In this article expresions used to simulate the trap stiffness k of the optical trap are derived. The influence of principle parameters as total energy, beam waist and duration of pulsed laser beam, radius of dielectric particle, and viscosity of surrounding medium on the stiffness  is simulated and discussed

Hai, Hoang Dinh

Nam, Hoang Van

Quy, Ho Quang

English

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 21, No. 1 (2011), pp. 71-76
INFLUENCE OF PRINCIPLE PARAMETERS
ON THE AVERAGE STIFFNESS OF OPTICAL TWEEZER
USING PULSED GAUSSIAN BEAMS
HO QUANG QUY
Journal of Military Sciences and Technology
HOANG DINH HAI
Nghe An Pedagogical College
HOANG VAN NAM
Ha Tinh Peoples Committee
Abstract. In this article expresions used to simulate the trap stiffness k of the optical trap are
derived. The influence of principle parameters as total energy, beam waist and duration of pulsed
laser beam, radius of dielectric particle, and viscosity of surrounding medium on the stiffness is
simulated and discussed.
I. INTRODUCTION
A particle trapped in an optical trap is not completely immobile. There are thermal
fluctuations that cause the particle to execute Brownian motion while trapped. It is the
fact that the trap has an inherent stiffness k, which is controlled by power of the laser
light. That is, an optical trap is taken to be a potential well with stiffness k. A spring
with stiffness k prevents the mass attached to it from escaping and pulls it (or pushes it)
back to the equilibrium position. Similarly, the optical trap tries to restore the particle
to its equilibrium position, the bottom of the traps potential well [1]. There are many
experimental works succesfully measured the stiffness of optical trap [2, 3, 4]. In those
works, there are some methods used and their results confirm that the stiffness depends on
laser intensity (or average laser input power). However, in medium the motion of particle
is under impact not only of optical force, but of Brownian force, which depends on radius
of particle and medium viscosity.
The goal of this article to simulate influence of principle parameters of optical trap
on the stiffness.
II. THEORY
A pulsed Gaussian beam (PGB) is considered to trap fluctuating dielectric nano
particles in the water plate (Fig.1). We consider the Gradient optical force is induced by
72 HO QUANG QUY et al.
Fig. 1. (a) The schematic of optical tweezer. (b) The motion with radial variance
of particle in the medium plate.
PGB acting on a Rayleigh dielectric particle. The polarization direction of the electric
field is assumed to be along the x axis.
The expression for the electric field of the above PGB is defined by [5]
~E (ρ, z, t) = xˆE0
ikw2
0
ikw2
0
+2z
exp {−i [k (z)− ω0t]}
× exp
[
−i 2kzρ2
(kw20)
2
+4z2
]
× exp
[
− (kw
2
0)
2
ρ2
(kw20)
2
+4z2
]
× exp
{
− [t−z/c]2τ2
} (1)
where w0 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, τ is the duration of laser pulse. For the fixed input energy U
of a single beam, the constant E0 is determined by E
2
0 = 4
√
2U/
[
w20 (pi)
3/2 τ
]
.
From the definition of the Poynting vector, we can readily obtain the intensity
distribution
I (ρ, z, t) =
〈
~S (ρ, z, t)
〉
= zˆ
P
1 + 4z2
exp
[
− 2ρ˜
2
1 + 4z˜2
]
exp
[
−2
(
t˜− 2z˜kw
2
0
cτ
)]
(2)
where P = 2cε0
√
2U/
[
(pi)3/2w20τ
]
. Herez˜ = z/kw20,t˜ = t/τ ind ρ˜ = ρ/w0 are normalazea
parameters.
For simplicity, we assume that the radius of the particle is much smaller than the
laser wavelength (a << λ), ( ), that means the Rayleigh approximation is satisfied. 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, where n2 is a refractive index of medium.
As shown in Fig.1 and definition of optical force [6], on the particle acts only the
gradient force, ~Fgrad, which is given by
~Fgrad,ρ (ρ, z, t) = − ρˆ2βI (ρ, z, t) ρ˜
cn2ε0w0 (1 + 4z2)
(3)
INFLUENCE OF PRINCIPLE PARAMETERS ON THE AVERAGE STIFFNESS ... 73
where β = 4pin22ε0a
3
[(
m2 − 1) / (m2 + 2)] is the polarizability, and m = n1/n2[5, 7].
Moreover, on the particle acts also Brownian force, which causes the particle moves
randomly under the Brownian law. Assuming a low Reynold’s number regime [8], the
Brownian motion of the dielectric particle in the optical trap is described by a set of
Langevin equations as follows:
γ˙~ρ (t) +K~ρ (t) =
√
2Dγ~h (t) (4)
where ~ρ (t) = [x (t) , y (t)] is the dielectric particles position in the medium plate, γ = 6piaη
is its friction coefficient, η is the medium viscosity,
√
2Dγ~h (t) =
√
2Dγ [hx (t) , hy (t)] is
a vector describing the Brownian force, D = kBT/γ is the diffusion coefficient, T is the
absolute temperature, and kB is the Boltzmann constant, K is the stiffness matrix.
On the other hand, in the presence of the optical force, equation (4) can be written
in the form [9]:
γ˙~ρ (t) = Fgrad,ρ(~ρ(t)) +
√
2Dγ~h (t) (5)
From Eqs.(4) and (5) the total force acting on the particle
Fgrad,ρ(~ρ(t)) = K~ρ (t) (6)
depends on the position of the particle itself.
We limit our analysis to a bidimensional system (i.e. considering z = 0), because it
is the most interesting from the applied point of view. Once the stiffness is known (Kρ),
the position of the particle yields, for small displacements (∆ρ(t)), the force (Fgrad,ρ), i.e.
Fgrad,ρ (ρ, t) = Kρ∆ρ (t) (7)
ffor bidimension system, where ∆ρ(t) is the mean value of radial variances experimentally
measured in time interval ∆t. For given particle (with radius a) and fluid (with constant
drag coefficient η), stiffness Kρincreases linearly with increasing laser power of the optical
trap. Using Eq. (7)) for our model, in which the laser field is the Gaussian pulse, the
average stiffness Kρ over durating time of the laser pulse can be expressed by:
Kρ =
〈Fgrad,ρ (ρ, t)〉t
〈∆ρ (t)〉t
≈ 〈Fgrad,ρ (ρ, t)〉t〈ρ (t)〉t
=
1
2τ
∞∫
−∞
Fgrad,ρ(ρ (t) , t)dt
1
2τ
∞∫
−∞
ρ(t)dt
≈
1
2τ
τ∫
−τ
Fgrad,ρ(ρ (t) , t)dt
1
2τ
τ∫
−τ
ρ(t)dt
=
τ∫
−τ
Fgrad,ρ(ρ (t) , t)dt
τ∫
−τ
ρ(t)dt
(8)
As shown in previous work [10] the particle oscillates around the equilibrium position
(ρ = 0) with an radial variance is about its radius. Thus in Eq. (8) the term 〈ρ (t)〉t can
replace the term 〈∆ρ (t)〉t.
Using Eqs. (3), (4) and (8), and numerical simulation, we can find the average
stiffness, Kρ, which depends on total energy (U), beam waist (w0), and on duration (τ)
of the Gaussian pulse, and on radius of particle a with given parameters of surrounding
medium η, n2, T .
74 HO QUANG QUY et al.
III. SIMULATED RESULTS AND DISCUSSION
We compute the two-dimensional motion and the radial variance (position) of a glass
particle in water (specimen at z = 0) using the Brownian dynamic simulation method. A
particle/bead- 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
× δt+
√
2D × ⇀h (t) × δt (9)
where δt is the time increment of the simulation, ~h (t) is a random vector whose com-
ponents are chosen from the range [-1, 1] in each time step. ~Fgrad,ρ (~ρ (t)) in Eq. (9)
describes the gradient optical force acting on the particle located at position ρ (t). For
example, at beginning time t=0, the glass particle is assumed to locate at the position
ρ (0) = w0, then we understand that the gradient optical force Fgrad,ρ (w0, 0) acts on the
particle, which will move to position ρ (0 + δt) = w0 + ∆ρ after a time increment δt. .
At this position the gradient force has new value Fgrad,ρ (w0 +∆ρ, 0+ δt). Continously,
after having Fgrad,ρ (ρ(ti), ti) and ρ (ti), using Newton- Contes integration method for two
integrationss in (8) we have average stiffness Kρ.
For numerical simulation we choose principle parameters as follows: λ = 1.064µm,
m = n1/n2 = 1.592/1.332, η = 7.797×10−1Pa.s, kbT = 4.14×10−21 (the small glass par-
ticle and water, for instance)[5], w0 = (0.5÷ 9.5) µm,a = (10÷ 50) nm, τ = (1÷ 20) ps,
and the input total energy is changed by U = (1÷ 200)µJ.
The influence of the total energy on average stiffness is simulated and illustrated
in Fig.2. The results confirm again that stiffness of optical trap increases with increasing
of the laser total energy. From Eqs.(2) and (3) can see that it is in good agreement to
that presented in work of A. Kumar [2], in which the experimentally measured stiffness is
directly proportional to the optical force.
Figures 3, 4, and 5 illustrate the influence of beam waist of laser beam, duration
of laser pulse and radius of particle on stiffness, respectively on the stiffness. They show
also that the stiffness depends not only on laser intensity, but mainly on the optical force,
which depends on the beam waist, duration of the laser pulse. Moreover, the optical
force depends on the polarizability, which depends on third power of particles radius, a3.
The dependence of the stiffness on principle parameters as shown in Fig.4 and Fig.5 is
completely logical except what shown in Fig.3, where the stiffness reaches an optimal at
w0 ≈ 1.06 = λ, increases and decreases with increasing of beam waist from 0.5 µm to 1.06
µm, and from 1.06 µm to higher one, respectively.
This phenomena can be explained as follows: The beam divergence is directly pro-
portional to the ratio between the wavelength λ, and the beam-waist diameter 2w0(i.e.
θ0 = 2λ/pi2w0). If the waist is squeezed, the beam diverges, which leads to obtain a low
directional beam. Consequently, the depth of focus is directly proportional to the area of
the beam at its waist, and inversely proportional to the wavelength (i.e. 2b = 2piw20/λ).
Thus when a beam is focused to a small spot size, the depth of focus is short and the plan
of focus must be located with greater accuracy [10]. The low direction of beam and the
short depth of focus causing by decreasing of beam waist lead to decreasing laser intensity
INFLUENCE OF PRINCIPLE PARAMETERS ON THE AVERAGE STIFFNESS ... 75
Fig. 2. Average stiffness vs total energy. Fig. 3. Average stiffness vs beam waist.
Fig. 4. Average stiffness vs. duration of pulse. Fig. 5. Average stiffness vs. radius of particle
gradient in the area of the beam at its waist, consequently lead to decreasing of optical
force.
IV. CONCLUSION
From above discussions, we can collude that: First, the stiffness of opticae trap
can be found by simulation using Langevin equation and expression of the gradient force;
second, the average stiffness of trap does not directly depend on laser intensity only, but
Sn the beam waist, total energy, duration of laser pulse, and on radius of trapped particle
76 HO QUANG QUY et al.
when using the laser Gaussian pulsed beam. The simulated results will be used to check
the optical force or the radial variance of the particle, which are experimentally obtained.
REFERENCES
[1] ME/CE96, “Optical tweezer- Measuring the Steffness k of the optical trap”,
http:/www.nbi.dk/∼tweezer/introduction.htm (2005).
[2] A. Kumar De, D. Roy, B. Saha, and D. Goswami, A simple method for constructing and calibrating
an optical tweezer, Current Science, 95 (2008) 723-724.
[3] D. Preece et al., Optics Express 17 (2009) 22718-22725.
[4] G. Honglian et al, Sciecce in China (Series A) 45 (2002) 919- 925.
[5] C. L. Zhao, L. G. Wang, X. H. Lo, Phys. Lett. A363 (2006) 502-506.
[6] A. Ashkin, J. M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288-290.
[7] L.G. Wang et al., Opt. Lett. 32 (2007) 1393-1395.
[8] J. Happel and H. Brenner, Low Reybnold Number Hydrodynamics, Springer, NEW York, 1983, Chap.
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[9] G. Volpe, G. Volpe, and D. Petrol, Phys. Rev. E76 (2007) 061118.
[10] H. Q. Quy et al., Chinese Optics Letters 8 (3) (2010) 332-334.
Received 13 July 2010.


Influence of  Principle Parameters on the Average Stiffness of Optical Tweezer Using Pulsed Gaussian Beams

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Influence of Principle Parameters on the Average Stiffness of Optical Tweezer Using Pulsed Gaussian Beams

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