obtained is closed to the maximum theoretical one but did not reach it as a consequence of Brownian effect. Figures 5(c) and 5(d) show the time and intensity dependence of the particles radial positions for conveyor configuration T2. As may be seen, the T2 conveyor configuration traps all particles located at the focusing plane in practically all cases except for low intensity values (dotted lines). As it has been previously mentioned the dynamic of the particles has been obtained by averaging one hundred individual paths. For simplicity, in Figs. 6 and 7 we show only the uncertainty of the conveyor T2 where the shaded areas correspond to ± mean deviation. As can be observed the uncertainty is lower at high intensity values and for high values of radius because the influence of Brownian effects is lower in these cases. It is interesting to remark that the behavior observed in the axial position dynamics shown in Figs. 4 and 8 can be explained by the transition from the so-called Brownian surfer mode to the Brownian Swimmer mode [18, 20].

 

Fig. 4 Conveyor configuration T1. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

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Fig. 5 Conveyor configuration T2. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

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Fig. 6 Conveyor configuration T2 for spherical particles with radius of a = λ/30. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

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Fig. 7 Conveyor configuration T2 for spherical particles with radius of a = λ/60. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

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Fig. 8 Conveyor configuration T3. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

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Finally, regarding to the results obtained for conveyor configuration T3, as can be observed Fig. 8, in this configuration, the particles located outside the origin are not trapped at short time, and as a consequence, none of them reach the maximum axial range. By comparing Figs. 4, 5 and 8, we can conclude that the optimal conveyor configuration for the intensity range analyzed is T2.

4. Conclusions

We have obtained analytical expressions for the optical forces generated by an optical conveyor by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is used. The analysis has been performed for radial polarization but the approximations can be used for any kind of polarization by using the appropriate integrals. The obtained expressions have been used to analyze the 3D dynamical behavior of particles in the Rayleigh approximation solving the Langevin equation for three different configurations of the optical conveyor. Our results have demonstrated that for a given intensity interval, conveyor configuration can be optimized in order to trap all the particles at the focal plane regardless of their initial position and axially transport them as much as possible.

References and links

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

2. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). [CrossRef]  

3. T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]  

4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007). [CrossRef]   [PubMed]  

5. M. Siler, P. Jakl, O. Brzobohaty, and P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012). [CrossRef]   [PubMed]  

6. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). [CrossRef]  

7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012). [CrossRef]   [PubMed]  

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

9. S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013). [CrossRef]   [PubMed]  

10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef]   [PubMed]  

11. Z. Chen and D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012). [CrossRef]   [PubMed]  

12. B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959). [CrossRef]  

13. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef]   [PubMed]  

14. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagentic fiel,” Opt. Lett. 25, 1065–1067 (2000). [CrossRef]  

15. M. Nieto-Vesperinas, J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010). [CrossRef]   [PubMed]  

16. V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013). [CrossRef]  

17. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

18. M. Borromeo and F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998). [CrossRef]  

19. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002). [CrossRef]  

20. M. Siler, T. Cizmar, A. Jonas, and P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008). [CrossRef]  

References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  2. T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
    [CrossRef]
  3. T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
    [CrossRef]
  4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007).
    [CrossRef] [PubMed]
  5. M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
    [CrossRef] [PubMed]
  6. N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
    [CrossRef]
  7. D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
    [CrossRef] [PubMed]
  8. T. A. Nieminen, N. R. Heckenberg, H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
    [CrossRef] [PubMed]
  9. S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013).
    [CrossRef] [PubMed]
  10. C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [CrossRef] [PubMed]
  11. Z. Chen, D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
    [CrossRef] [PubMed]
  12. B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  13. K. S. Youngworth, T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  14. P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagentic fiel,” Opt. Lett. 25, 1065–1067 (2000).
    [CrossRef]
  15. M. Nieto-Vesperinas, J. Sáenz, R. Gómez-Medina, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
    [CrossRef] [PubMed]
  16. V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
    [CrossRef]
  17. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
  18. M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
    [CrossRef]
  19. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
    [CrossRef]
  20. M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
    [CrossRef]

2013 (3)

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013).
[CrossRef] [PubMed]

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

2012 (3)

2010 (1)

2008 (2)

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

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

2007 (1)

2006 (1)

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

2005 (1)

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

2004 (1)

2002 (1)

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

2000 (2)

1998 (1)

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

1970 (1)

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

1959 (1)

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Ashkin, A.

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

Borromeo, M.

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

Brown, T.

Brzobohaty, O.

Chantada, L.

Chaumet, P. C.

Chen, J.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Chen, Z.

Choudhury, A.

Cizmar, T.

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Dholakia, H.

Dholakia, K.

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Ding, W.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Dogariu, A.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Garces-Chavez, V.

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Gómez-Medina, R.

Grier, D. G.

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Heckenberg, N. R.

Iati, M.

Jakl, P.

Jonas, A.

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

Jones, P. H.

Kajorndenukul, V.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Lin, Z.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Liu, S.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Marago, O.

Marchesoni, F.

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

McGloin, D.

Milne, G.

Nieminen, T. A.

Nieto-Vesperinas, M.

Qiu, C.-W.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

Reimann, P.

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Rubinstein-Dunlop, H.

Ruffner, D. B.

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Sáenz, J.

Saija, R.

Sergides, M.

Sheppard, C. J. R.

Siler, M.

M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
[CrossRef] [PubMed]

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

Skelton, S. E.

Sukhov, S.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Volke-Sepulveda, K.

Wang, N.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Youngworth, K. S.

Zemanek, P.

M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
[CrossRef] [PubMed]

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007).
[CrossRef] [PubMed]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Zhao, D.

Appl. Opt. (1)

Appl. Phys. B (1)

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Nat. Photonics (1)

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

New J. Phys. (1)

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (1)

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

Phys. Rep. (1)

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

Phys. Rev. A (1)

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

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

Proc. R. Soc. A (1)

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Other (1)

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

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Figures (8)

Fig. 1
Fig. 1

Main parameters used in Eqs. (1)(3). The inset, shows |lo(θ)T (θ)| (continuous line) compared to linearly polarized Gaussian beam as apodization function (dashed lines). Values of the used parameters are given in Table 1

Fig. 2
Fig. 2

Normalized Intensity I = ir + iz at time t = 0 as a function of coordinates r and z for the three conveyors configurations (T1,T2 and T3) described in Table 1

Fig. 3
Fig. 3

Normalized Axial Irradiance I(r = 0) = iz at time t = 0s versus z-coordinate for the three conveyors configurations (T1 (blue),T2 (red) and T3 (green)) described in Table 1.

Fig. 4
Fig. 4

Conveyor configuration T1. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Fig. 5
Fig. 5

Conveyor configuration T2. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Fig. 6
Fig. 6

Conveyor configuration T2 for spherical particles with radius of a = λ/30. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

Fig. 7
Fig. 7

Conveyor configuration T2 for spherical particles with radius of a = λ/60. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

Fig. 8
Fig. 8

Conveyor configuration T3. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Tables (1)

Tables Icon

Table 1 Values of the parameters used in the numerical simulations and theoretical axial range

Equations (23)

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e r = A 0 α h r ( θ ) l 0 ( θ ) T ( θ ) J 1 ( k r Sin ( θ ) ) Exp ( i k z Cos ( θ ) ) d θ e ψ = 0 e z = i A 0 α h z ( θ ) l 0 ( θ ) T ( θ ) J 0 ( k r Sin ( θ ) ) Exp ( i k z Cos ( θ ) ) d θ
l 0 ( θ ) = Exp ( β 2 Sin ( θ ) 2 Sin ( α ) 2 ) β Sin ( θ ) Sin ( α )
T ( θ ) = { g 1 θ 1 δ 1 2 θ θ 1 + δ 1 2 g 2 θ 2 δ 2 2 θ θ 2 + δ 2 2 0 otherwise
Exp ( i k z Cos ( θ ) ) Exp ( i k z Cos ( θ l ) Sin ( θ l ) ( θ θ l ) ) )
e r ( r , z ) = A l = 1 2 g l h ^ r l J 1 ( k s l r ) Exp ( i k c l z ) Sin c ( k s l z δ l 2 ) δ l e ψ = 0 e z ( r , z ) = i A l = 1 2 g l h ^ z l J 0 ( k s l r ) Exp ( i k c l z ) Sin c ( k s l z δ l 2 ) δ l
e r ( r , z ) = i r ( r , z ) Exp ( i φ r ( r , z ) ) e ψ = 0 e z ( r , z ) = i z ( r , z ) Exp ( i φ z ( r , z ) ) ,
I = | e r | 2 + | e z | 2 = i r + i z
i r = ( δ 1 2 J 1 ( k s 1 r ) 2 h ^ r 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 J 1 ( k s 2 r ) 2 h ^ r 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 δ 1 δ 2 J 1 ( k s 1 r ) J 1 ( k s 2 r ) Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ r 1 h ^ r 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
i z = ( δ 1 2 J 0 ( k s 1 r ) 2 h ^ z 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 J 0 ( k s 2 r ) 2 h ^ z 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 δ 1 δ 2 J 0 ( k s 1 r ) J 0 ( k s 2 r ) Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ z 1 h ^ z 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
φ r = Tan 1 ( δ 1 J 1 ( k s 1 r ) h ^ r 1 Sin ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 1 ( k s 2 r ) h ^ r 2 Sin ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) δ 1 J 1 ( k s 1 r ) h ^ r 1 Cos ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 1 ( k s 2 r ) h ^ r 2 Cos ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) )
φ z = Tan 1 ( δ 1 J 0 ( k s 1 r ) h ^ z 1 Cos ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 0 ( k s 2 r ) h ^ z 2 Cos ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) δ 1 J 0 ( k s 1 r ) h ^ z 1 Sin ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 0 ( k s 2 r ) h ^ z 2 Sin ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) )
F r = 1 2 [ α ^ ( e r e r * r + e ψ e ψ * r + e z e z * r ) ] F ψ = 1 2 [ α ^ ( e r e r * r ψ + e ψ e ψ * r ψ + e z e z * r ψ ) ] F z = 1 2 [ α ^ ( e r e r * z + e ψ e ψ * z + e z e z * z ) ]
α R = 9 a 3 ( ε p ε ) ε ( ε p + 2 ε ) ( 9 + 4 a 6 k 0 6 ) ε p 2 + 4 ( 9 2 a 6 k 0 6 ) ε ε p + 4 ( 9 + a 6 k 0 6 ) ε 2 α I = 6 a 6 k 0 3 ( ε p ε ) 2 ε ( 9 + 4 a 6 k 0 6 ) ε p 2 + 4 ( 9 2 a 6 k 0 6 ) ε ε p + 4 ( 9 + a 6 k 0 6 ) ε 2
F r = 1 4 α R I r + α I 2 ( φ r r i r + φ ψ r i ψ + φ z r i z ) F ψ = 0 F z = 1 4 α R I z + α I 2 ( φ r z i r + φ ψ z i ψ + φ z z i z )
F = α R 4 I + α I 2 u i u φ u
m d 2 R d t 2 = F ( R ( t ) ) γ d R d t + ( t )
I ( z ) = ( δ 1 2 h ^ z 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 h ^ z 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ z 1 h ^ z 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
δ 2 e = h ^ z 1 δ 1 h ^ z 2 = δ 1 l 0 ( θ 1 ) Cos ( θ 1 ) Sin ( θ 1 ) 2 l 0 ( θ 2 ) Cos ( θ 2 ) Sin ( θ 2 ) 2
I ( z ) = δ 1 2 h ^ z 1 2 ( Sin c ( δ 1 k s 1 z 2 ) 2 + Sin c ( δ 2 e k s 2 z 2 ) 2 ) + δ 1 2 h ^ z 1 2 ( 2 Cos ( ξ t + ( k s 2 k s 1 ) z ) Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 e k s 2 z 2 ) )
Δ z = λ Max ( δ 2 e Sin ( θ 2 ) , δ 1 Sin ( θ 1 ) )
θ 1 = Sin 1 ( Sin ( α ) 2 β )
β = Sin ( α ) 2 Sin ( α δ 1 2 )
Δ z = λ δ 2 e Sin ( θ 2 )

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