Abstract

We report near-optimal tracking of freely diffusing fluorescent particles in a quasi-two-dimensional geometry via photon counting and real-time feedback. We present a quantitative statistical model of our feedback network and find excellent agreement with the experiment. We monitor the motion of a single fluorescent particle with a sensitivity of 15nmHz while collecting fewer than 5000fluorescencephotonss. Fluorescent microspheres (diffusion coefficient 1.3μm2s) are tracked with a root-mean-square tracking error of 170nm, within a factor of 2 of the theoretical limit set by photon counting shot noise.

© 2006 Optical Society of America

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References

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  1. T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
    [CrossRef]
  2. J. Enderlein, Appl. Phys. B 71, 773 (2000).
    [CrossRef]
  3. A. J. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004).
    [CrossRef]
  4. V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
    [CrossRef] [PubMed]
  5. A. J. Berglund and H. Mabuchi, Opt. Express 13, 8069 (2005).
    [CrossRef] [PubMed]
  6. S. B. Andersson, Appl. Phys. B 80, 809 (2005).
    [CrossRef]
  7. A. E. Cohen and W. E. Moerner, Proc. Natl. Acad. Sci. U.S.A. 103, 4362 (2006).
    [CrossRef] [PubMed]
  8. A. J. Berglund and H. Mabuchi, Appl. Phys. B 83, 127 (2006).
    [CrossRef]
  9. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).
  10. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science, 2001).
  11. O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford U. Press, 1996).

2006 (2)

A. E. Cohen and W. E. Moerner, Proc. Natl. Acad. Sci. U.S.A. 103, 4362 (2006).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, Appl. Phys. B 83, 127 (2006).
[CrossRef]

2005 (3)

V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, Opt. Express 13, 8069 (2005).
[CrossRef] [PubMed]

S. B. Andersson, Appl. Phys. B 80, 809 (2005).
[CrossRef]

2004 (1)

A. J. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004).
[CrossRef]

2000 (1)

J. Enderlein, Appl. Phys. B 71, 773 (2000).
[CrossRef]

1997 (1)

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Andersson, S. B.

S. B. Andersson, Appl. Phys. B 80, 809 (2005).
[CrossRef]

Berglund, A. J.

A. J. Berglund and H. Mabuchi, Appl. Phys. B 83, 127 (2006).
[CrossRef]

A. J. Berglund and H. Mabuchi, Opt. Express 13, 8069 (2005).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004).
[CrossRef]

Chemla, D. S.

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Cohen, A. E.

A. E. Cohen and W. E. Moerner, Proc. Natl. Acad. Sci. U.S.A. 103, 4362 (2006).
[CrossRef] [PubMed]

Enderle, T.

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Enderlein, J.

J. Enderlein, Appl. Phys. B 71, 773 (2000).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).

Gratton, E.

V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
[CrossRef] [PubMed]

Ha, T.

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Jacobs, O. L. R.

O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford U. Press, 1996).

Levi, V.

V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
[CrossRef] [PubMed]

Mabuchi, H.

A. J. Berglund and H. Mabuchi, Appl. Phys. B 83, 127 (2006).
[CrossRef]

A. J. Berglund and H. Mabuchi, Opt. Express 13, 8069 (2005).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004).
[CrossRef]

Moerner, W. E.

A. E. Cohen and W. E. Moerner, Proc. Natl. Acad. Sci. U.S.A. 103, 4362 (2006).
[CrossRef] [PubMed]

Ruan, Q.

V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
[CrossRef] [PubMed]

Van Kampen, N. G.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science, 2001).

Weiss, S.

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Appl. Phys. B (4)

J. Enderlein, Appl. Phys. B 71, 773 (2000).
[CrossRef]

A. J. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004).
[CrossRef]

A. J. Berglund and H. Mabuchi, Appl. Phys. B 83, 127 (2006).
[CrossRef]

S. B. Andersson, Appl. Phys. B 80, 809 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, Appl. Phys. Lett. 70, 782 (1997).
[CrossRef]

Biophys. J. (1)

V. Levi, Q. Ruan, and E. Gratton, Biophys. J. 88, 2919 (2005).
[CrossRef] [PubMed]

Opt. Express (1)

Proc. Natl. Acad. Sci. U.S.A. (1)

A. E. Cohen and W. E. Moerner, Proc. Natl. Acad. Sci. U.S.A. 103, 4362 (2006).
[CrossRef] [PubMed]

Other (3)

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science, 2001).

O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford U. Press, 1996).

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Figures (4)

Fig. 1
Fig. 1

Data from a typical run of the experiment. The upper plot shows the rate of photon detections. In the lower plot, the upper curves are the x and y positions of the sample stage during the same trajectory. The lower curve on the lower plot shows the excitation laser power (in arbitrary units) necessary to lock the number of detected photons to 5600 s 1 . The plots show three regions, in which no particle is present (until 17 s ), a 60 nm diameter particle is tracked (until 74 s ), and a 210 nm diameter particle is tracked (until 90 s ).

Fig. 2
Fig. 2

MSD ( Δ t ) 2 Δ t as defined by Eq. (1) for two typical [ 210 nm diameter (upper) and 60 nm diameter (lower) microspheres] tracking trajectories together with fits to Eq. (5). At long times, the curves approach the particle diffusion coefficient, while the short-time behavior depends on the tracking feedback performance.

Fig. 3
Fig. 3

Block diagram used in the text to represent the particle tracking control system.

Fig. 4
Fig. 4

Measured localization L versus D for 62 individual tracking trajectories (dark dots). The lighter dots are the results of simulations described in the text. The dashed curves are the localization limit based on optical shot noise. Typical fit parameters for the 210 nm beads (red online, dark dots clustered in the lower left corner of the plot) are D = 1.3 μ m 2 s , n = 15 nm Hz , γ c = 111 Hz , γ p = 343 Hz . The fitted value of γ c is commensurate with the bandwidth of the analog integrator used for tracking control, while γ p serves primarily to represent the phase delay of our piezoelectric stage at high frequencies ( 100 Hz ) .

Equations (6)

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MSD ( Δ t ) = [ X ( t + Δ t ) X ( t ) ] 2 ,
U ( t ) = 2 D d W d t ,
d ξ ( t ) = A ξ ( t ) d t + U ( t ) d t , X ̇ ( t ) = C ξ ( t ) ,
A = ( γ p γ p γ c 0 ) , C = ( 0 γ p ) .
MSD ( Δ t ) = 2 D Δ t 2 D γ c γ p C A 2 ( I n 2 A 2 2 D ) ( I e A Δ t ) ( 1 γ c 0 0 1 γ p ) C T .
L = D ( 1 γ c + 1 γ p ) + n 2 γ c 2 .

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