Angular and position stability of a nanorod trapped in an optical tweezers

Angular and position stability
of a nanorod trapped in an optical tweezers
Paul B. Bareil and Yunlong Sheng*
Center for Optics, Photonics and Lasers, Laval University, Quebec, G1K 7P4, Canada
*sheng@phy.ulaval.ca
Abstract: We analyze the trap stiffness and trapping force potential for a
nano-cylinder trapped in the optical tweezers against its axial and lateral
shift and tilt associated to the natural Brownian motion. We explain the
physical properties of the optical trapping by computing and integrating the
radiation stress distribution on the nano-cylinder surfaces using the T-matrix
approach. Our computation shows that the force stiffness to the lateral shift
is several times higher than that to the axial shift of the nano-cylinder, and
lateral torque due to the stress on the side-face is 1-2 orders of magnitude
higher than that on the end-faces of a nano-cylinder with the aspect ratio of
2 – 20. The torque due to the stress on the nano-cylinder surface is 2-3
orders of magnitude higher than the spin torque. We explain why a nano-
cylinder of low aspect ratio is trapped and aligned normal to the trapping
beam axis.
©2010 Optical Society of America
OCIS codes: (140.7010) Trapping; (160.4236) Nanomaterials; (260.2110) Electromagnetic
optics; (350.4238) Nanophotonics and photonic crystals; (350.4855) Optical tweezers or optical
manipulation; (000.0000) General.
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#136650 - $15.00 USD Received 15 Oct 2010; revised 16 Nov 2010; accepted 17 Nov 2010; published 1 Dec 2010
(C) 2010 OSA 6 December 2010 / Vol. 18, No. 25 / OPTICS EXPRESS 26388

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asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14),
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1. Introduction
Manipulating nanoparticles using optical tweezers has attracted intensive research interest
recently [1]. Stable trapping of a nanoparticle is challenging because as the particle size is
reduced to the nano-scale the radiation force is reduced proportionally, while the Brownian
motion increases. Stable trapping of a single crystal KNbO3 nanorod of sub-wavelength cross
section and micrometer length has been demonstrated experimentally [2], showing a potential
for application to microscopy of nano-scale resolution using the second harmonic of the
tapping beam as a scanning-source. In this application fluctuations in position and orientation
of the trapped nanorod can greatly affect the imaging resolution. Thus, the stiffness of the
trap, the angular and position stability of the trapped nanorod become an important issue of
study. Borghese et al investigated the trapping of a linear nanostructure by modelling the
nanorod as a linear chain of identical nano-spheres [3]. However, a significant difference
exists in the geometry of a nanorod and that of a chain of nano-spheres, in which the aspect
ratio has to be discrete and larger than two.
In this paper we compute the optical radiation forces and torques on a nano-cylinder in an
optical tweezers using the T-matrix approach with the point matching method. The T-matrix
approach for calculation of the trapping force is an established method and has been applied to
non-spherical particles [4] for analyzing the optical rotation of elongated particles by rotating
the trapping beam polarization in the micro-machine applications [5]. Our motivation is the
nano-resolution imaging with the emphasis put on the angular and position stability against
the natural Brownian motion of the trapped nanorod in aqueous medium without space
constraint. In this paper the optical force and torque are not only calculated by analytical
expressions of integrals of the Maxwell stress tensor [6], but also by integrating the radiation
stress distribution on the nano-cylinder surfaces, including the side-face and end-faces. The
integrations take more time to compute than the existing analytical formula, but gains physical
insight into the optical trapping of nanorod, so that we can explain the sign change of the
torque as a function of the aspect ratio of the nanorod, and the consequent stable trapping
orientation along or normal to the trapping beam axis. We also compute the position and
angular stiffness of the trap as a function of the size, length and tilt angle of the nano-cylinder
and the beam numerical aperture.
2. Model and calculation
The T-Matrix approach based on the formalism provided in Ref. [4]. was used to compute the
scattering of a nano-cylinder in an optical tweezers. First the incident and scattered fields, Einc
and Escat, and the interior field in the scatter, Eint, are expanded into the vector spherical wave
function (VSWF) basis
max
1
max
int
1
max
(1) (1)
1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
N n
inc nm nm nm nm
n m n
N n
nm nm nm nm
n m n
N n
scat nm nm nm nm
n m n
E r a RgM kr b RgN kr
E r c RgM kr d RgN kr
E r p M kr q N kr
 
 
 
 
 
 
 
 
 
(1)
where the VSWFs, Mnm and Nnm, and the regular VSWFs, RgMnm and RgNnm, are defined as in
Ref. [4]. We expanded the incident field Einc into RgMnm and RgNnm instead of Mnm and Nnm
because when computing in the scale of nano-particles the divergence of Mnm and Nnm at the
origin can slow down the convergence of the T-Matrix solver and requires a higher number of
modes, Nmax.

The VSWFs are a complete and orthogonal set of the solutions to the vector Helmholtz
equation but the Gaussian laser beam is a solution of the paraxial scalar Helmholtz equation,
so that the expansion of a highly focused Gaussian trapping beam on the VSWF basis is an
approximation. The point matching method was used to fit the amplitude distributions in the
far field of the expanded trapping beam and the TEM00 beam in order to determine the
coefficients [5,7], anm and bnm in Eq. (1).
The T-Matrix is defined such that the expansion coefficients of the scattered and the
interior fields can be obtained from the anm and bnm by the T-Matrix as
11 12
21 22
11 12
21 22
p aT T
q bT T
c aTI TI
d bTI TI
    
     
    
    
     
    
(2)
On substituting Eq. (1) into the boundary conditions for the electric and magnetic field
components parallel to the interface of the nano-cylinder as
// // int //
// // int //
inc scat
inc scat
E E E
H H H
  
  
(3)
the T-matrix coefficients can be computed using the point matching method to build and solve
an over-determined linear system of equations with a number of matching points higher than
the number of unknown coefficients.
In the T-matrix approach, we put the nano-cylinder always centered at the origin and
aligned with the z-axis, as shown in Fig. 1, so that the T-matrix needs to be computed only
once for a given nano-cylinder. We let the trapping beam translate and tilt with respect to the
nano-cylinder. By using the translation matrix [8] and the Wigner rotation matrix [9] the
expansion coefficients of the shifted and rotated beams can be computed from the anm and bnm
of the original beam, which was aligned along the z-axis with the focal point at the origin.
The radiation forces and torque were computed by the integrals as
1
Re
2
1
Re
2
S
S
F dSn T
dSn T r
 
  


(4)
where T is the Maxwell stress tensor, r is the position vector, n is the norm to surface and
S is any closed surface enclosing the nano-cylinder. By choosing a spherical S and performing
the integrations in the far field, we obtain the axial and transversal forces by the analytical
expression in SI units as
 
* *
1 1
* *
0 1 1
0 * *
2 ( 2)( 1)( 1)
Im
( 1) (2 1)(2 3)
2
2
Im
( 1)
nm n m nm n m
nm n m nm n m
z
nm nmnm nm
p p q qn n n m n m
n n n a a b bF
k
m
iq p ib a
n n

 
 
        
         
 
  
 
(5)

* *
1 1 1 1
* *
1 1 1 1
* *
1 1 1 10
* *
0 1 1 1 1
1 ( 2)( 1)( 2)
Im
( 1) (2 1)(2 3)
1 ( 2)( )( 1)
Im
2 ( 1) (2 1)(2 3)
nm n m nm n m
nm n m nm n m
n m nm n m nm
x
n m nm n m nm
p p q qn n n m n m
n n n a a b b
p p q qn n n m n m
F
k n n n a a b b

   
   
   
   
       
      
      
    
  
 * * * *
1 11 1
1
Im
( 1)
nm nm nm nmnm nm nm nm
n m n m
ib a i p q iq p ia b
n n
  
 
 
 
 
 
 
 
 
 
        
 
(6)
The spin torque around the z-axis [10] is given as
 
max 2 2 2 2
0
102
N n
nm nmZ nm nm
n m n
m a b p q
k


 
     (7)
where 2nm nma a , 2nm nmb b , nm nmnmp a p  and nm nmnmq b q  , k0 = 2π/λ.
We also computed the stress distribution on the surface of the nano-cylinder and choose
the S as the surface of the nano-cylinder to perform the numerical integrations of Eq. (4) in
order to gain physical insight into the optical trapping of nano-cylinder, although the
computed total force and torque remain the same as that computed with Eqs. (5) and (6). The
torque was calculated by Eq. (4) written as
( , )
S
dS z r n    (8)
where the stress is computed by
 
2
2 2 2 21
2 1 1 12
2
1
2
n t
n
n n E E n
n

 
   
 
(9)
and is considered as always perpendicular to the surface [11] where n is the outwards normal
of the surface, as shown in Fig. 1.
Fig. 1. Nano-cylinder fixed in the spherical coordinate system for calculating forces and
torques.
When the refraction index n1 = 1.33 for the water buffer, n2 = 1.57 for the dielectric nano-
cylinder, the Fresnel reflection coefficient of the nano-cylinder/buffer interface remains less
than 3% for the incident angle less than the Brewster angle of 50°, so that most light is
transmitted. According to the Minkowski model the photon’s momentum P = nE/c where E is
the beam energy and c the speed of light, so that the light entering to the nano-cylinder gains

momentum due to the higher refraction index inside, resulting in the radiation stress in the
direction opposite to the light propagation. On the other hand the light exiting from the nano-
cylinder losses momentum, resulting in the radiation stress in the direction along the light
propagation. In both cases, the stress  is along the outwards surface normal n of the nano-
cylinder.
In our calculation, the trapping beam is of circular polarization. The nanorod is of a
cylindrical shape and is a dielectric and isotropic medium. We do not consider birefringent
medium in this paper. We neglect the gravity force and the floating force, when the nano-
cylinder is in an aqueous buffer. Both forces are constant forces and their difference is small.
Neglecting the two forces can change the equilibrium position in z-axis of the trapped nano-
cylinder and the torques when the nano-cylinder is tilted with respect to the vertical axis.
3. Position stability
To investigate the stability of the optical trapping of a nano-cylinder we computed the
radiation forces and torques when the nano-cylinder is shifted and tilted from the equilibrium
position by the random Brownian motion in the aquatic buffer. We first considered a nano-
cylinder trapped along the z-axis and computed its equilibrium position and the trapping
position stability. In all the calculations in this paper the trapping beam NA = 1.25 and power
P = 10 mW. The nano-cylinder was fixed in the coordinate system along the z-axis and
centered at the origin. The nano-cylinder has a reasonable aspect ratio H/2R<<20 where H is
the length and R the radius, as when H/2R>20 the scattering can be dramatically different and
the T-matrix method may be no longer valid [12]. The trapping beam was shifted and tilted
relative to the nano-cylinder. Without loss of generality we assume the trapping beam axis
always lies in the x-z plane with the azimuth angle φ = 0. In this case the total force is null
along the y-axis, Fy = 0, as both the beam and the nano-cylinder are symmetric with respect to
the x-z plane.
We computed the equilibrium position of the nano-cylinder trapped and aligned along the
beam axis. When the beam axis is the z-axis the stress distribution on the side-face of the
nano-cylinder is axially symmetric, as shown in Fig. 2(a), so that the total lateral force Fx = 0.
The equilibrium position in z is where the forces on the top and bottom end-faces of the nano-
cylinder are balanced Fz = 0 and the gradient Fz/z < 0, i.e. a force opposite to displacement
is generated when the nano-cylinder is displaced in z from the equilibrium position. The nano-
cylinder in the equilibrium position is shifted by zeq relative to the beam focus in the beam
propagating direction and this shift increases with the length of the nano-cylinder. The total
force Fz as a function of the beam focus position in z is shown in Fig. 2(c) for nano-cylinders
of radius R = 300 nm. When the nano-cylinder length H = 1.2 μm, we have zeq = + 0.13 μm
and when H = 3 μm, zeq = + 0.41 μm.
The gradient of the forces |Fz/z| and |Fx/x| is defined as the axial and lateral trapping
stiffness respectively. ( ) zU z F dz  and ( ) xU x F dx  is defined respectively as axial and
lateral trapping force potential, which represents the energy required for the nano-cylinder to
escape from the trap. U(z), corresponding to the area under the curve Fz(z) in Fig. 2(c), is
asymmetric with respect to the equilibrium position Fz = 0. The energy barrier for the nano-
cylinder to escape is lower in the beam propagating direction. Comparison of the blue and
green curves in Fig. 2(c) shows that the longer nano-cylinder has a longer shift of the
equilibrium position from the beam focus, a lower axial stiffness and a more pronounced
asymmetry of trapping potential, and therefore, is easier to escape from the trapping than the
shorter nano-cylinder.
When the trapping beam is parallel to the nano-cylinder, the lateral equilibrium position is
at x = 0 where Fx = 0 and Fx/x<0. The lateral trapping force potential ( ) xU x F dx  is
symmetrical with respect to x = 0. When the trapping beam is parallel to the z-axis but shifted
by x, the stress is concentrated on one side of the nano-cylinder as shown in Fig. 2(b), so that
the total force would attract the nano-cylinder back to the beam axis. The stress distribution is

Fig. 2. Stress distribution on a nano-cylinder of radius R = 50 nm at equilibrium position with
(a) Beam along the z-axis; (b) Beam shifted to x = 50nm; (c) Axial force and axial stiffness as a
function of position of the focus in the z-axis for radius R = 300 nm; (d) Lateral foce and lateral
stiffness as a function of shift distance from the x-axis for a length H = 1 μm.
not uniform in the z-direction, as shown in Fig. 2(b), resulting in a torque, which tends to tilt
the nano-cylinder. The orientation stability of the trapping will be discussed in the next
section. The lateral trap stiffness |Fx/x| is larger for a nano-cylinder of R = 70 nm than that
of R = 50 nm, as shown in Fig. 2(d).
In Figs. 3(a) and 3(b) we show the axial trap stiffness, |Fz/z|, as a function of the nano-
cylinder length and radius. At the equilibrium position the axial trap stiffness increases with
the nano-cylinder radius. The stiffness increases with the length until a maximum value and
then tends to a constant value after the aspect ratio reaches about H/2R>2.5. The value of the
maximum stiffness increases with the nano-cylinder radius. For nano-cylinders of small
radius, the stiffness does not increase much with the length, as shown in Fig. 3(a) the curve
for R = 50 nm. The length for the maximum stiffness increases with the radius. For example,
for R = 100 nm the maximum stiffness is achieved when H = 500 nm, while for R = 300 nm
the maximum stiffness occurs for H = 2 μm. The lateral trap stiffness showed similar
dependence on the nano-cylinder length and radius, as shown in Figs. 4(a) and 4(b). However,
the lateral trap stiffness is several times higher than the axial trap stiffness.
The axial and lateral stiffness increases with the nano-cylinder radius, as show in Fig. 3(c)
and 4(c), respectively. More precisely, the axial and lateral stiffness remains almost constant
for 20 nm < R < 60 nm but increases dramatically by more than 8 times in the range 60 nm <R
<140 nm.
The other important parameter affecting the trap stiffness is the trapping beam numerical
aperture. We show in Figs. 3(d) and 4(d) that the stiffness increases with the increase of NA
from 1.05 to 1.29 for a nano-cylinder with length H = 1 μm and radius R = 100 nm. There is a
sharp increase of the stiffness for the NA > 1.2. For instance, the lateral stiffness increases
only 0.3 pN/μm when the NA increases from 1.05 to 1.2, but increases more than 7 times
when the NA increases from 1.2 to 1.29.

Fig. 3. Axial stiffness as a function of length for (a) radius R = 50 – 100 nm; (b) R = 300nm; as
a function of (c) radius for length H = 200–1000 nm; and (d) NA for R = 100nm and H = 1μm.
Fig. 4. Lateral stiffness as a function of length for (a) radius R = 50-100nm; and (b) R = 300nm;
as a function of (c) radius for length H = 200-1000 nm; and (d) NA for R = 100nm and H =
1μm.
4. Orientation stability
To investigate the stability of the trapped nano-cylinder orientation we consider the case
where the nano-cylinder is tilted from its equilibrium trapping position along the beam axis by
an angle θ0 due to the Brownian motion, or equivalently the trapping beam is tilted from the z-
axis by a polar angle β = -θ0 in the x-z plane with the fixed nano-cylinder, as shown in
Fig. 5(b). The tilt destroys the axial symmetry with the z-axis of the stress distribution,

resulting in the lateral torque τy. The negative τy tends to rotate the nano-cylinder back to align
with the beam axis, and the positive τy tends to rotate the nano-cylinder further to align with
the normal of the beam axis.
Fig. 5. (a) Spin torque τz, (b) Lateral torque τy. (c) Stress distribution in the situation of (b) with
nano-cylinder R = 100 nm and H = 1 μm; (d) Stress distribution on a nano-cylinder aligned
along the x-axis with the beam tilted by β = 40° with respect to the nano-cylinder axis, R = 25
nm and H = 100 nm.
The gradient of the lateral torque |τy/β| is defined as the angular stiffness of the trap. In
Fig. 5(c) we show the stress distribution when the beam is rotated by β = 40°. As the stress is
outwards from the nano-cylinder, this stress distribution generates negative lateral torque.
When the beam is tilted, the symmetry with respect to the x-axis and z-axis are lost, the
scattering of the beam by the parts of the nano-cylinder of positive and negative x and z
become different, so that the equilibrium position in the x-direction for Fx = 0 is no longer at x
= 0 as shown in Fig. 6(a). The lateral trapping force potential becomes asymmetric with
respect to the x-axis, as shown in Fig. 6(a) for H = 2.8 μm and R = 300 nm and tilt angle β = ±
40°. When β is negative, the energy barrier for the nano-cylinder to escape the trap is lower in
the –x direction. The asymmetry of the trapping force potential and the shift of the equilibrium
position in x is small for the nano-cylinder of a small radius R = 70 nm, as shown in of
Fig. 6(b). The equilibrium position in z for Fz = 0 of the trapped nano-cylinder is also shifted
by its tilt.
We computed the lateral torque τy as a function of the tilt angle β for the nano-cylinders of
radius 50 nm and length varying from 100 nm to 1.2 μm, as shown in Fig. 7. In the vicinity of
the equilibrium state β = 0°, y increases linearly with β until a maximal value with the
angular stiffness |τy/β| increasing with the length and radius of the nano-cylinder.
After the peak value at β ~ 35-45°, y starts to decrease with the increase of the tilt angle,
but the lateral torque is always negative, except for the nano-cylinder of small aspect ratios,
such as when the length H = 0.1 μm and the aspect ratio H/2R = 1, the τy is positive.
Our calculation revealed that the stress on the side-face of the nano-cylinder contributes to
the lateral torque much more significantly than does the stress on the two end-faces for the
aspect ratio >2. Consider the top end-face z = H/2. On the part of end-face with x>0 the stress
gives rise negative τy, while on the part with x<0 the stress gives rice positive τy, as 1y r n  
and the stresses are normal to the end-face in the + z-direction, as shown in Fig. 1.

Fig. 6. Lateral force Fx and grandient dFx/dx for a nano-cylinder tilted by ± 40° for H = 2.8μm
and (a) R = 300 nm, (b) R = 70 nm.
Fig. 7. Lateral torque as a function of tilt angle for a nano-cylinder of radius R = 50 nm.
The total torque τy is the difference between the two. As the part with x<0 is closer to the
tilted beam, as shown in Fig. 5(b), the total torque is positive on the two end-faces, but its
absolute value is almost 2 orders of magnitude lower than that of the negative torque
associated to the stress on the side of the nano-cylinder, especially for nano-cylinder of high
aspect ratio, because on the side the area to apply the force is larger than that on the end-faces,
and the lever arm on the side rcosθ can be much larger than r1sinθ = ρ on the end-face, as can
be seen in Fig. 1. The lateral torque as a function of the tilt angle and aspect ratio is shown in
Fig. 7. Thus, modeling the nanorod as a linear chain of spheres for computing radiation force
and torque in Ref. [3]. could result in errors in the calculated lateral torque and the total torque
due to the inaccuracy in the presentation of the side-face of nano-cylinders.
Figure 8(a) shows the lateral torque τy and the gradient τy /β for the tilt angle 20°< β
<110° for a nano-cylinder of R = 50 nm, H = 900 nm. We see that whatever the positive or
negative tilt angles the equilibrium orientation is at β = 0°, where τy = 0 and τy /β is
negative. The nano-cylinder is trapped along the beam axis. Even the perturbation tilt β = 90°
the torque τy = 0, but the nano-cylinder cannot be trapped stably normal to the beam axis
because τy /β >0.
On the other hand, for a nano-cylinder of R = 25 nm, H = 100 nm with the aspect ratio
H/2R = 2, we see from Fig. 8(b) that the equilibrium position is at β = 90° where τy = 0 and τy
/β <0 for a wide range of the tilt angle β, so that the nano-cylinder is trapped and aligned
normal to the beam axis. Furthermore, the lateral torque is positive τy>0 for all tilt angles 0°<β
< 90°, the trapping of the nano-cylinder along the z-axis is not stable. Any tilt of a trapped
nano-cylinder will lead to its continuous rotation until reaching the orientation normal to the
beam axis.

Fig. 8. Lateral torque τy and its gradient τy /β as a function of tilt angle for a nano-cylinder of:
(a) R = 50 nm and H = 900 nm; (b) R = 25 nm and H = 100 nm.
It is well known from experiments that an elongated particle in general tends to align with
its long axis along the trapping beam axis [5]. Our calculation showed why the nano-cylinder
of aspect ratio of H/2R<2 is aligned with its end-faces to the beam axis. In fact, when the
nano-cylinder aspect ratio decreases, the torque associated to the stress on the end-faces
becomes more important than that on the side-face, because of the increase of both the relative
surface and the lever arms on the end-faces, so that the torque becomes positive. Figure 5(d)
shows the stress distribution on a nano-cylinder of R = 25 nm and H = 100 nm aligned with
the x-axis and the beam axis is tilted by 40°. In this case the stress distribution on the side-face
is similar to that on the top-end face of a nano-cylinder aligned to the z-axis, and the torque is
positive. Note that in this case the range of variation of the stress on the nano-cylinder side-
face is small (24 – 26 N/m2
).
Fig. 9. (a) Spin torque and (b) lateral torque as a function of tilt angle for R = 50nm and H =
500 nm to1.1 μm; (c) lateral and spin torques for R = 25nm and H = 100nm.
We calculated the critical length of the nano-cylinder for the lateral torque changing from
negative to positive. The critical length is H<100 nm for R = 25 nm and H<250 nm for R = 50
nm. These results are different from that of H<400 nm for both R = 25 and R = 50 nm given in
Ref. [3], where the nano-cylinder is modeled as a linear chain of spheres.

The spin torque τz, resulting from the transfer of the angular momentum from the trapping
beam to the nano-cylinder, induces rotation of the nano-cylinder around the z-axis, as
illustrated in Fig. 5(a) and reported in Ref [13]. for nanofiber. However, the nanofibers in [13]
can have high aspect ratio H/2R>>20 exceeding the limit of our model. Our calculation
showed that the spin torque is at least 2-3 orders of magnitude smaller than the lateral torque
for the nano-cylinders of high aspect ratio, as shown in Fig. 9(a) and Fig. 9(b). However, for
small aspect ratio H/2R = 2, the lateral torque is small and is comparable with the spin torque,
as shown in Fig. 9(c).
5. Conclusion
We have used the T-Matrix approach to analyze the optical trapping of a nano-cylinder with
the point-matching method to compute the vector spherical wave expansion coefficients of the
highly focused incident beam and the T-Matrix of the nano-cylinder. In order to gain a
physical insight into this trapping we computed the stress distribution on the nano-cylinder
surface by the Maxwell stress tensor and computed the total forces and torques by integrating
the stress over the nano-cylinder surface. We studied the trap stability for the nano-cylinder,
which is shifted and tilted from its equilibrium position with the Brownian motion, by
computing the stiffness and potential of the trapping force and torque. We showed the
dependence of the trap stiffness on the nano-cylinder length and radius for both axial and
lateral forces. The energy barrier for a nano-cylinder to escape from the trapping decreases
with increase of the length, so that too long nano-cylinder cannot be trapped. We found that
the force stiffness against the lateral shift of the nano-cylinder is several times higher than that
against the axial shift. We showed that the force stiffness increases with the nano-cylinder
radius and the numerical aperture of the trapping beam. The stability of the orientation of the
trapped nano-cylinder depends mostly on the lateral torque, to which the contribution of the
stress distribution on the side-face is in general higher than that on the two end-faces by 1-2
orders of magnitudes. The former is negative torque against the tilt caused by the Brownian
motion, while the latter is positive torque. For the nano-cylinder of low aspect ratio, as H/2R <
2, the positive torque applied on the two end-faces may exceed that applied on the side-
surface, such that the lateral torque is positive and the nano-cylinder continues to rotate until
being aligned to the normal of the beam axis. The spin torque is usually 2-3 orders of
magnitude smaller than the lateral torque. However, for the nano-cylinder of small aspect ratio
as H/2R < 2 the spin torque can be large and comparable with the lateral torque. This result is
obtained for dielectric isotropic nano-cylinders with the aspect ration less than 20 and with the
circular polarized trapping beam. However, it is well known that the spin torque and the
rotation speed of the nanorod with various aspect ratios can be controlled and enhanced by
enlarging the material birefringence and changing the polarization states of the trapping beam
as reported in the experiments in [14–16].

Angular and position stability of a nanorod trapped in an optical tweezers

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