The influence of the self-focusing effect arised from Kerr effect on the optical force acting on the dielecric particle embedded in the Kerr medium, which is irradiated by the Gaussian beam, is proposed to concern. The expressions of the optical forces with the nonlinear refractive index and nonlinear focal length are derived. Using them, the distribution of the optical forces in the trapping region of the optical tweezer is simulated and discussed for same distinguished case of the Kerr medium with different nonlinear coefficients. The results show that the stabe region of the optical tweezer depends on the nonlinear coefficient of refractive index. Moreover, the stable region could be brokendown with a critical value of the nonlinear coefficient of refractive index of the surrounding medium irradiated by Gaussian laser pulse described by given parameters as  intensity, duration and radius of beam waist.

Le, Cao Thanh

Nam, Hoang Van

Quy, Ho Quang

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 23, No. 2 (2013), pp. 155-161
THE INFLUENCE OF THE SELF-FOCUSING EFFECT ON THE
OPTICAL FORCE ACTING ON DIELECTRIC PARTICLE
EMBEDDED IN KERR MEDIUM
HOANG VAN NAM
Council of Science and Technology, Ha Tinh, Vietnam
CAO THANH LE
Ha Tinh University
No. 447, 26/3 Road, Dai Nai, Ha Tinh City, Ha Tinh, Vietnam
HO QUANG QUY
Academy of Military Science and Technology
17 Hoang Sam Street, Nghia Do, Cau Giay, Hanoi, Vietnam
Received 30 January 2013; revised manuscript received 31 May 2013
Accepted for publication 10 April 2013
Abstract. The influence of the self-focusing effect arised from Kerr effect on the optical force
acting on the dielecric particle embedded in the Kerr medium, which is irradiated by the Gaussian
beam, is proposed to concern. The expressions of the optical forces with the nonlinear refractive
index and nonlinear focal length are derived. Using them, the distribution of the optical forces in
the trapping region of the optical tweezer is simulated and discussed for same distinguished case
of the Kerr medium with different nonlinear coefficients. The results show that the stabe region of
the optical tweezer depends on the nonlinear coefficient of refractive index. Moreover, the stable
region could be brokendown with a critical value of the nonlinear coefficient of refractive index
of the surrounding medium irradiated by Gaussian laser pulse described by given parameters as
intensity, duration and radius of beam waist.
I. INTRODUCTION
The optical tweezer used CW laser beam is used to trap the micro dielectric particle,
which is embedded in a medium as gas, fluid [1-8]. Up to now, the optical trap using pulsed
Gaussian beam and the optical trap using counter-propagating pulsed Gaussian beams has
been paid attention to trap the nanoparticle [9-11]. Unfotunately, in the mentioned works,
there was attention to thermal effect of the medium only (the Brownian motion), but not
to other effect as random or Kerr. In work [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.
156 THE INFLUENCE OF THE SELF-FOCUSING EFFECT ON THE OPTICAL FORCE ACTING ...
Therefore, in this paper, the influence of the both effects, the Kerr in medium and
the self-focusing 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 derive the expressions of the optical forces concerning self-focusing, which is
arised from the Kerr effect in the Kerr medium irradiated by the Gaussian pulse; in Sec.3
we present the influence of the nonlinear coefficient of refractive index on the optical forces
and discuss about the stability of the optical 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]:
~E (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)
as a Gaussian beam, where W0 is the radius of the beam waist at 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 pulse duration, c is the light speed.
Assuming that this pulse causes the Kerr effect in the medium surrounding the
particle. So after propagating through medium the Gaussian beam will be self-focused by
the nonlinear lens of focal length (see Fig. 1) which is approximately given by [14]:
fnl(P, nnl) = W
2
0
√(
pin2
2nnlP
)
(2)
where n2, nnl are the linear refractive index and nonlinear refractive coefficient of the
Kerr medium, respectively, and P = 2
√
2U/
[
(pi)3/2W 20 τ
]
is the total power of the input
Gaussian beam.
It can be considered that the Kerr medium with distance of z′ plays as the thin
lens with focal length of fnl. Such that, after propagating through the Kerr medium, the
Gaussian beam is modified and its waist radius, W0 is modified into [15]:
W ′0(P, nnl) = M(P, nnl)W0, (3)
where M(P, nnl) = Mr/
√
1 + r2, r = z0/(z
′−fnl), Mr = |fnl/z′ − fnl|, z0 = piW 20 /λ 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.
HOANG VAN NAM, CAO THANH LE, AND HO QUANG QUY 157
From (1), (3) and the definition of the Pointing vector, we can readily obtain the
intensity distribution for the redistributed pulse in the tweezer as follows:
I (ρ, z, t) =
P
1 + 4z˜2
exp
[
− 2ρ˜
2
1 + 4z˜2
]
× exp
−2(t˜− z˜kW ′02
cτ
)22 (4)
where z˜ = z/kW ′20 ρ˜ = ρ/W ′0 and t˜ = t/τ .
Fig. 1. Sketch of optical tweezer with Kerr medium.
For simplicity, we assume that the radius (a) of the linear particle (considering the
particle is not sensitive to Kerr effect) 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. 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
nm(ρ, z, t) = n2 + nnlI(ρ, z, t) (5)
and the relative refractive index is given by
mm(ρ, z, t) = n1/nm(ρ, z, t) (6)
Using (6) and as shown in Refs. [7, 8], we have the scattering cross sections (α) and
polarizabilities (β) as follows:
αm(ρ, z, t) =
(
128pi5a6
3λ4
)[(
mm(ρ, z, t)
2 − 1)
(mm(ρ, z, t)2 + 2)
]2
(7)
158 THE INFLUENCE OF THE SELF-FOCUSING EFFECT ON THE OPTICAL FORCE ACTING ...
and
βm(ρ, z, t) = 4pin
2
m(ρ, z, t)ε0a
3
(
mm(ρ, z, t)
2 − 1)
(mm(ρ, z, t)2 + 2)
(8)
To show influence of self-focusing effect on the optical force, as example for simplic-
ity, we present here expressions of the longitudinal force along the beam axis (at ρ = 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:
~Fgrad,ρ (ρ, 0, 0) = −ρˆ 2βm(ρ, 0.0)I (ρ, 0, 0) ρ˜
cnm(ρ, 0, 0)ε0W ′0(P, nnl)
(9)
and the longitudinal force is given by:
~Fz = −−→z 2βm(0,z,0)I(0,z,0)
nm(0,z,0)ε0ckw
′2
0 (P,nnl)
[
z˜k2w
′4
0 (P,nnl)
c2τ2
+
2z˜(1+4z˜2)
(1+4z˜2)2
]
+−→z nm(0,z,0)c αm(0, z, 0)I (0, z, 0)
(10)
The general Exps. (9) and (10) describe the redistribution of gradient force in speci-
ment plane (ρ) and longitudinal force in beam axis (s) caused by both effects: self-focusing
(d) and refractive index’s nonlinearity (nnl). If fnl =∞, from (3) results M(P, nl) = 1 or
W ′0(P, nnl) = W0, i.e. the self-focusing effect is ignored, Exps. (9) and (10) coincide with
Exps. (9) and (10), respectively, in the our previous work [13].
III. SIMULATED DISTRIBUTION OF OPTICAL FORCES
In following numerical simulation we choose parameters as:U = 10µJ , W0 = 2µm,τ =
10ns,λ = 0.53µm (for Gaussian pulse); the refractive index of the glass particle with ra-
dius a = 20nm is n1 = 1.592; this particle is embedded on water of refractive index
n2 = 1.332. Moreover, we considered the water is dissolved by the nonlinear material, so
their nonlinear coefficient should be changed.
The optical gradient force Fgrad,ρ and Fz are calculated by expression (9) and (10)
in the ranges: ρ = (−2÷ 2) µm and z = (−5÷ 5) µm (consider the beam waist of pulsed
Gaussian beam located in the traping plane z = 0 and the beam axis goes through point
ρ = 0), respectively. 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.
It can be seen that, if the medium is not Kerr medium or the nonlinear refractive
coefficient of it is small (nnl = 1× 10−11cm2/W), the particle will be trapped in a certain
stable region with a radial radius of about W0/2 in the speciment plane (Fig. 2a) and
a length of about 3 µm along beam axis on the medium (Fig. 3a). If the nonlinear
coefficient increases, the optical force decreases and the stable region is larger. Especially,
if the nonlinear coefficient is large enough, inside the trapping region appears a small
unstable region, where the particle moves along the direction of the force (Figs. 2b, 2c,
3b, and 3c), because of the changing sign of βm. This phenomenon appears due to the
Gaussian distribution of the laser intensity, i.e. the intensity in the center of the laser
beam is larger than one in the edge.
HOANG VAN NAM, CAO THANH LE, AND HO QUANG QUY 159
Fig. 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 unstable region is more and more large with increasing of the nonlinear refrac-
tive coefficient (Fig. 2d, Fig. 3d). This phenomenon can be explained by the decreasing
of the relative refractive index mm, consequently, reducing the refraction of laser rays
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 will be larger than that of particle. In this case, mn < 1 (i.e.αm, βm < 0),
consequence sign~F ≡ sign~ρ, where ~ρ is the radial variance of particle [14], the particle will
be pushed ouside the centre of tweezer (Figs. 2c, 2d and Figs. 3c, 3d), and the tweezer
will be brokendown.
160 THE INFLUENCE OF THE SELF-FOCUSING EFFECT ON THE OPTICAL FORCE ACTING ...
Fig. 3. Distribution of longitudinal force (N) on beam axis (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.
From Exps. (2), (5), (9) and (10), we can see that, the distribution of optical forces
and consequence, the stable region depend not on n2n only, but also on energy U , beam
waist W0 and duration τ of the pulsed Gaussian beam.
IV. CONCLUSION
The distribution optical forces depend on the nonlinear refractive coefficient of the
surrounding medium. If the medium is a nonlinear one (the particle is linear), the Kerr
effect and the relating self-focusing effect reduce the trapping forces, so far brokendown
the trapping. The dimension of the stable region depend not on the nonlinear coefficient
nnl only, but also on the total energy U , beam waist W0 and duration τ of the laser pulse
and other parameters of the particle and medium. As shown above, there is a collection
of parameters: nnl = 7 × 10−11cm2/W , W0 = 2 µm, U = 10 µJ, τ = 10 ns, . . . , with
it the optical tweezer is brokendown. To reduce the Kerr effect and relating self-focusing
HOANG VAN NAM, CAO THANH LE, AND HO QUANG QUY 161
effect, it is necessary to choice the laser beam in accordance with the nonlinear coeficient
of refractive index of the medium embedding the particle, so that the trapping condition
mn > 1 is satisfied always.
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The Influence of the Self-focusing Effect on the Optical Force Acting on Dielectric Particle  Embedded in Kerr Medium

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