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.
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
e

,
z
e

, and e


are the unit vectors in the radial,
azimuthal, and propagating directions, respectively. ,
r
E
z
E, and E

are amplitudes of the three orthogonal
components and can be expressed as
  
12
0
,coscos sin2
r
Erz AP


  

 
1sin expcos
J
krikz d

  (2)
  
12
2
0
,2coscos sin
z
Erz iAP


  

 
0sin expcos
J
krikz d

  (3)
  
12
0
,2 sincossinErz AP



  

 
1sin expcos
J
krikz d

  (4)

X. M. Gao et al. / J. Biomedical Science and Engineering 3 (2010) 304-307
305
JBiSE
where r and z are the radial and z coordinates of obser-
vation point in focal region, respectively. k is wave
number. Here

P

is the pupil apodization function
[15],

2
12
1
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
sin
C




  (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],
C

23 22
2
1,,
22
b
grad
nr m
F
Er z
m



 




(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
rb n
m
p
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.
b
n
p
b
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
1
A
, , and 0.95NA 21


k
. 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
Vr
0
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
0
5
10
Vr
Vz
(a)
Vz
-20 -10 010 20
-10
-5
0
5
10
Vr
Vz
(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
C
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.
C
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
C
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
0
5
10
V
z
Vr
(a)
Vz
-20 -10010 20
-10
-5
0
5
10
Vr
Vz
(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
0
5
10
Vr
Vz
(a)
Vz
-20 -1001020
-1 0
-5
0
5
10
Vr
Vz
(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
0
5
10
Vz
Vr
(a)
-20 -1001020
-1 0
-5
0
5
10
Vz
Vr
(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).

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307
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