Gián án nhip quang hoc

The Influence of Parameters of the Dielectric and Fluid
on Usefulness of the Optical Trap
Ho Quang Quy
1
, Hoang Dinh Hai
2
,
Phan Si Chau
3
, Do Ich Tinh
3
, Doan Hoai Son
3
, Mai Van Luu
3
1
Institute of Applied Physics, NEWTECHPRO, VAST, 8 Hoang Quoc Viet, Hano, Vietnam.

2
Pedagogical College of Nghe An, Vietnam.
3
Faculty of Physics, Vinh University, Vietnam.
Abstract: In this article the stable process of the dielectric particle in optical trap is
investigated. The influence of the mechanical parameters as the radius of dielectric
particle, viscosity of surrounding fluid on the usefulness of optical trap using pulsed
Gaussian beam is simulated and discussed.
Key words: Optical trap, Pulsed Gaussian beam, Optical force, Brownian motion.
1. Introduction
In 1970, Ashkin [1] first demonstrated the optical trapping of particles using the
radiation force produced by the focused continuous-wave (CW) Gaussian beam. Since then
the optical trap and tweezers has been a powerful tool for manipulating dielectric particles

[2, 3].
In works [4-7], the discussions about stability of the optical trap and the tweezers as
well as the effectively controlling dielectric particles like gold nanoparticles and live
membrane, have been conducted taking into account of Brownian force. But, the stabilizing
process during the pulsing of the optical beam and the absolutely-stable conditions of
dielectric particles were not clear. Therefore, it is desirable to advance the studies of above
questions for the pulsing optical trap.
Up to now, we have paid attention to optical trap using counter-propagating pulsed
Gaussian beams [8] and after discussing about its stability in general, but not in detail.
So, in this paper the stable process during pulsing time of optical tweezer using pulsed
Gaussian beam and influence of mechanical parameters on its stability and usefulness are
interested. This paper is organized as follows: In Sec. II we introduce the set of Langevin
equations concerning thermal fluctuations of the probe with appearance of the Gradient
optical force acting on dielectric nano particles in the optical tweezer pulsed Gaussian
beams (PGB),; In section III, we present the radial variances of glass nano particles in
water, which are trapped by picoseconds PGB and the discussions about stability of them
and usefulness of tweezer.
2. Theory
Two PGBs with the total energy, U, duration time, τ, and beam waist, W
0
, are
considered to trap fluctuating dielectric nano particles with radius, a, refractive index, n
1
in
the fluid with viscosity, γ, and refractive index n
2
, which is placed at z=0 (Fig.1). We
consider the gradient optical force is induced by PGB acting on a Rayleigh dielectric
particle. The polarization direction of the electric field is assumed to be along the
x

axis.
For simplicity, we assume that the radius (a) of the particle is much smaller than the
wavelength of the laser (i.e.,
λ
<<a
), in this case we can treat the dielectric particle as a
1
point dipole. We also assume that the refractive index of the dielectric particle is
1
n
and
21
nn
>>
.
Assuming a low Reynold's number regime [9], the Brownian motion of the dielectric in
the optical force field is described by a set of Langevin equations as follows:
( ) ( ) ( )
,
2
grad
t F t D h t
ρ
γρ ρ γ
+ =
r
r
r r
&
(1)

where
( ) ( ) ( )
,t x t y t
ρ
=  
 
r
is the dielectric particle’s position in the fluid plate,
6 a
γ π η
=
is
its friction coefficient,
η
is the fluid viscosity,
( ) ( ) ( )
2 2 ,
x y
D h t D h t h t
γ γ
 
=
 
r
is a vector
of independent white Gaussian random processes describing the Brownian force forces,
/
B
D k T
γ

=
is the diffusion coefficient, T is the absolute temperature, and k
B
is the
Boltzmann constant,
ρ
,grad
F

is the gradient optical force acting on dielectric particle,
which is given by [4,8]
( ) ( )
[ ]
002,
/
~
,2
ˆ
, WcntItF
grad
ερρβρρ
ρ
−=

(2)
where is the scattering cross section,
( ) ( )
[ ]
2/14
223

0
2
2
+−=
mman
επβ
is the polarizability,
ρ
ˆ
is the unit radial vector,
0
/W
ρ ρ
=
%
,
0
W
is the waist radius of Gaussian beam at the plane
0z =
,
21
/ nnm
=
, and I is the intensity distribution for the PGB as follows [4]:
[ ] [ ]
22
~
2exp
~

2exp)( tPtI
−−=
ρρ
(3)
where
( )
3/ 2
2
0
2 2 /P U W
π τ
 
=
 
, and
τ
/
~
tt
=
, ,
λπ
/2
=
k
is the wave number.
We compute the two-dimensional motion and the radial variance (position) of a
dielectric particle in fluid using the Brownian dynamic simulation method. A particle/bead-
spring model is employed to represent the dielectric particle, and the following equation of
motion is computed for each particle:

2
Fig. 1 (a) The schematic of optical trap.
(b) The motion with radial variance of
particle in the fluid plate.
Pulsed laser beam
Focus lens
dielectric particle
Plate with Fluid
Radial Variance ρ
a
b
Z
Z=0
X
Y

( ) ( )
( )
( )
( ) ( )
,
2
grad
F t
t t t t t D t h t
ρ
ρ
ρ δ ρ ρ δ δ
γ
+ − = − × × + × ×

r
r
r
r r r
(4)
where
t
δ
is the time increment of the simulation,
( )
h t
r
is a random vector whose
components are chosen from the range [-1, 1] in each time step.
( )
( )
,grad
F t
ρ
ρ
r
r
in Eq. (2)
describes the gradient optical force acting on the particle located at position ρ at time t. For
example, at beginning time t=0, the dielectric particle is assumed to locate at the position
( )
0
0t W
ρ
= =

, where W
0
is the beam waist, then we understand that the gradient optical
force
( )
tWF
grad
,
0,
ρ

acts on the particle, which will be located at position
W
ρ
+ ∆
after a
time increment
t
δ
.
3. Simulated Results
We interest only on the radial variance of glass particle in the pulsing time (this
parameter describes the stability of particle), so the simulation will be computed from
beginning moment t=-3τ (or t=0) to ending moment t=3τ (or t=6τ) of the optical pulse. In
following numerical simulation we choose fixed parameters as follows: The laser beam
with wavelength,
m
µλ
064.1
=

, duration,
ps1
=
τ
, beam waist,
JU
µ
1.0
=
, total energy,
JU
µ
1.0
=
; The glass particle with refractive index,
592.1
1
=
n
, radius,
nma )202(
÷=
;
The fluids have been changed from alcohol methyl with viscosity,
,
0.59 .
alcohol methyl
mPa s
η
=

, to water,
1.00( . )
water
mPa s
η
=
, alcohol ethyl,
,
1.1( . )
alcohol ethyl
mPa s
η
=
, and
,
2.4( . )
alcohol ethyl
mPa s
η
=
. The refractive index of the fluid at temperature of 20
o
C is
determined by [11]
( )
2
2
5 6
4
0 1 2 3 7

2 2 2 2 2
2
1
1/
2
UV IR
a a
a
n
a a a T a T a
n
ρ ρ λ ρ
λ λ λ λ λ
−
= + + + + + + +
+
− −
(5)
where
*
T
T
T
=
,
*
ρ
ρ
ρ
=

,
*
λ
λ
λ
=

0
0.24425773a =
,
1
0.00974634476a =
,
2
0.00373234996a = −
,
3
0.000268678472a =
,
4
0.0015892057a =
,
5
0.00245934259a =
,
6
0.90070492a =
,
7
0.0166626219a = −

,
0.229202
UV
λ
=
,
*
273.15T K=
,
* 3
1000 .kg m
ρ
−
=
,
*
589nm
λ
=
,
IR
5.432937
λ
=
.
Considering the temperature of the fluid is fixed at 20
o
C and mass density is chosen
to be
3

1000 .kg m
ρ
−
=
, and wave length of laser
m
µλ
064.1
=
, using (5) we have n
2
=
1.289.
From curves in Fig.2, which discribes the dependence of the stable radius and
stable time of patricle in trap, we can see that, with fixed optical parameters, the stability of
particle in trap is higher when the radius of particle is shorter. In detail, when trapped
particle has radius of
( )
nm2014
÷
, the stable radius
nm
s
25
≈
ρ
and stable time
ps
s
3.3

≈
τ
. It means that the particle oscillates around its center, and the designed optical
trap is useful with the particle of
( )
nm2014
÷
radius.
3
However, the above-mentioned stable conditions changes in different fluid. From
Fig.3, we can see the dependence of stable radius and stable time on viscosity of fluid. The
same particle, which is mixed in fluid with high viscosity, will be more stable than one,
which is mixed in fluid with low viscosity. It is of course that in the fluid with high
viscosity, the vividity of particle is lower, and then the Brownian force is lower.


In general, to have the stability of dielectric particle in given optical trap, it is
convenient to choose a suitable particle, and to mix it in a suitable fluid.

4. Conclusion
From above results, we can conlude that: First, the stability of dielectric particle in
optical trap depends not only on optical parameters, but also on mechanical parameters as
radius of particle, viscosity of fluid,..; Second, one designed optical trap will be useful with
the suitable particle, which is mixed in suitable fuid.
4
a [nm]
ρ
s [nm]
Fig.2. Left: ρ
s

vs a; Right: τ
s
vs a
.
with: , ,, , T=20
o
C
a [nm]
τ
s[ps]
ρ
s [10
-
8 m
]
γ [mPa.s] γ [mPa.s]
Fig.3. Left: τ
s
vs γ; Right: ρ
s
vs γ.
with: , ,, , T=20
o
C
τ
s[ps]
References
1. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev.
Lett. 24 (1970), pp. 156-159.
2. A. Ashkin, J. M. Dziedzic, J.E. Bjorkholm, S. Chu, “Observation of a Single-beam

Gradient Force Optical Trap for Dielectric Particles,” Opt. Lett. 11 (1986), pp. 288-290.
3. S.C. Kuo, M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2 (1992),
pp. 16-24.
4. H. Q. Quy, M. V. Luu, “Stable Manipulation Dielectric Sphere of Optical Trapping by
two Counter-propagating Gaussian Pulsed Beams,” IWP&A, Nha trang, Sept.10-14,
(2008), pp.181-186.
5. H. Kress, Ernest H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of Relative
Radiation Pressure in Optical Traps: Application to Phagocyte Membrane binding
studies,” Phys. Rev. E 71, 061927 (2005).
6. Y. Seol, A. E. Carpenter, and T.T. Perkins, “Gold Nanoparticles: Enhanced Optical
Trapping and Sensitivity Coupled with Significant Heating,” Opt. Lett., Vol. 31, No. 16
(2006), pp. 2429-2431.
7. G. Volpe, G. Volpe, and D. Petrol, “Brownian Motion in a Non-homogeneous Force
Field and Photonic Force Microscope,” Phys. Rev. E76, 061118 (2007).
8. C. L. Zhao, L. G. Wang, X. H. Lu, “Radiation Forces on a Dielectric Sphere produced
by Highly Focused Hollow Gaussian Beams,” Phys. Let. A (2006), pp. 502-506.
9. J. Happel and H. Brenner, “Low Reybnold Number Hydrodynamics,” Springer, NEW
York, 1983.
10. L.G. Wang et al, “Effect of Spatial Coherence on Radiation Forces acting on a
Rayleigh Dielectric Sphere,” Opt. Lett. 32 (2007), pp. 1393-1395.
5