Abstract

Holographic optical tweezing permits the trapping of objects with less than spherical symmetry in appropriately distributed sets of beams thereby permitting control to be exerted over both the orientation and position. In contrast to the familiar case of the singly trapped sphere, the stiffness and strength of such compound traps will have rotational components. We investigate this for a simple model system consisting of multiply trapped dielectric cylinder. Optically induced forces and torques are evaluated using the discrete dipole approximation and the resulting trap stiffnesses are presented. A variety of configurations of trapping beams are considered. Hydrodynamic resistances for the cylinder are also calculated and used to estimate translation and rotation rates. A number of conclusions are reached concerning the optimal trapping and dragging conditions for the rod. In particular, it is clear that it is advantageous to drag a rod in a direction perpendicular rather than parallel to its length. In addition, it is observed that the polarization of the incident light plays a significant role. Finally, it is noted that the non-conservative nature of the optical force field manifests itself directly in the stiffness of the trapped cylinder. The consequences of this last point are discussed.

© 2010 Optical Society of America

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2009 (7)

S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 173–183 (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]

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

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]

T. L. Min, P. J. Mears, L. M. Chubiz, I. Golding, Y. R. Chemla, and C. V. Rao, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–835 (2009).
[CrossRef] [PubMed]

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]

J. Trojek, V. Karasek, and P. Zemanek, “Extreme axial optical force in a standing wave achieved by optimized object shape,” Opt. Express 17, 10472–10488 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (8)

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

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]

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

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[CrossRef]

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A, Pure Appl. Opt. 9, S228–S234 (2007).
[CrossRef]

A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106, 417–436 (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]

2006 (2)

N. B. Viana, A. Mazolli, P. A. M. Neto, H. M. Nussenzveig, M. S. Rocha, and O. N. Mesquita, “Absolute calibration of optical tweezers,” Appl. Phys. Lett. 88, 131110 (2006).
[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]

2005 (4)

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]

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95, 168102 (2005).
[CrossRef] [PubMed]

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

2004 (3)

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

H. Kress, E. H. K. Stelzer, and A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84, 4271–4273 (2004).
[CrossRef]

T. Yu, F. Cheong, and C. Sow, “The manipulation and assembly of CuO nanorods with line optical tweezers,” Nanotechnology 15, 1732–1736 (2004).
[CrossRef]

2002 (1)

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

2000 (2)

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Comput. Phys. 159, 13–37 (2000).
[CrossRef]

1999 (3)

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

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

D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed. (Oxford U. Press, 1999).

1998 (1)

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

1997 (3)

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

L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (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 (Soc. Ind. Appl. Math.) J. Sci. Stat. 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)

1980 (2)

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

V.K.Varadan and V.V.Varadan, eds., Acoustic, Electromagnetic and Elastic Wave Scattering: Focus on theT-matrix Approach (Pergamon, 1980).

1979 (1)

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

1977 (1)

P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. 25, 373–381 (1977).
[CrossRef]

1973 (1)

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

1965 (1)

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1965).

Agarwal, R.

Ashkin, A.

Barber, P. W.

P. W. Barber, “Resonance electromagnetic absorption by nonspherical dielectric objects,” IEEE Trans. Microwave Theory Tech. 25, 373–381 (1977).
[CrossRef]

Benito, D.

Berg-Sorensen, K.

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[CrossRef]

Bernal, J. M. García

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

Bian, R. X.

L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[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]

Bjorkholm, J. E.

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]

Bonessi, D.

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A, Pure Appl. Opt. 9, S228–S234 (2007).
[CrossRef]

Bonin, K.

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A, Pure Appl. Opt. 9, S228–S234 (2007).
[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.

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. García 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]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

Chemla, Y. R.

T. L. Min, P. J. Mears, L. M. Chubiz, I. Golding, Y. R. Chemla, and C. V. Rao, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–835 (2009).
[CrossRef] [PubMed]

Chen, J.

Cheong, F.

T. Yu, F. Cheong, and C. Sow, “The manipulation and assembly of CuO nanorods with line optical tweezers,” Nanotechnology 15, 1732–1736 (2004).
[CrossRef]

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]

Chu, S.

Chubiz, L. M.

T. L. Min, P. J. Mears, L. M. Chubiz, I. Golding, Y. R. Chemla, and C. V. Rao, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–835 (2009).
[CrossRef] [PubMed]

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]

Cullen, A. L.

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. 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]

de la Torre, J. García

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

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

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.

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.

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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).
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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).
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N. B. Viana, A. Mazolli, P. A. M. Neto, H. M. Nussenzveig, M. S. Rocha, and O. N. Mesquita, “Absolute calibration of optical tweezers,” Appl. Phys. Lett. 88, 131110 (2006).
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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).
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Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
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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).
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A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106, 417–436 (2007).
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Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
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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]

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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).
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A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106, 417–436 (2007).
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H. Kress, E. H. K. Stelzer, and A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84, 4271–4273 (2004).
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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).
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Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
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Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
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N. B. Viana, A. Mazolli, P. A. M. Neto, H. M. Nussenzveig, M. S. Rocha, and O. N. Mesquita, “Absolute calibration of optical tweezers,” Appl. Phys. Lett. 88, 131110 (2006).
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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).
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Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
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M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
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Figures (7)

Fig. 1
Fig. 1

Schematic showing a typical initial simulation geometry with a cylinder held in a parallel linear array of Gaussian beams.

Fig. 2
Fig. 2

Calculations of (a) the forces and (b) the torques acting on a displaced cylinder with a radius of 0.2 μ m and a length of 5 μ m , held in two parallel traps, as functions of the DDA lattice parameter δ. In (a) the cylinder is displaced by 0.1 μ m along the x- and y-axes, while in (b) it is rotated by 0.1 rad about each axis.

Fig. 3
Fig. 3

Calculations of induced forces and torques on the cylinder in two parallel Gaussian beams as functions of (a)–(d) translational and (e)–(h) rotational displacements from its initial position.

Fig. 4
Fig. 4

Restoring forces acting on a cylinder in two traps for displacement in the (a) x- and (b) y-directions, for different separations between the beams. The beam polarization is parallel to the cylinder axis.

Fig. 5
Fig. 5

Calculations of (a) the equilibrium trapping height and (b)–(e) the stiffness coefficients of a 5 μ m long 0.2 μ m radius cylinder trapped in two parallel beams as functions of the beam separation.

Fig. 6
Fig. 6

The torque about the beam axis, τ z , experienced by a cylinder with a length of 5 μ m and a radius of 0.25 μ m symmetrically trapped in a linear parallel array of beams. The beams are polarized perpendicular to the cylinder axis.

Fig. 7
Fig. 7

Induced (a) forces and (b) torques as functions of y-displacement, for a cylinder trapped in an asymmetric arrangement of three parallel beams with collinear foci. The displacement is measured relative to the trapping coordinates (see text).

Equations (17)

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f = 1 4 | E | 2 Δ ϵ δ ( n ) n ̂ ,
t = r × f ,
K = ( K t t K t r K r t K r r ) .
α = α CM 1 i 2 3 k 3 α CM ,
E j = E j inc k = 1 , . . , N ; k j A j k P k ,
P k = α E k .
E j inc = k = 1 , . . , N A ¯ j k P k ,
A ¯ j k = { A j k , j k 1 / α , j = k , }
ψ ( r ) R ψ ( R 1 r ) + r 0 ,
ψ ( r ) R ψ ( R 1 r ) R 1 + r 0 .
f ( x ) f ( x + 2 Δ ) + 8 f ( x + Δ ) 8 f ( x Δ ) + f ( x 2 Δ ) 12 Δ .
K t t = ( K x x t t 0 0 0 K y y t t 0 0 0 K z z t t ) ,     K r r = ( 0 0 0 0 K y y r r 0 0 0 K z z r r ) ,
K t r = ( 0 K x y t r 0 0 0 0 0 0 0 ) ,     K r t = ( 0 0 0 K y x r t 0 0 0 0 0 ) .
E = ( g ( k R ) + [ f ( k R ) g ( k R ) ] x 2 R 2 + i 2 f ( k R ) k z ) x ̂ + ( [ f ( k R ) g ( k R ) ] x y R 2 ) y ̂ + ( [ f ( k R ) g ( k R ) ] x z R 2 i 2 f ( k R ) k x ) z ̂ ,
R = ( x 2 + y 2 + z 2 ) 1 / 2 .
f ( k R ) = j 0 ( k R ) + j 2 ( k R ) ,
g ( k R ) = j 0 ( k R ) 1 2 j 2 ( k R ) ,

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