In this article the gradient force of optical trap using two counter-propagating pulsed Gaussian beam and the Brownian motion in optical force field are investigated. The influence of the energy and duration time of optical pulsed Gaussian beams on satability of nano-particle in trap is simulated and discussed

Hai, Hoang Dinh

Luu, Mai Van

Quy, Ho Quang

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 1 (2010), pp. 37-43
INFLUENCE OF ENERGY AND DURATION OF LASER PULSES
ON STABILITY OF DIELECTRIC NANOPARTICLES
IN OPTICAL TRAP
HO QUANG QUY
Academy of Military Science and Technology
MAI VAN LUU
Faculty of Physics, Vinh University
HOANG DINH HAI
Nghe An Pedagogical College
Abstract. In this article the gradient force of optical trap using two counter-propagating pulsed
Gaussian beam and the Brownian motion in optical force field are investigated. The influence of
the energy and duration time of optical pulsed Gaussian beams on satability of nano-particle in
trap is simulated and discussed.
I. 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
that the optical trap and tweezer have been a powerful tool for manipulating dielectric
particles [2, 3].
Usually, the optical traps or tweezer in many experiments are constructed by 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. Recently, Ambardekar et al. [4], Deng et al. [5], Zhao [6] and Wang et
al. [7] used a pulsed laser to generate the large gradient force, up to 2500 pN within a
short duration, about ps. Up to now, we have paid attention on optical trap using pulsed
Gaussian beam [7] and on optical trap using counter-propagating pulsed Gaussian beams
[8]. In works [8-11], the discussions about stability of the optical trap and the tweezer as
well as the effective controlling dielectric particles as gold nanoparticles, live membrane,
have been discussed into account in 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, above questions become an urgent one for the pulsing optical
trap.
This paper is organized as follows: In Sec. II we introduce the Gradient Optical force
acting on dielectric nanoparticles in the optical trap using two counter-propagating pulsed
Gaussian beam (PGB) and the set of Langevin equations concerning thermal fluctuations
of the probe, discribing the Brownian motion in the optical force field; Sec. III presents a
38 HO QUANG QUY et al.
simulated method; Sec. IV presents the radial variances of glass nanoparticles in waters,
which are traped by picosecond PGB and the discussions about influence of energy and
duration time of pulses on stability of them.
II. THEORY
A PGB is considered to trap fluctuating dielectic nanoparticles in the water plate
(Fig.1). We consider the Gradient optical force is induced by two counter-propagating
PGBs acting on a Rayleigh dielectric particle. The polarization direction of the electric
field is assumed to be along the x axis.
Laser pulsed beam
Focuslens
Glass particle Plate with Water
Radial Variance ρ
a b
Z=0 X
Y
Fig. 1. Optical trap’s sketch using the pulsed Gaussian beams (a) and motion
with radial variance of glass particle in the water plate (b).
The expression for the electric field of the above PGB is defined by [6]
~Eabove (ρ, z, t) =xˆE0
ikW 20
ikW 20 + 2z
exp [−i (k (z)− ω0t)]
× exp
[
−i 2kzρ
2(
kW 20
)
+ 2z2
]
· exp
[
−
(
kW 20
)2
ρ2(
kW 20
)2
+ 4z2
]
× exp
[
− (t− zc)2 /τ2
]
,
(1)
and for the below PGB
~Ebelow (ρ, z, t) =xˆE0
ikW 20
ikW 20 + 2z
exp [−i (k (z)− ω0t)]
× exp
[
−i 2kzρ
2(
kW 20
)
+ 2z2
]
· exp
[
−
(
kW 20
)2
ρ2(
kW 20
)2
+ 4z2
]
× exp
[
− (t+ zc)2 /τ2
]
,
(2)
INFLUENCE OF ENERGY AND DURATION OF LASER PULSES ON STABILITY ... 39
whereW0 is the spot radius of the beam waist at the plane z = 0, ρ is the radial coordinate,
xˆ is the unit vector of the polarization along the x direction, k = 2pi/λ is the wave number,
ω0 is the carrier frequency, and τ is the pulse duration. For the fixed input energy U of
a single pulsed beam, the constant E0 is determined by E
2
0 = 4
√
2U/
[
n2ε0cW
2
0 (pi)
3/2 τ
]
.
Here n2 is the refractive index of the surrounding medium. From the definition of the
Poynting vector, we can readily obtain the intensity distribution for the above PGB as
follows:
Iabove (ρ, z, t) =
〈
~Sabove (ρ, z, t)
〉
=
P
1 + 4z˜2
exp
[
− 2ρ˜
2
1 + 4z˜2
]
× exp
[
−2
(
t˜− z˜kW
2
0
cτ
)2] (3)
Ibelow (ρ, z, t) =
〈
~Sbelow (ρ, z, t)
〉
=
P
1 + 4z˜2
exp
[
− 2ρ˜
2
1 + 4z˜2
]
× exp
[
−2
(
t˜+
z˜kW 20
cτ
)2] (4)
where P = 2
√
2U/
[
(pi)3/2W 20 τ
]
, z˜ = z/kW 20 , ρ˜ = ρ/W0 and t˜ = t/τ .
For simplicity, we assume that the radius (a) of the particle is much smaller than
the wavelength of the laser (i.e., a << λ), in this case we can treat the dielectric particle
as a point dipole. Assume that the refractive index of the particle is n1 and n1 > n2.
By argument similar to that shown in the work of Zhao [6] for one PGB and of
Isomura for counter-PGB, the gradient optical force acting on dielectric particle of two
counter-propagating PGBs is given by
~Fgrad,ρ (ρ, z, t) = − ρˆ2β [Iabove (ρ, z, t) + Ibelow (ρ, z, t)] ρ˜[
cn2ε0W0
(
1 + 4 (z˜)2
)] (5)
where β = 4pin22ε0a
3
[(
m2 − 1) / (m2 + 2)] is the polarizability, and m = n1/n2 [6, 7].
Assuming a low Reynold number regime [12], the Brownian motion of the dielectric
in the optical force field (in the optical trap) is described by a set of Langevin equations
as follows:
γ~˙ρ (t) + ~Fgrad,ρ~ρ (t) =
√
2Dγ~h (t) (6)
where ~ρ (t) = [x (t) , y (t)] is the dielectric particle’s position in the water plate, γ = 6piaη
is its friction coefficient,η is the medium viscosity,
√
2Dγ~h (t) =
√
2Dγ [hx (t) , hy (t)] is a
vector of independent white Gaussian random processes describing the Brownian force,
D = kBT/γ is the diffusion coefficient, T is the absolute temperature, and kB is the
Boltzmann constant.
III. SIMULATION METHOD
We compute the two-dimensional motion and the radial variance (position) of a
glass particle in water using the Brownian dynamic simulation method. A particle/bead-
40 HO QUANG QUY et al.
spring model is employed to represent the glass particle, and the following equation of
motion is computed for each particle:
~ρ (t+ δt)− ~ρ (t) = −
~Fgrad,ρ (~ρ (t))
γ
× ~ρ (t)× δt+
√
2D × δt×~h (t) (7)
where δt is the time increment of the simulation,~h (t) is a random vector whose components
are chosen from the range [-1, 1] in each time step. ~Fgrad,ρ (~ρ (t))in Eq.(6) describes the
gradient optical force acting on particle located at position ~ρ at time t. For example, at
beginning time t=0, the glass particle is assumed to locate at the position|ρ (t = 0)| =
W0, where W0 is the beam waist, then we understand that the gradient optical force
~Fgrad,ρ (W0, z, 0) acts on particle, which will be located at position W +∆ρ after a time
increment δt.
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 parameters as follows: λ = 1.064µm, m =
n1/n2 = 1.592/1.332, η = 7.797 × 10−1Pa.s (the small glass particle and water, for
instance) [6], W0 = 1µm, a = 10nm, τ = (0.1÷ 100) ps, and the input power is changed
by U = (0.01÷ 100)µJ[7], T=25oC. The gradient optical force Fgrad,ρ is calculated by
expression (5) in the ranges: t = (−3÷ 3) τ , and ρ = (−2÷ 2)W0 at z = 0 µm (consider
the beam waist of pulsed Gaussian beam located in the traping plane z = 0).
IV. RESULTS
IV.1. Distribution of optical force
The simulations show that the gradient optical forces acting on the glass particle
in water are divided into two parts whose directions are opposite to each other (Fig. 2a)
and magnitudes are distributed as Gaussian function of radial distance (Fig. 2b), and of
pulsing time (Fig. 2c). This distribution of gradient optical force is in good agreement to
that presented in the work [6], it means gradient forces are centripetal.
Fig. 2. Distribution of optical force in phase plane (ρ, t) (a), in direction ρ (b),
and in pulsing time (c).
INFLUENCE OF ENERGY AND DURATION OF LASER PULSES ON STABILITY ... 41
IV.2. Particle’s motion in trap
The motions of two glass particles, whose beginning position |ρ (t = 0)| = W0
√
2
(for example: x(0) =W0, y(0) =W0) and |ρ (t = 0)| = W0 (for example: x(0) = 0, y(0) =
−W0), during pulsing time are simulated and discribed in Fig.3f. We can see that, at
beginning and ending time of the pulse when energy is low, the particle moves randomly
(Figs. 3a, d and e). With increasing of energy beginning from second picosecond of the
pulse, i.e. the gradient optical force increases, the particles are pulled into, “called“ stable
circle with radius W0 of the trap (Fig. 3b). When energy is high enough the optical force
predominates over Brownian force, particle is trapped in center of the trap (Fig. 3c). In
this case we can affirm that particle is absolutely stable.
Fig. 3. Position of glass particle in trap during pulsing time of the pulse τ = 1ps
a- first ps, b- second ps, c- third and fourth ps, d- firth ps, e- sixth ps, f)- all pulse
IV.3. Influence of duration time and energy of pulse
With given particle and fluid, i.e. with the given Brownian force, stability of particle
in optical trap depends on optical force. It means stability of particle depends on duration
time and energy of the pulse. It is proved in Fig. 4 and Fig. 5. If energy (U) is too low
or duration time (τ) is too long, particle moves randomly (Fig. 4a and Fig. 5a), even the
particle in center of the trap. But when duration time descreases or energy increases, the
particle is forced by optical gradient force, so it is pulled into center of the trap (Fig. 4b,
Fig. 4c, Fig. 5b, and Fig. 5c). With shorter duration time or higher energy, stability
of the particle is more reached and stable time is longer (Fig. 4c, Fig. 4d, Fig. 5c, and
Fig. 5d). Therefore, to reach stability of the particle in optical trap, we must to choose
pulsed beams with energy and duration time optimal, appropriate to dielectric particle
and its fluid. Moreover, to long stable time, we must to use pulsed beam so that optical
force is always predominant over Brownian force. These questions will be solved when
radius, refractive index of the particle and viscovity of the fluid are known.
42 HO QUANG QUY et al.
1
0,1
-1
R
a
id
a
l
V
a
ri
a
n
ce
[μ
m
]
0 1 2 3 4 5 6
Pulsingtime [ps]
1
0,1
-1
R
a
d
ia
l
V
a
ri
a
n
ce
[μ
m
]
0 1 2 3 4 5 6
Pulsing time [ps]
a
b
R
a
id
a
l
V
a
ri
a
n
ce
[μ
m
]
1
0,1
-1
0 1 2 3 4 5 6
Pulsing time [ps]
R
a
d
ia
l
V
a
ri
a
n
ce
[μ
m
]
1
0,1
-1
0 1 2 3 4 5 6
Pulsing time [ps]
c d
Fig. 4. Radial variance of glass particle during pulsing time a) -τ=100 ps, b)
τ=10 ps, c) τ=1 ps, d) τ=0.1 ps.
1
0,1
-1
R
a
id
a
l
V
a
ri
a
n
ce
[μ
m
]
0 1 2 3 4 5 6
Pulsing time [ps]
1
0,1
-1
R
a
d
ia
l
V
a
ri
a
n
ce
[μ
m
]
0 1 2 3 4 5 6
Pulsing time [ps]
a
b
R
a
id
a
l
V
a
ri
a
n
ce
[μ
m
]
1
0,1
-1
0 1 2 3 4 5 6
Pulsing time [ps]
R
a
d
ia
l
V
a
ri
a
n
ce
[μ
m
]
1
0,1
-1
0 1 2 3 4 5 6
Pulsing time [ps]
c d
Fig. 5. Radial variance of glass particle during pulsing time a) U = 0.01 µJ, b)
U = 0.1 µJ, c) U = 1 µJ, d) U = 10 µJ.
INFLUENCE OF ENERGY AND DURATION OF LASER PULSES ON STABILITY ... 43
V. CONCLUSION
Using the Langevin equation describing the Brownian motion in force field for opti-
cal trap, stability of the dielectric particle in fluid is investigated and discussed. From all
results and discussions, we can conlude that: first, the Brownian force influences on stabi-
lizing process of the glass particles in water by the optical trap using the pulsed Gaussian
optical beams; second, energy and duration of the pulses influence on stability of the trap;
third, to have absolute stability of the trap for application, an appropriateness between
parameters of the pulse, particle and fluid must be chosen.
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[4] A. A. Ambardekar, Y. Q. Li, Opt. Lett. 30 (2005) 1797-1799.
[5] J. L. Deng, Q. Wei, Y. Z. Wang, Y. Q. Li, Opt. Express. 13 (2006) 3673-3680.
[6] C. L. Zhao, L. G. Wang, X. H. Lu, Phys. Lett. A (2006) 502-506.
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[8] H.Q.Quy, M.V. Luu, IWP&A, Nhatrang, Sept.10-14, (2008) 181-186.
[9] H. Kress, Ernest H. K. Stelzer, G. Griffiths, and A. Rohrbach, Phys. Rev. E71, (2005) 061927.
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[12] J. Happel and H. Brenner, “Low Reybnold Number Hydrodynamic”, Springer, New York, (1983).
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321- 325.
Received 05 May 2009.


Influence of Energy and Duration of Laser Pulses on Stability of Dielectric Nanoparticles in Optical Trap

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Influence of Energy and Duration of Laser Pulses on Stability of Dielectric Nanoparticles in Optical Trap

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