## Abstract

Chu *et al*. constructed a kind of Ince-Gaussian modes (IGM)-based vortex array laser beams consisting of *p* x *p* embedded optical vortexes from Ince-Gaussian modes, *IG ^{e}_{p,p}* modes [Opt. Express

**16**, 19934 (2008)]. Such an IGM-based vortex array laser beams maintains its vortex array profile during both propagation and focusing, and is applicable to optical tweezers. This study uses the discrete dipole approximation (DDA) method to study the properties of the IGM-based vortex array laser tweezers while it traps dielectric particles. This study calculates the resultant force exerted on the spherical dielectric particles of different sizes situated at the IGM-based vortex array laser beam waist. Numerical results show that the number of trapping spots of a structure light (i.e. IGM-based vortex laser beam), is depended on the relation between the trapped particle size and the structure light beam size. While the trapped particle is small comparing to the beam size of the IGM-based vortex array laser beams, the IGM-based vortex array laser beams tweezers are suitable for multiple traps. Conversely, the tweezers is suitable for single traps. The results of this study is useful to the future development of the vortex array laser tweezers applications.

© 2013 Optical Society of America

## 1. Introduction

After Ashkin showed the possibility of trapping a particle by focusing a light beam, the manipulation of particles with optical tweezers has been extensively discussed by many researchers, including topics such as particle acceleration, two-beam trapping [1] and levitation trapping [2]. The capability to control micro/nano-sized particles makes optical tweezers an important tool, and optical tweezers have been widely applied in regard to levitated nonspherical particles [3], biological cells [4], etc. Recently, multiple traps have also been discussed, for example, using evanescent waves formed at interfaces by counter-propagating laser beams to trap multiple particles [5], and using an inclined dual-fiber optical tweezers setup to create multiple optical traps [6].

Optical vortices possess several special properties, including carrying optical angular momentum (OAM) and exhibiting zero intensity. As a result, vortex laser beams are widely used as optical tweezers [7, 8], and in the study of the transfer of angular momentum to micro particles or atoms [9–12]. Recently, Chu *et al*. construct a kind of IGM-based vortex array laser beams consisting *p* x *p* embedded optical vortexes from Ince-Gaussian modes, *IG ^{e}_{p,p}* modes [13]. The vortex laser beams can be generated by using a Dove prism-embedded unbalanced Mach-Zehnder interferometer, including a variable phase retarder [13]. Such an IGM-based vortex array laser beams maintains its vortex array profile during both propagation and focusing, and is applicable for use as optical tweezers.

The discrete dipole approximation (DDA) method has been developed for a long time; it is a good approximation method to describe the electromagnetic fields scattered by a nanometer/micrometer-sized particle with arbitrary shape [14–21]. This study uses the DDA method to study the properties of IGM-based vortex array laser tweezers while it traps dielectric particles. This study finds the resultant force on the spherical dielectric particles of different sizes situated at the IGM-based vortex array laser beam waist. This study assumes the material of the dielectric trapped particle to be silicon, and the surrounding medium of the trapped particle is water. Numerical results show that the number of trapping spots of a structure light (i.e. IGM-based vortex laser beam), depends on the relation between the trapped particle size and the structure light beam size. While the trapped particle is small compared to the beam size of the IGM-based vortex array laser beams, the IGM-based vortex array laser beams tweezers are suitable for multiple traps. On the contrary, the tweezers are suitable for single traps.

The paper is organized as following. Section 2 briefly reviews the IGM-based vortex array laser beams, while section 3 addresses the main concept of the DDA method and shows how this paper estimates DDA simulation code precision using Mie’s theory. Section 4 shows the numerical results of this study, i.e. the properties of the IGM-based vortex array laser tweezers. Section 5 gives a more in-depth discussion on the IGM-based vortex array laser tweezers. Lastly, section 6 gives a brief conclusion.

## 2. IGM-based vortex array laser beams [13]

An IGM-based vortex array laser beam is a combination of two Ince-Gaussian (IG) beams, which both have the same mode, but with one rotating by 90 degrees. There is a π/2 relative phase delay between two IG beams [13]. The expression of any order IGM-based vortex array laser beams is given by:

where*IG*is the electric field distribution of the even Ince-Gaussian mode, and the degree and order are both

^{e}_{p,p}*p*. The notation [ ]

^{T}denotes a transpose operation, i.e. the field in the square bracket rotates by 90 degrees. The expression for even Ince-Gaussian modes (IGMs) that propagate along the z axis is written as [22, 23]:

**r**= (

*𝜉*,

*𝜂*,

*z*) is in the elliptic coordinate system, with x =

*f*(

*z*) cos

*𝜉*cos

*𝜂*, y =

*f*(

*z*) sin

*𝜉*sin

*𝜂*, and

*𝜉*∈ [0, ∞],

*𝜂*∈ [0, 2π].

*f*(

*z*) =

*f*

_{0}

*w*(

*z*)/

*w*

_{0}is the semifocal separation of IGMs defined as the Gaussian beam width, where

*f*

_{0}and

*w*

_{0}are the semifocal separation and the beam width at the

*z*= 0 plane, respectively. Here, the IG beam waist is situated at the

*z*= 0 plane.

*w*(

*z*) =

*w*

_{0}(1 +

*z*

^{2}/

*z*

_{R}

*)*

^{2}^{1/2}is the beam width, where

*z*

_{R}=

*kw*

_{0}

^{2}/2 describes the Rayleigh length, and $k$ is the wave number of the beam. The superscript

*e*refers to even modes, and the term

*C*is the normalization constant.

*C*(., ε) is the even Ince polynomial [23] of degree

_{p}^{m}*m*, order

*p*, and ellipticity parameter

*ε*, respectively. Note that the parameters of ellipticity parameter

*ε*, semifocal separation

*f*

_{0}and beam width

*w*

_{0}are not independent, but are related by

*ε*= 2

*f*

_{0}

^{2}/

*w*

_{0}

^{2}. In Eq. (2),

*r*is the radial distance from the mode optical axis, and

*R*(

*z*) =

*z*+

*z*

_{R}

^{2}/

*z*is the curvature radius of the beam phase front, and

*ψ*= arctan(

_{z}*z*/

*z*

_{R}). Figures 1(a), 1(b), and 1(c) show the amplitude distributions of

*ε*= 10 IG beams and the corresponding IGM-based vortex array laser beams with mode order

*p*= 2,

*p*= 4, and

*p*= 6, respectively.

## 3. Approach to calculate optical force of IGM-based vortex array laser tweezers

#### 3.1 Discrete-dipole approximation (DDA) method [14–21]

The DDA method has been developed over a long period of time; it is a good approximation method to describe the electromagnetic fields scattered by a particle with arbitrary shape. In the DDA method, a particle trapped in optical tweezers is decomposed into many dipoles at first. After that, the DDA method calculates the force of each dipole. The DDA method then finds the sum of the force on each dipole; the sum of the forces on each dipole gives the total force acting on the trapped particle. Suppose that the particle is decomposed into $N$ dipoles, the dipole moment ${P}_{i}$ of the dipole $i$ is expressed by [17, 18]:

where ${E}_{tot,i}$ and ${E}_{inc,i}$ are the total and incident electric field at the place of dipole $i$, ${\alpha}_{i}$ is the polarizability of the dipole $i$, ${G}_{ij}$ is the dyadic Green’s function [15, 17].Substituting Eq. (3) into Eq. (4) gives

where ${A}_{ii}={\alpha}_{i}^{-1}$, ${A}_{i,j\ne i}={G}_{ij}$. Equation (5) can be solved by just multiplying the inverse of $A$ on both sides, but this approach costs a great deal of time. Previous results [18, 19] suggested how to solve Eq. (5) in a quick way using a fast Fourier transform (FFT) technique and conjugate-gradient (CG) algorithm. This study adopts this fast approach. For a harmonic field, time averaging of the resultant force of a dipole in the electromagnetic field is written as [16]:*> can be separated into two components, incident force <*

_{i}**F**

*> and scattering force <*

_{inc,i}**F**

*>:where <*

_{sca,i}**F**

*> due to the gradient of the incident field and <*

_{inc,i}**F**

*> due to the fields radiated by all other dipoles except dipole $i$. The approaches to find the values of <*

_{sca,i}**F**

*> and <*

_{inc,i}**F**

*> can be founded in Ref [17].*

_{sca,i}#### 3.2 Relations between cutting dipole number in the DDA method and simulation precision

This study drafted DDA simulation codes and performed all of the simulations using MATLAB software [24]. Before using the drafted DDA code for this study, it was important to know the relation between the cutting dipole number in the DDA method and the simulation precision of the DDA calculated resultant force on the particle. Here, the “cutting dipole number” denotes the lattice number of the particle-decomposing space. To solve the problem, this study referred to the Lorentz-Mie scattering theory [25, 26], which gives the exact solutions of “*resultant force on a spherical dielectric particle due to an incident plane wave*.*”* This study used the drafted DDA code to simulate the same situation, and then compared the DDA results with the exact solutions of the Lorentz-Mie scattering theory. It gave us the estimations of how many *cutting dipole numbers* should be used in the coming simulations of section 4.

The calculation of the Lorentz-Mie scattering theory is briefly summarized here. In the Lorentz-Mie scattering theory, the total incident force <**F*** _{inc}*> and the total scattering force <

**F**

*> acting on the particle are given by:*

_{sca}*C*

_{ext}and

*C*

_{sca}are

*extinction*and

*scattering cross section,*respectively. Their value are given by

*C*

_{ext}= π

*a*

^{2}

*Q*

_{ext}and

*C*

_{sca}= π

*a*

^{2}

*Q*

_{sca};

*a*is the radius of the sphere, and the

*extinction efficiency*${Q}_{ext}$ and the

*scattering efficiency*${Q}_{sca}$ are related by the

*radiation pressure efficiency*${Q}_{pr}$, written by [25]:where $g\equiv \u3008\mathrm{cos}\theta \u3009$ indicates the

*scattering asymmetry parameter*[6].

Here, we calculated “*resultant force on a spherical dielectric particle due to an incident plane wave*” using both the drafted DDA code and the Lorentz-Mie scattering theory. Comparing the results from two approaches gave us “the relation between the *particle cutting rate* and the *relative error* of the DDA calculations”. Figure 2(a) shows the extinction efficiency *Q*_{ext} at different *size parameter x*( = 2*πa*/*λ*), where *λ* is the incident plane wave wavelength. Figure 2(b) shows the relative error of DDA-calculations in *Q*_{ext} at different size parameter *x*. Figure 2(c) shows the scattering efficiency times the asymmetry parameter, *gQ*_{sca}, at different *x*. Figure 2(d) shows the relative error of DDA-calculations in *gQ*_{sca} at different *x*. In Fig. 2, the black dotted line shows the results of the Lorentz-Mie scattering theory. The red, blue, purple and green dotted lines show the results of DDA calculations with cutting dipole number N = 512, 4096, 13824 and 32768, respectively. In Fig. 2, when N = 4096, the error in the extinction efficiency *Q*_{ext} and the error in the scattering efficiency times the asymmetry parameter, *gQ*_{sca}, are both smaller than 4%. When N = 32768, both errors are smaller than 2%. This study used *particle-decomposing space of (lattice) number* N = 32768 in all of the following DDA calculations to ensure that the relative errors of all results are smaller than 2%. It is because the accuracy of the calculated values, *Q*_{ext} and *gQ*_{sca}, of three situations (i.e., with cutting dipole number N = 4096, N = 13824, and N = 32768) are both very high, three curves in both Figs. 2(a) and 2(c) are almost overlapped and are hard to be visually distinguished.

## 4. Numerical results and discussions

This study used the DDA method to calculate the resultant force on the different-size silicon particles from an incident IGM-based vortex array laser beam while the surrounding medium is water. The parameters used in simulations are addressed as follows. The laser beam wavelength *λ* is 1.064μm, beam waist *w*_{0} is 1.0μm. In all of the simulations, the vortex array laser beams propagate along the *z*-axis with linear polarization along *x*-axis. The particles are silicon spheres with refractive index *n* = 1.59, and the surrounding medium is water with refractive index *n* = 1.33. Sections 4.1 and 4.2 show the simulation results of several different-size particles trapped by *p* = 2 and *p* = 4 IGM-based vortex array laser tweezers, respectively. This study uses the symbol *a* to denote the trapped particle radius. The following numerical results of section 4 are all shown in a 6μm x 6μm window situated at the beam waist *x*-*y* plane.

#### 4.1 p = 2 Vortex array laser beam

Figure 3 shows the simulation results of several different-size silicon particles trapped by *p* = 2 IGM-based vortex array laser tweezers while the surrounding media is water. Figure 3(a) shows the intensity distribution of the *p* = 2 vortex array laser beam. The trapped particle radius in results, in Figs. 3(b)–3(f), are 0.1µm, 0.3µm, 0.5µm, 0.7µm and 1.0µm, respectively. In Figs. 3(b)–3(f), the color background plots the distributions of the normalized absolute value of the resultant force on the trapped particle with particle transverse *x*-*y* position, where the force values are normalized by the resultant peak force value in the calculated region. The dark arrows in Figs. 3(b)–3(f) show the *vector of resultant force* on the particle while the particle is situated at the position of the arrow tail.

In Figs. 3(b)–3(d), for the most part of the resultant force vectors, point to five spots, i.e. the bright spot position of Fig. 3(a). The results imply that particles can be trapped at the five bright spots of the *p* = 2 IGM-based vortex array laser beam while the trapped particle radius are smaller than 0.5µm. In Fig. 3(e), the most part of the resultant force vectors still point to five spots, but the resultant forces on the particle in the region between the center trapping spot and outer four trapping spots are relative weak. Besides, the center trapping spot is much larger than the four outer trapping spots. That is, the probability that the particle be trapped in the center trapping spot is much higher than the particle be trapped at four outer trapping spots as the particle radius is 0.7µm. However, in Fig. 3(f), the resultant forces on the particle only point to the center of the vortex array laser beam. The results shows that the *p* = 2 IGM-based vortex array laser tweezers tend to trap particles at the beam center when the trapped particle has a large radius *a.* In this situation (i.e. *a* = 1.0µm), the IGM-based vortex array laser tweezers behave likes a conventional linear polarized lowest-order Gaussian beam tweezers, i.e. the tweezers tend to trap particles at the beam center.

Note that while the particle radius changes from 0.1μm to 1.0μm, the number of trapping spots of a *p* = 2 IGM-based vortex array laser tweezers changes from five into one. The “number of trapping spots” means the number of regions that particle could be trapped by the vortex array laser beams. Note that the number of trapping spots of a vortex array laser beam does not suddenly change at a specific particle radius. Figure 3 shows that, as the trapped particle size is increasing, the resultant force vectors gradually change their directions from “toward multiple trapping spots” to “toward single center trapping spot.” Firstly, the results shows that *the number of trapping spots of a structure light (here, p = 2 IGM-based vortex laser beam), depends on the trapped particle size.* Fig. 3 also shows that *the particle-trapped position of a same structure light is dependent on the trapped particle size*. It implies that once we put some different-size particles into the IGM-based vortex laser beam tweezers, only the smaller particles can be trapped at the positions apart from the beam center. Secondly, Fig. 3 also shows that “*different-size particles suffer different resultant force distributions while being situated under a same IGM-based vortex laser beam field.*” Both interesting behaviors of the vortex array laser beam can be applied to the particle-separation microfluidics system. To integrate the vortex array laser beam with the microchip fluidics system [27–29] can construct an optical sorting system to separate the different-size particles. Note that to build a continuous or discontinuous particle-sorting system, vortex array laser beams have to be integrated with microchip fluidics system. Otherwise, the vortex array laser beam can only separate smaller particles at the outer trapping spots. The detail design of the vortex-array laser beam-based sorting system will be detailed in other place in the near future.

#### 4.2 p = 4 Vortex array laser beam

Figure 4 shows the simulation results of several different size silicon particles trapped by IGM-based vortex array laser tweezers while the surrounding media is water. The format of the Fig. 4 is same as the Fig. 3. Figure 4(a) shows the intensity distribution of the *p* = 4 vortex array laser beam. The trapped particle radius in results, in Figs. 4(b)–4(f), are 0.1µm, 0.3µm, 0.5µm, 1.0µm and 1.5µm, respectively.

In Fig. 4(b), the resultant force vectors point to nine bright spots on the square structure near the beam center and four bright spots outside the square structure of the *p* = 4 IGM-based vortex array laser beam, shown in Fig. 4(a). That is, the trapped particles tend to be trapped at the nine bright spots on the square structure and the four bright spots outside the square structure of the *p* = 4 GM-based vortex array laser tweezers. It implies that the tweezers can trap at least thirteen particles when the trapped particle radii are smaller than 0.1µm. In Fig. 4(c), the resultant force vectors also point to the thirteen spots, but part of resultant force become weaker, i.e. the trapping forces of some particle-trapped spots become weaker when the trapped particle radius is 0.3µm. In Fig. 4(d), the resultant force vectors inside the beam square structure point to the center of the square structure. It shows that the number of trapping spots of the tweezers is now only five, while the trapped particle radii are 0.5µm. The situation of Fig. 4(e) is similar to Fig. 3(e). As the particle size growing larger (i.e. *a* = 1.0µm), though the particle can still be trapped at five particle-trapped spots, the possibility for the particle to be trapped at the outer four particle-trapped spots become small. The particle of radius 1.0µm tends to be trapped at the beam center. In Fig. 4(f), the results shows that the *p* = 4 IGM-based vortex array laser tweezers can only trap particles at the beam center when the trapped particle have a large radius: *a* = 1.5µm. Similarly, in this situation (i.e. *a* = 1.5µm), the *p* = 4 IGM-based vortex array laser tweezers behave like a conventional linear polarized lowest-order Gaussian beam tweezers, i.e. the tweezers tend to trap particles at the beam center.

Note that while the particle radius is changing from 0.1μm to 1.5μm, the number of trapping spots of a *p* = 4 IGM-based vortex array laser tweezers is changing from thirteen to one. The result is similar to what we observed from the Fig. 3, i.e. the number of spots that the particle will be trapped by a *structure light* (here, *p* = 4 IGM-based vortex laser beam) is related to the size of the particle. Both Figs. 3 and 4 show that the vortex array laser beam tend to trap multi-particles at the beam bright spots while the trapped particles are small, and tend to trap particle at the beam center while the particle grow larger. It also implies that *the particle-trapped spots of a structure light are dependent on the trapped particle size*. Similarly, the property of the *p* = 4 vortex array laser beam can be applied to do the optical sorting in the near future.

## 5. Discussion

#### 5.1 The change in the ratio between the scattering force and the incident force

The variation in the number of trapping spots from the change of the trapped particle size is discussed as follows. Figure 5 plots the relation between the ratio of peak scattering force to peak incident force on the trapped particle from two kinds of vortex array laser tweezers, F_{sca,max}/F_{inc,max}, with the trapped particle radius *a*. As the particle radius increases, the ratio F_{sca,max}/F_{inc,max} increases, and the change in the ratio F_{sca,max}/F_{inc,max} is at a great scale. It means that when the ratio of the particle radius to the beam waist *a/w*_{0} increases, the portion of incident field that interacts with the particle will increase, i.e. the incident field will be strongly scattered by the trapped particle. Equation (7) shows that total resultant force on a trapped particle is the sum of the incident force and the scattering force. At the same time, Fig. 5 shows that when the trapped particle radius is 10% of the beam waist, the ratio F_{sca,max}/F_{inc,max} is smaller than 0.01., i.e. in this situation, most of the resultant force on the trapped particle is from the incident force. In means that the force acting on the particle is very close to the gradient of the incident light field in this situation. Thus, while trapping small particles (the value *a/w*_{0} is small) by IGM-based vortex array laser tweezers, the particle-trapping position will be the bright spots of the IGM-based vortex array laser beams. Since there are multiple bright spots in the IGM-based vortex array laser beam distribution, the vortex array laser tweezers are suitable as multiple traps for small particles. On the contrary, when the trapped particle radius becomes larger, the ratio F_{sca,max}/F_{inc,max} grows larger, i.e. the resultant force on the trapped particle will gradually depart from the gradient of the incident light field. Simulation results of section 4 show that as the trapped particle radius increases, the number of trapping spots of the vortex array laser tweezers will decrease. Finally, the vortex array laser tweezers can only trap particles at the beam center, which is similar to the conventional linear polarized lowest-order Gaussian beam tweezers.

#### 5.2 The influence on the resultant force of a particle while an additional particle exists

Whether the vortex array laser beam can trap several particles at the same time is closely related to a question. That is, if the existence of other trapped particles will violate the resultant force distributions that section 4 calculates to a large degree? To solve the question, this study compares the resultant force on a particle of two situations. One situation is there is only one particle situated at the vortex array laser beam waist, and the other situation is an additional trapped particle also exists except the first particle. Figures 6(a), 6(b), and 6(c) show the resultant force on a particle of different x-position along the x-axis, while two particles radii are (*a*_{1} = 0.1 μm, *a*_{2} = 1.5 μm), (*a*_{1} = 0.1 μm, *a*_{2} = 1.0 μm), and (*a*_{1} = 0.3 μm, *a*_{2} = 1.0 μm), respectively. Symbols *a*_{1} denotes the radius of the particle that we calculate resultant force, and symbols *a*_{2} denotes the radius of the additional trapped particle. The optical beam used in Fig. 6 is the *p* = 4 Vortex array laser beam. With the discussions of section 5.1, we knew that the scattering force/field from a large particle is much noticeable than a small particle. Thus, this study lets the additional trapped particle of large radius (i.e., *a*_{2}). And the large trapped particle can only be trapped at the beam center, as section 4 obtains. Thus, Fig. 6 investigates into that *will the existence of a large trapped particle frustrate the trapping of the smaller particle at the outer trapping spots.* The horizontal axis of Fig. 6 starts from the closest x-position that the smaller particle may exist.

All results of the Fig. 6 show that *the existence of a large particle at the beam center will not frustrate the trapping of the smaller particle at the outer trapping spots.* The green arrows in Fig. 6 indicate the original trapped location of the smaller particle while there is only the small particle under the vortex array laser beam. Comparing Figs. 6(a) and 6(b) shows that the existence of an additional large trapped particle may only shift the trapped location of other small particle to a little degree. And a lager additional particle leads to more influence on the small particle than a smaller additional particle. The results are not surprised, since the scattering field of large particle is larger than a small particle. Besides, as what we already knew, the scattering force/field from a large particle is much larger than a small particle. Thus, Figs. 6(a) and 6(b) have shown enough proof that, *no matter the additional trapped particle at the beam center is large or small, the vortex array laser beam still can trap smaller particle at the outer trapping spots at the same time.* Fig. 6(c) is just an auxiliary proof, which shows that above statements will not be violated as increasing the size of trapping particle at the outer trapping spots. The results of this section also implies that the resultant force distributions this study find (i.e., Figs. 4 and 5) are of great value for the reference in situation that trapping multiple particles by vortex array laser beams at the same time.

#### 5.3 The similarity of as the dimension scaling

This study further checks the properties of the vortex array laser tweezers while changing the dimension of particle radius *a* and beam waist *w*_{0}. It is interesting to learn that while keeping the ratio *a/w*_{0}, the resultant force distribution on the trapped particle is almost the same for all IGM-based vortex array laser tweezers. For example, Fig. 7 shows two resultant force distributions of *p* = 4 IGM-based vortex array laser tweezers. In two groups of simulations, “Figs. 7(a) and 7(b)” and “Figs. 7(c) and 7(d)”, we keep the laser beam waist and trapped particle radius different, but keep the ratio *a/w*_{0} = 0.1 and the ratio *a/w*_{0} = 0.3, respectively. In Figs. 7(a)–7(d), the particle radius and beam waist are: (*w*_{0} = 1.0µm, *a* = 0.1µm), (*w*_{0} = 10µm, *a* = 1µm), (*w*_{0} = 1.0µm, *a* = 0.3µm) and (*w*_{0} = 3.0µm, *a* = 0.9µm), respectively. The resultant force vector (dark arrows) in Figs. 7(a) and 7(b) are almost same. Also, the resultant force vectors (dark arrows) in Figs. 7(c) and 7(d) are almost same. In other words, as in discussing the property of the IGM-based vortex array laser tweezers, the ratio *a/w*_{0} is the key parameter, i.e. the number of trapping spots of IGM-based vortex laser tweezers is closely dependent on the relation between the trapped particle size and the trapping beam size.

## 6. Conclusions

This study uses the discrete dipole approximation (DDA) method to study the property of the IGM-based vortex array laser tweezers. This study calculates the resultant force on the spherical dielectric particles of different sizes situated at the IGM-based vortex array laser beam waist. Firstly, this study finds that the number of trapping spots of the IGM-based vortex array laser tweezers is dependent on the ratio of the trapped particle radius *a* to the trapped beam waist *w*_{0}. While the trapped particle is small compared to the beam size of the IGM-based vortex array laser beams, the IGM-based vortex array laser beam tweezers are suitable for multiple traps. Contrarily, the tweezers are suitable for single traps. For example, the results of this study show that the *p* = 2 IGM-based vortex array laser tweezers are capable of multiple trapping when the particle radius is smaller than 0.5μm as the beam waist of the Ince-Gaussian beams is 1.0µm. The *p* = 4 IGM-based vortex array laser tweezers are capable of multiple trapping when the particle radius is smaller than 0.5μm as the beam waist of the Ince-Gaussian beams is 1.0µm. On the contrary, the optical IGM-based vortex array laser tweezers is suitable for single traps as the particle’s radius is larger than 1.0μm and 1.5μm as the IGM order are *p* = 2 and *p* = 4, respectively. The results indicate that while trapping different-size particles with the same IGM-based vortex laser tweezers, the number of trapping spots is related to the particle radius. It also means that the particle-trapped position of the IGM-based vortex laser tweezers is related to the trapped particle size. Secondly, this study finds that “different-size particles suffer different resultant force distributions while being situated at a same IGM-based vortex laser beam field.” To integrate the vortex array laser beam with the microchip fluidics system can construct an optical sorting system to separate the different-size particles. We currently apply the IGM-based vortex laser tweezers to particle-separation microfluidics systems, and the study results will be presented in the near future.

## Acknowledgments

This work was supported by the Advanced Optoelectronic Technology Center, National Cheng Kung University, under projects from the Ministry of Education and the National Science Council (NSC 99-2112-M-006-007-MY3, and NSC 102-2112-M-006 −006 -MY3) of Taiwan.

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