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

doi:10.1364/OE.18.026388

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

Paul B. Bareil and Yunlong Sheng »View Author Affiliations

Optics Express, Vol. 18, Issue 25, pp. 26388-26398 (2010)
http://dx.doi.org/10.1364/OE.18.026388

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▸ Article Outline
– Abstract
1. Introduction
2. Model and calculation
3. Position stability
4. Orientation stability
5. Conclusion
– References and links

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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.

1. Introduction

Manipulating nanoparticles using optical tweezers has attracted intensive research interest recently [1

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008). [CrossRef]

]. 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

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007). [CrossRef] [PubMed]

], 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

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]

]. 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

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. Soc. Am. A 9, S196–S203 (2007).

] for analyzing the optical rotation of elongated particles by rotating the trapping beam polarization in the micro-machine applications [5

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of optically trapped nonspherical birefringent particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 021911 (2006). [CrossRef] [PubMed]

]. 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

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]

], 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, E_inc and E_scat , and the interior field in the scatter, E_int , are expanded into the vector spherical wave function (VSWF) basis

\begin{array}{l} E_{i n c} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} a_{n m} R g M_{n m}^{} (k r) + b_{n m} R g N_{n m}^{} (k r) \\ E_{int} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} c_{n m} R g M_{n m}^{} (k r) + d_{n m} R g N_{n m}^{} (k r) \\ E_{s c a t} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} p_{n m} M_{n m}^{(1)} (k r) + q_{n m} N_{n m}^{(1)} (k r) \end{array}

(1)

where the VSWFs, M _nm and N _nm , and the regular VSWFs, RgM _nm and RgN _nm , are defined as in Ref. [4

]. We expanded the incident field E_inc into RgM _nm and RgN _nm instead of M _nm and N _nm because when computing in the scale of nano-particles the divergence of M _nm and N _nm at the origin can slow down the convergence of the T-Matrix solver and requires a higher number of modes, N_max .

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

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003). [CrossRef]

], a_nm and b_nm 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 a_nm and b_nm by the T-Matrix as

\begin{array}{l} [\begin{array}{l} p \\ q \end{array}] = [\begin{matrix} T^{11} & T^{12} \\ T^{21} & T^{22} \end{matrix}] [\begin{matrix} a \\ b \end{matrix}] \\ [\begin{matrix} c \\ d \end{matrix}] = [\begin{matrix} T I^{11} & T I^{12} \\ T I^{21} & T I^{22} \end{matrix}] [\begin{matrix} a \\ b \end{matrix}] \end{array}

(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

\begin{array}{l} E_{i n c / /} = - E_{s c a t / /} + E_{int / /} \\ H_{i n c / /} = - H_{s c a t / /} + H_{int / /} \end{array}

(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

G. Videen, Light Scattering from a Sphere Near a Plane Interface (Springer, 2000).

] and the Wigner rotation matrix [9

C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111(19), 8825 (1999). [CrossRef]

] the expansion coefficients of the shifted and rotated beams can be computed from the a_nm and b_nm of the original beam, which was aligned along the z-axis with the focal point at the origin.

Fig. 1 Nano-cylinder fixed in the spherical coordinate system for calculating forces and torques.

The radiation forces and torque were computed by the integrals as

\begin{array}{l} 〈 \vec{F} 〉 = \frac{1}{2} Re ∯_{S} d S \vec{n} • 〈 \vec{T} 〉 \\ 〈 \vec{τ} 〉 = \frac{1}{2} Re ∯_{S} d S \vec{n} • 〈 \vec{T} 〉 \times \vec{r} \end{array}

(4)

where

〈 \vec{T} 〉

is the Maxwell stress tensor,

\vec{r}

is the position vector,

\vec{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

〈 \vec{F_{z}} 〉 = \frac{ε_{0}}{2 k_{0}} (\begin{array}{l} \frac{2}{(n + 1)} \sqrt{\frac{n (n + 2) (n - m + 1) (n + m + 1)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n m} {\tilde{p}}_{n + 1 m}^{*} + {\tilde{q}}_{n m} {\tilde{q}}_{n + 1 m}^{*} \\ - {\tilde{a}}_{n m} {\tilde{a}}_{n + 1 m}^{*} - {\tilde{b}}_{n m} {\tilde{b}}_{n + 1 m}^{*} \end{array}) \\ - \frac{2 m}{n (n + 1)} Im (i {\tilde{q}}_{n m} {\tilde{p}}_{n m}^{*} - i {\tilde{b}}_{n m} {\tilde{a}}_{n m}^{*}) \end{array})

(5)

〈 \vec{F_{x}} 〉 = \frac{ε_{0}}{2 k_{0}} (\begin{array}{l} \frac{1}{(n + 1)} \sqrt{\frac{n (n + 2) (n + m + 1) (n + m + 2)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n m} {\tilde{p}}_{n + 1 m + 1}^{*} + {\tilde{q}}_{n m} {\tilde{q}}_{n + 1 m + 1}^{*} \\ - {\tilde{a}}_{n m} {\tilde{a}}_{n + 1 m + 1}^{*} - {\tilde{b}}_{n m} {\tilde{b}}_{n + 1 m + 1}^{*} \end{array}) \\ + \frac{1}{(n + 1)} \sqrt{\frac{n (n + 2) (n - m) (n - m + 1)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n + 1 m} {\tilde{p}}_{n m + 1}^{*} + {\tilde{q}}_{n + 1 m} {\tilde{q}}_{n m + 1}^{*} \\ - {\tilde{a}}_{n + 1 m} {\tilde{a}}_{n m + 1}^{*} - {\tilde{b}}_{n + 1 m} {\tilde{b}}_{n m + 1}^{*} \end{array}) \\ - \frac{\sqrt{(n - m) (n + m + 1)}}{n (n + 1)} Im (i {\tilde{b}}_{n m} {\tilde{a}}_{n m + 1}^{*} - i {\tilde{p}}_{n m} {\tilde{q}}_{n m + 1}^{*} + i {\tilde{q}}_{n m} {\tilde{p}}_{n m + 1}^{*} - i {\tilde{a}}_{n m} {\tilde{b}}_{n m + 1}^{*}) \end{array})

(6)

The spin torque around the z-axis [10

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Differ. Equations Conf. 04, 37 (2000).

] is given as

τ_{Z} = \frac{ε_{0}}{2 k_{0}} \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} m ({| {\tilde{a}}_{n m} |}^{2} + {| {\tilde{b}}_{n m} |}^{2} - {| {\tilde{p}}_{n m} |}^{2} - {| {\tilde{q}}_{n m} |}^{2})

(7)

where

{\tilde{a}}_{n m} = a_{n m} / 2

{\tilde{b}}_{n m} = b_{n m} / 2

{\tilde{p}}_{n m} = {\tilde{a}}_{n m} + p_{n m}

and

{\tilde{q}}_{n m} = {\tilde{b}}_{n m} + q_{n m}

, 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

\vec{τ} = \iint_{S} d S σ (ϕ, z) \vec{r} \times \vec{n}

(8)

where the stress is computed by

\vec{σ} = \frac{1}{2} (n_{2}^{2} - n_{1}^{2}) (\frac{n_{1}^{2}}{n_{2}^{2}} E_{1 n}^{2} + E_{1 t}^{2}) \overset{⇀}{n}

(9)

and is considered as always perpendicular to the surface [11

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]

] where

\vec{n}

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

\vec{σ}

is along the outwards surface normal

\vec{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

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002), p. 307–319.

]. 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, F_y = 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 F_x = 0. The equilibrium position in z is where the forces on the top and bottom end-faces of the nano-cylinder are balanced F_z = 0 and the gradient ∂F_z/∂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 z_eq relative to the beam focus in the beam propagating direction and this shift increases with the length of the nano-cylinder. The total force F_z 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.

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.

The gradient of the forces |∂F_z/∂z| and |∂F_x/∂x| is defined as the axial and lateral trapping stiffness respectively.

U (z) = - \int F_{z} d z

and

U (x) = - \int F_{x} d x

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 F_z (z) in Fig. 2(c), is asymmetric with respect to the equilibrium position F_z = 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 F_x = 0 and ∂F_x/∂x<0. The lateral trapping force potential

U (x) = - \int F_{x} d x

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 |∂F_x/∂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, |∂F_z/∂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.

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.

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 τ_y . The negative τ_y tends to rotate the nano-cylinder back to align with the beam axis, and the positive τ_y tends to rotate the nano-cylinder further to align with the normal of the beam axis.

Fig. 5 (a) Spin torque τ_z, (b) Lateral torque τ_y. (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 |∂τ_y/∂ 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 F_x = 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 F_z = 0 of the trapped nano-cylinder is also shifted by its tilt.

Fig. 6 Lateral force F_x and grandient dF_x /dx for a nano-cylinder tilted by ± 40° for H = 2.8μm and (a) R = 300 nm, (b) R = 70 nm.

We computed the lateral torque τ_y 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°,

| τ_{y} |

increases linearly with β until a maximal value with the angular stiffness |∂τ_y/∂β| increasing with the length and radius of the nano-cylinder.

Fig. 7 Lateral torque as a function of tilt angle for a nano-cylinder of radius R = 50 nm.

After the peak value at β ~ 35-45°,

| τ_{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 τ_y 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 τ_y , while on the part with x<0 the stress gives rice positive τ_y , as

{\vec{τ}}_{y} = {\vec{r}}_{1} \times \vec{n}

and the stresses are normal to the end-face in the + z-direction, as shown in Fig. 1.

The total torque τ_y 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 τ_y and the gradient ∂τ_y /∂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 τ_y = 0 and ∂τ_y /∂β is negative. The nano-cylinder is trapped along the beam axis. Even the perturbation tilt β = 90° the torque τ_y = 0, but the nano-cylinder cannot be trapped stably normal to the beam axis because ∂τ_y /∂β >0.

Fig. 8 Lateral torque τ_y and its gradient ∂τ_y /∂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.

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 τ_y = 0 and ∂τ_y /∂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 τ_y >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 [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).

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 τ_z , 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

A. A. R. Neves, A. Camposeo, S. Pagliara, R. Saija, F. Borghese, P. Denti, M. A. Iatì, R. Cingolani, O. M. Maragò, and D. Pisignano, “Rotational dynamics of optically trapped nanofibers,” Opt. Express 18(2), 822–830 (2010). [CrossRef] [PubMed]

]. 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).

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.

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

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998). [CrossRef]

–16

F.-W. Sheu, T.-K. Lan, Y.-C. Lin, S. Chen, and C. Ay, “Stable trapping and manually controlled rotation of an asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14), 14724–14729 (2010). [CrossRef] [PubMed]

References and links

1.	M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008). [CrossRef]
2.	Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007). [CrossRef] [PubMed]
3.	F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]
4.	T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. Soc. Am. A 9, S196–S203 (2007).
5.	W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of optically trapped nonspherical birefringent particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 021911 (2006). [CrossRef] [PubMed]
6.	J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]
7.	T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003). [CrossRef]
8.	G. Videen, Light Scattering from a Sphere Near a Plane Interface (Springer, 2000).
9.	C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111(19), 8825 (1999). [CrossRef]
10.	J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Differ. Equations Conf. 04, 37 (2000).
11.	K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]
12.	M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002), p. 307–319.
13.	A. A. R. Neves, A. Camposeo, S. Pagliara, R. Saija, F. Borghese, P. Denti, M. A. Iatì, R. Cingolani, O. M. Maragò, and D. Pisignano, “Rotational dynamics of optically trapped nanofibers,” Opt. Express 18(2), 822–830 (2010). [CrossRef] [PubMed]
14.	M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998). [CrossRef]
15.	M. Rodriguez-Otazo, A. Augier-Calderin, J.-P. Galaup, J.-F. Lamère, and S. Fery-Forgues, “High rotation speed of single molecular microcrystals in an optical trap with elliptically polarized light,” Appl. Opt. 48(14), 2720–2730 (2009). [CrossRef] [PubMed]
16.	F.-W. Sheu, T.-K. Lan, Y.-C. Lin, S. Chen, and C. Ay, “Stable trapping and manually controlled rotation of an asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14), 14724–14729 (2010). [CrossRef] [PubMed]

OCIS Codes
(000.0000) General : General
(140.7010) Lasers and laser optics : Laser trapping
(260.2110) Physical optics : Electromagnetic optics
(160.4236) Materials : Nanomaterials
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: October 15, 2010
Revised Manuscript: November 16, 2010
Manuscript Accepted: November 17, 2010
Published: December 1, 2010

Citation
Paul B. Bareil and Yunlong Sheng, "Angular and position stability  of a nanorod trapped in an optical tweezers," Opt. Express 18, 26388-26398 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-26388

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References

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008). [CrossRef]
Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007). [CrossRef] [PubMed]
F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]
T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. Soc. Am. A 9, S196–S203 (2007).
W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of optically trapped nonspherical birefringent particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 021911 (2006). [CrossRef] [PubMed]
J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]
T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003). [CrossRef]
G. Videen, Light Scattering from a Sphere Near a Plane Interface (Springer, 2000).
C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111(19), 8825 (1999). [CrossRef]
J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Differ. Equations Conf. 04, 37 (2000).
K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]
M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002), p. 307–319.
A. A. R. Neves, A. Camposeo, S. Pagliara, R. Saija, F. Borghese, P. Denti, M. A. Iatì, R. Cingolani, O. M. Maragò, and D. Pisignano, “Rotational dynamics of optically trapped nanofibers,” Opt. Express 18(2), 822–830 (2010). [CrossRef] [PubMed]
M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998). [CrossRef]
M. Rodriguez-Otazo, A. Augier-Calderin, J.-P. Galaup, J.-F. Lamère, and S. Fery-Forgues, “High rotation speed of single molecular microcrystals in an optical trap with elliptically polarized light,” Appl. Opt. 48(14), 2720–2730 (2009). [CrossRef] [PubMed]
F.-W. Sheu, T.-K. Lan, Y.-C. Lin, S. Chen, and C. Ay, “Stable trapping and manually controlled rotation of an asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14), 14724–14729 (2010). [CrossRef] [PubMed]

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