ZHOU Zhe-hai,TAN Qiao-feng,J IN Guo-fan

(1.State Key Laboratory of Precision M easurement Technology and Instruments,

Tsinghua University,B eijing100084,China;

2.Depa r tment of Optical Info rm ation Science&Technology,B eijing Info rm ation Science&Technology University,Beijing100192,China)

1 Introduction

Opticaltrapping,or optical tweezer[1,2],is a noncontact technique formanipulating microparticles using the radiation pressure force from a tightly focused laser beam,and has proved to be a powerful tool for many applications in numerous areas of science,such as biology[3,4]and colloid chemistry.As a result,many varieties of optical tweezers have been developed.To extend the capability of optical tweezers,some multiple optical trapping schemes have also been proposed relying on very different techniques,including diffractive optical elements[5],interfering beams[6],VCSEL arrays[7],microlens arrays[8]or opticalfibre bundles[9].Certain optical trapping schemes even allow for the generation ofmultiple traps that are computer-reconfigurable using laser scanning[10]and spatial light modulators[11].

In general,single-beam optical tweezers are only used for individual particle trapping.Further,the very limited field-of-view of high numerical aperture objective lenses commonly employed for optical trapping restricts the number and the size of particles that can be trapped simultaneously,while some single-beam optical tweezers with several trapping sites have also been presented[12-14]. In this paper,we purpose a multiple optical trapping scheme based on the single-beam configuration but using higher polarization order axially-symmetric polarized beams in an aplanatic focusing system.We study the high numerical aperture focusing properties of such beams,and calculate numerically the intensity distribution near the focus.We find some unique focusing properties of multi-focus-spot patterns,which provide the possibility ofmultiple 3D optical trapping.Meanwhile,the number and sizes of spots can easily be changed in order to satisfy different applications bymodifying several parameters of the system,such as the polarization order number of the incident beams and the numerical aperture of the lens,which provides a solution formassively parallel trapping of nanometersized particles.In addition,more trapping flexibility is achieved in combination with Diffractive Optical Elements(DOEs). Finally,we present a typical three-dimensional optical chain.

2 Basic theory

Axially-symmetric Polarized Beams(ASPB) are space-variant polarized beams with axial symmetry where the symmetric axis is the propagation axis of the light beam.For an ASPB,as shown in Fig.1,the polarization orientation is the same for two arbitrary axially-symmetric points of the beam profile,SandS′,and the polarization orientation angleΦ (r,φ)of the electric field only depends on the azimuthal angleφ asΦ(r,φ)=Pφ +φ0,wherePis the polarization order number,andφ0is the initial polarization orientation forφ=0.Well-known radially polarized beams and azimuthally polarized beams are axially-symmetric polarized beams with polarization order 1.

Fig.1 Polarization orientation of an ASPB.

Fig.2 Focusing of an ASPB.In the diagram,fis the focal length of the focusing lens,S(rs,φs,zs)is an observation point near the focusing plane.

Fig.2 shows the focusing of an ASPB. The incident field is an ASPB,which is assumed to have a planar phase front andfis the focal length of the focusing lens.S(rs,φs,zs)is an arbitrary observation point near the focus,φsdenotes the az imuthal angle with respect to thex-axis,andθrepresents the polar angle. Following the theory of Richards&Wolf[15],the electric field at the pointScan be written as

whereer,eφandezare unit vectors in the radial,azimuthally and longitudinal directions,respectively.Er,EφandEzare the amplitudes of the three orthogonal components that can be expressed as

wherel(θ)is the pupil apodization function that denotes the relative amplitude and phase of the field,andkis the wavelength number.θmaxandθminare the maximum and minimum polar angles deter mined by the numerical aperture of the objective lens.

Based on above equations,we can calculate the intensity and amplitude distributions corresponding to different components aswell as the total field near the focus for different polarization orderASPBs.

3 Numerical simulations

Fig.3 Intensity distribution of total field for different polarization order numbers at focus(the left column)and through focus(the right column).

We calculate numerically the intensity distribution of the total field near the focus for the ASPB with different polarization order numberP.Fig.3 shows the intensity distribution for NA=0.90 andP=4 and 10 respectively,where we assume the refractive indexnof the medium and the incident wavelength are 1. Obviously,the focusing field presents a multi-focus-spot pattern which is different from that of radially polarized beams. The number of focal spots is related to the polarization order number as 2×(P-1).The multi-focus-spot property provides the possibility of multiple parallel manipulations of particles,such as trapping,rotation,and acceleration.

Fig.4 Gradient force at focus form=4,φ0=0.

When the trapped particle ismuch smaller than the wavelength of the trapping lasers,i.e.,r?λ,the conditions for Raleigh scattering are satisfied,and the scattering and gradient force components are expressed as[2],

whereI0is the intensity of the incident beam,nmis the refractive index of the medium,ris the radius of the particle,cis the speed of light in vacuum andmis the ratio of the refractive index of the particle to the refractive index of the medium(np/nm). The scattering forceFscattis in the direction of propagation of the incident light and is proportional to the intensity,while the gradient forceFgradis proportional to the intensity gradient and is parallel to the increasing gradientwhenm＞1.Fig.4 shows numerically the normalized gradient force on the particle corresponding tom=4,and we choosenp=1.59,λ=1,r=0.1λ.

In addition,we can use theDOEs to control the 3D focusing field distribution near the focus more freely,such as a three-d imensional optical chain.The transmission function of the DOE is expressed as,

where,aiandφiare the trans mission efficiency and phase of theith belt of the DOE respectively.We use a five-belt structure as

whereθmaxis the maximal polar angle determined by the NA of objective lens.Fig.5 presents the field distribution of the three-dimensional optical chain through the focus.It can carry outmultiple 3D optical trapping.

Fig.5 Intensity distribution of designed optical chain through focus.

4 Conclusions

We demonstrate a multiple 3D optical trapping scheme based on higher polarization order ASPBs,which overcomes the traditional limits of single-beam configuration on the number and size of trapped particles because of the l imited field-of-view of high numerical aperture objective lenses.We can easily change the number and sizes of spots by modifying the polarization order number of incident beams and numerical aperture of the lens. In addition,some interestingmultiple optical trapping structures can be produced in combination with a diffractive optical element,such as a three-dimensional optical chain,which provides a solution formassively parallel trapping of nanometer-sized particles.

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