Abstract

Optical traps have been used in a multitude of applications requiring the sensing and application of forces. However, optical traps also have the ability to accurately apply and sense torques. Birefringent particles experience a torque when trapped in elliptically polarized light. By measuring the frequency content of the exiting beam, the rotational rates can be set up in a feedback loop and actively controlled. Here we describe an optical trap with feedback torque control to maintain constant rotational rates despite the introduction of an increased drag on the particle. As a result, this research has the potential to advance the understanding of rotary motor proteins such as F1 ATPase.

© 2008 Optical Society of America

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    [CrossRef]
  2. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1987).
    [CrossRef] [PubMed]
  4. M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
    [CrossRef] [PubMed]
  5. S. Chu, “Laser manipulation of atoms and particles,” Science 253, 861-866 (1991).
    [CrossRef] [PubMed]
  6. T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
    [CrossRef] [PubMed]
  7. S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
    [CrossRef] [PubMed]
  8. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
    [CrossRef] [PubMed]
  9. G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
    [CrossRef] [PubMed]
  10. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
    [CrossRef] [PubMed]
  11. T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
    [CrossRef]
  12. J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
    [CrossRef] [PubMed]
  13. C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
    [CrossRef] [PubMed]
  14. K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
    [CrossRef] [PubMed]
  15. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-127(1936).
    [CrossRef]
  16. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
    [CrossRef]
  17. M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
    [CrossRef]
  18. A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
    [CrossRef]
  19. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
    [CrossRef] [PubMed]
  20. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  22. S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678(1994).
    [CrossRef]
  23. V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
    [CrossRef]
  24. A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).
  25. A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
    [CrossRef] [PubMed]
  26. A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
    [CrossRef] [PubMed]
  27. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).
  28. E. Higurashi, H. Ukita, H. Tanaka, and O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209-2210(1994).
    [CrossRef]
  29. 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]
  30. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
    [CrossRef]
  31. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
    [CrossRef]

2004 (3)

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
[CrossRef] [PubMed]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

2003 (3)

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

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

2002 (1)

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

2001 (4)

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, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

2000 (1)

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

1999 (2)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

1998 (2)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
[CrossRef]

1997 (2)

1996 (1)

S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
[CrossRef] [PubMed]

1995 (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

1994 (3)

T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
[CrossRef] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678(1994).
[CrossRef]

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

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1991 (1)

S. Chu, “Laser manipulation of atoms and particles,” Science 253, 861-866 (1991).
[CrossRef] [PubMed]

1989 (1)

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

1987 (2)

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

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

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-127(1936).
[CrossRef]

Allemand, J. F. O.

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678(1994).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Andrews, J. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Ashkin, A.

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

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

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[CrossRef]

Barnett, S. M.

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678(1994).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Bensimon, D.

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

Berns, M. W.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Berry, R. M.

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-127(1936).
[CrossRef]

Bishop, A. I.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

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

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Burns, J. E.

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Bustamante, C.

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
[CrossRef] [PubMed]

Chávez-Cerda, S.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Chu, S.

T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
[CrossRef] [PubMed]

S. Chu, “Laser manipulation of atoms and particles,” Science 253, 861-866 (1991).
[CrossRef] [PubMed]

Croquette, V.

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

Cui, Y. J.

S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
[CrossRef] [PubMed]

Davenport, R. J.

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Dholakia, K.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

Dziedzic, J. M.

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

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

Friese, M. E. J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Garcés-Chávez, V.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Gelles, J.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Gold, J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

Hagberg, P.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

Hanstorp, D.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[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, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Higurashi, E.

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

Kendrickjones, J.

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Kendrick-Jones, J.

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

La Porta, A.

A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
[CrossRef] [PubMed]

Landick, R.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Leake, M. C.

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Molloy, J. E.

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Morgan, H.

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Nieminen, T. A.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

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

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).

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]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

Ohguchi, O.

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

Padgett, M. J.

Perkins, T. T.

T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
[CrossRef] [PubMed]

Profeta, G. A.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Rappaport, A.

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Rowe, A. D.

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

Rubinszteindunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

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

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (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]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1-3 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

Schmitz, S.

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

Schnitzer, M. J.

K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

Sibbett, W.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Simpson, N. B.

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Smith, D. E.

T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
[CrossRef] [PubMed]

Smith, S. B.

S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Strick, T.

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

Tanaka, H.

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

Tregear, R. T.

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Tromberg, B. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Ukita, H.

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

Veigel, C.

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

Visscher, K.

K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

Volke-Sepulveda, K.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Walter, R. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Wang, M. D.

A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
[CrossRef] [PubMed]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

White, D. C. S.

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Wright, W. H.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Wuite, G. J. L.

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Yamane, T.

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

Yin, H.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. 78, 547-549 (2001).
[CrossRef]

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

Biophys. J. (2)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335-1346 (1997).
[CrossRef] [PubMed]

G. J. L. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155-1167 (2000).
[CrossRef] [PubMed]

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. Mod. Opt. (2)

A. D. Rowe, M. C. Leake, H. Morgan, and R. M. Berry, “Rapid rotation of micron and submicron dielectric particles measured using optical tweezers,” J. Mod. Opt. 50, 1539-1554(2003).

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405-413 (2001).

Nat. Cell Biol. (1)

C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-Jones, “Load-dependent kinetics of force production by smooth muscle myosin measured with optical tweezers,” Nat. Cell Biol. 5, 980-986 (2003).
[CrossRef] [PubMed]

Nature (4)

K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

J. E. Molloy, J. E. Burns, J. Kendrickjones, R. T. Tregear, and D. C. S. White, “Movement and force produced by a single myosin head,” Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

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

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348-350(1998).
[CrossRef]

Opt. Commun. (1)

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678(1994).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-127(1936).
[CrossRef]

Phys. Rev. A (3)

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

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104(2004).
[CrossRef] [PubMed]

A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszteindunlop, “Direct observation of transfer of angular-momentum to absorptive particles from a laser-beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Phys. Today (1)

T. Strick, J. F. O. Allemand, V. Croquette, and D. Bensimon, “The manipulation of single biomolecules,” Phys. Today 54 (10), 46-51 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 4539-4543 (1989).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Science (5)

S. Chu, “Laser manipulation of atoms and particles,” Science 253, 861-866 (1991).
[CrossRef] [PubMed]

T. T. Perkins, D. E. Smith, and S. Chu, “Direct observation of tube-like motion of a single polymer-chain,” Science 264, 819-822 (1994).
[CrossRef] [PubMed]

S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795-799(1996).
[CrossRef] [PubMed]

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

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

Other (1)

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Rotation process. Linearly polarized light enters a quarter-wave plate, is converted to circularly polarized light, and then passes through a birefringent particle (vaterite). Angular moment is transferred to the particle, causing it to rotate. The resulting light exiting the particle is elliptically polarized.

Fig. 2
Fig. 2

Schematic of the optical trapping setup for rotating birefringent particles. The linearly polarized laser beam is converted to elliptically polarized light through a quarter-wave plate mounted in a rotational stage, which rotates trapped birefringent particles. The forward scattered laser is projected onto a quadrant photodiode to track the rotational frequency of the particle.

Fig. 3
Fig. 3

Intensity fluctuations of a rotating particle. The particles do not rotate perfectly about the axis of the laser. This produces a sinusoidal response signal at the frequency of rotation, which can be traced in the measured intensity of the forward scattered laser. This intensity fluctuation is critical to the rotational tracking. This particle is rotating at 24 Hz .

Fig. 4
Fig. 4

Frequency error block diagram. The block diagram determines the difference between a desired frequency and the actual frequency. The cutoff frequency of the high- and low-pass filters equals the desired frequency.

Fig. 5
Fig. 5

Error signal versus input frequency for a simulated response of the rotational error sensor. The cutoff frequencies of the low- and high-pass filters were both set at 25 Hz .

Fig. 6
Fig. 6

Experimental results of the rotational frequency of a 4 μm vaterite particle versus the angle (ϕ) of the quarter-wave plate.

Fig. 7
Fig. 7

Block diagram representation of the rotational control system with feedback control. The rotating particle as well as the sensor dynamics are represented as G, while K represents the controller. G A and K A represent the rotational stage and its controller.

Fig. 8
Fig. 8

Step response of a 4 μm vaterite particle from 25 to 30 Hz . The rise time is 0.77 s , and the controller stabilizes the system to within 1 Hz of the desired frequency.

Equations (10)

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

Γ = k r d ( n o n e ) .
τ = ϵ 2 ω E 0 2 sin Γ cos 2 ϕ sin 2 θ + ϵ 2 ω E 0 2 { 1 cos Γ } sin 2 ϕ ,
C 1 = ϵ E 0 2 2 ω sin Γ cos 2 ϕ ,
C 2 = ϵ E 0 2 2 ω { 1 cos Γ } sin 2 ϕ .
I θ ¨ + β θ ˙ = C 1 sin 2 θ + C 2 ,
β θ ˙ = C 1 sin 2 θ + C 2 .
d y a + b sin y = t + C = 2 a 2 b 2 tan 1 ( a tan y 2 + b a 2 b 2 ) ,
y a 2 b 2 t .
θ = C 2 2 C 1 2 β t .
θ ˙ = C 2 2 C 1 2 β = P ω β ( { 1 cos Γ } 2 sin 2 2 ϕ sin 2 Γ cos 2 2 ϕ ) 1 / 2 .

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