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OPTICAL FORCE ACTING ON DIELECTRIC PARTICLE
EMBEDDED IN KERR MEDIUM
HOANG VAN NAM
Council of Science and Technology, Hatinh
CAO THANH LE
Hatinh University
HO QUANG QUY
Academy of Military Science and Technology
Abstract. Basing on the necessary condition for the trapping dielecric particle by the
Gaussian beam, the Kerr effect in the tweezer with the nonlinear medium is proposed to
concern. The expressions of the optical forces concerned with the Kerr effect, which affects
the refractive index change of the medium and the self-focusing, are presented. The
distribution of the optical forces in the trapping region is simulated and discussed. The results
show that the stability of the tweezer depends on the nonlinear coefficient of refractive index,
and the optical tweezer could be brokendown with a critical value of the nonlinear coefficient
of refractive index of the surrounding medium, or with a critical value of the laser intensity,
duration of laser pulse, and radius of beam waist. Moreover, these results are used to simulate
the dynamic of the optical tweezer, and consequently to explain the stability of the optical
tweezer used for the trapped object (the biological molecule for example) embedded in the
fluid, which is sensitive with Kerr effect.
Keywords: Optical tweezer, Optical force, Kerr effect, Nonlinear medium, Self-focusing.
I. INTRODUCTION
The optical tweezer is used to trap the micro dielectric particle, which is embedded in a
medium as gas, fluid [1, 2, 3]. Usually, the tweezers in many experiments are conducted using
the CW laser. It is well known that the CW laser with the power of a few milliwatt can only
produce the radiation force with an order of a few pN to manipulate the micro-sized particles
and bological molecule [4]. Recently, Ambardekar et al. [5], Deng et al. [6], Zhao [7] and
Wang et al. [8] used a pulsed laser to generate the large gradient force, up to 2500 pN within
a short duration, about picoseconds. Up to now, the optical trap using pulsed Gaussian beam
and to optical trap using counter-propagating pulsed Gaussian beams has been paid attention
[9,11]. Unfotunately, in previous works, there was attention to thermal effect of the medium
only (the Brownian motion), but not to other effect as random or Kerr. In works [12], the
optical force acting on dielectric particle embedded in the random medium, which causes the
partial coherence of laser beam is discussed. In work [13], the Kerr effect appeared in the
medium or inside the particle, which is irradiated by intense laser pulse is concerned for the
optical tweezer. However, in this work, the self-focusing of the laser beam, which is an
important effect relating to Kerr effect, has been not concerned.
Therefore, in this paper, the influence of the both effects, the Kerr in medium and the selffocusing of the laser beam in optical tweezer on the distribution of optical force acting on
dielectric nanoparticle are investigated. This article is organized as follows: in Sec.2 we
present expressions concering the Kerr effect and self-focusing, and the optical forces acting
on dielectric nano particles in optical tweezer; in Sec.3 we present the influence of the
nonlineart coefficient of refractive index on the optical forces and discuss about the stability
of the tweezer.
II. OPTICAL FORCES
As a example, we consider an optical tweezer to trap a dielectric nano particles in the Kerr
medium (see Fig.1). A pulsed plane laser TEM00 beam focused by a lens, and the electric field
can be approximately expressed by [7]
ˆ 0
E   , z, t   xE
ikW02
exp  i  k  z   0t  
ikW02  2 z
  kW 2 2  2 

2kz  2 
0

  exp  
 exp  i
2
2
2
  kW 2  4 z 2 
  kW0   2 z 
0


(1)
 exp    t  zc  /  2  ,


beam, where W0
2
as a Gaussian
is the radius of the beam waist at z=0,  is the radial
coordinate, x̂ is the unit vector of the polarization along the x direction, k  2 /  is the
wave number,  0 is the carrier frequency,  is the pulse duration, c is the light speed.
After propagating through the Kerr medium, this Gaussian beam will be redistributed by
the self-focusing effect and then it is refocused by the nonlinear lens of focal length, which is
approximately [14]
  n2 
(2)
f nl ( P, nnl )  W02 

 2nnl P 
where n2 , nnl are the refractive index of medium and nonlinear refractive coefficient of the
Kerr medium, respectively, and P is the total power of the input Gaussian beam.
It can be considered that the Kerr medium with distance of f nl plays as the thin lens
with focal length of f nl . Such that, after propagating through the Kerr medium, the Gaussian
beam is modified and the its waist radius is given by [15]
W0' ( P, nnl )  M ( P, nnl )W0 ,
(3)
where M ( P, n nl )  M r / 1  r 2 , r  z0 / ( z ' f nl ) , M r  f nl / z ' f nl , z0   W02 /  is the
Rayleigh range, z’ is the distance from beam waist to thin lens, which has minus sign for
considered optical tweezer in Fig.1.
From (1), (3) and the definition of the Pointing vector, we can readily obtain the
intensity distribution for the light pulse in the tweezer as follows:
  zkW '2 2 
 2 2 
P
0
2  t 
I   , z, t  
exp


exp


2
c

1  4z 2
1

4
z
 


 
~
where z  z / kW0' 2    / W0 ' and t  t /  .
.
2
(4)
Fig.1 Sketch of optical tweezer with Kerr medium.
For simplicity, we assume that the radius (a) of the particle is much smaller than the
wavelength of the laser (i.e., a   ), in this case we can treat the dielectric particle as a point
dipole. We also assume that the refractive index of the dielectric particle is n1 and n1  n2 .
This is the necessary condition for trapping operation of the tweezer using Gaussian beam
[16].
The medium is Kerr, then its refractive index will be
(5)
nm (  , z, t )  n2  nnl I (  , z, t )
and the relative refractive index is given by (considering the particle is not sensitive to Kerr
effect)
mm (  , z, t )  n1 / nm (  , z, t )
(6)
Using (6) and as shown in work [7, 8], we have the scattering cross sections (  ) and
polarizabilities (  ) as follows:
2
 128 5 a 6    m(  , z, t )m  1 

 m (  , z, t )  

4
2
 3
   m(  , z, t )m  2  
2
(7)
and
 m (  , z, t )  4 n (  , z, t ) 0 a
2
m
3
 m(  , z , t )
 m(  , z , t )
2
m
2
m
 1
 2
(8)
To show influence of Kerr effect on the optical force, as example for simplicity, we
present here expressions of the longitudinal force along the beam axis (at   0 ) and
transverse gradient force in the specimen plane (at z=0) when t=0 (at moment laser pulse
reaches peak .
As shown in previous works [9, 10, 13, 17], the transverse gradient force is reduced to:
2 (  ,0.0) I   ,0,0  
Fgrad ,    ,0,0    ˆ m
(9)
cnm (  ,0,0) 0W0 '( P, nnl )
and the longitudinal force is given by:
.
2
2 m (0, z,0) I  0, z,0   zk 2 w0'4 ( P, nnl ) 2 z 1  4 z  


Fz   z

2 2 
nm (0, z,0) 0ckw0'2 ( P, nnl ) 
c 2 2
1  4 z  
(10)

n (0, z,0)
z m
 m (0, z,0) I  0, z,0 
c
Using exps. (2)(10), the distribution of optical forces in speciment plane (  ) and
along beam axis (z) can be simulated.
III. IFLUENCE OF NONLINEAR REFRACTIVE COEFFICIENT
ON OPTICAL FORCES
In
following
numerical
simulation
we
choose
parameters
as: U  10 J ,
W0  2 m ,   10ns ,   0.53m (for SHG of YAG:Nd laser), the refractive index of glass
particle with radius a  20nm is n1  1.592 , and the refractive index of water is n2  1.332 .
Moreover, we considered the water is dissolved by nonlinear material, so their nonlinear
coefficient should be changed.
a)
b)
c)
d)
Figure 2. Distribution of gradient force [N] on radical radius [μm] with different
value of the nonlinear coefficient of the medium:
a) nnl=1.10-11cm2/W; b) nnl=3.10-11cm2/W;
c) nnl=5.10-11cm2/W ; d) nnl=7.10-11cm2/W .
.
The optical gradient force Fgrad ,  and Fz are calculated by expression (9) and (10) in the
ranges:   (2  2) m and z   5  5  m (consider the beam waist of pulsed Gaussian beam
located in the traping plane z=0 and the beam axis goes through point   0 ), respectively.
For the case of Kerr medium, the distributions of the gradient force in the specimen plane
and of the longitudinal force in the beam axis are simulated and shown in Fig. 2 and Fig. 3,
respectively.
This region is more and more large with increasing of the nonlinear coefficient (Figure 2d
and Figure 3d). This phenomenom can be explained by the changing of the refractive index
of the Kerr medium and the self-focusing of the laser beam. The increasing of the nonlinear
coefficient results the increasing of the refractive index of medium, consequently, the
decreasing of the relative refractive
index m  n p / nm reduces the b)
refraction of laser rays
a)
through the particle. It means the trapping forces are reduced.
If the nonlinear coefficient is too large, because of the Kerr effect the Gaussian beam will
be self-focused, since that, the refractive index of the medium inside defined stable region wil
be larger then that of particle. In this case,  ,   0 , consequence F   , where  is the
radial variance of particle [14], the particle will be pushed ouside the centre of tweezer
(Figure.2c,d and Figure.3c,d), and the tweezer will be brokendown.
From the necessary condition for trapping by the Gaussian beam, m  1 , we can derive the
condition for tweezer concerning the Kerr effect as follows:
n2  n1  n2 n I (  , z, t )
(11a)
and critical value of the nonlinear coefficient as:
c)
d)
n 2  n1
3.
Distribution
of
longitudinal
force
(N)
on
beam
axis
(μm)
with different value
nFigure

(11b)
2n
 , z, t )
of theI (nonlinear
coefficient of the medium:
a) nnl=1.10-11cm2/W; b) nnl=3.10-11cm2/W;
c) nnl=5.10-11cm2/W ; d) nnl=7.10-11cm2/W .
We can see that, if the medium is not Kerr or the nonlinear refractive coefficient is small
(nnl1=1.10-11cm2/W), the particle will be trapped in a certain stable region with a radial radius of
about W0 / 2 (Fig. 2a) and a length of about z  6  m (Fig. 3a), and consequently, the
particle is absolutely stable. If the nonlinear refractive coefficient increases, the optical force
decreases and the stable region is larger. Expecially, if the nonlinear refractive coefficient is
large enougth, inside the stable region appears a small unstable region, where the particle
moves along the direction of the force (Fig. 2b,c and Fig. 3b,c). In this sase, we can say that
the particle is partially stable. If the nonlinear refractive coefficient is too high, the trapping
condition is not satisfied, i.e.
n1  n2  nnl I
(11).
It results the optical forces have direction same as that of the particle (Fig.2d and Fig.3d).
So, the particle is fulled ouside from the center of the tweezer, and consequently, the trap will
.
be brokendown.
From Exp.(1) (4), we can see that, the condition (11) depends not on n 2 n only, but also
on energy, U , beam waist, W0 and duration,  of the pulsed Gaussian beam.
IV. CONCLUSION
From above discussions, there are conclusions: First, if the medium is a nonlinear one (the
particle is linear), the Kerr effect reduces the trapping forces, so far brokendown the trapping;
Second, the necessary condition for trapping particle in the nonlinear medium depends not
only on the nonlinear coefficient, but also on quality characters of the pulsed Gaussian beam
as energy, beam wast and duration; Third, with given value of nonlinear refractive coefficient
of the surrounding medium, there are critical value of quality characters of the pulsed
Gaussian beam to keep the optical tweezer stable.
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