Abstract

We present a general computational method of determining radiation pressure forces and torques exerted on small particles by a converging beam of light. This method, based on a ray optics model of optical trapping, allows time-series dynamic motion analysis to be performed on nonspherical objects that are initially positioned off the optical axis with arbitrary orientation. Comparison tests of computer simulation with experimental results prove that the proposed model can be used to predict complicated trapping behavior of microfabricated objects.

© 2000 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
    [CrossRef] [PubMed]
  4. A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
    [CrossRef]
  5. S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
    [CrossRef]
  6. S. Sato, H. Inaba, “Second-harmonic and sum-frequency generation from optically trapped KTiOPO4 microscopic particles by use of Nd:YAG and Ti:Al2O3 lasers,” Opt. Lett. 19, 927–929 (1994).
    [CrossRef] [PubMed]
  7. E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
    [CrossRef]
  8. E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
    [CrossRef]
  9. E. Higurashi, O. Ohguchi, H. Ukita, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
    [CrossRef] [PubMed]
  10. L. P. Ghislain, W. W. Webb, “Scanning-force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
    [CrossRef] [PubMed]
  11. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  12. G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
    [CrossRef]
  13. W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
    [CrossRef]
  14. R. C. Gauthier, S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equation,” J. Opt. Soc. Am. B 12, 1680–1686 (1995).
    [CrossRef]
  15. R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micro-motor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
    [CrossRef]
  16. R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
    [CrossRef]
  17. 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 14, 782–789 (1997).
    [CrossRef]
  18. R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B 14, 3323–3333 (1997).
    [CrossRef]
  19. R. C. Gauthier, M. Ashman, “Simulated dynamic behavior of single and multiple spheres in the trap region of focused laser beams,” Appl. Opt. 37, 6421–6431 (1998).
    [CrossRef]
  20. W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
    [CrossRef] [PubMed]
  21. J. J. Craig, Introduction to Robotics (Addison-Wesley, Reading, Mass., 1989).
  22. T. R. Kane, D. A. Levinson, Dynamics: Theory and Applications (McGraw-Hill, New York, 1985).
  23. M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
    [CrossRef]
  24. E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
    [CrossRef]
  25. E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
    [CrossRef]

1998 (2)

E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
[CrossRef]

R. C. Gauthier, M. Ashman, “Simulated dynamic behavior of single and multiple spheres in the trap region of focused laser beams,” Appl. Opt. 37, 6421–6431 (1998).
[CrossRef]

1997 (3)

1996 (1)

R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
[CrossRef]

1995 (4)

R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micro-motor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
[CrossRef]

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

R. C. Gauthier, S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equation,” J. Opt. Soc. Am. B 12, 1680–1686 (1995).
[CrossRef]

E. Higurashi, O. Ohguchi, H. Ukita, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
[CrossRef] [PubMed]

1994 (3)

1993 (1)

1992 (2)

M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
[CrossRef]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

1990 (1)

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

1989 (1)

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

1987 (2)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

1986 (1)

1976 (1)

G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

1970 (1)

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

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

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

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

Ashman, M.

Berg, H. C.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Berns, M. W.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Bjorkholm, J. E.

Blair, D. F.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Block, S. M.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Chu, S.

Craig, J. J.

J. J. Craig, Introduction to Robotics (Addison-Wesley, Reading, Mass., 1989).

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

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

Gauthier, R. C.

Ghislain, L. P.

Higurashi, E.

E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
[CrossRef]

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

E. Higurashi, O. Ohguchi, H. Ukita, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
[CrossRef] [PubMed]

E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

Imbert, C.

G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

Inaba, H.

Itao, K.

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

Ito, T.

E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
[CrossRef]

Kane, T. R.

T. R. Kane, D. A. Levinson, Dynamics: Theory and Applications (McGraw-Hill, New York, 1985).

Levinson, D. A.

T. R. Kane, D. A. Levinson, Dynamics: Theory and Applications (McGraw-Hill, New York, 1985).

Matsuura, T.

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

Mehregany, M.

M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
[CrossRef]

Mullen, R. L.

M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
[CrossRef]

Ohguchi, O.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

E. Higurashi, O. Ohguchi, H. Ukita, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
[CrossRef] [PubMed]

E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

Omar, M. P.

M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
[CrossRef]

Roosen, G.

G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

Sato, S.

Sawada, R.

E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
[CrossRef]

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

Sonek, G. J.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Tadir, Y.

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Tamamura, T.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

Tanaka, H.

E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

Ukita, H.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

E. Higurashi, O. Ohguchi, H. Ukita, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
[CrossRef] [PubMed]

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

Wallace, S.

Webb, W. W.

Wright, W. H.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Yamane, T.

A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

E. Higurashi, R. Sawada, T. Ito, “Optically induced rotation of a trapped micro-object about an axis perpendicular to the laser beam axis,” Appl. Phys. Lett. 72, 2951–2953 (1998).
[CrossRef]

R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micro-motor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
[CrossRef]

R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
[CrossRef]

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

J. Appl. Phys. (1)

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

J. Jpn. Soc. Prec. Eng. (1)

E. Higurashi, H. Ukita, O. Ohguchi, T. Matsuura, K. Itao, “Fabrication and optical rotation characteristics of anisotropically shaped micro-objects made of fluorinated polyimide,” J. Jpn. Soc. Prec. Eng. 61, 1021–1025 (1995).
[CrossRef]

J. Microelectromech. Syst. (1)

M. P. Omar, M. Mehregany, R. L. Mullen, “Electric and fluid field analysis of side-drive micromotors,” J. Microelectromech. Syst. 1, 130–140 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nature (London) (2)

A. Ashkin, J. M. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. A (1)

G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

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

Science (1)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

Other (2)

J. J. Craig, Introduction to Robotics (Addison-Wesley, Reading, Mass., 1989).

T. R. Kane, D. A. Levinson, Dynamics: Theory and Applications (McGraw-Hill, New York, 1985).

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Figures (12)

Fig. 1
Fig. 1

Ray optics model for optical trapping of the upward beam. A cylindrical micro-object is shown to represent nonspherical micro-objects.

Fig. 2
Fig. 2

Schematic of the experimental apparatus.

Fig. 3
Fig. 3

Scanning electron microscope photograph of the fabricated micro-objects.

Fig. 4
Fig. 4

Computed z-axis force in frame {A} versus x-rotation angle θ x and y-rotation angle θ y at position (1,1,1) µm.

Fig. 5
Fig. 5

Computed x-axis force in frame {A} versus x-rotation angle θ x and y-rotation angle θ y at position (1,1,1) µm.

Fig. 6
Fig. 6

Computed x-axis torque in frame {B} versus x-rotation angle θ x and y-rotation angle θ y at position (1,1,1) µm.

Fig. 7
Fig. 7

Computed z-axis force in frame {A} versus x-rotation angle θ x and y-rotation angle θ y at position (4,0,2) µm.

Fig. 8
Fig. 8

Computed x-axis force in frame {A} versus x-rotation angle θ x and y-rotation angle θ y at position (4,0,2) µm.

Fig. 9
Fig. 9

Computed x-axis torque in frame {B} versus x-rotation angle θ x and y-rotation angle θ y at position (4,0,2) µm.

Fig. 10
Fig. 10

Dynamic simulation results for initial conditions of x = 1 µm, y = 1 µm, z = 1 µm, θ x = 0, θ y = 0, and θ z = 0: (a) position trajectories and (b) orientation trajectories.

Fig. 11
Fig. 11

Dynamic simulation results for initial conditions of x = 4 µm, y = 0 µm, z = 2 µm, θ x = 0, θ y = 0, and θ z = 0: (a) position trajectories and (b) orientation trajectories.

Fig. 12
Fig. 12

Comparison of animated simulation results with experimental results for initial conditions of x = 4 µm, y = 0 µm, z = 2 µm, θ x = 0, θ y = 0, and θ z = 0 performed in water. A 30-mW laser beam was irradiated perpendicular to the plane of the photograph with the focal points indicated by the crosses. Simulation results of the time that elapsed after trapping began is (a) 0.0 s, (b) 0.6 s, (c) 1.0 s, (d) 1.2 s. The ring cylinder is represented by a solid model for animation. Experimental results (e) before trapping and after (f) 0.6 s, (g) 1.0 s, (h) 1.2 s time has elapsed.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

AP=Ax, y, zT=BAR BP+APBorg,
BP=Bx, y, zT.
BAR=θ˜x2X+Cθ˜xθ˜yX-θ˜zSθ˜xθ˜zX+θ˜ySθ˜yθ˜yX+θ˜zSθ˜y2X+Cθ˜yθ˜zX-θ˜xSθ˜xθ˜zX-θ˜ySθ˜yθ˜zX+θ˜xSθ˜z2X+C,
AP=Ax, y, zT=Adi×s,
BO=x, y, z|x, y, zF1=0 Fm=0.
BP=Bx, y, zT=Bdi×s+BPo,
Fx=mv˙x+bxvx, Fy=mv˙y+byvy, Fz=mv˙z+bzvz+m-mmg,
Tx=Ixω˙x-Iy-Izωyωz+bτxωx, Ty=Iyω˙y-Iz-Ixωzωx+bτyωy, Tz=Izω˙z-Ix-Iyωxωy+bτzωz,
Bk+1AR=BkARBk+1BkR,

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