## 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 [3–5], 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 $\mathrm{Im}[({\stackrel{\rightharpoonup}{E}}^{*}\cdot \nabla )\overrightarrow{E}]$, which is zero only when $\overrightarrow{E}$ 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

## 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]

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 $\phi $ varies and gradually rotates as the beam propagates along the $z$axis. Substituting Eq. (4) into Eq. (6), after tedious calculations, we can analytically obtain the components of optical force due to different contributions as follows:

In the following section, we will show how the order number $m$ affects on the trapping efficiency and how important the optical curl forces (${F}_{c,x}$ and ${F}_{c,y}$) 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: $\lambda =500$nm, $m=2$µm, $P=1$W, ${\epsilon}_{m}=1$ (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.

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 $L{G}_{01}$ beam for $z=0$ and $\gamma =\pi /2$ 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 $L{G}_{01}$ 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 $y$and $x$ 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 $L{G}_{01}$ 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 $L{G}_{01}$ beam.

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 $z=-0.5{Z}_{R}$ to $z=0.5{Z}_{R}$, 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.

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 $a=10$nm. From Fig. 8, we clear see that that a slight absorption will destroy the stability of the longitudinal trapping effect. For example, when $\mathrm{Im}[{\epsilon}_{p}]=0.01$, the depth of the negative ${F}_{z}$ becomes very weak, compared with the non-absorbing case; when $\mathrm{Im}[{\epsilon}_{p}]=0.05$, the value of ${F}_{z}$ completely becomes positive, which provides the pushing force along the z direction. In fact, from Eq. (10c), it is seen that ${F}_{z}$ 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 ${F}_{z}$.

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 ${F}_{c}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$, ${F}_{s}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$, and $|{\overrightarrow{F}}_{c}+{\overrightarrow{F}}_{s}{|}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$ on the imaginary part of the particle’s permittivity. Here all these forces are in the $x-y$ plane at $z=0$. It is found that with the increasing of $\mathrm{Im}[{\epsilon}_{p}]$, 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 $\mathrm{Im}[{\epsilon}_{p}]$.

Figure 10 shows the effect of the order number $m$ on the ratios of ${F}_{c}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$, ${F}_{s}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$, and $|{\overrightarrow{F}}_{c}+{\overrightarrow{F}}_{s}{|}^{\mathrm{max}}/{F}_{g}^{\mathrm{max}}$, under the fixed parameters of $\mathrm{Im}[{\epsilon}_{p}]$ 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 ${w}_{0}$with two independent adjustable parameters${w}_{0x}$ and ${w}_{0y}$ in Eqs. (2) and (3). Then the result of the trapping effect can be further improved.

In the above calculations, we have used the large value of $\mathrm{Im}[{\epsilon}_{p}]$. Actually, for the nano-scaled metallic particles, such large values of $\mathrm{Im}[{\epsilon}_{p}]$ 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 $a=50$nm, see Fig. 11 . For an Al particle, its permittivity is ${\epsilon}_{p}=-36.373+i9.412$ at the wavelength $\lambda =$500nm [25]; while for an Au particle, its permittivity is ${\epsilon}_{p}=-2.81+i3.18$ at the same wavelength [10]. From Fig. 11, we can see that, for the metallic nano particles with the larger negative value of $\mathrm{Re}[{\epsilon}_{p}]$ 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 $\mathrm{Re}[{\epsilon}_{p}]$, 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 $z=-0.5{Z}_{R}$, the resultant optical force field will drive the particle rotating into the center; while at $z=0.5{Z}_{R}$, 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.

## 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 $\mathrm{Re}[{\epsilon}_{p}]$ and positive $\mathrm{Im}[{\epsilon}_{p}]$, 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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