Abstract

We present a holographic optical tweezers system capable of position clamping multiple particles. Moving an optical trap in response to the trapped object’s motion is a powerful technique for optical control and force measurement. We have now realised this experimentally using a Boulder Nonlinear Systems Spatial Light Modulator (SLM) with a refresh rate of 203Hz. We obtain a reduction of 44% in the variance of the bead’s position, corresponding to an increase in effective trap stiffness of 77%. This reduction relies on the generation of holograms at high speed. We present software capable of calculating holograms in under 1ms using a graphics processor unit.

© 2009 Optical Society of America

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References

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  1. A. Ashkin, "Accelleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. D. Grier, "A revolution in optical manipulation," Nat. Photonics 424, 810-816 (2003).
  3. G. Sinclair, P. Jordan, J. Leach, M. Padgett, and J. Cooper, "Defining the trapping limits of holographical optical tweezers," J. Modern Opt. 51, 409-414 (2004).
    [CrossRef]
  4. G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, "Assembly of 3-dimentional structures using programmable holographic optical tweezers," Opt. Express 12, 5475-5480 (2004).
    [CrossRef] [PubMed]
  5. K. D. Wulff, D. G. Cole, and R. L. Clark, "Servo control of an optical trap," Appl. Opt. 46, 4923-4931 (2007).
    [CrossRef] [PubMed]
  6. A. E. Wallin, H. Ojala, E. Haeggstrom, and R. Tuma, "Stiffer optical tweezers through real-time feedback control," Appl. Phys. Lett. 92, 224104 (2008).
    [CrossRef]
  7. G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
    [CrossRef] [PubMed]
  8. M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
    [CrossRef] [PubMed]
  9. J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
    [CrossRef] [PubMed]
  10. C. O. Mejean, A.W. Schaefer, E. A. Millman, P. Forscher, and E. R. Dufresne, "Multiplexed force measurements on live cells with holographic optical tweezers," Opt. Express 17, 6209-6217 (2009).
    [CrossRef] [PubMed]
  11. O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, "Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis," Rev. Sci. Instrum. 79, 023710 (pages 6) (2008).
    [CrossRef] [PubMed]
  12. G. M. Gibson, S. J. Leach, Keen, A. J.  Wright, and M. J. Padgett, "Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy," Opt. Express 16, 14,56114,570 (2008).
    [CrossRef]
  13. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, and J. Gl¨uckstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13, 6899-6904 (2005).
    [CrossRef] [PubMed]
  14. L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
    [CrossRef]
  15. K. Berg-Sorensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004).
    [CrossRef]
  16. C. Schmitz, J. Spatz, and J. Curtis, "High-precision steering of multiple holographic optical traps," Opt. Express 13, 8678-8685 (2005).
    [CrossRef] [PubMed]
  17. D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
    [CrossRef]
  18. T. H. J. Liesener, M. Reicherter and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
    [CrossRef]
  19. J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. Laczik, and M. Padgett, "Interactive approach to optical tweezers control," Appl. Opt. 45, 897-903 (2006).
    [CrossRef] [PubMed]
  20. R. D. Leonardo, F. Ianni, and G. Ruocco, "Computer generation of optimal holograms for optical trap arrays," Opt. Express 15, 1913-1922 (2007).
    [CrossRef] [PubMed]
  21. M. Reicherter, S. Zwick, T. Haist, C. Kohler, H. Tiziani, and W. Osten, "Fast digital hologram generation and adaptive force measurement in liquid-crystal-display-based holographic tweezers," Appl. Opt. 45, 888-896 (2006).
    [CrossRef] [PubMed]
  22. URL http://www.physics.gla.ac.uk/Optics/projects/tweezers/software/.
  23. J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Scie. 179, 298-310 (1996).
    [CrossRef]
  24. Z. Zhang and C.-H. Menq, "Three-dimensional particle tracking with subnanometer resolution using off-focus images," Appl. Opt. 47, 2361-2370 (2008).
    [CrossRef] [PubMed]
  25. C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
    [CrossRef]

2009

C. O. Mejean, A.W. Schaefer, E. A. Millman, P. Forscher, and E. R. Dufresne, "Multiplexed force measurements on live cells with holographic optical tweezers," Opt. Express 17, 6209-6217 (2009).
[CrossRef] [PubMed]

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

2008

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

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
[CrossRef]

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

2007

2006

2005

2004

K. Berg-Sorensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004).
[CrossRef]

G. Sinclair, P. Jordan, J. Leach, M. Padgett, and J. Cooper, "Defining the trapping limits of holographical optical tweezers," J. Modern Opt. 51, 409-414 (2004).
[CrossRef]

G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, "Assembly of 3-dimentional structures using programmable holographic optical tweezers," Opt. Express 12, 5475-5480 (2004).
[CrossRef] [PubMed]

2003

D. Grier, "A revolution in optical manipulation," Nat. Photonics 424, 810-816 (2003).

2000

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

T. H. J. Liesener, M. Reicherter and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

1997

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

1996

J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Scie. 179, 298-310 (1996).
[CrossRef]

1995

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

1970

A. Ashkin, "Accelleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Ashkin, A.

A. Ashkin, "Accelleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Bengtsson, J.

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
[CrossRef]

Bergander, A.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

Berg-Sorensen, K.

K. Berg-Sorensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004).
[CrossRef]

Block, S.

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Bøggild, P.

Bowman, R.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

Burns, J. E.

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Bustamante, C.

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Carberry, D.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

Carberry, D. M.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Clark, R. L.

Cole, D. G.

Cooper, J.

Courtial, J.

Crocker, J. C.

J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Scie. 179, 298-310 (1996).
[CrossRef]

Curtis, J.

Davenport, R.

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Dufresne, E. R.

Engstrom, D.

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
[CrossRef]

Eriksson, E.

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
[CrossRef]

Flyvbjerg, H.

K. Berg-Sorensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004).
[CrossRef]

Forscher, P.

Gammelgaard, L.

Gelles, J.

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Gibson, G.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. Laczik, and M. Padgett, "Interactive approach to optical tweezers control," Appl. Opt. 45, 897-903 (2006).
[CrossRef] [PubMed]

Gibson, G. M.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Gl¨uckstad, J.

Goksor, M.

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, "Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator," Opt. Express 16, 18,275-18,287 (2008).
[CrossRef]

Grier, D.

D. Grier, "A revolution in optical manipulation," Nat. Photonics 424, 810-816 (2003).

Grier, D. G.

J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Scie. 179, 298-310 (1996).
[CrossRef]

Haeggstrom, E.

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

Haist, T.

Haliyo, S.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

Ianni, F.

Ikin, L.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Jordan, P.

Karunwi, K.

Kendrick-jones, J.

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Kohler, C.

Laczik, Z.

Landick, R.

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Leach, J.

Leonardo, R. D.

Liesener, T. H. J.

T. H. J. Liesener, M. Reicherter and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

Mejean, C. O.

Menq, C.-H.

Miles, M. J.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Millman, E. A.

Molloy, J. E.

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Ojala, H.

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

Osten, W.

Pacoret, C.

C. Pacoret, R. Bowman, G. Gibson, S. Haliyo, D. Carberry, A. Bergander, S. Regnier, and M. Padgett, "Touching the microworld with force-feedback optical tweezers," Opt. Express 17, 10,259-10,264 (2009).
[CrossRef]

Padgett, M.

Padgett, M. J.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Perch-Nielsen, I.

Rappaport, A.

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Reicherter, M.

Rodrigo, P. J.

Ruocco, G.

Schaefer, A.W.

Schmitz, C.

Sinclair, G.

Spatz, J.

Thomson, L.

Tiziani, H.

Tiziani, H. J.

T. H. J. Liesener, M. Reicherter and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

Tregear, R. T.

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Tuma, R.

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

Wallin, A. E.

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

Wang, M.

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

White, D. C. S.

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Wuite, G.

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

Wulff, K.

Wulff, K. D.

Yin, H.

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

Zhang, Z.

Zwick, S.

Appl. Opt.

Appl. Phys. Lett.

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

Biophys. J

G. Wuite, R. Davenport, A. Rappaport, and C. Bustamante, "An integrated laser trap/flow control video microscope for the study of single biomolecules," Biophys. J 79, 1155-1167 (2000).
[CrossRef] [PubMed]

M. Wang, H. Yin, R. Landick, J. Gelles, and S. Block, "Stretching DNA with optical tweezers," Biophys. J 72, 1335-1346 (1997).
[CrossRef] [PubMed]

J. Colloid Interface Scie.

J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Scie. 179, 298-310 (1996).
[CrossRef]

J. Modern Opt.

G. Sinclair, P. Jordan, J. Leach, M. Padgett, and J. Cooper, "Defining the trapping limits of holographical optical tweezers," J. Modern Opt. 51, 409-414 (2004).
[CrossRef]

N. J. Phys.

L. Ikin, D. M. Carberry, G. M. Gibson, M. J. Padgett, and M. J. Miles, "Assembly and force measurement with SPM-like probes in holographic optical tweezers," N. J. Phys. 11, 023102 (2009).
[CrossRef]

Nat. Photonics

D. Grier, "A revolution in optical manipulation," Nat. Photonics 424, 810-816 (2003).

Nature

J. E. Molloy, J. E. Burns, J. Kendrick-jones, R. T. Tregear, and D. C. S. White, "Movement and force produced by a single myosin head," Nature 378, 209-212 (1995).
[CrossRef] [PubMed]

Opt. Commun.

T. H. J. Liesener, M. Reicherter and H. J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

A. Ashkin, "Accelleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Rev. Sci. Instrum.

K. Berg-Sorensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004).
[CrossRef]

Other

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, "Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis," Rev. Sci. Instrum. 79, 023710 (pages 6) (2008).
[CrossRef] [PubMed]

G. M. Gibson, S. J. Leach, Keen, A. J.  Wright, and M. J. Padgett, "Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy," Opt. Express 16, 14,56114,570 (2008).
[CrossRef]

URL http://www.physics.gla.ac.uk/Optics/projects/tweezers/software/.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Experimental setup; a 532nm laser beam is expanded then steered via an SLM onto the back aperture of the microscope objective. This same objective is then used to view the sample with bright-field illumination. The Gantt chart to the right outlines the steps in one iteration of the loop, along with approximate timings and transfer functions used to calculate the theoretical power spectrum in (6).

Fig. 2.
Fig. 2.

Intensity at two points, when the SLMis used to switch the laser spot between them. The solid line shows the modelled response of Section 2, with τ R=2ms.

Fig. 3.
Fig. 3.

Experimentally measured power spectra (points) and theoretical curves (lines) from (6) for a 5µm bead in a trap with stiffness κ≈2.1×10-6Nm-1. [inset] The variance of the particle’s position as a function of feedback gain, with experimental data as points and the theoretical curve as a line. Model parameters and experimental data are the same for both plots.

Fig. 4.
Fig. 4.

Three 5µm beads in optical traps (a), along with a histogram of the position distributions (b) and the power spectra of their motion (c) for no gain and a gain of 1.7. The variances with and without clamping, and the percentage decrease in 〈x 2〉 and 〈y 2〉 are shown in the legend.

Equations (9)

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m + γ + k x = ζ ( t )
( ω 2 m + i γ + κ ) = ζ̃ ( ω )
S x = γ k B T π ( ( κ m ω 2 ) 2 + γ 2 ω 2 ) 1
m + γ + κ ( x x trap ( t ) ) = ζ ( t ) .
trap G sinc ( ω τ SLM 2 ) e ( τ SLM 2 + τ lag ) 1 1 + i τ r ω
S x = γk B T π ω 2 m + iγω + κ + κ G sin c ( πω ω SLM ) e ( τ SLM 2 + τ lag ) ( 1 + i τ R ω ) 2
x trap = G x
ϕ i ( x , y ) = k x x + k y y + k z ( x 2 + y 2 ) .
ϕ T = Arg [ Σ i exp ( i ϕ i ( x , y ) ) ] = Arg [ Σ i exp ( i [ k x x + k y y + k z ( x 2 + y 2 ) ] ) ]

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