## Abstract

Based on a hybrid discrete dipole approximation (DDA) and T-matrix method, a powerful dynamic simulation model is used to find plausible equilibrium orientation landscapes of micro- and nano-spheroids of varying size and aspect ratio. Orientation landscapes of spheroids are described in both linearly and circularly polarized Gaussian beams. It’s demonstrated that the equilibrium orientations of the prolate and oblate spheroids have different performances. Effect of beam polarization on orientation landscapes is revealed as well as new orientation of oblate spheroids. The torque efficiencies of spheroids at equilibrium are also studied as functions of tilt angle, from which the orientations of the spheroids can be affirmed. This investigation elucidates a solid background in both the function and properties of micro-and nano-spheroidal particles trapped in optical tweezers.

© 2014 Optical Society of America

## 1. Introduction

Since Ashkin’s work for optical tweezers in the 1970s [1], optical trapping and rotation have been widely used in many fields [2–5]. Spherical particles have few degrees of freedom; it’s therefore much easier to understand the trapping and the effect of manipulating of such particles. Theoretical calculation for optical trapping of spheres, such as the axial and radial trap strengths and spring constants, can be easily done by generalized Lorenz-Mie theory [6–8]. Microspheres have thus found widespread in use as handles or probes and continued to be popular objects of optical trapping. However, many other non-spherical particles, such as crystals, single cells, carbon nanotube or some fabricated non-spherical particles, have been trapped or manipulated in optical tweezers. Some non-spherical particles have been used as probes in the experiment of optical manipulation as well [9–11]. Thus, it is necessary to investigate the trapping mechanism of microstructures or nano-sized non-spherical particles.

It is of great interest to discuss their orientations and rotations of non-spherical particles in optical traps [12,13]. One more interesting class of non-spherical particles is spheroidal particles which have reduced particle symmetries [14]. For example, some biological cells (such as the red blood cell and Prokaryotic cell) can be considered as spheroids by themselves. What’s more, for deformable cells, they may be elongated by the stretching force of two optical traps moving in opposite directions, which forms spheroids with tunable aspect ratio [15]. A spheroidal particle is defined as an ellipsoid with uniaxial symmetry or derived by simple deformations of a sphere. The aspect ratio of a spheroid *δ* is determined by$\delta =a/b$, where *a* and *b* are the lengths of semi-primary and semi-secondary symmetry axes, respectively. A spheroid can therefore be replaced by a prolate or oblate spheroid when$\delta >1$or$\delta <1.$ It was predicted that prolate spheroids would tend to align with their primary axes parallel to the beam axis in a linear polarized beam, while oblate spheroids would orient with their secondary axes perpendicular to both the beam axis and the polarization direction [16]. However, we will reveal some spheroids can be trapped stably with a tilt angle between 0 ~90 degrees. Orientations of some oblate spheroids can rotate 90 degrees when the beam’s polarization changes from linear to circular polarizations.

In this paper, a model for single non-spherical particles to simulate the trapping process is applied to find the equilibrium orientations of micro- and nano-spheroids in single tightly focused Gaussian beams. As it affects the interaction with the environment, the optical force and torque, and the translational and rotational viscous tensors are required to be calculated for this purpose. We investigated the orientation landscapes of spheroids with different sizes and aspect ratios, trapped in beams with different polarizations. The orientations are drawn from the torque efficiency curves of the prolate and oblate spheroids located at their equilibrium positions.

## 2. Theory

In optical tweezers, the electromagnetic fields can be expanded in terms of incident and scattered field potentials, which can be written in terms of vector spherical wave functions (VSWFs) [17]. The relationship between the incident and scattered fields can be written as $P=TA$ where **T** is the transition matrix or commonly referred to as the T-matrix. **A** and **P** are vectors of the beam shape coefficients of incident and scattered fields, respectively. Therefore, the beam coefficients of the scattered field, **p** and **q** (the TE/TM modes of **P**), can be found when the beam shape coefficients of the incident field are obtained. The force and torque acting on the particle, **F** and **Г**, can then be calculated using optical tweezers computational toolbox provided by T. A. Nieminen [7].

The T-matrix is calculated based on one of the most flexible modeling methods, the discrete dipole approximation (DDA) [18–20], also known as the coupled-dipole method, is used to solve the scattering problem. The dynamic simulation model includes hydrodynamic calculations. All our micro- or nanospheroid calculations are performed in low Reynolds number regime [21]; we therefore consider the viscous drag exerted on the particle as opposing the optical force **F**, and that the viscous torque opposes the optical torque **Г**. Given the optical force, torque, position and orientation at time *t*, the particle’s next position and orientation after a short time interval can be obtained using the following equations

**and**

*v***are velocity and angular velocity of the particle at time**

*ω**t*,

**is the position of the center of mass of the particle,**

*r***γ**

_{t}and

**γ**

_{r}are the translational and rotational viscous friction tensors for the particle given by M. A. Charsooghi [22], respectively. d

*t*is generally a time interval between 10

^{−4}~10

^{−3}s.

**R**

*and*

_{t}**R**

_{t}_{+d}

*are the rotation matrices of the particle at time*

_{t}*t*and

*t*+ d

*t*, which represent the orientations of the particle [23]. $\Delta R$is found using the Euler-Rodrigues formula for axis angle rotations [24].

## 3. Results

In our calculations, the single tightly focused Gaussian beam was focused by a high numerical aperture lens with NA = 1.25, which propagated along the + z direction with their focuses located at the origin of the Cartesian coordinate system as shown in Fig. 1(a). The wavelength chosen for the beams was *λ*_{0} = 1064 nm in free space. The power going through the focal plane was *P*_{inc} = 1 mW.

The trajectories and equilibrium orientations of spheroids of varying sizes and aspect ratios can be easily obtained using the model outlined in section 2. For instance, trajectories of prolate and oblate common glass spheroids with the refractive index of 1.57 (*n* = 1.57) in circularly polarized Gaussian beams are shown in Fig. 1(b) and 1(c), from which their stable orientations can also be obtained. We performed calculations for micro- and nano-spheroids with primary axes less than 2000 nm and secondary axes less than 500 nm. The orientation landscapes of micro- and nano-spheroids with the refractive index of 1.57 in both linearly and circularly polarized beams were shown in Fig. 2. The dot lines are the spheres with different diameters, which were ignored here. The fitting curves were used to distinguish different orientation regions. The regime of prolate spheroids ($\delta >1$) was divided into three regions in both linearly and circularly polarized beams: a vertical region, horizontal region, and an intermediate region between the vertical and horizontal regions. The capital characters “H”, “V” and “I” represent the horizontal, vertical and intermediate regions, respectively, in which the spheroids were trapped with their primary axes were perpendicular to the beam axes, along the beam axes and with a tilt angle between 0 and 90 degrees from the beam axes. The orientation performance of prolate spheroids trapped in differently polarized beams is completely the same as the orientation landscapes of cylinders [23]. It’s therefore not necessary to elaborate that performance again.

For oblate spheroids ($\delta <1$), there is only a horizontal region for linear polarization. However, the orientation landscape of oblate spheroids was divided into three regions for circular polarization. The oblate spheroids with small sizes can be rotated vertically or with a tilt angle for circular polarization, while the oblate spheroids with large size orient horizontally. Meanwhile, another difference between orientations of prolate and oblate spheroids in linearly polarized beams was found in our calculations as well. The prolate spheroids in horizontal regions orients horizontally with the primary axes along the polarization direction. However, the primary-axis orientations of the oblate spheroids in horizontal regions are perpendicular to the polarization direction. Thus, both beam polarization and the aspect ratio of spheroids play very important roles in their orientation landscapes.

The equilibrium orientations of spheroids can be estimated using the torque efficiency curves of the objects located at the equilibrium positions as functions of tilt angle. The torque efficiencies of oblate spheroids with secondary axis of 450 nm and with primary axes of 120, 240 and 360 nm, for instance, were calculated as functions of tilt angle from the beam axes in linearly and circularly polarized beams were shown in Fig. 3(a) and 3(b) separately. These calculations were performed for the spheroids located at equilibrium positions for both y-linearly and circularly polarized beams. Since the equilibrium orientations of prolate and oblate spheroids are different, as mentioned above, the primary-axis orientations of these oblate spheroids were changed from + z to –x axes. The torque efficiency, *Q*_{ty}, is the torque along y- axis. The orientation angle decreases when *Q*_{ty} > 0 and increases when *Q*_{ty} < 0. A stable orientation is achieved when a deviation either side of *Q*_{ty} = 0 returns the object its previous orientation.

According to the curves shown in Fig. 3, the oblate spheroids with thicknesses along the primary axes of 120, 240 and 360 nm orient horizontally in y-linearly polarized beams, but vertically, with a tilt angle (*θ* = 52°) and horizontally in circularly polarized beams, respectively. In Ref [25], Simpson & Hanna only used the linearly polarized beam and therefore the secondary axis was always aligned with the beam axis, which is consistent with the results obtained in this paper. Comparing to the linearly polarized Gaussian beams without spinning angular momentum, the circularly polarized beams do have spinning angular momentum [26,27], which will translate to the particle during optical trapping process. According to the components of total torque efficiencies, it’s demonstrated that *Q*_{ty} is a key factor to make oblate spheroids with smaller sizes have vertical orientation. Based on the rectangular coordinate system in particle frame, the torque provided by spinning angular momentum of the beam has three components *Γ*_{L1}, *Γ*_{L2} and *Γ*_{S}, where *Γ*_{S} and *Γ*_{L} are the torque components along the short and long axes, respectively. The torque component along symmetry axis cannot determine the object’s orientation, but helpful to the spinning of the particle. Since the aspect ratio of spheroids plays an important role in their orientations as mentioned above, a tentative inference on this result is therefore obtained. In the case of unconsidering of the spinning of the spheroids, the effect of torque components along the long axes of oblate spheroids become weaker when their volumes increase.

## 4. Conclusions

In summary, equilibrium orientations of micro- and nano-spheroids (n = 1.57) with varying sizes and aspect ratios were numerically investigated systematically. The spheroids are illuminated by a Gaussian beam with a vertical beam axial and being tightly focused by a lens with a numerical aperture of 1.25. According to the alignment of the primary axis of the spheroids to the beam axis, three orientation landscapes of horizontal region, vertical region and intermediate region are found. For the oblate spheroids, they can be trapped vertically or with a tilt angle in circular polarization beams. In linearly polarized beams, however, they orient horizontally only. It’s therefore demonstrated that the stable orientations of spheroids is strongly depend on the polarization of the trapping beam. In circularly polarized beams, it’s also demonstrated that the equilibrium orientations of prolate and oblate spheroids have quite different performances. The prolate spheroids could stand up, while the oblate spheroids could be orientated vertically (with their primary axes along the beam axes). According to the orientation difference of oblate spheroids in different polarization beams (i.e. orient horizontally in linearly polarized beams and rotate with the primary axes along the beam axes in circular polarization), they might be used as switches in microfluidics. For future work, we are interested in the investigation of translation and rotation of other geometries and demonstrating their applications based on their properties.

## Acknowledgments

We thank the anonymous reviewer very much for the valuable comments on this paper. This work is supported by Program for Innovation Research of Science in Harbin Institute of Technology (B201407 and A201411).

## References and links

**1. **A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**(4), 156–159 (1970). [CrossRef]

**2. **M. Gu, S. Kuriakose, and X. S. Gan, “A single beam near-field laser trap for optical stretching, folding and rotation of erythrocytes,” Opt. Express **15**(3), 1369–1375 (2007). [CrossRef] [PubMed]

**3. **G. Whyte, G. Gibson, J. Leach, M. Padgett, D. Robert, and M. Miles, “An optical trapped microhand for manipulating micron-sized objects,” Opt. Express **14**(25), 12497–12502 (2006). [CrossRef] [PubMed]

**4. **M. Tassieri, R. M. L. Evans, R. L. Warren, N. J. Bailey, and J. M. Cooper, “Microrheology with optical tweezers: data analysis,” New J. Phys. **14**(11), 115032 (2012). [CrossRef]

**5. **O. Toader, S. John, and K. Busch, “Optical trapping, field enhancement and laser cooling in photonic crystals,” Opt. Express **8**, 217–222 (2001).

**6. **A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. Lond. A **459**(2040), 3021–3041 (2003). [CrossRef]

**7. **T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. **9**(8), S196–S203 (2007). [CrossRef]

**8. **A. A. R. Neves, A. Fontes, L. Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express **14**(26), 13101–13106 (2006). [CrossRef] [PubMed]

**9. **D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express **20**(28), 29679–29693 (2012). [CrossRef] [PubMed]

**10. **M. R. Pollard, S. W. Botchway, B. Chichkov, E. Freeman, R. N. J. Halsall, D. W. K. Jenkins, I. Loader, A. Ovsianikov, A. W. Parker, R. Stevens, R. Turchetta, A. D. Ward, and M. Towrie, “Optically trapped probes with nanometer-scale tips for femto-newton force measurement,” New J. Phys. **12**(11), 113056 (2010). [CrossRef]

**11. **L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, “Assembly and measurement with SPM-like probes in holographic optical tweezers,” New J. Phys. **11**(2), 023012 (2009). [CrossRef]

**12. **S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of biological cells using plane-polarized Gaussian beam optical tweezers,” J. Mod. Opt. **50**(10), 1581–1590 (2003). [CrossRef]

**13. **T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. **142**(1-3), 468–471 (2001). [CrossRef]

**14. **S. H. Simpson and S. Hanna, “Optical trapping of dielectric ellipsoids,” Proc. SPIE **7762**, 77621B (2010). [CrossRef]

**15. **M. Wojdyla, S. Raj, and D. Petrov, “Absorption spectroscopy of single red blood cells in the presence of mechanical deformations induced by optical traps,” J. Biomed. Opt. **17**(9), 097006 (2012). [CrossRef] [PubMed]

**16. **S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A **24**(2), 430–443 (2007). [CrossRef] [PubMed]

**17. **A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A **68**(3), 033802 (2003). [CrossRef]

**18. **T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. **58**(5-6), 528–544 (2011). [CrossRef]

**19. **V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. **110**(14-16), 1460–1471 (2009). [CrossRef]

**20. **S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express **19**(17), 16526–16541 (2011). [CrossRef] [PubMed]

**21. **E. M. Purcell, “Life at low Reynolds-number,” Am. J. Phys. **45**(1), 3–11 (1977). [CrossRef]

**22. **M. A. Charsooghi, E. A. Akhlaghi, S. Tavaddod, and H. R. Khalesifard, “A MATLAB program to calculate translational and rotational diffusion coefficients of a single particle,” Comput. Phys. Commun. **182**(2), 400–408 (2011). [CrossRef]

**23. **Y. Y. Cao, A. B. Stilgoe, L. X. Chen, T. A. Nieminen, and H. Rubinsztein-Dunlop, “Equilibrium orientations and positions of non-spherical particles in optical traps,” Opt. Express **20**(12), 12987–12996 (2012). [CrossRef] [PubMed]

**24. **O. A. Bauchau and L. Trainelli, “The vectorial parameterization of rotation,” Nonlinear Dyn. **32**(1), 71–92 (2003). [CrossRef]

**25. **S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A **84**(5), 053808 (2011). [CrossRef]

**26. **S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. **102**(11), 113602 (2009). [CrossRef] [PubMed]

**27. **T. Asavei, V. L. Y. Loke, M. Barbieri, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical angular momentum transfer to microrotors fabricated by two-photon photopolymerization,” New J. Phys. **11**(9), 093021 (2009). [CrossRef]