Abstract

An experimental strategy for post-eliminating thermal noise on position measurements of optically trapped particles is presented. Using a nanosecond pulsed laser, synchronized to the detection system, to exert a periodic driving force on an optically trapped 10 μm polystyrene bead, the laser pulse-bead interaction is repeated hundreds of times. Traces with the bead position following the prompt displacement from equilibrium, induced by each laser pulse, are averaged and reveal the underlying deterministic motion of the bead, which is not visible in a single trace due to thermal noise. The motion of the bead is analyzed from the direct time-dependent position measurements and from the power spectrum. The results show that the bead is on average displaced 208 nm from the trap center and exposed to a force amplitude of 71 nanoNewton, more than five orders of magnitude larger than the trapping forces. Our experimental method may have implications for microrheology.

© 2013 Optical Society of America

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  1. R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
    [Crossref] [PubMed]
  2. F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Meth. Cell. Biol,  55, 129–156 (1998).
    [Crossref]
  3. E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
    [Crossref]
  4. A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
    [Crossref]
  5. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [Crossref]
  6. T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
    [Crossref]
  7. F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
    [Crossref]
  8. T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
    [Crossref]
  9. S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
    [Crossref] [PubMed]
  10. K. D. Wulff, D. G. Cole, and R. L. Clark, “Servo control of an optical trap,” Appl. Opt. 46, 4923 (2007).
    [Crossref] [PubMed]
  11. A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
    [Crossref]
  12. D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express 17, 22718–22725 (2009).
    [Crossref]
  13. S. Tauro, A. Bañas, D. Palima, and J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18, 18217–18222 (2010).
    [Crossref] [PubMed]
  14. R. Bowman, A. Jesacher, G. Thalhammer, G. Gibson, M. Ritsch-Marte, and M. Padgett, “Position clamping in a holographic counterpropagating optical trap,” Opt. Express 19, 9908–9914 (2011).
    [Crossref] [PubMed]
  15. Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
    [Crossref] [PubMed]
  16. Y. Huang, Z. Zhang, and C.-H. Menq, “Minimum-variance Brownian motion control of an optically trapped probe,” Appl. Opt. 48, 5871–5880 (2009).
    [Crossref] [PubMed]
  17. D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
    [Crossref]
  18. F. C. Mackintosh and C. F. Schmidt, “Active cellular materials,” Curr. Opin. Cell Biol. 22, 29–35 (2010).
    [Crossref] [PubMed]
  19. T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
    [Crossref]
  20. H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
    [Crossref]
  21. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
    [Crossref]
  22. S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express 14, 13095–13100 (2006).
    [Crossref] [PubMed]
  23. S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
    [Crossref] [PubMed]
  24. P. S. Alves and M. S. Rocha, “Videomicroscopy calibration of optical tweezers by position autocorrelation function analysis,” Appl. Phys. B 107, 375–378 (2012).
    [Crossref]
  25. “The experiment was only carried out at one power of the pulsed laser beam. If the power is reduced (increased) we expect a qualitative similar motion of the bead, following the laser pulse, but with a reduced (increased) maximum displacement from equilibrium due to the weaker (stronger) push from the laser pulse. Beyond a certain power the action on the bead will be so strong that it is pushed out of the optical trap.”
  26. T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
    [Crossref] [PubMed]
  27. F. Czerwinski, A. C. Richardson, and L. B. Oddershede, “Quantifying noise in optical tweezers by allan variance,” Opt. Express 17, 13255–13269 (2009).
    [Crossref] [PubMed]
  28. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
    [Crossref]
  29. F. Harris, “On the use of windows for harmonic analysis with the discrete fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [Crossref]
  30. J. Wan, Y. Huang, S. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832–4841 (2009).
    [Crossref] [PubMed]
  31. Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
    [Crossref]

2012 (1)

P. S. Alves and M. S. Rocha, “Videomicroscopy calibration of optical tweezers by position autocorrelation function analysis,” Appl. Phys. B 107, 375–378 (2012).
[Crossref]

2011 (4)

Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
[Crossref]

R. Bowman, A. Jesacher, G. Thalhammer, G. Gibson, M. Ritsch-Marte, and M. Padgett, “Position clamping in a holographic counterpropagating optical trap,” Opt. Express 19, 9908–9914 (2011).
[Crossref] [PubMed]

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

2010 (3)

F. C. Mackintosh and C. F. Schmidt, “Active cellular materials,” Curr. Opin. Cell Biol. 22, 29–35 (2010).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

S. Tauro, A. Bañas, D. Palima, and J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18, 18217–18222 (2010).
[Crossref] [PubMed]

2009 (6)

2008 (2)

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

2007 (1)

2006 (2)

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

S. C. Chapin, V. Germain, and E. R. Dufresne, “Automated trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express 14, 13095–13100 (2006).
[Crossref] [PubMed]

2004 (1)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

2000 (2)

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
[Crossref] [PubMed]

1998 (3)

F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Meth. Cell. Biol,  55, 129–156 (1998).
[Crossref]

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

1997 (2)

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

1996 (2)

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[Crossref]

1978 (1)

F. Harris, “On the use of windows for harmonic analysis with the discrete fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

Alves, P. S.

P. S. Alves and M. S. Rocha, “Videomicroscopy calibration of optical tweezers by position autocorrelation function analysis,” Appl. Phys. B 107, 375–378 (2012).
[Crossref]

Bañas, A.

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Bowman, R.

Chapin, S. C.

Cheng, M.-C.

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Cheng, P.

Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
[Crossref]

Chu, S.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

Clark, R. L.

Cole, D. G.

Cooper, J.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Cooper, J. M.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Crocker, J. C.

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[Crossref]

Czerwinski, F.

Dam, J. S.

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Di Leonardo, R.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Dufresne, E. R.

Evans, R. M. L.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Finer, J. T.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

Florin, E.-L.

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

Flyvbjerg, H.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Frey, E.

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Ganesan, K.

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

Germain, V.

Gibson, G.

Gibson, G. M.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Gisler, T.

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Gittes, F.

F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Meth. Cell. Biol,  55, 129–156 (1998).
[Crossref]

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Glückstad, J.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

S. Tauro, A. Bañas, D. Palima, and J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18, 18217–18222 (2010).
[Crossref] [PubMed]

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Grier, D. G.

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[Crossref]

Hæggström, E.

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

Harris, F.

F. Harris, “On the use of windows for harmonic analysis with the discrete fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

Hörber, J.

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

Howard, J.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Huang, Y.

Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
[Crossref]

J. Wan, Y. Huang, S. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832–4841 (2009).
[Crossref] [PubMed]

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Y. Huang, Z. Zhang, and C.-H. Menq, “Minimum-variance Brownian motion control of an optically trapped probe,” Appl. Opt. 48, 5871–5880 (2009).
[Crossref] [PubMed]

Jesacher, A.

Jhiang, S.

Jhiang, S. M.

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Jülicher, F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Keen, S.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Keiding, S. R.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Kheifets, S.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Kristensen, M. V.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

Kroy, K.

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Kuo, S.

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

Kuo, S. C.

S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
[Crossref] [PubMed]

Kylling, A. P.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

Leach, J.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Li, T.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Lindballe, T. B.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

Linnenberger, A.

Love, G.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

MacKintosh, F.

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Mackintosh, F. C.

F. C. Mackintosh and C. F. Schmidt, “Active cellular materials,” Curr. Opin. Cell Biol. 22, 29–35 (2010).
[Crossref] [PubMed]

Mason, T.

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

Mason, T. G.

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Medellin, D.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Menq, C.-H.

Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
[Crossref]

J. Wan, Y. Huang, S. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832–4841 (2009).
[Crossref] [PubMed]

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Y. Huang, Z. Zhang, and C.-H. Menq, “Minimum-variance Brownian motion control of an optically trapped probe,” Appl. Opt. 48, 5871–5880 (2009).
[Crossref] [PubMed]

Oddershede, L. B.

Ojala, H.

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

Olmsted, P.

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Padgett, M.

Padgett, M. J.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Palima, D.

Palima, D. Z.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Pavone, F. S.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Perch-Nielsen, I. R.

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Pralle, A.

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

Preece, D.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express 17, 22718–22725 (2009).
[Crossref]

Raizen, M. G.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Richardson, A. C.

Ritsch-Marte, M.

Rocha, M. S.

P. S. Alves and M. S. Rocha, “Videomicroscopy calibration of optical tweezers by position autocorrelation function analysis,” Appl. Phys. B 107, 375–378 (2012).
[Crossref]

Saunter, C.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Schäffer, E.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Schmidt, C.

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Schmidt, C. F.

F. C. Mackintosh and C. F. Schmidt, “Active cellular materials,” Curr. Opin. Cell Biol. 22, 29–35 (2010).
[Crossref] [PubMed]

F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Meth. Cell. Biol,  55, 129–156 (1998).
[Crossref]

Schnurr, B.

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Serati, S.

Simmons, R. M.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

Spudich, J. A.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

Stapelfeldt, H.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Stelzer, E.

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

Tassieri, M.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Tauro, S.

Thalhammer, G.

Thøgersen, J.

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

Tolic-Nørrelykke, S. F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Tuma, R.

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

Ulriksen, H.-U.

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

van Zanten, J.

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

Wallin, A. E.

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

Wan, J.

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

J. Wan, Y. Huang, S. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832–4841 (2009).
[Crossref] [PubMed]

Warren, R.

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

Weitz, D. A.

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Wirtz, D.

S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
[Crossref] [PubMed]

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

Wulff, K. D.

Yamada, S.

S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
[Crossref] [PubMed]

Yao, A.

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Zhang, Z.

Y. Huang, Z. Zhang, and C.-H. Menq, “Minimum-variance Brownian motion control of an optically trapped probe,” Appl. Opt. 48, 5871–5880 (2009).
[Crossref] [PubMed]

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Appl. Opt. (3)

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

E.-L. Florin, A. Pralle, E. Stelzer, and J. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A Mater. Sci. 66, S75–S78 (1998).
[Crossref]

A. Pralle, E.-L. Florin, E. Stelzer, and J. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A Mater. Sci. 66, S71–S73 (1998).
[Crossref]

Appl. Phys. B (1)

P. S. Alves and M. S. Rocha, “Videomicroscopy calibration of optical tweezers by position autocorrelation function analysis,” Appl. Phys. B 107, 375–378 (2012).
[Crossref]

Appl. Phys. Lett. (1)

A. E. Wallin, H. Ojala, E. Hæggström, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[Crossref]

Biophys. J. (2)

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822 (1996).
[Crossref] [PubMed]

S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000).
[Crossref] [PubMed]

Curr. Opin. Cell Biol. (1)

F. C. Mackintosh and C. F. Schmidt, “Active cellular materials,” Curr. Opin. Cell Biol. 22, 29–35 (2010).
[Crossref] [PubMed]

IEEE ASME Trans. Mechatron. (1)

Y. Huang, P. Cheng, and C.-H. Menq, “Dynamic Force Sensing Using an Optically Trapped Probing System,” IEEE ASME Trans. Mechatron. 16, 1145–1154 (2011).
[Crossref]

J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[Crossref]

J. Eur. Opt. Soc-Rapid (1)

H.-U. Ulriksen, J. Thøgersen, S. R. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc-Rapid 3, 08034 (2008).
[Crossref]

J. Opt. (1)

D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J. Padgett, J. M. Cooper, and M. Tassieri, “Optical tweezers: wideband microrheology,” J. Opt. 13, 044022 (2011).
[Crossref]

J. Rheol. (1)

T. G. Mason, T. Gisler, K. Kroy, E. Frey, and D. A. Weitz, “Rheology of f-actin solutions determined from thermally driven tracer motion,” J. Rheol. 44, 917–928 (2000).
[Crossref]

Journal of J. Eur. Opt. Soc-Rapid (1)

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” Journal of J. Eur. Opt. Soc-Rapid 6, 11057 (2011).
[Crossref]

Lab Chip (1)

S. Keen, A. Yao, J. Leach, R. Di Leonardo, C. Saunter, G. Love, J. Cooper, and M. Padgett, “Multipoint viscosity measurements in microfluidic channels using optical tweezers,” Lab Chip 9, 2059 (2009).
[Crossref] [PubMed]

Meth. Cell. Biol (1)

F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Meth. Cell. Biol,  55, 129–156 (1998).
[Crossref]

Opt. Express (5)

Phys. Rev. Lett. (2)

T. Mason, K. Ganesan, J. van Zanten, D. Wirtz, and S. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[Crossref]

F. Gittes, B. Schnurr, P. Olmsted, F. MacKintosh, and C. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Proc. IEEE (1)

F. Harris, “On the use of windows for harmonic analysis with the discrete fourier transform,” Proc. IEEE 66, 51–83 (1978).
[Crossref]

Rev. Sci. Instrum. (3)

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Y. Huang, J. Wan, M.-C. Cheng, Z. Zhang, S. M. Jhiang, and C.-H. Menq, “Three-axis rapid steering of optically propelled micro/nanoparticles,” Rev. Sci. Instrum. 80, 063107 (2009).
[Crossref] [PubMed]

Science (1)

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the Instantaneous Velocity of a Brownian Particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Other (1)

“The experiment was only carried out at one power of the pulsed laser beam. If the power is reduced (increased) we expect a qualitative similar motion of the bead, following the laser pulse, but with a reduced (increased) maximum displacement from equilibrium due to the weaker (stronger) push from the laser pulse. Beyond a certain power the action on the bead will be so strong that it is pushed out of the optical trap.”

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Figures (5)

Fig. 1
Fig. 1

Schematics showing the pulsed laser setup. The zoom-in on the upper left part shows a cartoon of the experimental procedure described in the main text. This is also the point of view of the motion-camera which monitors the experiment through one of the trapping objectives (not shown). The two counter-propagating trapping beams are aligned along the z-axis and are indicated by the red coloring in the cartoon. The inset shows the camera trigger setup.

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Fig. 2
Fig. 2

Bead position as a function of time with and without the pulsed laser included. The vertical dashed lines indicate the positions of the pulse-bead interaction events. The first pulse is located at time t = 0 s.

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Fig. 3
Fig. 3

Plot of the bead’s averaged y-position as a function of time (blue dots). Averaging 450 events like the seven shown in Fig. 2 gave this result. The inset shows the same for the average of 45 events including error bars (shown for every tenth data point), which indicate the standard deviation of the 45 measurements, i. e. the magnitude of the thermal motion. The fits of Eq. (3) are shown by the red dashed lines. They are almost indistinguishable from the data.

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Fig. 4
Fig. 4

Example of the position correlation function (dots) of a 17 second long data section without the laser pulses. The fit of Eq. (1) (red dashed line) was performed for 0 < τ < 1.2τt = 0.1 s.

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Fig. 5
Fig. 5

(a) The measured power spectrum of the bead motion when subject to the laser pulses (blue line). The red dashed line is a fit of Eq. (8). The black line is a plot of the periodic pulsed part of the fit. (b)–(c) Closeup of the peaks at 1 Hz and 50 Hz respectively (connected blue dots). The pulse rate was found to be fp = 1.0002 Hz.

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Equations (11)

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

〈 y ( t + τ ) y ( t ) 〉 = A e − κ corr γ τ .
m y ¨ = − κ y − γ y ˙ + ξ ( t ) + Θ t p ( t ) ,
y ( t ) = Δ y exp ( − κ det γ t ) ,
Θ t p ( t ) = F p ∑ k = 0 N − 1 h ( t − k t p ) ,                 h ( t ) ≈ exp ( − t 2 2 σ p 2 ) .
〈 S y ( f ) 〉 ≃ Y ( f ) Y ( f ) * t msr .
Θ ^ t p ( f ) = F p ∫ − ∞ ∞ ∑ k = 0 N − 1 h ( t − k t p ) e − i 2 π f t d t = F p ( ∑ k = 0 N − 1 e − i 2 π f k t p ) ∫ − ∞ ∞ h ( t ) e − i 2 π f t d t = F p 2 π σ p 2 e − 2 π 2 σ p 2 f 2 ∑ k = 0 N − 1 e − i 2 π f k t p .
Θ ^ t p ( f ) ≃ F p 2 π σ p 2 ∑ k = 0 N − 1 e − i 2 π f k t p .
〈 S y ( f ) 〉 ≃ 1 2 π 2 D + π σ p 2 F p 2 γ 2 t msr sin 2 ( π N t p f ) sin 2 ( π t p f ) f c 2 + f 2 .
Δ y 2 ≈ 2 π σ p 2 F p 2 γ 2 .
〈 S y ( f peak , i ) 〉 ≈ 1 2 π 2 D + π N σ p 2 F p 2 γ 2 t p f c 2 + f peak , i 2
Δ y min = 2 D t p N ,

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