Abstract

The lateral position of an optically trapped object in a microscope can be monitored with a quadrant photodiode to within nanometers or better by measurement of intensity shifts in the back focal plane of the lens that is collimating the outgoing laser light. This detection is largely independent of the position of the trap in the field of view. We provide a model for the essential mechanism of this type of detection, giving a simple, closed-form analytic solution with simplifying assumptions. We identify intensity shifts as first-order far-field interference between the outgoing laser beam and scattered light from the trapped particle, where the latter is phase advanced owing to the Gouy phase anomaly. This interference also reflects momentum transfer to the particle, giving the spring constant of the trap. Our response formula is compared with the results of experiments.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dziedzic, J. C. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
    [CrossRef]
  2. A. Ashkin, Proc. Nat. Acad. Sci. USA 94, 4853 (1997).
    [CrossRef]
  3. K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
    [CrossRef]
  4. W. Denk and W. W. Webb, Appl. Opt. 29, 2382 (1990).
    [CrossRef] [PubMed]
  5. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
    [CrossRef]
  6. N. H. Dekkers and H. de Lang, Philips Tech. Rev. 37, 1 (1977).
  7. N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).
  8. T. Wilson, Optik 80, 167 (1988).
  9. L. P. Ghislain, N. A. Switz, and W. W. Webb, Rev. Sci. Instrum. 65, 2762 (1994).
    [CrossRef]
  10. L. P. Ghislain and W. W. Webb, Opt. Lett. 18, 1678 (1993).
    [CrossRef] [PubMed]
  11. S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
    [CrossRef] [PubMed]
  12. K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
    [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  14. J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
    [CrossRef]
  15. J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
    [CrossRef]
  16. J. A. Lock and G. Gouesbet, J. Opt. Soc. Am. A 11, 2503 (1994).
    [CrossRef]
  17. G. Gouesbet and J. A. Lock, J. Opt. Soc. Am. A 11, 2516 (1994).
    [CrossRef]
  18. K. F. Ren, G. Gréhan, and G. Gouesbet, Appl. Opt. 35, 2702 (1996).
    [CrossRef] [PubMed]
  19. Y. Harada and T. Asakura, Opt. Commun. 124, 529 (1996).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1989).
  21. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  22. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  23. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  24. R. C. Gauthier and S. Wallace, J. Opt. Soc. Am. B 12, 1680 (1995).
    [CrossRef]

1997 (1)

A. Ashkin, Proc. Nat. Acad. Sci. USA 94, 4853 (1997).
[CrossRef]

1996 (4)

Y. Harada and T. Asakura, Opt. Commun. 124, 529 (1996).
[CrossRef]

S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
[CrossRef] [PubMed]

K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, Appl. Opt. 35, 2702 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (4)

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, J. Opt. Soc. Am. A 11, 2503 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, J. Opt. Soc. Am. A 11, 2516 (1994).
[CrossRef]

L. P. Ghislain, N. A. Switz, and W. W. Webb, Rev. Sci. Instrum. 65, 2762 (1994).
[CrossRef]

1993 (2)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

L. P. Ghislain and W. W. Webb, Opt. Lett. 18, 1678 (1993).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

1988 (1)

T. Wilson, Optik 80, 167 (1988).

1986 (1)

1977 (1)

N. H. Dekkers and H. de Lang, Philips Tech. Rev. 37, 1 (1977).

1974 (1)

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

Alexander, D. R.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, Opt. Commun. 124, 529 (1996).
[CrossRef]

Ashkin, A.

Barton, J. P.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

Bjorkholm, J. C.

Block, S. M.

K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
[CrossRef]

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1989).

Bustamante, C.

S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
[CrossRef] [PubMed]

Chu, S.

Cui, Y. J.

S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
[CrossRef] [PubMed]

de Lang, H.

N. H. Dekkers and H. de Lang, Philips Tech. Rev. 37, 1 (1977).

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

Dekkers, N. H.

N. H. Dekkers and H. de Lang, Philips Tech. Rev. 37, 1 (1977).

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

Denk, W.

Dziedzic, J. M.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Gauthier, R. C.

Ghislain, L. P.

L. P. Ghislain, N. A. Switz, and W. W. Webb, Rev. Sci. Instrum. 65, 2762 (1994).
[CrossRef]

L. P. Ghislain and W. W. Webb, Opt. Lett. 18, 1678 (1993).
[CrossRef] [PubMed]

Gouesbet, G.

Gréhan, G.

Gross, S. P.

K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
[CrossRef]

Harada, Y.

Y. Harada and T. Asakura, Opt. Commun. 124, 529 (1996).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Lock, J. A.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Ren, K. F.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

Schmidt, C. F.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

Schnapp, B. J.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Smith, S. B.

S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
[CrossRef] [PubMed]

Svoboda, K.

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

Switz, N. A.

L. P. Ghislain, N. A. Switz, and W. W. Webb, Rev. Sci. Instrum. 65, 2762 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Visscher, K.

K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
[CrossRef]

Wallace, S.

Webb, W. W.

Wilson, T.

T. Wilson, Optik 80, 167 (1988).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1989).

Annu. Rev. Biophys. Biomol. Struct. (1)

K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994).
[CrossRef]

Appl. Opt. (2)

IEEE J. Select. Topics Quantum Electron. (1)

K. Visscher, S. P. Gross, and S. M. Block, IEEE J. Select. Topics Quantum Electron. 2, 1066 (1996).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Nature (London) (1)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, Nature (London) 365, 721 (1993).
[CrossRef]

Opt. Commun. (1)

Y. Harada and T. Asakura, Opt. Commun. 124, 529 (1996).
[CrossRef]

Opt. Lett. (2)

Optik (2)

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

T. Wilson, Optik 80, 167 (1988).

Philips Tech. Rev. (1)

N. H. Dekkers and H. de Lang, Philips Tech. Rev. 37, 1 (1977).

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

A. Ashkin, Proc. Nat. Acad. Sci. USA 94, 4853 (1997).
[CrossRef]

Rev. Sci. Instrum. (1)

L. P. Ghislain, N. A. Switz, and W. W. Webb, Rev. Sci. Instrum. 65, 2762 (1994).
[CrossRef]

Science (1)

S. B. Smith, Y. J. Cui, and C. Bustamante, Science 271, 795 (1996).
[CrossRef] [PubMed]

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1989).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

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 (3)

Fig. 1
Fig. 1

Optical trapping setup in a microscope with displacement detection. A dichroic mirror diverts the laser from the imaging path, and a quadrant diode (10-mm diameter) is placed in a plane conjugate to the BFP of the 1.4-N.A.  oil immersion condenser. In practice, light is not exactly collimated by the condenser so that an intermediate image is formed (this angle is exaggerated in the figure). Analog and digital electronics convert current from the diode quadrants into normalized X and Y signals, related as explained in the text to the displacement of a trapped particle from the focus in the specimen plane.

Fig. 2
Fig. 2

Focus geometry as discussed in the text. A refractive particle, located in the focal plane of a Gaussian beam, is displaced laterally by x. Interference of scattered and unscattered light is considered at a large distance r and is observed in the BFP at a radius f sin θ (the collimating lens, of focal length f, obeys the sine condition). The angular radius of the Gaussian focus (30°) is exaggerated.

Fig. 3
Fig. 3

Experimental calibration data compared with model response. Solid curve, a silica bead fixed upon a cover slip was moved by a piezoelectric stage through the sensitive range of the detector. Dashed curve, expression  (7) with no adjustable parameters besides the nominal sphere size 0.5 µm and a focus size w00.53 µm estimated from the input beam width. The function Gx/w0 in expression  (7) has a slope of w0-1 at x=0 and extrema Gx/w0=±0.334 at x=±0.552 w0.

Equations (8)

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

Er-ikw0Itot1/2rπscs1/2expikr-k2w02θ2/4.
Ers=Ex=2Itot1/2w0πscs1/2exp-x2/w02.
α=a3nr2-1nr2+2=0.0074d3,
Erk2αrExexpikr-rs, k2αrExexpikr-x sin θ cos ϕ
δI=scs2E+E2-E2scs ReEE*.
δIxItot=2k3απr2exp-x2/w02×sinkx sin θ cos ϕexp-k2w02θ2/4.
I+-I-I++I-16πkαw02Gx/w0, Gu=exp-2u20uexpy2dy.
Fx/x16αItotcsw04,

Metrics