## Equilibrium orientations and positions of non-spherical particles in optical traps |

Optics Express, Vol. 20, Issue 12, pp. 12987-12996 (2012)

http://dx.doi.org/10.1364/OE.20.012987

Acrobat PDF (1599 KB)

### Abstract

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

## 1. Introduction

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**(5), 288–290 (1986). [PubMed]

3. J. Harris and G. McConnell, “Optical trapping and manipulation of live T cells with a low numerical aperture lens,” Opt. Express **16**(18), 14036–14043 (2008). [PubMed]

5. 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). [PubMed]

8. A. A. R. Neves, A. Fontes, Lde. 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). [PubMed]

8. A. A. R. Neves, A. Fontes, Lde. 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). [PubMed]

10. A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express **16**(19), 15039–15051 (2008). [PubMed]

14. K. Ramser and D. Hanstorp, “Optical manipulation for single-cell studies,” J Biophoton. **3**(4), 187–206 (2010). [PubMed]

15. R. C. Gauthier and A. Frangioudakis, “Theoretical investigation of the optical trapping properties of a micro-optic cubic glass structure,” Appl. Opt. **39**(18), 3060–3070 (2000). [PubMed]

16. H. Ukita and H. Kawashima, “Optical rotor capable of controlling clockwise and counterclockwise rotation in optical tweezers by displacing the trapping position,” Appl. Opt. **49**(10), 1991–1996 (2010). [PubMed]

17. P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragó, “Rotation Detection in Light-Driven Nanorotors,” ACS Nano **3**(10), 3077–3084 (2009). [PubMed]

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

25. 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). [PubMed]

26. 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). [PubMed]

27. S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A **27**(6), 1255–1264 (2010). [PubMed]

## 2. Theory

### 2.1. T-matrix calculation

*n*and

*k*are the radial mode indices,

*a*and

_{n}*p*are the expansion coefficients of incident and scattered fields respectively. The relationship between the incident and scattered fields can be written as a matrix equationwhere

_{k}**T**is the transition matrix.

### 2.2. Translation and rotation of beam coefficients

**a**

_{0}and

**b**

_{0}, coefficients in a coordinate system at any position can be found using a linear transformation [18,31], Where

**a**and

**b**are the TE/TM modes of

**R**is the rotation of beam shape coefficients from the x

_{1}_{1}y

_{1}z

_{1}frame such that z

_{1}would point along

**O**, as shown in Fig. 2 .

_{1}O_{2}**A**and

**B**are the translations of beam coefficients in the rotated coordinate system along the direction of

**O**.

_{1}O_{2}**R**is the rotation of beam coefficients from the Cartesian coordinate system x

_{2}_{2}y

_{2}z

_{2}centered at the particle position to the coordinate system x

_{3}y

_{3}z

_{3}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

**R**

_{1}is calculated using the direction vector

**O**

_{1}

**O**

_{2},

**A**and

**B**move the beams between O

_{1}and O

_{2}.

**R**

_{2}orients the translated field into a particle orientated frame, which depends on a 3 × 3 rotation matrix

**R**of particle.

_{2,3}**R**is considered as a function of time, which can be written asWhereis the Euler-Rodrigues formula for axis angle rotations.

_{2,3}**u.**( u × ) 2 is the skew matrix [32]. If we define the angular velocity of cylinder at time

*t*as

*ω**,*

_{t}### 2.3. Translation and rotation of micro- and nano-particles

**and**

*Q***, are calculated with optical tweezers computational toolbox [7], which uses sums of products of the expansion coefficients of incident and scattered fields (**

*τ**a*,

_{nm}*b*,

_{nm}*p*and

_{nm}*q*) 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 Where

_{nm}*n*

_{m}is refractive index of medium,

*P*

_{inc}is the incident power of laser beam, and

*ω*is the optical frequency.

*t*, the equations of motion we solve for a particle are where

**γ**and

_{t}**γ**are the translational and rotational friction tensors for a particle [34

_{r}34. J. G. Garcia de la Torre and V. A. Bloomfield, “Hydrodynamic properties of complex, rigid, biological macromolecules: theory and applications,” Q. Rev. Biophys. **14**(1), 81–139 (1981). [PubMed]

**is the position of the center of the object. Since the translational and rotational friction tensors given by Delatorre and Bloomfield [34**

*r*34. J. G. Garcia de la Torre and V. A. Bloomfield, “Hydrodynamic properties of complex, rigid, biological macromolecules: theory and applications,” Q. Rev. Biophys. **14**(1), 81–139 (1981). [PubMed]

*α*and

*β*are constants, but they can be changed for different cylinders and different beams. New orientations can be obtained by 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.

## 3. Results

_{1}direction with its focus located at (x

_{1}, y

_{1}, z

_{1}) = (0, 0, 0). The wavelength of the beam was

*λ*

_{0}= 1064 nm in vacuum and

*λ*= 800 nm in water, the power of the beam going through the focal plane was

*P*

_{inc}= 1 mW. The immersed cylinder was made of common glass with a refractive index

*n*

_{p}= 1.57. The origin of the cylinder was located at the centre of mass.

*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 [35]. 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.

*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.

36. 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). [PubMed]

*Q*

_{τt,}is the torque in these planes. When

*Q*

_{τt}< 0 the orientation angle decreases and increases when

*Q*

_{τt}> 0. A stable orientation is achieved when a deviation either side of

*Q*

_{τt}= 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 [26

26. 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). [PubMed]

## 4. Conclusions

## Acknowledgments

## References and links

1. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

3. | J. Harris and G. McConnell, “Optical trapping and manipulation of live T cells with a low numerical aperture lens,” Opt. Express |

4. | M. C. Zhong, J. H. Zhou, Y. X. Ren, Y. M. Li, and Z. Q. Wang, “Rotation of birefringent particles in optical tweezers with spherical aberration,” Appl. Opt. |

5. | 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. |

6. | A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. Lond. A |

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 |

8. | A. A. R. Neves, A. Fontes, Lde. 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 |

9. | G. Knöner, T. A. Nieminen, S. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Calculation of optical trapping landscapes,” Proc. SPIE |

10. | A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express |

11. | T. A. Nieminen, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Symmetry and the generation and measurement of optical torque,” J. Quant. Spectrosc. Radiat. Transf. |

12. | R. C. Gauthier, “Trapping model for the low-index ring-shaped micro-object in a focused, lowest-order Gaussian laser-beam profile,” J. Opt. Soc. Am. B |

13. | R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B |

14. | K. Ramser and D. Hanstorp, “Optical manipulation for single-cell studies,” J Biophoton. |

15. | R. C. Gauthier and A. Frangioudakis, “Theoretical investigation of the optical trapping properties of a micro-optic cubic glass structure,” Appl. Opt. |

16. | H. Ukita and H. Kawashima, “Optical rotor capable of controlling clockwise and counterclockwise rotation in optical tweezers by displacing the trapping position,” Appl. Opt. |

17. | P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragó, “Rotation Detection in Light-Driven Nanorotors,” ACS Nano |

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. |

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. |

20. | S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express |

21. | P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. |

22. | S. D. Tan, H. A. Lopez, C. W. Cai, and Y. G. Zhang, “Optical trapping of single-walled carbon nanotubes,” Nano Lett. |

23. | J. L. Zhang, T. G. Kim, S. C. Jeoung, F. F. Yao, H. Lee, and X. D. Sun, “Controlled trapping and rotation of carbon nanotube bundle with optical tweezers,” Opt. Commun. |

24. | Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature |

25. | 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 |

26. | 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. |

27. | S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A |

28. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. |

29. | T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. |

30. | 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 |

31. | 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. |

32. | O. A. Bauchau and L. Trainelli, “The vectorial parameterization of rotation,” Nonlinear Dyn. |

33. | E. M. Purcell, “Life at low Reynolds-number,” Am. J. Phys. |

34. | J. G. Garcia de la Torre and V. A. Bloomfield, “Hydrodynamic properties of complex, rigid, biological macromolecules: theory and applications,” Q. Rev. Biophys. |

35. | T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. |

36. | 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. |

37. | H. K. Moffat, “Six Lectures on General Fluid Dynamics and Two on Hydromagnetic Dynamo Theory,” in |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: March 7, 2012

Revised Manuscript: May 3, 2012

Manuscript Accepted: May 4, 2012

Published: May 24, 2012

**Virtual Issues**

Vol. 7, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Yongyin Cao, Alexander B Stilgoe, Lixue Chen, Timo A Nieminen, and Halina Rubinsztein-Dunlop, "Equilibrium orientations and positions of non-spherical particles in optical traps," Opt. Express **20**, 12987-12996 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12987

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### References

- A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24, 156–159 (1970).
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986). [PubMed]
- J. Harris and G. McConnell, “Optical trapping and manipulation of live T cells with a low numerical aperture lens,” Opt. Express16(18), 14036–14043 (2008). [PubMed]
- M. C. Zhong, J. H. Zhou, Y. X. Ren, Y. M. Li, and Z. Q. Wang, “Rotation of birefringent particles in optical tweezers with spherical aberration,” Appl. Opt.48(22), 4397–4402 (2009). [PubMed]
- 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). [PubMed]
- A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. Lond. A459, 3021–3041 (2003).
- 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. A9, S196–S203 (2007).
- A. A. R. Neves, A. Fontes, Lde. 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. Express14(26), 13101–13106 (2006). [PubMed]
- G. Knöner, T. A. Nieminen, S. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Calculation of optical trapping landscapes,” Proc. SPIE6326, U119–U127 (2006).
- A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express16(19), 15039–15051 (2008). [PubMed]
- T. A. Nieminen, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Symmetry and the generation and measurement of optical torque,” J. Quant. Spectrosc. Radiat. Transf.110, 1472–1482 (2009).
- R. C. Gauthier, “Trapping model for the low-index ring-shaped micro-object in a focused, lowest-order Gaussian laser-beam profile,” J. Opt. Soc. Am. B14, 782–789 (1997).
- R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B14, 3323–3333 (1997).
- K. Ramser and D. Hanstorp, “Optical manipulation for single-cell studies,” J Biophoton.3(4), 187–206 (2010). [PubMed]
- R. C. Gauthier and A. Frangioudakis, “Theoretical investigation of the optical trapping properties of a micro-optic cubic glass structure,” Appl. Opt.39(18), 3060–3070 (2000). [PubMed]
- H. Ukita and H. Kawashima, “Optical rotor capable of controlling clockwise and counterclockwise rotation in optical tweezers by displacing the trapping position,” Appl. Opt.49(10), 1991–1996 (2010). [PubMed]
- P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragó, “Rotation Detection in Light-Driven Nanorotors,” ACS Nano3(10), 3077–3084 (2009). [PubMed]
- 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, 528–544 (2011).
- 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, 1460–1471 (2009).
- S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express19(17), 16526–16541 (2011). [PubMed]
- P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys.101, 023106 (2007).
- S. D. Tan, H. A. Lopez, C. W. Cai, and Y. G. Zhang, “Optical trapping of single-walled carbon nanotubes,” Nano Lett.4, 1415–1419 (2004).
- J. L. Zhang, T. G. Kim, S. C. Jeoung, F. F. Yao, H. Lee, and X. D. Sun, “Controlled trapping and rotation of carbon nanotube bundle with optical tweezers,” Opt. Commun.267, 260–263 (2006).
- Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature447(7148), 1098–1101 (2007). [PubMed]
- 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. Express18(2), 822–830 (2010). [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). [PubMed]
- S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A27(6), 1255–1264 (2010). [PubMed]
- M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton.2, 021875 (2008).
- T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf.79–80, 1019–1029 (2003).
- 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. A68, 033802 (2003).
- 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, 8825–8831 (1999).
- O. A. Bauchau and L. Trainelli, “The vectorial parameterization of rotation,” Nonlinear Dyn.32, 71–92 (2003).
- E. M. Purcell, “Life at low Reynolds-number,” Am. J. Phys.45, 3–11 (1977).
- J. G. Garcia de la Torre and V. A. Bloomfield, “Hydrodynamic properties of complex, rigid, biological macromolecules: theory and applications,” Q. Rev. Biophys.14(1), 81–139 (1981). [PubMed]
- T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142, 468–471 (2001).
- 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). [PubMed]
- H. K. Moffat, “Six Lectures on General Fluid Dynamics and Two on Hydromagnetic Dynamo Theory,” in Fluid Dynamics, R. Balian and J.-L. Peube eds., (Gordon and Breach, 1977), pp. 149–234.

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