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
Manipulating nanoparticles using optical tweezers has attracted intensive research interest recently . 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 , 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 . 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  for analyzing the optical rotation of elongated particles by rotating the trapping beam polarization in the micro-machine applications . 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 , 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. . 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) basis4]. 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 asEq. (1) into the boundary conditions for the electric and magnetic field components parallel to the interface of the nano-cylinder as
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  and the Wigner rotation matrix  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 as10] is given as
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 as11] where is the outwards normal of the surface, as shown in Fig. 1.
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 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 . 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 z eq = + 0.13 μm and when H = 3 μm, z eq = + 0.41 μm.
The gradient of the forces |∂Fz/∂z| and |∂Fx/∂x| is defined as the axial and lateral trapping stiffness respectively. and 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 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 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.
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.
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°, 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°, 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 and the stresses are normal to the end-face in the + z-direction, as shown in Fig. 1.
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. . 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.
It is well known from experiments that an elongated particle in general tends to align with its long axis along the trapping beam axis . 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).
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. , 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 . for nanofiber. However, the nanofibers in  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).
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].
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