Abstract

Optical traps can be characterized in terms of two simple parameters: the stiffness, given by the gradient of the force at mechanical equilibrium, and the strength, as expressed by the maximum restoring force available for displacement in a given direction. We present numerical calculations of these quantities for dielectric microrods of varying radius and refractive index held horizontally in pairs of holographically generated Gaussian beams. The resulting variations are seen to be influenced by optical resonances, as well as by the relative sizes of the beam waist and rod diameter. In addition, it is shown that trapping in these systems is sensitive to the polarization state of the incident field; i.e., for certain rods, trapping will occur for beams polarized perpendicular to the long axis of the rod, but not for beams polarized parallel to the long axis. Finally, friction coefficients are evaluated and used to estimate the maximum rates at which the rods may be dragged through the ambient medium.

© 2011 Optical Society of America

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    [CrossRef] [PubMed]
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  33. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Series A 366, 155–171 (1979).
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  38. L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
    [CrossRef]
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    [CrossRef]
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2011 (1)

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

2010 (4)

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[CrossRef]

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

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

2009 (5)

C. Song, N.-T. Nguyen, and A. K. Asundi, “Optical alignment of a cylindrical object,” J. Opt. A Pure Appl. Opt. 11, 034008 (2009).
[CrossRef]

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

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[CrossRef]

J.-Q. Qin, X.-L. Wang, D. Jia, J. Chen, Y.-X. Fan, J. Ding, and H.-T. Wang, “FDTD approach to optical forces of tightly focused vector beams on metal particles,” Opt. Express 17, 8407–8416 (2009).
[CrossRef] [PubMed]

S. H. Simpson and S. Hanna, “Thermal motion of a holographically trapped SPM-like probe,” Nanotechnology 20, 395710(2009).
[CrossRef] [PubMed]

2008 (2)

2007 (6)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

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

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

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

2006 (1)

2005 (3)

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express 13, 8906–8912 (2005).
[CrossRef] [PubMed]

2003 (1)

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

2001 (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

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).
[CrossRef]

2000 (1)

V. A. Bloomfield, “Survey of biomolecular hydrodynamics,” in Biophysics Textbook Online, Separations and Hydrodynamics, T.M.Schuster, ed. (Biophysical Society, 2000), Chap. 1, pp. 1–16.

1999 (3)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

B. Carrasco and J. Garcia de la Torre, “Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures,” Biophys. J. 76, 3044–3057 (1999).
[CrossRef] [PubMed]

C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[CrossRef]

1998 (1)

A.Taflove, ed., Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1998).

1997 (2)

A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, 1997).
[CrossRef]

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

1994 (1)

1992 (1)

R. W. Freund, “Conjugate gradient-type methods for linear-systems with complex symmetrical coefficient matrices,” SIAM J. Sci. Statist. Comput. 13, 425–448 (1992).
[CrossRef]

1991 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986 (1)

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Univ. Press, 1986).

1980 (1)

J. M. Garcia Bernal and J. G. Garcia de la Torre, “Transport-properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes,” Biopolymers 19, 751–766 (1980).
[CrossRef]

1979 (1)

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Series A 366, 155–171 (1979).
[CrossRef]

1973 (2)

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff International, 1973).

E. H. Hauge and A. Martin-Löf, “Fluctuating hydrodynamics and Brownian motion,” J. Stat. Phys. 7, 259–281 (1973).
[CrossRef]

1969 (1)

J. Rotne and S. Prager, “Variational treatment of hydrodynamic interactions in polymers,” J. Chem. Phys. 50, 4831–4837 (1969).
[CrossRef]

1934 (1)

F. Perrin, “Mouvement Brownien d’un ellipsoide—I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium 5, 497–511 (1934).
[CrossRef]

Agarwal, R.

Asundi, A. K.

C. Song, N.-T. Nguyen, and A. K. Asundi, “Optical alignment of a cylindrical object,” J. Opt. A Pure Appl. Opt. 11, 034008 (2009).
[CrossRef]

Benito, D. C.

D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008).
[CrossRef] [PubMed]

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Billaudeau, C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

Bingelyte, V.

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

Bishop, A. I.

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).
[CrossRef]

Blackford, L. S.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Bloomfield, V. A.

V. A. Bloomfield, “Survey of biomolecular hydrodynamics,” in Biophysics Textbook Online, Separations and Hydrodynamics, T.M.Schuster, ed. (Biophysical Society, 2000), Chap. 1, pp. 1–16.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

Bowman, R.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

Branczyk, A. M.

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

Brenner, H.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff International, 1973).

Bryant, G. W.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Carberry, D. M.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

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

Carrasco, B.

B. Carrasco and J. Garcia de la Torre, “Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures,” Biophys. J. 76, 3044–3057 (1999).
[CrossRef] [PubMed]

Chaumet, P. C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Chen, J.

Choi, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Cleary, A.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Courtial, J.

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

Cullen, A. L.

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Series A 366, 155–171 (1979).
[CrossRef]

D’Azevedo, E.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Demmel, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Denz, C.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

Dhillon, I.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Ding, J.

Doi, M.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Univ. Press, 1986).

Dongarra, J.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Draine, B. T.

Edwards, S. F.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Univ. Press, 1986).

Fan, Y.-X.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

Flatau, P. J.

Freund, R. W.

R. W. Freund, “Conjugate gradient-type methods for linear-systems with complex symmetrical coefficient matrices,” SIAM J. Sci. Statist. Comput. 13, 425–448 (1992).
[CrossRef]

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

Garcia Bernal, J. M.

J. M. Garcia Bernal and J. G. Garcia de la Torre, “Transport-properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes,” Biopolymers 19, 751–766 (1980).
[CrossRef]

Garcia de la Torre, J.

B. Carrasco and J. Garcia de la Torre, “Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures,” Biophys. J. 76, 3044–3057 (1999).
[CrossRef] [PubMed]

Garcia de la Torre, J. G.

J. M. Garcia Bernal and J. G. Garcia de la Torre, “Transport-properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes,” Biopolymers 19, 751–766 (1980).
[CrossRef]

Gibson, G. M.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

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

Goodman, J. J.

Greenbaum, A.

A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, 1997).
[CrossRef]

Grier, D. G.

Hammarling, S.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Hanna, S.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[CrossRef]

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

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[CrossRef]

S. H. Simpson and S. Hanna, “Thermal motion of a holographically trapped SPM-like probe,” Nanotechnology 20, 395710(2009).
[CrossRef] [PubMed]

D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008).
[CrossRef] [PubMed]

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

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419–1431 (2006).
[CrossRef]

Happel, J.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff International, 1973).

Hauge, E. H.

E. H. Hauge and A. Martin-Löf, “Fluctuating hydrodynamics and Brownian motion,” J. Stat. Phys. 7, 259–281 (1973).
[CrossRef]

Heckenberg, N. R.

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

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

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

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).
[CrossRef]

Henry, G.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Hörner, F.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

Huang, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

Ikin, L.

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

Jia, D.

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

Knoener, G.

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

Knoner, G.

Ladavac, K.

Leach, J.

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

Li, Z.-Y.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

Lieber, C. M.

Ling, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

Loke, V. L. Y.

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

Maier, B.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

Martin-Löf, A.

E. H. Hauge and A. Martin-Löf, “Fluctuating hydrodynamics and Brownian motion,” J. Stat. Phys. 7, 259–281 (1973).
[CrossRef]

Mazolli, A.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Mesquita, O. N.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Miles, M. J.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

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

Müller, S.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

Neto, P. A. M.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Nguyen, N.-T.

C. Song, N.-T. Nguyen, and A. K. Asundi, “Optical alignment of a cylindrical object,” J. Opt. A Pure Appl. Opt. 11, 034008 (2009).
[CrossRef]

Nieminen, T. A.

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

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

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).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

Nussenzveig, H. M.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Padgett, M. J.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

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

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

Perrin, F.

F. Perrin, “Mouvement Brownien d’un ellipsoide—I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium 5, 497–511 (1934).
[CrossRef]

Petitet, A.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Phillips, D. B.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

Prager, S.

J. Rotne and S. Prager, “Variational treatment of hydrodynamic interactions in polymers,” J. Chem. Phys. 50, 4831–4837 (1969).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

Qin, J.-Q.

Rahmani, A.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Rocha, M. S.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Roichman, Y.

Rotne, J.

J. Rotne and S. Prager, “Variational treatment of hydrodynamic interactions in polymers,” J. Chem. Phys. 50, 4831–4837 (1969).
[CrossRef]

Rubinsztein-Dunlop, H.

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

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

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

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).
[CrossRef]

Saghafi, S.

Schäfer, H.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

Sentenac, A.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Sheppard, C. J. R.

Simpson, S. H.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[CrossRef]

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

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[CrossRef]

S. H. Simpson and S. Hanna, “Thermal motion of a holographically trapped SPM-like probe,” Nanotechnology 20, 395710(2009).
[CrossRef] [PubMed]

D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008).
[CrossRef] [PubMed]

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

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419–1431 (2006).
[CrossRef]

Song, C.

C. Song, N.-T. Nguyen, and A. K. Asundi, “Optical alignment of a cylindrical object,” J. Opt. A Pure Appl. Opt. 11, 034008 (2009).
[CrossRef]

Stanley, K.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Steinhart, M.

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

Stilgoe, A. B.

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

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

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

Viana, N. B.

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

Walker, D.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Wang, H.-T.

Wang, X.-L.

Whaley, R. C.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

Woerdemann, M.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

Yu, G. H.

Yu, P. K.

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Series A 366, 155–171 (1979).
[CrossRef]

Zhou, F.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003).
[CrossRef]

Astrophys. J. (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Biophys. J. (1)

B. Carrasco and J. Garcia de la Torre, “Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures,” Biophys. J. 76, 3044–3057 (1999).
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[CrossRef]

Comput. Phys. Commun. (1)

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).
[CrossRef]

J. Appl. Phys. (2)

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[CrossRef]

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

J. Biophotonics (1)

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3d translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophotonics 3, 468–475 (2010).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

J. Rotne and S. Prager, “Variational treatment of hydrodynamic interactions in polymers,” J. Chem. Phys. 50, 4831–4837 (1969).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

C. Song, N.-T. Nguyen, and A. K. Asundi, “Optical alignment of a cylindrical object,” J. Opt. A Pure Appl. Opt. 11, 034008 (2009).
[CrossRef]

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

J. Opt. Soc. Am. A (6)

J. Optics (1)

D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schäfer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Optics 13, 044023 (2011).
[CrossRef]

J. Phys. Radium (1)

F. Perrin, “Mouvement Brownien d’un ellipsoide—I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium 5, 497–511 (1934).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

J. Stat. Phys. (1)

E. H. Hauge and A. Martin-Löf, “Fluctuating hydrodynamics and Brownian motion,” J. Stat. Phys. 7, 259–281 (1973).
[CrossRef]

Nanotechnology (1)

S. H. Simpson and S. Hanna, “Thermal motion of a holographically trapped SPM-like probe,” Nanotechnology 20, 395710(2009).
[CrossRef] [PubMed]

New J. Phys. (1)

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

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (1)

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Phys. Rev. E (3)

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75, 021914(2007).
[CrossRef]

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[CrossRef]

Proc. IEEE (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

Proc. R. Soc. London Series A (1)

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London Series A 366, 155–171 (1979).
[CrossRef]

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R. W. Freund, “Conjugate gradient-type methods for linear-systems with complex symmetrical coefficient matrices,” SIAM J. Sci. Statist. Comput. 13, 425–448 (1992).
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[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Programming(Cambridge Univ. Press, 2007), 3rd ed.

V. A. Bloomfield, “Survey of biomolecular hydrodynamics,” in Biophysics Textbook Online, Separations and Hydrodynamics, T.M.Schuster, ed. (Biophysical Society, 2000), Chap. 1, pp. 1–16.

L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users Guide (SIAM, 1997).
[CrossRef]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Univ. Press, 1986).

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, 2nd ed. (Noordhoff International, 1973).

A.Taflove, ed., Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1998).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

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Figures (9)

Fig. 1
Fig. 1

Schematic showing a typical configuration of a dielectric cylinder held in a pair of Gaussian beams. The orientation of the coordinate axes is shown. The origin of coordinates is taken in the focal plane.

Fig. 2
Fig. 2

Calculations of (a) the translational and (b) the rotational friction coefficients of a cylinder of radius 0.25 μm and length 5 μm , as a function of the lattice parameter δ, given as a fraction of the cylinder radius, r.

Fig. 3
Fig. 3

Calculations of the maximum restoring force f z , max and the vertical position z max at which it occurs, for cylinders of varying refractive index and radius held symmetrically in pairs of Gaussian beams with the electric field polarized (a) parallel and (b) perpendicular to the cylinder axis. White regions indicate the absence of an extremum in the restoring force.

Fig. 4
Fig. 4

Calculations of the maximum restoring force in the x y plane, and the position at which it occurs, for cylinders of varying refractive index and radius held symmetrically in pairs of beams with the electric field polarized parallel to the cylinder axis: (a)  f x , max and x max ; (b)  f y , max and y max .

Fig. 5
Fig. 5

(a) Vertical equilibrium trapping positions, z eqm , (b) translational trap stiffnesses, K x x t t , K y y t t , and K y y t t [in pN / ( mW . μm ) ], and (c) rotational trap stiffnesses, K y y r r and K z z r r [in pN . μm / ( mW . rad ) ], for dielectric rods of varying refractive index and radius held symmetrically in two Gaussian beams. Missing points correspond to untrappable parameter combinations. The electric field is polarized parallel to the cylinder axis.

Fig. 6
Fig. 6

(a) Vertical equilibrium trapping positions, (b) translational trap stiffnesses, and (c) rotational trap stiffnesses, as shown in Fig. 5, but with the electric field polarized perpendicular to the cylinder axis.

Fig. 7
Fig. 7

Coupling coefficients, K x y t r (in pN / mW ) and K y x r t [in pN / ( mW . rad ) ] for rods trapped in beams polarized (a) parallel and (b) perpendicular to the axes of the rods.

Fig. 8
Fig. 8

Calculations of the (a) translational and (b) rotational friction coefficients for cylinders of varying aspect ratio (length/diameter), normalized by the friction terms for a sphere of equivalent volume. (c) Comparison of the numerical calculation of ξ y y t with analytical results for prolate spheroids, and with results obtained from Kirkwood theory.

Fig. 9
Fig. 9

Maximum possible drag rates for motion of the cylinder parallel to the x and y axes, as determined from the calculated trap strengths and friction coefficients.

Equations (4)

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q ˙ + ξ 1 Kq = ξ 1 f L ( t ) .
f L ( t ) f L ( t ) = 2 k B T ξ δ ( t t ) .
K = ( K tt K tr K rt K rr ) ,
K tt = ( K x x t t 0 0 0 K y y t t 0 0 0 K z z t t ) , K tr = ( 0 K x y t r 0 0 0 0 0 0 0 ) , K rt = ( 0 0 0 K y x r t 0 0 0 0 0 ) , K rr = ( 0 0 0 0 K y y r r 0 0 0 K z z r r ) .

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