Dynamic simulation is a powerful tool to observe the behavior of arbitrary shaped particles trapped in a focused laser beam. Here we develop a method to find equilibrium positions and orientations using dynamic simulation. This general method is applied to micro- and nano-cylinders as a demonstration of its predictive power. Orientation landscapes for particles trapped with beams of differing polarisation are presented. The torque efficiency of micro-cylinders at equilibrium in a plane is also calculated as a function of tilt angle. This systematic investigation elucidates in both the function and properties of micro- and nano-cylinders trapped in optical tweezers.
© 2012 OSA
In the early part of 1970s , Ashkin and associates reported that a single microsphere can be suspended against gravity by a focused laser beam, and also showed that the sphere can be trapped in the focal region of two perfectly aligned counter-propagating laser beams. In 1986, Ashkin et al.  demonstrated that a single tightly focused laser beam can trap a microsphere in three dimensions in the waist of a highly focused beam. Since the pioneering work of Ashkin, laser-based trapping and manipulation of particles has found widespread use in fields of biology, physics, and chemistry [3–5]. Microspheres have continued to be popular objects of optical trapping, being widely used as handles or probes. The simple geometry of microspheres allows computational modelling of optical tweezers using generalized Lorenz-Mie theory [6–8]. Where such a particle is trapped using a Gaussian beam, it is known beforehand that the equilibrium will lie on the beam axis, and the most important properties of the trap, such as the axial and radial trap strengths and spring constants can be calculated. Such calculations can be performed efficiently enough to explore the dependence on parameters such as particle size, refractive index, the numerical aperture of the focussing lens, or other parameters [6,8–10]. More general calculations can be performed, for example to find the optical force as a function of position. This can be carried out over a closely-spaced three-dimensional grid (or two-dimensional for circularly polarized beams).
However, not all particles that are trapped or manipulated in optical traps are spherical. Particles such as crystals and bacteria or other single cells can be non-spherical, or we can choose to fabricate non-spherical particles for specific purposes, such as to exert optical torques . Non-spherical particles that have been investigated include: rings , cylinders , single cells , cubes , and microrotors [16,17]. Characterization of the trapping of such particles by computational modelling is significantly more difficult than for spherical particles. The first difficulty is that, unlike the analytical solution for spheres, it is necessary to resort to numerical methods to solve the scattering problem. A number of methods are known to be useful for modelling optical trapping, for example, using the T-matrix method. One of the most flexible, as far as particle geometry is concerned, is the discrete dipole approximation, or DDA (also known as the coupled-dipole method) [18–21]. The second difficulty is that the non-spherical shape introduces two (for axisymmetric particles) or three degrees of freedom, required to describe the orientation of the particle. Since, for a single particle, this results in a total of 4 (for axisymmetric particles in circularly polarized Gaussian beams), 5, or 6 degrees of freedom, the computational time required to calculate the force over a sufficiently fine grid becomes excessive. In addition, it is no longer certain that the particle is centred on the beam axis in the equilibrium position. An alternative approach is required. Noting that if a particle is introduced into an optical trap, it will “fall” into its equilibrium position and orientation, we see that if we model this process, we can readily find the equilibrium position and orientation of a particle. This requires us to calculate the optical force and torque, and the translational and rotational viscous drag force and torque, and model the motion of the particle as it moves into the trap. This can be done with far fewer calculations than required to characterize the trap as a function of the degrees of freedom (particle position and orientation).
One interesting class of non-spherical particles is elongated nanoparticles, such as carbon nanotubes [22,23], semiconductor nanowires , and nanofibers . Such particles often have subwavelength cross-sections. Optical trapping and manipulation of non-spherical nanoparticles have been observed in aqueous environments. Recently, radiation torque and trapping force on optically trapped linear nanostructures has been studied by F. Borghese et al. . The holographic optical trapping of nanowires has been calculated by Stephen H. Simpson et al.  using the T-matrix method. Nanoparticle trapping is challenging with optical tweezers and they offer ample new phenomena to explore , thus, it is of great importance to investigate the behavior of trapped nanostructures and to discuss their orientation and rotation. Dynamic simulation can rapidly provide information on beam-particle systems without having to interpret conventional force and torque versus position and orientation curves. Here we develop this model and apply it to micro- and nano-cylinders. Figure 1 shows the dynamic simulation for a microcylinder trapped in a single tightly Gaussian beam propagating along z-axis. The cylinder for all starting positions and orientations considered here is finally trapped at the same position with the orientation along the beam axis. A movie of those translations and rotations of the cylinder has been made and is included as supplementary material, from which one can easily observe motion of the cylinder. In this paper, we calculate the force and torque on dielectric cylinders illuminated by a tightly focused laser beam, and study their translation and rotation dynamics. The trapping force and torque exerted on dielectric cylinders are calculated using a numerical method based on discrete dipole approximation (DDA) and tangential field point-matching . The equilibrium positions and stable orientations of micro- and nano-cylinders are obtained using a dynamic simulation model incorporating the translational and rotational friction tensors.
For the purpose of explanation of the dynamic simulation, in this section we will present the theory of our calculations.
2.1. T-matrix calculation
In optical tweezers, the optical forces and torques exerted on particles can be considered as the result of the transfer of linear and angular momentum from the electromagnetic field to the particle. In other words, the linear and angular momentum flux of the beam are changed by light scattering off the particle. Thus, the problem of calculating the forces and torques acting on particles is light scattering problem. In our calculations, optical fields are expanded in terms of incident and scattered field potentials, which can be written in terms of a discrete basis set of functions and , such that,30]. If the incident field and T-matrix of the particle are given, the coefficients of the scattered field can be found. There are a number different ways to calculate T matrix. Our T-matrices are constructed using DDA and point-matching.
2.2. Translation and rotation of beam coefficients
If the beam expansion coefficients in the Cartesian coordinate system centered on the focus of the beam are a0 and b0, coefficients in a coordinate system at any position can be found using a linear transformation [18,31],Fig. 2 . A and B are the translations of beam coefficients in the rotated coordinate system along the direction of O1O2. R2 is the rotation of beam coefficients from the Cartesian coordinate system x2y2z2 centered at the particle position to the coordinate system x3y3z3 in particle orientated frame. According to Eqs. (4) – (5), as long as the beam coefficients in the coordinate system centered at the focus of the beam are given, the coefficients in an arbitrary coordinate system centered at any position can be found using rotations and translations of beam coefficients. Then the coefficients of the scattered field, p and q (the TE/TM modes of ), can be found by Eq. (3).
Note that, R1 is calculated using the direction vector O1O2, A and B move the beams between O1 and O2. R2 orients the translated field into a particle orientated frame, which depends on a 3 × 3 rotation matrix R2,3 of particle. R2,3 is considered as a function of time, which can be written as32]. If we define the angular velocity of cylinder at time t as ωt, is the rotation angle around the axis . When , and the orientation is at equilibrium.
2.3. Translation and rotation of micro- and nano-particles
A particle near the focal region, as a result of optical force and torque, will be forced to translate to an equilibrium position and rotate to a stable orientation. The force and torque efficiencies exerted on the particle, Q and τ, are calculated with optical tweezers computational toolbox , which uses sums of products of the expansion coefficients of incident and scattered fields (anm, bnm, pnm and qnm) to calculate the change in linear momentum. The force and torque acting on the particle, currently in normalized form, can be converted to SI with
The dynamic simulation model is also based on hydrodynamic equations. For a particle at low Reynolds number, when an exerted force is removed, it will take less than a few microseconds to slow to the flow speed . Our calculations are in the regime of low Reynolds number, therefore, we consider that the viscous drag exerted opposes the optical force, and that the viscous torque opposes the radiation torque. So we can write and , by Newton’s third law of motion.
The effect of force and torque exerted on a particle are obtained with the following equations, which is based on computing the particle’s next position and orientation a short time interval later. Given the force, torque, position and orientation at time t, the equations of motion we solve for a particle are34], and r is the position of the center of the object. Since the translational and rotational friction tensors given by Delatorre and Bloomfield  only apply to elongated cylinders (aspect ratio > 1), we did not perform calculations for disks (aspect ratio < 1).Eq. (6). Based on this model, we can dynamically simulate the motion of micro- and nano-structures near the focal region. By changing the starting position and orientation, we can exclude local minima from the simulation.
The trapping and rotation dynamics of cylindrical structures was determined from Eqs. (8) – (9). The laser beam we used in the calculations was a single tightly focused Gaussian beam, focused by high numerical aperture lens (NA = 1.25). The beam propagates along the z1 direction with its focus located at (x1, y1, z1) = (0, 0, 0). The wavelength of the beam was λ0 = 1064 nm in vacuum and λ = 800 nm in water, the power of the beam going through the focal plane was Pinc = 1 mW. The immersed cylinder was made of common glass with a refractive index np = 1.57. The origin of the cylinder was located at the centre of mass.
A glass nanowire with diameter of 50 nm and length of 2000 nm has been simulated using dynamic simulation model. Figure 3 shows the trajectories of the nanowire as functions of time in linearly polarized and circularly polarized beams. The nanowire was trapped at (0, 0, −0.03)λ in a linearly polarized beam and at (0, 0, −0.1)λ in a circularly polarized beam. For some particles, the equilibrium position is possibly before the focal plane . That is, the center of the particle is before the focal plane (but part of the particle lies beyond the focal plane as well). The nanowire orients with a tilt angle φ = 86.6 degrees in the linear polarized beam, but along the beam axis in the circularly polarized beam. Differences between the final orientations of the nanowire in beams of different polarisations mean that the polarisation of focused laser beam plays an important role in optical trapping of the nanowire.
The microcylinder in Fig. 1 and nanowire in Fig. 3 behave in very different manners. In a linearly polarised beam, the microcylinder is trapped along the beam axis, whereas the nanowire is trapped with a large tilt angle (φ = 86.6 degrees). Nanowire orientation is sensitive to polarisations, it is therefore reasonable to believe that the nanowires and microcylinders with other sizes may also be trapped with different orientations both in linearly and circularly polarized beams. Knowing this behavior is crucial when considering using such a particle as a probe. A cylinder’s size and the beam polarisation change the equilibrium orientation. We look at stable orientations of micro- and nano-cylinders with diameters less than 500 nm and lengths less than 3000 nm. Figure 4 shows the orientation landscapes for these nanowires and microcylinders trapped in linearly and circularly polarized beams. There are four regimes in the orientation landscapes for cylinders with aspect ratio larger than one: an untrapped region, vertical region, horizontal region, and an intermediate region between the vertical and horizontal regions. In general, larger cylinders can’t be trapped. The cylinders in vertical and horizontal regions are trapped along the beam axis and transverse to the beam axis respectively. The intermediate region is the region where a cylinder can be trapped with a tilt angle between 0 and 90 degrees from the beam axis.
There is a big difference between the orientation landscapes of cylinders trapped with beams of different polarisations. The vertical region is larger for circular polarisation and the intermediate and horizontal regions become smaller when compared to corresponding regions for linear polarisation. This means that the rods trapped with stable horizontal orientation or with a tilt angle in linear polarisation may be trapped with vertical orientation in a circularly polarized beam. A similar strong dependence on polarization has been noted for non-conservative forces on nanoparticles , and it is possible that such forces contribute here. However, our T-matrix method only gives the total force, rather than conservative and non-conservative components separately, so this remains an interesting open question. However, the untrapped region doesn’t change from linearly to circularly polarized beams. Polarisation is therefore not a key factor to determine whether a particle can be trapped or not.
The torque efficiencies of cylinders with diameters of 300 nm with different lengths were calculated as functions of tilt angle from the beam axis and are shown in Fig. 5 . The lengths of cylinders depicted are 600, 1200 and 1800 nm. This calculation was performed for cylinders located at equilibrium positions in the polarisation plane of a linearly polarized beam. In the circularly polarized beam, the plane of tilt is simply a plane parallel to the beam axis. The torque efficiency, Qτt, is the torque in these planes. When Qτt < 0 the orientation angle decreases and increases when Qτt > 0. A stable orientation is achieved when a deviation either side of Qτt = 0 returns the object its previous orientation. According to the curves shown in Fig. 5, one can obtain the stable orientations of these cylinders. The cylinder with length of 600 nm orients horizontally (φ = 90°) in a linearly polarized beam, but vertically (φ = 0°) in a circularly polarized beam. The cylinder with length of 1200 nm orients with a tilt angle φ = 43 degrees in a linearly polarized beam, but vertically in a circularly polarized beam. The cylinder with length of 1800 nm is aligned vertically both in a linearly polarized beam and in a circularly polarized beam. According to our calculation, shorter cylinders are stably aligned transverse to the beam axis, whereas longer cylinders dispose themselves toward the beam axis. This behavior is consistent with other theoretical predictions for particles with linear structures subjected to tightly focused laser beams . In the intermediate region, cylinders may orient to quite different orientations with small changes in their dimensions and material. The results demonstrate that the polarisation of focused laser beam plays an important role the stable orientation of a cylinder.
The translation and rotation of microcylinders and nanowires have been studied under dynamic simulation. Dynamic modelling can show how an object (such as the cylinders discussed here) will travel and rotate as they find equilibrium positions and stable orientations. We have shown a range of microcylinders and nanowires which can be stably trapped and rotate in the single tightly focused Gaussian beam. A microcylinder with diameter of 500 nm and length of 2000 nm at different starting positions and orientations is trapped at the same equilibrium position along the beam axis. A nanowire with diameter of 50 nm and length of 2000 nm has been simulated, showing that it can be trapped with a large tilt angle in a linearly beam and along the beam axis in a circularly polarized beam and may rotate with handedness of polarisation. The orientation landscapes of glass cylinders with diameters between 0 and 500 nm and lengths between 0 and 3000 nm have been presented. There are four regimes in the orientation landscapes of glass cylinders: an untrapped region, vertical region, horizontal region, and an intermediate region between the vertical and horizontal regions. The orientation landscapes of the cylinders demonstrate that the stable orientations of cylinders can strongly depend on the trapping beam polarisation. The torque efficiencies on cylinders with diameter D = 300 nm with different lengths in linearly polarized and circularly polarized beams have been calculated. The orientations of microcylinders and nanowires depend on the competition between polarisation and gradient force. Apart from these specific results for cylindrical particles, including nanowires, it is important to realise that the method is quite general, and can be used to find equilibrium positions and orientations for any particle for which the optical force can be calculated. Ideally, it should also be possible to calculate the translational and rotational viscous drag tensors, and coupling tensors between rotation and translation as can be required for chiral particles and linear structures near an interface .
However, if the particle falls into an equilibrium position in the trap, it is possible to approximate these drag tensors very simply by using the tensors for a sphere. In this case, the trajectories will not be correct, but the final equilibrium position will be. In the case that the particle does not reach an equilibrium position and orientation, but instead approaches a steady-state cyclic trajectory, this trajectory will be affected by approximations made in these tensors. Even with these limitations, the method presented in this paper provides a useful method for the determination of equilibrium positions of many kinds of particles. More generally, the method used here can be used to simulate the motion, both translational and rotational, of many kinds of particles in optical traps. For future work, we are interest in the investigation of translation and rotation of other geometries and materials, and demonstrating their applications based on their properties.
This work was supported by the Australian Research Council.
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