Abstract

A method of adaptive system identification for the modeling of an optical trap’s system dynamics is presented. The system dynamics can be used to locate the corner frequency for trapping stiffness calibration using the power spectral method. The method is based on an adaptive least-mean-square (LMS) algorithm, which adjusts weights of a tapped delay line filter using a gradient descent method. The identified model is the inverse of the high order finite impulse response (FIR) filter. The model order is reduced using balanced model reduction, giving the corner frequency which can be used to calibrate the trapping stiffness. This method has an advantage over other techniques in that it is quick, does not explicitly require operator interaction, and can be acquired in real time. It is also a necessary step for an adaptive controller that can automatically update the controller for changes in the trap (e.g., power fluctuations) and for particles of different sizes and refractive indices.

© 2008 Optical Society of America

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References

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  1. A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987).
    [CrossRef] [PubMed]
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  3. 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]
  4. 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]
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    [CrossRef]
  6. M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
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  10. M. Green and D. J. N. Limebeer, Linear robust control (Prentice Hall, Englewood Cliffs, N.J., 1995).
  11. F. Gittes and C. F. Schmidt, "Interference model for back-focal-plane displacement detection in optical tweezers," Opt. Lett. 23, 7-9 (1998).
    [CrossRef]
  12. L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
    [CrossRef]

2007 (1)

2003 (1)

M. J. Lang and S. M. Block, "Resource letter: LBOT-1: Laser-based optical tweezers," Am. J. Phys. 71, 201-215 (2003).
[CrossRef]

2002 (1)

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

1999 (1)

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 (1)

1997 (1)

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]

1995 (1)

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

1994 (1)

L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
[CrossRef]

1987 (1)

A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Ballerini, R.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Block, S. M.

M. J. Lang and S. M. Block, "Resource letter: LBOT-1: Laser-based optical tweezers," Am. J. Phys. 71, 201-215 (2003).
[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]

Block, S.M.

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. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

Capitanio, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Clark, R. L.

Cole, D. G.

Dunlap, D.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Finzi, L.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (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]

Ghislain, L. P.

L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
[CrossRef]

Gittes, F.

Giuntini, M.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Kendrickjones, J.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

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]

Lang, M. J.

M. J. Lang and S. M. Block, "Resource letter: LBOT-1: Laser-based optical tweezers," Am. J. Phys. 71, 201-215 (2003).
[CrossRef]

Molloy, J. E.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

Pavone, F. S.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Romano, G.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Schmidt, C. F.

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]

Sparrow, J. C.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

Switz, N. A.

L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
[CrossRef]

Tregear, R. T.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

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]

Wang, M. D.

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]

Webb, W. W.

L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
[CrossRef]

White, D. C. S.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

Wulff, K. D.

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]

Am. J. Phys. (1)

M. J. Lang and S. M. Block, "Resource letter: LBOT-1: Laser-based optical tweezers," Am. J. Phys. 71, 201-215 (2003).
[CrossRef]

Appl. Opt. (1)

Biophys. J. (2)

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).

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]

Nature (1)

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]

Opt. Lett. (1)

Rev. Sci. Instrum. (2)

L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994).
[CrossRef]

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002).
[CrossRef]

Science (1)

A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Other (3)

B. Widrow and S. D. Stearns, Adaptive signal processing, Prentice-Hall signal processing series (Prentice-Hall, Englewood Cliffs, N.J., 1985).

S. S. Haykin, Adaptive filter theory, Prentice Hall information and system sciences series, 2nd ed. (Prentice Hall, Englewood Cliffs, NJ, 1991).

M. Green and D. J. N. Limebeer, Linear robust control (Prentice Hall, Englewood Cliffs, N.J., 1995).

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Figures (3)

Fig. 1.
Fig. 1.

Examples of changing plant dynamics depending on trapping power and particle size. (a) A 1 µmpolystyrene sphere trapped at power levels varying from20 mWto 80 mW. (b) Multiple particle sizes (0.5,1,5 µm) and indices of refraction (polystyrene and silica) trapped with the same power.

Fig. 2.
Fig. 2.

Transversal or tapped delay line filter where u(n) is the input, y(n) is the output, z -1 is a time step delay, and w is the tap weights.

Fig. 3.
Fig. 3.

Examples of changing plant dynamics depending on trapping power and particle size. (a) 1 µm silica sphere trapped in 20 mW, Ω=282 Hz. (b) 0.5 µm polystyrene sphere trapped in 20 mW, Ω=740 Hz

Tables (1)

Tables Icon

Table 1. Resultant corner frequencies and stiffnesses for various particle diameters.

Equations (4)

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S xx ( ω ) = ( Ω 2 ω 2 + Ω 2 ) 2 γ k B T k 2 ,
w ( n + 1 ) = w ( n ) + µ u ( n ) e * ( n ) ,
G ̂ ( z ) = F ( z ) 1 = 1 w 0 + w 1 z 1 + w 2 z 2 + . . . + w m 1 z 1 m ,
1 s + a 1 1 e aT z 1 .

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