Tunable optical gradient trap by radial varying polarization Bessel-Gauss beam

doi:10.4236/jbise.2010.33041

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J. Biomedical Science and Engineering, 2010, 3, 304-307

doi:10.4236/jbise.2010.33041 Published Online March 2010 (http://www.SciRP.org/journal/jbise/

JBiSE

Published Online March 2010 in SciRes. http://www.scirp.org/journal/jbise

Tunable optical gradient trap by radial varying polarization

Bessel-Gauss beam

Xiu-Min Gao1,2, Song Hu1, Jin-Song Li3, Zuo-Hong Ding2, Han-Ming Guo2, Song-Lin Zhuang2

1Electronics & Information College, Hangzhou Dianzi University, Hangzhou, China;

2Optics & Electronics College, University of shanghai for Science and Technology, Shanghai, China;

3Optics & Electronics College, China Jiliang University, Hangzhou, China.

Email: [email protected]

Received 28 December 2009; revised 10 January 2010; accepted 12 January 2010.

ABSTRACT

Optical tweezers play an important role in many

domains, especially in life science. And optical gradi-

ent force is necessary for constructing optical tweez-

ers. In this paper, the optical gradient force in the

focal region of radial varying polarization Bessel-

Gauss beam is investigated numerically by means of

vector diffraction theory. Results show that the beam

parameter and vary rate parameter that indicates the

change speed of polarization rotation angle affect the

optical gradient force pattern very considerably, and

some novel force distributions may come into being,

such as multiple force minimums, force ring, and

force crust. Therefore, the focusing of radial varying

polarization Bessel-Gauss beam can be used to con-

struct optical traps.

Keywords: Optical Gradient Force; Bessel-Gauss Beam;

Radial Varying Polarization; Vector Diffraction Theory

1. INTRODUCTION

Optical tweezers technique has accelerated many major

advances in numerous areas of science, especially in life

science, since Ashkin developed optical tweezers system

1980s [1,2,3,4]. Optical tweezers can offer a very con-

venient, noninvasive, and non-contact access to processes

at the microscopic scale [5], and a number of approaches

have been proposed to constructing optical trap, such as

generalized phase-contrast technique and holographic

optical tweezers arrays [6,7]. In optical trapping system,

it is usually deemed that the forces exerted on the parti-

cles in light field include two kinds of forces, one is the

gradient force, which is proportional to the intensity gra-

dient; the other is the scattering force, which is propor-

tional to the optical intensity [8]. Therefore, optical gra-

dient force is necessary for constructing optical tweezers,

and the tunable focal intensity distribution predicts that

the position of optical trap may be controllable.

It is well known that Bessel beams provide valid solu-

tions to Helmholtz equation, and have attracted a lot of

attention [9,10,11] for their non-diffracting property.

And these beams are easily generated external to the

laser cavity by illuminating an axicon with a Gaussian

beam [12]. In this paper, the optical gradient force in the

focal region of radial varying polarization Bessel-Gauss

beam is investigated numerically by means of vector

diffraction theory. The principle of the focusing this

non-spiral vortex Gaussian beam is given in Section 2.

Section 3 shows the simulation results and discussions.

The conclusions are summarized in Section 4.

2. PRINCIPLE OF FOCUSING RADIAL

VARYING POLARIZATION

BESSEL-GAUSS BEAM PRINCIPLE OF

THE FOCUSING GAUSSIAN

According the vector diffraction theory [13,14], the

electric field in focal region of the radial varying polari-

zation Bessel-Gauss beam can be written in the form as,





,, rr zz

ErzEe Ee Ee











(1)

where r



, and e





are the unit vectors in the radial,

azimuthal, and propagating directions, respectively. ,

E, and E



are amplitudes of the three orthogonal

components and can be expressed as

  

,coscos sin2

Erz AP





 





 

1sin expcos

krikz d





 (2)

  

,2coscos sin

Erz iAP





 





 

0sin expcos

krikz d





 (3)

  

,2 sincossinErz AP







 





 

1sin expcos

krikz d





 (4)

X. M. Gao et al. / J. Biomedical Science and Engineering 3 (2010) 304-307

305

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where r and z are the radial and z coordinates of obser-

vation point in focal region, respectively. k is wave

number. Here





is the pupil apodization function

[15],



2sin sin

expPJ

NA NA

 



























(5)



arcsin NA



, which practically indicates the radius

corresponding to each section zone of the cylindrical

vector beam.



is the polarization rotation angle from

radial direction. As the function of convergence angle



, and



is in the form of,







sin









  (6)

where is viable rate parameter that indicates the

change speed of polarization rotation angle. Based on

the optical intensity distribution in focal region, the gra-

dient force trap can be expressed as [1,16],



23 22

1,,

grad

nr m

Er z







 









(7)

where is the radius of trapped particles, is the

refraction index of the surrounding medium, and , the

relative index of refraction, equals to the ratio of the

refraction index of the particle

n to the refraction in-

dex of the surrounding medium . Gradient force

points in the direction of the light intensity gradient

when the diffractive index of particles is bigger than that

of surrounding medium, i.e.

nn. Therefore, the

gradient force pattern can be computed numerically by

substituting Eq. 1 into Eq. 6.

3. RESULTS AND DISCUSSIONS

Without losing generality and validity, it is supposed that

, , and 0.95NA 21





. It should be noted that

in this paper and Vz denote radial and axial coor-

dinates, and the distance is , where is wave

number of the incident BG beam. The intensity distribu-

tion and corresponding optical gradient force pattern for

1k

12.



 and are firstly calculated and illus-

trated in Figure 1. Arrows in this figure indicate the

force direction under condition of the diffractive index

of particles is bigger than that of surrounding medium. It

can be seen that the intensity distribution turns on the

focal ring, as shown in Figure 1(a), which can be used

to construct ring-shape focal trap, given in Figure 1(b).

1.0C

Now the effect of the parameter 1



on optical gradient

is investigated. It is chosen that 1



= 2.5 in the follow-

ing calculation. The corresponding intensity distribu-

-20 -10 010 20

-10

-5

(a)

-20 -10 010 20

-10

-5

(b)

Figure 1. The (a) intensity distribution and

corresponding (b) optical gradient force pat-

tern for 12.0





and , respectively.

Arrows indicate the force direction.

1.0C

tion and gradient force pattern are given in Figure 2. We

can see that the focal ring extends along optical axis, and

there three weak on-axis peaks. From Figure 2(b), it can

be seen that there is one cylindrical crust trap, and mul-

tiple weak traps on axis. Therefore, parameter 1



af-

fects the Bessel-Gauss beam, in turn can alter the optical

gradient force pattern considerably.

In order to get insight into the optical gradient force in

the focal region of radial varying polarization Bes-

sel-Gauss beam more deeply, different is also con-

sidered in calculation. From Figure 3(a), it can be seen

that one optical intensity crust comes into being under

condition of

0.2C



, namely, one local intensity mini-

mum occurs. Figure 3(b) illustrates the corresponding

optical gradient force pattern that is in practice force

crust pattern. So, the parameter can be used to alter

optical gradient force pattern in focal region of the Bessel-

Gauss beam.

In our theoretical investigation, more values of 1



and are studied. And many novel optical gradient

force patterns can occur. This paper only gives several

typical cases. Figure 4 illustrates the intensity distribu-

tion and corresponding optical gradient force pattern-

for

13.5





and 1.0C



. One intensity distorted cylinder

306 X. M. Gao et al. / J. Biomedical Science and Engineering 3 (2010) 304-307

-20 -1001020

-10

-5

(a)

-20 -10010 20

-10

-5

(b)

Figure 2. The (a) intensity distribution and

corresponding (b) optical gradient force pat-

tern for 12.5



 and , respec-

tively. Arrows indicate the force direction.

1.0C

-20 -1001020

-10

-5

(a)

-20 -1001020

-1 0

-5

(b)

Figure 3. The (a) intensity distribution and

corresponding (b) optical gradient force pat-

tern for 12.5



 and 0.2C



, respectively.

Arrows indicate the force direction.

-20 -1001020

-10

-5

(a)

-20 -1001020

-1 0

-5

(b)

Figure 4. The (a) intensity distribution and

corresponding (b) optical gradient force pat-

tern for 12.5





and , respectively.

Arrows indicate the force direction.

1.0C

appears outside of center main center intensity peak. Fig-

ure 4(b) shows that center optical trap comes into being,

and simultaneously, more complicate force pattern also

occur outside of this main trap. From above all optical

gradient evolution process, it can be given that the beam

parameter and vary rate parameter can be used to alter

intensity and corresponding optical gradient force dis-

tributions in focal region of Bessel-Gauss beam re-

markably.

4. CONCLUSIONS

The optical gradient force in the focal region of radial

varying polarization Bessel-Gauss beam is investigated

numerically by means of vector diffraction theory.

Simulation results show that the beam parameter and

vary rate parameter affect the optical gradient force pat-

tern very considerably, and some novel force distribution

patterns may come into being, which indicates that the

focusing of radial varying polarization Bessel-Gauss

beam can be used to construct optical traps.

5. ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of

China (60708002, 60878024, 60778022), China Postdoctoral Science

Foundation (20080430086), Shanghai Postdoctoral Science Foundation

of China (08R214141), and the Innovation Fund Project For Graduate

Student of Shanghai (JWCXSL1002).

X. M. Gao et al. / J. Biomedical Science and Engineering 3 (2010) 304-307

307

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