Abstract

Single-beam optical gradient force traps created by focusing helical modes of light are known as optical vortices. Modulating the helical pitch of such a mode’s wave front yields a new class of optical traps whose dynamically reconfigurable intensity distributions provide new opportunities for controlling motion in mesoscopic systems. An implementation of modulated optical vortices based on the dynamic holographic optical tweezer technique is described.

© 2003 Optical Society of America

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  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
    [CrossRef]
  2. A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6, 841 (2000).
    [CrossRef]
  3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
    [CrossRef]
  4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
    [CrossRef] [PubMed]
  5. N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
    [CrossRef]
  6. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
    [CrossRef] [PubMed]
  7. K. T. Gahagan and G. A. Swartzlander, Opt. Lett. 21, 827 (1996).
    [CrossRef] [PubMed]
  8. H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
    [CrossRef]
  9. K. T. Gahagan and G. A. Swartzlander, J. Opt. Soc. Am. B 16, 533 (1999).
    [CrossRef]
  10. L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
    [CrossRef]
  11. A. T. O’Neil and M. J. Padgett, Opt. Commun. 185, 139 (2000).
    [CrossRef]
  12. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
    [CrossRef]
  13. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
    [CrossRef] [PubMed]
  14. W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).
  15. M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
    [CrossRef]
  16. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
    [CrossRef]
  17. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  18. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef]
  19. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef]
  20. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
    [CrossRef]
  21. Standard results for Laguerre–Gaussian modes suggest that Rl∝l. However, they do not account for diffraction by the objective lens’s aperture. See, for example, Refs. 10 and 17.
  22. Hamamatsu Model X7550 parallel-aligned nematic liquid-crystal SLM.
  23. S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
    [CrossRef] [PubMed]
  24. P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
    [CrossRef]
  25. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
    [CrossRef]

2003

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

2002

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef]

2001

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

2000

A. T. O’Neil and M. J. Padgett, Opt. Commun. 185, 139 (2000).
[CrossRef]

A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6, 841 (2000).
[CrossRef]

1999

K. T. Gahagan and G. A. Swartzlander, J. Opt. Soc. Am. B 16, 533 (1999).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

1998

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

1997

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

1996

N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
[CrossRef]

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

K. T. Gahagan and G. A. Swartzlander, Opt. Lett. 21, 827 (1996).
[CrossRef] [PubMed]

1995

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

1986

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

Ashkin, A.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Berns, M. W.

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

Bjorkholm, J. E.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Chu, S.

Curtis, J. E.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

Dziedzic, J. M.

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

Gahagan, K. T.

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

Gauthier, R. C.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

Gold, J.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

Grier, D. G.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef]

Grover, C. P.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

Grover, S. C.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

Hagberg, P.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

Hanstorp, D.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

He, W.

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

Heckenberg, N. R.

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

Korda, P. T.

P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Liu, Y. G.

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

A. T. O’Neil and M. J. Padgett, Opt. Commun. 185, 139 (2000).
[CrossRef]

Padgett, M. J.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

A. T. O’Neil and M. J. Padgett, Opt. Commun. 185, 139 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Simpson, N. B.

N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
[CrossRef]

Skirtach, A. G.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

Smith, M.

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Taylor, M. B.

P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Adv. Quantum Chem.

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, Adv. Quantum Chem. 30, 469 (1998).
[CrossRef]

Appl. Phys. Lett.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, Appl. Phys. Lett. 78, 547 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6, 841 (2000).
[CrossRef]

J. Biomed. Opt.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, J. Biomed. Opt. 6, 14 (2001).
[CrossRef] [PubMed]

J. Mod. Opt.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

N. B. Simpson, L. Allen, and M. J. Padgett, J. Mod. Opt. 43, 2485 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Microsc. Microanal.

W. He, Y. G. Liu, M. Smith, and M. W. Berns, Microsc. Microanal. 3, 47 (1997).

Nature

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

Opt. Commun.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

A. T. O’Neil and M. J. Padgett, Opt. Commun. 185, 139 (2000).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, Opt. Commun. 197, 239 (2001).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef] [PubMed]

P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89, 128301 (2002).
[CrossRef]

Prog. Opt.

L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. 39, 291 (1999).
[CrossRef]

Science

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Other

Standard results for Laguerre–Gaussian modes suggest that Rl∝l. However, they do not account for diffraction by the objective lens’s aperture. See, for example, Refs. 10 and 17.

Hamamatsu Model X7550 parallel-aligned nematic liquid-crystal SLM.

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Figures (4)

Fig. 1
Fig. 1

Creating optical vortices with dynamic holographic optical tweezers. (a) Schematic diagram of the experimental apparatus. A reflective spatial light modulator (SLM) imprints the phase modulation φr onto the wave front of a TEM00 laser beam. The transformed beam is relayed by a telescope to the back aperture of a microscope objective lens that focuses it into an optical trap. A conventional illuminator and video camera create images of objects in the trap. (b) Phase modulations encoding an l=40 optical vortex. (c) Resulting optical vortex’s intensity in the focal plane. (d) Trajectory of a single 800-nm-diameter silica sphere traveling around the optical vortex’s circumference, measured at 1/6-s intervals over 5 s.

Fig. 2
Fig. 2

Modulated optical vortex with m=5, α=0.1: (a) phase modulation, (b) predicted radial profile Rθ, (c) experimental intensity distribution.

Fig. 3
Fig. 3

Modulated optical vortices at (top row) α=0.1 with different m values and (bottom row) m=4 with different α values. Additional lobes appear in the bottom patterns for α>αc0.25, with the direction of tangential forces indicated by arrows. All patterns were created with l=60. Scale bars indicate 1 µm.

Fig. 4
Fig. 4

Two particles’ transit around a modulated optical vortex. Data points show the positions of two 800-nm-diameter polystyrene spheres measured at 1/10-s intervals over 10 s. The two spheres, indicated by arrows, move along a trap with l=60, m=3, and α=0.1 at 300 mW, in the direction indicated by the curved arrow. Two additional spheres are trapped motionlessly in the undiffracted central spot.

Equations (3)

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

RlaλNA1+ll0,
Rθ=aλNA1+1l0dφθdθ,
φθ=lθ+α sinmθ+β

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