## 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]:

*l*

_{0}(

*θ*) is the apodization function that we have assumed that is an order one Hermite-Gauss mode:

*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:

*g*1 = 1 and

*g*2 =

*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

*θ*

_{2}= 0.6, the red lines to

*θ*

_{2}= 0.7 and finally green lines correspond to

*θ*

_{2}= 0.8. As can be observed, the first ring is the same in all cases.

If $\frac{{\delta}_{1}}{2}\ll 1$ and $\frac{{\delta}_{2}}{2}\ll 1$, then, analytical solutions to integrals 1 can be obtained if we also assume that:

- The dependence on
*θ*of the amplitude components of integrals 1 can be approximated to their constant value evaluated at the middle point (*θ*;_{l}*l*= 1, 2) of each annular ring described by Eq. (3).

*θ*variable of Eq. (1) can be analytically solved, so Eq. (1) can be written as:

*k*=

_{sl}*kSin*(

*θ*),

_{l}*k*=

_{cl}*kCos*(

*θ*), and

_{l}*ĥ*=

_{ul}*l*

_{0}(

*θ*)

_{l}*h*(

_{u}*θ*) being

_{l}*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

*J*

_{0}beams and the axial field is a superposition of Bessel

*J*

_{1}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:

*i*= |

_{r}*e*|

_{r}^{2}, and

*i*= |

_{z}*e*|

_{z}^{2}as the radial and axial intensity of the electric field, respectively. In the same way, we have introduced

*φ*, and

_{r}*φ*as the radial and axial phases of the electric field, respectively.

_{z}The intensity at the focal region is given by:

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:#### 2.1. Optical forces acting on a nanoparticle

We focus here on the optical forces acting on a particle in the Rayleigh regime (radius *r _{p}* <<

*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 (

*e*,

_{r}*e*,

_{ψ}*e*) exists) can be expressed as:

_{z}^{*}is a complex conjugate value and

*α̂*=

*α*+

_{R}*iα*is the complex value of the polarizability particle, which for the dielectric particles considered can be obtained by [15]:

_{I}*k*

_{0}the wavenumber in vacuum,

*a*the particle radius and

*ε*the particle dielectric permittivity and

_{p}*ε*the dielectric permittivity of the medium where the dipolar particle is embedded. Introducing Eq. (5) into Eq. (11), we obtain that:

*α*are the gradient force and the terms proportional to

_{R}*α*are the scattering force, so Eq. (13) can be written in a compact form as:

_{I}*R⃗*(

*t*) is the position vector of the particle at time

*t*, m is the particle mass, $-\gamma \frac{d\overrightarrow{R}}{dt}$ 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 <

*ℱ*(

_{i}*t*),

*ℱ*(

_{j}*t′*)>= 2

*γK*(

_{B}Tδ_{i,j}δ*t*−

*t′*), where

*k*is Boltzmann’s constant and T is the temperature. Coefficient

_{B}*γ*= 6

*πηa*, 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 *i _{z}*, because

*i*is null at

_{r}*r*= 0. Thus, the axial intensity is described by:

*δ*

_{2}=

*δ*

_{2}

*according to:*

_{e}*δ*

_{2}

*in Eq. (16), the axial intensity when the beam ratio is 1:1 at*

_{e}*t*= 0 is given by:

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 Δ* _{z}* 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:

*l*(

*θ*) shows the maximum value: Moreover, we choose: so that the first ring gives us the maximum aperture angle of the system. By taking these parameter values, it is accomplished that

*θ*

_{1}>

*θ*

_{2}and as consequence

*δ*

_{2}

*(*

_{e}Cos*θ*

_{2}) >

*δ*

_{1}

*Cos*(

*θ*

_{1}). Then, according to Eq. (19), the maximum theoretical axial range of the optical conveyor will be given by: Since, in order to optimize the tractor beam, we have two degrees of freedom,

*δ*

_{1}and

*θ*

_{2}.

#### 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 *δ*_{1} = 0.06. For this, the NA system was 1.1, and the refractive index between the lens and the sample is
$n=\sqrt{\epsilon}=1.33$ (water). Using these numerical values, according to Eqs. (20) and (21), we have that *θ*_{1} = 0.942.

Then, *δ*_{2}* _{e}* values for each conveyor configuration can be obtained by introducing in Eq. (17) Eq. (2) together with the numerical values of

*δ*

_{1}and

*θ*

_{1}. In Table 1, the obtained results for

*θ*

_{2}values between 0.6 to 0.8 are shown. This analysis is limited to this

*θ*

_{2}-range because at lower values than 0.6,

*δ*

_{2}

*increases and condition*

_{e}*δ*

_{2}<< 1 is not fulfilled. On the other hand, higher

*θ*

_{2}-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 *θ*_{2} value raises (Fig. 3). However, at higher *θ*_{2} 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.

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* = 10*Hz* and *λ* = 633*nm*. The particle dielectric constant was *ε _{p}* = 2.223 (PMMA), the constants

*γ*= 6

*πηa*and $m=\frac{4}{3}\rho \pi {a}^{3}$ have been obtained using the water viscosity coefficient

*η*= 8.9 × 10

^{−4}(Pa s) and the material density (PMMA)

*ρ*= 1.19 × 10

^{3}(Kg

*m*

^{−3}). The corresponding dynamics for the conveyors previously described are shown in Figs. 4 – 8. 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 Δ

*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].*

_{z}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.

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