## Abstract

Here we propose a new method for trapping the resonant metallic particles with the 4π tight focusing (high numerical-aperture (NA)) system, which is illuminated by radial polarization light. Numerical simulations have indicated the maximum total optical force is 16.1pN while with nearly zero scattering force under axis trapping, which keeps the gradient force predominant. Furthermore, the distribution of total force is centrosymmetric and odd. We also gain stable 3D trap with an equilibrium point along z axis and r axis as in normal optical tweezers. What’s more, we obtain the nearly pure longitudinal field. The maximum transverse intensity is only 2.3 × 10^{−3} and the transverse spot size reaches 0.36λ, which is below Abbe’s diffraction limit.

© 2016 Optical Society of America

## 1. Introduction

Optical trapping or optical tweezers have recently attracted much attention and become powerful tools for the trapping and manipulations of metal nanoparticles [1–3], semiconductor quantum dots [4,5], and dielectric particles [6]. Besides, there have been plenty of significant advances in the underlying theory and experimental studies about forming a stable three-dimensional (3D) optical trap. Many theoretical and experimental studies show that optical tweezers with radial polarization (RP) can stably trap metallic particles in 3D space [7–9]. The extremely strong axial component of a tightly focused RP beam can generate large gradient force and zero scattering force along the optical axis for its axial time average Poynting vector is zero [10]. Thus, a stable 3D optical trap for metallic particles can be generated.

However, the previous theoretical and experimental studies concerning the trapping with RP beams are limited in the trapping wavelength which is away from the plasmonic resonance wavelength of the metallic particles where metal behaves dramatically different from dielectric, therefore trapping resonant metallic particle is challenging. Under the electronic resonance condition, the induced polarization is resonantly enhanced, and so is the induced scattering force. Metallic particles are generally considered difficult to trap due to strong scattering force [11]. It is difficult to trap resonant particles with a single focused laser beam because the resonantly enhanced scattering force strongly pushes particles away from the foci. Meanwhile, resonant illuminations of metallic nanoparticles give rise to strong heating effects owing to strong light absorption. Recently many ideas have been proposed for supporting stable 3D trapping at resonance to increase the trapping efficiency of metallic nanoparticles [12,13].

In this paper, we propose a new method for forming a stable 3D trap of metallic particles even under resonant conditions with the 4π high NA focusing system illuminated by RP beams. In order to gain pure longitudinal field and small scattering force, we utilize a non-uniform amplitude distribution of the illumination which increases with increasing convergence angle, letting the outer rays pass through but attenuating rays that are close to the optical axis.

## 2. Theory

We show the 4π focusing system (corresponding to solid angle 4π) which is comprised of two high NA objective (aplanatic lens (AL)) illuminated by counter-propagating RP beams in Fig. 1, with the instaneous electric field vector represented by the short arrows. In order to illustrate the advantage of 4π system, we investigate the situation of 2π system (the single AL).

According to the Debye vector integral theory, the field distribution in the vicinity of the focus for the single high NA focusing system can be expressed as [14]:

Where the maximum convergence angle *α* = arcsin(NA/n), NA is the numerical-aperture, n and *k* are the refractive index and the wave number in image space, respectively; while *A* is a constant related to the power of incident beam, *J*_{m} is the first kind of Bessel function of order m, and *l*(*θ*) is the amplitude of the incident beam at the pupil plane. The apodization factor for AL is cos^{1/2}*θ* . In this paper, the amplitude distribution of the illumination *l*(θ) is given by [15]:

The NA of the objective is chosen in order that the maximum convergence angle from the edge of the lens is 89° to make sure *l*(*θ*) cannot reach infinity and gain the gradient force as lager as possible.

In order to achieve the perfect destructive interference of the transverse component in the focal plane so as to increase the intensity of electric field while enhancing the gradient force and reducing scattering force at the same time, two counter-propagating RP beams must have opposite polarization directions [16,17], the electric fields near the foci of the 4π high NA focusing system can be expressed as [18,19]:

Where **E _{1}** and

**E**denote the focal electric field of the left and right incident beam in Fig. 1, respectively. The negative sign of the

_{2}*r*in

**E**indicates the opposite direction of polarization of two incident beams, while the negative sign of the

_{2}*z*in

**E**means their counter- propagating directions.

_{2}Assume that a metallic Rayleigh nanoparticle suspended in the medium whose dielectric constant is *ε*. The optical properties of a spherical particle with radius a much smaller than the incident wavelength, are characterized by the polarizability *α* [20]:

Where *α*_{0} = 4πa^{3}[*ε*_{m}(ω)-*ε*]/[*ε*_{m}(ω) + 2*ε*],*ε*_{m}(ω) is the relative permittivity of the metal from bulk material, and ω is the angular frequency. In the presence of the optical focal field described by Eqs. (1) and (4), the particle moves under the influence of a time averaged light-induced force. The axial optical forces exerted on the particle can be written as the sum of three terms [15]:

Where σ = k*Im*(*α*) is the total cross section of the particle, *c* is the speed of light in vacuum, *n* is the refractive index of the surrounding medium. In Eq. (5), the first term is the gradient force which is proportional to the gradient of the intensity of the focal field, which can give rise to the 3D confinement in optical tweezers so long as it is predominant to the second and third terms. The second term is identified as the radiation force proportional to the time averaged Poynting vector, and the third term is a force arising from the presence of spatial polarization gradients [19]. The sum of the second and the third term is the total scattering force.

## 3. Results and discussion

To illustrate the advantage of this innovative trapping method, in this paper, the gold nanoparticle with radius 50 nm is suspended in water (*ε* = 1.77, *n* = 1.333), the illumination wavelength is 532 nm. Please notice that this chosen wavelength is corresponding to the peak of the absorption cross section, which is proportional to the imaginary part of the polarizability. It is quite different from the plasmonic resonance wavelength of the particle. The index of refraction of gold is complex number, 0.467 + 2.4083i at the wavelength of 532 nm [20,21], where the imaginary part is attributed to optical absorption. The laser power is assumed to be 100 mW. The numerical aperture (NA) is 1.

According to the Eq. (6), simulation results of optical forces in the case of 2π focusing system and 4π focusing system illuminated by RP beams are shown in Figs. 2(a)-2(c) it can be seen that the scattering force is positive all through the focal region and represents a dominant contribution to the total force. The scattering force has a wide and large peak while with weak gradient force near the focus. The nanoparticle will be pushed away from the focal point due to the lack of equilibrium position and no negative force along the optical axis. Accordingly, the optical trapping cannot trap the gold particle stably by the 2π focusing system. Compared with the 2π focusing system, the advantages of utilizing 4π focusing system illuminated by radially polarized beams are shown in Figs. 2(d)-2(f). It can be seen that the 4π focusing system provides lager gradient force as well as extremely small scattering force. The gradient force has a wide and large peak compared with weak gradient force near the focus. There is an equilibrium point at z = 0. The maximum positive total force 16.1pN is at −0.42λ and the maximum negative total force −16.1pN is at 0.42λ.

Along r axis, there is still an equilibrium point at r = 0 for 4π system in Fig. 3, thus a stable 3D optical trap is formed for the gold nanoparticle in both directions even under the resonant condition. In Fig. 2 and Fig. 3, we can see that *the distribution of total force is centrosymmetric and odd along z axis and r axis.*

To understand these results better, the relationship between the characteristics of the optical focal field and the optical forces are further investigated below. Figures 4(a) and 4(d) show the electric field intensity distributions near the focus for two cases: the 2π focusing system and the 4π focusing system, both illuminated by RP beams. Compared with Fig. 4(a), the introduction of 4π focusing system leads to the focal spot smaller, which is due to that the 4π focusing system illuminated by two counter-propagating RP beams with opposite polarization can lead to perfect destructive interference of transverse component in the focal plane and sharpen the axial extend [16, 19].

The total intensity, transverse and longitudinal components in rz plane for 4π focusing system are illustrated in Figs. 4(a)-4(c), respectively. The intensity has been normalized. The maximum of transverse components is only 2.3 × 10^{−3}, far smaller than 1. The focal transverse spot size is about 0.36λ, which is below Abbe’s limit. Thus we gain nearly pure longitudinal field with small transverse spot size 0.36λ. It can be also used for super-resolution imaging, micro-processing and data storage. The total intensity in 2π focusing system is shown in Fig. 4(d). We illustrate both intensity profile along the z axis and r axis in Figs. 4(e)-4(f) respectively for comparison. The intensity has been normalized for the 4π system while the intensity for 2π system is relative to 4π system. The focal peak intensity of the 4π focusing system is larger for both directions. The longitudinal spot size is 2λ and 4λ for the 4π and 2π focusing system respectively. The transverse spot size are both same about 0.36λ.

To better evaluate the optical trapping performance in terms of stability, the potential depth and escape time are also numerically calculated. Traditionally an optical trap with potential depth U larger than k_{B}T or escape time t longer than 1s is considered as stable [15]. According to the calculation method and parameters of the Ref [15], for the case of 4π focusing system illuminated on 50 nm (radius) gold nanoparticle, the potential depth and escape time are 492 × k_{B}T, 6.2 × 10^{183} s in the longitudinal direction, respectively. It is worthy to note that the potential depth and the escape time are much larger than that reported previously [15].

It is well known that it is possible to trap the gold 90nm nanoparticle under 1064nm in experiment study [22] and the research has dealt with trapping nanoparticle (gold 38.2nm) away from the resonant wavelength under 1047nm in theoretical study [10]. Now, we simulate the distributions of the optical total forces on 50 nm (radius) gold nanoparticle under 1047 nm away from the resonant wavelength of 532nm with the proposed method. In Fig. 5, there is still an equilibrium point along r direction and z direction. The maximum positive total force 20.3pN is at −0.42λ and the maximum negative total force −20.3pN is at 0.42λ along z axis, thus a stable 3D optical trap is formed for the gold nanoparticle in both directions under the non-resonant condition.

## 4. Conclusion

In summary, we have theoretically studied optical forces distribution of gold nanoparticle at the wavelength of 532 nm near the focus by means of transmitting two counter-propagating radially polarized beams in a 4π high NA focusing configuration. From numerical simulations we have found that there is a perfect equilibrium point in the both directions in optical tweezers. Thus a stable 3D optical trap is formed for the gold nanoparticle under the resonant condition. The simulation results provide new insights into how to implement optical micromanipulation for nano-sized metal particles, such as Red cells, nanorods, nanowires, nano self-assembly. Besides, we obtain the nearly pure longitudinal field with transverse 0.36λ. This versatile method may open up new avenues for optical manipulation and their applications in other scientific fields, such as super-resolution imaging [23], particle accelerating [24–26], and near-field scanning optical microscopy [27].

## Acknowledgments

This work is supported by the National Basic Research Program (973 Program) of China (No. 2012CB921900), the National Natural Science Foundation of China (NSFC) (No. 11274180), National Nature Science Foundation of China (NSFC) (No. 61505024, 61435003), Fundamental Research Funds for the Central Universities (No.ZYGX2015KYQD015) and Science and Technology Project of Sichuan Province (No.2016JY0102).

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