## Abstract

The radiation force of highly focused Lorentz-Gauss beams (LG beam) on a dielectric sphere in the Rayleigh scattering regime is theoretically studied. The numerical results show that the Lorentz-Gauss beam can be used to trap particles with the refractive index larger than that of the ambient. The radiation force distribution has been studied under different beam widths of the Lorentz part. The trapping stability under different conditions is also analyzed.

©2011 Optical Society of America

## 1. Introduction

Recently, a new type of optical beam called Lorentz-Gauss beam has attracted a great deal of interest. The existence of Lorentz-Gauss beam is demonstrated both in theory and in experiment. In theory, the Lorentz-Gauss beam is proved a closed-form solution of the paraxial wave equation [1,2]; in experiment, the Lorentz-Gauss beam can be realized by certain double heterojunction lasers [3,4]. The characteristics and applications of Lorentz-Gauss beams have been investigated [5–9].

In 1986, Ashkin reported that optical trapping of dielectric particles by a single-beam gradient force trap was demonstrated for the first time [10]. Since then, this new technology has found wide applications in manipulating various particles such as micro-sized dielectric particles, neutral atoms, cells, DNA molecules, and living biological cells [11–16]. The conventional optical traps or tweezers are constructed mainly by fundamental Gaussian beams. However, many researches have demonstrated that other beams are also useful in trapping particles. The trapping characteristics of different beams, such as Laguerre Gaussian beams [17], hollow Gaussian beams [18], Bessel Gaussian beams [19], cylindrical vector beams [20], Gaussian Schell model beams [21] and flat- topped Gaussian beams [22] have been studied. It has been found that the radiation forces produced by a laser beam are mainly related to its beam characteristics such as beam profile and polarization. In this paper, we investigate the radiation force produced by highly focused Lorentz Gauss beam on a dielectric spherical particle in the Rayleigh scattering regime. By comparing the optical trap of the Lorentz-Gauss beam with that of the conventional Gaussian beam, We find some interesting and useful results.

## 2. Radiation force produced by the Lorentz-Gauss beam

Using the Cartesian coordinate system, whose origin is the center of the beam, we can represent the electric field of the Lorentz-Gauss beam at the input plane as [1]:

*x*and

_{0}*y*is the coordinates in the input plane;

_{0}*w*and

_{x}*w*are beam widths of the Lorentz part;$k=2\pi /\lambda $is the wave number,${q}_{0}=i\pi {w}_{0}^{2}/\lambda $is the radius of curvature,

_{y}*λ*is the wavelength,

*w*is the waist of the Gaussian part;

_{0}*A*is a constant.

_{0}Under paraxial approximation, the electric field of Lorentz-Gauss beam that passes through an ABCD optical system can be expressed as [1,6]:

*z*is the axial distance, $q=(A{q}_{0}+B)/(C{q}_{0}+D)$is the radius of curvature at the output plane, and

*j*stands for

*x*or

*y*, $G=A+B/{q}_{0}$, erf(

*x*) is the error function.We consider the Lorentz-Gauss beam propagating through a lens as shown in Fig. 1 , so the transfer matrix for this system is:

*z*

_{1}is the axial distance between the focus plane and the output plane,

*s*is the axial distance between the input plane and the lens,

*f*is the focal length. Substituting Eq. (4) into Eq. (2), we can get the distribution of the electric field at the output plane. In our paper, we choose

*s*= 100mm,

*f*= 10 mm,

*λ*= 1064nm,

*w*= 10mm,

_{0}*w*=

_{x}*w*, and the input power

_{y}*P*= 1W.

In Fig. 2
, the intensity distribution of the Lorentz-Gauss beam is compared with the fundamental Gaussian beam in the *x* direction. We can see that when the two types of beams possess the same power and *w _{0}*, the maximum intensity of the Lorentz-Gauss beam is greater than that of the Gaussian beam at the input plane; but this relationship reverses at the focus plane.

As is well known, the Rayleigh particle, whose radius is much smaller than the wavelength, can be treated as a point dipole in the light fields. And the polarisability*α*for the point dipole in SI units is [23]:

*a*is the radius of the particle,${\epsilon}_{p}$and${\epsilon}_{m}$are dielectric functions of the particle and the medium surrounding the particle, respectively. So the gradient force ${F}_{grad}$ and the scattering force ${F}_{sca}$ can be calculated by [24,25]:

*a*= 30nm, the refractive index of the particle

*n*= 1.59 (i.e., glass) and the refractive index of the surrounding medium

_{p}*n*= 1.33 (i.e., water), so${\epsilon}_{p}={n}_{p}^{2}$, ${\epsilon}_{m}={n}_{m}^{2}$.

_{m}We plot distributions of the transverse gradient force ${F}_{grad,x}$, the longitudinal gradient force ${F}_{grad,z}$, the scattering force ${F}_{sca}$ exerted on the dielectric particles in Fig. 3
. For simplicity, we only investigate the radiation force distribution in the *x* direction; the force distribution in other transverse directions can be obtained by analogy. From Figs. 3(a), 3(d) and 3(g), we can see that a Rayleigh particle whose refractive index is larger than that of the ambient can be trapped at the focus point by the highly focused Lorentz-Gauss beam, because there are stable equilibrium points in Figs. 3(a) and 3(d). From Fig. 3, we can also see that the force distribution of Lorentz-Gauss beam gradually approaches to that of the fundamental Gaussian beam as *w _{x}* increases: ${F}_{grad,x}$ in the focus plane and ${F}_{grad,z}$ increase with

*w*, but ${F}_{grad,x}$ near the focus plane (

_{x}*z*= 2um) decreases as

_{1}*w*increase. From Figs. 3(e) and 3(f), we can find that the Lorentz-Gauss beam could stably trap the particle at

_{x}*x*= 0.25um, but the fundamental Gaussian beam could not. When

*x*= 0.5um, neither the two beams could trap the particle. So we can conclude that using Lorentz-Gauss beam instead of Gaussian beam, the trapping stability for the Rayleigh dielectric particle will be better in the trapping region except at the focal point. From Figs. 3(g) and 3(h), it is also found that the Lorentz-Gauss beam with small

*w*results in small scattering force, which is of benefit to trapping.

_{x}## 3. Analysis of trapping stability

For stably trapping, the gradient force must be large enough to overcome the scattering force, the Brownian force and the gravity of the particle [10]. The Brownian force, which describes the influence of the Brownian motion, can be calculated by the expression [26]:

where*η*is the viscosity of the ambient, which is 8.0 × 10

^{−4}Pa·s at

*T*= 300K;

*a*is the radius and

*k*is the Boltzmann constant.

_{B}We plot the change of the magnitude of all the forces in Fig. 4
, where ${F}_{grad,x}^{m}$ is maximum transverse gradient force, ${F}_{grad,z}^{m}$ is maximum longitudinal gradient force, ${F}_{sca}^{m}$ is maximum scattering force, ${F}_{b}$ is the Brownian force, ${F}_{g}$ is the gravity. We can see that the gravity of the particle could be neglected comparing with the gradient force. In Fig. 4(a), it is found that the particle can be stably trapped when 1mm<*w _{x}*<20mm under the conditions that

*f*=10 mm,

*a*=30nm,

*w*=10mm; if

_{0}*w*is too small, the Brownian force will be too strong. In Fig. 4(b), it is found that the particle can be stably trapped when 10nm<

_{x}*a*<50nm under the conditions that

*w*=10mm,

_{x}*f*=10mm,

*w*=10mm. When

_{0}*a*is too small, the disturbance is mainly from the Brownian motion; but when

*a*is greater than 27nm, the disturbance is mainly from the scattering force. However, trapping for even larger particles with Lorentz-Gauss beam could not be analyzed by the theory of Rayleigh approximation, so alternative theories must be applied, which deserves our further investigations.

## 4. Conclusion

In summary, we have studied the radiation force on a dielectric spherical particle produced by the highly focused Lorentz-Gauss beam in the Rayleigh scattering regime. We have also investigated the effect of the beam width of the Lorentz part *w _{x}* on the radiation force. It is found that the gradient force at the focus plane increases as

*w*increases, but decreases at the planes near the focus plane. So the Lorentz Gauss beam with a small

_{x}*w*offers advantage over the beam with a great

_{x}*w*and the Gaussian beam while trapping particles at the planes near the focus plane. Finally, the analysis of trapping stability shows that it is necessary to choose suitable

_{x}*w*and radius

_{x}*a*for effective trapping with the Lorentz-Gauss beam. Our results are interesting and useful for particle trapping.

## Acknowledgments

This work was supported by National Nature Science Foundation of China (Grant No. 10974177, 10874012) and the program of International S&T Cooperation of China (Grant No. 2010DFA04690).

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