Abstract

Multi-dimensional, correlated particle tracking is a key technology to reveal dynamic processes in living and synthetic soft matter systems. In this paper we present a new method for tracking micron-sized beads in parallel and in all three dimensions – faster and more precise than existing techniques. Using an acousto-optic deflector and two quadrant-photo-diodes, we can track numerous optically trapped beads at up to tens of kHz with a precision of a few nanometers by back-focal plane interferometry. By time-multiplexing the laser focus, we can calibrate individually all traps and all tracking signals in a few seconds and in 3D. We show 3D histograms and calibration constants for nine beads in a quadratic arrangement, although trapping and tracking is easily possible for more beads also in arbitrary 2D arrangements. As an application, we investigate the hydrodynamic coupling and diffusion anomalies of spheres trapped in a 3 × 3 arrangement.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
    [CrossRef] [PubMed]
  2. H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
    [CrossRef] [PubMed]
  3. J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73(2), 352–355 (1994).
    [CrossRef] [PubMed]
  4. A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
    [CrossRef] [PubMed]
  5. J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
    [CrossRef] [PubMed]
  6. H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
    [CrossRef] [PubMed]
  7. A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
    [CrossRef] [PubMed]
  8. J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
    [CrossRef]
  9. N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
    [CrossRef] [PubMed]
  10. M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
    [CrossRef] [PubMed]
  11. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
    [CrossRef] [PubMed]
  12. H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
    [CrossRef] [PubMed]
  13. I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
    [CrossRef] [PubMed]
  14. J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
    [CrossRef]
  15. R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
    [CrossRef] [PubMed]
  16. J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. 71(3), 487–493 (2005).
    [CrossRef]
  17. M. Speidel, A. Jonás, and E. L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003).
    [CrossRef] [PubMed]
  18. Z. Zhang and C.-H. Menq, “Three-dimensional particle tracking with subnanometer resolution using off-focus images,” Appl. Opt. 47(13), 2361–2370 (2008).
    [CrossRef] [PubMed]
  19. S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18275–18282 (2007).
    [CrossRef] [PubMed]
  20. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18(13), 13563–13573 (2010).
    [CrossRef] [PubMed]
  21. M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010).
    [CrossRef] [PubMed]
  22. R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express 18(11), 11785–11790 (2010).
    [CrossRef] [PubMed]
  23. A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
    [CrossRef] [PubMed]
  24. A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
    [CrossRef]
  25. R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
    [CrossRef]
  26. L. Friedrich and A. Rohrbach, “Improved interferometric tracking of trapped particles using two frequency-detuned beams,” Opt. Lett. 35(11), 1920–1922 (2010).
    [CrossRef] [PubMed]
  27. M. Speidel, L. Friedrich, and A. Rohrbach, “Interferometric 3D tracking of several particles in a scanning laser focus,” Opt. Express 17(2), 1003–1015 (2009).
    [CrossRef] [PubMed]
  28. K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
    [CrossRef] [PubMed]
  29. A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
    [CrossRef] [PubMed]
  30. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
    [CrossRef] [PubMed]
  31. A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005).
    [CrossRef] [PubMed]
  32. A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
    [CrossRef]
  33. J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).
  34. E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
    [CrossRef] [PubMed]
  35. J.-C. Meiners and S. R. Quake, “Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
    [CrossRef]
  36. S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
    [CrossRef] [PubMed]
  37. R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
    [CrossRef] [PubMed]
  38. G. K. Batchelor, “Diffusion in a dilute polydisperse system of interacting spheres,” J. Fluid Mech. 131(-1), 155–175 (1983).
    [CrossRef]

2011

A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
[CrossRef] [PubMed]

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

2010

2009

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

M. Speidel, L. Friedrich, and A. Rohrbach, “Interferometric 3D tracking of several particles in a scanning laser focus,” Opt. Express 17(2), 1003–1015 (2009).
[CrossRef] [PubMed]

2008

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Z. Zhang and C.-H. Menq, “Three-dimensional particle tracking with subnanometer resolution using off-focus images,” Appl. Opt. 47(13), 2361–2370 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

2007

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18275–18282 (2007).
[CrossRef] [PubMed]

2005

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005).
[CrossRef] [PubMed]

J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. 71(3), 487–493 (2005).
[CrossRef]

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

2004

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

2003

M. Speidel, A. Jonás, and E. L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003).
[CrossRef] [PubMed]

A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
[CrossRef] [PubMed]

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

2002

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
[CrossRef]

2001

S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
[CrossRef] [PubMed]

2000

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

1999

J.-C. Meiners and S. R. Quake, “Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

1998

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

1994

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73(2), 352–355 (1994).
[CrossRef] [PubMed]

1983

G. K. Batchelor, “Diffusion in a dilute polydisperse system of interacting spheres,” J. Fluid Mech. 131(-1), 155–175 (1983).
[CrossRef]

Allersma, M. W.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Altmann, S. M.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

Bartlett, P.

S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
[CrossRef] [PubMed]

Batchelor, G. K.

G. K. Batchelor, “Diffusion in a dilute polydisperse system of interacting spheres,” J. Fluid Mech. 131(-1), 155–175 (1983).
[CrossRef]

Baumgartl, J.

J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. 71(3), 487–493 (2005).
[CrossRef]

Bausch, A. R.

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

Bechinger, C.

J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. 71(3), 487–493 (2005).
[CrossRef]

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

Becker, N. B.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

Berg-Sørensen, K.

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

Bewersdorf, J.

M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010).
[CrossRef] [PubMed]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Bowman, R.

Brenner, M. P.

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

Brunner, M.

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

Buss, F.

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

Bustamante, C.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Candelli, A.

A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
[CrossRef] [PubMed]

Chavez, I.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Cheong, F. C.

Clemen, A. E. M.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Crocker, J. C.

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73(2), 352–355 (1994).
[CrossRef] [PubMed]

Curtis, J. E.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

deCastro, M. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Di Leonardo, R.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Dinsmore, A. D.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

Dobnikar, J.

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

Dogterom, M.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

Dufresne, E. R.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

Dumont, S.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Fahmy, T. M.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Florin, E. L.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

M. Speidel, A. Jonás, and E. L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003).
[CrossRef] [PubMed]

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Florin, E.-L.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Forster, J. D.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Friedrich, L.

Gardel, M. L.

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

Gibson, G.

Gittes, F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Grier, D. G.

F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18(13), 13563–13573 (2010).
[CrossRef] [PubMed]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18275–18282 (2007).
[CrossRef] [PubMed]

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73(2), 352–355 (1994).
[CrossRef] [PubMed]

Griffiths, G.

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

Haraszti, T.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Henderson, S.

S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
[CrossRef] [PubMed]

Holzer, D.

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

Hoogenboom, J. P.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

Hörber, J. K.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Huang, R.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Ianni, F.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Jeney, S.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Jonás, A.

Juette, M. F.

M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010).
[CrossRef] [PubMed]

Keen, S.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Kim, S.-H.

Kress, H.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
[CrossRef] [PubMed]

Krishnatreya, B. J.

Kurre, R.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Leach, J.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Lee, S.-H.

Liphardt, J.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Lukic, B.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Matteo, J. A.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

Meiners, J. C.

J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

Meiners, J.-C.

J.-C. Meiners and S. R. Quake, “Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

Mejean, C. O.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Menq, C.-H.

Mitchell, S.

S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
[CrossRef] [PubMed]

Munteanu, E. L.

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Neumayer, D.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Oddershede, L.

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

Padgett, M.

Padgett, M. J.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Park, J.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Park, J. G.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Peterman, E. J. G.

A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
[CrossRef] [PubMed]

Pralle, A.

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Prummer, M.

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Quake, S. R.

J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

J.-C. Meiners and S. R. Quake, “Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

Raizen, M. G.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Rohrbach, A.

L. Friedrich and A. Rohrbach, “Improved interferometric tracking of trapped particles using two frequency-detuned beams,” Opt. Lett. 35(11), 1920–1922 (2010).
[CrossRef] [PubMed]

M. Speidel, L. Friedrich, and A. Rohrbach, “Interferometric 3D tracking of several particles in a scanning laser focus,” Opt. Express 17(2), 1003–1015 (2009).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
[CrossRef]

Roichman, Y.

Ruocco, G.

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Schmidt, C. F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Schmitz, C.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Scholz, T.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

Smith, S. B.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Spatz, J. P.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Speidel, M.

Squires, T. M.

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

Stelzer, E. H.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Stelzer, E. H. K.

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
[CrossRef]

Stewart, R. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Taute, K. M.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Thon, G.

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

Tinoco, I.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Tischer, C.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Tolic-Nørrelykke, I. M.

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

Uhrig, K.

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Valentine, M. T.

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

van Blaaderen, A.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18275–18282 (2007).
[CrossRef] [PubMed]

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

van der Horst, A.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

van Oostrum, P.

Visscher, K.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

von Grünberg, H.-H.

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

Vossen, D. L. J.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

Walse, S. S.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Weiner, O. D.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Weitz, D. A.

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

Wu, D. Q.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Wuite, G. J. L.

A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
[CrossRef] [PubMed]

Yang, S.-M.

Yethiraj, A.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

Yi, G.-R.

Yodh, A. G.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

Zhang, Y.

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Zhang, Z.

Appl. Opt.

Biophys. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, and C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74(2), 1074–1085 (1998).
[CrossRef] [PubMed]

Europhys. Lett.

J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. 71(3), 487–493 (2005).
[CrossRef]

Faraday Discuss.

A. van Blaaderen, J. P. Hoogenboom, D. L. J. Vossen, A. Yethiraj, A. van der Horst, K. Visscher, and M. Dogterom, “Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization,” Faraday Discuss. 123, 107–119, discussion 173–192, 419–421 (2003).
[CrossRef] [PubMed]

J. Appl. Phys.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
[CrossRef]

J. Fluid Mech.

G. K. Batchelor, “Diffusion in a dilute polydisperse system of interacting spheres,” J. Fluid Mech. 131(-1), 155–175 (1983).
[CrossRef]

Lab Chip

K. Uhrig, R. Kurre, C. Schmitz, J. E. Curtis, T. Haraszti, A. E. M. Clemen, and J. P. Spatz, “Optical force sensor array in a microfluidic device based on holographic optical tweezers,” Lab Chip 9(5), 661–668 (2009).
[CrossRef] [PubMed]

Microsc. Res. Tech.

A. Pralle, M. Prummer, E. L. Florin, E. H. Stelzer, and J. K. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

Nano Lett.

M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010).
[CrossRef] [PubMed]

Nat. Methods

H. Kress, J. G. Park, C. O. Mejean, J. D. Forster, J. Park, S. S. Walse, Y. Zhang, D. Q. Wu, O. D. Weiner, T. M. Fahmy, and E. R. Dufresne, “Cell stimulation with optically manipulated microsources,” Nat. Methods 6(12), 905–909 (2009).
[CrossRef] [PubMed]

Nat. Phys.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Chem. Chem. Phys.

A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, “Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,” Phys. Chem. Chem. Phys. 13(16), 7263–7272 (2011).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

N. B. Becker, S. M. Altmann, T. Scholz, J. K. Hörber, E. H. Stelzer, and A. Rohrbach, “Three-dimensional bead position histograms reveal single-molecule nanomechanics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 021907 (2005).
[CrossRef] [PubMed]

J. Dobnikar, M. Brunner, H.-H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 031402 (2004).
[CrossRef] [PubMed]

H. Kress, E. H. K. Stelzer, G. Griffiths, and A. Rohrbach, “Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 061927 (2005).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

S. Henderson, S. Mitchell, and P. Bartlett, “Direct measurements of colloidal friction coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 061403 (2001).
[CrossRef] [PubMed]

R. Di Leonardo, S. Keen, F. Ianni, J. Leach, M. J. Padgett, and G. Ruocco, “Hydrodynamic interactions in two dimensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 031406 (2008).
[CrossRef] [PubMed]

Phys. Rev. Lett.

E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two brownian spheres to a planar surface,” Phys. Rev. Lett. 85(15), 3317–3320 (2000).
[CrossRef] [PubMed]

J.-C. Meiners and S. R. Quake, “Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73(2), 352–355 (1994).
[CrossRef] [PubMed]

I. M. Toli?-Nørrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93(7), 078102 (2004).
[CrossRef] [PubMed]

J. C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82(10), 2211–2214 (1999).
[CrossRef]

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled F-actin solutions,” Phys. Rev. Lett. 91(15), 158302 (2003).
[CrossRef] [PubMed]

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82(21), 4352–4355 (1999).
[CrossRef]

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95(16), 168102 (2005).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

H. Kress, E. H. K. Stelzer, D. Holzer, F. Buss, G. Griffiths, and A. Rohrbach, “Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity,” Proc. Natl. Acad. Sci. U.S.A. 104(28), 11633–11638 (2007).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a Photonic Force Microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Science

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality,” Science 296(5574), 1832–1835 (2002).
[CrossRef] [PubMed]

Other

J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Principle of setup and laser displacement scheme. (a) The aperture of a 2D-accousto-optic deflector (AOD) is imaged telecentrically on the BFP of the objective lens (OL), leading to a displacement of the trapping and tracking beam in the focal plane of the OL. For each beam center position rL(t) and particle position b, the DL projects scattered and unscattered light for a short time onto the QPD. The sequence of beam displacements across all 9 particles begins with the central particle #1 (r08 indicates the center of the 8-th focus). (b) Brightfield image of four 1.16 µm and five 0.62µm trapped glass beads separated by adjustable distances dx and dy. Bead numbers are indicated nearby each bead image.

Fig. 2
Fig. 2

Principles of sequential particle tracking. (a) Sketch of nine particles of different sizes arranged in the xy-plane. The exemplary diffusion paths (red and black) represent experimentally obtained data. (b) Upper graph: Focus displacements xL(t) and yL(t) as a function of time for the bead arrangement displayed in a). Lower Graph: Corresponding QPD signals for the particle z-positions. The piece of signal trajectory of the center particle #1 is drawn in black. (c) The sample points (red) i, i + 1, i + 2 of the particle trajectory #1 are derived by averaging (blue) over ten centered signal points of the raw signal (black) to eliminate high frequency fluctuations (>70 kHz) of the AOD. (d) Resulting axial position trajectory of particle #1, where the inset shows i = 9 single positions. Three positions with red markers, which are derived from the periods i, i + 1, i + 2 in (c), are highlighted.

Fig. 3
Fig. 3

Projections of particle trajectories measured in parallel in a point cloud representation. Projections are shown in x-, y- and z-direction for all 9 particles trajectories. The symmetric shape and nearly identical extensions of the clouds reflect the well-defined adjustment of individual trapping strengths in the trap array. All glass spheres have a diameter of 1.16µm.

Fig. 5
Fig. 5

(a) Increase in the effective auto-correlation (AC) time (left) and the rms-displacement of the particle (right). The AC-time increases linearly with the off-time τoff = Nτon (number of traps N = 0..120). (b) The experimental data can be well approximated by the particle’s expected free diffusion.

Fig. 4
Fig. 4

Measured trap stiffnesses κj (a,c) and detector sensitivities gj (b,d) (j = x,y,z) from four 1.16µm (big markers) and five 0.62µm (small markers) trapped glass beads. The axial components are in the top row, the lateral in the bottom row. The markers are colored in red for the x-, in green for the y- and in blue for the z-direction. a,c) The trap stiffnesses are the same of each bead size in all three directions. Differences between x- and y- direction result from the polarization of the incident field. b,d) The detector sensitivities clearly reveal the two different bead radii in all three directions.

Fig. 6
Fig. 6

Cross-correlation functions CC(τ) for two 1.16 µm glass spheres (marked with red circles) in various distances r = dy to each other. (a) The strongest coupling is for longitudinal motions CCyy(τ), weaker coupling for transversal motions CCxx(τ) (b) and negligible small coupling for motions in perpendicular directions CCxy(τ) (c). Coupling is strongest for short distances dy = 2µm with a time delay of τmin ≈2ms. The particle arrangement is quadratic as pointed out in the insets.

Fig. 7
Fig. 7

(Color online) Cross-correlation functions CC(τ) for two beads in a distance dy = 4µm to each other with and without mediator bead in between. The longitudinal coupling strength is increased without the mediator bead, but the delay time is reduced with mediator bead. The auto-correlation time of a single trapped bead was τy = 7 ms, and τy = 5 ms, respectively.

Fig. 8
Fig. 8

Self-diffusion of the center bead as a function of distance to the beads. (a) Decay of the normalized diffusion coefficient as a function of volume ratio corresponding to decreasing distances dy. (b) Normalized auto-correlation function AC(τ) of the center bead for 4 bead to bead distances r = dy.

Equations (23)

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

S ^ m ( b i , r L )= A m | E ˜ 0,n ( k x , k y , r L ,)+ E ˜ s,n ( k x , k y , b n , r L ) | 2 H( k x , k y )d k x d k y
S n (t)= S n raw (t)/ c n S 0n (t) ( ( g n,x b n,x (t) ),( g n,y b n,y (t) ),( g n,z b n,z (t)+ S 0n (t) ) ) S 0n (t) g n b n (t)
x rel,n = x 0n x L (t)= d x + n=0 N' d x Θ( tnΔt )
τ scan =NΔt= τ on + τ off =1/ f AOD +(N1)Δt
γ b ˙ n,x (t)+ F opt,n,x ( x L , b n,x (t) )= F th (t)
S ¯ n,j (i τ scan +n τ on ) = S ¯ n,j ( t n )= 1 τ on /2 t n + τ on /4 t n +3 τ on /4 S n,j ( t + t n )d t = g n,j b n,j (t)
κ j b j 2 = k B T
AC( S j ( t ) )= S j ( t )× S j ( t+τ ) =  ( σ j sig ) 2 ·exp( τ× κ j /γ ),  κ j =γ/ τ j
p( b j )   p 0 exp( 1 2 κ j b j 2 / k B T ) =  p 0 exp( 1 2 b j 2 / σ j 2 )
g j = σ j sig σ j = σ j sig κ j k B T
F grad ( b x )= αn 2cV V(bead) I(r b x e x )dV = αn 2cV I(r b x e x )s(r)dV αn 2c b x ( I 0 exp( ( b x / Δ x ) 2 ) ) b x Δx κ x b x
F grad ( b x , x rel (t)){ V 0 b x exp( ( b x x rel (t) ) 2 Δ x 2 ) if| b x x rel (t) | Δ x 0 otherwise
F eff ( b x )= 1 NΔt ( 2 V 0 Δ x 2 ) 0 Δt b x exp( ( b x 0 ) 2 Δ x 2 )dt =constΔt + 1 NΔt ( 2 V 0 Δ x 2 ) 0 (N1)Δt ( b x x rel (t) ) d x > Δ x exp( ( b x (t) x rel (t) ) 2 Δ x 2 ) =0 dt
F eff ( b x )= 1 N ( 2 V 0 Δ x 2 ) b x exp( b x 2 Δ x 2 ) = 1 N F grad ( b x )= τ on τ on + τ off F grad ( b x ) b x Δ x κ x,eff b x
κ eff = γ τ eff = 1 N κ= τ on τ on + τ off I 0 αn c Δ x 2
p eff ( b j )    p 0 exp( 1 2 b j 2 / σ eff,j 2 )and σ eff,j = N σ j
f scan = 1 τ on + τ off = f AOD N q f c or τ scan τ 0 /q
τ eff,x =( 1+ τ off / τ on ) τ x =N τ x 2 τ off
σ eff,x = k B T τ eff,x /γ = 1+ τ off / τ on σ x = N σ x
( γ 11 (r) γ 12 (r) γ 1n (r) γ 21 (r) γ 22 (r) γ 2n (r) γ m1 (r) γ m2 (r) γ mn (r) )( r ˙ 1 (t) r ˙ 2 (t) r ˙ n (t) )+( κ 11 0 0 0 κ 22 0 0 0 κ nn )( r 1 (t) r 2 (t) r n (t) )=( F th, 1 (t) F th, 2 (t) F th, n (t) )
µ mn (r)= µ mn (r) rr r 2 + µ mn (r)( 1 rr r 2 )
C C i ( τ )= k B T 2 κ i ( e τ µ 0 κ i (1+ 3 2 R/r) e τ µ 0 κ i (1 3 2 R/r) )
C C i ( τ )= k B T 2 κ i ( e τ µ 0 κ i (1+ 3 4 R/r) e τ µ 0 κ i (1 3 4 R/r) )

Metrics