Abstract

We demonstrate, what is to the best of our knowledge, a new method for studying the motion of a particle trapped by optical tweezers; in this method the trapping beam itself is used as a confocal probe. By studying the response of the particle to periodic motion of the tweezers, we obtain information about the medium viscosity, particle properties, and trap stiffness. We develop the mathematical model, demonstrate experimentally its validity for our system, and discuss advantages of using this method as a new form of scanning photonic force microscopy for applications in which a high spatial and temporal resolution of the medium viscosity is desired.

© 2003 Optical Society of America

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  1. K. Francis, B. O. Palsson, “Effective intercellular communication distances are determined by relative time constants for cyto/chemokine secretion and diffusion,” Proc. Natl. Acad. Sci. USA 94, 12258–12262 (1997).
    [CrossRef]
  2. F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
    [CrossRef]
  3. B. A. Nemet, Y. Shabtai, M. Cronin-Golomb, “Imaging microscopic viscosity with confocal scanning optical tweezers,” Opt. Lett. 27, 264–266 (2002).
    [CrossRef]
  4. K. Svoboda, S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
    [CrossRef] [PubMed]
  5. Some axial offset is required since the particle is trapped slightly after the geometric focal point of the laser. In one realization of the optical design, the pinhole is replaced by a single-mode fiber, and the lens in front of it is replaced with a fiber collimator. Use of an optical fiber instead of a pinhole is sometimes more convenient since it allows us to switch from an avalanche photodiode, which gives an analog signal, to a photomultiplier tube with a built-in pulse discriminator, which gives the digital photon count signal needed for the autocorrelator board during DLS measurements. Thus the switching involves no realignment of the optics.
  6. Instruments: AOD (Model AOM 1205C-2, Isomet Corporation, Springfield, Va.); objective lens (oil immersion 100×, 1.25 NA semiplan, Edmund Industrial Optics, Barrington, N.J.); digital LIA (Model SR850, Stanford Research Systems, Sunnyvale, Calif.); galvanometer SMs (Model Z1913), servo controller (Model DX2003), and control bus (Model DG1003) (GSI Lumonics, Ottawa, Ontario); laser (cw Ti-Sapphire laser, Model 3900S, pumped by Argon Ion laser stabilite 2017, Spectra Physics, Mountain View, Calif.); APD (Model C5460–01, Hamamatsu Bridgewater, N.J.); CCD camera (Model 4915-2000, COHU, Inc., San Diego, Calif.) frame grabber (Model DT3155, Data Translation, Inc., Marlboro, Mass.); microscope cover glass (No. 1 Fisherfinest, Fisher, Pittsburgh, Pa.); Microspheres—uniform silica microspheres (Bangs Laboratories, Inc., Fishers, Indiana); and dc motor actuator (Model 850, Newport, Inc., Irvine, Calif.).
  7. J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Plenum, New York, 1995).
  8. C. J. R. Sheppard, D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, New York, 1997).
  9. The back focal plane that is the desired plane is ∼20 mm inside most objective lens packages (unless an infinity-corrected lens is used); therefore, a compromise must be made for steering without light loss and vignetting effects.
  10. Similar analysis can be applied to the axial direction if the beam is set to oscillate axially. Since the confocal detection is sensitive to displacements in z as well as in x and y, the axial trap stiffness can be determined.
  11. A. J. Levine, T. C. Lubensky, “One and two-particle micro-rheology,” Phys. Rev. Lett. 85, 1774–1777 (2000).
    [CrossRef] [PubMed]
  12. M. Doi, Introduction to Polymer Physics (Oxford University Press, New York, 1996).
  13. Before any analysis, the phase data was corrected for the overall system time delay that exists regardless of the motion of the particle. By observing the signal from a static reflective edge, we found experimentally that the fundamental phase decreases linearly as we increase the frequency with a proportionality time constant of 4.017 μs, which is the time delay of the system (the response of the AOD and the rest of the electronics). At low frequencies this proportionality time constant adds an unnoticeable phase delay, but as we go to high frequencies, this additional delay becomes large, and a systematic deviation from the theoretical line appears. The second-harmonic phase was corrected as follows ϕ2corr = ϕ2meas + 2ω0(4.017 × 10-6) where ϕ2 is in radians. Also note that in the model we use a sin function as the reference, but the LIA measures the in-phase component as the cos function; we therefore add π/2 to the measured phase before we apply the analysis using our convention.
  14. K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
    [CrossRef]
  15. Both the time constant and the slope of the low-pass filter at the reference frequency determine the actual NEBW. A sharper slope results in a smaller NEBW, allowing faster measurement for a given LIA time constant.
  16. Sampling faster than 64 Hz is, of course, possible but under the conditions of the experiment would lead to underestimating the standard deviation since we would be sampling much faster than the response of the LIA, and the sampled points would not be reflecting the randomness of the noise. This underestimation, however, would not affect the result for the mean value.
  17. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  18. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  19. In the graph of the corrected κ we decided to omit the endpoints that, owing to the uncertainty in the exact z position (∼1 μm), give unreasonably large values close to the singularity at z = 0 (and at z = 95, the top surface).
  20. A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  21. W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
    [CrossRef] [PubMed]
  22. K. Visscher, G. J. Brakenhoff, “Single beam optical trapping integrated in a confocal microscope for biological applications,” Cytometry 12, 486–491 (1991).
    [CrossRef] [PubMed]
  23. E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
    [CrossRef]
  24. A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
    [CrossRef]
  25. G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
    [CrossRef]
  26. M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
    [CrossRef]
  27. E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
    [CrossRef]
  28. A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
    [CrossRef] [PubMed]
  29. P. D. Kaplan, V. Trappe, D. A. Weitz, “Light-scattering microscope,” Appl. Opt. 38, 4151–4157 (1999).
    [CrossRef]
  30. B. J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).
  31. B. A. Nemet, M. Cronin-Golomb, “Microscopic flow measurements with optically trapped microprobes,” Opt. Lett. 27, 1357–1359 (2002).
    [CrossRef]
  32. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  33. S. Haykin, Communication Systems, 2nd ed. (Wiley, New York, 1983).

2002 (2)

2000 (2)

F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
[CrossRef]

A. J. Levine, T. C. Lubensky, “One and two-particle micro-rheology,” Phys. Rev. Lett. 85, 1774–1777 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (4)

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
[CrossRef]

1997 (1)

K. Francis, B. O. Palsson, “Effective intercellular communication distances are determined by relative time constants for cyto/chemokine secretion and diffusion,” Proc. Natl. Acad. Sci. USA 94, 12258–12262 (1997).
[CrossRef]

1996 (3)

K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
[CrossRef]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

1994 (2)

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef] [PubMed]

1992 (1)

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

1991 (1)

K. Visscher, G. J. Brakenhoff, “Single beam optical trapping integrated in a confocal microscope for biological applications,” Cytometry 12, 486–491 (1991).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Bar-Ziv, R.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

Berns, M. W.

Block, S. M.

K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef] [PubMed]

Brakenhoff, G. J.

K. Visscher, G. J. Brakenhoff, “Single beam optical trapping integrated in a confocal microscope for biological applications,” Cytometry 12, 486–491 (1991).
[CrossRef] [PubMed]

Brenner, H.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Cronin-Golomb, M.

Dewalt, L. E.

M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
[CrossRef]

Doi, M.

M. Doi, Introduction to Polymer Physics (Oxford University Press, New York, 1996).

Florin, E. L.

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Francis, K.

K. Francis, B. O. Palsson, “Effective intercellular communication distances are determined by relative time constants for cyto/chemokine secretion and diffusion,” Proc. Natl. Acad. Sci. USA 94, 12258–12262 (1997).
[CrossRef]

Gross, S. P.

K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

Happel, J.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, Englewood Cliffs, N.J., 1965).

Haykin, S.

S. Haykin, Communication Systems, 2nd ed. (Wiley, New York, 1983).

Horber, J. K. H.

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Horiike, K.

F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
[CrossRef]

Kaplan, P. D.

Levine, A. J.

A. J. Levine, T. C. Lubensky, “One and two-particle micro-rheology,” Phys. Rev. Lett. 85, 1774–1777 (2000).
[CrossRef] [PubMed]

Libchaber, A.

G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
[CrossRef]

Lubensky, T. C.

A. J. Levine, T. C. Lubensky, “One and two-particle micro-rheology,” Phys. Rev. Lett. 85, 1774–1777 (2000).
[CrossRef] [PubMed]

Meller, A.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Moses, E.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Nemet, B. A.

Ou Yang, H. D.

M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
[CrossRef]

Palsson, B. O.

K. Francis, B. O. Palsson, “Effective intercellular communication distances are determined by relative time constants for cyto/chemokine secretion and diffusion,” Proc. Natl. Acad. Sci. USA 94, 12258–12262 (1997).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

Pralle, A.

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

Safran, S. A.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Shabtai, Y.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Sheppard, C. J. R.

C. J. R. Sheppard, D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, New York, 1997).

ShiPing, H.

F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
[CrossRef]

Shivashankar, G. V.

G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
[CrossRef]

Shotton, D. M.

C. J. R. Sheppard, D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, New York, 1997).

Sonek, G. J.

Stavans, J.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Stelzer, E. H. K.

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Stolovitzky, G.

G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
[CrossRef]

Svoboda, K.

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef] [PubMed]

Tlusty, T.

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

Trappe, V.

Valentine, M. T.

M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
[CrossRef]

Visscher, K.

K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

K. Visscher, G. J. Brakenhoff, “Single beam optical trapping integrated in a confocal microscope for biological applications,” Cytometry 12, 486–491 (1991).
[CrossRef] [PubMed]

Weitz, D. A.

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Wright, W. H.

Yoshida, F.

F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
[CrossRef]

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

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. A: Mater. Sci. Process. (2)

E. L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A: Mater. Sci. Process. 66, S75–S78 (1998).
[CrossRef]

A. Pralle, E. L. Florin, E. H. K. Stelzer, J. K. H. Horber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A: Mater. Sci. Process. 66, S71–S73 (1998).
[CrossRef]

Appl. Phys. Lett. (2)

G. V. Shivashankar, G. Stolovitzky, A. Libchaber, “Backscattering from tethered bead as a probe of DNA flexibility,” Appl. Phys. Lett. 73, 291–293 (1998).
[CrossRef]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Biophys. J. (2)

A. Meller, R. Bar-Ziv, T. Tlusty, E. Moses, J. Stavans, S. A. Safran, “Localized dynamic light scattering: a new approach to dynamic measurements in optical microscopy,” Biophys. J. 74, 1541–1548 (1998).
[CrossRef] [PubMed]

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Cytometry (1)

K. Visscher, G. J. Brakenhoff, “Single beam optical trapping integrated in a confocal microscope for biological applications,” Cytometry 12, 486–491 (1991).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

K. Visscher, S. P. Gross, S. M. Block, Construction of multiple-beam optical traps with nanometer-resolution position sensing. IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

J. Phys. Condens. Matter (1)

M. T. Valentine, L. E. Dewalt, H. D. Ou Yang, “Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys. Condens. Matter 8, 9477–9482 (1996).
[CrossRef]

J. Phys. Soc. Jpn. (1)

F. Yoshida, K. Horiike, H. ShiPing, “Time-dependent concentration profile of secreted molecules in the intercellular signaling,” J. Phys. Soc. Jpn. 69, 3736–3743 (2000).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. J. Levine, T. C. Lubensky, “One and two-particle micro-rheology,” Phys. Rev. Lett. 85, 1774–1777 (2000).
[CrossRef] [PubMed]

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

K. Francis, B. O. Palsson, “Effective intercellular communication distances are determined by relative time constants for cyto/chemokine secretion and diffusion,” Proc. Natl. Acad. Sci. USA 94, 12258–12262 (1997).
[CrossRef]

Other (16)

Both the time constant and the slope of the low-pass filter at the reference frequency determine the actual NEBW. A sharper slope results in a smaller NEBW, allowing faster measurement for a given LIA time constant.

Sampling faster than 64 Hz is, of course, possible but under the conditions of the experiment would lead to underestimating the standard deviation since we would be sampling much faster than the response of the LIA, and the sampled points would not be reflecting the randomness of the noise. This underestimation, however, would not affect the result for the mean value.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, Englewood Cliffs, N.J., 1965).

In the graph of the corrected κ we decided to omit the endpoints that, owing to the uncertainty in the exact z position (∼1 μm), give unreasonably large values close to the singularity at z = 0 (and at z = 95, the top surface).

M. Doi, Introduction to Polymer Physics (Oxford University Press, New York, 1996).

Before any analysis, the phase data was corrected for the overall system time delay that exists regardless of the motion of the particle. By observing the signal from a static reflective edge, we found experimentally that the fundamental phase decreases linearly as we increase the frequency with a proportionality time constant of 4.017 μs, which is the time delay of the system (the response of the AOD and the rest of the electronics). At low frequencies this proportionality time constant adds an unnoticeable phase delay, but as we go to high frequencies, this additional delay becomes large, and a systematic deviation from the theoretical line appears. The second-harmonic phase was corrected as follows ϕ2corr = ϕ2meas + 2ω0(4.017 × 10-6) where ϕ2 is in radians. Also note that in the model we use a sin function as the reference, but the LIA measures the in-phase component as the cos function; we therefore add π/2 to the measured phase before we apply the analysis using our convention.

Some axial offset is required since the particle is trapped slightly after the geometric focal point of the laser. In one realization of the optical design, the pinhole is replaced by a single-mode fiber, and the lens in front of it is replaced with a fiber collimator. Use of an optical fiber instead of a pinhole is sometimes more convenient since it allows us to switch from an avalanche photodiode, which gives an analog signal, to a photomultiplier tube with a built-in pulse discriminator, which gives the digital photon count signal needed for the autocorrelator board during DLS measurements. Thus the switching involves no realignment of the optics.

Instruments: AOD (Model AOM 1205C-2, Isomet Corporation, Springfield, Va.); objective lens (oil immersion 100×, 1.25 NA semiplan, Edmund Industrial Optics, Barrington, N.J.); digital LIA (Model SR850, Stanford Research Systems, Sunnyvale, Calif.); galvanometer SMs (Model Z1913), servo controller (Model DX2003), and control bus (Model DG1003) (GSI Lumonics, Ottawa, Ontario); laser (cw Ti-Sapphire laser, Model 3900S, pumped by Argon Ion laser stabilite 2017, Spectra Physics, Mountain View, Calif.); APD (Model C5460–01, Hamamatsu Bridgewater, N.J.); CCD camera (Model 4915-2000, COHU, Inc., San Diego, Calif.) frame grabber (Model DT3155, Data Translation, Inc., Marlboro, Mass.); microscope cover glass (No. 1 Fisherfinest, Fisher, Pittsburgh, Pa.); Microspheres—uniform silica microspheres (Bangs Laboratories, Inc., Fishers, Indiana); and dc motor actuator (Model 850, Newport, Inc., Irvine, Calif.).

J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Plenum, New York, 1995).

C. J. R. Sheppard, D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, New York, 1997).

The back focal plane that is the desired plane is ∼20 mm inside most objective lens packages (unless an infinity-corrected lens is used); therefore, a compromise must be made for steering without light loss and vignetting effects.

Similar analysis can be applied to the axial direction if the beam is set to oscillate axially. Since the confocal detection is sensitive to displacements in z as well as in x and y, the axial trap stiffness can be determined.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

S. Haykin, Communication Systems, 2nd ed. (Wiley, New York, 1983).

B. J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

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Figures (5)

Fig. 1
Fig. 1

Experimental layout. The pinhole (P) is placed in an image plane of the object, forming the confocal detection. These conjugate planes are marked with #. The AOD is placed at an optically conjugate plane of the back aperture of the objective lens (OL). The conjugate planes are marked with *. Two galvanometer SMs are placed in another such conjugate plane (*). The LIA drives the AOD with a sinusoidal waveform and measures the magnitude and phase of the APD signal at the second-harmonic frequency. The scene is viewed with a CCD camera in transmission through the dichroic mirror (DM) that has high reflectivity in the IR. SL1 is the scan lens.

Fig. 2
Fig. 2

Experimentally measured second-harmonic phase for 1.9-μm diameter silica microsphere in water, for varying laser power (15 to 60 mW at the sample) at a constant frequency of 600 Hz (□) and for varying frequency (200 to 2000 Hz) at a constant power of 46 mW (○). Each measurement of ϕ2 is the average of 512 samples from the LIA (sampled at 64 Hz). The LIA time constant was 100 ms, corresponding to NEBW of 0.78 Hz. The errors in ϕ2 from the standard deviations are smaller or equal to the size of the symbols and are omitted for clarity. Both sets of measurements are plotted together after a linear regression fit was done on each set separately to obtain b. The abscissa parameter ω0τ is calculated from ω0 b/ P. The solid curve shows the theoretical curve for which the x axis is cot(ϕ2/2). Inset: Linear regression fit to the frequency data, τ = slope = 1.290 ± 0.003 ms with a correlation coefficient 0.99974. From a similar linear regression (not shown), we obtain for the power measurements b = 59.48 ± 0.27 μJ with a correlation coefficient 0.999.

Fig. 3
Fig. 3

SNR of the viscosity measurement for 0.9-μm-diameter silica microsphere in water with 18 mW of laser power at 815 nm and a constant NEBW of 2.6 Hz. The characteristic frequency fc = 585 Hz is indicated by the dotted line. The solid curve is the theoretical curve (see text for details).

Fig. 4
Fig. 4

Trap stiffness as a function of distance from the cover glass. κ is calculated by means of γ/τ(□) and by means of γ g(z)/τ (○) with Brenner’s formula for the correction near the two surfaces (see text). We obtain τ from measurements of the second-harmonic phase for the following conditions: 1.9-μm-diameter silica microsphere in water γ = 6πηr = 1.79 × 10-8 kg/s, constant laser power of 29 mW at the sample, 815-nm wavelength, constant frequency of oscillations of 500 Hz, objective lens 100× magnification, and 1.25 NA oil immersion. Solid lines are drawn to guide the eye. Each data point is the mean of 512 samples (sampled at 64 Hz where the time constant of the LIA was 300 ms and 0.26 Hz NEBW). Error bars obtained from the standard deviations.

Fig. 5
Fig. 5

Comparison of the theoretical SNR of the viscosity measurement for the confocal detection (our device) and for the split-photodiode position detection (Valentine’s device). The measurement bandwidth was taken to be proportional to the oscillation frequency. The SNR of the two methods are equal at the characteristic frequency, which was taken to be f c = 585 Hz (as used in Section 5). The confocal SNR peaks at 3 f c = 1013 Hz, whereas Valentine’s SNR peaks at 1/3 f c = 338 Hz. The three frequencies are marked with dotted lines. From left to right these are the frequency of peak SNR for Valentine’s device, the frequency where the SNR of Valentine’s device is equivalent to ours, and the frequency of peak SNR for our device.

Equations (44)

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γ dxdt+κx-pt=Lt,
γ dudt+κu=-γaω0 cos ω0t+Lt.
ut=-u0 sinω0t+ϕ1,
ϕ2=2ϕ1=2 cot-1 ω0τ.
SNRη=ω0τ2ω0τ2+1akBTκ1/218Δωτ1/2.
SNRη=SNRη1+ω0τ-2,
SNRη= 12 κa2γkBT2Δω1/2 =FspringrmsFLangevinrms.
SNRη=ηΔη=γΔγ=κΔκ.
gz=1-916rz+18rz3-45256rz4-116rz5-1.
SNRηValentine=ω0ωcω02+ωc2akBTκ1/2ωc8Δω1/2.
SNRηConfocalSNRηValentine=ω0τ.
RNt=Nt N0¯=kBTκexp-ωc|t|, where ωc=1/τ
N2t¯=RN0=kBT/κ.
Ryyt=yt y0¯=kBTκ21+2 exp-2 |t|τ,
It  1-αu2t,
dutdt+ωcut=Ltγ-aω0 cos ω0t.
SLω=|L˜ω|2¯=q.
ut=-u0 sinω0t+ϕ1+Nt,
u0=aω0/ωc2+ω021/2 ϕ1=cot-1ω0/ωc
SNω=|Ñω|2¯=qγ2ω2+ωc2=2kBTγω2+ωc2=kBTκ2ωcω2+ωc2,
RNt=Nt N0¯=kBTκexp-ωc|t|,
N2t¯=RN0=kBTκ.
yt=u2t=u02 sin2ω0t+ϕ1+N2t-2Nt×u0 sinω0t+ϕ1.
y˜Sω=u0242δω+δω-2ω0exp j2ϕ1+δω+2ω0exp-j2ϕ1.
SNNω=|y˜NNω|2¯=kBTκ2δω+8ωcω2+2ωc2,
RNNt=yNt yN0¯=kBTκ21+2 exp-2ωc|t|.
SSNω=u02|Ñω-ω0|2¯=u02kBTκ2ωcω-ω02+ωc2
SP2ω0=u048=aω048ω02+ωC22,
NP2ω0=4kBTκ2ωcΔωω02+ωc2,
SNP2ω0=kBTκ4a2ωcω02Δωω02+ωc22.
SNRy=SP2ω0NP2ω0+SNP2ω01/2.
SNRySP2ω0SNP2ω01/2=akBTκ1/2ω0ωcωc32Δω1/2.
Δϕ2Δy2ω0y2ω0=1SNRy, where Δϕ2 is in radians).
SNRη=SNRκ=SNRγ=γΔγ=sin ϕ2Δϕ2=2ω0ωcω02+ωc21Δϕ2=2ω0ωcω02+ωc2SNRy,
SNRη=ω02ω02+ωc2akBT/κ1/2ωc8Δω1/2=SNRη1+ωc/ω02,
dxtdt+ωcxt=Ltγ+aωc sin ω0t,
xt=x0 sinω0t+θ1+Nt,
x0=aωcω02+ωc21/2
θ1=-cot-1ωcω0.
NPω0=kBTκ4ωcΔωω02+ωc2.
SPω0V=x022=ωc2a22ωc2+ω02.
SNRxV=SPω0VNPω01/2=akBTκ1/2ωc8Δω1/2,
SNRηValentine=γΔγ=sin 2θ12Δθ1=ω0ωcω02+ωc21Δθ1=ω0ωcω02+ωc2SNRxV,
SNRηValentine=ω0ωcω02+ωc2akBTκ1/2ωc8Δω1/2.

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