Abstract

We characterize a single beam supercontinuum “white light” trap and determine the trap stiffness in the transverse trapping plane. We realize a holographic white light trapping system using a spatial light modulator, and explore the generation of a dual beam trap and characterize its performance. We also demonstrate optical trapping and rotation of particles using a supercontinuum vortex beam. It is shown that orbital angular momentum can be transferred to spheres trapped in a supercontinuum vortex. Quantified rotation rates are demonstrated.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dzeidzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  2. K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
    [CrossRef] [PubMed]
  3. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
    [CrossRef]
  4. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Pattern formation and flow control of fine particles by laser-scanning micromanipulation," Opt. Lett. 16, 1463-1465 (1991).
    [CrossRef] [PubMed]
  5. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
    [CrossRef] [PubMed]
  6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "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]
  7. 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]
  8. V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]
  9. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  10. V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
    [CrossRef] [PubMed]
  11. P. Li, K. Shi, and Z. Liu, "Manipulation and spectroscopy of a single particle by use of white-light optical tweezers," Opt. Lett. 30, 156-158 (2005).
    [CrossRef] [PubMed]
  12. P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Enhanced optical guiding of colloidal particles using a supercontinuum light source," Opt. Express 14, 5792-5802 (2006).
    [CrossRef] [PubMed]
  13. M. Guillon, K. Dholakia, and D. McGloin, "Optical trapping and spectral analysis of aerosols with a supercontinuum laser source," Opt. Express 16, 7655-7664 (2008).
    [CrossRef] [PubMed]
  14. A. E. Siegman, Lasers (University Science Books, 1986).
  15. J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Third Ed., p. 261.
  17. R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
    [CrossRef]
  18. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]
  19. J. W. Goodman, Statistical Optics (Wiley-Interscience 1985), First Ed., p. 224.
  20. J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
    [CrossRef]
  21. P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
    [CrossRef]
  22. W. P. Wong, and K. Halvorsen, "The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur," Opt. Express 14, 12517 (2006).
    [CrossRef] [PubMed]
  23. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
    [CrossRef]
  24. J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
    [CrossRef]
  25. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
    [CrossRef]
  26. 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]
  27. K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
    [CrossRef]
  28. H. I. Sztul, V. Kartazayev, and R. R. Alfano, "Laguerre-Gaussian supercontinuum," Opt. Lett. 31, 2725-2727 (2006).
    [CrossRef] [PubMed]
  29. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
    [CrossRef]
  30. P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
    [CrossRef]
  31. A. V. Filippov, "Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number," J. Colloid Interface Sci. 229, 184-195 (2000).
    [CrossRef] [PubMed]
  32. J. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999).
    [CrossRef]

2008 (3)

K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

M. Guillon, K. Dholakia, and D. McGloin, "Optical trapping and spectral analysis of aerosols with a supercontinuum laser source," Opt. Express 16, 7655-7664 (2008).
[CrossRef] [PubMed]

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

2007 (2)

R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

2006 (3)

2005 (1)

2003 (1)

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

2002 (5)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

2001 (2)

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
[CrossRef]

2000 (2)

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

A. V. Filippov, "Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number," J. Colloid Interface Sci. 229, 184-195 (2000).
[CrossRef] [PubMed]

1999 (2)

J. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

1998 (2)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (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]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "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]

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)

1989 (1)

J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1986 (1)

1984 (1)

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Alexander, D. R.

J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Alfano, R. R.

Allen, L.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
[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]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Ashkin, A.

Bartlett, P.

P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
[CrossRef]

Barton, J. P.

J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[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]

Bjorkholm, J. E.

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

Brown, C. T. A.

Carruthers, A. E.

Chavez-Cerda, S.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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.

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Dearing, M. T.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Dholakia, K.

K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

M. Guillon, K. Dholakia, and D. McGloin, "Optical trapping and spectral analysis of aerosols with a supercontinuum laser source," Opt. Express 16, 7655-7664 (2008).
[CrossRef] [PubMed]

P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Enhanced optical guiding of colloidal particles using a supercontinuum light source," Opt. Express 14, 5792-5802 (2006).
[CrossRef] [PubMed]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Dufresne, E. R.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Dultz, W.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Dzeidzic, J. M.

Filippov, A. V.

A. V. Filippov, "Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number," J. Colloid Interface Sci. 229, 184-195 (2000).
[CrossRef] [PubMed]

Fischer, P.

Friese, M. E. J.

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. Rubinsztein-Dunlop, "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]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Gu, M.

K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Guillon, M.

Haist, T.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Halvorsen, K.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "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.

R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

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. Rubinsztein-Dunlop, "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]

Henderson, S. I.

P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
[CrossRef]

Kamgar-Parsi, B.

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Kartazayev, V.

Kestin, J.

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Kitamura, N.

Knoner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

Koshioka, M.

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Leburn, C. G.

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

Levelt Sengers, J. M. H.

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Li, P.

Liesener, J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

Liu, Z.

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

MacVicar, I.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Masuhara, H.

McGloin, D.

M. Guillon, K. Dholakia, and D. McGloin, "Optical trapping and spectral analysis of aerosols with a supercontinuum laser source," Opt. Express 16, 7655-7664 (2008).
[CrossRef] [PubMed]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Meiners, J. C.

J. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999).
[CrossRef]

Misawa, H.

Mitchell, S. J.

P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[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]

O'Neil, A. T.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

Pfeifer, R. N. C.

R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

Quake, S. R.

J. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999).
[CrossRef]

Reece, P.

K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Reicherter, M.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

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. Rubinsztein-Dunlop, "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]

Sasaki, K.

Schmitzer, H.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

Sengers, J. V.

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Sheets, S. A.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Shi, K.

Sibbett, W.

P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Enhanced optical guiding of colloidal particles using a supercontinuum light source," Opt. Express 14, 5792-5802 (2006).
[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

Skelton, S. E.

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

Spalding, G. C.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

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]

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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]

Streuber, C. T.

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

Sztul, H. I.

Tiziani, H. J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Volke-Sepulveda, K.

P. Fischer, A. E. Carruthers, K. Volke-Sepulveda, E. M. Wright, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Enhanced optical guiding of colloidal particles using a supercontinuum light source," Opt. Express 14, 5792-5802 (2006).
[CrossRef] [PubMed]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

Wagemann, E. U.

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]

Wong, W. P.

Wright, E. M.

Chem. Soc. Rev. (1)

K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. P. Barton, and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. Colloid Interface Sci. (1)

A. V. Filippov, "Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number," J. Colloid Interface Sci. 229, 184-195 (2000).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, "The production of multiringed Laguerre-Gaussian modes by computer-generated holograms," J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, 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. B (1)

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B 4, S82-S89 (2002).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

J. Kestin, J. V. Sengers, B. Kamgar-Parsi, J. M. H. Levelt Sengers, "Thermophysical properties of fluid H2O," J. Phys. Chem. Ref. Data 13, 175-183 (1984).
[CrossRef]

Nature (1)

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. (2)

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A (2)

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]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-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]

Phys. Rev. Lett. (4)

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "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. C. Meiners, and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999).
[CrossRef]

Physica C (1)

P. Fischer, S. E. Skelton, C. G. Leburn, C. T. Streuber, E. M. Wright, and K. Dholakia, "Propagation and diffraction of Optical Vortices," Physica C 468, 514-517 (2008).
[CrossRef]

Rev. Mod. Phys. (1)

R. N. C. Pfeifer, T. A. Nieminen, and N. R. Heckenberg, "Colloquium: Momentum of an electromagnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Science (1)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

Ser. A (1)

P. Bartlett, S. I. Henderson, and S. J. Mitchell, "Measurement of the hydrodynamic forces between two polymer-coated spheres," Philos. Trans. R. Soc. London, Ser. A 359, 883-895 (2001).
[CrossRef]

Other (3)

J. W. Goodman, Statistical Optics (Wiley-Interscience 1985), First Ed., p. 224.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), Third Ed., p. 261.

A. E. Siegman, Lasers (University Science Books, 1986).

Supplementary Material (1)

» Media 1: MOV (450 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1.

Trap stiffness values for the direction parallel (kp ) and perpendicular (ks ) to the beam’s polarization as a function of wavelength for a 1µm polymer particle trapped in the focal plane of a Gaussian beam with a power of 11mW and a waist w0 =1.75µm.

Fig. 2.
Fig. 2.

The visible component of the SC beam was selected using the infrared filter (IR filter) and expanded through an achromatic lens relay system, f1=100mm and f2=250mm. A dichroic beam splitter, DBS, reflected the visible light onto the back aperture of objective 1, 100x (Nikon NA 1.4) apochromatic microscope objective. A LED emitting at 950nm was used in a Köhler illumination scheme. This consisted of four f=50mm lenses, and two apertures, a, used to adjust the LED brightness and the field of view. A long working distance microscope objective (Mitutoyo, 100x, NA 0.7), objective 2, delivered the illumination to the sample plane. Light is reflected onto a CCD camera (Basler pl640-210gm) via an infrared mirror (M) and a 950nm interference filter, IF, (Comar 950IL25) to remove residual visible light. The spectrum of the supercontinuum source, shown as an insert, was recorded in the trapping plane immediately after objective 1.

Fig. 3.
Fig. 3.

A trapped 1.28µm diameter silica sphere (white arrow) is shown with a stuck sphere (black arrow) for reference, to demonstrate successful tweezing while visualizing the particle using our illumination setup.

Fig. 4.
Fig. 4.

Centered histograms of the position of the 1µm polymer particle in the supercontinuum trap. Red curves correspond to Gaussian fits using Eq. (9).

Fig. 5.
Fig. 5.

(a). The set up used for holographic trapping was adapted from Fig. 2 as shown. The IR filter selected the visible wavelength component of the beam, and was expanded using achromatic lenses, f1=50mm and f2=250mm, to fill the SLM-spatial light modulator display area. Lenses f3 and f4 were used to image the reflected beam, and the first order was selected using an aperture positioned a focal distance away from f3 and f4. A 10° prism was used in the conjugate plane of the SLM, placed a focal length away from f4, to image the hologram produced from the SLM. (b) The dual white light spots (f3=630mm and f4=500mm), and (c) the supercontinuum LG beam (f3=200mm and f4=160mm), created are shown. Images were taken with a Watec-250D CCD.

Fig. 6.
Fig. 6.

Graphs (a) and (b) show the red, green and blue parts of the spectrum for the respective vertical and horizontal beam profiles of the upper trap, cross sections were taken using a camera plot, and the intensities have been normalized for ease of comparison. Graphs (c) and (d) show the vertical and horizontal beam profiles for the lower trap, again, the red, green and blue parts of the spectrum are shown. Figure 5(b) shows a picture of the dual traps.

Fig. 7.
Fig. 7.

Holographically generated dual optical white light trap is shown trapping two 2µm spheres (white circle) with respect to a reference sphere (black circle).

Fig. 8.
Fig. 8.

Centered histograms of the x and y position of the two 2 µm polymer particles in the dual supercontinuum trap. Graphs (a) and (b) show the upper trap seen in Fig. 5, and graphs (c) and (d) show the lower trap. Red curves correspond to Gaussian fits using Eq. (9).

Fig. 9.
Fig. 9.

Rotation rate for three touching 1µm diameter spheres optically trapped in the first bright annular ring of the focal spot of supercontinuum LG beam (l=3, p=0). The LG beam has some higher order components, Fig. 5(c). The three trapped spheres and the tracking reconstruction are shown in the inset (see multimedia file 450kB).[Media 1]

Fig. 10.
Fig. 10.

(a). The LG beam waists as a function of l are shown with pictures of the supercontinuum LG beam at l=2, 4, and 5. (b) Particle rotation rates, Ω, for three touching 1µm polymer spheres are shown in black with increasing l. Theoretical points are shown in red with a line to guide the eye.

Fig. 11.
Fig. 11.

(a). The calculated rotation rates, Ω (Hz), per incident power P0 (mW) for three polymer spheres (Ø=1µm, refractive index n=1.59) due to OAM transfer from an LG beam (Eq. (1) with l=3, p=0, w0 =640nm), is shown. The red line represents the SC spectrum weighted total rotation rate. (b). The centripetal force Fc (nN) is shown per incident power P0 (mW). Again, the red line represents the SC spectrum weighted centripetal force.

Equations (10)

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

u p = 0 l ( x , y , z ) = u 0 ( z + i z r ) l + 1 ( x i y ) l exp ( i n 0 k 0 x 2 + y 2 2 ( z + i z r ) )
E = 1 n 0 2 k 0 2 ( · A ) + A
H = i Z 0 k 0 × A
u 0 = P 0 Z π w 0 2 ( 1 ) l + 1 s 2 2 l z r 2 ( 1 + s 2 ( l + 1 ) ( l s 2 + 2 s 2 2 ) ) l ! 2 l 2 .
A = d ω A ( ω ) ( u p = 0 l ( x , y , z ) exp ( i n 0 k 0 z ) , 0 , 0 ) ,
T ij = E i D j * + H i B j * 1 2 ( E k D k * + H k B k * ) .
< F i > = 1 Δ t f i d V dt = 1 Δ t f i d V T ij d s j d t ,
< F i > = A ( ω ) 2 T ij n j d s d ω ,
< N i > = A ( ω ) 2 ε ijk x j T kl d s l d ω .
p ( x ) exp ( U ( x ) k B T ) = exp ( k x 2 2 k B T )

Metrics