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Computational toolbox for optical tweezers in geometrical optics
Agnese Callegari,∗Mite Mijalkov, A. Burak G¨ok¨oz, and Giovanni Volpe†
Soft Matter Lab, Department of Physics, Bilkent University, Cankaya, Ankara 06800, Turkey.
(Dated: March 4, 2014)
Optical tweezers have found widespread application in many fields, from physics to biology. Here,
we explain in detail how optical forces and torques can be described within the geometrical op-
tics approximation and we show that this approximation provides reliable results in agreement
with experiments for particles whose characteristic dimensions are larger than the wavelength of
the trapping light. Furthermore, we provide an object-oriented software package implemented in
MatLab for the calculation of optical forces and torques in the geometrical optics regime: OTGO -
Optical Tweezers in Geometrical Optics. We provide all source codes for OTGO as well as the
documentation and code examples – e.g., standard optical tweezers, optical tweezers with elongated
particle, windmill effect, Kramers transitions between two optical traps – necessary to enable users
to effectively employ it in their research and teaching.
INTRODUCTION
Optical tweezers are tightly focused laser beams capa-
ble of holding and manipulating microscopic particles in
three dimensions. Since their invention in 1986 [1], opti-
cal tweezers have been increasing and consolidating their
importance in several fields, from physics to biology [2–
7]. In the last fifteen years, thanks to the development
of relatively simple and cheap setups, optical tweezers
have also started to be employed in undergraduate and
graduate laboratories as a tool to introduce students to
advanced experimental techniques [8–11].
Part of the reason for the success of optical tweez-
ers lies in that the forces they can exert – from tens of
piconewtons down to tens of femtonewtons – are just
in the correct order of magnitude for a gentle but ef-
fective manipulation of colloidal particles and biologi-
cal samples [2–7]. An accurate mathematical descrip-
tion of these forces requires the use of electromagnetic
theory in order to model the interaction between an in-
coming electromagnetic wave and a microscopic particle
[12–14]. However, this can be a daunting task. There-
fore, it comes handy that simpler theoretical approaches
have been shown to deliver accurate results in the lim-
its where the particle characteristic dimensions are much
smaller or much larger than the wavelength of the trap-
ping light [15], which is typically between 532 nm and
1064 nm for optical tweezing applications. For particles
much smaller than the wavelength, one can make use of
the dipole approximation, which has already been ex-
tensively described and employed to describe the trap-
ping of nanoparticles [7]. For particles much larger than
the wavelength, such as cells and large colloidal particles,
whose size is typically significantly larger than one mi-
crometer, one can make use of geometrical optics for the
calculation of optical forces [16]. This approach has been
successfully employed, for example, to describe optical
forces acting on cells [17], the deformation of microscopic
bubbles in a optical field [18], the optical lift effect [19]
and the emerging of negative optical forces [20].
In this paper, we explain in detail how geometrical
optics can be employed in order to study the optical
forces and torques arising in an optical tweezers. We
will first introduce how optical tweezers can be modeled
in geometrical optics. Then, we will study in detail the
forces associated to the scattering of a ray and of an
optical beam by a spherical particle, distinguishing be-
tween scattering and gradient forces. Finally, we will
explore some more complex situations, such as the aris-
ing of torque on non-spherical objects and the emergence
of Kramers transitions between two optical tweezers. As
an integral part of this article, we provide a complete
MatLab software package – OTGO - Optical Tweezers
in Geometrical Optics – to perform the calculation of
optical forces and torques within the geometrical optics
approach [21]. OTGO is fully documented, accompanied by
code examples and ready to be employed to explore more
complex situations, both in research and in teaching. In
fact, we have implemented OTGO using an object-oriented
approach so that it can be easily extended and adapted
to the specific needs of users; for example, it is possible
to create more complex optically trappable particles by
extending the objects provided for spherical, cylindrical
and ellipsoidal particles. In particular, we have used OTGO
to obtain all the results presented in this article.
GEOMETRICAL OPTICS MODEL OF OPTICAL
TWEEZERS
A schematic of a typical optical tweezers is shown in
Fig. 1(a) and in supplementary movies 1, 2 and 3 [21].
A laser beam is focused by a high-NA objective (O1) in
order to create a high-intensity focal spot where a micro-
scopic particle can be trapped. Typically, the particle is
a dielectric sphere with refractive index npimmersed in a
liquid medium with refractive index nm. The scattering
of the focused beam on the particle generates some opti-
cal restoring forces that keep the particle near the focus.
The sum of the incoming and scattered electromagnetic
arXiv:1402.5439v1 [physics.optics] 21 Feb 2014
2
fields can be collected by a second objective (O2) and
projected onto a screen placed in the back-focal plane.
The position of the optically trapped particle can be de-
tected by using the image on the screen [22], as shown in
Figs. 1(b) and 1(c). Note that Fig. 1 is not to scale by
a factor ∼100 because, in an actual setup, the objective
focal length is ∼170 µm and the particle size is typically
∼2µm.
In the geometrical optics approach [16], the incoming
laser beam, whose intensity profile is shown on the left
of Figs. 1(a)-1(c), is decomposed into a set of optical
rays, which are then focused by the objective O1. As the
rays reach the particle, they get partially reflected and
partially transmitted. The direction of the reflected and
transmitted rays are different from those of the incoming
rays. This change of direction entails a change of mo-
mentum and, because of the action-reaction law, a force
acting on the sphere. As we will see, if np> nm, these
optical forces tend to pull the sphere towards the equilib-
rium position near the focal point. As the scattered rays
reach the objective O2, they are collected and projected
onto the back-focal plane.
FORCES BY A RAY ON A PLANAR SURFACE
The energy flux transported by a monochromatic elec-
tromagnetic field, such as the one of a laser beam, is given
by its Poynting vector
S=1
2µRe {E×B∗},(1)
where Eand Bare the complex electric and magnetic
fields. In order to describe how this energy is transported,
a series of rays can be associated with the electromagnetic
field [23]. These rays are lines perpendicular to the elec-
tromagnetic wavefronts and pointing in the direction of
the electromagnetic energy flow.
When a light ray impinges on a flat surface between
two media with different refractive indices, it is partly
reflected and partly transmitted. Given an incidence an-
gle θi, i.e., the angle between the incoming ray riand
the normal nto the surface at the incidence point, the
reflection angle θris given by the reflection law
θr=θi(2)
and the transmission angle θtis given by Snell’s law
θt= asin ni
nt
sin θi,(3)
where niis the refractive index of the medium of the
incident ray rrand ntis the one of the medium of the
transmitted ray rt. Both rrand rtlie in the plane of
incidence, i.e., the plane that contains riand n. Because
of energy conservation, the power Piof rimust be equal
(a)
O1O2
nm= 1.33
np= 1.50
NA = 1.00
F
(b)
O1O2
F
(c)
O1O2
F
FIG. 1. (color online). Schematic of an optical tweezers setup
(not to scale). A Gaussian laser beam, whose intensity is
shown on the left, is divided into a set of optical rays (lines).
The rays that can cross an aperture stop, whose radius is
equal to the beam waist in this case, are then focused by an
objective (O1, NA = 1.00 in water). Near the focal point, a
dielectric spherical particle (refractive index np= 1.50) im-
mersed in a fluid (refractive index nm= 1.33) scatters the
rays (for clarity, the reflection of the incoming beam and the
internally scattered rays are omitted) and, therefore, experi-
ences a restoring recoil optical force F(black arrow). The
scattered rays are collected by a second objective (O2) and
projected onto a screen placed in the back-focal plane. The
position of the particle can be tracked by monitoring the back-
focal plane image, which sensitively depends on the position
of the particle, e.g., (a) at the focal point, (b) displaced in
the transverse plane and (c) displaced along the longitudinal
direction. The distance between the objectives and the size
of the particle are not to scale by a factor ∼100. See also
supplementary movies 1, 2 and 3 [21].
to the sum of the power Prof rrand the power Ptof rt,
3
i.e.,
Pi=Pr+Pt.(4)
How the power is split can be calculated by using
Maxwell’s equations with the appropriate boundary con-
ditions [24]. The result is expressed by Fresnel’s equa-
tions and depends on the polarization of the incoming
ray, as we must distinguish the case when the electric
field of the ray oscillates in the plane of incidence (p-
polarization) from the one when it oscillates in a plane
perpendicular to the plane of incidence (s-polarization).
The Fresnel’s reflection and transmission coefficients for
p-polarized light are
Rp=
nicos θt−ntcos θi
nicos θt+ntcos θi
2
,(5)
Tp=4nintcos θicos θt
|nicos θt+ntcos θi|2(6)
and for s-polarized light
Rs=
nicos θi−ntcos θt
nicos θi+ntcos θt
2
,(7)
Ts=4nintcos θicos θt
|nicos θi+ntcos θt|2.(8)
For unpolarized and circularly polarized light, one can
use the average of the previous coefficients, i.e.,
R=Rp+Rs
2,(9)
T=Tp+Ts
2.(10)
The recoil optical forces are equal and opposite to the
rate of change of linear momentum of the light. Since for
a ray of power Pin a medium of refractive index n, the
momentum flux is nP/c, where cis the speed of light in
vacuum, the optical force is [16]
Fpl =niPi
cˆ
ui−niPr
cˆ
ur−ntPt
cˆ
ut,(11)
where ˆ
uiis the unit vector of ri,ˆ
uris the unit vec-
tor of rrand ˆ
utis the unit vector of rt. We must
note that the definition of the momentum of light in a
medium is a thorny issue, which is often referred to as
the Abraham-Minkowski dilemma after the works of Her-
mann Minkowski [25] and Max Abraham [26]. This issue
is discussed in detail, e.g., in Refs. [27, 28]. Since most re-
sults in optical trapping and manipulation do not depend
qualitatively on the momentum definition, in this work
we employ the Minkowski momentum definition which in
fact is the most often employed in optical tweezers stud-
ies [16, 29]. However, we remark that all results can be
easily adapted to the Abraham momentum definition by
changing the definition of the force in Eq. (11) [27].
i
r
i
θ
(1 )
r
r
(2 )
t
r
(3 )
t
r
(4 )
t
r
(5 )
t
r
(6 )
t
r
(1 )
t
r
(2 )
r
r
(3 )
r
r
(4 )
r
r
Qray
Qray,g
Qray,s
30◦60◦90◦
0
0.2
0.4
0.6
θi
Q(θi)
(a)
(b)
FIG. 2. (color online). (a) Scattering of a ray impinging on a
sphere. The incident ray riimpinges on a glass spherical parti-
cle (np= 1.50) immersed in water (nm= 1.33). The reflected
(r(j)
r) and transmitted (r(j)
t) rays for the first seven scattering
events are represented. Because of the spherical symmetry of
the particle, all rays lie in the plane of incidence. See also
supplementary movies 4 and 5 [21]. (b) Corresponding trap-
ping efficiencies as a function of the incidence angle θi. See
also supplementary movies 6 and 7 [21].
FORCES BY A RAY ON A SPHERE
We now consider a ray riof power Piimpinging from
a medium with refractive index nmon a dielectric sphere
with refractive index npat an incidence angle θi, as shown
in Fig. 2(a) and in supplementary movies 4 and 5 [21]. As
soon as rihits the sphere, a small amount of its power,
P(1)
r, is diverted into the reflected ray r(1)
r, while most
power, P(1)
t, goes into the transmitted ray r(1)
t. The ray
r(1)
tcrosses the sphere until it reaches the opposite sur-
face, where again a large portion of its power, P(2)
t, is
transmitted outside the sphere into the ray r(2)
t, while a
small amount of its power, P(2)
r, is reflected inside the
sphere into the ray r(2)
r. The ray r(2)
rundergoes another
scattering event as soon as it reaches the sphere bound-
ary, and the process continues until all light has escaped
from the sphere. The force Fray produced on the sphere
by this series of scattering events can be calculated by
4
using repeatedly Eq. (11), i.e.,
Fray =nmPi
cˆ
ui−nmP(1)
r
cˆ
u(1)
r−
∞
X
j=2
nmP(j)
t
cˆ
u(j)
t,(12)
where ˆ
ui,ˆ
u(1)
rand ˆ
u(j)
tare the unit vectors of the incident
ray, the first reflected ray and the j-th transmitted ray,
respectively. We note that the dependence of Eq. (12)
on npis hidden in the dependence of the quantities P(1)
r
and P(j)
ton the Fresnel’s coefficients [Eqs. (7), (8), (5)
and (6)]. Furthermore, we can notice that the absolute
value of the force does not depend on the dimension of
the particle.
Since all the reflected and transmitted rays are con-
tained in the plane of incidence, as can be seen in
Fig. 2(a) and in the supplementary movies 4 and 5 [21],
also the force Fray in Eq. (12) has components only
within the incidence plane. We can, therefore, split Fray
into a component along the direction of the incoming ray,
i.e., the scattering force Fray,s= (Fray ·ˆ
ui)ˆ
ui=Fray,sˆ
ui,
and a component perpendicular to the direction of the in-
coming ray, i.e., the gradient force Fray,g=Fray −(Fray ·
ˆ
ui)ˆ
ui=Fray,gˆ
u⊥:
Fray =Fray,s+Fray,g=Fray,sˆ
ui+Fray,gˆ
u⊥,(13)
where ˆ
u⊥is the unit vector perpendicular to ˆ
uiand con-
tained in the incidence plane. Interestingly, the gradient
force is a conservative force, while the scattering force is
nonconservative. If np> nm, the particle is attracted
towards the ray [supplementary movie 4 [21]], while, if
np< nm, the particle is pushed away from the ray [sup-
plementary movie 5 [21]].
In order to quantify the effectiveness of the transfer of
momentum from the ray to the particle, we can introduce
the trapping efficiency, i.e., the ratio between the mod-
ulus of the optical force and the momentum per second
of the incoming ray in a medium with refraction index
ni. The trapping efficiency is bound to lie between 0,
corresponding to a ray that is not deflected, and 2, cor-
responding to a ray that is reflected back on its path [16].
For example, for a 1 mW ray, the maximum optical force
is 7 ·10−12 N, i.e., 7 piconewtons. Albeit small, this force
is comparable to the forces that are relevant in the mi-
croscopic and nanoscopic world, e.g., the forces generated
by molecular motors [30], and gives us a first impression
of the potential of optical manipulation. In particular,
we can define the scattering trapping efficiency
Qray,s=c
niPi
Fray,s,(14)
the gradient trapping efficiency
Qray,g=c
niPi
Fray,g(15)
and the total scattering efficiency
Qray =qQ2
ray,g+Q2
ray,s.(16)
Fig. 2(b) and supplementary movie 6 [21] show the trap-
ping efficiencies as a function of θifor a circularly po-
larized ray impinging on a glass sphere (np= 1.50) im-
mersed in water (nm= 1.33); supplementary movie 7 [21]
shows the trapping efficiencies for a circularly polarized
ray impinging on an air bubble (np= 1.00) immersed in
water (nm= 1.33). In both cases, the major contribu-
tion to the total trapping efficiency is given by Qray,g,
while only for very large incidence angles Qray,sbecomes
appreciable.
FORCES BY A FOCUSED BEAM ON A SPHERE
It is not possible to achieve a stable trapping using
a single ray because the particle is permanently pushed
by the scattering force in the direction of the incom-
ing ray, as we have seen in Fig. 2(b). A possible ap-
z
x
x
y
z
Qbe am
Qbe am,g
Qbe am,s-.3
-.2
-.1
.1
.2
.3
R
x
-.3
-.2
-.1
.1
.2
.3
R
(a) (b)
(c) (d)
FIG. 3. (color online). Optical force fields in an optical tweez-
ers. The arrows represent the direction and magnitude of the
force exerted on a glass spherical particle (np= 1.50) in water
(nm= 1.33) illuminated by a highly focused Gaussian beam
(NA = 1.30, beam waist equal to the aperture stop radius)
propagating along the z-direction as a function of the parti-
cle position (a) in the longitudinal (zx) plane and (b) in the
transverse (xy) plane. The shaded area represents the dimen-
sion of the particle. The corresponding trapping efficiencies
are shown in (c) and (d) for particle displacements along the
z-axis and x-axis, respectively. Note that for displacements
along the z-axis both the scattering and the gradient force are
directed along z, while for displacements along the x-axis the
gradient force is directed along x, but the scattering force is
directed along z.
5
(a)
z
x
x
y
(b)
z
x
x
y
(c)
z
x
x
y
FIG. 4. (color online). Force fields in an optical tweezers
generated using various kinds of beams. (a) A Gaussian beam
with a waist much smaller than the radius of the aperture stop
can trap a particle in the transverse plane but cannot confine
the particle along the longitudinal direction. (b) A Laguerre-
Gaussian beam, or doughnut beam, improves the trapping
along the longitudinal direction because of the presence of
more power at large angles. (c) A Hermite-Gaussian beam
clearly breaks the cylindrical symmetry of the trap, as can be
seen from the force field in the transverse plane. In all cases,
the force fields are calculated for a glass spherical particle
(np= 1.50) in water (nm= 1.33). The circle on the left
corresponds to NA = 1.30.
proach to achieve a stable trap is to use a second counter-
propagating light ray. In fact, such a configuration using
two laser beams was amongst the first ones to be em-
ployed in order to trap and manipulate microscopic par-
ticles [31] and a modern version has been obtained us-
ing the light emerging from two optical fibers facing each
other [32]. This approach works also if the two beams are
not perfectly counter-propagating, but they are arranged
with a sufficiently large angle.
A more convenient alternative to using several counter-
propagating light beams is to use a single highly-focused
light beam. In fact, rays originating from diametrically
opposite points of a high-NA focusing lens produce in
practice a set of rays that converge at a very large angle,
as can be seen in Fig. 1.
The most commonly employed laser beam is a Gaus-
sian beam. Its intensity profile at the waist is given by
IG(ρ) = I0e−ρ2
2w2
0,(17)
where ρis the radial coordinate, w0is the beam waist,
I0=1
2cε0nmE2
0is the beam intensity at ρ= 0, ε0is the
dielectric permittivity of vacuum, and E0is the modulus
of the electric field magnitude at ρ= 0. Such a beam
can be approximated by a set of rays parallel to the op-
tical axis (z) each endowed with a power proportional to
the local intensity of the beam. The resulting rays are
then focused by an objective lens, which has the effect of
bending the light rays towards the focal point, as shown
in Fig. 1 and supplementary movies 1, 2 and 3 [21]. Each
one of these rays produces a force F(m)
ray on the sphere
given by Eq. (12). The total optical force exerted by the
focused beam on the sphere is then the sum of all the
rays’ contributions, i.e.,
Fbeam =X
m
F(m)
ray (18)
In Fig. 3(a) and 3(b), the force fields in the longitudinal
(zx) and transverse (xy) planes are represented as a func-
tion of the distance of the high-refractive index spherical
particle (np= 1.50, nm= 1.33) from the focal point, in
the case of an objective with NA = 1.30 and a circularly
polarized beam.
To have a comparison of the force obtained from our
simulations with the forces usually found in experiments,
we can compare our prediction with the results in Ref.
[30]. We take, for comparison, the measured trapping
stiffnesses for a polystyrene sphere (np= 1.57) with a
1.66 µm diameter in a trap generated using a Gaussian
beam with power P= 10 mW focused by an objective
with NA = 1.20. Performing a calculation with the cited
parameters, we obtain a trapping stiffness along the lon-
gitudinal direction (z) equal to kOTGO
z= 6.45 pN/µm, and
a trapping stiffness along the transversal direction (x)
equal to kOTGO
x= 12.66 pN/µm, in reasonable agreement
with the experimental values found in Ref. [30], which are
respectively kexp
z= 3.85 pN/µm and kexp
x= 11.0 pN/µm.
The optical force field is cylindrically symmetric
around the z-axis. The equilibrium position lies on z-
axis, i.e., (x, y) = (0,0), and is slightly displaced towards
positive zbecause of the presence of scattering forces,
as is commonly observed in experiments [33]. In fact, a
Brownian particle in an optical trap is in dynamic equi-
librium with the thermal noise pushing it out of the trap
and the optical forces driving it toward the center of the
trap [11], as can be seen from the Brownian motion that
the particles experience in supplementary movies 1, 2 and
3 [21]. The maximum value of the force is achieved when
the particle displacement is about equal to the particle
radius R. We can notice again that, like in the case of a
single ray, the value of the force does not depend on R;
however, the trap stiffness, which is given by the force
divided by the displacement is inversely proportional to
R.
The force Fbeam in Eq. (17) can be split into a scat-
tering force Fbeam,s, given by the sum of the scattering
6
forces of each ray, and a gradient force Fbeam,g, given by
the sum of the gradient forces of each ray [16], i.e.,
Fbeam,s=X
m
F(m)
ray,s(19)
and
Fbeam,g=X
m
F(m)
ray,g.(20)
Like in the case of a single ray, while the gradient force
Fbeam,gis conservative, the scattering force Fbeam,scan
give rise to nonconservative effects, as has been shown in
various experiments [33]. These nonconservative effects
are, however, small [34] as can be seen from the small
displacement along the z-axis of the equilibrium position.
It is now possible to define the scattering efficiencies
for the focused beam as
Qbeam,s=c
niPbeam
Fbeam,s,(21)
Qbeam,g=c
niPbeam
Fbeam,g(22)
and
Qbeam =c
niPbeam
Fbeam,(23)
where Pbeam is the power of beam that contributes to the
focal fields, i.e., after the aperture stop. The scattering
coefficients are shown in Fig. 3(c) and 3(d) for a sphere
displaced along the longitudinal and transverse direction,
respectively. If the sphere is on the z-axis, i.e., the prop-
agation axis of the beam, both the scattering force and
the gradient force act only along the z-direction because
of symmetry. For displacements of the particle along the
x-direction, the gradient force is along the x-direction
and the scattering force along the z-direction.
Other kinds of beams can also be used in optical trap-
ping experiments. In particular, Laguerre-Gaussian and
Hermite-Gaussian beams [24] have been widely exploited.
An accurate description of these beams requires one to
take into account their orbital angular momentum [35],
which can have major effects on their trapping proper-
ties. However, the features related to the presence of spin
angular momentum and of a non-uniform phase profile in
the beam, which lead, e.g., to the presence of orbital an-
gular momentum, cannot be accurately modeled within
the geometrical optics approach. Nevertheless, some fea-
tures connected to the different intensity distribution in
Laguerre-Gaussian and Hermite-Gaussian beams can be
explored, as shown in Fig. 4.
FURTHER NUMERICAL EXPERIMENTS
This section provides some guidelines and examples
on how readers can use geometrical optics and OTGO to
explore more complex situations both in the lab and in
the classroom, going beyond the basic optical tweezers
case of a microscopic sphere optically trapped in a highly
focused laser beam.
(a) t = 10 ms
(b) t = 30 ms
(c) t = 70 ms
FIG. 5. (color online). Simulation of the motion of a pro-
late ellipsoidal Brownian particle (np= 1.50, short semiaxes
2.00 µm, long semiaxis 3.33 µm) in water (nm= 1.33) un-
der the action both of Brownian motion and of the optical
forces and torques arising from a highly focused Gaussian
beam (NA = 1.30, Pi= 1 mW), whose rays are coming from
the bottom. Because of the presence of optical torque, after
∼70 ms the particle’s long axis gets aligned along the longi-
tudinal direction. See also supplementary video 8 [21].
We will first consider the case of a non-spherical parti-
cle. If the particle is convex, one can still use the formula
in Eq. (13) to calculate the forces due to a single ray.
However, in general, the scattered rays and the force will
not lie all on incidence plane and, therefore, apart from
the optical force also an optical torque can arise:
T= (P1−C)×nmPi
cˆ
ui−(P1−C)×nmP(1)
r
cˆ
u(1)
r−
−
∞
X
j=2
(Pj−C)×nmP(j)
t
cˆ
u(j)
t(24)
where Cis center of mass of the particle and Pjis the po-
sition where the j-th scattering event takes place. This
is different from the case of a spherical particle, such
as the one shown in Fig. 2(a), where the torque is null
[16]. The typical order of magnitude of the torque on a
particle with characteristic dimension of ∼1µm is ap-
proximatively 10−18 Nm to 10−21 Nm for a ray of power
≈1mW, as shown in experiments [30, 36–38]. For ex-
ample, we can consider the case of an elongated parti-
cle, which can be modelled as a prolate ellipsoidal glass
particle (short semiaxes 2.00 µm, long semiaxis 3.33 µm,
np= 1.50, nm= 1.33), as shown in Fig. 5. Elongated
particles are known to get aligned with their longer axis
along the longitudinal direction because of the presence
of an optical torque [39]. We simulated the optical forces
using OTGO and the Brownian motion using the approach
described in Ref. [40] using the freeware HYDRO++ to cal-
culate the diffusion tensor [41], assuming the particle to
be a room temperature (T= 300 K). We start from a
7
Tto t
ri
rr
F
Mirror 1
Mirror 2
Mirror 3
Mirro r 4
FIG. 6. (color online). The windmill effect. An asymmetric
object illuminated by a plane wave, i.e., by a set of paral-
lel rays (coming from the top), undergoes an optical torque
that can set it into rotation. The object in the illustration is
characterized by a rotational symmetry that permits the can-
cellation of the optical forces in the transverse plane, while
an optical torque is still present.
configuration where the particle center of mass is at the
focal point, but the longer semiaxis lies in the transverse
plane, as shown in Fig. 5(a). Because of the presence of
the optical torque due to a focused Gaussian laser beam
of power 1 mW, the particle gets aligned with the long
semiaxis along the longitudinal direction in about 70 ms,
as shown in Figs. 5(b) and 5(c), which is a result compa-
rable to experiments [42].
Closely related to the optical torque, the windmill ef-
fect [43], where an asymmetric object illuminated by a
plane wave, i.e., a series of parallel rays, can start rotating
around its axis, can also be reproduced using OTGO. In the
simulation, we took a set of parallel rays and shone them
onto a perfectly reflecting object reproducing the shape
of a windmill wheel, i.e., four circular mirrors oriented
as shown in Fig. 6. In the presence of an illuminating
electromagnetic field, this object starts rotating. Simi-
lar structures have been indeed experimentally realized
[43, 44].
Another interesting effect that can be reproduced using
OTGO is the emergence of Kramers’ transitions [45, 46].
We simulated the motion of a Brownian spherical par-
ticle with radius R= 1 µm in the presence of a dou-
ble trap obtained by focalizing two Gaussian beams each
with power 0.25 mW so that their focal points laid at a
distance d= 1.7µm in the transverse plane, as shown in
Fig. 7(a). Letting the system free to evolve under the
(a)
(b)
0.85
−0.85
0
x[µm]
t[s]
20 40 60 80 100 120 140 16 0
FIG. 7. (color online). (a) Kramers’ transitions of a spherical
particle of radius R= 1 µm (np= 1.50) in water (nm= 1.33)
held in the optical potential generated by a double optical
tweezers, i.e., two highly focused Gaussian beams (NA=1.30,
Pi= 0.2 mW) whose focal points are separated in the trans-
verse plane by a distance d= 1.7µm and whose rays are
coming from the bottom. The position of the particle is il-
lustrated by the solid line and shows that the particle jumps
between two equilibrium positions. (b) Particle position along
the transverse axis that joins the two traps centers as a func-
tion of time. See also supplementary movie 8 [21].
action of the Brownian motion and the optical forces,
the particle jumps from one potential well to the other
one, as shown by the trajectory in Fig. 7(b). The rel-
atively low value of the power of the trapping beams,
necessary in order to be able to observe the transitions
at room temperature T= 300 K within a relatively short
time frame, is comparable with the one in actual experi-
ments [46]. Changing the parameters of the system, e.g.,
distance between the focal spots, beam power, temper-
ature of the system, one can alter the transition rates
and, moreover, an additional local minimum may arise
between the two traps (e.g., for d= 1.5R), which has
8
ri
r(1c )
r
r(1c )
t
r(1n )
r
r(1n )
t
r(2c )
t
r(2n )
r
r(2n )
tr(3c )
t
r(3n )
t
r(4c )
t
FIG. 8. (color online). A biological cell can be modeled by
using two spherical particles with different refractive indices
placed one inside the other to represent the cytoplasm and
the nucleus. As the cytoplasm is a non-convex shape, the
scattering process is more complex than in the case of a sphere
and for a given ray multiple scattering events may need to be
taken into consideration.
indeed been observed in experiments [47].
It is also possible to extend the computational capabili-
ties of OTGO to non-convex and/or non-simply-connected
shapes. For example, in Fig. 8 we show the case of a
simple optical model for a biological cell [17]: in first ap-
proximation, a cell containing a nucleus can be modeled
by a sphere (the cytoplasm) containing a smaller sphere
of different refractive index (the nucleus). It is interest-
ing to notice that in a scattering event a ray can now be
split into multiple rays that may not necessarily be able
to escape the particle; this is typical of all non-convex
shapes and can lead to a steep increase of the number of
rays to be taken into account.
Acknowledgements
We would like to thank Sevgin Sakıcı for her help in the
early stages of the development of the code, and Onofrio
M. Marag´o, Philip H. Jones and Rosalba Saija for useful
discussions and suggestions. This work has been partially
financially supported by the Scientific and Technological
Research Council of Turkey (TUBITAK) under Grants
111T758 and 112T235, Marie Curie Career Integration
Grant (MC-CIG) under Grant PCIG11 GA-2012-321726,
and COST Actions MP-1205 and IP-1208.
∗agnese.callegari@fen.bilkent.edu.tr
†giovanni.volpe@fen.bilkent.edu.tr;
http://www.softmatter.bilkent.edu.tr
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