Download Influence of the Brownian movement on the motion of nano

OPTO−ELECTRONICS REVIEW 20(3), 247–254
DOI: 10.2478/s11772−012−0030−1
Influence of the Brownian movement on the motion of nanoand micro-particles in the inhomogeneous optical field
C.Yu. ZENKOVA*, M.P. GORSKY, I.V. SOLTYS, and P.O. ANGELSKY
Chernivtsi National University, 2 Kotsubinsky Str., 58−012 Chernivtsi, Ukraine
The motion of light scattering particles of the Mie and Rayleigh micro− and nano−range type in the inhomogeneously−pola−
rized optical field, with allowance made for the Brownian movement, is analysed in the paper. The spatial modulation of po−
larization in the observation plane determines the spatial modulation of the volume energy density. That is why the velocity
and the resulting optical force, which cause the motion of the testing particles, change according to the degree of coherence
of the interacting fields. The influence of the forces which arise in the viscous medium and cause the Brownian movement
upon the mechanisms of manipulating and trapping testing particles by the optical field is studied.
Keywords: polarization, Poynting vector, scattering objects, degree of coherence.
1. Introduction
One of the most actual tasks of the present day biomedicine
is to control micro−particles and nano−particles of different
nature with help of laser radiation. A controlled object can
be a colloid particle, a molecule, an atom or a cell, including
a bio−object, e.g., a DNA molecule, an organelle cell. One
can say, that the technology of particle controlling is a po−
werful tool for scientists working with micro−objects of dif−
ferent nature. The process of trapping and controlling parti−
cles depends on characteristics of the laser beam. At pre−
sent, the possibilities to use beams of various nature, such as
Gaussian beams [1,2], bottle beams [3], zero−order Bessel
beams [4], self−focused laser beams [5] and even evanescent
fields [6,7] are investigated. One of the principal prerequi−
sites for an effective particle manipulation with help of these
laser beams is their complete coherence. However, in actual
practice, any laser fields are always only partially coherent
[8]. Moreover, the possibility for highly accurate focusing
and, then, for producing a force component capable to
change the motion direction of an object is determined by
the degree of corresponding beams coherence [9,10]. The
performed analysis of controlling Rayleigh dielectric sphe−
res [11] in the field of partially coherent radiation shows that
all optical forces are greatly affected by light coherence. As
coherence decreases, the magnitude of the optical forces on
a dielectric sphere near the focus point greatly decreases, as
well. For the light beam with good coherence optical force
may be used to trap a particle and for the light beam with
intermediate coherence optical force may be used to guide
and accelerate a particle.
*e−mail:
[email protected]
Opto−Electron. Rev., 20, no. 3, 2012
The aim of the paper is to investigate the peculiarities of
a particle motion in a spatial periodically−modulated pola−
rized field and to study the possibilities of using the pola−
rization mechanisms for controlling the motion of micro−
and nano−objects.
The results of these investigations can find their practi−
cal use in applied biomedicine [12] as well as in various
directions of singular polarization optics [13–15].
The offered paper is a continuation of investigations
[16,17] of the influence of spatial modulations of the aver−
aged Poynting vector values on nano−size particles where
the action of corresponding optical flow is not connected to
a certain component of the optical force. Various compo−
nents of the optical force and their e ffect on the tested parti−
cles of nano− and micro−sizes are analysed in this paper.
The proposed paper also investigates the influence of
a coherence degree of superposing mutually orthogonal line−
arly−polarized waves on the peculiarities of particle motion
in a field of resulting distribution. In order to obtain more
general pattern of particle motion in the viscous medium
which is caused by the resulting optical field we shall ana−
lyse the influence of forces determining the Brownian
motion upon particle motion velocity.
Thus, the aim of the given paper is to establish a connec−
tion between interacting optical fields and motion velocity
of tested particles positioned in energy inhomogeneous
optical field.
2. Formation of optical forces affecting particles
of various light scattering mechanisms
The ways of optical trapping and manipulating of micro−
−particles with help of a laser beam depend on light pressure
r
r
force. Every photon carries a momentum equal to p = hk ,
C.Yu. Zenkova
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Influence of the Brownian movement on the motion of nano− and micro−particles in the inhomogeneous optical field
r
r
where k is the wave vector, k = k = 2p l, h is the Plank
constant. The interaction of irradiation with substance results
in a change of a photon momentum. The motion mechanism
is described with help of the second and third Newton’s laws.
Hence, light possesses a momentum and the change of the di−
rection of light propagation determines the force involved in
the direction change [18]. The motion value imparted to the
particles follows from the Newton’s law:
r
r
r
r r
dp d
dv
= (mv) = m ×
= ma = Fopt .
dt dt
dt
For example, while passing through a transparent parti−
cle light deflects. The refraction of light on a dielectric sur−
face changes the direction of the light momentum (a particle
acts as a micro−lens). The momentum change leads to the
force arising which propels the particle either into a region
of the maximum light intensity (transverse trapping) or into
a beam focal region which is the maximum intensity zone,
as well (longitudinal trapping).
Consequently, the force which arises as a result of
a direction change of light propagation makes a particle
move into the direction of the greatest light intensity.
The aim of the paper was to establish a link between the
degree of coherence of interacting fields and the particle
motion velocity. In order to do this, we chose two groups of
particles arbitrarily called “nano−particles” and “micro−par−
ticles”. For the first group of particles it is possible to use
Rayleigh theory to describe the light scattering mechanism,
while for the second group we use Mie theory, which is also
known as Lorenz−Mie solution or Lorenz−Mie−Debye solu−
tion or Mie scattering. To simplify matters, in the case of
micro−particles the approximations of Ray optics were used
[19].
Let us determine the resulting optical force affecting
particles in the laser radiation field. By the term “resulting
optical force” is meant a force which causes a directional
particle motion. It can be divided into the following com−
ponents:
scattering force (Kepler light pressure force). It is pro−
portional to the particle scattering surface and to the
light intensity of paraxial Poynting vector−directed
beams. This force arises because of the change of the
electromagnetic wave
r impulse with the dipole induced.
The dipole moment
is proportional to the vector of the
d
r
electric field E. The proportional coefficient is taken as
the polarizabilityr of the
r molecule a. The dipole moment
is calculated as d = aE.
gradient force. This force arises in the field of inhomo−
geneously distributed laser radiation intensity. The grea−
test intensity is on the beam axis. In this case, the gradi−
ent force will be directed towards the beam centre. It is
often regarded as an optical force turning a micro−parti−
cle and nano−particle into a multi−pole. The gradient
force is usually called the dipole force for Rayleigh
(submicron) particles. The gradient force is equal to the
Lorenz force, affecting the created (induced) dipole and
depends on the distribution of the laser radiation
intensity.
For absorbing and reflecting particles the suggested list
of forces is complemented by forces which show the effect
of the optical field on particles with corresponding proper−
ties. In this case each component of the optical force can be
counted using the averaged value of the Poynting vector.
The instantaneous value of the Poynting vector sets, by
definition, direction and density of the energy
at the
r r current
r
point and is determined by the relation S = E ´ H. The den−
sity value of the energy current is S = EH = w × 1 mem 0 e 0 ,
where w is the energy of electromagnetic field; e, e 0 ( m, m 0 )
are the values of dielectric (magnetic) penetrability of
medium and vacuum, correspondingly.
As it is known, w = 2wE = 2wH , where
r the
r energy den−
sity of the electric field is wE = (1 2)ee 0 ( E, E) and the
r ener−
r
gy density of the magnetic field is wH = (1 2) mm 0 ( H, H).
The averaged value of the Poynting vector is the time−
−averaged distribution of instantaneous value taken over the
whole period of time.
Let us consider the superposition of two plane waves
W1, W2 of equal amplitudes polarized in the plane of inci−
dence (Fig. 1) and determine the averaged value of the elec−
tric (magnetic) energy density and
direction
of the
r the
r
r
r
resulting
Let E (1) ( r1 , t )[ H (1) ( r1 , t )] and
r (2 ) r wave
r (2 )propagation.
r
E ( r2 , t )[ H ( r2 , t )] be the vectors of the electric (mag−
netic) fields for the first and the second interacting waves. In
this case, the vector of the resulting electric (magnetic)
r rfield
r
in
registration
plane
at therpoint r isrdefined as E( r ) =
r (1the
r
r
r
r
r
r
r
E ) ( r ) + E (2 ) ( r ) and H ( r ) = H (1) ( r ) + H (2 ) ( r ). Correspon−
dingly, the
r direction
r
r of the Poynting
r
rvector of the resulting
wave is S = S (1) + S (2 ) , where S (1) ( S (2 ) ) are the Poynting
vectors of the interacting waves.
Let us introduce the following definitions of the ana−
lyzed planes: as far as the z−component of the field is actual
in this interference scheme, the definition of planes is con−
nected to the orientation of a z−component. Thus, an inci−
dence plane and an observation plane coincide with each
other and with the plane of the figure, e. g., XOZ. The regis−
l
l
248
Fig. 1. Superposition of plane waves of equal amplitudes linearly
polarized at incidence plane with an interference angle of 90°. Perio−
dical spatial modulation of polarization takes place in plane of inci−
dence.
Opto−Electron. Rev., 20, no. 3, 2012
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tration plane is the plane that analysis the intensity dis−
tribution, e. g., XOY.
We define the time−averaged density of the energy cur−
r
rent at the point of observation as r [17]
S=
ee 0
mm 0
å { (1ij) ( r) + (2ij) ( r) + 2
r
r
In general case, the forces exerted on the particles by
a laser beam depend on the properties of both beam and par−
ticles.
}
r r
r r
tr[W ( r1 , r1 , 0)]tr[W ( r2 , r2 , 0)] × h ij(1,2 ) cos(a ij(1,2 ) ) × cos(d e )
i,j
r
r
r
where jij( m ) ( r ) =< Ei( m ) ( r , t ) E j*( m ) ( r , t ) > and m = 1,2; i,j =
( m ,n )
x, z, h ij
determines the degree of mutual coherence of
the corresponding fields (m, n = 1,2) and is defined by
Eq. (2); a ij( m ,n ) is the argument of h ij( m ,n ) and determines
the phase difference between i– and j– components of the
field, d e is the phase difference between the beams (in this
case their electrical parts) at the registration plane. The mu−
r r
tual coherence matrix W ( r1 , r2 , t ) describes the coherence
properties of the vector optical fields and characterizes the
r
correlation of the fields at two different spatial points: r1 and
r r
r
(1) r
r2 [20,21]. It is defined as W ( r1 , r2 , t ) =< Ei ( r1 , t ) E (j2 )*
r
( r2 , t ) >, where i, j = x, z. Within the framework of such an
approach the degree of mutual coherence of the field [22] is
defined as
r r
h ij ( r1 , r2 , t ) =
(1)
Rayleigh particles
Let us consider the formation of the optical force affecting
Rayleigh particles.
Depending on the interaction mechanism of the electro−
magnetic wave and Rayleigh particles of various optical
properties, the optical force can be imagined as a decompo−
sition of a traditional gradient, scattering, absorbing and
reflecting components of the optical force [27]. The direc−
tion of an optical force gradient component is stipulated by
a distribution gradient of energy volume density. The direc−
tion of scattering and absorbing components is set by the
direction of energy flow propagation. The direction of
a reflecting component is set by the orientation of a particle
surface. In order to trap particles of a given class it is enough
r r
Wij ( r1 , r2 , t )
=
r r
r r
tr[W ( r1 , r1 , 0)] × tr[W ( r2 , r2 , 0)]
r r
Wij ( r1 , r2 , t )
r r
r r
å Wii ( r1 , r1 , 0)Wij ( r2 , r2 , 0)
(2)
ij
The distribution of the time−averaged density of the
energy current in space according to Eq. (1) determines the
current value at different points of the observation plane and
is unambiguously determined by the degree of coherence of
the superposing waves h ij . The direction of the resulting
current is set by the direction of the Poynting vectors of
these waves.
When accounting the superposition of two mutually
orthogonal linearly−polarized fields, the modulation of the
averaged Poynting vector value, e. g., the spatial modula−
tion of the energy density, is noted in the observation
plane. The gradient force, expressed by the Poynting vec−
tor, forms the particle distribution in this plane. Moreover,
the change of the degree of coherence leads to the change
of the modulation depth of the energy bulk density, which,
in its turn, determines gradient force value and peculiari−
ties of particle motion. This type of modulation of the ave−
raged Poynting vector values is especially actual for calcu−
lating optical forces affecting Rayleigh scattering
particles.
As it was shown in papers, changes of the depth of
polarization modulation in the analyzed scheme depend on
a degree of coherence of superposing waves [23–26]. Mie
scattering is polarization dependent. It can be supposed that
the distribution of polarization states in the observation
plane determines the optical force value causing the particle
motion.
Opto−Electron. Rev., 20, no. 3, 2012
to take into account only a gradient component while calcu−
lating the optical force in the first approximation. Then the
optical force can be presented as
r
sn r
1
Ftr = - n m aÑE 2 + m S,
2
c
r
1
where Fgrad = - n m aÑE 2 is responsible for the effect of
2
the optical force gradient component.
As it is known
r
c
S=
4p
e 2r
E s,
m
r
where s is an unit vector in the direction of propagation [28].
c e 2
Or S =
E .
4p m
r
2pn m a
4pS m
1
T h e n E2 =
a n d Fgrad = - n m aÑE 2 = c e
2
c
m
ÑS
e
Thus, the effect of the gradient force is linked with the
gradient of energy volume density and, correspondingly,
with the value of the coherence degree of interacting fields.
Temperature effects influencing the motion of particles
arranged in the optical field take place in the medium sur−
rounding the particle. In certain cases, the force which is
249
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Influence of the Brownian movement on the motion of nano− and micro−particles in the inhomogeneous optical field
determined by temperature effects can exceed the gradient
force value and hinder the capture of nano−particles. Here
only a correct choice of laser radiation power allows to cre−
ate such temperature conditions that make the force caused
by temperature change insignificantly in comparison with
the gradient force which determines the capture of Rayleigh
nano−particles.
r
r In the first approximation we will consider that Fopt =
Fgrad , which allows us to connect particle trapping velocity
with energy inhomogeneity value of the optical field and
a coherence degree of interacting initial fields.
Mie scattering particles
Models for Mie scattering particles are based on a computer
simulation of optical forces caused by reflection and refrac−
tion from a spherical surface neglecting diffraction effects.
Using conservation of the momentum, the total force acting
onto the sphere will be the summation of all the force acting
onto it. The scattering force ( Fsc ) and the gradient force
( Fgrad ) can be simplified into the following equations [29]
n P ïì
T 2 [cos( 2q -2e ) + R cos( 2q )] ïü
Fsc = m í 1 + R cos( 2q ) ý (3)
c îï
1 + R 2 + 2 R cos 2e
þï
and
Fgrad =
n m P ìï
T 2 [cos( 2q -2e ) + R sin( 2q )] üï
í 1 + R sin( 2q ) ý, (4)
c ïî
ïþ
1 + R 2 + 2 R cos 2e
where q is the angle of incidence, e is the angle of refraction
and R and T are the Fresnel reflection and the refraction coeffi−
cients, P is the power of the incident light, n m is the index of
refraction of the surrounding medium. R and T are polarization
dependent and, therefore, so is the optical force. The resulting
force includes the effects of all internally reflected and re−
fracted beams. Hence, it is exact within the corresponding ap−
proximation. Absorption of laser energy power by Mie parti−
cles can take place as well. The overall force exerted by a beam
with a given profile is simply a vector sum of forces resulting
from the ensemble of rays that comprise the beam. Thus, in
this case Fopt = Fsc + Fgrad .
The modulation of polarization in the observation plane,
considered in the proposed interference scheme, causes the
change of both optical force components ( Fsc , Fgrad ) and the
redistribution of particles in motion in the analyzed field.
The computer simulation of the Mie particle motion in
the inhomogeneously−polarized optical field makes it possi−
ble to determine the influence of the polarization modulation
depth and the coherence degree of interacting linearly−pola−
rized plane waves upon the velocity of particle motion and
the resulting force pre−setting this motion [23–25].
3. Simulation of mechanical forces which
determine particles behaviour in viscous
medium
May the analyzed particles (Rayleigh or Mie ones) be situ−
ated in a viscous medium. Let us assume, that the mixture
250
consists of a viscous incompressible carrier phase and sphe−
rical particles whose radius is r and whose mass is m.
While moving in liquid a particle experiences friction
r
force, which at therconstant value of the velocity v is defined
r
r
by Stokes law as Fst = 6phrv = mgv, where g is the friction
coefficient, h is the dispersive medium viscosity. In this
case, it is not enough to introduce the friction force (Stokes
force) to describe Brownian motion of this object. Such an
approximation corresponds to an approximation of the con−
tinuous medium. In order to take into account the atomic
structure of the liquid, Langevin introduced a corresponding
additional force FL = my(t ) into the motion equation which
is a random time function. The directions of such a random
force are on equal term and the averaged values of the force
are equal to zero. The correlation time of the Langevin force
values is equal to zero. In this case, the correlation time of
L
the Langevin force values t cor
is much less than the relaxa−
tion time of particles at the expense of the friction force
L
t rel = 1 g , e. g., t cor
<< t rel . Such random sources are
d−correlated and the random process y(t) can be considered
as Gauss process [30]. Thus, the random process, which is
preset by the Brownian motion, is approximated with the
use of the Gauss distribution.
Then, the equation of particle motion, under the effect of
the optical force in the viscous medium and with allowance
made for the Brownian movement, is put down as
r
r
r
r
dv r
= Fopt + Fst + F0 + FL ,
m
dt
r
where F0 is the external force, e. g., gravity force.
4. Simulation results
The computer simulation was performed within the frame−
work of the approach set forth in Introduction. The contribu−
tion of gradient force and light scattering force to the result−
ing optical force which conditions the particles motion is
studied, as well.
The investigation was performed in a spatial periodi−
cally−polarized optical field, which was formed as a result of
interference of two plane−polarized equal in intensity waves
in the incidence plane [23–25]. Both lasers and partially
coherent sources were used as radiators.
In the offered interference scheme with inhomogeneous
intensity in a registration plane, the modulation of the
energy volume density is realized in the observation plane.
The formed inhomogeneous optical field is able to influence
particles position. The optical force influence on particles
depends on coherence properties of superposing waves
which form the energy inhomogeneous field.
Let us consider behaviour of the tested particles in the
inhomogeneous optical field, which, according to the
experiment proposed in Ref. 16, are particles of hydrosol
of gold of 1− and 100−nm size and are located at room tem−
perature.
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We analyze the motion of Mie and Rayleigh particles in
the resulting optical field which is formed by the interaction
of two waves of different values of the coherence degree.
On a viscous medium side, a place of particle localization,
a complex force is affecting particles. It essentially changes
the resulting force. Thus, when changing randomly the loca−
tion of particles by the force Brownian component, time dis−
tribution of the optical force changes, as well. In the process
of simulating forces which pre−set the Brownian motion the
order of force values is chosen commensurable with the
order of the resulting force value.
Let us analyse Mie particles motion
Time dependence of the resulting optical force which con−
sists of gradient and scattering forces is presented in Fig. 2.
In this case, the Brownian force component can influence
the optical force distribution.
The coherence degree of interacting waves influences
the optical force value. The change of the coherence degree
leads to the change of depth of a spatial modulation of pola−
rization and, correspondingly, to the change of depth of
a spatial modulation of the averaged Poynting vector value
which essentially influences the gradient component of the
resulting optical force. If we neglect the influence of the
medium on a particle motion, we obtain the maximum re−
sulting optical force provided that the optical field is formed
by a coherent interaction of the initial waves (an optical
force gradient component is at its maximum and a scattering
component of the optical force is present). When the inter−
action of the initial waves is incoherent, only the scattering
component of the resulting optical force will take part in the
formation of the resulting optical force.
The Brownian particle motion influences time distribu−
tion of the resulting optical force. Thus, the fluctuations of
the optical force are explained by random chaotic particle
motion, connected to chaotic influence of the Brownian
force (Fig. 2, curves 2 and 3). The optical force gradient
Fig. 2. Time dependence of resulting optical force, affecting Mie
particle, with allowance made for random fluctuations in motion
caused by influence of the Brownian forces.
Opto−Electron. Rev., 20, no. 3, 2012
component moves the particle into a region of the maximum
energy density, correspondingly the Brownian component
of the force randomly removes the particle from the maxi−
mum. The result of changing the particle position will be the
optical force change which will again return it to the region
of the maximum. Let the optical field be formed by an inco−
herent interaction of superposing fields with the absence of
the optical force gradient component (Fig. 2, curve 1). Then,
the random particle motion, caused by the Brownian force
effect in the homogeneous optical field does not influence
any optical force and the force remains constant.
The resulting force is changed equally by random force
components, arising from the side of the liquid during the
particles motion and from the resulting optical force. Only
under the effect of the optical force, particles redistribute in
the optical field and concentrate in the region of the energy
density maximum value, continuing their directional
motion.
Mechanical forces, induced by the medium and sur−
rounding particles, try to push them out of their location into
which the gradient optical force directs them.
We shall compare the particle motion velocity when the
Brownian motion is not taken into consideration in the
resulting force (Fig. 3) and the velocity of particle motion
when the Brownian force is taken into consideration in the
resulting force (Fig. 4).
The plots which describe time dependence of the parti−
cle motion velocity and are presented in Fig. 3 show, that
the increase of the coherence degree leads to the particle
motion velocity increase in the regions of the maximum
energy density value. The energy density maximum gradi−
ent in the case of absolutely coherent waves determines the
particle motion maximum velocity.
Figure 4 shows the motion velocity change of Mie parti−
cles which are localized in the viscous medium. The forces
which cause the motion velocity change possess a compo−
Fig. 3. Motion velocity of Mie particles under change of coherence
degree of interacting waves (h (1, 2 ) ):1 - h (1, 2 ) = 1; 2 - h (1, 2 ) = 0.5;
3 - h (1, 2 ) = 0. The normalized value of velocity motion relative to
the value, obtained in the optical field with two interacting abso−
lutely−coherent linearly−polarized plane waves.
251
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Influence of the Brownian movement on the motion of nano− and micro−particles in the inhomogeneous optical field
Fig. 4. Change of Mie particle motion velocity under the effect of
forces arising from optical field and viscous medium, including
forces caused by the Brownian movement.
Fig. 5. Change of resulting optical force, affecting Rayleigh parti−
cles in viscous medium with allowance made for the Brownian
forces.
nent induced by the optical force action and the action of
force on the liquid side. In this case, the Brownian compo−
nent of the resulting force determines random changes of
the velocity which is supported by the results of a computer
simulation and is shown in Fig.4. Irrespective of the cohe−
rence peculiarities of initial superposing waves, the optical
field influence on the velocity motion will be the same.
The gradient component of the optical force determines
the place of particle trapping. The force, acting on the mini−
mum side, e. g., the Brownian motion force, is responsible
for the position of particles in the place of trapping (Fig. 5,
curves 2 and 3).
Let us analyse the velocity change of Rayleigh particles,
positioned in the optical field, when the Brownian motion is
taken into account (Fig. 6) and when the Brownian motion
is not taken into account (Fig. 7).
In the case of an incoherent interaction of fields (Fig. 6,
curve 3) zero value of the optical force is responsible for the
absence of any influence upon the particle motion.
Any coherent interaction of optical fields causes the cre−
ation of such a resulting optical field which is able to change
the particles motion dynamics in this field (Fig. 6, curves 1
Let us analyze the motion of Rayleigh particles
The size of Rayleigh particles in our computer simulation
was 1000 times smaller than the size of Mie particles and
the mass of Rayleigh particles is 100 times smaller than the
mass of Mie particles.
Figure 5 shows the change of the resulting optical force
affecting Rayleigh particles in the viscous medium. The
scattering component of the optical force is much smaller
than the gradient force. That is why we can neglect the scat−
tering component for Rayleigh particles. If the optical field
is formed by incoherent interaction of the initial waves, the
resulting optical force practically zeroes. It is caused by the
absence of energy gradient density (Fig. 5, curve 1).
The increase of the coherence degree of the superposing
waves stipulates the increase of modulation depth of the
energy volume density distribution. This, in its turn, causes
a significant increase of the optical force gradient compo−
nent. The random change of the optical force value is
explained by the fact that the random effect of the Brownian
forces try to bring the particle out of the position into which
it is directed by the optical force gradient component. In
order to return the particle into its initial position, the optical
force must compensate for the Brownian force action. This
results in random changes of the optical force which is
shown in Fig. 5, curves 2 and 3. When the gradient force
does not arise and the gradient energy volume density is
absent, the Brownian forces random effects on the resulting
optical force are not significant (Fig. 5, curve 1).
252
Fig. 6. Change of motion velocity of Rayleigh particles with time
and with change of coherence degree of interacting waves (h (1, 2 ) ):
1 - h (1, 2 ) = 1; 2 - h (1, 2 ) = 0.5; 3 - h (1, 2 ) = 0. Normalized value of
velocity motion relative to value, obtained in optical field with two
interacting absolutely−coherent linearly−polarized plane waves, is
plotted along the ordinate.
Opto−Electron. Rev., 20, no. 3, 2012
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References
Fig. 7. Dependence of motion velocity of Rayleigh particles with
forces affecting particles from optical field and from viscous me−
dium, including forces causing the Brownian motion.
and 2). While comparing curves 1, 2 and Fig. 6, it is possible
to say, that the coherence degree of superposing optical
fields essentially influences the particles motion velocity.
The coherence degree is also set by the time during which
the particle changes its position and localizes in the maxi−
mum energy regions of the inhomogeneous optical field.
While analysing the time dependence of Rayleigh parti−
cle motion velocity (Fig. 7), we can see that the influence of
the optical field on the motion velocity is essential at the ini−
tial moments when particles redistribute in the inhomoge−
neous optical field under the effect of the resulting optical
force. In this case, the gradient force effect is sufficient for
changing the velocity. With time, when the particle is
trapped by the optical field and the motion velocity is zero
(Fig. 6), the availability of Brownian forces affects ran−
domly not only the particles location, but also the velocity
of their motion. Matching between the particle motion
velocity and the degree of coherence of superposing waves
forming the optical field for particle location disappears
with time as well.
General time necessary for a redistribution of particles in
the optical field is much shorter for Rayleigh particles than
for Mie particles.
5. Conclusions
When examining particles characterized by different light
scattering mechanisms, we analysed the resulting force
which causes the particle motion and consists of the optical
force and the forces arising on the liquid side. It includes the
force causing the Brownian motion of the particles under
investigation, as well. It is shown, that the optical force
serves to straighten up the particle motion and the trapping
of the above mentioned particles. The Brownian random
force fluctuations influence the resulting optical force and
change the particle motion velocity. In this case, informa−
tion about coherence features of the formed optical field,
causing the particle motion, is lost.
Opto−Electron. Rev., 20, no. 3, 2012
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